TPTP Problem File: DAT120^1.p
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%------------------------------------------------------------------------------
% File : DAT120^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Coinductive list 1483
% 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__1483.p [Bla16]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.25 v7.5.0, 1.00 v7.3.0, 0.67 v7.2.0, 0.75 v7.1.0
% Syntax : Number of formulae : 347 ( 128 unt; 56 typ; 0 def)
% Number of atoms : 794 ( 284 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 3537 ( 102 ~; 39 |; 49 &;3017 @)
% ( 0 <=>; 330 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 7 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 207 ( 207 >; 0 *; 0 +; 0 <<)
% Number of symbols : 56 ( 53 usr; 5 con; 0-6 aty)
% Number of variables : 991 ( 51 ^; 850 !; 40 ?; 991 :)
% ( 50 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:50:34.804
%------------------------------------------------------------------------------
%----Could-be-implicit typings (7)
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_Extended__Nat_Oenat,type,
extended_enat: $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (49)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ominus,type,
minus:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocomm__monoid__diff,type,
comm_monoid_diff:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1352612707id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel146912293up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
comple187826305attice:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ogen__llength,type,
coindu493225792length:
!>[A: $tType] : ( nat > ( coindu1593790203_llist @ A ) > extended_enat ) ).
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_Ollength,type,
coindu1018505716length:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > extended_enat ) ).
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_Olhd,type,
coindu1046438764le_lhd:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > A ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Ollist__all2,type,
coindu987187967t_all2:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ B ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Olmap,type,
coindu1062782156e_lmap:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ Aa ) ) ).
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_Oltl,type,
coindu1047225960le_ltl:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
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_Olstrict__prefix,type,
coindu574146665prefix:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Oltake,type,
coindu1802687541_ltake:
!>[A: $tType] : ( extended_enat > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
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_Olzip,type,
coindu847746867e_lzip:
!>[A: $tType,B: $tType] : ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ B ) > ( coindu1593790203_llist @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Extended__Nat_Oenat_Ocase__enat,type,
extended_case_enat:
!>[T: $tType] : ( ( nat > T ) > T > extended_enat > T ) ).
thf(sy_c_Extended__Nat_Othe__enat,type,
extended_the_enat: extended_enat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[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_Orderings_Oord__class_Omin,type,
ord_min:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Relation_Oinv__imagep,type,
inv_imagep:
!>[B: $tType,A: $tType] : ( ( B > B > $o ) > ( A > B ) > A > A > $o ) ).
thf(sy_c_Relation_Otransp,type,
transp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
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_m,type,
m: extended_enat ).
thf(sy_v_n,type,
n: extended_enat ).
thf(sy_v_xs,type,
xs: coindu1593790203_llist @ a ).
%----Relevant facts (252)
thf(fact_0_lprefix__refl,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ Xs @ Xs ) ).
% lprefix_refl
thf(fact_1_llist_Oleq__refl,axiom,
! [A: $tType,X: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ X @ X ) ).
% llist.leq_refl
thf(fact_2__092_060open_062n_A_092_060le_062_Am_A_092_060or_062_Allength_Axs_A_092_060le_062_Am_092_060close_062,axiom,
( ( ord_less_eq @ extended_enat @ n @ m )
| ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ a @ xs ) @ m ) ) ).
% \<open>n \<le> m \<or> llength xs \<le> m\<close>
thf(fact_3_ltake__is__lprefix,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) @ Xs ) ).
% ltake_is_lprefix
thf(fact_4_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_5_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_6_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_7_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_8_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_9_lstrict__prefix__def,axiom,
! [A: $tType] :
( ( coindu574146665prefix @ A )
= ( ^ [Xs2: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs2 @ Ys2 )
& ( Xs2 != Ys2 ) ) ) ) ).
% lstrict_prefix_def
thf(fact_10_ltake__eq__ltake__antimono,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,M: extended_enat] :
( ( ( coindu1802687541_ltake @ A @ N @ Xs )
= ( coindu1802687541_ltake @ A @ N @ Ys ) )
=> ( ( ord_less_eq @ extended_enat @ M @ N )
=> ( ( coindu1802687541_ltake @ A @ M @ Xs )
= ( coindu1802687541_ltake @ A @ M @ Ys ) ) ) ) ).
% ltake_eq_ltake_antimono
thf(fact_11_ltake__all,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,M: extended_enat] :
( ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ M )
=> ( ( coindu1802687541_ltake @ A @ M @ Xs )
= Xs ) ) ).
% ltake_all
thf(fact_12_ltake__lmap,axiom,
! [A: $tType,B: $tType,N: extended_enat,F: B > A,Xs: coindu1593790203_llist @ B] :
( ( coindu1802687541_ltake @ A @ N @ ( coindu1062782156e_lmap @ B @ A @ F @ Xs ) )
= ( coindu1062782156e_lmap @ B @ A @ F @ ( coindu1802687541_ltake @ B @ N @ Xs ) ) ) ).
% ltake_lmap
thf(fact_13_ltake__LNil,axiom,
! [A: $tType,N: extended_enat] :
( ( coindu1802687541_ltake @ A @ N @ ( coindu1598213697e_LNil @ A ) )
= ( coindu1598213697e_LNil @ A ) ) ).
% ltake_LNil
thf(fact_14_ltake__ltake,axiom,
! [A: $tType,N: extended_enat,M: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( coindu1802687541_ltake @ A @ N @ ( coindu1802687541_ltake @ A @ M @ Xs ) )
= ( coindu1802687541_ltake @ A @ ( ord_min @ extended_enat @ N @ M ) @ Xs ) ) ).
% ltake_ltake
thf(fact_15_lprefix__code_I1_J,axiom,
! [A: $tType,Ys: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ ( coindu1598213697e_LNil @ A ) @ Ys ) ).
% lprefix_code(1)
thf(fact_16_wlog__linorder__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,B2: A,A2: A] :
( ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% wlog_linorder_le
thf(fact_17_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_18_llist_Otransp__le,axiom,
! [A: $tType] : ( transp @ ( coindu1593790203_llist @ A ) @ ( coindu1696667936prefix @ A ) ) ).
% llist.transp_le
thf(fact_19_lprefix__lappend,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ Xs @ ( coindu268472904append @ A @ Xs @ Ys ) ) ).
% lprefix_lappend
thf(fact_20_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_21_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_22_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_23_lappend__code_I1_J,axiom,
! [A: $tType,Ys: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ ( coindu1598213697e_LNil @ A ) @ Ys )
= Ys ) ).
% lappend_code(1)
thf(fact_24_lappend__LNil2,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ Xs @ ( coindu1598213697e_LNil @ A ) )
= Xs ) ).
% lappend_LNil2
thf(fact_25_llength__lmap,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: coindu1593790203_llist @ B] :
( ( coindu1018505716length @ A @ ( coindu1062782156e_lmap @ B @ A @ F @ Xs ) )
= ( coindu1018505716length @ B @ Xs ) ) ).
% llength_lmap
thf(fact_26_llength__ltake,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( coindu1018505716length @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) )
= ( ord_min @ extended_enat @ N @ ( coindu1018505716length @ A @ Xs ) ) ) ).
% llength_ltake
thf(fact_27_llist_Osimps_I13_J,axiom,
! [B: $tType,A: $tType,F: A > B,X21: A,X22: coindu1593790203_llist @ A] :
( ( coindu1062782156e_lmap @ A @ B @ F @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) )
= ( coindu1121789889_LCons @ B @ ( F @ X21 ) @ ( coindu1062782156e_lmap @ A @ B @ F @ X22 ) ) ) ).
% llist.simps(13)
thf(fact_28_llist_Osimps_I12_J,axiom,
! [A: $tType,B: $tType,F: A > B] :
( ( coindu1062782156e_lmap @ A @ B @ F @ ( coindu1598213697e_LNil @ A ) )
= ( coindu1598213697e_LNil @ B ) ) ).
