TPTP Problem File: DAT126^1.p
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%------------------------------------------------------------------------------
% File : DAT126^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Coinductive list 2704
% 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__2704.p [Bla16]
% Status : Theorem
% Rating : 1.00 v7.2.0, 0.75 v7.1.0
% Syntax : Number of formulae : 349 ( 154 unt; 62 typ; 0 def)
% Number of atoms : 711 ( 341 equ; 0 cnn)
% Maximal formula atoms : 10 ( 2 avg)
% Number of connectives : 3211 ( 53 ~; 20 |; 54 &;2843 @)
% ( 0 <=>; 241 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 6 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 168 ( 168 >; 0 *; 0 +; 0 <<)
% Number of symbols : 60 ( 59 usr; 1 con; 0-5 aty)
% Number of variables : 921 ( 26 ^; 794 !; 48 ?; 921 :)
% ( 53 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:57:50.228
%------------------------------------------------------------------------------
%----Could-be-implicit typings (6)
thf(ty_t_Coinductive__List__Mirabelle__kmikjhschf_Ollist,type,
coindu1593790203_llist: $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 (56)
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_Num_Oneg__numeral,type,
neg_numeral:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Ocomm__semiring__1,type,
comm_semiring_1:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[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_Ocancel__semigroup__add,type,
cancel_semigroup_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_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
ordere216010020id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere779506340up_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_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere236663937imp_le:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
ordere516151231imp_le:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
semiri456707255roduct:
!>[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__llength,type,
coindu493225792length:
!>[A: $tType] : ( nat > ( coindu1593790203_llist @ A ) > extended_enat ) ).
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_Oldrop,type,
coindu191418589_ldrop:
!>[A: $tType] : ( extended_enat > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_OldropWhile,type,
coindu438612276pWhile:
!>[A: $tType] : ( ( A > $o ) > ( 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_Ollcp,type,
coindu731692203e_llcp:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > extended_enat ) ).
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_Olmap,type,
coindu1062782156e_lmap:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ Aa ) ) ).
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__Nat_OLe__enatp,type,
coinductive_Le_enatp: extended_enat > extended_enat > $o ).
thf(sy_c_Coinductive__Nat_Oenat__set,type,
coinductive_enat_set: set @ extended_enat ).
thf(sy_c_Complete__Partial__Order_Omonotone,type,
comple1396247847notone:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > B > $o ) > ( A > B ) > $o ) ).
thf(sy_c_Extended__Nat_OeSuc,type,
extended_eSuc: extended_enat > extended_enat ).
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_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_P,type,
p: a > $o ).
%----Relevant facts (253)
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_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_3_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_4_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_5_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_6_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_7_llist_Omonotone__if__bot,axiom,
! [B: $tType,A: $tType,Bound: coindu1593790203_llist @ A,G: ( coindu1593790203_llist @ A ) > B,Bot: B,F: ( coindu1593790203_llist @ A ) > B,Ord: B > B > $o] :
( ! [X2: coindu1593790203_llist @ A] :
( ( ( coindu1696667936prefix @ A @ X2 @ Bound )
=> ( ( G @ X2 )
= Bot ) )
& ( ~ ( coindu1696667936prefix @ A @ X2 @ Bound )
=> ( ( G @ X2 )
= ( F @ X2 ) ) ) )
=> ( ! [X2: coindu1593790203_llist @ A,Y2: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ X2 @ Y2 )
=> ( ~ ( coindu1696667936prefix @ A @ X2 @ Bound )
=> ( Ord @ ( F @ X2 ) @ ( F @ Y2 ) ) ) )
=> ( ! [X2: coindu1593790203_llist @ A] :
( ~ ( coindu1696667936prefix @ A @ X2 @ Bound )
=> ( Ord @ Bot @ ( F @ X2 ) ) )
=> ( ( Ord @ Bot @ Bot )
=> ( comple1396247847notone @ ( coindu1593790203_llist @ A ) @ B @ ( coindu1696667936prefix @ A ) @ Ord @ G ) ) ) ) ) ).
% llist.monotone_if_bot
thf(fact_8_monotoneD,axiom,
! [B: $tType,A: $tType,Orda: A > A > $o,Ordb: B > B > $o,F: A > B,X: A,Y: A] :
( ( comple1396247847notone @ A @ B @ Orda @ Ordb @ F )
=> ( ( Orda @ X @ Y )
=> ( Ordb @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% monotoneD
thf(fact_9_monotoneI,axiom,
! [B: $tType,A: $tType,Orda: A > A > $o,Ordb: B > B > $o,F: A > B] :
( ! [X2: A,Y2: A] :
( ( Orda @ X2 @ Y2 )
=> ( Ordb @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( comple1396247847notone @ A @ B @ Orda @ Ordb @ F ) ) ).
% monotoneI
thf(fact_10_monotone__def,axiom,
! [B: $tType,A: $tType] :
( ( comple1396247847notone @ A @ B )
= ( ^ [Orda2: A > A > $o,Ordb2: B > B > $o,F2: A > B] :
! [X3: A,Y3: A] :
( ( Orda2 @ X3 @ Y3 )
=> ( Ordb2 @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ) ) ).
% monotone_def
thf(fact_11_monotone__ltakeWhile,axiom,
! [A: $tType,P: A > $o] : ( comple1396247847notone @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) @ ( coindu1696667936prefix @ A ) @ ( coindu1696667936prefix @ A ) @ ( coindu721411036eWhile @ A @ P ) ) ).
% monotone_ltakeWhile
thf(fact_12_Coinductive__List__Mirabelle__kmikjhschf_Ofinite__lprefix__def,axiom,
! [A: $tType] :
( ( coindu1571841457prefix @ A )
= ( coindu1696667936prefix @ A ) ) ).
% Coinductive_List_Mirabelle_kmikjhschf.finite_lprefix_def
thf(fact_13_monotone__ldrop2,axiom,
! [A: $tType,N: extended_enat] : ( comple1396247847notone @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) @ ( coindu1696667936prefix @ A ) @ ( coindu1696667936prefix @ A ) @ ( coindu191418589_ldrop @ A @ N ) ) ).
% monotone_ldrop2
thf(fact_14_monotone__LCons,axiom,
! [A: $tType,X: A] : ( comple1396247847notone @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) @ ( coindu1696667936prefix @ A ) @ ( coindu1696667936prefix @ A ) @ ( coindu1121789889_LCons @ A @ X ) ) ).
