TPTP Problem File: DAT131^1.p
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%------------------------------------------------------------------------------
% File : DAT131^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Coinductive list 3781
% 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 : coinductive_list__3781.p [Bla16]
% Status : Theorem
% Rating : 0.33 v8.1.0, 1.00 v7.1.0
% Syntax : Number of formulae : 315 ( 136 unt; 46 typ; 0 def)
% Number of atoms : 736 ( 249 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 3937 ( 104 ~; 39 |; 72 &;3400 @)
% ( 0 <=>; 322 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 9 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 310 ( 310 >; 0 *; 0 +; 0 <<)
% Number of symbols : 48 ( 45 usr; 9 con; 0-8 aty)
% Number of variables : 1224 ( 74 ^;1043 !; 67 ?;1224 :)
% ( 40 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 15:03:12.383
%------------------------------------------------------------------------------
%----Could-be-implicit typings (5)
thf(ty_t_Coinductive__List__Mirabelle__kmikjhschf_Ollist,type,
coindu1593790203_llist: $tType > $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $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 (41)
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__Mirabelle__kmikjhschf_Ofinite__lprefix,type,
coindu1571841457prefix:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ogen__lset,type,
coindu1928975208n_lset:
!>[A: $tType] : ( ( set @ A ) > ( coindu1593790203_llist @ A ) > ( set @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olappend,type,
coindu268472904append:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olfinite,type,
coindu1213758845finite:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollast,type,
coindu2000965700_llast:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > A ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollexord,type,
coindu300403952lexord:
!>[A: $tType] : ( ( A > A > $o ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_OLCons,type,
coindu1121789889_LCons:
!>[A: $tType] : ( A > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_OLNil,type,
coindu1598213697e_LNil:
!>[A: $tType] : ( coindu1593790203_llist @ A ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Ocorec__llist,type,
coindu1244876290_llist:
!>[C: $tType,A: $tType] : ( ( C > $o ) > ( C > A ) > ( C > $o ) > ( C > ( coindu1593790203_llist @ A ) ) > ( C > C ) > C > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Olnull,type,
coindu335574135_lnull:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Opred__llist,type,
coindu44415597_llist:
!>[A: $tType] : ( ( A > $o ) > ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olmember,type,
coindu567634248member:
!>[A: $tType] : ( A > ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olprefix,type,
coindu1696667936prefix:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olsetp,type,
coindu1055736764_lsetp:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > A > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olstrict__prefix,type,
coindu574146665prefix:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_OltakeWhile,type,
coindu721411036eWhile:
!>[A: $tType] : ( ( A > $o ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Oord_Olsorted,type,
coindu543932918sorted:
!>[A: $tType] : ( ( A > A > $o ) > ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Oord__class_Olsorted,type,
coindu48241764sorted:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ounfold__llist,type,
coindu1599971794_llist:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > B ) > ( A > A ) > A > ( coindu1593790203_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_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_Ointernal__case__prod,type,
produc2004651681e_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
product_rec_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_r,type,
r: a > a > $o ).
thf(sy_v_thesis____,type,
thesis: $o ).
thf(sy_v_x____,type,
x: a ).
thf(sy_v_xs,type,
xs: coindu1593790203_llist @ a ).
thf(sy_v_xs_H____,type,
xs2: coindu1593790203_llist @ a ).
thf(sy_v_xsa____,type,
xsa: coindu1593790203_llist @ a ).
thf(sy_v_ys,type,
ys: coindu1593790203_llist @ a ).
thf(sy_v_ysa____,type,
ysa: coindu1593790203_llist @ a ).
%----Relevant facts (254)
thf(fact_0_local_Olinear,axiom,
! [X: a,Y: a] :
( ( r @ X @ Y )
| ( X = Y )
| ( r @ Y @ X ) ) ).
% local.linear
thf(fact_1__092_060open_062_092_060not_062_Allexord_Ar_Ays_Axs_092_060close_062,axiom,
~ ( coindu300403952lexord @ a @ r @ ysa @ xsa ) ).
% \<open>\<not> llexord r ys xs\<close>
thf(fact_2__092_060open_062_092_060not_062_Allexord_Ar_Ays_Axs_092_060close_062,axiom,
~ ( coindu300403952lexord @ a @ r @ ys @ xs ) ).
% \<open>\<not> llexord r ys xs\<close>
thf(fact_3_LCons,axiom,
( xsa
= ( coindu1121789889_LCons @ a @ x @ xs2 ) ) ).
% LCons
thf(fact_4_llist_Oinject,axiom,
! [A: $tType,X21: A,X22: coindu1593790203_llist @ A,Y21: A,Y22: coindu1593790203_llist @ A] :
( ( ( coindu1121789889_LCons @ A @ X21 @ X22 )
= ( coindu1121789889_LCons @ A @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% llist.inject
thf(fact_5_llexord__refl,axiom,
! [A: $tType,R: A > A > $o,Xs: coindu1593790203_llist @ A] : ( coindu300403952lexord @ A @ R @ Xs @ Xs ) ).
% llexord_refl
thf(fact_6_llexord__LCons__LCons,axiom,
! [A: $tType,R: A > A > $o,X: A,Xs: coindu1593790203_llist @ A,Y: A,Ys: coindu1593790203_llist @ A] :
( ( coindu300403952lexord @ A @ R @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ ( coindu1121789889_LCons @ A @ Y @ Ys ) )
= ( ( ( X = Y )
& ( coindu300403952lexord @ A @ R @ Xs @ Ys ) )
| ( R @ X @ Y ) ) ) ).
% llexord_LCons_LCons
thf(fact_7_llexord__trans,axiom,
! [A: $tType,R: A > A > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
( ( coindu300403952lexord @ A @ R @ Xs @ Ys )
=> ( ( coindu300403952lexord @ A @ R @ Ys @ Zs )
=> ( ! [A2: A,B2: A,C2: A] :
( ( R @ A2 @ B2 )
=> ( ( R @ B2 @ C2 )
=> ( R @ A2 @ C2 ) ) )
=> ( coindu300403952lexord @ A @ R @ Xs @ Zs ) ) ) ) ).
% llexord_trans
thf(fact_8_llexord__antisym,axiom,
! [A: $tType,R: A > A > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu300403952lexord @ A @ R @ Xs @ Ys )
=> ( ( coindu300403952lexord @ A @ R @ Ys @ Xs )
=> ( ! [A2: A,B2: A] :
( ( R @ A2 @ B2 )
=> ~ ( R @ B2 @ A2 ) )
=> ( Xs = Ys ) ) ) ) ).
% llexord_antisym
thf(fact_9_llexord__LCons__left,axiom,
! [A: $tType,R: A > A > $o,X: A,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu300403952lexord @ A @ R @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ Ys )
= ( ? [Y2: A,Ys2: coindu1593790203_llist @ A] :
( ( Ys
= ( coindu1121789889_LCons @ A @ Y2 @ Ys2 ) )
& ( ( ( X = Y2 )
& ( coindu300403952lexord @ A @ R @ Xs @ Ys2 ) )
| ( R @ X @ Y2 ) ) ) ) ) ).
% llexord_LCons_left
thf(fact_10_llexord__LCons__eq,axiom,
! [A: $tType,R: A > A > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,X: A] :
( ( coindu300403952lexord @ A @ R @ Xs @ Ys )
=> ( coindu300403952lexord @ A @ R @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ ( coindu1121789889_LCons @ A @ X @ Ys ) ) ) ).
% llexord_LCons_eq
thf(fact_11_llexord__LCons__less,axiom,
! [A: $tType,R: A > A > $o,X: A,Y: A,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( R @ X @ Y )
=> ( coindu300403952lexord @ A @ R @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ ( coindu1121789889_LCons @ A @ Y @ Ys ) ) ) ).
% llexord_LCons_less
thf(fact_12_llexord_I2_J,axiom,
~ ( coindu335574135_lnull @ a @ xsa ) ).
% llexord(2)
thf(fact_13_llexord_I1_J,axiom,
~ ( coindu300403952lexord @ a @ r @ ysa @ xsa ) ).
% llexord(1)
thf(fact_14_lmember__code_I2_J,axiom,
! [A: $tType,X: A,Y: A,Ys: coindu1593790203_llist @ A] :
( ( coindu567634248member @ A @ X @ ( coindu1121789889_LCons @ A @ Y @ Ys ) )
= ( ( X = Y )
| ( coindu567634248member @ A @ X @ Ys ) ) ) ).
% lmember_code(2)
thf(fact_15_lstrict__prefix__code_I4_J,axiom,
! [B: $tType,X: B,Xs: coindu1593790203_llist @ B,Y: B,Ys: coindu1593790203_llist @ B] :
( ( coindu574146665prefix @ B @ ( coindu1121789889_LCons @ B @ X @ Xs ) @ ( coindu1121789889_LCons @ B @ Y @ Ys ) )
= ( ( X = Y )
& ( coindu574146665prefix @ B @ Xs @ Ys ) ) ) ).
% lstrict_prefix_code(4)
thf(fact_16_llexord_Ocases,axiom,
! [A: $tType,R: A > A > $o,A1: coindu1593790203_llist @ A,A22: coindu1593790203_llist @ A] :
( ( coindu300403952lexord @ A @ R @ A1 @ A22 )
=> ( ! [Xs2: coindu1593790203_llist @ A,Ys3: coindu1593790203_llist @ A,X2: A] :
( ( A1
= ( coindu1121789889_LCons @ A @ X2 @ Xs2 ) )
=> ( ( A22
= ( coindu1121789889_LCons @ A @ X2 @ Ys3 ) )
=> ~ ( coindu300403952lexord @ A @ R @ Xs2 @ Ys3 ) ) )
=> ( ! [X2: A] :
( ? [Xs2: coindu1593790203_llist @ A] :
( A1
= ( coindu1121789889_LCons @ A @ X2 @ Xs2 ) )
=> ! [Y3: A] :
( ? [Ys3: coindu1593790203_llist @ A] :
( A22
= ( coindu1121789889_LCons @ A @ Y3 @ Ys3 ) )
=> ~ ( R @ X2 @ Y3 ) ) )
=> ~ ( ( A1
= ( coindu1598213697e_LNil @ A ) )
=> ! [Ys3: coindu1593790203_llist @ A] : A22 != Ys3 ) ) ) ) ).
