TPTP Problem File: DAT189^1.p
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%------------------------------------------------------------------------------
% File : DAT189^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Lazy list mirror 23
% Version : [Bla16] axioms : Especial.
% English :
% Refs : [Loc10] Lochbihler (2010), Coinductive
% : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% Source : [Bla16]
% Names : lmirror__23.p [Bla16]
% Status : Theorem
% Rating : 0.00 v7.1.0
% Syntax : Number of formulae : 306 ( 125 unt; 38 typ; 0 def)
% Number of atoms : 772 ( 265 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 3778 ( 138 ~; 45 |; 75 &;3175 @)
% ( 0 <=>; 345 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 8 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 254 ( 254 >; 0 *; 0 +; 0 <<)
% Number of symbols : 40 ( 37 usr; 4 con; 0-8 aty)
% Number of variables : 1074 ( 44 ^; 943 !; 53 ?;1074 :)
% ( 34 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:40:45.178
%------------------------------------------------------------------------------
%----Could-be-implicit typings (4)
thf(ty_t_Coinductive__List_Ollist,type,
coinductive_llist: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (34)
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_Oorder,type,
order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ofinite__lprefix,type,
coindu328551480prefix:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ogen__lset,type,
coinductive_gen_lset:
!>[A: $tType] : ( ( set @ A ) > ( coinductive_llist @ A ) > ( set @ A ) ) ).
thf(sy_c_Coinductive__List_Olappend,type,
coinductive_lappend:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Olfinite,type,
coinductive_lfinite:
!>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ollast,type,
coinductive_llast:
!>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
thf(sy_c_Coinductive__List_Ollexord,type,
coinductive_llexord:
!>[A: $tType] : ( ( A > A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ollist_OLCons,type,
coinductive_LCons:
!>[A: $tType] : ( A > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Ollist_OLNil,type,
coinductive_LNil:
!>[A: $tType] : ( coinductive_llist @ A ) ).
thf(sy_c_Coinductive__List_Ollist_Ocorec__llist,type,
coindu1259883913_llist:
!>[C: $tType,A: $tType] : ( ( C > $o ) > ( C > A ) > ( C > $o ) > ( C > ( coinductive_llist @ A ) ) > ( C > C ) > C > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Ollist_Olhd,type,
coinductive_lhd:
!>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
thf(sy_c_Coinductive__List_Ollist_Olnull,type,
coinductive_lnull:
!>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ollist_Oltl,type,
coinductive_ltl:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Olmember,type,
coinductive_lmember:
!>[A: $tType] : ( A > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Olprefix,type,
coinductive_lprefix:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Olstrict__prefix,type,
coindu1478340336prefix:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_OltakeWhile,type,
coindu501562517eWhile:
!>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Oord__class_Olsorted,type,
coindu63249387sorted:
!>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ounfold__llist,type,
coindu1441602521_llist:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > B ) > ( A > A ) > A > ( coinductive_llist @ B ) ) ).
thf(sy_c_HOL_Oundefined,type,
undefined:
!>[A: $tType] : A ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_LMirror__Mirabelle__wyovfcktfy_Olmirror,type,
lMirro427583474mirror:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_LMirror__Mirabelle__wyovfcktfy_Olmirror__aux,type,
lMirro999291890or_aux:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
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_Set_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_acc,type,
acc: coinductive_llist @ a ).
thf(sy_v_xs,type,
xs: coinductive_llist @ a ).
%----Relevant facts (254)
thf(fact_0_lmirror__aux_Odisc__iff_I2_J,axiom,
! [A: $tType,Acc: coinductive_llist @ A,Xs: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ ( lMirro999291890or_aux @ A @ Acc @ Xs ) ) )
= ( ~ ( coinductive_lnull @ A @ Xs )
| ~ ( coinductive_lnull @ A @ Acc ) ) ) ).
% lmirror_aux.disc_iff(2)
thf(fact_1_lmirror__aux_Odisc__iff_I1_J,axiom,
! [A: $tType,Acc: coinductive_llist @ A,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ ( lMirro999291890or_aux @ A @ Acc @ Xs ) )
= ( ( coinductive_lnull @ A @ Xs )
& ( coinductive_lnull @ A @ Acc ) ) ) ).
% lmirror_aux.disc_iff(1)
thf(fact_2_lmirror__aux_Odisc_I2_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Acc: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ Xs )
| ~ ( coinductive_lnull @ A @ Acc ) )
=> ~ ( coinductive_lnull @ A @ ( lMirro999291890or_aux @ A @ Acc @ Xs ) ) ) ).
% lmirror_aux.disc(2)
thf(fact_3_lmirror__aux_Odisc_I1_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Acc: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lnull @ A @ Acc )
=> ( coinductive_lnull @ A @ ( lMirro999291890or_aux @ A @ Acc @ Xs ) ) ) ) ).
% lmirror_aux.disc(1)
thf(fact_4_lmirror__aux_Oexhaust,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Acc: coinductive_llist @ A] :
( ( ( coinductive_lnull @ A @ Xs )
=> ~ ( coinductive_lnull @ A @ Acc ) )
=> ( ~ ( coinductive_lnull @ A @ Xs )
| ~ ( coinductive_lnull @ A @ Acc ) ) ) ).
% lmirror_aux.exhaust
thf(fact_5_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_6_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_7_lmirror__aux_Octr_I1_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Acc: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lnull @ A @ Acc )
=> ( ( lMirro999291890or_aux @ A @ Acc @ Xs )
= ( coinductive_LNil @ A ) ) ) ) ).
% lmirror_aux.ctr(1)
thf(fact_8_lmirror__aux__simps_I1_J,axiom,
! [A: $tType,Acc: coinductive_llist @ A] :
( ( lMirro999291890or_aux @ A @ Acc @ ( coinductive_LNil @ A ) )
= Acc ) ).
% lmirror_aux_simps(1)
thf(fact_9_lmirror__aux__simps_I2_J,axiom,
! [A: $tType,Acc: coinductive_llist @ A,Xa: A,X: coinductive_llist @ A] :
( ( lMirro999291890or_aux @ A @ Acc @ ( coinductive_LCons @ A @ Xa @ X ) )
= ( coinductive_LCons @ A @ Xa @ ( lMirro999291890or_aux @ A @ ( coinductive_LCons @ A @ Xa @ Acc ) @ X ) ) ) ).
% lmirror_aux_simps(2)
thf(fact_10_lmirror__def,axiom,
! [A: $tType] :
( ( lMirro427583474mirror @ A )
= ( lMirro999291890or_aux @ A @ ( coinductive_LNil @ A ) ) ) ).
% lmirror_def
thf(fact_11_llist_Ocorec__disc__iff_I2_J,axiom,
! [A: $tType,C: $tType,P: C > $o,G21: C > A,Q22: C > $o,G221: C > ( coinductive_llist @ A ),G222: C > C,A2: C] :
( ( ~ ( coinductive_lnull @ A @ ( coindu1259883913_llist @ C @ A @ P @ G21 @ Q22 @ G221 @ G222 @ A2 ) ) )
= ( ~ ( P @ A2 ) ) ) ).
% llist.corec_disc_iff(2)
thf(fact_12_llist_Ocorec__disc__iff_I1_J,axiom,
! [A: $tType,C: $tType,P: C > $o,G21: C > A,Q22: C > $o,G221: C > ( coinductive_llist @ A ),G222: C > C,A2: C] :
( ( coinductive_lnull @ A @ ( coindu1259883913_llist @ C @ A @ P @ G21 @ Q22 @ G221 @ G222 @ A2 ) )
= ( P @ A2 ) ) ).
% llist.corec_disc_iff(1)
thf(fact_13_unfold__llist_Odisc__iff_I2_J,axiom,
! [B: $tType,A: $tType,P: A > $o,G21: A > B,G22: A > A,A2: A] :
( ( ~ ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P @ G21 @ G22 @ A2 ) ) )
= ( ~ ( P @ A2 ) ) ) ).
% unfold_llist.disc_iff(2)
thf(fact_14_unfold__llist_Odisc__iff_I1_J,axiom,
! [B: $tType,A: $tType,P: A > $o,G21: A > B,G22: A > A,A2: A] :
( ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P @ G21 @ G22 @ A2 ) )
= ( P @ A2 ) ) ).
% unfold_llist.disc_iff(1)
thf(fact_15_llist_Oinject,axiom,
! [A: $tType,X21: A,X22: coinductive_llist @ A,Y21: A,Y22: coinductive_llist @ A] :
( ( ( coinductive_LCons @ A @ X21 @ X22 )
= ( coinductive_LCons @ A @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% llist.inject
thf(fact_16_unfold__llist__eq__LCons,axiom,
! [A: $tType,B: $tType,IS_LNIL: B > $o,LHD: B > A,LTL: B > B,B2: B,X: A,Xs: coinductive_llist @ A] :
( ( ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ B2 )
= ( coinductive_LCons @ A @ X @ Xs ) )
= ( ~ ( IS_LNIL @ B2 )
& ( X
= ( LHD @ B2 ) )
& ( Xs
= ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ ( LTL @ B2 ) ) ) ) ) ).