% llist.simps(12)
thf(fact_29_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_30_LNil__eq__lmap,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: coindu1593790203_llist @ B] :
( ( ( coindu1598213697e_LNil @ A )
= ( coindu1062782156e_lmap @ B @ A @ F @ Xs ) )
= ( Xs
= ( coindu1598213697e_LNil @ B ) ) ) ).
% LNil_eq_lmap
thf(fact_31_lmap__eq__LNil,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: coindu1593790203_llist @ B] :
( ( ( coindu1062782156e_lmap @ B @ A @ F @ Xs )
= ( coindu1598213697e_LNil @ A ) )
= ( Xs
= ( coindu1598213697e_LNil @ B ) ) ) ).
% lmap_eq_LNil
thf(fact_32_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_33_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_34_neq__LNil__conv,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( Xs
!= ( coindu1598213697e_LNil @ A ) )
= ( ? [X2: A,Xs3: coindu1593790203_llist @ A] :
( Xs
= ( coindu1121789889_LCons @ A @ X2 @ Xs3 ) ) ) ) ).
% neq_LNil_conv
thf(fact_35_lappend__LNil__LNil,axiom,
! [A: $tType] :
( ( coindu268472904append @ A @ ( coindu1598213697e_LNil @ A ) @ ( coindu1598213697e_LNil @ A ) )
= ( coindu1598213697e_LNil @ A ) ) ).
% lappend_LNil_LNil
thf(fact_36_lmap__eq__LCons__conv,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: coindu1593790203_llist @ B,Y: A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu1062782156e_lmap @ B @ A @ F @ Xs )
= ( coindu1121789889_LCons @ A @ Y @ Ys ) )
= ( ? [X2: B,Xs3: coindu1593790203_llist @ B] :
( ( Xs
= ( coindu1121789889_LCons @ B @ X2 @ Xs3 ) )
& ( Y
= ( F @ X2 ) )
& ( Ys
= ( coindu1062782156e_lmap @ B @ A @ F @ Xs3 ) ) ) ) ) ).
% lmap_eq_LCons_conv
thf(fact_37_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_38_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_39_lmap__lappend__distrib,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: coindu1593790203_llist @ B,Ys: coindu1593790203_llist @ B] :
( ( coindu1062782156e_lmap @ B @ A @ F @ ( coindu268472904append @ B @ Xs @ Ys ) )
= ( coindu268472904append @ A @ ( coindu1062782156e_lmap @ B @ A @ F @ Xs ) @ ( coindu1062782156e_lmap @ B @ A @ F @ Ys ) ) ) ).
% lmap_lappend_distrib
thf(fact_40_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_41_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 ) )
| ? [Xs3: coindu1593790203_llist @ A] :
( ( Xs
= ( coindu1121789889_LCons @ A @ Y @ Xs3 ) )
& ( coindu1696667936prefix @ A @ Xs3 @ Ys ) ) ) ) ).
% lprefix_LCons_conv
thf(fact_42_lprefix_Ocoinduct,axiom,
! [A: $tType,X3: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o,X: coindu1593790203_llist @ A,Xa: coindu1593790203_llist @ A] :
( ( X3 @ X @ Xa )
=> ( ! [X4: coindu1593790203_llist @ A,Xa2: coindu1593790203_llist @ A] :
( ( X3 @ X4 @ Xa2 )
=> ( ? [Xs4: coindu1593790203_llist @ A] :
( ( X4
= ( coindu1598213697e_LNil @ A ) )
& ( Xa2 = Xs4 ) )
| ? [Xs4: coindu1593790203_llist @ A,Ys3: coindu1593790203_llist @ A,Xb: A] :
( ( X4
= ( coindu1121789889_LCons @ A @ Xb @ Xs4 ) )
& ( Xa2
= ( coindu1121789889_LCons @ A @ Xb @ Ys3 ) )
& ( ( X3 @ Xs4 @ Ys3 )
| ( coindu1696667936prefix @ A @ Xs4 @ Ys3 ) ) ) ) )
=> ( coindu1696667936prefix @ A @ X @ Xa ) ) ) ).
% lprefix.coinduct
thf(fact_43_lprefix_Osimps,axiom,
! [A: $tType] :
( ( coindu1696667936prefix @ A )
= ( ^ [A1: coindu1593790203_llist @ A,A22: coindu1593790203_llist @ A] :
( ? [Xs2: coindu1593790203_llist @ A] :
( ( A1
= ( coindu1598213697e_LNil @ A ) )
& ( A22 = Xs2 ) )
| ? [Xs2: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A,X2: A] :
( ( A1
= ( coindu1121789889_LCons @ A @ X2 @ Xs2 ) )
& ( A22
= ( coindu1121789889_LCons @ A @ X2 @ Ys2 ) )
& ( coindu1696667936prefix @ A @ Xs2 @ Ys2 ) ) ) ) ) ).
% lprefix.simps
thf(fact_44_lprefix_Ocases,axiom,
! [A: $tType,A12: coindu1593790203_llist @ A,A23: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ A12 @ A23 )
=> ( ( ( A12
= ( coindu1598213697e_LNil @ A ) )
=> ! [Xs5: coindu1593790203_llist @ A] : A23 != Xs5 )
=> ~ ! [Xs5: coindu1593790203_llist @ A,Ys4: coindu1593790203_llist @ A,X4: A] :
( ( A12
= ( coindu1121789889_LCons @ A @ X4 @ Xs5 ) )
=> ( ( A23
= ( coindu1121789889_LCons @ A @ X4 @ Ys4 ) )
=> ~ ( coindu1696667936prefix @ A @ Xs5 @ Ys4 ) ) ) ) ) ).
% lprefix.cases
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X4: A] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X4: A] :
( ( F @ X4 )
= ( G @ X4 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_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_50_ltake__lappend1,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ord_less_eq @ extended_enat @ N @ ( coindu1018505716length @ A @ Xs ) )
=> ( ( coindu1802687541_ltake @ A @ N @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( coindu1802687541_ltake @ A @ N @ Xs ) ) ) ).
% ltake_lappend1
thf(fact_51_LCons__lprefix__conv,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ Ys )
= ( ? [Ys5: coindu1593790203_llist @ A] :
( ( Ys
= ( coindu1121789889_LCons @ A @ X @ Ys5 ) )
& ( coindu1696667936prefix @ A @ Xs @ Ys5 ) ) ) ) ).
% LCons_lprefix_conv
thf(fact_52_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_53_LNil__lprefix,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ ( coindu1598213697e_LNil @ A ) @ Xs ) ).
% LNil_lprefix
thf(fact_54_min_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ ( ord_min @ A @ B2 @ C ) )
= ( ( ord_less_eq @ A @ A2 @ B2 )
& ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% min.bounded_iff
thf(fact_55_min_Oidem,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_min @ A @ A2 @ A2 )
= A2 ) ) ).
% min.idem
thf(fact_56_min_Oleft__idem,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_min @ A @ A2 @ ( ord_min @ A @ A2 @ B2 ) )
= ( ord_min @ A @ A2 @ B2 ) ) ) ).
% min.left_idem
thf(fact_57_min_Oright__idem,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_min @ A @ ( ord_min @ A @ A2 @ B2 ) @ B2 )
= ( ord_min @ A @ A2 @ B2 ) ) ) ).
% min.right_idem
thf(fact_58_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_59_lfp_Otransp__le,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ( transp @ A @ ( ord_less_eq @ A ) ) ) ).
% lfp.transp_le
thf(fact_60_preorder__class_Otransp__le,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ( transp @ A @ ( ord_less_eq @ A ) ) ) ).
% preorder_class.transp_le
thf(fact_61_min__absorb1,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_min @ A @ X @ Y )
= X ) ) ) ).
% min_absorb1
thf(fact_62_min__absorb2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_min @ A @ X @ Y )
= Y ) ) ) ).
% min_absorb2
thf(fact_63_min__def,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ( ( ord_min @ A )
= ( ^ [A5: A,B4: A] : ( if @ A @ ( ord_less_eq @ A @ A5 @ B4 ) @ A5 @ B4 ) ) ) ) ).
% min_def
thf(fact_64_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_65_lfp_Oleq__antisym,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( X = Y ) ) ) ) ).