% monotone_LCons
thf(fact_15_monotone__ltake2,axiom,
! [A: $tType,N: extended_enat] : ( comple1396247847notone @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) @ ( coindu1696667936prefix @ A ) @ ( coindu1696667936prefix @ A ) @ ( coindu1802687541_ltake @ A @ N ) ) ).
% monotone_ltake2
thf(fact_16_monotone__lmap,axiom,
! [B: $tType,A: $tType,F: A > B] : ( comple1396247847notone @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ B ) @ ( coindu1696667936prefix @ A ) @ ( coindu1696667936prefix @ B ) @ ( coindu1062782156e_lmap @ A @ B @ F ) ) ).
% monotone_lmap
thf(fact_17_monotone__lappend2,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] : ( comple1396247847notone @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) @ ( coindu1696667936prefix @ A ) @ ( coindu1696667936prefix @ A ) @ ( coindu268472904append @ A @ Xs ) ) ).
% monotone_lappend2
thf(fact_18_monotone__ltl,axiom,
! [A: $tType] : ( comple1396247847notone @ ( coindu1593790203_llist @ A ) @ ( coindu1593790203_llist @ A ) @ ( coindu1696667936prefix @ A ) @ ( coindu1696667936prefix @ A ) @ ( coindu1047225960le_ltl @ A ) ) ).
% monotone_ltl
thf(fact_19_ldropWhile__LNil,axiom,
! [A: $tType,P: A > $o] :
( ( coindu438612276pWhile @ A @ P @ ( coindu1598213697e_LNil @ A ) )
= ( coindu1598213697e_LNil @ A ) ) ).
% ldropWhile_LNil
thf(fact_20_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_21_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_22_lprefix__code_I1_J,axiom,
! [A: $tType,Ys: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ ( coindu1598213697e_LNil @ A ) @ Ys ) ).
% lprefix_code(1)
thf(fact_23_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_24_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_25_lappend__code_I1_J,axiom,
! [A: $tType,Ys: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ ( coindu1598213697e_LNil @ A ) @ Ys )
= Ys ) ).
% lappend_code(1)
thf(fact_26_lappend__LNil2,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ Xs @ ( coindu1598213697e_LNil @ A ) )
= Xs ) ).
% lappend_LNil2
thf(fact_27_ltake__LNil,axiom,
! [A: $tType,N: extended_enat] :
( ( coindu1802687541_ltake @ A @ N @ ( coindu1598213697e_LNil @ A ) )
= ( coindu1598213697e_LNil @ A ) ) ).
% ltake_LNil
thf(fact_28_ldrop__LNil,axiom,
! [A: $tType,N: extended_enat] :
( ( coindu191418589_ldrop @ A @ N @ ( coindu1598213697e_LNil @ A ) )
= ( coindu1598213697e_LNil @ A ) ) ).
% ldrop_LNil
thf(fact_29_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_30_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_31_ldrop__lmap,axiom,
! [A: $tType,B: $tType,N: extended_enat,F: B > A,Xs: coindu1593790203_llist @ B] :
( ( coindu191418589_ldrop @ A @ N @ ( coindu1062782156e_lmap @ B @ A @ F @ Xs ) )
= ( coindu1062782156e_lmap @ B @ A @ F @ ( coindu191418589_ldrop @ B @ N @ Xs ) ) ) ).
% ldrop_lmap
thf(fact_32_ltakeWhile__LNil,axiom,
! [A: $tType,P: A > $o] :
( ( coindu721411036eWhile @ A @ P @ ( coindu1598213697e_LNil @ A ) )
= ( coindu1598213697e_LNil @ A ) ) ).
% ltakeWhile_LNil
thf(fact_33_ldropWhile__LCons,axiom,
! [A: $tType,P: A > $o,X: A,Xs: coindu1593790203_llist @ A] :
( ( ( P @ X )
=> ( ( coindu438612276pWhile @ A @ P @ ( coindu1121789889_LCons @ A @ X @ Xs ) )
= ( coindu438612276pWhile @ A @ P @ Xs ) ) )
& ( ~ ( P @ X )
=> ( ( coindu438612276pWhile @ A @ P @ ( coindu1121789889_LCons @ A @ X @ Xs ) )
= ( coindu1121789889_LCons @ A @ X @ Xs ) ) ) ) ).
% ldropWhile_LCons
thf(fact_34_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_35_ltl__simps_I2_J,axiom,
! [A: $tType,X21: A,X22: coindu1593790203_llist @ A] :
( ( coindu1047225960le_ltl @ A @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) )
= X22 ) ).
% ltl_simps(2)
thf(fact_36_ltl__simps_I1_J,axiom,
! [A: $tType] :
( ( coindu1047225960le_ltl @ A @ ( coindu1598213697e_LNil @ A ) )
= ( coindu1598213697e_LNil @ A ) ) ).
% ltl_simps(1)
thf(fact_37_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_38_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_39_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_40_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 ) )
| ? [Xs2: coindu1593790203_llist @ A] :
( ( Xs
= ( coindu1121789889_LCons @ A @ Y @ Xs2 ) )
& ( coindu1571841457prefix @ A @ Xs2 @ Ys ) ) ) ) ).
% Coinductive_List_Mirabelle_kmikjhschf.finite_lprefix_nitpick_simps(3)
thf(fact_41_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_42_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_43_ltl__ldrop,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( coindu1047225960le_ltl @ A @ ( coindu191418589_ldrop @ A @ N @ Xs ) )
= ( coindu191418589_ldrop @ A @ N @ ( coindu1047225960le_ltl @ A @ Xs ) ) ) ).
% ltl_ldrop
thf(fact_44_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_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X2: A] :
( ( F @ X2 )
= ( G @ X2 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_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_50_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_51_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_52_neq__LNil__conv,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( Xs
!= ( coindu1598213697e_LNil @ A ) )
= ( ? [X3: A,Xs2: coindu1593790203_llist @ A] :
( Xs
= ( coindu1121789889_LCons @ A @ X3 @ Xs2 ) ) ) ) ).
% neq_LNil_conv
thf(fact_53_lappend__LNil__LNil,axiom,
! [A: $tType] :
( ( coindu268472904append @ A @ ( coindu1598213697e_LNil @ A ) @ ( coindu1598213697e_LNil @ A ) )
= ( coindu1598213697e_LNil @ A ) ) ).