% llexord.cases
thf(fact_17_llexord_Osimps,axiom,
! [A: $tType] :
( ( coindu300403952lexord @ A )
= ( ^ [R2: A > A > $o,A12: coindu1593790203_llist @ A,A23: coindu1593790203_llist @ A] :
( ? [Xs3: coindu1593790203_llist @ A,Ys4: coindu1593790203_llist @ A,X3: A] :
( ( A12
= ( coindu1121789889_LCons @ A @ X3 @ Xs3 ) )
& ( A23
= ( coindu1121789889_LCons @ A @ X3 @ Ys4 ) )
& ( coindu300403952lexord @ A @ R2 @ Xs3 @ Ys4 ) )
| ? [X3: A,Y2: A,Xs3: coindu1593790203_llist @ A,Ys4: coindu1593790203_llist @ A] :
( ( A12
= ( coindu1121789889_LCons @ A @ X3 @ Xs3 ) )
& ( A23
= ( coindu1121789889_LCons @ A @ Y2 @ Ys4 ) )
& ( R2 @ X3 @ Y2 ) )
| ? [Ys4: coindu1593790203_llist @ A] :
( ( A12
= ( coindu1598213697e_LNil @ A ) )
& ( A23 = Ys4 ) ) ) ) ) ).
% llexord.simps
thf(fact_18_llexord_Ocoinduct,axiom,
! [A: $tType,X4: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o,X: coindu1593790203_llist @ A,Xa: coindu1593790203_llist @ A,R: A > A > $o] :
( ( X4 @ X @ Xa )
=> ( ! [X2: coindu1593790203_llist @ A,Xa2: coindu1593790203_llist @ A] :
( ( X4 @ X2 @ Xa2 )
=> ( ? [Xs4: coindu1593790203_llist @ A,Ys5: coindu1593790203_llist @ A,Xb: A] :
( ( X2
= ( coindu1121789889_LCons @ A @ Xb @ Xs4 ) )
& ( Xa2
= ( coindu1121789889_LCons @ A @ Xb @ Ys5 ) )
& ( ( X4 @ Xs4 @ Ys5 )
| ( coindu300403952lexord @ A @ R @ Xs4 @ Ys5 ) ) )
| ? [Xb: A,Y4: A,Xs4: coindu1593790203_llist @ A,Ys5: coindu1593790203_llist @ A] :
( ( X2
= ( coindu1121789889_LCons @ A @ Xb @ Xs4 ) )
& ( Xa2
= ( coindu1121789889_LCons @ A @ Y4 @ Ys5 ) )
& ( R @ Xb @ Y4 ) )
| ? [Ys5: coindu1593790203_llist @ A] :
( ( X2
= ( coindu1598213697e_LNil @ A ) )
& ( Xa2 = Ys5 ) ) ) )
=> ( coindu300403952lexord @ A @ R @ X @ Xa ) ) ) ).
% llexord.coinduct
thf(fact_19_llist_Opred__inject_I2_J,axiom,
! [A: $tType,P: A > $o,A3: A,Aa: coindu1593790203_llist @ A] :
( ( coindu44415597_llist @ A @ P @ ( coindu1121789889_LCons @ A @ A3 @ Aa ) )
= ( ( P @ A3 )
& ( coindu44415597_llist @ A @ P @ Aa ) ) ) ).
% llist.pred_inject(2)
thf(fact_20_llast__LCons2,axiom,
! [A: $tType,X: A,Y: A,Xs: coindu1593790203_llist @ A] :
( ( coindu2000965700_llast @ A @ ( coindu1121789889_LCons @ A @ X @ ( coindu1121789889_LCons @ A @ Y @ Xs ) ) )
= ( coindu2000965700_llast @ A @ ( coindu1121789889_LCons @ A @ Y @ Xs ) ) ) ).
% llast_LCons2
thf(fact_21_unfold__llist__eq__LCons,axiom,
! [A: $tType,B: $tType,IS_LNIL: B > $o,LHD: B > A,LTL: B > B,B3: B,X: A,Xs: coindu1593790203_llist @ A] :
( ( ( coindu1599971794_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ B3 )
= ( coindu1121789889_LCons @ A @ X @ Xs ) )
= ( ~ ( IS_LNIL @ B3 )
& ( X
= ( LHD @ B3 ) )
& ( Xs
= ( coindu1599971794_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ ( LTL @ B3 ) ) ) ) ) ).
% unfold_llist_eq_LCons
thf(fact_22_lsetp_Ointros_I2_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,X: A,X5: A] :
( ( coindu1055736764_lsetp @ A @ Xs @ X )
=> ( coindu1055736764_lsetp @ A @ ( coindu1121789889_LCons @ A @ X5 @ Xs ) @ X ) ) ).
% lsetp.intros(2)
thf(fact_23_lsetp_Ointros_I1_J,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] : ( coindu1055736764_lsetp @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ X ) ).
% lsetp.intros(1)
thf(fact_24_llexord__LNil__right,axiom,
! [A: $tType,Ys: coindu1593790203_llist @ A,R: A > A > $o,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Ys )
=> ( ( coindu300403952lexord @ A @ R @ Xs @ Ys )
= ( coindu335574135_lnull @ A @ Xs ) ) ) ).
% llexord_LNil_right
thf(fact_25_unfold__llist_Odisc__iff_I2_J,axiom,
! [B: $tType,A: $tType,P2: A > $o,G21: A > B,G22: A > A,A3: A] :
( ( ~ ( coindu335574135_lnull @ B @ ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ A3 ) ) )
= ( ~ ( P2 @ A3 ) ) ) ).
% unfold_llist.disc_iff(2)
thf(fact_26_unfold__llist_Odisc__iff_I1_J,axiom,
! [B: $tType,A: $tType,P2: A > $o,G21: A > B,G22: A > A,A3: A] :
( ( coindu335574135_lnull @ B @ ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ A3 ) )
= ( P2 @ A3 ) ) ).
% unfold_llist.disc_iff(1)
thf(fact_27_lstrict__prefix__code_I1_J,axiom,
! [A: $tType] :
~ ( coindu574146665prefix @ A @ ( coindu1598213697e_LNil @ A ) @ ( coindu1598213697e_LNil @ A ) ) ).
% lstrict_prefix_code(1)
thf(fact_28_llast__singleton,axiom,
! [A: $tType,X: A] :
( ( coindu2000965700_llast @ A @ ( coindu1121789889_LCons @ A @ X @ ( coindu1598213697e_LNil @ A ) ) )
= X ) ).
% llast_singleton
thf(fact_29_lstrict__prefix__code_I2_J,axiom,
! [B: $tType,Y: B,Ys: coindu1593790203_llist @ B] : ( coindu574146665prefix @ B @ ( coindu1598213697e_LNil @ B ) @ ( coindu1121789889_LCons @ B @ Y @ Ys ) ) ).
% lstrict_prefix_code(2)
thf(fact_30_lstrict__prefix__code_I3_J,axiom,
! [B: $tType,X: B,Xs: coindu1593790203_llist @ B] :
~ ( coindu574146665prefix @ B @ ( coindu1121789889_LCons @ B @ X @ Xs ) @ ( coindu1598213697e_LNil @ B ) ) ).
% lstrict_prefix_code(3)
thf(fact_31_llist_Odisc_I1_J,axiom,
! [A: $tType] : ( coindu335574135_lnull @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% llist.disc(1)
thf(fact_32_llist_OdiscI_I1_J,axiom,
! [A: $tType,Llist: coindu1593790203_llist @ A] :
( ( Llist
= ( coindu1598213697e_LNil @ A ) )
=> ( coindu335574135_lnull @ A @ Llist ) ) ).
% llist.discI(1)
thf(fact_33_lmember__code_I1_J,axiom,
! [A: $tType,X: A] :
~ ( coindu567634248member @ A @ X @ ( coindu1598213697e_LNil @ A ) ) ).
% lmember_code(1)
thf(fact_34_llist_Ocollapse_I1_J,axiom,
! [A: $tType,Llist: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Llist )
=> ( Llist
= ( coindu1598213697e_LNil @ A ) ) ) ).
% llist.collapse(1)
thf(fact_35_unfold__llist_Octr_I1_J,axiom,
! [A: $tType,B: $tType,P2: A > $o,A3: A,G21: A > B,G22: A > A] :
( ( P2 @ A3 )
=> ( ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ A3 )
= ( coindu1598213697e_LNil @ B ) ) ) ).
% unfold_llist.ctr(1)
thf(fact_36_llist_Opred__inject_I1_J,axiom,
! [A: $tType,P: A > $o] : ( coindu44415597_llist @ A @ P @ ( coindu1598213697e_LNil @ A ) ) ).
% llist.pred_inject(1)
thf(fact_37_unfold__llist_Odisc_I2_J,axiom,
! [B: $tType,A: $tType,P2: A > $o,A3: A,G21: A > B,G22: A > A] :
( ~ ( P2 @ A3 )
=> ~ ( coindu335574135_lnull @ B @ ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ A3 ) ) ) ).
% unfold_llist.disc(2)
thf(fact_38_unfold__llist_Odisc_I1_J,axiom,
! [B: $tType,A: $tType,P2: A > $o,A3: A,G21: A > B,G22: A > A] :
( ( P2 @ A3 )
=> ( coindu335574135_lnull @ B @ ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ A3 ) ) ) ).
% unfold_llist.disc(1)
thf(fact_39_lzip_Oexhaust,axiom,
! [A: $tType,B: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B] :
( ~ ( ( coindu335574135_lnull @ A @ Xs )
| ( coindu335574135_lnull @ B @ Ys ) )
=> ~ ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( coindu335574135_lnull @ B @ Ys ) ) ) ).
% lzip.exhaust
thf(fact_40_lappend_Oexhaust,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu335574135_lnull @ A @ Xs )
=> ~ ( coindu335574135_lnull @ A @ Ys ) )
=> ( ~ ( coindu335574135_lnull @ A @ Xs )
| ~ ( coindu335574135_lnull @ A @ Ys ) ) ) ).
% lappend.exhaust
thf(fact_41_lnull__def,axiom,
! [A: $tType] :
( ( coindu335574135_lnull @ A )
= ( ^ [Llist2: coindu1593790203_llist @ A] :
( Llist2
= ( coindu1598213697e_LNil @ A ) ) ) ) ).
% lnull_def
thf(fact_42_llist_Opred__True,axiom,
! [A: $tType] :
( ( coindu44415597_llist @ A
@ ^ [Uu: A] : $true )
= ( ^ [Uu: coindu1593790203_llist @ A] : $true ) ) ).