% lfp.leq_antisym
thf(fact_66_gfp_Oleq__antisym,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( X = Y ) ) ) ) ).
% gfp.leq_antisym
thf(fact_67_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C @ B2 )
=> ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_68_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: A,B3: A] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_69_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_70_lfp_Oleq__trans,axiom,
! [A: $tType] :
( ( comple187826305attice @ 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 ) ) ) ) ).
% lfp.leq_trans
thf(fact_71_gfp_Oleq__trans,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A,Z: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ Z @ Y )
=> ( ord_less_eq @ A @ Z @ X ) ) ) ) ).
% gfp.leq_trans
thf(fact_72_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_73_lfp_Oleq__refl,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% lfp.leq_refl
thf(fact_74_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_75_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% ord_le_eq_trans
thf(fact_76_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% ord_eq_le_trans
thf(fact_77_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_78_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_79_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% order.trans
thf(fact_80_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_81_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_82_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_83_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_84_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y2: A,Z2: A] : Y2 = Z2 )
= ( ^ [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
& ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ) ) ).
% eq_iff
thf(fact_85_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > B,C: B] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ord_less_eq @ B @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_86_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B2: B,C: B] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C )
=> ( ! [X4: B,Y4: B] :
( ( ord_less_eq @ B @ X4 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_87_order__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > C2,C: C2] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C2 @ ( F @ B2 ) @ C )
=> ( ! [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ord_less_eq @ C2 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_subst2
thf(fact_88_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B2: B,C: B] :
( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C )
=> ( ! [X4: B,Y4: B] :
( ( ord_less_eq @ B @ X4 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_subst1
thf(fact_89_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_90_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B] :
( ! [X4: A] : ( ord_less_eq @ B @ ( F @ X4 ) @ ( G @ X4 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_91_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_92_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_93_min_Oleft__commute,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( ord_min @ A @ B2 @ ( ord_min @ A @ A2 @ C ) )
= ( ord_min @ A @ A2 @ ( ord_min @ A @ B2 @ C ) ) ) ) ).
% min.left_commute
thf(fact_94_min_Ocommute,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ( ( ord_min @ A )
= ( ^ [A5: A,B4: A] : ( ord_min @ A @ B4 @ A5 ) ) ) ) ).
% min.commute
thf(fact_95_min_Oassoc,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_min @ A @ ( ord_min @ A @ A2 @ B2 ) @ C )
= ( ord_min @ A @ A2 @ ( ord_min @ A @ B2 @ C ) ) ) ) ).
% min.assoc
thf(fact_96_min__le__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ ( ord_min @ A @ X @ Y ) @ Z )
= ( ( ord_less_eq @ A @ X @ Z )
| ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% min_le_iff_disj
thf(fact_97_min_OcoboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [B2: A,C: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ C ) ) ) ).
% min.coboundedI2
thf(fact_98_min_OcoboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ C )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ C ) ) ) ).
% min.coboundedI1
thf(fact_99_min_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A5: A] :
( ( ord_min @ A @ A5 @ B4 )
= B4 ) ) ) ) ).
% min.absorb_iff2
thf(fact_100_min_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
( ( ord_min @ A @ A5 @ B4 )
= A5 ) ) ) ) ).
% min.absorb_iff1
thf(fact_101_min_Ocobounded2,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ B2 ) ) ).
% min.cobounded2
thf(fact_102_min_Ocobounded1,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ A2 ) ) ).
% min.cobounded1
thf(fact_103_min_Oorder__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
( A5
= ( ord_min @ A @ A5 @ B4 ) ) ) ) ) ).
% min.order_iff
thf(fact_104_min_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ A2 @ C )
=> ( ord_less_eq @ A @ A2 @ ( ord_min @ A @ B2 @ C ) ) ) ) ) ).
% min.boundedI
thf(fact_105_min_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ ( ord_min @ A @ B2 @ C ) )
=> ~ ( ( ord_less_eq @ A @ A2 @ B2 )
=> ~ ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% min.boundedE
thf(fact_106_min_Oabsorb2,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_min @ A @ A2 @ B2 )
= B2 ) ) ) ).
% min.absorb2
thf(fact_107_min_Oabsorb1,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_min @ A @ A2 @ B2 )
= A2 ) ) ) ).
% min.absorb1
thf(fact_108_min_OorderI,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( A2
= ( ord_min @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% min.orderI
thf(fact_109_min_OorderE,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2
= ( ord_min @ A @ A2 @ B2 ) ) ) ) ).
% min.orderE
thf(fact_110_min_Omono,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,B2: A,D: A] :
( ( ord_less_eq @ A @ A2 @ C )
=> ( ( ord_less_eq @ A @ B2 @ D )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ ( ord_min @ A @ C @ D ) ) ) ) ) ).
% min.mono
thf(fact_111_ltake__lappend2,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,N: extended_enat,Ys: coindu1593790203_llist @ A] :
( ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ N )
=> ( ( coindu1802687541_ltake @ A @ N @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( coindu268472904append @ A @ Xs @ ( coindu1802687541_ltake @ A @ ( minus_minus @ extended_enat @ N @ ( coindu1018505716length @ A @ Xs ) ) @ Ys ) ) ) ) ).
% ltake_lappend2
thf(fact_112_DEADID_Orel__transp,axiom,
! [A: $tType] :
( transp @ A
@ ^ [Y2: A,Z2: A] : Y2 = Z2 ) ).
% DEADID.rel_transp
thf(fact_113_transp__def,axiom,
! [A: $tType] :
( ( transp @ A )
= ( ^ [R: A > A > $o] :
! [X2: A,Y3: A,Z3: A] :
( ( R @ X2 @ Y3 )
=> ( ( R @ Y3 @ Z3 )
=> ( R @ X2 @ Z3 ) ) ) ) ) ).
% transp_def
thf(fact_114_transpI,axiom,
! [A: $tType,R2: A > A > $o] :
( ! [X4: A,Y4: A,Z4: A] :
( ( R2 @ X4 @ Y4 )
=> ( ( R2 @ Y4 @ Z4 )
=> ( R2 @ X4 @ Z4 ) ) )
=> ( transp @ A @ R2 ) ) ).
% transpI
thf(fact_115_transpE,axiom,
! [A: $tType,R2: A > A > $o,X: A,Y: A,Z: A] :
( ( transp @ A @ R2 )
=> ( ( R2 @ X @ Y )
=> ( ( R2 @ Y @ Z )
=> ( R2 @ X @ Z ) ) ) ) ).
% transpE
thf(fact_116_transpD,axiom,
! [A: $tType,R2: A > A > $o,X: A,Y: A,Z: A] :
( ( transp @ A @ R2 )
=> ( ( R2 @ X @ Y )
=> ( ( R2 @ Y @ Z )
=> ( R2 @ X @ Z ) ) ) ) ).
% transpD
thf(fact_117_llexord_Ocoinduct,axiom,
! [A: $tType,X3: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o,X: coindu1593790203_llist @ A,Xa: coindu1593790203_llist @ A,R2: A > A > $o] :
( ( X3 @ X @ Xa )
=> ( ! [X4: coindu1593790203_llist @ A,Xa2: coindu1593790203_llist @ A] :
( ( X3 @ X4 @ Xa2 )
=> ( ? [Xs4: coindu1593790203_llist @ A,Ys3: coindu1593790203_llist @ A,Xb: A] :
( ( X4
= ( coindu1121789889_LCons @ A @ Xb @ Xs4 ) )
& ( Xa2
= ( coindu1121789889_LCons @ A @ Xb @ Ys3 ) )
& ( ( X3 @ Xs4 @ Ys3 )
| ( coindu300403952lexord @ A @ R2 @ Xs4 @ Ys3 ) ) )
| ? [Xb: A,Y5: A,Xs4: coindu1593790203_llist @ A,Ys3: coindu1593790203_llist @ A] :
( ( X4
= ( coindu1121789889_LCons @ A @ Xb @ Xs4 ) )
& ( Xa2
= ( coindu1121789889_LCons @ A @ Y5 @ Ys3 ) )
& ( R2 @ Xb @ Y5 ) )
| ? [Ys3: coindu1593790203_llist @ A] :
( ( X4
= ( coindu1598213697e_LNil @ A ) )
& ( Xa2 = Ys3 ) ) ) )
=> ( coindu300403952lexord @ A @ R2 @ X @ Xa ) ) ) ).