% lappend_LNil_LNil
thf(fact_54_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 ) )
= ( ? [X3: B,Xs2: coindu1593790203_llist @ B] :
( ( Xs
= ( coindu1121789889_LCons @ B @ X3 @ Xs2 ) )
& ( Y
= ( F @ X3 ) )
& ( Ys
= ( coindu1062782156e_lmap @ B @ A @ F @ Xs2 ) ) ) ) ) ).
% lmap_eq_LCons_conv
thf(fact_55_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_56_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_57_lappend__ltake__ldrop,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) @ ( coindu191418589_ldrop @ A @ N @ Xs ) )
= Xs ) ).
% lappend_ltake_ldrop
thf(fact_58_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_59_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_60_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 ) )
| ? [Xs2: coindu1593790203_llist @ A] :
( ( Xs
= ( coindu1121789889_LCons @ A @ Y @ Xs2 ) )
& ( coindu1696667936prefix @ A @ Xs2 @ Ys ) ) ) ) ).
% lprefix_LCons_conv
thf(fact_61_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 )
=> ( ? [Xs3: coindu1593790203_llist @ A] :
( ( X2
= ( coindu1598213697e_LNil @ A ) )
& ( Xa2 = Xs3 ) )
| ? [Xs3: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A,Xb: A] :
( ( X2
= ( coindu1121789889_LCons @ A @ Xb @ Xs3 ) )
& ( Xa2
= ( coindu1121789889_LCons @ A @ Xb @ Ys2 ) )
& ( ( X4 @ Xs3 @ Ys2 )
| ( coindu1696667936prefix @ A @ Xs3 @ Ys2 ) ) ) ) )
=> ( coindu1696667936prefix @ A @ X @ Xa ) ) ) ).
% lprefix.coinduct
thf(fact_62_lprefix_Osimps,axiom,
! [A: $tType] :
( ( coindu1696667936prefix @ A )
= ( ^ [A1: coindu1593790203_llist @ A,A22: coindu1593790203_llist @ A] :
( ? [Xs4: coindu1593790203_llist @ A] :
( ( A1
= ( coindu1598213697e_LNil @ A ) )
& ( A22 = Xs4 ) )
| ? [Xs4: coindu1593790203_llist @ A,Ys3: coindu1593790203_llist @ A,X3: A] :
( ( A1
= ( coindu1121789889_LCons @ A @ X3 @ Xs4 ) )
& ( A22
= ( coindu1121789889_LCons @ A @ X3 @ Ys3 ) )
& ( coindu1696667936prefix @ A @ Xs4 @ Ys3 ) ) ) ) ) ).
% lprefix.simps
thf(fact_63_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,X2: A] :
( ( A12
= ( coindu1121789889_LCons @ A @ X2 @ Xs5 ) )
=> ( ( A23
= ( coindu1121789889_LCons @ A @ X2 @ Ys4 ) )
=> ~ ( coindu1696667936prefix @ A @ Xs5 @ Ys4 ) ) ) ) ) ).
% lprefix.cases
thf(fact_64_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_65_lprefix__ltlI,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( coindu1696667936prefix @ A @ ( coindu1047225960le_ltl @ A @ Xs ) @ ( coindu1047225960le_ltl @ A @ Ys ) ) ) ).
% lprefix_ltlI
thf(fact_66_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_67_lprefix__conv__lappend,axiom,
! [A: $tType] :
( ( coindu1696667936prefix @ A )
= ( ^ [Xs4: coindu1593790203_llist @ A,Ys3: coindu1593790203_llist @ A] :
? [Zs2: coindu1593790203_llist @ A] :
( Ys3
= ( coindu268472904append @ A @ Xs4 @ Zs2 ) ) ) ) ).
% lprefix_conv_lappend
thf(fact_68_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_69_lappend__lprefixE,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ ( coindu268472904append @ A @ Xs @ Ys ) @ Zs )
=> ~ ! [Zs3: coindu1593790203_llist @ A] :
( Zs
!= ( coindu268472904append @ A @ Xs @ Zs3 ) ) ) ).
% lappend_lprefixE
thf(fact_70_lprefix__lappend,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ Xs @ ( coindu268472904append @ A @ Xs @ Ys ) ) ).
% lprefix_lappend
thf(fact_71_lmap__lprefix,axiom,
! [B: $tType,A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,F: A > B] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( coindu1696667936prefix @ B @ ( coindu1062782156e_lmap @ A @ B @ F @ Xs ) @ ( coindu1062782156e_lmap @ A @ B @ F @ Ys ) ) ) ).
% lmap_lprefix
thf(fact_72_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_73_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_74_LNil__lprefix,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ ( coindu1598213697e_LNil @ A ) @ Xs ) ).
% LNil_lprefix
thf(fact_75_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_76_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_77_llexord_Ocases,axiom,
! [A: $tType,R: A > A > $o,A12: coindu1593790203_llist @ A,A23: coindu1593790203_llist @ A] :
( ( coindu300403952lexord @ A @ R @ A12 @ A23 )
=> ( ! [Xs5: coindu1593790203_llist @ A,Ys4: coindu1593790203_llist @ A,X2: A] :
( ( A12
= ( coindu1121789889_LCons @ A @ X2 @ Xs5 ) )
=> ( ( A23
= ( coindu1121789889_LCons @ A @ X2 @ Ys4 ) )
=> ~ ( coindu300403952lexord @ A @ R @ Xs5 @ Ys4 ) ) )
=> ( ! [X2: A] :
( ? [Xs5: coindu1593790203_llist @ A] :
( A12
= ( coindu1121789889_LCons @ A @ X2 @ Xs5 ) )
=> ! [Y2: A] :
( ? [Ys4: coindu1593790203_llist @ A] :
( A23
= ( coindu1121789889_LCons @ A @ Y2 @ Ys4 ) )
=> ~ ( R @ X2 @ Y2 ) ) )
=> ~ ( ( A12
= ( coindu1598213697e_LNil @ A ) )
=> ! [Ys4: coindu1593790203_llist @ A] : A23 != Ys4 ) ) ) ) ).