% llist.pred_True
thf(fact_43_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_44_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_45_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_46_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X2: A] :
( ( F @ X2 )
= ( G @ X2 ) )
=> ( F = G ) ) ).
% ext
thf(fact_47_llist__less__induct,axiom,
! [A: $tType,P: ( coindu1593790203_llist @ A ) > $o,Xs: coindu1593790203_llist @ A] :
( ! [Xs2: coindu1593790203_llist @ A] :
( ! [Ys5: coindu1593790203_llist @ A] :
( ( coindu574146665prefix @ A @ Ys5 @ Xs2 )
=> ( P @ Ys5 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% llist_less_induct
thf(fact_48_llast__LCons,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,X: A] :
( ( ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu2000965700_llast @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) )
= X ) )
& ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu2000965700_llast @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) )
= ( coindu2000965700_llast @ A @ Xs ) ) ) ) ).
% llast_LCons
thf(fact_49_unfold__llist_Ocode,axiom,
! [B: $tType,A: $tType] :
( ( coindu1599971794_llist @ A @ B )
= ( ^ [P3: A > $o,G212: A > B,G222: A > A,A5: A] : ( if @ ( coindu1593790203_llist @ B ) @ ( P3 @ A5 ) @ ( coindu1598213697e_LNil @ B ) @ ( coindu1121789889_LCons @ B @ ( G212 @ A5 ) @ ( coindu1599971794_llist @ A @ B @ P3 @ G212 @ G222 @ ( G222 @ A5 ) ) ) ) ) ) ).
% unfold_llist.code
thf(fact_50_not__lnull__conv,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( ~ ( coindu335574135_lnull @ A @ Xs ) )
= ( ? [X3: A,Xs5: coindu1593790203_llist @ A] :
( Xs
= ( coindu1121789889_LCons @ A @ X3 @ Xs5 ) ) ) ) ).
% not_lnull_conv
thf(fact_51_llist_OdiscI_I2_J,axiom,
! [A: $tType,Llist: coindu1593790203_llist @ A,X21: A,X22: coindu1593790203_llist @ A] :
( ( Llist
= ( coindu1121789889_LCons @ A @ X21 @ X22 ) )
=> ~ ( coindu335574135_lnull @ A @ Llist ) ) ).
% llist.discI(2)
thf(fact_52_llist_Odisc_I2_J,axiom,
! [A: $tType,X21: A,X22: coindu1593790203_llist @ A] :
~ ( coindu335574135_lnull @ A @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) ) ).
% llist.disc(2)
thf(fact_53_neq__LNil__conv,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( Xs
!= ( coindu1598213697e_LNil @ A ) )
= ( ? [X3: A,Xs5: coindu1593790203_llist @ A] :
( Xs
= ( coindu1121789889_LCons @ A @ X3 @ Xs5 ) ) ) ) ).
% neq_LNil_conv
thf(fact_54_llist_Oexhaust,axiom,
! [A: $tType,Y: coindu1593790203_llist @ A] :
( ( Y
!= ( coindu1598213697e_LNil @ A ) )
=> ~ ! [X212: A,X222: coindu1593790203_llist @ A] :
( Y
!= ( coindu1121789889_LCons @ A @ X212 @ X222 ) ) ) ).
% llist.exhaust
thf(fact_55_llist_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: coindu1593790203_llist @ A] :
( ( coindu1598213697e_LNil @ A )
!= ( coindu1121789889_LCons @ A @ X21 @ X22 ) ) ).
% llist.distinct(1)
thf(fact_56_lnull__llexord,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,R: A > A > $o,Ys: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( coindu300403952lexord @ A @ R @ Xs @ Ys ) ) ).
% lnull_llexord
thf(fact_57_llexord__LNil,axiom,
! [A: $tType,R: A > A > $o,Ys: coindu1593790203_llist @ A] : ( coindu300403952lexord @ A @ R @ ( coindu1598213697e_LNil @ A ) @ Ys ) ).
% llexord_LNil
thf(fact_58_unfold__llist_Octr_I2_J,axiom,
! [B: $tType,A: $tType,P2: A > $o,A3: A,G21: A > B,G22: A > A] :
( ~ ( P2 @ A3 )
=> ( ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ A3 )
= ( coindu1121789889_LCons @ B @ ( G21 @ A3 ) @ ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ ( G22 @ A3 ) ) ) ) ) ).
% unfold_llist.ctr(2)
thf(fact_59_lsetp_Oinducts,axiom,
! [A: $tType,X1: coindu1593790203_llist @ A,X23: A,P: ( coindu1593790203_llist @ A ) > A > $o] :
( ( coindu1055736764_lsetp @ A @ X1 @ X23 )
=> ( ! [X2: A,Xs2: coindu1593790203_llist @ A] : ( P @ ( coindu1121789889_LCons @ A @ X2 @ Xs2 ) @ X2 )
=> ( ! [Xs2: coindu1593790203_llist @ A,X2: A,X6: A] :
( ( coindu1055736764_lsetp @ A @ Xs2 @ X2 )
=> ( ( P @ Xs2 @ X2 )
=> ( P @ ( coindu1121789889_LCons @ A @ X6 @ Xs2 ) @ X2 ) ) )
=> ( P @ X1 @ X23 ) ) ) ) ).
% lsetp.inducts
thf(fact_60_lsetp_Osimps,axiom,
! [A: $tType] :
( ( coindu1055736764_lsetp @ A )
= ( ^ [A12: coindu1593790203_llist @ A,A23: A] :
( ? [X3: A,Xs3: coindu1593790203_llist @ A] :
( ( A12
= ( coindu1121789889_LCons @ A @ X3 @ Xs3 ) )
& ( A23 = X3 ) )
| ? [Xs3: coindu1593790203_llist @ A,X3: A,X7: A] :
( ( A12
= ( coindu1121789889_LCons @ A @ X7 @ Xs3 ) )
& ( A23 = X3 )
& ( coindu1055736764_lsetp @ A @ Xs3 @ X3 ) ) ) ) ) ).
% lsetp.simps
thf(fact_61_lsetp_Ocases,axiom,
! [A: $tType,A1: coindu1593790203_llist @ A,A22: A] :
( ( coindu1055736764_lsetp @ A @ A1 @ A22 )
=> ( ! [X2: A] :
( ? [Xs2: coindu1593790203_llist @ A] :
( A1
= ( coindu1121789889_LCons @ A @ X2 @ Xs2 ) )
=> ( A22 != X2 ) )
=> ~ ! [Xs2: coindu1593790203_llist @ A] :
( ? [X6: A] :
( A1
= ( coindu1121789889_LCons @ A @ X6 @ Xs2 ) )
=> ~ ( coindu1055736764_lsetp @ A @ Xs2 @ A22 ) ) ) ) ).
% lsetp.cases
thf(fact_62_Coinductive__List__Mirabelle__kmikjhschf_Ofinite__lprefix__nitpick__simps_I3_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Y: A,Ys: coindu1593790203_llist @ A] :
( ( coindu1571841457prefix @ A @ Xs @ ( coindu1121789889_LCons @ A @ Y @ Ys ) )
= ( ( Xs
= ( coindu1598213697e_LNil @ A ) )
| ? [Xs5: coindu1593790203_llist @ A] :
( ( Xs
= ( coindu1121789889_LCons @ A @ Y @ Xs5 ) )
& ( coindu1571841457prefix @ A @ Xs5 @ Ys ) ) ) ) ).
% Coinductive_List_Mirabelle_kmikjhschf.finite_lprefix_nitpick_simps(3)
thf(fact_63_llimit__induct,axiom,
! [A: $tType,P: ( coindu1593790203_llist @ A ) > $o,Xs: coindu1593790203_llist @ A] :
( ( P @ ( coindu1598213697e_LNil @ A ) )
=> ( ! [X2: A,Xs2: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ Xs2 )
=> ( ( P @ Xs2 )
=> ( P @ ( coindu1121789889_LCons @ A @ X2 @ Xs2 ) ) ) )
=> ( ( ! [Ys5: coindu1593790203_llist @ A] :
( ( coindu574146665prefix @ A @ Ys5 @ Xs )
=> ( P @ Ys5 ) )
=> ( P @ Xs ) )
=> ( P @ Xs ) ) ) ) ).
% llimit_induct
thf(fact_64_lsorted__code_I2_J,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: A] : ( coindu48241764sorted @ A @ ( coindu1121789889_LCons @ A @ X @ ( coindu1598213697e_LNil @ A ) ) ) ) ).
% lsorted_code(2)
thf(fact_65_gen__lset__code_I1_J,axiom,
! [A: $tType,A4: set @ A] :
( ( coindu1928975208n_lset @ A @ A4 @ ( coindu1598213697e_LNil @ A ) )
= A4 ) ).
% gen_lset_code(1)
thf(fact_66_llist_Ocorec__code,axiom,
! [A: $tType,C: $tType] :
( ( coindu1244876290_llist @ C @ A )
= ( ^ [P3: C > $o,G212: C > A,Q22: C > $o,G221: C > ( coindu1593790203_llist @ A ),G2222: C > C,A5: C] : ( if @ ( coindu1593790203_llist @ A ) @ ( P3 @ A5 ) @ ( coindu1598213697e_LNil @ A ) @ ( coindu1121789889_LCons @ A @ ( G212 @ A5 ) @ ( if @ ( coindu1593790203_llist @ A ) @ ( Q22 @ A5 ) @ ( G221 @ A5 ) @ ( coindu1244876290_llist @ C @ A @ P3 @ G212 @ Q22 @ G221 @ G2222 @ ( G2222 @ A5 ) ) ) ) ) ) ) ).
% llist.corec_code
thf(fact_67_Coinductive__List__Mirabelle__kmikjhschf_Ofinite__lprefix__nitpick__simps_I2_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] : ( coindu1571841457prefix @ A @ ( coindu1598213697e_LNil @ A ) @ Xs ) ).
% Coinductive_List_Mirabelle_kmikjhschf.finite_lprefix_nitpick_simps(2)
thf(fact_68_Coinductive__List__Mirabelle__kmikjhschf_Ofinite__lprefix__nitpick__simps_I1_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu1571841457prefix @ A @ Xs @ ( coindu1598213697e_LNil @ A ) )
= ( Xs
= ( coindu1598213697e_LNil @ A ) ) ) ).
% Coinductive_List_Mirabelle_kmikjhschf.finite_lprefix_nitpick_simps(1)
thf(fact_69_lprefix__LNil,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ ( coindu1598213697e_LNil @ A ) )
= ( coindu335574135_lnull @ A @ Xs ) ) ).