% unfold_llist_eq_LCons
thf(fact_17_llist_Ocorec_I2_J,axiom,
! [A: $tType,C: $tType,P: C > $o,A2: C,G21: C > A,Q22: C > $o,G221: C > ( coinductive_llist @ A ),G222: C > C] :
( ~ ( P @ A2 )
=> ( ( coindu1259883913_llist @ C @ A @ P @ G21 @ Q22 @ G221 @ G222 @ A2 )
= ( coinductive_LCons @ A @ ( G21 @ A2 ) @ ( if @ ( coinductive_llist @ A ) @ ( Q22 @ A2 ) @ ( G221 @ A2 ) @ ( coindu1259883913_llist @ C @ A @ P @ G21 @ Q22 @ G221 @ G222 @ ( G222 @ A2 ) ) ) ) ) ) ).
% llist.corec(2)
thf(fact_18_llist_Ocorec_I1_J,axiom,
! [C: $tType,A: $tType,P: C > $o,A2: C,G21: C > A,Q22: C > $o,G221: C > ( coinductive_llist @ A ),G222: C > C] :
( ( P @ A2 )
=> ( ( coindu1259883913_llist @ C @ A @ P @ G21 @ Q22 @ G221 @ G222 @ A2 )
= ( coinductive_LNil @ A ) ) ) ).
% llist.corec(1)
thf(fact_19_llist_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: coinductive_llist @ A] :
( ( coinductive_LNil @ A )
!= ( coinductive_LCons @ A @ X21 @ X22 ) ) ).
% llist.distinct(1)
thf(fact_20_unfold__llist_Octr_I2_J,axiom,
! [B: $tType,A: $tType,P: A > $o,A2: A,G21: A > B,G22: A > A] :
( ~ ( P @ A2 )
=> ( ( coindu1441602521_llist @ A @ B @ P @ G21 @ G22 @ A2 )
= ( coinductive_LCons @ B @ ( G21 @ A2 ) @ ( coindu1441602521_llist @ A @ B @ P @ G21 @ G22 @ ( G22 @ A2 ) ) ) ) ) ).
% unfold_llist.ctr(2)
thf(fact_21_unfold__llist_Octr_I1_J,axiom,
! [A: $tType,B: $tType,P: A > $o,A2: A,G21: A > B,G22: A > A] :
( ( P @ A2 )
=> ( ( coindu1441602521_llist @ A @ B @ P @ G21 @ G22 @ A2 )
= ( coinductive_LNil @ B ) ) ) ).
% unfold_llist.ctr(1)
thf(fact_22_llist_Oexhaust,axiom,
! [A: $tType,Y: coinductive_llist @ A] :
( ( Y
!= ( coinductive_LNil @ A ) )
=> ~ ! [X212: A,X222: coinductive_llist @ A] :
( Y
!= ( coinductive_LCons @ A @ X212 @ X222 ) ) ) ).
% llist.exhaust
thf(fact_23_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_24_llist_Ocorec__code,axiom,
! [A: $tType,C: $tType] :
( ( coindu1259883913_llist @ C @ A )
= ( ^ [P2: C > $o,G212: C > A,Q222: C > $o,G2212: C > ( coinductive_llist @ A ),G2222: C > C,A3: C] : ( if @ ( coinductive_llist @ A ) @ ( P2 @ A3 ) @ ( coinductive_LNil @ A ) @ ( coinductive_LCons @ A @ ( G212 @ A3 ) @ ( if @ ( coinductive_llist @ A ) @ ( Q222 @ A3 ) @ ( G2212 @ A3 ) @ ( coindu1259883913_llist @ C @ A @ P2 @ G212 @ Q222 @ G2212 @ G2222 @ ( G2222 @ A3 ) ) ) ) ) ) ) ).
% llist.corec_code
thf(fact_25_unfold__llist_Ocode,axiom,
! [B: $tType,A: $tType] :
( ( coindu1441602521_llist @ A @ B )
= ( ^ [P2: A > $o,G212: A > B,G223: A > A,A3: A] : ( if @ ( coinductive_llist @ B ) @ ( P2 @ A3 ) @ ( coinductive_LNil @ B ) @ ( coinductive_LCons @ B @ ( G212 @ A3 ) @ ( coindu1441602521_llist @ A @ B @ P2 @ G212 @ G223 @ ( G223 @ A3 ) ) ) ) ) ) ).
% unfold_llist.code
thf(fact_26_not__lnull__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ Xs ) )
= ( ? [X2: A,Xs2: coinductive_llist @ A] :
( Xs
= ( coinductive_LCons @ A @ X2 @ Xs2 ) ) ) ) ).
% not_lnull_conv
thf(fact_27_llist_OdiscI_I2_J,axiom,
! [A: $tType,Llist: coinductive_llist @ A,X21: A,X22: coinductive_llist @ A] :
( ( Llist
= ( coinductive_LCons @ A @ X21 @ X22 ) )
=> ~ ( coinductive_lnull @ A @ Llist ) ) ).
% llist.discI(2)
thf(fact_28_llist_Odisc_I2_J,axiom,
! [A: $tType,X21: A,X22: coinductive_llist @ A] :
~ ( coinductive_lnull @ A @ ( coinductive_LCons @ A @ X21 @ X22 ) ) ).
% llist.disc(2)
thf(fact_29_lnull__def,axiom,
! [A: $tType] :
( ( coinductive_lnull @ A )
= ( ^ [Llist2: coinductive_llist @ A] :
( Llist2
= ( coinductive_LNil @ A ) ) ) ) ).
% lnull_def
thf(fact_30_llist_Ocollapse_I1_J,axiom,
! [A: $tType,Llist: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Llist )
=> ( Llist
= ( coinductive_LNil @ A ) ) ) ).
% llist.collapse(1)
thf(fact_31_llist_OdiscI_I1_J,axiom,
! [A: $tType,Llist: coinductive_llist @ A] :
( ( Llist
= ( coinductive_LNil @ A ) )
=> ( coinductive_lnull @ A @ Llist ) ) ).
% llist.discI(1)
thf(fact_32_llist_Odisc_I1_J,axiom,
! [A: $tType] : ( coinductive_lnull @ A @ ( coinductive_LNil @ A ) ) ).
% llist.disc(1)
thf(fact_33_unfold__llist_Odisc_I1_J,axiom,
! [B: $tType,A: $tType,P: A > $o,A2: A,G21: A > B,G22: A > A] :
( ( P @ A2 )
=> ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P @ G21 @ G22 @ A2 ) ) ) ).
% unfold_llist.disc(1)
thf(fact_34_unfold__llist_Odisc_I2_J,axiom,
! [B: $tType,A: $tType,P: A > $o,A2: A,G21: A > B,G22: A > A] :
( ~ ( P @ A2 )
=> ~ ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P @ G21 @ G22 @ A2 ) ) ) ).
% unfold_llist.disc(2)
thf(fact_35_llist_Ocorec__disc_I1_J,axiom,
! [A: $tType,C: $tType,P: C > $o,A2: C,G21: C > A,Q22: C > $o,G221: C > ( coinductive_llist @ A ),G222: C > C] :
( ( P @ A2 )
=> ( coinductive_lnull @ A @ ( coindu1259883913_llist @ C @ A @ P @ G21 @ Q22 @ G221 @ G222 @ A2 ) ) ) ).
% llist.corec_disc(1)
thf(fact_36_llist_Ocorec__disc_I2_J,axiom,
! [A: $tType,C: $tType,P: C > $o,A2: C,G21: C > A,Q22: C > $o,G221: C > ( coinductive_llist @ A ),G222: C > C] :
( ~ ( P @ A2 )
=> ~ ( coinductive_lnull @ A @ ( coindu1259883913_llist @ C @ A @ P @ G21 @ Q22 @ G221 @ G222 @ A2 ) ) ) ).
% llist.corec_disc(2)
thf(fact_37_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_38_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_39_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_40_llast__singleton,axiom,
! [A: $tType,X: A] :
( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) )
= X ) ).
% llast_singleton
thf(fact_41_lmember__code_I1_J,axiom,
! [A: $tType,X: A] :
~ ( coinductive_lmember @ A @ X @ ( coinductive_LNil @ A ) ) ).
% lmember_code(1)
thf(fact_42_lmember__code_I2_J,axiom,
! [A: $tType,X: A,Y: A,Ys: coinductive_llist @ A] :
( ( coinductive_lmember @ A @ X @ ( coinductive_LCons @ A @ Y @ Ys ) )
= ( ( X = Y )
| ( coinductive_lmember @ A @ X @ Ys ) ) ) ).
% lmember_code(2)
thf(fact_43_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P3: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P3 ) )
= ( P3 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_44_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_45_Collect__cong,axiom,
! [A: $tType,P3: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P3 @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P3 )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_46_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_47_gen__lset__code_I1_J,axiom,
! [A: $tType,A4: set @ A] :
( ( coinductive_gen_lset @ A @ A4 @ ( coinductive_LNil @ A ) )
= A4 ) ).
% gen_lset_code(1)
thf(fact_48_llast__LCons,axiom,
! [A: $tType,Xs: coinductive_llist @ A,X: A] :
( ( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
= X ) )
& ( ~ ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coinductive_llast @ A @ Xs ) ) ) ) ).
% llast_LCons
thf(fact_49_lsorted__code_I2_J,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: A] : ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) ) ) ).
% lsorted_code(2)
thf(fact_50_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_51_lsorted__code_I1_J,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ( coindu63249387sorted @ A @ ( coinductive_LNil @ A ) ) ) ).
% lsorted_code(1)
thf(fact_52_llast__LCons2,axiom,
! [A: $tType,X: A,Y: A,Xs: coinductive_llist @ A] :
( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y @ Xs ) ) )
= ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y @ Xs ) ) ) ).