% llexord.coinduct
thf(fact_118_minus__apply,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B @ ( type2 @ B ) )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A6: A > B,B5: A > B,X2: A] : ( minus_minus @ B @ ( A6 @ X2 ) @ ( B5 @ X2 ) ) ) ) ) ).
% minus_apply
thf(fact_119_fun__diff__def,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B @ ( type2 @ B ) )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A6: A > B,B5: A > B,X2: A] : ( minus_minus @ B @ ( A6 @ X2 ) @ ( B5 @ X2 ) ) ) ) ) ).
% fun_diff_def
thf(fact_120_llexord__LCons__eq,axiom,
! [A: $tType,R2: A > A > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,X: A] :
( ( coindu300403952lexord @ A @ R2 @ Xs @ Ys )
=> ( coindu300403952lexord @ A @ R2 @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ ( coindu1121789889_LCons @ A @ X @ Ys ) ) ) ).
% llexord_LCons_eq
thf(fact_121_llexord__LCons__less,axiom,
! [A: $tType,R2: A > A > $o,X: A,Y: A,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( R2 @ X @ Y )
=> ( coindu300403952lexord @ A @ R2 @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ ( coindu1121789889_LCons @ A @ Y @ Ys ) ) ) ).
% llexord_LCons_less
thf(fact_122_llexord__LNil,axiom,
! [A: $tType,R2: A > A > $o,Ys: coindu1593790203_llist @ A] : ( coindu300403952lexord @ A @ R2 @ ( coindu1598213697e_LNil @ A ) @ Ys ) ).
% llexord_LNil
thf(fact_123_ltake__lappend,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1802687541_ltake @ A @ N @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( coindu268472904append @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) @ ( coindu1802687541_ltake @ A @ ( minus_minus @ extended_enat @ N @ ( coindu1018505716length @ A @ Xs ) ) @ Ys ) ) ) ).
% ltake_lappend
thf(fact_124_llexord_Ocases,axiom,
! [A: $tType,R2: A > A > $o,A12: coindu1593790203_llist @ A,A23: coindu1593790203_llist @ A] :
( ( coindu300403952lexord @ A @ R2 @ A12 @ A23 )
=> ( ! [Xs5: coindu1593790203_llist @ A,Ys4: coindu1593790203_llist @ A,X4: A] :
( ( A12
= ( coindu1121789889_LCons @ A @ X4 @ Xs5 ) )
=> ( ( A23
= ( coindu1121789889_LCons @ A @ X4 @ Ys4 ) )
=> ~ ( coindu300403952lexord @ A @ R2 @ Xs5 @ Ys4 ) ) )
=> ( ! [X4: A] :
( ? [Xs5: coindu1593790203_llist @ A] :
( A12
= ( coindu1121789889_LCons @ A @ X4 @ Xs5 ) )
=> ! [Y4: A] :
( ? [Ys4: coindu1593790203_llist @ A] :
( A23
= ( coindu1121789889_LCons @ A @ Y4 @ Ys4 ) )
=> ~ ( R2 @ X4 @ Y4 ) ) )
=> ~ ( ( A12
= ( coindu1598213697e_LNil @ A ) )
=> ! [Ys4: coindu1593790203_llist @ A] : A23 != Ys4 ) ) ) ) ).
% llexord.cases
thf(fact_125_llexord_Osimps,axiom,
! [A: $tType] :
( ( coindu300403952lexord @ A )
= ( ^ [R: A > A > $o,A1: coindu1593790203_llist @ A,A22: coindu1593790203_llist @ A] :
( ? [Xs2: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A,X2: A] :
( ( A1
= ( coindu1121789889_LCons @ A @ X2 @ Xs2 ) )
& ( A22
= ( coindu1121789889_LCons @ A @ X2 @ Ys2 ) )
& ( coindu300403952lexord @ A @ R @ Xs2 @ Ys2 ) )
| ? [X2: A,Y3: A,Xs2: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A] :
( ( A1
= ( coindu1121789889_LCons @ A @ X2 @ Xs2 ) )
& ( A22
= ( coindu1121789889_LCons @ A @ Y3 @ Ys2 ) )
& ( R @ X2 @ Y3 ) )
| ? [Ys2: coindu1593790203_llist @ A] :
( ( A1
= ( coindu1598213697e_LNil @ A ) )
& ( A22 = Ys2 ) ) ) ) ) ).
% llexord.simps
thf(fact_126_enat__diff__cancel__left,axiom,
! [M: extended_enat,X: extended_enat,Y: extended_enat] :
( ( ord_less_eq @ extended_enat @ M @ X )
=> ( ( ord_less_eq @ extended_enat @ M @ Y )
=> ( ( ( minus_minus @ extended_enat @ X @ M )
= ( minus_minus @ extended_enat @ Y @ M ) )
= ( X = Y ) ) ) ) ).
% enat_diff_cancel_left
thf(fact_127_enat__minus__mono1,axiom,
! [X: extended_enat,Y: extended_enat,M: extended_enat] :
( ( ord_less_eq @ extended_enat @ X @ Y )
=> ( ord_less_eq @ extended_enat @ ( minus_minus @ extended_enat @ X @ M ) @ ( minus_minus @ extended_enat @ Y @ M ) ) ) ).
% enat_minus_mono1
thf(fact_128_min__diff__distrib__left,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( minus_minus @ A @ ( ord_min @ A @ X @ Y ) @ Z )
= ( ord_min @ A @ ( minus_minus @ A @ X @ Z ) @ ( minus_minus @ A @ Y @ Z ) ) ) ) ).
% min_diff_distrib_left
thf(fact_129_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A,D: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C @ D ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
= ( ord_less_eq @ A @ C @ D ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_130_diff__right__commute,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,B2: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C ) @ B2 )
= ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C ) ) ) ).
% diff_right_commute
thf(fact_131_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A,D: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C @ D ) )
=> ( ( A2 = B2 )
= ( C = D ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_132_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,D: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ D @ C )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B2 @ D ) ) ) ) ) ).
% diff_mono
thf(fact_133_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C @ A2 ) @ ( minus_minus @ A @ C @ B2 ) ) ) ) ).
% diff_left_mono
thf(fact_134_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B2 @ C ) ) ) ) ).
% diff_right_mono
thf(fact_135_lmember__code_I1_J,axiom,
! [A: $tType,X: A] :
~ ( coindu567634248member @ A @ X @ ( coindu1598213697e_LNil @ A ) ) ).
% lmember_code(1)
thf(fact_136_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_137_lfinite__rev__induct,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: ( coindu1593790203_llist @ A ) > $o] :
( ( coindu1213758845finite @ A @ Xs )
=> ( ( P @ ( coindu1598213697e_LNil @ A ) )
=> ( ! [X4: A,Xs5: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ Xs5 )
=> ( ( P @ Xs5 )
=> ( P @ ( coindu268472904append @ A @ Xs5 @ ( coindu1121789889_LCons @ A @ X4 @ ( coindu1598213697e_LNil @ A ) ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lfinite_rev_induct
thf(fact_138_llist_Orel__transp,axiom,
! [A: $tType,R3: A > A > $o] :
( ( transp @ A @ R3 )
=> ( transp @ ( coindu1593790203_llist @ A ) @ ( coindu987187967t_all2 @ A @ A @ R3 ) ) ) ).
% llist.rel_transp
thf(fact_139_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_140_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_141_llist_Orel__inject_I2_J,axiom,
! [A: $tType,B: $tType,R3: A > B > $o,X21: A,X22: coindu1593790203_llist @ A,Y21: B,Y22: coindu1593790203_llist @ B] :
( ( coindu987187967t_all2 @ A @ B @ R3 @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) @ ( coindu1121789889_LCons @ B @ Y21 @ Y22 ) )
= ( ( R3 @ X21 @ Y21 )
& ( coindu987187967t_all2 @ A @ B @ R3 @ X22 @ Y22 ) ) ) ).