% llexord.cases
thf(fact_78_llexord_Osimps,axiom,
! [A: $tType] :
( ( coindu300403952lexord @ A )
= ( ^ [R2: A > A > $o,A1: coindu1593790203_llist @ A,A22: coindu1593790203_llist @ A] :
( ? [Xs4: coindu1593790203_llist @ A,Ys3: coindu1593790203_llist @ A,X3: A] :
( ( A1
= ( coindu1121789889_LCons @ A @ X3 @ Xs4 ) )
& ( A22
= ( coindu1121789889_LCons @ A @ X3 @ Ys3 ) )
& ( coindu300403952lexord @ A @ R2 @ Xs4 @ Ys3 ) )
| ? [X3: A,Y3: A,Xs4: coindu1593790203_llist @ A,Ys3: coindu1593790203_llist @ A] :
( ( A1
= ( coindu1121789889_LCons @ A @ X3 @ Xs4 ) )
& ( A22
= ( coindu1121789889_LCons @ A @ Y3 @ Ys3 ) )
& ( R2 @ X3 @ Y3 ) )
| ? [Ys3: coindu1593790203_llist @ A] :
( ( A1
= ( coindu1598213697e_LNil @ A ) )
& ( A22 = Ys3 ) ) ) ) ) ).
% llexord.simps
thf(fact_79_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 )
=> ( ? [Xs3: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A,Xb: A] :
( ( X2
= ( coindu1121789889_LCons @ A @ Xb @ Xs3 ) )
& ( Xa2
= ( coindu1121789889_LCons @ A @ Xb @ Ys2 ) )
& ( ( X4 @ Xs3 @ Ys2 )
| ( coindu300403952lexord @ A @ R @ Xs3 @ Ys2 ) ) )
| ? [Xb: A,Y4: A,Xs3: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A] :
( ( X2
= ( coindu1121789889_LCons @ A @ Xb @ Xs3 ) )
& ( Xa2
= ( coindu1121789889_LCons @ A @ Y4 @ Ys2 ) )
& ( R @ Xb @ Y4 ) )
| ? [Ys2: coindu1593790203_llist @ A] :
( ( X2
= ( coindu1598213697e_LNil @ A ) )
& ( Xa2 = Ys2 ) ) ) )
=> ( coindu300403952lexord @ A @ R @ X @ Xa ) ) ) ).
% llexord.coinduct
thf(fact_80_ltake__plus__conv__lappend,axiom,
! [A: $tType,N: extended_enat,M: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( coindu1802687541_ltake @ A @ ( plus_plus @ extended_enat @ N @ M ) @ Xs )
= ( coindu268472904append @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) @ ( coindu1802687541_ltake @ A @ M @ ( coindu191418589_ldrop @ A @ N @ Xs ) ) ) ) ).
% ltake_plus_conv_lappend
thf(fact_81_lmember__code_I1_J,axiom,
! [A: $tType,X: A] :
~ ( coindu567634248member @ A @ X @ ( coindu1598213697e_LNil @ A ) ) ).
% lmember_code(1)
thf(fact_82_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_83_gen__lset__code_I1_J,axiom,
! [A: $tType,A3: set @ A] :
( ( coindu1928975208n_lset @ A @ A3 @ ( coindu1598213697e_LNil @ A ) )
= A3 ) ).
% gen_lset_code(1)
thf(fact_84_ltake__0,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu1802687541_ltake @ A @ ( zero_zero @ extended_enat ) @ Xs )
= ( coindu1598213697e_LNil @ A ) ) ).
% ltake_0
thf(fact_85_ldrop__0,axiom,
! [B: $tType,Xs: coindu1593790203_llist @ B] :
( ( coindu191418589_ldrop @ B @ ( zero_zero @ extended_enat ) @ Xs )
= Xs ) ).
% ldrop_0
thf(fact_86_ldrop__ldrop,axiom,
! [A: $tType,N: extended_enat,M: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( coindu191418589_ldrop @ A @ N @ ( coindu191418589_ldrop @ A @ M @ Xs ) )
= ( coindu191418589_ldrop @ A @ ( plus_plus @ extended_enat @ N @ M ) @ Xs ) ) ).
% ldrop_ldrop
thf(fact_87_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_88_lstrict__prefix__code_I1_J,axiom,
! [A: $tType] :
~ ( coindu574146665prefix @ A @ ( coindu1598213697e_LNil @ A ) @ ( coindu1598213697e_LNil @ A ) ) ).
% lstrict_prefix_code(1)
thf(fact_89_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_90_llist__less__induct,axiom,
! [A: $tType,P: ( coindu1593790203_llist @ A ) > $o,Xs: coindu1593790203_llist @ A] :
( ! [Xs5: coindu1593790203_llist @ A] :
( ! [Ys2: coindu1593790203_llist @ A] :
( ( coindu574146665prefix @ A @ Ys2 @ Xs5 )
=> ( P @ Ys2 ) )
=> ( P @ Xs5 ) )
=> ( P @ Xs ) ) ).
% llist_less_induct
thf(fact_91_lstrict__prefix__def,axiom,
! [A: $tType] :
( ( coindu574146665prefix @ A )
= ( ^ [Xs4: coindu1593790203_llist @ A,Ys3: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs4 @ Ys3 )
& ( Xs4 != Ys3 ) ) ) ) ).
% lstrict_prefix_def
thf(fact_92_lmap__lstrict__prefix,axiom,
! [B: $tType,A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,F: A > B] :
( ( coindu574146665prefix @ A @ Xs @ Ys )
=> ( coindu574146665prefix @ B @ ( coindu1062782156e_lmap @ A @ B @ F @ Xs ) @ ( coindu1062782156e_lmap @ A @ B @ F @ Ys ) ) ) ).
% lmap_lstrict_prefix
thf(fact_93_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_94_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_95_llexord__LNil,axiom,
! [A: $tType,R: A > A > $o,Ys: coindu1593790203_llist @ A] : ( coindu300403952lexord @ A @ R @ ( coindu1598213697e_LNil @ A ) @ Ys ) ).
% llexord_LNil
thf(fact_96_ltake__ldrop,axiom,
! [A: $tType,N: extended_enat,M: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( coindu1802687541_ltake @ A @ N @ ( coindu191418589_ldrop @ A @ M @ Xs ) )
= ( coindu191418589_ldrop @ A @ M @ ( coindu1802687541_ltake @ A @ ( plus_plus @ extended_enat @ N @ M ) @ Xs ) ) ) ).