% llast_LCons2
thf(fact_53_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_54_lstrict__prefix__code_I1_J,axiom,
! [A: $tType] :
~ ( coindu1478340336prefix @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) ) ).
% lstrict_prefix_code(1)
thf(fact_55_llist__less__induct,axiom,
! [A: $tType,P3: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
( ! [Xs3: coinductive_llist @ A] :
( ! [Ys2: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Ys2 @ Xs3 )
=> ( P3 @ Ys2 ) )
=> ( P3 @ Xs3 ) )
=> ( P3 @ Xs ) ) ).
% llist_less_induct
thf(fact_56_LNil,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ( coindu63249387sorted @ A @ ( coinductive_LNil @ A ) ) ) ).
% LNil
thf(fact_57_Singleton,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: A] : ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) ) ) ).
% Singleton
thf(fact_58_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_59_lsorted_Ocoinduct,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X4: ( coinductive_llist @ A ) > $o,X: coinductive_llist @ A] :
( ( X4 @ X )
=> ( ! [X3: coinductive_llist @ A] :
( ( X4 @ X3 )
=> ( ( X3
= ( coinductive_LNil @ A ) )
| ? [Xa2: A] :
( X3
= ( coinductive_LCons @ A @ Xa2 @ ( coinductive_LNil @ A ) ) )
| ? [Xa2: A,Y2: A,Xs4: coinductive_llist @ A] :
( ( X3
= ( coinductive_LCons @ A @ Xa2 @ ( coinductive_LCons @ A @ Y2 @ Xs4 ) ) )
& ( ord_less_eq @ A @ Xa2 @ Y2 )
& ( ( X4 @ ( coinductive_LCons @ A @ Y2 @ Xs4 ) )
| ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y2 @ Xs4 ) ) ) ) ) )
=> ( coindu63249387sorted @ A @ X ) ) ) ) ).
% lsorted.coinduct
thf(fact_60_lsorted_Osimps,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ( ( coindu63249387sorted @ A )
= ( ^ [A3: coinductive_llist @ A] :
( ( A3
= ( coinductive_LNil @ A ) )
| ? [X2: A] :
( A3
= ( coinductive_LCons @ A @ X2 @ ( coinductive_LNil @ A ) ) )
| ? [X2: A,Y3: A,Xs5: coinductive_llist @ A] :
( ( A3
= ( coinductive_LCons @ A @ X2 @ ( coinductive_LCons @ A @ Y3 @ Xs5 ) ) )
& ( ord_less_eq @ A @ X2 @ Y3 )
& ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y3 @ Xs5 ) ) ) ) ) ) ) ).
% lsorted.simps
thf(fact_61_lsorted_Ocases,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: coinductive_llist @ A] :
( ( coindu63249387sorted @ A @ A2 )
=> ( ( A2
!= ( coinductive_LNil @ A ) )
=> ( ! [X3: A] :
( A2
!= ( coinductive_LCons @ A @ X3 @ ( coinductive_LNil @ A ) ) )
=> ~ ! [X3: A,Y4: A,Xs3: coinductive_llist @ A] :
( ( A2
= ( coinductive_LCons @ A @ X3 @ ( coinductive_LCons @ A @ Y4 @ Xs3 ) ) )
=> ( ( ord_less_eq @ A @ X3 @ Y4 )
=> ~ ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y4 @ Xs3 ) ) ) ) ) ) ) ) ).
% lsorted.cases
thf(fact_62_llimit__induct,axiom,
! [A: $tType,P3: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
( ( P3 @ ( coinductive_LNil @ A ) )
=> ( ! [X3: A,Xs3: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs3 )
=> ( ( P3 @ Xs3 )
=> ( P3 @ ( coinductive_LCons @ A @ X3 @ Xs3 ) ) ) )
=> ( ( ! [Ys2: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Ys2 @ Xs )
=> ( P3 @ Ys2 ) )
=> ( P3 @ Xs ) )
=> ( P3 @ Xs ) ) ) ) ).
% llimit_induct
thf(fact_63_lsorted__LCons__LCons,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Xs: coinductive_llist @ A] :
( ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y @ Xs ) ) )
= ( ( ord_less_eq @ A @ X @ Y )
& ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y @ Xs ) ) ) ) ) ).
% lsorted_LCons_LCons
thf(fact_64_gen__lset__code_I2_J,axiom,
! [A: $tType,A4: set @ A,X: A,Xs: coinductive_llist @ A] :
( ( coinductive_gen_lset @ A @ A4 @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coinductive_gen_lset @ A @ ( insert @ A @ X @ A4 ) @ Xs ) ) ).
% gen_lset_code(2)
thf(fact_65_lprefix__LNil,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ ( coinductive_LNil @ A ) )
= ( coinductive_lnull @ A @ Xs ) ) ).
% lprefix_LNil
thf(fact_66_ltakeWhile__LCons,axiom,
! [A: $tType,P3: A > $o,X: A,Xs: coinductive_llist @ A] :
( ( ( P3 @ X )
=> ( ( coindu501562517eWhile @ A @ P3 @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coinductive_LCons @ A @ X @ ( coindu501562517eWhile @ A @ P3 @ Xs ) ) ) )
& ( ~ ( P3 @ X )
=> ( ( coindu501562517eWhile @ A @ P3 @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coinductive_LNil @ A ) ) ) ) ).
% ltakeWhile_LCons
thf(fact_67_llist_Oleq__refl,axiom,
! [A: $tType,X: coinductive_llist @ A] : ( coinductive_lprefix @ A @ X @ X ) ).
% llist.leq_refl
thf(fact_68_lprefix__refl,axiom,
! [A: $tType,Xs: coinductive_llist @ A] : ( coinductive_lprefix @ A @ Xs @ Xs ) ).
% lprefix_refl
thf(fact_69_LCons__lprefix__LCons,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ Y @ Ys ) )
= ( ( X = Y )
& ( coinductive_lprefix @ A @ Xs @ Ys ) ) ) ).
% LCons_lprefix_LCons
thf(fact_70_lprefix__code_I1_J,axiom,
! [A: $tType,Ys: coinductive_llist @ A] : ( coinductive_lprefix @ A @ ( coinductive_LNil @ A ) @ Ys ) ).
% lprefix_code(1)
thf(fact_71_lfinite__code_I2_J,axiom,
! [B: $tType,X: B,Xs: coinductive_llist @ B] :
( ( coinductive_lfinite @ B @ ( coinductive_LCons @ B @ X @ Xs ) )
= ( coinductive_lfinite @ B @ Xs ) ) ).
% lfinite_code(2)
thf(fact_72_lfinite__LCons,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coinductive_lfinite @ A @ Xs ) ) ).
% lfinite_LCons
thf(fact_73_lfinite__code_I1_J,axiom,
! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% lfinite_code(1)
thf(fact_74_ltakeWhile__LNil,axiom,
! [A: $tType,P3: A > $o] :
( ( coindu501562517eWhile @ A @ P3 @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% ltakeWhile_LNil
thf(fact_75_Coinductive__List_Olprefix__nitpick__simps,axiom,
! [A: $tType] :
( ( coinductive_lprefix @ A )
= ( ^ [Xs5: coinductive_llist @ A,Ys3: coinductive_llist @ A] :
( ( ( coinductive_lfinite @ A @ Xs5 )
=> ( coindu328551480prefix @ A @ Xs5 @ Ys3 ) )
& ( ~ ( coinductive_lfinite @ A @ Xs5 )
=> ( Xs5 = Ys3 ) ) ) ) ) ).
% Coinductive_List.lprefix_nitpick_simps
thf(fact_76_not__lfinite__lprefix__conv__eq,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_lprefix @ A @ Xs @ Ys )
= ( Xs = Ys ) ) ) ).
% not_lfinite_lprefix_conv_eq
thf(fact_77_lprefix__down__linear,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Zs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ Zs )
=> ( ( coinductive_lprefix @ A @ Ys @ Zs )
=> ( ( coinductive_lprefix @ A @ Xs @ Ys )
| ( coinductive_lprefix @ A @ Ys @ Xs ) ) ) ) ).
% lprefix_down_linear
thf(fact_78_lprefix__ltakeWhile,axiom,
! [A: $tType,P3: A > $o,Xs: coinductive_llist @ A] : ( coinductive_lprefix @ A @ ( coindu501562517eWhile @ A @ P3 @ Xs ) @ Xs ) ).
% lprefix_ltakeWhile
thf(fact_79_llist_Oleq__antisym,axiom,
! [A: $tType,X: coinductive_llist @ A,Y: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ X @ Y )
=> ( ( coinductive_lprefix @ A @ Y @ X )
=> ( X = Y ) ) ) ).
% llist.leq_antisym
thf(fact_80_wlog__linorder__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P3: A > A > $o,B2: A,A2: A] :
( ! [A5: A,B3: A] :
( ( ord_less_eq @ A @ A5 @ B3 )
=> ( P3 @ A5 @ B3 ) )
=> ( ( ( P3 @ B2 @ A2 )
=> ( P3 @ A2 @ B2 ) )
=> ( P3 @ A2 @ B2 ) ) ) ) ).
% wlog_linorder_le
thf(fact_81_lprefix__lfiniteD,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ Ys )
=> ( ( coinductive_lfinite @ A @ Ys )
=> ( coinductive_lfinite @ A @ Xs ) ) ) ).
% lprefix_lfiniteD
thf(fact_82_lprefix__antisym,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ Ys )
=> ( ( coinductive_lprefix @ A @ Ys @ Xs )
=> ( Xs = Ys ) ) ) ).