% llist.rel_inject(2)
thf(fact_142_lfinite__code_I1_J,axiom,
! [A: $tType] : ( coindu1213758845finite @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% lfinite_code(1)
thf(fact_143_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_144_lfinite__lmap,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: coindu1593790203_llist @ B] :
( ( coindu1213758845finite @ A @ ( coindu1062782156e_lmap @ B @ A @ F @ Xs ) )
= ( coindu1213758845finite @ B @ Xs ) ) ).
% lfinite_lmap
thf(fact_145_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_146_llist_Orel__eq,axiom,
! [A: $tType] :
( ( coindu987187967t_all2 @ A @ A
@ ^ [Y2: A,Z2: A] : Y2 = Z2 )
= ( ^ [Y2: coindu1593790203_llist @ A,Z2: coindu1593790203_llist @ A] : Y2 = Z2 ) ) ).
% llist.rel_eq
thf(fact_147_llist_Orel__refl,axiom,
! [B: $tType,Ra: B > B > $o,X: coindu1593790203_llist @ B] :
( ! [X4: B] : ( Ra @ X4 @ X4 )
=> ( coindu987187967t_all2 @ B @ B @ Ra @ X @ X ) ) ).
% llist.rel_refl
thf(fact_148_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_149_lfinite__LNil,axiom,
! [A: $tType] : ( coindu1213758845finite @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% lfinite_LNil
thf(fact_150_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_151_llist_Octr__transfer_I1_J,axiom,
! [A: $tType,B: $tType,R3: A > B > $o] : ( coindu987187967t_all2 @ A @ B @ R3 @ ( coindu1598213697e_LNil @ A ) @ ( coindu1598213697e_LNil @ B ) ) ).
% llist.ctr_transfer(1)
thf(fact_152_llist_Orel__intros_I2_J,axiom,
! [A: $tType,B: $tType,R3: A > B > $o,X21: A,Y21: B,X22: coindu1593790203_llist @ A,Y22: coindu1593790203_llist @ B] :
( ( R3 @ X21 @ Y21 )
=> ( ( coindu987187967t_all2 @ A @ B @ R3 @ X22 @ Y22 )
=> ( coindu987187967t_all2 @ A @ B @ R3 @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) @ ( coindu1121789889_LCons @ B @ Y21 @ Y22 ) ) ) ) ).
% llist.rel_intros(2)
thf(fact_153_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_154_llist_Orel__mono,axiom,
! [B: $tType,A: $tType,R3: A > B > $o,Ra: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ R3 @ Ra )
=> ( ord_less_eq @ ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ B ) > $o ) @ ( coindu987187967t_all2 @ A @ B @ R3 ) @ ( coindu987187967t_all2 @ A @ B @ Ra ) ) ) ).
% llist.rel_mono
thf(fact_155_llist_Orel__distinct_I2_J,axiom,
! [A: $tType,B: $tType,R3: A > B > $o,Y21: A,Y22: coindu1593790203_llist @ A] :
~ ( coindu987187967t_all2 @ A @ B @ R3 @ ( coindu1121789889_LCons @ A @ Y21 @ Y22 ) @ ( coindu1598213697e_LNil @ B ) ) ).
% llist.rel_distinct(2)
thf(fact_156_llist_Orel__distinct_I1_J,axiom,
! [A: $tType,B: $tType,R3: A > B > $o,Y21: B,Y22: coindu1593790203_llist @ B] :
~ ( coindu987187967t_all2 @ A @ B @ R3 @ ( coindu1598213697e_LNil @ A ) @ ( coindu1121789889_LCons @ B @ Y21 @ Y22 ) ) ).
% llist.rel_distinct(1)
thf(fact_157_llist_Orel__cases,axiom,
! [A: $tType,B: $tType,R3: A > B > $o,A2: coindu1593790203_llist @ A,B2: coindu1593790203_llist @ B] :
( ( coindu987187967t_all2 @ A @ B @ R3 @ A2 @ B2 )
=> ( ( ( A2
= ( coindu1598213697e_LNil @ A ) )
=> ( B2
!= ( coindu1598213697e_LNil @ B ) ) )
=> ~ ! [X1: A,X23: coindu1593790203_llist @ A] :
( ( A2
= ( coindu1121789889_LCons @ A @ X1 @ X23 ) )
=> ! [Y1: B,Y23: coindu1593790203_llist @ B] :
( ( B2
= ( coindu1121789889_LCons @ B @ Y1 @ Y23 ) )
=> ( ( R3 @ X1 @ Y1 )
=> ~ ( coindu987187967t_all2 @ A @ B @ R3 @ X23 @ Y23 ) ) ) ) ) ) ).
% llist.rel_cases
thf(fact_158_lfinite_Ocases,axiom,
! [A: $tType,A2: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ A2 )
=> ( ( A2
!= ( coindu1598213697e_LNil @ A ) )
=> ~ ! [Xs5: coindu1593790203_llist @ A] :
( ? [X4: A] :
( A2
= ( coindu1121789889_LCons @ A @ X4 @ Xs5 ) )
=> ~ ( coindu1213758845finite @ A @ Xs5 ) ) ) ) ).
% lfinite.cases
thf(fact_159_lfinite_Osimps,axiom,
! [A: $tType] :
( ( coindu1213758845finite @ A )
= ( ^ [A5: coindu1593790203_llist @ A] :
( ( A5
= ( coindu1598213697e_LNil @ A ) )
| ? [Xs2: coindu1593790203_llist @ A,X2: A] :
( ( A5
= ( coindu1121789889_LCons @ A @ X2 @ Xs2 ) )
& ( coindu1213758845finite @ A @ Xs2 ) ) ) ) ) ).
% lfinite.simps
thf(fact_160_lfinite_Oinducts,axiom,
! [A: $tType,X: coindu1593790203_llist @ A,P: ( coindu1593790203_llist @ A ) > $o] :
( ( coindu1213758845finite @ A @ X )
=> ( ( P @ ( coindu1598213697e_LNil @ A ) )
=> ( ! [Xs5: coindu1593790203_llist @ A,X4: A] :
( ( coindu1213758845finite @ A @ Xs5 )
=> ( ( P @ Xs5 )
=> ( P @ ( coindu1121789889_LCons @ A @ X4 @ Xs5 ) ) ) )
=> ( P @ X ) ) ) ) ).
% lfinite.inducts
thf(fact_161_llength__LNil,axiom,
! [A: $tType] :
( ( coindu1018505716length @ A @ ( coindu1598213697e_LNil @ A ) )
= ( zero_zero @ extended_enat ) ) ).
% llength_LNil
thf(fact_162_in__inv__imagep,axiom,
! [B: $tType,A: $tType] :
( ( inv_imagep @ A @ B )
= ( ^ [R: A > A > $o,F2: B > A,X2: B,Y3: B] : ( R @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) ) ) ).
% in_inv_imagep
thf(fact_163_LNil__eq__ltake__iff,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( ( coindu1598213697e_LNil @ A )
= ( coindu1802687541_ltake @ A @ N @ Xs ) )
= ( ( Xs
= ( coindu1598213697e_LNil @ A ) )
| ( N
= ( zero_zero @ extended_enat ) ) ) ) ).
% LNil_eq_ltake_iff
thf(fact_164_ltake__0,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu1802687541_ltake @ A @ ( zero_zero @ extended_enat ) @ Xs )
= ( coindu1598213697e_LNil @ A ) ) ).
% ltake_0
thf(fact_165_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_166_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_167_diff__zero,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% diff_zero
thf(fact_168_zero__diff,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% zero_diff
thf(fact_169_diff__0__right,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% diff_0_right
thf(fact_170_diff__self,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% diff_self
thf(fact_171_diff__ge__0__iff__ge,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% diff_ge_0_iff_ge
thf(fact_172_enat__min__eq__0__iff,axiom,
! [A2: extended_enat,B2: extended_enat] :
( ( ( ord_min @ extended_enat @ A2 @ B2 )
= ( zero_zero @ extended_enat ) )
= ( ( A2
= ( zero_zero @ extended_enat ) )
| ( B2
= ( zero_zero @ extended_enat ) ) ) ) ).