% lprefix_LNil
thf(fact_70_ltakeWhile__LCons,axiom,
! [A: $tType,P: A > $o,X: A,Xs: coindu1593790203_llist @ A] :
( ( ( P @ X )
=> ( ( coindu721411036eWhile @ A @ P @ ( coindu1121789889_LCons @ A @ X @ Xs ) )
= ( coindu1121789889_LCons @ A @ X @ ( coindu721411036eWhile @ A @ P @ Xs ) ) ) )
& ( ~ ( P @ X )
=> ( ( coindu721411036eWhile @ A @ P @ ( coindu1121789889_LCons @ A @ X @ Xs ) )
= ( coindu1598213697e_LNil @ A ) ) ) ) ).
% ltakeWhile_LCons
thf(fact_71_ord_OSingleton,axiom,
! [A: $tType,Less_eq: A > A > $o,X: A] : ( coindu543932918sorted @ A @ Less_eq @ ( coindu1121789889_LCons @ A @ X @ ( coindu1598213697e_LNil @ A ) ) ) ).
% ord.Singleton
thf(fact_72_llist_Oleq__refl,axiom,
! [A: $tType,X: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ X @ X ) ).
% llist.leq_refl
thf(fact_73_lprefix__refl,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ Xs @ Xs ) ).
% lprefix_refl
thf(fact_74_ltakeWhile__K__True,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu721411036eWhile @ A
@ ^ [Uu: A] : $true
@ Xs )
= Xs ) ).
% ltakeWhile_K_True
thf(fact_75_LCons__lprefix__LCons,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A,Y: A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ ( coindu1121789889_LCons @ A @ Y @ Ys ) )
= ( ( X = Y )
& ( coindu1696667936prefix @ A @ Xs @ Ys ) ) ) ).
% LCons_lprefix_LCons
thf(fact_76_lprefix__code_I1_J,axiom,
! [A: $tType,Ys: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ ( coindu1598213697e_LNil @ A ) @ Ys ) ).
% lprefix_code(1)
thf(fact_77_lfinite__code_I2_J,axiom,
! [B: $tType,X: B,Xs: coindu1593790203_llist @ B] :
( ( coindu1213758845finite @ B @ ( coindu1121789889_LCons @ B @ X @ Xs ) )
= ( coindu1213758845finite @ B @ Xs ) ) ).
% lfinite_code(2)
thf(fact_78_lfinite__LCons,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) )
= ( coindu1213758845finite @ A @ Xs ) ) ).
% lfinite_LCons
thf(fact_79_lfinite__code_I1_J,axiom,
! [A: $tType] : ( coindu1213758845finite @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% lfinite_code(1)
thf(fact_80_ltakeWhile__LNil,axiom,
! [A: $tType,P: A > $o] :
( ( coindu721411036eWhile @ A @ P @ ( coindu1598213697e_LNil @ A ) )
= ( coindu1598213697e_LNil @ A ) ) ).
% ltakeWhile_LNil
thf(fact_81_lsorted__code_I1_J,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ( coindu48241764sorted @ A @ ( coindu1598213697e_LNil @ A ) ) ) ).
% lsorted_code(1)
thf(fact_82_llist_Ocorec__disc__iff_I1_J,axiom,
! [A: $tType,C: $tType,P2: C > $o,G21: C > A,Q222: C > $o,G2212: C > ( coindu1593790203_llist @ A ),G2223: C > C,A3: C] :
( ( coindu335574135_lnull @ A @ ( coindu1244876290_llist @ C @ A @ P2 @ G21 @ Q222 @ G2212 @ G2223 @ A3 ) )
= ( P2 @ A3 ) ) ).
% llist.corec_disc_iff(1)
thf(fact_83_llist_Ocorec__disc__iff_I2_J,axiom,
! [A: $tType,C: $tType,P2: C > $o,G21: C > A,Q222: C > $o,G2212: C > ( coindu1593790203_llist @ A ),G2223: C > C,A3: C] :
( ( ~ ( coindu335574135_lnull @ A @ ( coindu1244876290_llist @ C @ A @ P2 @ G21 @ Q222 @ G2212 @ G2223 @ A3 ) ) )
= ( ~ ( P2 @ A3 ) ) ) ).
% llist.corec_disc_iff(2)
thf(fact_84_ltakeWhile__K__False,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu721411036eWhile @ A
@ ^ [Uu: A] : $false
@ Xs )
= ( coindu1598213697e_LNil @ A ) ) ).
% ltakeWhile_K_False
thf(fact_85_not__lfinite__lprefix__conv__eq,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ~ ( coindu1213758845finite @ A @ Xs )
=> ( ( coindu1696667936prefix @ A @ Xs @ Ys )
= ( Xs = Ys ) ) ) ).
% not_lfinite_lprefix_conv_eq
thf(fact_86_lsorted__lprefixD,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( ( coindu48241764sorted @ A @ Ys )
=> ( coindu48241764sorted @ A @ Xs ) ) ) ) ).
% lsorted_lprefixD
thf(fact_87_Coinductive__List__Mirabelle__kmikjhschf_Olprefix__nitpick__simps,axiom,
! [A: $tType] :
( ( coindu1696667936prefix @ A )
= ( ^ [Xs3: coindu1593790203_llist @ A,Ys4: coindu1593790203_llist @ A] :
( ( ( coindu1213758845finite @ A @ Xs3 )
=> ( coindu1571841457prefix @ A @ Xs3 @ Ys4 ) )
& ( ~ ( coindu1213758845finite @ A @ Xs3 )
=> ( Xs3 = Ys4 ) ) ) ) ) ).
% Coinductive_List_Mirabelle_kmikjhschf.lprefix_nitpick_simps
thf(fact_88_ord_Olsorted__lprefixD,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Less_eq: A > A > $o] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( ( coindu543932918sorted @ A @ Less_eq @ Ys )
=> ( coindu543932918sorted @ A @ Less_eq @ Xs ) ) ) ).
% ord.lsorted_lprefixD
thf(fact_89_lprefix__down__linear,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Zs )
=> ( ( coindu1696667936prefix @ A @ Ys @ Zs )
=> ( ( coindu1696667936prefix @ A @ Xs @ Ys )
| ( coindu1696667936prefix @ A @ Ys @ Xs ) ) ) ) ).
% lprefix_down_linear
thf(fact_90_lprefix__ltakeWhile,axiom,
! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) @ Xs ) ).
% lprefix_ltakeWhile
thf(fact_91_Coinductive__List__Mirabelle__kmikjhschf_Ofinite__lprefix__def,axiom,
! [A: $tType] :
( ( coindu1571841457prefix @ A )
= ( coindu1696667936prefix @ A ) ) ).
% Coinductive_List_Mirabelle_kmikjhschf.finite_lprefix_def
thf(fact_92_llist_Oleq__antisym,axiom,
! [A: $tType,X: coindu1593790203_llist @ A,Y: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ X @ Y )
=> ( ( coindu1696667936prefix @ A @ Y @ X )
=> ( X = Y ) ) ) ).
% llist.leq_antisym
thf(fact_93_lprefix__lfiniteD,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( ( coindu1213758845finite @ A @ Ys )
=> ( coindu1213758845finite @ A @ Xs ) ) ) ).
% lprefix_lfiniteD
thf(fact_94_lprefix__antisym,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( ( coindu1696667936prefix @ A @ Ys @ Xs )
=> ( Xs = Ys ) ) ) ).
% lprefix_antisym
thf(fact_95_llist_Oleq__trans,axiom,
! [A: $tType,X: coindu1593790203_llist @ A,Y: coindu1593790203_llist @ A,Z: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ X @ Y )
=> ( ( coindu1696667936prefix @ A @ Y @ Z )
=> ( coindu1696667936prefix @ A @ X @ Z ) ) ) ).
% llist.leq_trans
thf(fact_96_lprefix__trans,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( ( coindu1696667936prefix @ A @ Ys @ Zs )
=> ( coindu1696667936prefix @ A @ Xs @ Zs ) ) ) ).
% lprefix_trans
thf(fact_97_lfinite__LConsI,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,X: A] :
( ( coindu1213758845finite @ A @ Xs )
=> ( coindu1213758845finite @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) ) ) ).
% lfinite_LConsI
thf(fact_98_lnull__imp__lfinite,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( coindu1213758845finite @ A @ Xs ) ) ).
% lnull_imp_lfinite
thf(fact_99_lfinite__LNil,axiom,
! [A: $tType] : ( coindu1213758845finite @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% lfinite_LNil
thf(fact_100_Le__LCons,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,X: A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( coindu1696667936prefix @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ ( coindu1121789889_LCons @ A @ X @ Ys ) ) ) ).
% Le_LCons
thf(fact_101_LCons__lprefix__conv,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ Ys )
= ( ? [Ys2: coindu1593790203_llist @ A] :
( ( Ys
= ( coindu1121789889_LCons @ A @ X @ Ys2 ) )
& ( coindu1696667936prefix @ A @ Xs @ Ys2 ) ) ) ) ).
% LCons_lprefix_conv
thf(fact_102_lprefix__not__lnullD,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ~ ( coindu335574135_lnull @ A @ Ys ) ) ) ).
% lprefix_not_lnullD
thf(fact_103_lprefix__lnullD,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( ( coindu335574135_lnull @ A @ Ys )
=> ( coindu335574135_lnull @ A @ Xs ) ) ) ).
% lprefix_lnullD
thf(fact_104_lprefix__lnull,axiom,
! [A: $tType,Ys: coindu1593790203_llist @ A,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Ys )
=> ( ( coindu1696667936prefix @ A @ Xs @ Ys )
= ( coindu335574135_lnull @ A @ Xs ) ) ) ).
% lprefix_lnull
thf(fact_105_lnull__lprefix,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( coindu1696667936prefix @ A @ Xs @ Ys ) ) ).
% lnull_lprefix
thf(fact_106_LNil__lprefix,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ ( coindu1598213697e_LNil @ A ) @ Xs ) ).
% LNil_lprefix
thf(fact_107_lprefix__imp__llexord,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,R: A > A > $o] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( coindu300403952lexord @ A @ R @ Xs @ Ys ) ) ).
% lprefix_imp_llexord
thf(fact_108_lstrict__prefix__lfinite1,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu574146665prefix @ A @ Xs @ Ys )
=> ( coindu1213758845finite @ A @ Xs ) ) ).