% lprefix_antisym
thf(fact_83_llist_Oleq__trans,axiom,
! [A: $tType,X: coinductive_llist @ A,Y: coinductive_llist @ A,Z: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ X @ Y )
=> ( ( coinductive_lprefix @ A @ Y @ Z )
=> ( coinductive_lprefix @ A @ X @ Z ) ) ) ).
% llist.leq_trans
thf(fact_84_lprefix__trans,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ Ys )
=> ( ( coinductive_lprefix @ A @ Ys @ Zs )
=> ( coinductive_lprefix @ A @ Xs @ Zs ) ) ) ).
% lprefix_trans
thf(fact_85_lfinite__LConsI,axiom,
! [A: $tType,Xs: coinductive_llist @ A,X: A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X @ Xs ) ) ) ).
% lfinite_LConsI
thf(fact_86_lnull__imp__lfinite,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( coinductive_lfinite @ A @ Xs ) ) ).
% lnull_imp_lfinite
thf(fact_87_lfinite__LNil,axiom,
! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% lfinite_LNil
thf(fact_88_Le__LCons,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,X: A] :
( ( coinductive_lprefix @ A @ Xs @ Ys )
=> ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ X @ Ys ) ) ) ).
% Le_LCons
thf(fact_89_LCons__lprefix__conv,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X @ Xs ) @ Ys )
= ( ? [Ys4: coinductive_llist @ A] :
( ( Ys
= ( coinductive_LCons @ A @ X @ Ys4 ) )
& ( coinductive_lprefix @ A @ Xs @ Ys4 ) ) ) ) ).
% LCons_lprefix_conv
thf(fact_90_lprefix__not__lnullD,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ Ys )
=> ( ~ ( coinductive_lnull @ A @ Xs )
=> ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% lprefix_not_lnullD
thf(fact_91_lprefix__lnullD,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ Ys )
=> ( ( coinductive_lnull @ A @ Ys )
=> ( coinductive_lnull @ A @ Xs ) ) ) ).
% lprefix_lnullD
thf(fact_92_lprefix__lnull,axiom,
! [A: $tType,Ys: coinductive_llist @ A,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Ys )
=> ( ( coinductive_lprefix @ A @ Xs @ Ys )
= ( coinductive_lnull @ A @ Xs ) ) ) ).
% lprefix_lnull
thf(fact_93_lnull__lprefix,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( coinductive_lprefix @ A @ Xs @ Ys ) ) ).
% lnull_lprefix
thf(fact_94_LNil__lprefix,axiom,
! [A: $tType,Xs: coinductive_llist @ A] : ( coinductive_lprefix @ A @ ( coinductive_LNil @ A ) @ Xs ) ).
% LNil_lprefix
thf(fact_95_lsorted__lprefixD,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ Ys )
=> ( ( coindu63249387sorted @ A @ Ys )
=> ( coindu63249387sorted @ A @ Xs ) ) ) ) ).
% lsorted_lprefixD
thf(fact_96_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_97_lstrict__prefix__def,axiom,
! [A: $tType] :
( ( coindu1478340336prefix @ A )
= ( ^ [Xs5: coinductive_llist @ A,Ys3: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs5 @ Ys3 )
& ( Xs5 != Ys3 ) ) ) ) ).
% lstrict_prefix_def
thf(fact_98_Coinductive__List_Ofinite__lprefix__def,axiom,
! [A: $tType] :
( ( coindu328551480prefix @ A )
= ( coinductive_lprefix @ A ) ) ).
% Coinductive_List.finite_lprefix_def
thf(fact_99_lfinite_Ocases,axiom,
! [A: $tType,A2: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ A2 )
=> ( ( A2
!= ( coinductive_LNil @ A ) )
=> ~ ! [Xs3: coinductive_llist @ A] :
( ? [X3: A] :
( A2
= ( coinductive_LCons @ A @ X3 @ Xs3 ) )
=> ~ ( coinductive_lfinite @ A @ Xs3 ) ) ) ) ).
% lfinite.cases
thf(fact_100_lfinite_Osimps,axiom,
! [A: $tType] :
( ( coinductive_lfinite @ A )
= ( ^ [A3: coinductive_llist @ A] :
( ( A3
= ( coinductive_LNil @ A ) )
| ? [Xs5: coinductive_llist @ A,X2: A] :
( ( A3
= ( coinductive_LCons @ A @ X2 @ Xs5 ) )
& ( coinductive_lfinite @ A @ Xs5 ) ) ) ) ) ).
% lfinite.simps
thf(fact_101_lfinite_Oinducts,axiom,
! [A: $tType,X: coinductive_llist @ A,P3: ( coinductive_llist @ A ) > $o] :
( ( coinductive_lfinite @ A @ X )
=> ( ( P3 @ ( coinductive_LNil @ A ) )
=> ( ! [Xs3: coinductive_llist @ A,X3: A] :
( ( coinductive_lfinite @ A @ Xs3 )
=> ( ( P3 @ Xs3 )
=> ( P3 @ ( coinductive_LCons @ A @ X3 @ Xs3 ) ) ) )
=> ( P3 @ X ) ) ) ) ).
% lfinite.inducts
thf(fact_102_lprefix__code_I2_J,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
~ ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LNil @ A ) ) ).
% lprefix_code(2)
thf(fact_103_lprefix_Ocases,axiom,
! [A: $tType,A1: coinductive_llist @ A,A22: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ A1 @ A22 )
=> ( ( ( A1
= ( coinductive_LNil @ A ) )
=> ! [Xs3: coinductive_llist @ A] : A22 != Xs3 )
=> ~ ! [Xs3: coinductive_llist @ A,Ys5: coinductive_llist @ A,X3: A] :
( ( A1
= ( coinductive_LCons @ A @ X3 @ Xs3 ) )
=> ( ( A22
= ( coinductive_LCons @ A @ X3 @ Ys5 ) )
=> ~ ( coinductive_lprefix @ A @ Xs3 @ Ys5 ) ) ) ) ) ).
% lprefix.cases
thf(fact_104_lprefix_Osimps,axiom,
! [A: $tType] :
( ( coinductive_lprefix @ A )
= ( ^ [A12: coinductive_llist @ A,A23: coinductive_llist @ A] :
( ? [Xs5: coinductive_llist @ A] :
( ( A12
= ( coinductive_LNil @ A ) )
& ( A23 = Xs5 ) )
| ? [Xs5: coinductive_llist @ A,Ys3: coinductive_llist @ A,X2: A] :
( ( A12
= ( coinductive_LCons @ A @ X2 @ Xs5 ) )
& ( A23
= ( coinductive_LCons @ A @ X2 @ Ys3 ) )
& ( coinductive_lprefix @ A @ Xs5 @ Ys3 ) ) ) ) ) ).
% lprefix.simps
thf(fact_105_lprefix_Ocoinduct,axiom,
! [A: $tType,X4: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,X: coinductive_llist @ A,Xa: coinductive_llist @ A] :
( ( X4 @ X @ Xa )
=> ( ! [X3: coinductive_llist @ A,Xa3: coinductive_llist @ A] :
( ( X4 @ X3 @ Xa3 )
=> ( ? [Xs4: coinductive_llist @ A] :
( ( X3
= ( coinductive_LNil @ A ) )
& ( Xa3 = Xs4 ) )
| ? [Xs4: coinductive_llist @ A,Ys2: coinductive_llist @ A,Xb: A] :
( ( X3
= ( coinductive_LCons @ A @ Xb @ Xs4 ) )
& ( Xa3
= ( coinductive_LCons @ A @ Xb @ Ys2 ) )
& ( ( X4 @ Xs4 @ Ys2 )
| ( coinductive_lprefix @ A @ Xs4 @ Ys2 ) ) ) ) )
=> ( coinductive_lprefix @ A @ X @ Xa ) ) ) ).
% lprefix.coinduct
thf(fact_106_lprefix__LCons__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) )
= ( ( Xs
= ( coinductive_LNil @ A ) )
| ? [Xs2: coinductive_llist @ A] :
( ( Xs
= ( coinductive_LCons @ A @ Y @ Xs2 ) )
& ( coinductive_lprefix @ A @ Xs2 @ Ys ) ) ) ) ).
% lprefix_LCons_conv
thf(fact_107_LCons__LCons,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Xs: coinductive_llist @ A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y @ Xs ) )
=> ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y @ Xs ) ) ) ) ) ) ).
% LCons_LCons
thf(fact_108_insertCI,axiom,
! [A: $tType,A2: A,B4: set @ A,B2: A] :
( ( ~ ( member @ A @ A2 @ B4 )
=> ( A2 = B2 ) )
=> ( member @ A @ A2 @ ( insert @ A @ B2 @ B4 ) ) ) ).
% insertCI
thf(fact_109_insert__iff,axiom,
! [A: $tType,A2: A,B2: A,A4: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B2 @ A4 ) )
= ( ( A2 = B2 )
| ( member @ A @ A2 @ A4 ) ) ) ).
% insert_iff
thf(fact_110_insert__subset,axiom,
! [A: $tType,X: A,A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A4 ) @ B4 )
= ( ( member @ A @ X @ B4 )
& ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% insert_subset
thf(fact_111_insert__absorb2,axiom,
! [A: $tType,X: A,A4: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ X @ A4 ) )
= ( insert @ A @ X @ A4 ) ) ).