% enat_min_eq_0_iff
thf(fact_173_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_174_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ( ( ^ [Y2: A,Z2: A] : Y2 = Z2 )
= ( ^ [A5: A,B4: A] :
( ( minus_minus @ A @ A5 @ B4 )
= ( zero_zero @ A ) ) ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_175_zero__le,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% zero_le
thf(fact_176_le__iff__diff__le__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A5 @ B4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% le_iff_diff_le_0
thf(fact_177_ltake__eq__LNil__iff,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( ( coindu1802687541_ltake @ A @ N @ Xs )
= ( coindu1598213697e_LNil @ A ) )
= ( ( Xs
= ( coindu1598213697e_LNil @ A ) )
| ( N
= ( zero_zero @ extended_enat ) ) ) ) ).
% ltake_eq_LNil_iff
thf(fact_178_min__enat__simps_I3_J,axiom,
! [Q2: extended_enat] :
( ( ord_min @ extended_enat @ ( zero_zero @ extended_enat ) @ Q2 )
= ( zero_zero @ extended_enat ) ) ).
% min_enat_simps(3)
thf(fact_179_min__enat__simps_I2_J,axiom,
! [Q2: extended_enat] :
( ( ord_min @ extended_enat @ Q2 @ ( zero_zero @ extended_enat ) )
= ( zero_zero @ extended_enat ) ) ).
% min_enat_simps(2)
thf(fact_180_idiff__0__right,axiom,
! [N: extended_enat] :
( ( minus_minus @ extended_enat @ N @ ( zero_zero @ extended_enat ) )
= N ) ).
% idiff_0_right
thf(fact_181_idiff__0,axiom,
! [N: extended_enat] :
( ( minus_minus @ extended_enat @ ( zero_zero @ extended_enat ) @ N )
= ( zero_zero @ extended_enat ) ) ).
% idiff_0
thf(fact_182_i0__lb,axiom,
! [N: extended_enat] : ( ord_less_eq @ extended_enat @ ( zero_zero @ extended_enat ) @ N ) ).
% i0_lb
thf(fact_183_ile0__eq,axiom,
! [N: extended_enat] :
( ( ord_less_eq @ extended_enat @ N @ ( zero_zero @ extended_enat ) )
= ( N
= ( zero_zero @ extended_enat ) ) ) ).
% ile0_eq
thf(fact_184_le__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% le_numeral_extra(3)
thf(fact_185_the__enat__0,axiom,
( ( extended_the_enat @ ( zero_zero @ extended_enat ) )
= ( zero_zero @ nat ) ) ).
% the_enat_0
thf(fact_186_case__enat__0,axiom,
! [A: $tType,F: nat > A,I: A] :
( ( extended_case_enat @ A @ F @ I @ ( zero_zero @ extended_enat ) )
= ( F @ ( zero_zero @ nat ) ) ) ).
% case_enat_0
thf(fact_187_llength__code,axiom,
! [A: $tType] :
( ( coindu1018505716length @ A )
= ( coindu493225792length @ A @ ( zero_zero @ nat ) ) ) ).
% llength_code
thf(fact_188_ltake_Octr_I1_J,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( ( N
= ( zero_zero @ extended_enat ) )
| ( coindu335574135_lnull @ A @ Xs ) )
=> ( ( coindu1802687541_ltake @ A @ N @ Xs )
= ( coindu1598213697e_LNil @ A ) ) ) ).
% ltake.ctr(1)
thf(fact_189_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_190_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_191_llist_Omap__disc__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A2: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ B @ ( coindu1062782156e_lmap @ A @ B @ F @ A2 ) )
= ( coindu335574135_lnull @ A @ A2 ) ) ).
% llist.map_disc_iff
thf(fact_192_lprefix__LNil,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ ( coindu1598213697e_LNil @ A ) )
= ( coindu335574135_lnull @ A @ Xs ) ) ).
% lprefix_LNil
thf(fact_193_ltake_Odisc__iff_I2_J,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( ~ ( coindu335574135_lnull @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) ) )
= ( ( N
!= ( zero_zero @ extended_enat ) )
& ~ ( coindu335574135_lnull @ A @ Xs ) ) ) ).
% ltake.disc_iff(2)
thf(fact_194_ltake_Odisc__iff_I1_J,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) )
= ( ( N
= ( zero_zero @ extended_enat ) )
| ( coindu335574135_lnull @ A @ Xs ) ) ) ).
% ltake.disc_iff(1)
thf(fact_195_lnull__ltake,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) )
= ( ( coindu335574135_lnull @ A @ Xs )
| ( N
= ( zero_zero @ extended_enat ) ) ) ) ).
% lnull_ltake
thf(fact_196_llength__lnull,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu1018505716length @ A @ Xs )
= ( zero_zero @ extended_enat ) ) ) ).
% llength_lnull
thf(fact_197_llength__eq__0,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( ( coindu1018505716length @ A @ Xs )
= ( zero_zero @ extended_enat ) )
= ( coindu335574135_lnull @ A @ Xs ) ) ).
% llength_eq_0
thf(fact_198_ltake_Oexhaust,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ~ ( ( N
= ( zero_zero @ extended_enat ) )
| ( coindu335574135_lnull @ A @ Xs ) )
=> ~ ( ( N
!= ( zero_zero @ extended_enat ) )
=> ( coindu335574135_lnull @ A @ Xs ) ) ) ).
% ltake.exhaust
thf(fact_199_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_200_ltake_Odisc_I1_J,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( ( N
= ( zero_zero @ extended_enat ) )
| ( coindu335574135_lnull @ A @ Xs ) )
=> ( coindu335574135_lnull @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) ) ) ).
% ltake.disc(1)
thf(fact_201_ltake_Odisc_I2_J,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( N
!= ( zero_zero @ extended_enat ) )
=> ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ~ ( coindu335574135_lnull @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) ) ) ) ).
% ltake.disc(2)
thf(fact_202_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_203_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_204_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_205_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_206_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_207_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_208_lnull__def,axiom,
! [A: $tType] :
( ( coindu335574135_lnull @ A )
= ( ^ [Llist: coindu1593790203_llist @ A] :
( Llist
= ( coindu1598213697e_LNil @ A ) ) ) ) ).
% lnull_def
thf(fact_209_llist_Ocollapse_I1_J,axiom,
! [A: $tType,Llist2: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Llist2 )
=> ( Llist2
= ( coindu1598213697e_LNil @ A ) ) ) ).
% llist.collapse(1)
thf(fact_210_llist_OdiscI_I1_J,axiom,
! [A: $tType,Llist2: coindu1593790203_llist @ A] :
( ( Llist2
= ( coindu1598213697e_LNil @ A ) )
=> ( coindu335574135_lnull @ A @ Llist2 ) ) ).
% llist.discI(1)
thf(fact_211_llist_Odisc_I1_J,axiom,
! [A: $tType] : ( coindu335574135_lnull @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% llist.disc(1)
thf(fact_212_not__lnull__conv,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( ~ ( coindu335574135_lnull @ A @ Xs ) )
= ( ? [X2: A,Xs3: coindu1593790203_llist @ A] :
( Xs
= ( coindu1121789889_LCons @ A @ X2 @ Xs3 ) ) ) ) ).
% not_lnull_conv
thf(fact_213_llist_OdiscI_I2_J,axiom,
! [A: $tType,Llist2: coindu1593790203_llist @ A,X21: A,X22: coindu1593790203_llist @ A] :
( ( Llist2
= ( coindu1121789889_LCons @ A @ X21 @ X22 ) )
=> ~ ( coindu335574135_lnull @ A @ Llist2 ) ) ).
% llist.discI(2)
thf(fact_214_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_215_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_216_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_217_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_218_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_219_lnull__imp__lfinite,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( coindu1213758845finite @ A @ Xs ) ) ).