% ltake_ldrop
thf(fact_97_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_98_add__cancel__right__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( A2
= ( plus_plus @ A @ A2 @ B2 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_right
thf(fact_99_add__cancel__right__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( A2
= ( plus_plus @ A @ B2 @ A2 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_left
thf(fact_100_add__cancel__left__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= A2 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_right
thf(fact_101_add__cancel__left__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= A2 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_left
thf(fact_102_double__zero__sym,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ A2 @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% double_zero_sym
thf(fact_103_double__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ( plus_plus @ A @ A2 @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% double_zero
thf(fact_104_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C ) )
= ( B2 = C ) ) ) ).
% add_left_cancel
thf(fact_105_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C @ A2 ) )
= ( B2 = C ) ) ) ).
% add_right_cancel
thf(fact_106_add_Oleft__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add.left_neutral
thf(fact_107_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.right_neutral
thf(fact_108_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_109_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_110_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus @ A @ I @ K )
= ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_111_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% add.assoc
thf(fact_112_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C ) )
= ( B2 = C ) ) ) ).
% add.left_cancel
thf(fact_113_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C @ A2 ) )
= ( B2 = C ) ) ) ).
% add.right_cancel
thf(fact_114_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type2 @ A ) )
=> ( ( plus_plus @ A )
= ( ^ [A4: A,B3: A] : ( plus_plus @ A @ B3 @ A4 ) ) ) ) ).
% add.commute
thf(fact_115_add_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C ) )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% add.left_commute
thf(fact_116_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C ) )
=> ( B2 = C ) ) ) ).
% add_left_imp_eq
thf(fact_117_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C @ A2 ) )
=> ( B2 = C ) ) ) ).
% add_right_imp_eq
thf(fact_118_comm__monoid__add__class_Oadd__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% comm_monoid_add_class.add_0
thf(fact_119_add_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.comm_neutral
thf(fact_120_add_Ogroup__left__neutral,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add.group_left_neutral
thf(fact_121_add__eq__0__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_122_plus__enat__eq__0__conv,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ( plus_plus @ extended_enat @ M @ N )
= ( zero_zero @ extended_enat ) )
= ( ( M
= ( zero_zero @ extended_enat ) )
& ( N
= ( zero_zero @ extended_enat ) ) ) ) ).
% plus_enat_eq_0_conv
thf(fact_123_semiring__normalization__rules_I6_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% semiring_normalization_rules(6)
thf(fact_124_semiring__normalization__rules_I5_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% semiring_normalization_rules(5)
thf(fact_125_add__0__iff,axiom,
! [A: $tType] :
( ( semiri456707255roduct @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( B2
= ( plus_plus @ A @ B2 @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% add_0_iff
thf(fact_126_llcp__LNil1,axiom,
! [A: $tType,Ys: coindu1593790203_llist @ A] :
( ( coindu731692203e_llcp @ A @ ( coindu1598213697e_LNil @ A ) @ Ys )
= ( zero_zero @ extended_enat ) ) ).
% llcp_LNil1
thf(fact_127_llcp__LNil2,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu731692203e_llcp @ A @ Xs @ ( coindu1598213697e_LNil @ A ) )
= ( zero_zero @ extended_enat ) ) ).
% llcp_LNil2
thf(fact_128_llcp__commute,axiom,
! [A: $tType] :
( ( coindu731692203e_llcp @ A )
= ( ^ [Xs4: coindu1593790203_llist @ A,Ys3: coindu1593790203_llist @ A] : ( coindu731692203e_llcp @ A @ Ys3 @ Xs4 ) ) ) ).
% llcp_commute
thf(fact_129_semiring__normalization__rules_I20_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A,D: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ C @ D ) )
= ( plus_plus @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ D ) ) ) ) ).
% semiring_normalization_rules(20)
thf(fact_130_semiring__normalization__rules_I21_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% semiring_normalization_rules(21)
thf(fact_131_semiring__normalization__rules_I22_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,D: A] :
( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ C @ D ) )
= ( plus_plus @ A @ C @ ( plus_plus @ A @ A2 @ D ) ) ) ) ).
% semiring_normalization_rules(22)
thf(fact_132_semiring__normalization__rules_I23_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
= ( plus_plus @ A @ ( plus_plus @ A @ A2 @ C ) @ B2 ) ) ) ).
% semiring_normalization_rules(23)
thf(fact_133_semiring__normalization__rules_I24_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ( ( plus_plus @ A )
= ( ^ [A4: A,C2: A] : ( plus_plus @ A @ C2 @ A4 ) ) ) ) ).
% semiring_normalization_rules(24)
thf(fact_134_semiring__normalization__rules_I25_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,D: A] :
( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ C @ D ) )
= ( plus_plus @ A @ ( plus_plus @ A @ A2 @ C ) @ D ) ) ) ).
% semiring_normalization_rules(25)
thf(fact_135_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_136_llcp__lappend__same,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
( ( coindu731692203e_llcp @ A @ ( coindu268472904append @ A @ Xs @ Ys ) @ ( coindu268472904append @ A @ Xs @ Zs ) )
= ( plus_plus @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ ( coindu731692203e_llcp @ A @ Ys @ Zs ) ) ) ).
% llcp_lappend_same
thf(fact_137_llcp__LCons,axiom,
! [A: $tType,X: A,Y: A,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ( X = Y )
=> ( ( coindu731692203e_llcp @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ ( coindu1121789889_LCons @ A @ Y @ Ys ) )
= ( extended_eSuc @ ( coindu731692203e_llcp @ A @ Xs @ Ys ) ) ) )
& ( ( X != Y )
=> ( ( coindu731692203e_llcp @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ ( coindu1121789889_LCons @ A @ Y @ Ys ) )
= ( zero_zero @ extended_enat ) ) ) ) ).
% llcp_LCons
thf(fact_138_llimit__induct,axiom,
! [A: $tType,P: ( coindu1593790203_llist @ A ) > $o,Xs: coindu1593790203_llist @ A] :
( ( P @ ( coindu1598213697e_LNil @ A ) )
=> ( ! [X2: A,Xs5: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ Xs5 )
=> ( ( P @ Xs5 )
=> ( P @ ( coindu1121789889_LCons @ A @ X2 @ Xs5 ) ) ) )
=> ( ( ! [Ys2: coindu1593790203_llist @ A] :
( ( coindu574146665prefix @ A @ Ys2 @ Xs )
=> ( P @ Ys2 ) )
=> ( P @ Xs ) )
=> ( P @ Xs ) ) ) ) ).