% insert_absorb2
thf(fact_112_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_113_lsorted__LCons_H,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: A,Xs: coinductive_llist @ A] :
( ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( ~ ( coinductive_lnull @ A @ Xs )
=> ( ( ord_less_eq @ A @ X @ ( coinductive_lhd @ A @ Xs ) )
& ( coindu63249387sorted @ A @ Xs ) ) ) ) ) ).
% lsorted_LCons'
thf(fact_114_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_115_subsetI,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( member @ A @ X3 @ B4 ) )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% subsetI
thf(fact_116_subset__antisym,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ).
% subset_antisym
thf(fact_117_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_118_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_119_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_120_lappend__LNil2,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ Xs @ ( coinductive_LNil @ A ) )
= Xs ) ).
% lappend_LNil2
thf(fact_121_lappend__code_I1_J,axiom,
! [A: $tType,Ys: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ Ys )
= Ys ) ).
% lappend_code(1)
thf(fact_122_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_123_lprefix__lappend__same,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) @ ( coinductive_lappend @ A @ Xs @ Zs ) )
= ( ( coinductive_lfinite @ A @ Xs )
=> ( coinductive_lprefix @ A @ Ys @ Zs ) ) ) ).
% lprefix_lappend_same
thf(fact_124_lhd__lappend,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lhd @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coinductive_lhd @ A @ Ys ) ) )
& ( ~ ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lhd @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% lhd_lappend
thf(fact_125_ltakeWhile_Odisc__iff_I2_J,axiom,
! [A: $tType,P3: A > $o,Xs: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P3 @ Xs ) ) )
= ( ~ ( coinductive_lnull @ A @ Xs )
& ( P3 @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% ltakeWhile.disc_iff(2)
thf(fact_126_ltakeWhile_Odisc__iff_I1_J,axiom,
! [A: $tType,P3: A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P3 @ Xs ) )
= ( ( coinductive_lnull @ A @ Xs )
| ~ ( P3 @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% ltakeWhile.disc_iff(1)
thf(fact_127_lnull__ltakeWhile,axiom,
! [A: $tType,P3: A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P3 @ Xs ) )
= ( ~ ( coinductive_lnull @ A @ Xs )
=> ~ ( P3 @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% lnull_ltakeWhile
thf(fact_128_set__mp,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( member @ A @ X @ A4 )
=> ( member @ A @ X @ B4 ) ) ) ).
% set_mp
thf(fact_129_in__mono,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( member @ A @ X @ A4 )
=> ( member @ A @ X @ B4 ) ) ) ).
% in_mono
thf(fact_130_subsetD,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( member @ A @ C2 @ A4 )
=> ( member @ A @ C2 @ B4 ) ) ) ).
% subsetD
thf(fact_131_subsetCE,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( member @ A @ C2 @ A4 )
=> ( member @ A @ C2 @ B4 ) ) ) ).
% subsetCE
thf(fact_132_equalityE,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( A4 = B4 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ) ).
% equalityE
thf(fact_133_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B5: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ A6 )
=> ( member @ A @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_134_equalityD1,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( A4 = B4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% equalityD1
thf(fact_135_equalityD2,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( A4 = B4 )
=> ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ).
% equalityD2
thf(fact_136_set__rev__mp,axiom,
! [A: $tType,X: A,A4: set @ A,B4: set @ A] :
( ( member @ A @ X @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( member @ A @ X @ B4 ) ) ) ).
% set_rev_mp
thf(fact_137_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B5: set @ A] :
! [T: A] :
( ( member @ A @ T @ A6 )
=> ( member @ A @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_138_rev__subsetD,axiom,
! [A: $tType,C2: A,A4: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( member @ A @ C2 @ B4 ) ) ) ).
% rev_subsetD
thf(fact_139_subset__refl,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ A4 ) ).
% subset_refl
thf(fact_140_Collect__mono,axiom,
! [A: $tType,P3: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P3 @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P3 ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_141_subset__trans,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% subset_trans
thf(fact_142_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y5: set @ A,Z2: set @ A] : Y5 = Z2 )
= ( ^ [A6: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
& ( ord_less_eq @ ( set @ A ) @ B5 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_143_contra__subsetD,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ~ ( member @ A @ C2 @ B4 )
=> ~ ( member @ A @ C2 @ A4 ) ) ) ).
% contra_subsetD
thf(fact_144_Collect__mono__iff,axiom,
! [A: $tType,P3: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P3 ) @ ( collect @ A @ Q ) )
= ( ! [X2: A] :
( ( P3 @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_145_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_146_lprefix__lappend,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] : ( coinductive_lprefix @ A @ Xs @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ).
% lprefix_lappend
thf(fact_147_lappend__lprefixE,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) @ Zs )
=> ~ ! [Zs2: coinductive_llist @ A] :
( Zs
!= ( coinductive_lappend @ A @ Xs @ Zs2 ) ) ) ).
% lappend_lprefixE
thf(fact_148_lprefix__lappendD,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ ( coinductive_lappend @ A @ Ys @ Zs ) )
=> ( ( coinductive_lprefix @ A @ Xs @ Ys )
| ( coinductive_lprefix @ A @ Ys @ Xs ) ) ) ).
% lprefix_lappendD
thf(fact_149_lprefix__conv__lappend,axiom,
! [A: $tType] :
( ( coinductive_lprefix @ A )
= ( ^ [Xs5: coinductive_llist @ A,Ys3: coinductive_llist @ A] :
? [Zs3: coinductive_llist @ A] :
( Ys3
= ( coinductive_lappend @ A @ Xs5 @ Zs3 ) ) ) ) ).
% lprefix_conv_lappend
thf(fact_150_lprefix__lappend__sameI,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ Ys )
=> ( coinductive_lprefix @ A @ ( coinductive_lappend @ A @ Zs @ Xs ) @ ( coinductive_lappend @ A @ Zs @ Ys ) ) ) ).
% lprefix_lappend_sameI
thf(fact_151_lhd__LCons,axiom,
! [A: $tType,X21: A,X22: coinductive_llist @ A] :
( ( coinductive_lhd @ A @ ( coinductive_LCons @ A @ X21 @ X22 ) )
= X21 ) ).
% lhd_LCons
thf(fact_152_ltakeWhile_Oexhaust,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P3: A > $o] :
( ~ ( ( coinductive_lnull @ A @ Xs )
| ~ ( P3 @ ( coinductive_lhd @ A @ Xs ) ) )
=> ~ ( ~ ( coinductive_lnull @ A @ Xs )
=> ~ ( P3 @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% ltakeWhile.exhaust
thf(fact_153_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_154_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_155_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_156_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_157_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_158_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_159_lappend__LNil__LNil,axiom,
! [A: $tType] :
( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% lappend_LNil_LNil
thf(fact_160_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_161_unfold__llist_Osimps_I3_J,axiom,
! [B: $tType,A: $tType,P: A > $o,A2: A,G21: A > B,G22: A > A] :
( ~ ( P @ A2 )
=> ( ( coinductive_lhd @ B @ ( coindu1441602521_llist @ A @ B @ P @ G21 @ G22 @ A2 ) )
= ( G21 @ A2 ) ) ) ).
% unfold_llist.simps(3)
thf(fact_162_llist_Ocorec__sel_I1_J,axiom,
! [A: $tType,C: $tType,P: C > $o,A2: C,G21: C > A,Q22: C > $o,G221: C > ( coinductive_llist @ A ),G222: C > C] :
( ~ ( P @ A2 )
=> ( ( coinductive_lhd @ A @ ( coindu1259883913_llist @ C @ A @ P @ G21 @ Q22 @ G221 @ G222 @ A2 ) )
= ( G21 @ A2 ) ) ) ).
% llist.corec_sel(1)
thf(fact_163_lprefix__lhdD,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ Ys )
=> ( ~ ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lhd @ A @ Xs )
= ( coinductive_lhd @ A @ Ys ) ) ) ) ).
% lprefix_lhdD
thf(fact_164_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_165_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_166_ltakeWhile_Odisc_I2_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P3: A > $o] :
( ~ ( coinductive_lnull @ A @ Xs )
=> ( ( P3 @ ( coinductive_lhd @ A @ Xs ) )
=> ~ ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P3 @ Xs ) ) ) ) ).
% ltakeWhile.disc(2)
thf(fact_167_ltakeWhile_Odisc_I1_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P3: A > $o] :
( ( ( coinductive_lnull @ A @ Xs )
| ~ ( P3 @ ( coinductive_lhd @ A @ Xs ) ) )
=> ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P3 @ Xs ) ) ) ).
% ltakeWhile.disc(1)
thf(fact_168_lhd__ltakeWhile,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P3: A > $o] :
( ~ ( coinductive_lnull @ A @ Xs )
=> ( ( P3 @ ( coinductive_lhd @ A @ Xs ) )
=> ( ( coinductive_lhd @ A @ ( coindu501562517eWhile @ A @ P3 @ Xs ) )
= ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% lhd_ltakeWhile
thf(fact_169_ltakeWhile__eq__LNil__iff,axiom,
! [A: $tType,P3: A > $o,Xs: coinductive_llist @ A] :
( ( ( coindu501562517eWhile @ A @ P3 @ Xs )
= ( coinductive_LNil @ A ) )
= ( ( Xs
!= ( coinductive_LNil @ A ) )
=> ~ ( P3 @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% ltakeWhile_eq_LNil_iff
thf(fact_170_lfinite__rev__induct,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P3: ( coinductive_llist @ A ) > $o] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ( P3 @ ( coinductive_LNil @ A ) )
=> ( ! [X3: A,Xs3: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs3 )
=> ( ( P3 @ Xs3 )
=> ( P3 @ ( coinductive_lappend @ A @ Xs3 @ ( coinductive_LCons @ A @ X3 @ ( coinductive_LNil @ A ) ) ) ) ) )
=> ( P3 @ Xs ) ) ) ) ).