% lstrict_prefix_lfinite1
thf(fact_109_LNil,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ( coindu48241764sorted @ A @ ( coindu1598213697e_LNil @ A ) ) ) ).
% LNil
thf(fact_110_ord_OLCons__LCons,axiom,
! [A: $tType,Less_eq: A > A > $o,X: A,Y: A,Xs: coindu1593790203_llist @ A] :
( ( Less_eq @ X @ Y )
=> ( ( coindu543932918sorted @ A @ Less_eq @ ( coindu1121789889_LCons @ A @ Y @ Xs ) )
=> ( coindu543932918sorted @ A @ Less_eq @ ( coindu1121789889_LCons @ A @ X @ ( coindu1121789889_LCons @ A @ Y @ Xs ) ) ) ) ) ).
% ord.LCons_LCons
thf(fact_111_ord_Olsorted__LCons__LCons,axiom,
! [A: $tType,Less_eq: A > A > $o,X: A,Y: A,Xs: coindu1593790203_llist @ A] :
( ( coindu543932918sorted @ A @ Less_eq @ ( coindu1121789889_LCons @ A @ X @ ( coindu1121789889_LCons @ A @ Y @ Xs ) ) )
= ( ( Less_eq @ X @ Y )
& ( coindu543932918sorted @ A @ Less_eq @ ( coindu1121789889_LCons @ A @ Y @ Xs ) ) ) ) ).
% ord.lsorted_LCons_LCons
thf(fact_112_ord_OLNil,axiom,
! [A: $tType,Less_eq: A > A > $o] : ( coindu543932918sorted @ A @ Less_eq @ ( coindu1598213697e_LNil @ A ) ) ).
% ord.LNil
thf(fact_113_ord_Olsorted__code_I1_J,axiom,
! [A: $tType,Less_eq: A > A > $o] : ( coindu543932918sorted @ A @ Less_eq @ ( coindu1598213697e_LNil @ A ) ) ).
% ord.lsorted_code(1)
thf(fact_114_lstrict__prefix__def,axiom,
! [A: $tType] :
( ( coindu574146665prefix @ A )
= ( ^ [Xs3: coindu1593790203_llist @ A,Ys4: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs3 @ Ys4 )
& ( Xs3 != Ys4 ) ) ) ) ).
% lstrict_prefix_def
thf(fact_115_llist_Ocorec_I2_J,axiom,
! [A: $tType,C: $tType,P2: C > $o,A3: C,G21: C > A,Q222: C > $o,G2212: C > ( coindu1593790203_llist @ A ),G2223: C > C] :
( ~ ( P2 @ A3 )
=> ( ( coindu1244876290_llist @ C @ A @ P2 @ G21 @ Q222 @ G2212 @ G2223 @ A3 )
= ( coindu1121789889_LCons @ A @ ( G21 @ A3 ) @ ( if @ ( coindu1593790203_llist @ A ) @ ( Q222 @ A3 ) @ ( G2212 @ A3 ) @ ( coindu1244876290_llist @ C @ A @ P2 @ G21 @ Q222 @ G2212 @ G2223 @ ( G2223 @ A3 ) ) ) ) ) ) ).
% llist.corec(2)
thf(fact_116_llist_Ocorec__disc_I1_J,axiom,
! [A: $tType,C: $tType,P2: C > $o,A3: C,G21: C > A,Q222: C > $o,G2212: C > ( coindu1593790203_llist @ A ),G2223: C > C] :
( ( P2 @ A3 )
=> ( coindu335574135_lnull @ A @ ( coindu1244876290_llist @ C @ A @ P2 @ G21 @ Q222 @ G2212 @ G2223 @ A3 ) ) ) ).
% llist.corec_disc(1)
thf(fact_117_llist_Ocorec__disc_I2_J,axiom,
! [A: $tType,C: $tType,P2: C > $o,A3: C,G21: C > A,Q222: C > $o,G2212: C > ( coindu1593790203_llist @ A ),G2223: C > C] :
( ~ ( P2 @ A3 )
=> ~ ( coindu335574135_lnull @ A @ ( coindu1244876290_llist @ C @ A @ P2 @ G21 @ Q222 @ G2212 @ G2223 @ A3 ) ) ) ).
% llist.corec_disc(2)
thf(fact_118_llist_Ocorec_I1_J,axiom,
! [C: $tType,A: $tType,P2: C > $o,A3: C,G21: C > A,Q222: C > $o,G2212: C > ( coindu1593790203_llist @ A ),G2223: C > C] :
( ( P2 @ A3 )
=> ( ( coindu1244876290_llist @ C @ A @ P2 @ G21 @ Q222 @ G2212 @ G2223 @ A3 )
= ( coindu1598213697e_LNil @ A ) ) ) ).
% llist.corec(1)
thf(fact_119_llexord__empty,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu300403952lexord @ A
@ ^ [X3: A,Y2: A] : $false
@ Xs
@ Ys )
= ( coindu1696667936prefix @ A @ Xs @ Ys ) ) ).
% llexord_empty
thf(fact_120_lfinite_Ocases,axiom,
! [A: $tType,A3: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ A3 )
=> ( ( A3
!= ( coindu1598213697e_LNil @ A ) )
=> ~ ! [Xs2: coindu1593790203_llist @ A] :
( ? [X2: A] :
( A3
= ( coindu1121789889_LCons @ A @ X2 @ Xs2 ) )
=> ~ ( coindu1213758845finite @ A @ Xs2 ) ) ) ) ).
% lfinite.cases
thf(fact_121_lfinite_Osimps,axiom,
! [A: $tType] :
( ( coindu1213758845finite @ A )
= ( ^ [A5: coindu1593790203_llist @ A] :
( ( A5
= ( coindu1598213697e_LNil @ A ) )
| ? [Xs3: coindu1593790203_llist @ A,X3: A] :
( ( A5
= ( coindu1121789889_LCons @ A @ X3 @ Xs3 ) )
& ( coindu1213758845finite @ A @ Xs3 ) ) ) ) ) ).
% lfinite.simps
thf(fact_122_lfinite_Oinducts,axiom,
! [A: $tType,X: coindu1593790203_llist @ A,P: ( coindu1593790203_llist @ A ) > $o] :
( ( coindu1213758845finite @ A @ X )
=> ( ( P @ ( coindu1598213697e_LNil @ A ) )
=> ( ! [Xs2: coindu1593790203_llist @ A,X2: A] :
( ( coindu1213758845finite @ A @ Xs2 )
=> ( ( P @ Xs2 )
=> ( P @ ( coindu1121789889_LCons @ A @ X2 @ Xs2 ) ) ) )
=> ( P @ X ) ) ) ) ).
% lfinite.inducts
thf(fact_123_lprefix__code_I2_J,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] :
~ ( coindu1696667936prefix @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ ( coindu1598213697e_LNil @ A ) ) ).
% lprefix_code(2)
thf(fact_124_lprefix_Ocases,axiom,
! [A: $tType,A1: coindu1593790203_llist @ A,A22: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ A1 @ A22 )
=> ( ( ( A1
= ( coindu1598213697e_LNil @ A ) )
=> ! [Xs2: coindu1593790203_llist @ A] : A22 != Xs2 )
=> ~ ! [Xs2: coindu1593790203_llist @ A,Ys3: coindu1593790203_llist @ A,X2: A] :
( ( A1
= ( coindu1121789889_LCons @ A @ X2 @ Xs2 ) )
=> ( ( A22
= ( coindu1121789889_LCons @ A @ X2 @ Ys3 ) )
=> ~ ( coindu1696667936prefix @ A @ Xs2 @ Ys3 ) ) ) ) ) ).
% lprefix.cases
thf(fact_125_lprefix_Osimps,axiom,
! [A: $tType] :
( ( coindu1696667936prefix @ A )
= ( ^ [A12: coindu1593790203_llist @ A,A23: coindu1593790203_llist @ A] :
( ? [Xs3: coindu1593790203_llist @ A] :
( ( A12
= ( coindu1598213697e_LNil @ A ) )
& ( A23 = Xs3 ) )
| ? [Xs3: coindu1593790203_llist @ A,Ys4: coindu1593790203_llist @ A,X3: A] :
( ( A12
= ( coindu1121789889_LCons @ A @ X3 @ Xs3 ) )
& ( A23
= ( coindu1121789889_LCons @ A @ X3 @ Ys4 ) )
& ( coindu1696667936prefix @ A @ Xs3 @ Ys4 ) ) ) ) ) ).
% lprefix.simps
thf(fact_126_lprefix_Ocoinduct,axiom,
! [A: $tType,X4: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o,X: coindu1593790203_llist @ A,Xa: coindu1593790203_llist @ A] :
( ( X4 @ X @ Xa )
=> ( ! [X2: coindu1593790203_llist @ A,Xa2: coindu1593790203_llist @ A] :
( ( X4 @ X2 @ Xa2 )
=> ( ? [Xs4: coindu1593790203_llist @ A] :
( ( X2
= ( coindu1598213697e_LNil @ A ) )
& ( Xa2 = Xs4 ) )
| ? [Xs4: coindu1593790203_llist @ A,Ys5: coindu1593790203_llist @ A,Xb: A] :
( ( X2
= ( coindu1121789889_LCons @ A @ Xb @ Xs4 ) )
& ( Xa2
= ( coindu1121789889_LCons @ A @ Xb @ Ys5 ) )
& ( ( X4 @ Xs4 @ Ys5 )
| ( coindu1696667936prefix @ A @ Xs4 @ Ys5 ) ) ) ) )
=> ( coindu1696667936prefix @ A @ X @ Xa ) ) ) ).
% lprefix.coinduct
thf(fact_127_lprefix__LCons__conv,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Y: A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ ( coindu1121789889_LCons @ A @ Y @ Ys ) )
= ( ( Xs
= ( coindu1598213697e_LNil @ A ) )
| ? [Xs5: coindu1593790203_llist @ A] :
( ( Xs
= ( coindu1121789889_LCons @ A @ Y @ Xs5 ) )
& ( coindu1696667936prefix @ A @ Xs5 @ Ys ) ) ) ) ).
% lprefix_LCons_conv
thf(fact_128_corec__llist__never__stop,axiom,
! [A: $tType,B: $tType,IS_LNIL: B > $o,LHD: B > A,MORE: B > ( coindu1593790203_llist @ A ),LTL: B > B,X: B] :
( ( coindu1244876290_llist @ B @ A @ IS_LNIL @ LHD
@ ^ [Uu: B] : $false
@ MORE
@ LTL
@ X )
= ( coindu1599971794_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ X ) ) ).