% llimit_induct
thf(fact_139_co_Oenat_Oinject,axiom,
! [X23: extended_enat,Y23: extended_enat] :
( ( ( extended_eSuc @ X23 )
= ( extended_eSuc @ Y23 ) )
= ( X23 = Y23 ) ) ).
% co.enat.inject
thf(fact_140_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_141_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_142_lfinite__code_I1_J,axiom,
! [A: $tType] : ( coindu1213758845finite @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% lfinite_code(1)
thf(fact_143_lfinite__ltl,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ ( coindu1047225960le_ltl @ A @ Xs ) )
= ( coindu1213758845finite @ A @ Xs ) ) ).
% lfinite_ltl
thf(fact_144_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_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_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_147_llcp__same__conv__length,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu731692203e_llcp @ A @ Xs @ Xs )
= ( coindu1018505716length @ A @ Xs ) ) ).
% llcp_same_conv_length
thf(fact_148_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_149_llength__LNil,axiom,
! [A: $tType] :
( ( coindu1018505716length @ A @ ( coindu1598213697e_LNil @ A ) )
= ( zero_zero @ extended_enat ) ) ).
% llength_LNil
thf(fact_150_llength__LCons,axiom,
! [B: $tType,X: B,Xs: coindu1593790203_llist @ B] :
( ( coindu1018505716length @ B @ ( coindu1121789889_LCons @ B @ X @ Xs ) )
= ( extended_eSuc @ ( coindu1018505716length @ B @ Xs ) ) ) ).
% llength_LCons
thf(fact_151_ltake__eSuc__LCons,axiom,
! [A: $tType,N: extended_enat,X: A,Xs: coindu1593790203_llist @ A] :
( ( coindu1802687541_ltake @ A @ ( extended_eSuc @ N ) @ ( coindu1121789889_LCons @ A @ X @ Xs ) )
= ( coindu1121789889_LCons @ A @ X @ ( coindu1802687541_ltake @ A @ N @ Xs ) ) ) ).
% ltake_eSuc_LCons
thf(fact_152_ldrop__eSuc__LCons,axiom,
! [B: $tType,N: extended_enat,X: B,Xs: coindu1593790203_llist @ B] :
( ( coindu191418589_ldrop @ B @ ( extended_eSuc @ N ) @ ( coindu1121789889_LCons @ B @ X @ Xs ) )
= ( coindu191418589_ldrop @ B @ N @ Xs ) ) ).
% ldrop_eSuc_LCons
thf(fact_153_llength__lappend,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1018505716length @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( plus_plus @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ ( coindu1018505716length @ A @ Ys ) ) ) ).
% llength_lappend
thf(fact_154_llcp__lprefix1,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( ( coindu731692203e_llcp @ A @ Xs @ Ys )
= ( coindu1018505716length @ A @ Xs ) ) ) ).
% llcp_lprefix1
thf(fact_155_llcp__lprefix2,axiom,
! [A: $tType,Ys: coindu1593790203_llist @ A,Xs: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Ys @ Xs )
=> ( ( coindu731692203e_llcp @ A @ Xs @ Ys )
= ( coindu1018505716length @ A @ Ys ) ) ) ).
% llcp_lprefix2
thf(fact_156_neq__zero__conv__eSuc,axiom,
! [N: extended_enat] :
( ( N
!= ( zero_zero @ extended_enat ) )
= ( ? [N2: extended_enat] :
( N
= ( extended_eSuc @ N2 ) ) ) ) ).
% neq_zero_conv_eSuc
thf(fact_157_co_Oenat_Oexhaust,axiom,
! [Y: extended_enat] :
( ( Y
!= ( zero_zero @ extended_enat ) )
=> ~ ! [X24: extended_enat] :
( Y
!= ( extended_eSuc @ X24 ) ) ) ).
% co.enat.exhaust
thf(fact_158_enat__coexhaust,axiom,
! [N: extended_enat] :
( ( N
!= ( zero_zero @ extended_enat ) )
=> ~ ! [N3: extended_enat] :
( N
!= ( extended_eSuc @ N3 ) ) ) ).
% enat_coexhaust
thf(fact_159_co_Oenat_OdiscI,axiom,
! [Enat: extended_enat,X23: extended_enat] :
( ( Enat
= ( extended_eSuc @ X23 ) )
=> ( Enat
!= ( zero_zero @ extended_enat ) ) ) ).
% co.enat.discI
thf(fact_160_eSuc__plus,axiom,
! [M: extended_enat,N: extended_enat] :
( ( plus_plus @ extended_enat @ ( extended_eSuc @ M ) @ N )
= ( extended_eSuc @ ( plus_plus @ extended_enat @ M @ N ) ) ) ).
% eSuc_plus
thf(fact_161_co_Oenat_Odistinct_I1_J,axiom,
! [X23: extended_enat] :
( ( zero_zero @ extended_enat )
!= ( extended_eSuc @ X23 ) ) ).
% co.enat.distinct(1)
thf(fact_162_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_163_lappend__eq__lappend__conv,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Us: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Vs: coindu1593790203_llist @ A] :
( ( ( coindu1018505716length @ A @ Xs )
= ( coindu1018505716length @ A @ Us ) )
=> ( ( ( coindu268472904append @ A @ Xs @ Ys )
= ( coindu268472904append @ A @ Us @ Vs ) )
= ( ( Xs = Us )
& ( ( coindu1213758845finite @ A @ Xs )
=> ( Ys = Vs ) ) ) ) ) ).
% lappend_eq_lappend_conv
thf(fact_164_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_165_lfinite__LNil,axiom,
! [A: $tType] : ( coindu1213758845finite @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% lfinite_LNil
thf(fact_166_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_167_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_168_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_169_lprefix__llength__eq__imp__eq,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( ( ( coindu1018505716length @ A @ Xs )
= ( coindu1018505716length @ A @ Ys ) )
=> ( Xs = Ys ) ) ) ).
% lprefix_llength_eq_imp_eq
thf(fact_170_lfinite_Ocases,axiom,
! [A: $tType,A2: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ A2 )
=> ( ( A2
!= ( coindu1598213697e_LNil @ A ) )
=> ~ ! [Xs5: coindu1593790203_llist @ A] :
( ? [X2: A] :
( A2
= ( coindu1121789889_LCons @ A @ X2 @ Xs5 ) )
=> ~ ( coindu1213758845finite @ A @ Xs5 ) ) ) ) ).