% lfinite_rev_induct
thf(fact_171_ltakeWhile_Octr_I1_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P3: A > $o] :
( ( ( coinductive_lnull @ A @ Xs )
| ~ ( P3 @ ( coinductive_lhd @ A @ Xs ) ) )
=> ( ( coindu501562517eWhile @ A @ P3 @ Xs )
= ( coinductive_LNil @ A ) ) ) ).
% ltakeWhile.ctr(1)
thf(fact_172_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% dual_order.antisym
thf(fact_173_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_174_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P3: A > A > $o,A2: A,B2: A] :
( ! [A5: A,B3: A] :
( ( ord_less_eq @ A @ A5 @ B3 )
=> ( P3 @ A5 @ B3 ) )
=> ( ! [A5: A,B3: A] :
( ( P3 @ B3 @ A5 )
=> ( P3 @ A5 @ B3 ) )
=> ( P3 @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_175_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_176_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% order_trans
thf(fact_177_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ) ).
% order_class.order.antisym
thf(fact_178_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_179_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_180_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_181_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z ) )
=> ( ( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less_eq @ A @ X @ Z ) )
=> ( ( ( ord_less_eq @ A @ X @ Z )
=> ~ ( ord_less_eq @ A @ Z @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z )
=> ~ ( ord_less_eq @ A @ Z @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z @ X )
=> ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_182_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% order.trans
thf(fact_183_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_184_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_185_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_186_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_187_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y5: A,Z2: A] : Y5 = Z2 )
= ( ^ [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
& ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ) ) ).
% eq_iff
thf(fact_188_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > B,C2: B] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X3: A,Y4: A] :
( ( ord_less_eq @ A @ X3 @ Y4 )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_189_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B2: B,C2: B] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X3: B,Y4: B] :
( ( ord_less_eq @ B @ X3 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_190_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C @ ( type2 @ C ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > C,C2: C] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
=> ( ! [X3: A,Y4: A] :
( ( ord_less_eq @ A @ X3 @ Y4 )
=> ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_191_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B2: B,C2: B] :
( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X3: B,Y4: B] :
( ( ord_less_eq @ B @ X3 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_192_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_193_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_194_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_195_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_196_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_197_mk__disjoint__insert,axiom,
! [A: $tType,A2: A,A4: set @ A] :
( ( member @ A @ A2 @ A4 )
=> ? [B6: set @ A] :
( ( A4
= ( insert @ A @ A2 @ B6 ) )
& ~ ( member @ A @ A2 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_198_subset__insertI2,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B2 @ B4 ) ) ) ).
% subset_insertI2
thf(fact_199_subset__insertI,axiom,
! [A: $tType,B4: set @ A,A2: A] : ( ord_less_eq @ ( set @ A ) @ B4 @ ( insert @ A @ A2 @ B4 ) ) ).
% subset_insertI
thf(fact_200_insert__commute,axiom,
! [A: $tType,X: A,Y: A,A4: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ Y @ A4 ) )
= ( insert @ A @ Y @ ( insert @ A @ X @ A4 ) ) ) ).
% insert_commute
thf(fact_201_subset__insert,axiom,
! [A: $tType,X: A,A4: set @ A,B4: set @ A] :
( ~ ( member @ A @ X @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ X @ B4 ) )
= ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% subset_insert
thf(fact_202_insert__eq__iff,axiom,
! [A: $tType,A2: A,A4: set @ A,B2: A,B4: set @ A] :
( ~ ( member @ A @ A2 @ A4 )
=> ( ~ ( member @ A @ B2 @ B4 )
=> ( ( ( insert @ A @ A2 @ A4 )
= ( insert @ A @ B2 @ B4 ) )
= ( ( ( A2 = B2 )
=> ( A4 = B4 ) )
& ( ( A2 != B2 )
=> ? [C4: set @ A] :
( ( A4
= ( insert @ A @ B2 @ C4 ) )
& ~ ( member @ A @ B2 @ C4 )
& ( B4
= ( insert @ A @ A2 @ C4 ) )
& ~ ( member @ A @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_203_insert__absorb,axiom,
! [A: $tType,A2: A,A4: set @ A] :
( ( member @ A @ A2 @ A4 )
=> ( ( insert @ A @ A2 @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_204_insert__ident,axiom,
! [A: $tType,X: A,A4: set @ A,B4: set @ A] :
( ~ ( member @ A @ X @ A4 )
=> ( ~ ( member @ A @ X @ B4 )
=> ( ( ( insert @ A @ X @ A4 )
= ( insert @ A @ X @ B4 ) )
= ( A4 = B4 ) ) ) ) ).
% insert_ident
thf(fact_205_Set_Oinsert__mono,axiom,
! [A: $tType,C3: set @ A,D: set @ A,A2: A] :
( ( ord_less_eq @ ( set @ A ) @ C3 @ D )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A2 @ C3 ) @ ( insert @ A @ A2 @ D ) ) ) ).
% Set.insert_mono
thf(fact_206_Set_Oset__insert,axiom,
! [A: $tType,X: A,A4: set @ A] :
( ( member @ A @ X @ A4 )
=> ~ ! [B6: set @ A] :
( ( A4
= ( insert @ A @ X @ B6 ) )
=> ( member @ A @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_207_insertI2,axiom,
! [A: $tType,A2: A,B4: set @ A,B2: A] :
( ( member @ A @ A2 @ B4 )
=> ( member @ A @ A2 @ ( insert @ A @ B2 @ B4 ) ) ) ).
% insertI2
thf(fact_208_insertI1,axiom,
! [A: $tType,A2: A,B4: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B4 ) ) ).
% insertI1
thf(fact_209_insertE,axiom,
! [A: $tType,A2: A,B2: A,A4: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B2 @ A4 ) )
=> ( ( A2 != B2 )
=> ( member @ A @ A2 @ A4 ) ) ) ).
% insertE
thf(fact_210_insert__subsetI,axiom,
! [A: $tType,X: A,A4: set @ A,X4: set @ A] :
( ( member @ A @ X @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ X4 @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ X4 ) @ A4 ) ) ) ).
% insert_subsetI
thf(fact_211_llexord__conv,axiom,
! [A: $tType] :
( ( coinductive_llexord @ A )
= ( ^ [R: A > A > $o,Xs5: coinductive_llist @ A,Ys3: coinductive_llist @ A] :
( ( Xs5 = Ys3 )
| ? [Zs3: coinductive_llist @ A,Xs2: coinductive_llist @ A,Y3: A,Ys4: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Zs3 )
& ( Xs5
= ( coinductive_lappend @ A @ Zs3 @ Xs2 ) )
& ( Ys3
= ( coinductive_lappend @ A @ Zs3 @ ( coinductive_LCons @ A @ Y3 @ Ys4 ) ) )
& ( ( Xs2
= ( coinductive_LNil @ A ) )
| ( R @ ( coinductive_lhd @ A @ Xs2 ) @ Y3 ) ) ) ) ) ) ).
% llexord_conv
thf(fact_212_ltakeWhile_Ocode,axiom,
! [A: $tType] :
( ( coindu501562517eWhile @ A )
= ( ^ [P4: A > $o,Xs5: coinductive_llist @ A] :
( if @ ( coinductive_llist @ A )
@ ( ( coinductive_lnull @ A @ Xs5 )
| ~ ( P4 @ ( coinductive_lhd @ A @ Xs5 ) ) )
@ ( coinductive_LNil @ A )
@ ( coinductive_LCons @ A @ ( coinductive_lhd @ A @ Xs5 ) @ ( coindu501562517eWhile @ A @ P4 @ ( coinductive_ltl @ A @ Xs5 ) ) ) ) ) ) ).
% ltakeWhile.code
thf(fact_213_llast__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 ) ) ) ) ) ) ).
% llast_lappend
thf(fact_214_llexord__refl,axiom,
! [A: $tType,R2: A > A > $o,Xs: coinductive_llist @ A] : ( coinductive_llexord @ A @ R2 @ Xs @ Xs ) ).
% llexord_refl
thf(fact_215_lfinite__ltl,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( coinductive_ltl @ A @ Xs ) )
= ( coinductive_lfinite @ A @ Xs ) ) ).
% lfinite_ltl
thf(fact_216_llexord__LCons__LCons,axiom,
! [A: $tType,R2: A > A > $o,X: A,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R2 @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ Y @ Ys ) )
= ( ( ( X = Y )
& ( coinductive_llexord @ A @ R2 @ Xs @ Ys ) )
| ( R2 @ X @ Y ) ) ) ).
% llexord_LCons_LCons
thf(fact_217_llexord__LNil__right,axiom,
! [A: $tType,Ys: coinductive_llist @ A,R2: A > A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Ys )
=> ( ( coinductive_llexord @ A @ R2 @ Xs @ Ys )
= ( coinductive_lnull @ A @ Xs ) ) ) ).
% llexord_LNil_right
thf(fact_218_llexord__code_I1_J,axiom,
! [A: $tType,R2: A > A > $o,Ys: coinductive_llist @ A] : ( coinductive_llexord @ A @ R2 @ ( coinductive_LNil @ A ) @ Ys ) ).