% corec_llist_never_stop
thf(fact_129_Singleton,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: A] : ( coindu48241764sorted @ A @ ( coindu1121789889_LCons @ A @ X @ ( coindu1598213697e_LNil @ A ) ) ) ) ).
% Singleton
thf(fact_130_ord_Olsorted__code_I2_J,axiom,
! [A: $tType,Less_eq: A > A > $o,X: A] : ( coindu543932918sorted @ A @ Less_eq @ ( coindu1121789889_LCons @ A @ X @ ( coindu1598213697e_LNil @ A ) ) ) ).
% ord.lsorted_code(2)
thf(fact_131_ord_Olsorted_Ocases,axiom,
! [A: $tType,Less_eq: A > A > $o,A3: coindu1593790203_llist @ A] :
( ( coindu543932918sorted @ A @ Less_eq @ A3 )
=> ( ( A3
!= ( coindu1598213697e_LNil @ A ) )
=> ( ! [X2: A] :
( A3
!= ( coindu1121789889_LCons @ A @ X2 @ ( coindu1598213697e_LNil @ A ) ) )
=> ~ ! [X2: A,Y3: A,Xs2: coindu1593790203_llist @ A] :
( ( A3
= ( coindu1121789889_LCons @ A @ X2 @ ( coindu1121789889_LCons @ A @ Y3 @ Xs2 ) ) )
=> ( ( Less_eq @ X2 @ Y3 )
=> ~ ( coindu543932918sorted @ A @ Less_eq @ ( coindu1121789889_LCons @ A @ Y3 @ Xs2 ) ) ) ) ) ) ) ).
% ord.lsorted.cases
thf(fact_132_ord_Olsorted_Osimps,axiom,
! [A: $tType] :
( ( coindu543932918sorted @ A )
= ( ^ [Less_eq2: A > A > $o,A5: coindu1593790203_llist @ A] :
( ( A5
= ( coindu1598213697e_LNil @ A ) )
| ? [X3: A] :
( A5
= ( coindu1121789889_LCons @ A @ X3 @ ( coindu1598213697e_LNil @ A ) ) )
| ? [X3: A,Y2: A,Xs3: coindu1593790203_llist @ A] :
( ( A5
= ( coindu1121789889_LCons @ A @ X3 @ ( coindu1121789889_LCons @ A @ Y2 @ Xs3 ) ) )
& ( Less_eq2 @ X3 @ Y2 )
& ( coindu543932918sorted @ A @ Less_eq2 @ ( coindu1121789889_LCons @ A @ Y2 @ Xs3 ) ) ) ) ) ) ).
% ord.lsorted.simps
thf(fact_133_ord_Olsorted_Ocoinduct,axiom,
! [A: $tType,X4: ( coindu1593790203_llist @ A ) > $o,X: coindu1593790203_llist @ A,Less_eq: A > A > $o] :
( ( X4 @ X )
=> ( ! [X2: coindu1593790203_llist @ A] :
( ( X4 @ X2 )
=> ( ( X2
= ( coindu1598213697e_LNil @ A ) )
| ? [Xa3: A] :
( X2
= ( coindu1121789889_LCons @ A @ Xa3 @ ( coindu1598213697e_LNil @ A ) ) )
| ? [Xa3: A,Y4: A,Xs4: coindu1593790203_llist @ A] :
( ( X2
= ( coindu1121789889_LCons @ A @ Xa3 @ ( coindu1121789889_LCons @ A @ Y4 @ Xs4 ) ) )
& ( Less_eq @ Xa3 @ Y4 )
& ( ( X4 @ ( coindu1121789889_LCons @ A @ Y4 @ Xs4 ) )
| ( coindu543932918sorted @ A @ Less_eq @ ( coindu1121789889_LCons @ A @ Y4 @ Xs4 ) ) ) ) ) )
=> ( coindu543932918sorted @ A @ Less_eq @ X ) ) ) ).
% ord.lsorted.coinduct
thf(fact_134_llast__lappend__LCons,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Y: A,Ys: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ Xs )
=> ( ( coindu2000965700_llast @ A @ ( coindu268472904append @ A @ Xs @ ( coindu1121789889_LCons @ A @ Y @ Ys ) ) )
= ( coindu2000965700_llast @ A @ ( coindu1121789889_LCons @ A @ Y @ Ys ) ) ) ) ).
% llast_lappend_LCons
thf(fact_135_lsorted_Ocoinduct,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X4: ( coindu1593790203_llist @ A ) > $o,X: coindu1593790203_llist @ A] :
( ( X4 @ X )
=> ( ! [X2: coindu1593790203_llist @ A] :
( ( X4 @ X2 )
=> ( ( X2
= ( coindu1598213697e_LNil @ A ) )
| ? [Xa3: A] :
( X2
= ( coindu1121789889_LCons @ A @ Xa3 @ ( coindu1598213697e_LNil @ A ) ) )
| ? [Xa3: A,Y4: A,Xs4: coindu1593790203_llist @ A] :
( ( X2
= ( coindu1121789889_LCons @ A @ Xa3 @ ( coindu1121789889_LCons @ A @ Y4 @ Xs4 ) ) )
& ( ord_less_eq @ A @ Xa3 @ Y4 )
& ( ( X4 @ ( coindu1121789889_LCons @ A @ Y4 @ Xs4 ) )
| ( coindu48241764sorted @ A @ ( coindu1121789889_LCons @ A @ Y4 @ Xs4 ) ) ) ) ) )
=> ( coindu48241764sorted @ A @ X ) ) ) ) ).
% lsorted.coinduct
thf(fact_136_lsorted_Osimps,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ( ( coindu48241764sorted @ A )
= ( ^ [A5: coindu1593790203_llist @ A] :
( ( A5
= ( coindu1598213697e_LNil @ A ) )
| ? [X3: A] :
( A5
= ( coindu1121789889_LCons @ A @ X3 @ ( coindu1598213697e_LNil @ A ) ) )
| ? [X3: A,Y2: A,Xs3: coindu1593790203_llist @ A] :
( ( A5
= ( coindu1121789889_LCons @ A @ X3 @ ( coindu1121789889_LCons @ A @ Y2 @ Xs3 ) ) )
& ( ord_less_eq @ A @ X3 @ Y2 )
& ( coindu48241764sorted @ A @ ( coindu1121789889_LCons @ A @ Y2 @ Xs3 ) ) ) ) ) ) ) ).
% lsorted.simps
thf(fact_137_lsorted_Ocases,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A3: coindu1593790203_llist @ A] :
( ( coindu48241764sorted @ A @ A3 )
=> ( ( A3
!= ( coindu1598213697e_LNil @ A ) )
=> ( ! [X2: A] :
( A3
!= ( coindu1121789889_LCons @ A @ X2 @ ( coindu1598213697e_LNil @ A ) ) )
=> ~ ! [X2: A,Y3: A,Xs2: coindu1593790203_llist @ A] :
( ( A3
= ( coindu1121789889_LCons @ A @ X2 @ ( coindu1121789889_LCons @ A @ Y3 @ Xs2 ) ) )
=> ( ( ord_less_eq @ A @ X2 @ Y3 )
=> ~ ( coindu48241764sorted @ A @ ( coindu1121789889_LCons @ A @ Y3 @ Xs2 ) ) ) ) ) ) ) ) ).
% lsorted.cases
thf(fact_138_lstrict__prefix__lappend__conv,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu574146665prefix @ A @ Xs @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( ( coindu1213758845finite @ A @ Xs )
& ~ ( coindu335574135_lnull @ A @ Ys ) ) ) ).
% lstrict_prefix_lappend_conv
thf(fact_139_lprefixI,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,X4: set @ ( product_prod @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) )] :
( ( member @ ( product_prod @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) ) @ ( product_Pair @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) @ Xs @ Ys ) @ X4 )
=> ( ! [Xs2: coindu1593790203_llist @ A,Ys3: coindu1593790203_llist @ A] :
( ( member @ ( product_prod @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) ) @ ( product_Pair @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) @ Xs2 @ Ys3 ) @ X4 )
=> ( ( coindu335574135_lnull @ A @ Xs2 )
| ? [X8: A,Xs6: coindu1593790203_llist @ A,Ys6: coindu1593790203_llist @ A] :
( ( Xs2
= ( coindu1121789889_LCons @ A @ X8 @ Xs6 ) )
& ( Ys3
= ( coindu1121789889_LCons @ A @ X8 @ Ys6 ) )
& ( ( member @ ( product_prod @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) ) @ ( product_Pair @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) @ Xs6 @ Ys6 ) @ X4 )
| ( coindu1696667936prefix @ A @ Xs6 @ Ys6 ) ) ) ) )
=> ( coindu1696667936prefix @ A @ Xs @ Ys ) ) ) ).
% lprefixI
thf(fact_140_lappend__code_I2_J,axiom,
! [A: $tType,Xa: A,X: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ ( coindu1121789889_LCons @ A @ Xa @ X ) @ Ys )
= ( coindu1121789889_LCons @ A @ Xa @ ( coindu268472904append @ A @ X @ Ys ) ) ) ).
% lappend_code(2)
thf(fact_141_lappend_Odisc__iff_I2_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ~ ( coindu335574135_lnull @ A @ ( coindu268472904append @ A @ Xs @ Ys ) ) )
= ( ~ ( coindu335574135_lnull @ A @ Xs )
| ~ ( coindu335574135_lnull @ A @ Ys ) ) ) ).
% lappend.disc_iff(2)
thf(fact_142_lnull__lappend,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( ( coindu335574135_lnull @ A @ Xs )
& ( coindu335574135_lnull @ A @ Ys ) ) ) ).
% lnull_lappend
thf(fact_143_lappend__code_I1_J,axiom,
! [A: $tType,Ys: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ ( coindu1598213697e_LNil @ A ) @ Ys )
= Ys ) ).
% lappend_code(1)
thf(fact_144_lappend__LNil2,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ Xs @ ( coindu1598213697e_LNil @ A ) )
= Xs ) ).
% lappend_LNil2
thf(fact_145_lfinite__lappend,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( ( coindu1213758845finite @ A @ Xs )
& ( coindu1213758845finite @ A @ Ys ) ) ) ).
% lfinite_lappend
thf(fact_146_lsorted__LCons__LCons,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Xs: coindu1593790203_llist @ A] :
( ( coindu48241764sorted @ A @ ( coindu1121789889_LCons @ A @ X @ ( coindu1121789889_LCons @ A @ Y @ Xs ) ) )
= ( ( ord_less_eq @ A @ X @ Y )
& ( coindu48241764sorted @ A @ ( coindu1121789889_LCons @ A @ Y @ Xs ) ) ) ) ) ).