% lfinite.cases
thf(fact_171_lfinite_Osimps,axiom,
! [A: $tType] :
( ( coindu1213758845finite @ A )
= ( ^ [A4: coindu1593790203_llist @ A] :
( ( A4
= ( coindu1598213697e_LNil @ A ) )
| ? [Xs4: coindu1593790203_llist @ A,X3: A] :
( ( A4
= ( coindu1121789889_LCons @ A @ X3 @ Xs4 ) )
& ( coindu1213758845finite @ A @ Xs4 ) ) ) ) ) ).
% lfinite.simps
thf(fact_172_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,X2: A] :
( ( coindu1213758845finite @ A @ Xs5 )
=> ( ( P @ Xs5 )
=> ( P @ ( coindu1121789889_LCons @ A @ X2 @ Xs5 ) ) ) )
=> ( P @ X ) ) ) ) ).
% lfinite.inducts
thf(fact_173_ltake__ltl,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( coindu1802687541_ltake @ A @ N @ ( coindu1047225960le_ltl @ A @ Xs ) )
= ( coindu1047225960le_ltl @ A @ ( coindu1802687541_ltake @ A @ ( extended_eSuc @ N ) @ Xs ) ) ) ).
% ltake_ltl
thf(fact_174_ldrop__ltl,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( coindu191418589_ldrop @ A @ N @ ( coindu1047225960le_ltl @ A @ Xs ) )
= ( coindu191418589_ldrop @ A @ ( extended_eSuc @ N ) @ Xs ) ) ).
% ldrop_ltl
thf(fact_175_ldrop__eSuc__conv__ltl,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( coindu191418589_ldrop @ A @ ( extended_eSuc @ N ) @ Xs )
= ( coindu1047225960le_ltl @ A @ ( coindu191418589_ldrop @ A @ N @ Xs ) ) ) ).
% ldrop_eSuc_conv_ltl
thf(fact_176_Coinductive__List__Mirabelle__kmikjhschf_Olprefix__nitpick__simps,axiom,
! [A: $tType] :
( ( coindu1696667936prefix @ A )
= ( ^ [Xs4: coindu1593790203_llist @ A,Ys3: coindu1593790203_llist @ A] :
( ( ( coindu1213758845finite @ A @ Xs4 )
=> ( coindu1571841457prefix @ A @ Xs4 @ Ys3 ) )
& ( ~ ( coindu1213758845finite @ A @ Xs4 )
=> ( Xs4 = Ys3 ) ) ) ) ) ).
% Coinductive_List_Mirabelle_kmikjhschf.lprefix_nitpick_simps
thf(fact_177_lfinite__rev__induct,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: ( coindu1593790203_llist @ A ) > $o] :
( ( coindu1213758845finite @ A @ Xs )
=> ( ( P @ ( coindu1598213697e_LNil @ A ) )
=> ( ! [X2: A,Xs5: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ Xs5 )
=> ( ( P @ Xs5 )
=> ( P @ ( coindu268472904append @ A @ Xs5 @ ( coindu1121789889_LCons @ A @ X2 @ ( coindu1598213697e_LNil @ A ) ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lfinite_rev_induct
thf(fact_178_eSuc__inject,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ( extended_eSuc @ M )
= ( extended_eSuc @ N ) )
= ( M = N ) ) ).
% eSuc_inject
thf(fact_179_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus @ nat @ M @ N )
= ( zero_zero @ nat ) )
= ( ( M
= ( zero_zero @ nat ) )
& ( N
= ( zero_zero @ nat ) ) ) ) ).
% add_is_0
thf(fact_180_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% Nat.add_0_right
thf(fact_181_iadd__Suc__right,axiom,
! [M: extended_enat,N: extended_enat] :
( ( plus_plus @ extended_enat @ M @ ( extended_eSuc @ N ) )
= ( extended_eSuc @ ( plus_plus @ extended_enat @ M @ N ) ) ) ).
% iadd_Suc_right
thf(fact_182_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus @ nat @ M @ N )
= M )
=> ( N
= ( zero_zero @ nat ) ) ) ).
% add_eq_self_zero
thf(fact_183_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% plus_nat.add_0
thf(fact_184_zero__ne__eSuc,axiom,
! [N: extended_enat] :
( ( zero_zero @ extended_enat )
!= ( extended_eSuc @ N ) ) ).
% zero_ne_eSuc
thf(fact_185_ind__euclid,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A5: nat,B4: nat] :
( ( P @ A5 @ B4 )
= ( P @ B4 @ A5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ ( zero_zero @ nat ) )
=> ( ! [A5: nat,B4: nat] :
( ( P @ A5 @ B4 )
=> ( P @ A5 @ ( plus_plus @ nat @ A5 @ B4 ) ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% ind_euclid
thf(fact_186_Le__enatp_Ocases,axiom,
! [A12: extended_enat,A23: extended_enat] :
( ( coinductive_Le_enatp @ A12 @ A23 )
=> ( ( ( A12
= ( zero_zero @ extended_enat ) )
=> ! [N4: extended_enat] : A23 != N4 )
=> ~ ! [M2: extended_enat] :
( ( A12
= ( extended_eSuc @ M2 ) )
=> ! [N4: extended_enat,K2: extended_enat] :
( ( A23
= ( plus_plus @ extended_enat @ N4 @ K2 ) )
=> ( ( coinductive_Le_enatp @ M2 @ N4 )
=> ( K2
= ( zero_zero @ extended_enat ) ) ) ) ) ) ) ).
% Le_enatp.cases
thf(fact_187_Le__enatp_Osimps,axiom,
( coinductive_Le_enatp
= ( ^ [A1: extended_enat,A22: extended_enat] :
( ? [N5: extended_enat] :
( ( A1
= ( zero_zero @ extended_enat ) )
& ( A22 = N5 ) )
| ? [M3: extended_enat,N5: extended_enat,K3: extended_enat] :
( ( A1
= ( extended_eSuc @ M3 ) )
& ( A22
= ( plus_plus @ extended_enat @ N5 @ K3 ) )
& ( coinductive_Le_enatp @ M3 @ N5 )
& ( K3
!= ( zero_zero @ extended_enat ) ) ) ) ) ) ).
% Le_enatp.simps
thf(fact_188_Le__enatp_OLe__enat__zero,axiom,
! [N: extended_enat] : ( coinductive_Le_enatp @ ( zero_zero @ extended_enat ) @ N ) ).