% llexord_code(1)
thf(fact_219_unfold__llist__id,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coindu1441602521_llist @ ( coinductive_llist @ A ) @ A @ ( coinductive_lnull @ A ) @ ( coinductive_lhd @ A ) @ ( coinductive_ltl @ A ) @ Xs )
= Xs ) ).
% unfold_llist_id
thf(fact_220_ltl__lappend,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_ltl @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coinductive_ltl @ A @ Ys ) ) )
& ( ~ ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_ltl @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coinductive_lappend @ A @ ( coinductive_ltl @ A @ Xs ) @ Ys ) ) ) ) ).
% ltl_lappend
thf(fact_221_lhd__LCons__ltl,axiom,
! [A: $tType,Llist: coinductive_llist @ A] :
( ~ ( coinductive_lnull @ A @ Llist )
=> ( ( coinductive_LCons @ A @ ( coinductive_lhd @ A @ Llist ) @ ( coinductive_ltl @ A @ Llist ) )
= Llist ) ) ).
% lhd_LCons_ltl
thf(fact_222_llexord__coinduct,axiom,
! [A: $tType,X4: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,R2: A > A > $o] :
( ( X4 @ Xs @ Ys )
=> ( ! [Xs3: coinductive_llist @ A,Ys5: coinductive_llist @ A] :
( ( X4 @ Xs3 @ Ys5 )
=> ( ~ ( coinductive_lnull @ A @ Xs3 )
=> ( ~ ( coinductive_lnull @ A @ Ys5 )
& ( ~ ( coinductive_lnull @ A @ Ys5 )
=> ( ( R2 @ ( coinductive_lhd @ A @ Xs3 ) @ ( coinductive_lhd @ A @ Ys5 ) )
| ( ( ( coinductive_lhd @ A @ Xs3 )
= ( coinductive_lhd @ A @ Ys5 ) )
& ( ( X4 @ ( coinductive_ltl @ A @ Xs3 ) @ ( coinductive_ltl @ A @ Ys5 ) )
| ( coinductive_llexord @ A @ R2 @ ( coinductive_ltl @ A @ Xs3 ) @ ( coinductive_ltl @ A @ Ys5 ) ) ) ) ) ) ) ) )
=> ( coinductive_llexord @ A @ R2 @ Xs @ Ys ) ) ) ).
% llexord_coinduct
thf(fact_223_llexord__append__right,axiom,
! [A: $tType,R2: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] : ( coinductive_llexord @ A @ R2 @ Xs @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ).
% llexord_append_right
thf(fact_224_llexord__lappend__leftI,axiom,
! [A: $tType,R2: A > A > $o,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A,Xs: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R2 @ Ys @ Zs )
=> ( coinductive_llexord @ A @ R2 @ ( coinductive_lappend @ A @ Xs @ Ys ) @ ( coinductive_lappend @ A @ Xs @ Zs ) ) ) ).
% llexord_lappend_leftI
thf(fact_225_lprefix__ltlI,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ Ys )
=> ( coinductive_lprefix @ A @ ( coinductive_ltl @ A @ Xs ) @ ( coinductive_ltl @ A @ Ys ) ) ) ).
% lprefix_ltlI
thf(fact_226_ltl__simps_I2_J,axiom,
! [A: $tType,X21: A,X22: coinductive_llist @ A] :
( ( coinductive_ltl @ A @ ( coinductive_LCons @ A @ X21 @ X22 ) )
= X22 ) ).
% ltl_simps(2)
thf(fact_227_lnull__ltlI,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( coinductive_lnull @ A @ ( coinductive_ltl @ A @ Xs ) ) ) ).
% lnull_ltlI
thf(fact_228_ltl__simps_I1_J,axiom,
! [A: $tType] :
( ( coinductive_ltl @ A @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% ltl_simps(1)
thf(fact_229_lsorted__ltlI,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [Xs: coinductive_llist @ A] :
( ( coindu63249387sorted @ A @ Xs )
=> ( coindu63249387sorted @ A @ ( coinductive_ltl @ A @ Xs ) ) ) ) ).
% lsorted_ltlI
thf(fact_230_lprefix__imp__llexord,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,R2: A > A > $o] :
( ( coinductive_lprefix @ A @ Xs @ Ys )
=> ( coinductive_llexord @ A @ R2 @ Xs @ Ys ) ) ).
% lprefix_imp_llexord
thf(fact_231_llexord__LCons__less,axiom,
! [A: $tType,R2: A > A > $o,X: A,Y: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( R2 @ X @ Y )
=> ( coinductive_llexord @ A @ R2 @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ Y @ Ys ) ) ) ).
% llexord_LCons_less
thf(fact_232_llexord__LCons__eq,axiom,
! [A: $tType,R2: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,X: A] :
( ( coinductive_llexord @ A @ R2 @ Xs @ Ys )
=> ( coinductive_llexord @ A @ R2 @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ X @ Ys ) ) ) ).
% llexord_LCons_eq
thf(fact_233_llexord__LCons__left,axiom,
! [A: $tType,R2: A > A > $o,X: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R2 @ ( coinductive_LCons @ A @ X @ Xs ) @ Ys )
= ( ? [Y3: A,Ys4: coinductive_llist @ A] :
( ( Ys
= ( coinductive_LCons @ A @ Y3 @ Ys4 ) )
& ( ( ( X = Y3 )
& ( coinductive_llexord @ A @ R2 @ Xs @ Ys4 ) )
| ( R2 @ X @ Y3 ) ) ) ) ) ).
% llexord_LCons_left
thf(fact_234_llexord__code_I3_J,axiom,
! [A: $tType,R2: A > A > $o,X: A,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R2 @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ Y @ Ys ) )
= ( ( R2 @ X @ Y )
| ( ( X = Y )
& ( coinductive_llexord @ A @ R2 @ Xs @ Ys ) ) ) ) ).
% llexord_code(3)
thf(fact_235_lnull__llexord,axiom,
! [A: $tType,Xs: coinductive_llist @ A,R2: A > A > $o,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( coinductive_llexord @ A @ R2 @ Xs @ Ys ) ) ).
% lnull_llexord
thf(fact_236_llexord__LNil,axiom,
! [A: $tType,R2: A > A > $o,Ys: coinductive_llist @ A] : ( coinductive_llexord @ A @ R2 @ ( coinductive_LNil @ A ) @ Ys ) ).
% llexord_LNil
thf(fact_237_unfold__llist_Osimps_I4_J,axiom,
! [B: $tType,A: $tType,P: A > $o,A2: A,G21: A > B,G22: A > A] :
( ~ ( P @ A2 )
=> ( ( coinductive_ltl @ B @ ( coindu1441602521_llist @ A @ B @ P @ G21 @ G22 @ A2 ) )
= ( coindu1441602521_llist @ A @ B @ P @ G21 @ G22 @ ( G22 @ A2 ) ) ) ) ).
% unfold_llist.simps(4)
thf(fact_238_llist_Ocorec__sel_I2_J,axiom,
! [A: $tType,C: $tType,P: C > $o,A2: C,Q22: C > $o,G21: C > A,G221: C > ( coinductive_llist @ A ),G222: C > C] :
( ~ ( P @ A2 )
=> ( ( ( Q22 @ A2 )
=> ( ( coinductive_ltl @ A @ ( coindu1259883913_llist @ C @ A @ P @ G21 @ Q22 @ G221 @ G222 @ A2 ) )
= ( G221 @ A2 ) ) )
& ( ~ ( Q22 @ A2 )
=> ( ( coinductive_ltl @ A @ ( coindu1259883913_llist @ C @ A @ P @ G21 @ Q22 @ G221 @ G222 @ A2 ) )
= ( coindu1259883913_llist @ C @ A @ P @ G21 @ Q22 @ G221 @ G222 @ ( G222 @ A2 ) ) ) ) ) ) ).
% llist.corec_sel(2)
thf(fact_239_llexord__antisym,axiom,
! [A: $tType,R2: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R2 @ Xs @ Ys )
=> ( ( coinductive_llexord @ A @ R2 @ Ys @ Xs )
=> ( ! [A5: A,B3: A] :
( ( R2 @ A5 @ B3 )
=> ~ ( R2 @ B3 @ A5 ) )
=> ( Xs = Ys ) ) ) ) ).
% llexord_antisym
thf(fact_240_llexord__linear,axiom,
! [A: $tType,R2: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ! [X3: A,Y4: A] :
( ( R2 @ X3 @ Y4 )
| ( X3 = Y4 )
| ( R2 @ Y4 @ X3 ) )
=> ( ( coinductive_llexord @ A @ R2 @ Xs @ Ys )
| ( coinductive_llexord @ A @ R2 @ Ys @ Xs ) ) ) ).
% llexord_linear
thf(fact_241_llexord__trans,axiom,
! [A: $tType,R2: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R2 @ Xs @ Ys )
=> ( ( coinductive_llexord @ A @ R2 @ Ys @ Zs )
=> ( ! [A5: A,B3: A,C5: A] :
( ( R2 @ A5 @ B3 )
=> ( ( R2 @ B3 @ C5 )
=> ( R2 @ A5 @ C5 ) ) )
=> ( coinductive_llexord @ A @ R2 @ Xs @ Zs ) ) ) ) ).