% lnull_imp_lfinite
thf(fact_220_lprefixI,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,X3: 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 ) @ X3 )
=> ( ! [Xs5: coindu1593790203_llist @ A,Ys4: coindu1593790203_llist @ A] :
( ( member @ ( product_prod @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) ) @ ( product_Pair @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) @ Xs5 @ Ys4 ) @ X3 )
=> ( ( coindu335574135_lnull @ A @ Xs5 )
| ? [X5: A,Xs6: coindu1593790203_llist @ A,Ys6: coindu1593790203_llist @ A] :
( ( Xs5
= ( coindu1121789889_LCons @ A @ X5 @ Xs6 ) )
& ( Ys4
= ( coindu1121789889_LCons @ A @ X5 @ Ys6 ) )
& ( ( member @ ( product_prod @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) ) @ ( product_Pair @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) @ Xs6 @ Ys6 ) @ X3 )
| ( coindu1696667936prefix @ A @ Xs6 @ Ys6 ) ) ) ) )
=> ( coindu1696667936prefix @ A @ Xs @ Ys ) ) ) ).
% lprefixI
thf(fact_221_ltake_Osimps_I3_J,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( N
!= ( zero_zero @ extended_enat ) )
=> ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu1046438764le_lhd @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) )
= ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% ltake.simps(3)
thf(fact_222_lhd__lappend,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu1046438764le_lhd @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( coindu1046438764le_lhd @ A @ Ys ) ) )
& ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu1046438764le_lhd @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% lhd_lappend
thf(fact_223_llist_Omap__sel_I1_J,axiom,
! [B: $tType,A: $tType,A2: coindu1593790203_llist @ A,F: A > B] :
( ~ ( coindu335574135_lnull @ A @ A2 )
=> ( ( coindu1046438764le_lhd @ B @ ( coindu1062782156e_lmap @ A @ B @ F @ A2 ) )
= ( F @ ( coindu1046438764le_lhd @ A @ A2 ) ) ) ) ).
% llist.map_sel(1)
thf(fact_224_lhd__ltake,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( N
!= ( zero_zero @ extended_enat ) )
=> ( ( coindu1046438764le_lhd @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) )
= ( coindu1046438764le_lhd @ A @ Xs ) ) ) ).
% lhd_ltake
thf(fact_225_ltakeWhile_Oexhaust,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
( ~ ( ( coindu335574135_lnull @ A @ Xs )
| ~ ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) )
=> ~ ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ~ ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% ltakeWhile.exhaust
thf(fact_226_lprefix__lhdD,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu1046438764le_lhd @ A @ Xs )
= ( coindu1046438764le_lhd @ A @ Ys ) ) ) ) ).
% lprefix_lhdD
thf(fact_227_lhd__LCons,axiom,
! [A: $tType,X21: A,X22: coindu1593790203_llist @ A] :
( ( coindu1046438764le_lhd @ A @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) )
= X21 ) ).
% lhd_LCons
thf(fact_228_subrelI,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ! [X4: A,Y4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R2 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ S ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S ) ) ).
% subrelI
thf(fact_229_lzip_Osimps_I3_J,axiom,
! [A: $tType,B: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B] :
( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ~ ( coindu335574135_lnull @ B @ Ys )
=> ( ( coindu1046438764le_lhd @ ( product_prod @ A @ B ) @ ( coindu847746867e_lzip @ A @ B @ Xs @ Ys ) )
= ( product_Pair @ A @ B @ ( coindu1046438764le_lhd @ A @ Xs ) @ ( coindu1046438764le_lhd @ B @ Ys ) ) ) ) ) ).
% lzip.simps(3)
thf(fact_230_ltakeWhile_Octr_I1_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
( ( ( coindu335574135_lnull @ A @ Xs )
| ~ ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) )
=> ( ( coindu721411036eWhile @ A @ P @ Xs )
= ( coindu1598213697e_LNil @ A ) ) ) ).
% ltakeWhile.ctr(1)
thf(fact_231_lzip_Odisc__iff_I2_J,axiom,
! [A: $tType,B: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B] :
( ( ~ ( coindu335574135_lnull @ ( product_prod @ A @ B ) @ ( coindu847746867e_lzip @ A @ B @ Xs @ Ys ) ) )
= ( ~ ( coindu335574135_lnull @ A @ Xs )
& ~ ( coindu335574135_lnull @ B @ Ys ) ) ) ).
% lzip.disc_iff(2)
thf(fact_232_lzip_Odisc__iff_I1_J,axiom,
! [A: $tType,B: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B] :
( ( coindu335574135_lnull @ ( product_prod @ A @ B ) @ ( coindu847746867e_lzip @ A @ B @ Xs @ Ys ) )
= ( ( coindu335574135_lnull @ A @ Xs )
| ( coindu335574135_lnull @ B @ Ys ) ) ) ).
% lzip.disc_iff(1)
thf(fact_233_ltakeWhile_Odisc__iff_I1_J,axiom,
! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) )
= ( ( coindu335574135_lnull @ A @ Xs )
| ~ ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% ltakeWhile.disc_iff(1)
thf(fact_234_ltakeWhile_Odisc__iff_I2_J,axiom,
! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
( ( ~ ( coindu335574135_lnull @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) )
= ( ~ ( coindu335574135_lnull @ A @ Xs )
& ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% ltakeWhile.disc_iff(2)
thf(fact_235_lzip_Octr_I1_J,axiom,
! [B: $tType,A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B] :
( ( ( coindu335574135_lnull @ A @ Xs )
| ( coindu335574135_lnull @ B @ Ys ) )
=> ( ( coindu847746867e_lzip @ A @ B @ Xs @ Ys )
= ( coindu1598213697e_LNil @ ( product_prod @ A @ B ) ) ) ) ).
% lzip.ctr(1)
thf(fact_236_lzip_Odisc_I1_J,axiom,
! [A: $tType,B: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B] :
( ( ( coindu335574135_lnull @ A @ Xs )
| ( coindu335574135_lnull @ B @ Ys ) )
=> ( coindu335574135_lnull @ ( product_prod @ A @ B ) @ ( coindu847746867e_lzip @ A @ B @ Xs @ Ys ) ) ) ).
% lzip.disc(1)
thf(fact_237_lzip_Odisc_I2_J,axiom,
! [A: $tType,B: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B] :
( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ~ ( coindu335574135_lnull @ B @ Ys )
=> ~ ( coindu335574135_lnull @ ( product_prod @ A @ B ) @ ( coindu847746867e_lzip @ A @ B @ Xs @ Ys ) ) ) ) ).
% lzip.disc(2)
thf(fact_238_ltakeWhile_Osimps_I3_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( P @ ( coindu1046438764le_lhd @ A @ Xs ) )
=> ( ( coindu1046438764le_lhd @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) )
= ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% ltakeWhile.simps(3)
thf(fact_239_ltakeWhile_Odisc_I1_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
( ( ( coindu335574135_lnull @ A @ Xs )
| ~ ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) )
=> ( coindu335574135_lnull @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) ) ).
% ltakeWhile.disc(1)
thf(fact_240_ltakeWhile_Odisc_I2_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( P @ ( coindu1046438764le_lhd @ A @ Xs ) )
=> ~ ( coindu335574135_lnull @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) ) ) ).
% ltakeWhile.disc(2)
thf(fact_241_ltakeWhile_Ocode,axiom,
! [A: $tType] :
( ( coindu721411036eWhile @ A )
= ( ^ [P2: A > $o,Xs2: coindu1593790203_llist @ A] :
( if @ ( coindu1593790203_llist @ A )
@ ( ( coindu335574135_lnull @ A @ Xs2 )
| ~ ( P2 @ ( coindu1046438764le_lhd @ A @ Xs2 ) ) )
@ ( coindu1598213697e_LNil @ A )
@ ( coindu1121789889_LCons @ A @ ( coindu1046438764le_lhd @ A @ Xs2 ) @ ( coindu721411036eWhile @ A @ P2 @ ( coindu1047225960le_ltl @ A @ Xs2 ) ) ) ) ) ) ).