% lsorted_LCons_LCons
thf(fact_147_lprefix__lappend__same,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ ( coindu268472904append @ A @ Xs @ Ys ) @ ( coindu268472904append @ A @ Xs @ Zs ) )
= ( ( coindu1213758845finite @ A @ Xs )
=> ( coindu1696667936prefix @ A @ Ys @ Zs ) ) ) ).
% lprefix_lappend_same
thf(fact_148_lappend__assoc,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ ( coindu268472904append @ A @ Xs @ Ys ) @ Zs )
= ( coindu268472904append @ A @ Xs @ ( coindu268472904append @ A @ Ys @ Zs ) ) ) ).
% lappend_assoc
thf(fact_149_wlog__linorder__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,B3: A,A3: A] :
( ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% wlog_linorder_le
thf(fact_150_lprefix__lappend,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ Xs @ ( coindu268472904append @ A @ Xs @ Ys ) ) ).
% lprefix_lappend
thf(fact_151_lappend__lprefixE,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ ( coindu268472904append @ A @ Xs @ Ys ) @ Zs )
=> ~ ! [Zs2: coindu1593790203_llist @ A] :
( Zs
!= ( coindu268472904append @ A @ Xs @ Zs2 ) ) ) ).
% lappend_lprefixE
thf(fact_152_lprefix__lappendD,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ ( coindu268472904append @ A @ Ys @ Zs ) )
=> ( ( coindu1696667936prefix @ A @ Xs @ Ys )
| ( coindu1696667936prefix @ A @ Ys @ Xs ) ) ) ).
% lprefix_lappendD
thf(fact_153_lprefix__conv__lappend,axiom,
! [A: $tType] :
( ( coindu1696667936prefix @ A )
= ( ^ [Xs3: coindu1593790203_llist @ A,Ys4: coindu1593790203_llist @ A] :
? [Zs3: coindu1593790203_llist @ A] :
( Ys4
= ( coindu268472904append @ A @ Xs3 @ Zs3 ) ) ) ) ).
% lprefix_conv_lappend
thf(fact_154_lprefix__lappend__sameI,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( coindu1696667936prefix @ A @ ( coindu268472904append @ A @ Zs @ Xs ) @ ( coindu268472904append @ A @ Zs @ Ys ) ) ) ).
% lprefix_lappend_sameI
thf(fact_155_lappend_Odisc_I2_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ~ ( coindu335574135_lnull @ A @ Xs )
| ~ ( coindu335574135_lnull @ A @ Ys ) )
=> ~ ( coindu335574135_lnull @ A @ ( coindu268472904append @ A @ Xs @ Ys ) ) ) ).
% lappend.disc(2)
thf(fact_156_lappend_Odisc_I1_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu335574135_lnull @ A @ Ys )
=> ( coindu335574135_lnull @ A @ ( coindu268472904append @ A @ Xs @ Ys ) ) ) ) ).
% lappend.disc(1)
thf(fact_157_lappend__lnull1,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu268472904append @ A @ Xs @ Ys )
= Ys ) ) ).
% lappend_lnull1
thf(fact_158_lappend__lnull2,axiom,
! [A: $tType,Ys: coindu1593790203_llist @ A,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Ys )
=> ( ( coindu268472904append @ A @ Xs @ Ys )
= Xs ) ) ).
% lappend_lnull2
thf(fact_159_lappend__LNil__LNil,axiom,
! [A: $tType] :
( ( coindu268472904append @ A @ ( coindu1598213697e_LNil @ A ) @ ( coindu1598213697e_LNil @ A ) )
= ( coindu1598213697e_LNil @ A ) ) ).
% lappend_LNil_LNil
thf(fact_160_LNil__eq__lappend__iff,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu1598213697e_LNil @ A )
= ( coindu268472904append @ A @ Xs @ Ys ) )
= ( ( Xs
= ( coindu1598213697e_LNil @ A ) )
& ( Ys
= ( coindu1598213697e_LNil @ A ) ) ) ) ).
% LNil_eq_lappend_iff
thf(fact_161_lappend__eq__LNil__iff,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu268472904append @ A @ Xs @ Ys )
= ( coindu1598213697e_LNil @ A ) )
= ( ( Xs
= ( coindu1598213697e_LNil @ A ) )
& ( Ys
= ( coindu1598213697e_LNil @ A ) ) ) ) ).
% lappend_eq_LNil_iff
thf(fact_162_lappend__inf,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ~ ( coindu1213758845finite @ A @ Xs )
=> ( ( coindu268472904append @ A @ Xs @ Ys )
= Xs ) ) ).
% lappend_inf
thf(fact_163_llexord__lappend__leftI,axiom,
! [A: $tType,R: A > A > $o,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A,Xs: coindu1593790203_llist @ A] :
( ( coindu300403952lexord @ A @ R @ Ys @ Zs )
=> ( coindu300403952lexord @ A @ R @ ( coindu268472904append @ A @ Xs @ Ys ) @ ( coindu268472904append @ A @ Xs @ Zs ) ) ) ).
% llexord_lappend_leftI
thf(fact_164_llexord__append__right,axiom,
! [A: $tType,R: A > A > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] : ( coindu300403952lexord @ A @ R @ Xs @ ( coindu268472904append @ A @ Xs @ Ys ) ) ).
% llexord_append_right
thf(fact_165_lappend__snocL1__conv__LCons2,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Y: A,Ys: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ ( coindu268472904append @ A @ Xs @ ( coindu1121789889_LCons @ A @ Y @ ( coindu1598213697e_LNil @ A ) ) ) @ Ys )
= ( coindu268472904append @ A @ Xs @ ( coindu1121789889_LCons @ A @ Y @ Ys ) ) ) ).
% lappend_snocL1_conv_LCons2
thf(fact_166_lappend_Octr_I1_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu335574135_lnull @ A @ Ys )
=> ( ( coindu268472904append @ A @ Xs @ Ys )
= ( coindu1598213697e_LNil @ A ) ) ) ) ).
% lappend.ctr(1)
thf(fact_167_LCons__LCons,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Xs: coindu1593790203_llist @ A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( coindu48241764sorted @ A @ ( coindu1121789889_LCons @ A @ Y @ Xs ) )
=> ( coindu48241764sorted @ A @ ( coindu1121789889_LCons @ A @ X @ ( coindu1121789889_LCons @ A @ Y @ Xs ) ) ) ) ) ) ).
% LCons_LCons
thf(fact_168_lfinite__rev__induct,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: ( coindu1593790203_llist @ A ) > $o] :
( ( coindu1213758845finite @ A @ Xs )
=> ( ( P @ ( coindu1598213697e_LNil @ A ) )
=> ( ! [X2: A,Xs2: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ Xs2 )
=> ( ( P @ Xs2 )
=> ( P @ ( coindu268472904append @ A @ Xs2 @ ( coindu1121789889_LCons @ A @ X2 @ ( coindu1598213697e_LNil @ A ) ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lfinite_rev_induct
thf(fact_169_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X23: B,Y1: A,Y23: B] :
( ( ( product_Pair @ A @ B @ X1 @ X23 )
= ( product_Pair @ A @ B @ Y1 @ Y23 ) )
= ( ( X1 = Y1 )
& ( X23 = Y23 ) ) ) ).
% prod.inject
thf(fact_170_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A3: A,B3: B,A6: A,B4: B] :
( ( ( product_Pair @ A @ B @ A3 @ B3 )
= ( product_Pair @ A @ B @ A6 @ B4 ) )
= ( ( A3 = A6 )
& ( B3 = B4 ) ) ) ).
% old.prod.inject
thf(fact_171_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_172_pred__subset__eq2,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( A > B > $o )
@ ^ [X3: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y2 ) @ R3 )
@ ^ [X3: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y2 ) @ S ) )
= ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S ) ) ).
% pred_subset_eq2
thf(fact_173_predicate2I,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
( ! [X2: A,Y3: B] :
( ( P @ X2 @ Y3 )
=> ( Q @ X2 @ Y3 ) )
=> ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).
% predicate2I
thf(fact_174_predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,X: A,Y: B] :
( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( ( P @ X @ Y )
=> ( Q @ X @ Y ) ) ) ).
% predicate2D
thf(fact_175_rev__predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,X: A,Y: B,Q: A > B > $o] :
( ( P @ X @ Y )
=> ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( Q @ X @ Y ) ) ) ).
% rev_predicate2D
thf(fact_176_pred__subset__eq,axiom,
! [A: $tType,R3: set @ A,S: set @ A] :
( ( ord_less_eq @ ( A > $o )
@ ^ [X3: A] : ( member @ A @ X3 @ R3 )
@ ^ [X3: A] : ( member @ A @ X3 @ S ) )
= ( ord_less_eq @ ( set @ A ) @ R3 @ S ) ) ).
% pred_subset_eq
thf(fact_177_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ) ).
% dual_order.antisym
thf(fact_178_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A3: A,C3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ C3 @ B3 )
=> ( ord_less_eq @ A @ C3 @ A3 ) ) ) ) ).
% dual_order.trans
thf(fact_179_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,A3: A,B3: A] :
( ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: A,B2: A] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% linorder_wlog
thf(fact_180_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% dual_order.refl
thf(fact_181_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_182_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ) ).
% order_class.order.antisym
thf(fact_183_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( B3 = C3 )
=> ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_184_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A,C3: A] :
( ( A3 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C3 )
=> ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_185_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_186_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_187_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C3 )
=> ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% order.trans
thf(fact_188_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_189_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_190_linorder__class_Olinear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linorder_class.linear
thf(fact_191_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_192_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y5: A,Z2: A] : Y5 = Z2 )
= ( ^ [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
& ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ) ) ).
% eq_iff
thf(fact_193_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A3: A,B3: A,F: A > B,C3: B] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C3 )
=> ( ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ B @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_194_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A3: A,F: B > A,B3: B,C3: B] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C3 )
=> ( ! [X2: B,Y3: B] :
( ( ord_less_eq @ B @ X2 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_195_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C @ ( type2 @ C ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A3: A,B3: A,F: A > C,C3: C] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ C @ ( F @ B3 ) @ C3 )
=> ( ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ord_less_eq @ C @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ C @ ( F @ A3 ) @ C3 ) ) ) ) ) ).