% llexord_trans
thf(fact_242_llist_Ocoinduct__strong,axiom,
! [A: $tType,R3: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,Llist: coinductive_llist @ A,Llist3: coinductive_llist @ A] :
( ( R3 @ Llist @ Llist3 )
=> ( ! [Llist4: coinductive_llist @ A,Llist5: coinductive_llist @ A] :
( ( R3 @ Llist4 @ Llist5 )
=> ( ( ( coinductive_lnull @ A @ Llist4 )
= ( coinductive_lnull @ A @ Llist5 ) )
& ( ~ ( coinductive_lnull @ A @ Llist4 )
=> ( ~ ( coinductive_lnull @ A @ Llist5 )
=> ( ( ( coinductive_lhd @ A @ Llist4 )
= ( coinductive_lhd @ A @ Llist5 ) )
& ( ( R3 @ ( coinductive_ltl @ A @ Llist4 ) @ ( coinductive_ltl @ A @ Llist5 ) )
| ( ( coinductive_ltl @ A @ Llist4 )
= ( coinductive_ltl @ A @ Llist5 ) ) ) ) ) ) ) )
=> ( Llist = Llist3 ) ) ) ).
% llist.coinduct_strong
thf(fact_243_llist_Ocoinduct,axiom,
! [A: $tType,R3: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,Llist: coinductive_llist @ A,Llist3: coinductive_llist @ A] :
( ( R3 @ Llist @ Llist3 )
=> ( ! [Llist4: coinductive_llist @ A,Llist5: coinductive_llist @ A] :
( ( R3 @ Llist4 @ Llist5 )
=> ( ( ( coinductive_lnull @ A @ Llist4 )
= ( coinductive_lnull @ A @ Llist5 ) )
& ( ~ ( coinductive_lnull @ A @ Llist4 )
=> ( ~ ( coinductive_lnull @ A @ Llist5 )
=> ( ( ( coinductive_lhd @ A @ Llist4 )
= ( coinductive_lhd @ A @ Llist5 ) )
& ( R3 @ ( coinductive_ltl @ A @ Llist4 ) @ ( coinductive_ltl @ A @ Llist5 ) ) ) ) ) ) )
=> ( Llist = Llist3 ) ) ) ).
% llist.coinduct
thf(fact_244_llist_Oexpand,axiom,
! [A: $tType,Llist: coinductive_llist @ A,Llist3: coinductive_llist @ A] :
( ( ( coinductive_lnull @ A @ Llist )
= ( coinductive_lnull @ A @ Llist3 ) )
=> ( ( ~ ( coinductive_lnull @ A @ Llist )
=> ( ~ ( coinductive_lnull @ A @ Llist3 )
=> ( ( ( coinductive_lhd @ A @ Llist )
= ( coinductive_lhd @ A @ Llist3 ) )
& ( ( coinductive_ltl @ A @ Llist )
= ( coinductive_ltl @ A @ Llist3 ) ) ) ) )
=> ( Llist = Llist3 ) ) ) ).
% llist.expand
thf(fact_245_lappend__ltl,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ~ ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lappend @ A @ ( coinductive_ltl @ A @ Xs ) @ Ys )
= ( coinductive_ltl @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ) ) ).
% lappend_ltl
thf(fact_246_lfinite__induct,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P3: ( coinductive_llist @ A ) > $o] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ! [Xs3: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs3 )
=> ( P3 @ Xs3 ) )
=> ( ! [Xs3: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs3 )
=> ( ~ ( coinductive_lnull @ A @ Xs3 )
=> ( ( P3 @ ( coinductive_ltl @ A @ Xs3 ) )
=> ( P3 @ Xs3 ) ) ) )
=> ( P3 @ Xs ) ) ) ) ).
% lfinite_induct
thf(fact_247_ltl__unfold__llist,axiom,
! [A: $tType,B: $tType,IS_LNIL: B > $o,A2: B,LHD: B > A,LTL: B > B] :
( ( ( IS_LNIL @ A2 )
=> ( ( coinductive_ltl @ A @ ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ A2 ) )
= ( coinductive_LNil @ A ) ) )
& ( ~ ( IS_LNIL @ A2 )
=> ( ( coinductive_ltl @ A @ ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ A2 ) )
= ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ ( LTL @ A2 ) ) ) ) ) ).
% ltl_unfold_llist
thf(fact_248_llexord__code_I2_J,axiom,
! [A: $tType,R2: A > A > $o,X: A,Xs: coinductive_llist @ A] :
~ ( coinductive_llexord @ A @ R2 @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LNil @ A ) ) ).
% llexord_code(2)
thf(fact_249_llexord_Ocases,axiom,
! [A: $tType,R2: A > A > $o,A1: coinductive_llist @ A,A22: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R2 @ A1 @ A22 )
=> ( ! [Xs3: coinductive_llist @ A,Ys5: coinductive_llist @ A,X3: A] :
( ( A1
= ( coinductive_LCons @ A @ X3 @ Xs3 ) )
=> ( ( A22
= ( coinductive_LCons @ A @ X3 @ Ys5 ) )
=> ~ ( coinductive_llexord @ A @ R2 @ Xs3 @ Ys5 ) ) )
=> ( ! [X3: A] :
( ? [Xs3: coinductive_llist @ A] :
( A1
= ( coinductive_LCons @ A @ X3 @ Xs3 ) )
=> ! [Y4: A] :
( ? [Ys5: coinductive_llist @ A] :
( A22
= ( coinductive_LCons @ A @ Y4 @ Ys5 ) )
=> ~ ( R2 @ X3 @ Y4 ) ) )
=> ~ ( ( A1
= ( coinductive_LNil @ A ) )
=> ! [Ys5: coinductive_llist @ A] : A22 != Ys5 ) ) ) ) ).
% llexord.cases
thf(fact_250_llexord_Osimps,axiom,
! [A: $tType] :
( ( coinductive_llexord @ A )
= ( ^ [R: A > A > $o,A12: coinductive_llist @ A,A23: coinductive_llist @ A] :
( ? [Xs5: coinductive_llist @ A,Ys3: coinductive_llist @ A,X2: A] :
( ( A12
= ( coinductive_LCons @ A @ X2 @ Xs5 ) )
& ( A23
= ( coinductive_LCons @ A @ X2 @ Ys3 ) )
& ( coinductive_llexord @ A @ R @ Xs5 @ Ys3 ) )
| ? [X2: A,Y3: A,Xs5: coinductive_llist @ A,Ys3: coinductive_llist @ A] :
( ( A12
= ( coinductive_LCons @ A @ X2 @ Xs5 ) )
& ( A23
= ( coinductive_LCons @ A @ Y3 @ Ys3 ) )
& ( R @ X2 @ Y3 ) )
| ? [Ys3: coinductive_llist @ A] :
( ( A12
= ( coinductive_LNil @ A ) )
& ( A23 = Ys3 ) ) ) ) ) ).
% llexord.simps
thf(fact_251_llexord_Ocoinduct,axiom,
! [A: $tType,X4: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,X: coinductive_llist @ A,Xa: coinductive_llist @ A,R2: A > A > $o] :
( ( X4 @ X @ Xa )
=> ( ! [X3: coinductive_llist @ A,Xa3: coinductive_llist @ A] :
( ( X4 @ X3 @ Xa3 )
=> ( ? [Xs4: coinductive_llist @ A,Ys2: coinductive_llist @ A,Xb: A] :
( ( X3
= ( coinductive_LCons @ A @ Xb @ Xs4 ) )
& ( Xa3
= ( coinductive_LCons @ A @ Xb @ Ys2 ) )
& ( ( X4 @ Xs4 @ Ys2 )
| ( coinductive_llexord @ A @ R2 @ Xs4 @ Ys2 ) ) )
| ? [Xb: A,Y2: A,Xs4: coinductive_llist @ A,Ys2: coinductive_llist @ A] :
( ( X3
= ( coinductive_LCons @ A @ Xb @ Xs4 ) )
& ( Xa3
= ( coinductive_LCons @ A @ Y2 @ Ys2 ) )
& ( R2 @ Xb @ Y2 ) )
| ? [Ys2: coinductive_llist @ A] :
( ( X3
= ( coinductive_LNil @ A ) )
& ( Xa3 = Ys2 ) ) ) )
=> ( coinductive_llexord @ A @ R2 @ X @ Xa ) ) ) ).
% llexord.coinduct
thf(fact_252_llast__LNil,axiom,
! [A: $tType] :
( ( coinductive_llast @ A @ ( coinductive_LNil @ A ) )
= ( undefined @ A ) ) ).
% llast_LNil
thf(fact_253_llast__linfinite,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_llast @ A @ Xs )
= ( undefined @ A ) ) ) ).
% llast_linfinite
%----Type constructors (10)
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A7: $tType,A8: $tType] :
( ( preorder @ A8 @ ( type2 @ A8 ) )
=> ( preorder @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A7: $tType,A8: $tType] :
( ( order @ A8 @ ( type2 @ A8 ) )
=> ( order @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A7: $tType,A8: $tType] :
( ( ord @ A8 @ ( type2 @ A8 ) )
=> ( ord @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_1,axiom,
! [A7: $tType] : ( preorder @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_2,axiom,
! [A7: $tType] : ( order @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_3,axiom,
! [A7: $tType] : ( ord @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_4,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_5,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_6,axiom,
ord @ $o @ ( type2 @ $o ) ).
%----Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P3: $o] :
( ( P3 = $true )
| ( P3 = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $true @ X @ Y )
= X ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( ( coinductive_lnull @ a @ ( lMirro999291890or_aux @ a @ acc @ xs ) )
= ( ( coinductive_lnull @ a @ xs )
& ( coinductive_lnull @ a @ acc ) ) ) ).
%------------------------------------------------------------------------------