% ltakeWhile.code
thf(fact_242_ltakeWhile_Octr_I2_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( P @ ( coindu1046438764le_lhd @ A @ Xs ) )
=> ( ( coindu721411036eWhile @ A @ P @ Xs )
= ( coindu1121789889_LCons @ A @ ( coindu1046438764le_lhd @ A @ Xs ) @ ( coindu721411036eWhile @ A @ P @ ( coindu1047225960le_ltl @ A @ Xs ) ) ) ) ) ) ).
% ltakeWhile.ctr(2)
thf(fact_243_lfinite__ltl,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ ( coindu1047225960le_ltl @ A @ Xs ) )
= ( coindu1213758845finite @ A @ Xs ) ) ).
% lfinite_ltl
thf(fact_244_ltl__lmap,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: coindu1593790203_llist @ B] :
( ( coindu1047225960le_ltl @ A @ ( coindu1062782156e_lmap @ B @ A @ F @ Xs ) )
= ( coindu1062782156e_lmap @ B @ A @ F @ ( coindu1047225960le_ltl @ B @ Xs ) ) ) ).
% ltl_lmap
thf(fact_245_ltl__lappend,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu1047225960le_ltl @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( coindu1047225960le_ltl @ A @ Ys ) ) )
& ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu1047225960le_ltl @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( coindu268472904append @ A @ ( coindu1047225960le_ltl @ A @ Xs ) @ Ys ) ) ) ) ).
% ltl_lappend
thf(fact_246_lhd__LCons__ltl,axiom,
! [A: $tType,Llist2: coindu1593790203_llist @ A] :
( ~ ( coindu335574135_lnull @ A @ Llist2 )
=> ( ( coindu1121789889_LCons @ A @ ( coindu1046438764le_lhd @ A @ Llist2 ) @ ( coindu1047225960le_ltl @ A @ Llist2 ) )
= Llist2 ) ) ).
% lhd_LCons_ltl
thf(fact_247_llist_Oexpand,axiom,
! [A: $tType,Llist2: coindu1593790203_llist @ A,Llist3: coindu1593790203_llist @ A] :
( ( ( coindu335574135_lnull @ A @ Llist2 )
= ( coindu335574135_lnull @ A @ Llist3 ) )
=> ( ( ~ ( coindu335574135_lnull @ A @ Llist2 )
=> ( ~ ( coindu335574135_lnull @ A @ Llist3 )
=> ( ( ( coindu1046438764le_lhd @ A @ Llist2 )
= ( coindu1046438764le_lhd @ A @ Llist3 ) )
& ( ( coindu1047225960le_ltl @ A @ Llist2 )
= ( coindu1047225960le_ltl @ A @ Llist3 ) ) ) ) )
=> ( Llist2 = Llist3 ) ) ) ).
% llist.expand
thf(fact_248_llist_Ocoinduct,axiom,
! [A: $tType,R3: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o,Llist2: coindu1593790203_llist @ A,Llist3: coindu1593790203_llist @ A] :
( ( R3 @ Llist2 @ Llist3 )
=> ( ! [Llist4: coindu1593790203_llist @ A,Llist5: coindu1593790203_llist @ A] :
( ( R3 @ Llist4 @ Llist5 )
=> ( ( ( coindu335574135_lnull @ A @ Llist4 )
= ( coindu335574135_lnull @ A @ Llist5 ) )
& ( ~ ( coindu335574135_lnull @ A @ Llist4 )
=> ( ~ ( coindu335574135_lnull @ A @ Llist5 )
=> ( ( ( coindu1046438764le_lhd @ A @ Llist4 )
= ( coindu1046438764le_lhd @ A @ Llist5 ) )
& ( R3 @ ( coindu1047225960le_ltl @ A @ Llist4 ) @ ( coindu1047225960le_ltl @ A @ Llist5 ) ) ) ) ) ) )
=> ( Llist2 = Llist3 ) ) ) ).
% llist.coinduct
thf(fact_249_llist_Ocoinduct__strong,axiom,
! [A: $tType,R3: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o,Llist2: coindu1593790203_llist @ A,Llist3: coindu1593790203_llist @ A] :
( ( R3 @ Llist2 @ Llist3 )
=> ( ! [Llist4: coindu1593790203_llist @ A,Llist5: coindu1593790203_llist @ A] :
( ( R3 @ Llist4 @ Llist5 )
=> ( ( ( coindu335574135_lnull @ A @ Llist4 )
= ( coindu335574135_lnull @ A @ Llist5 ) )
& ( ~ ( coindu335574135_lnull @ A @ Llist4 )
=> ( ~ ( coindu335574135_lnull @ A @ Llist5 )
=> ( ( ( coindu1046438764le_lhd @ A @ Llist4 )
= ( coindu1046438764le_lhd @ A @ Llist5 ) )
& ( ( R3 @ ( coindu1047225960le_ltl @ A @ Llist4 ) @ ( coindu1047225960le_ltl @ A @ Llist5 ) )
| ( ( coindu1047225960le_ltl @ A @ Llist4 )
= ( coindu1047225960le_ltl @ A @ Llist5 ) ) ) ) ) ) ) )
=> ( Llist2 = Llist3 ) ) ) ).
% llist.coinduct_strong
thf(fact_250_lzip_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B] :
( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ~ ( coindu335574135_lnull @ B @ Ys )
=> ( ( coindu1047225960le_ltl @ ( product_prod @ A @ B ) @ ( coindu847746867e_lzip @ A @ B @ Xs @ Ys ) )
= ( coindu847746867e_lzip @ A @ B @ ( coindu1047225960le_ltl @ A @ Xs ) @ ( coindu1047225960le_ltl @ B @ Ys ) ) ) ) ) ).
% lzip.simps(4)
thf(fact_251_lprefix__expand,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ~ ( coindu335574135_lnull @ A @ Ys )
& ( ( coindu1046438764le_lhd @ A @ Xs )
= ( coindu1046438764le_lhd @ A @ Ys ) )
& ( coindu1696667936prefix @ A @ ( coindu1047225960le_ltl @ A @ Xs ) @ ( coindu1047225960le_ltl @ A @ Ys ) ) ) )
=> ( coindu1696667936prefix @ A @ Xs @ Ys ) ) ).
% lprefix_expand
%----Type constructors (35)
thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A7: $tType,A8: $tType] :
( ( comple187826305attice @ A8 @ ( type2 @ A8 ) )
=> ( comple187826305attice @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
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_fun___Groups_Ominus,axiom,
! [A7: $tType,A8: $tType] :
( ( minus @ A8 @ ( type2 @ A8 ) )
=> ( minus @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
cancel1352612707id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom,axiom,
linordered_semidom @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
comm_monoid_diff @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
preorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
order @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
ord @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ominus_4,axiom,
minus @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ozero,axiom,
zero @ nat @ ( type2 @ nat ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_5,axiom,
! [A7: $tType] : ( comple187826305attice @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_6,axiom,
! [A7: $tType] : ( preorder @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_7,axiom,
! [A7: $tType] : ( order @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_8,axiom,
! [A7: $tType] : ( ord @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Groups_Ominus_9,axiom,
! [A7: $tType] : ( minus @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_10,axiom,
comple187826305attice @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_11,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder_12,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_13,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_14,axiom,
ord @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Groups_Ominus_15,axiom,
minus @ $o @ ( type2 @ $o ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_16,axiom,
canoni770627133id_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__lattice_17,axiom,
comple187826305attice @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Opreorder_18,axiom,
preorder @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_19,axiom,
linorder @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_20,axiom,
order @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oord_21,axiom,
ord @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Ominus_22,axiom,
minus @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Ozero_23,axiom,
zero @ extended_enat @ ( type2 @ extended_enat ) ).
%----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 (1)
thf(conj_0,conjecture,
coindu1696667936prefix @ a @ ( coindu1802687541_ltake @ a @ n @ xs ) @ ( coindu1802687541_ltake @ a @ m @ xs ) ).
%------------------------------------------------------------------------------