% order_subst2
thf(fact_196_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A3: A,F: B > A,B3: B,C3: B] :
( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C3 )
=> ( ! [X2: B,Y3: B] :
( ( ord_less_eq @ B @ X2 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C3 ) ) ) ) ) ) ).
% order_subst1
thf(fact_197_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
! [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% le_fun_def
thf(fact_198_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B] :
( ! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_199_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_200_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_201_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A2: A,B2: B] : ( P @ ( product_Pair @ A @ B @ A2 @ B2 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_202_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A2: A,B2: B] :
( Y
!= ( product_Pair @ A @ B @ A2 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_203_prod__induct7,axiom,
! [G3: $tType,F3: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) )] :
( ! [A2: A,B2: B,C2: C,D2: D,E2: E,F4: F3,G4: G3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G3 ) @ E2 @ ( product_Pair @ F3 @ G3 @ F4 @ G4 ) ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct7
thf(fact_204_prod__induct6,axiom,
! [F3: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
( ! [A2: A,B2: B,C2: C,D2: D,E2: E,F4: F3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E2 @ F4 ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct6
thf(fact_205_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
( ! [A2: A,B2: B,C2: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct5
thf(fact_206_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A2: A,B2: B,C2: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B2 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
=> ( P @ X ) ) ).
% prod_induct4
thf(fact_207_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A2: A,B2: B,C2: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A2 @ ( product_Pair @ B @ C @ B2 @ C2 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_208_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,G3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) )] :
~ ! [A2: A,B2: B,C2: C,D2: D,E2: E,F4: F3,G4: G3] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G3 ) @ E2 @ ( product_Pair @ F3 @ G3 @ F4 @ G4 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_209_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
~ ! [A2: A,B2: B,C2: C,D2: D,E2: E,F4: F3] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E2 @ F4 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_210_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
~ ! [A2: A,B2: B,C2: C,D2: D,E2: E] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B2 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_211_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A2: A,B2: B,C2: C,D2: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A2 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B2 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_212_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A2: A,B2: B,C2: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A2 @ ( product_Pair @ B @ C @ B2 @ C2 ) ) ) ).
% prod_cases3
thf(fact_213_Pair__inject,axiom,
! [A: $tType,B: $tType,A3: A,B3: B,A6: A,B4: B] :
( ( ( product_Pair @ A @ B @ A3 @ B3 )
= ( product_Pair @ A @ B @ A6 @ B4 ) )
=> ~ ( ( A3 = A6 )
=> ( B3 != B4 ) ) ) ).
% Pair_inject
thf(fact_214_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
( ! [A2: A,B2: B] : ( P @ ( product_Pair @ A @ B @ A2 @ B2 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_215_surj__pair,axiom,
! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
? [X2: A,Y3: B] :
( P2
= ( product_Pair @ A @ B @ X2 @ Y3 ) ) ).
% surj_pair
thf(fact_216_subrelI,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ! [X2: A,Y3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ R )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ S2 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S2 ) ) ).
% subrelI
thf(fact_217_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X3: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y2 ) @ R3 ) )
= ( ^ [X3: A,Y2: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y2 ) @ S ) ) )
= ( R3 = S ) ) ).
% pred_equals_eq2
thf(fact_218_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A3: A,B3: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
= ( F1 @ A3 @ B3 ) ) ).
% old.prod.rec
thf(fact_219_subsetI,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] :
( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( member @ A @ X2 @ B5 ) )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B5 ) ) ).
% subsetI
thf(fact_220_subset__antisym,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A4 )
=> ( A4 = B5 ) ) ) ).
% subset_antisym
thf(fact_221_predicate1I,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% predicate1I
thf(fact_222_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_223_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_224_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_225_contra__subsetD,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,C3: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
=> ( ~ ( member @ A @ C3 @ B5 )
=> ~ ( member @ A @ C3 @ A4 ) ) ) ).
% contra_subsetD
thf(fact_226_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y5: set @ A,Z2: set @ A] : Y5 = Z2 )
= ( ^ [A7: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
& ( ord_less_eq @ ( set @ A ) @ B6 @ A7 ) ) ) ) ).
% set_eq_subset
thf(fact_227_subset__trans,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ C4 ) ) ) ).
% subset_trans
thf(fact_228_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_229_subset__refl,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ A4 ) ).
% subset_refl
thf(fact_230_rev__subsetD,axiom,
! [A: $tType,C3: A,A4: set @ A,B5: set @ A] :
( ( member @ A @ C3 @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
=> ( member @ A @ C3 @ B5 ) ) ) ).
% rev_subsetD
thf(fact_231_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
! [T2: A] :
( ( member @ A @ T2 @ A7 )
=> ( member @ A @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_232_set__rev__mp,axiom,
! [A: $tType,X: A,A4: set @ A,B5: set @ A] :
( ( member @ A @ X @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
=> ( member @ A @ X @ B5 ) ) ) ).
% set_rev_mp
thf(fact_233_equalityD2,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] :
( ( A4 = B5 )
=> ( ord_less_eq @ ( set @ A ) @ B5 @ A4 ) ) ).
% equalityD2
thf(fact_234_equalityD1,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] :
( ( A4 = B5 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B5 ) ) ).
% equalityD1
thf(fact_235_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
! [X3: A] :
( ( member @ A @ X3 @ A7 )
=> ( member @ A @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_236_equalityE,axiom,
! [A: $tType,A4: set @ A,B5: set @ A] :
( ( A4 = B5 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A4 ) ) ) ).
% equalityE
thf(fact_237_subsetCE,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,C3: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
=> ( ( member @ A @ C3 @ A4 )
=> ( member @ A @ C3 @ B5 ) ) ) ).
% subsetCE
thf(fact_238_subsetD,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,C3: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
=> ( ( member @ A @ C3 @ A4 )
=> ( member @ A @ C3 @ B5 ) ) ) ).
% subsetD
thf(fact_239_in__mono,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
=> ( ( member @ A @ X @ A4 )
=> ( member @ A @ X @ B5 ) ) ) ).
% in_mono
thf(fact_240_set__mp,axiom,
! [A: $tType,A4: set @ A,B5: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B5 )
=> ( ( member @ A @ X @ A4 )
=> ( member @ A @ X @ B5 ) ) ) ).
% set_mp
thf(fact_241_less__eq__set__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
( ord_less_eq @ ( A > $o )
@ ^ [X3: A] : ( member @ A @ X3 @ A7 )
@ ^ [X3: A] : ( member @ A @ X3 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_242_eq__subset,axiom,
! [A: $tType,P: A > A > $o] :
( ord_less_eq @ ( A > A > $o )
@ ^ [Y5: A,Z2: A] : Y5 = Z2
@ ^ [A5: A,B7: A] :
( ( P @ A5 @ B7 )
| ( A5 = B7 ) ) ) ).
% eq_subset
thf(fact_243_conj__subset__def,axiom,
! [A: $tType,A4: set @ A,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A4
@ ( collect @ A
@ ^ [X3: A] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( collect @ A @ P ) )
& ( ord_less_eq @ ( set @ A ) @ A4 @ ( collect @ A @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_244_predicate2D__conj,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,R3: $o,X: A,Y: B] :
( ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
& R3 )
=> ( R3
& ( ( P @ X @ Y )
=> ( Q @ X @ Y ) ) ) ) ).
% predicate2D_conj
thf(fact_245_subset__Collect__iff,axiom,
! [A: $tType,B5: set @ A,A4: set @ A,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ A4 )
& ( P @ X3 ) ) ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ B5 )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_246_subset__CollectI,axiom,
! [A: $tType,B5: set @ A,A4: set @ A,Q: A > $o,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ A4 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ B5 )
=> ( ( Q @ X2 )
=> ( P @ X2 ) ) )
=> ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ B5 )
& ( Q @ X3 ) ) )
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ A4 )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_247_prop__restrict,axiom,
! [A: $tType,X: A,Z3: set @ A,X4: set @ A,P: A > $o] :
( ( member @ A @ X @ Z3 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z3
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ X4 )
& ( P @ X3 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_248_Collect__restrict,axiom,
! [A: $tType,X4: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ X4 )
& ( P @ X3 ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_249_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R: A,S2: B,R3: set @ ( product_prod @ A @ B ),S3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S2 ) @ R3 )
=> ( ( S3 = S2 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S3 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_250_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A3: B,B3: C] :
( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A3 @ B3 ) )
= ( C3 @ A3 @ B3 ) ) ).
% internal_case_prod_conv
thf(fact_251_llast__lappend,axiom,
! [A: $tType,Ys: coindu1593790203_llist @ A,Xs: coindu1593790203_llist @ A] :
( ( ( coindu335574135_lnull @ A @ Ys )
=> ( ( coindu2000965700_llast @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( coindu2000965700_llast @ A @ Xs ) ) )
& ( ~ ( coindu335574135_lnull @ A @ Ys )
=> ( ( ( coindu1213758845finite @ A @ Xs )
=> ( ( coindu2000965700_llast @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( coindu2000965700_llast @ A @ Ys ) ) )
& ( ~ ( coindu1213758845finite @ A @ Xs )
=> ( ( coindu2000965700_llast @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( undefined @ A ) ) ) ) ) ) ).
% llast_lappend
thf(fact_252_llast__linfinite,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ~ ( coindu1213758845finite @ A @ Xs )
=> ( ( coindu2000965700_llast @ A @ Xs )
= ( undefined @ A ) ) ) ).
% llast_linfinite
thf(fact_253_llast__LNil,axiom,
! [A: $tType] :
( ( coindu2000965700_llast @ A @ ( coindu1598213697e_LNil @ A ) )
= ( undefined @ A ) ) ).
% llast_LNil
%----Type constructors (10)
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_Oord,axiom,
! [A8: $tType,A9: $tType] :
( ( ord @ A9 @ ( type2 @ A9 ) )
=> ( ord @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_1,axiom,
! [A8: $tType] : ( preorder @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_2,axiom,
! [A8: $tType] : ( order @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_3,axiom,
! [A8: $tType] : ( ord @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
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,
! [P: $o] :
( ( P = $true )
| ( P = $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 (2)
thf(conj_0,hypothesis,
! [Y4: a,Ys6: coindu1593790203_llist @ a] :
( ( ysa
= ( coindu1121789889_LCons @ a @ Y4 @ Ys6 ) )
=> ( ~ ( r @ Y4 @ x )
=> ( ( ( Y4 != x )
| ~ ( coindu300403952lexord @ a @ r @ Ys6 @ xs2 ) )
=> thesis ) ) ) ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------