% Le_enatp.Le_enat_zero
thf(fact_189_Le__enatp_OLe__enat__add,axiom,
! [M: extended_enat,N: extended_enat,K: extended_enat] :
( ( coinductive_Le_enatp @ M @ N )
=> ( ( K
!= ( zero_zero @ extended_enat ) )
=> ( coinductive_Le_enatp @ ( extended_eSuc @ M ) @ ( plus_plus @ extended_enat @ N @ K ) ) ) ) ).
% Le_enatp.Le_enat_add
thf(fact_190_Le__enatp_Ocoinduct,axiom,
! [X4: extended_enat > extended_enat > $o,X: extended_enat,Xa: extended_enat] :
( ( X4 @ X @ Xa )
=> ( ! [X2: extended_enat,Xa2: extended_enat] :
( ( X4 @ X2 @ Xa2 )
=> ( ? [N6: extended_enat] :
( ( X2
= ( zero_zero @ extended_enat ) )
& ( Xa2 = N6 ) )
| ? [M4: extended_enat,N6: extended_enat,K4: extended_enat] :
( ( X2
= ( extended_eSuc @ M4 ) )
& ( Xa2
= ( plus_plus @ extended_enat @ N6 @ K4 ) )
& ( ( X4 @ M4 @ N6 )
| ( coinductive_Le_enatp @ M4 @ N6 ) )
& ( K4
!= ( zero_zero @ extended_enat ) ) ) ) )
=> ( coinductive_Le_enatp @ X @ Xa ) ) ) ).
% Le_enatp.coinduct
thf(fact_191_llength__code,axiom,
! [A: $tType] :
( ( coindu1018505716length @ A )
= ( coindu493225792length @ A @ ( zero_zero @ nat ) ) ) ).
% llength_code
thf(fact_192_the__enat__0,axiom,
( ( extended_the_enat @ ( zero_zero @ extended_enat ) )
= ( zero_zero @ nat ) ) ).
% the_enat_0
thf(fact_193_ldrop__eq__LNil,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( ( coindu191418589_ldrop @ A @ N @ Xs )
= ( coindu1598213697e_LNil @ A ) )
= ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ N ) ) ).
% ldrop_eq_LNil
thf(fact_194_lprefix__ltake__same,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A,M: extended_enat] :
( ( coindu1696667936prefix @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) @ ( coindu1802687541_ltake @ A @ M @ Xs ) )
= ( ( ord_less_eq @ extended_enat @ N @ M )
| ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ M ) ) ) ).
% lprefix_ltake_same
thf(fact_195_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_196_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_cancel_right
thf(fact_197_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [C: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_cancel_left
thf(fact_198_eSuc__ile__mono,axiom,
! [N: extended_enat,M: extended_enat] :
( ( ord_less_eq @ extended_enat @ ( extended_eSuc @ N ) @ ( extended_eSuc @ M ) )
= ( ord_less_eq @ extended_enat @ N @ M ) ) ).
% eSuc_ile_mono
thf(fact_199_add__le__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel1
thf(fact_200_add__le__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel2
thf(fact_201_le__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% le_add_same_cancel1
thf(fact_202_le__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% le_add_same_cancel2
thf(fact_203_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_204_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_205_monotone__llength,axiom,
! [A: $tType] : ( comple1396247847notone @ ( coindu1593790203_llist @ A ) @ extended_enat @ ( coindu1696667936prefix @ A ) @ ( ord_less_eq @ extended_enat ) @ ( coindu1018505716length @ A ) ) ).
% monotone_llength
thf(fact_206_lprefix__llength__le,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ ( coindu1018505716length @ A @ Ys ) ) ) ).
% lprefix_llength_le
thf(fact_207_not__eSuc__ilei0,axiom,
! [N: extended_enat] :
~ ( ord_less_eq @ extended_enat @ ( extended_eSuc @ N ) @ ( zero_zero @ extended_enat ) ) ).
% not_eSuc_ilei0
thf(fact_208_ile__eSuc,axiom,
! [N: extended_enat] : ( ord_less_eq @ extended_enat @ N @ ( extended_eSuc @ N ) ) ).
% ile_eSuc
thf(fact_209_i0__lb,axiom,
! [N: extended_enat] : ( ord_less_eq @ extended_enat @ ( zero_zero @ extended_enat ) @ N ) ).
% i0_lb
thf(fact_210_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_211_add__le__imp__le__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_imp_le_right
thf(fact_212_add__le__imp__le__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [C: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_imp_le_left
thf(fact_213_le__iff__add,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B3: A] :
? [C2: A] :
( B3
= ( plus_plus @ A @ A4 @ C2 ) ) ) ) ) ).
% le_iff_add
thf(fact_214_add__right__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% add_right_mono
thf(fact_215_add__left__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) ) ) ) ).
% add_left_mono
thf(fact_216_add__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A,D: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C @ D )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ D ) ) ) ) ) ).
% add_mono
thf(fact_217_add__mono__thms__linordered__semiring_I1_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_218_add__mono__thms__linordered__semiring_I2_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_219_add__mono__thms__linordered__semiring_I3_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( K = L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_220_enat__le__plus__same_I2_J,axiom,
! [X: extended_enat,Y: extended_enat] : ( ord_less_eq @ extended_enat @ X @ ( plus_plus @ extended_enat @ Y @ X ) ) ).
% enat_le_plus_same(2)
thf(fact_221_enat__le__plus__same_I1_J,axiom,
! [X: extended_enat,Y: extended_enat] : ( ord_less_eq @ extended_enat @ X @ ( plus_plus @ extended_enat @ X @ Y ) ) ).
% enat_le_plus_same(1)
thf(fact_222_add__nonpos__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
=> ( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_223_add__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_224_add__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_nonpos
thf(fact_225_add__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_226_add__increasing2,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [C: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% add_increasing2
thf(fact_227_add__decreasing2,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [C: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ C @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ B2 ) ) ) ) ).
% add_decreasing2
thf(fact_228_add__increasing,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% add_increasing
thf(fact_229_add__decreasing,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ C @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ B2 ) ) ) ) ).
% add_decreasing
thf(fact_230_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_231_wlog__linorder__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,B2: A,A2: A] :
( ! [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
=> ( P @ A5 @ B4 ) )
=> ( ( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% wlog_linorder_le
thf(fact_232_zero__le,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% zero_le
thf(fact_233_eSuc__le__iff,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ord_less_eq @ extended_enat @ ( extended_eSuc @ X ) @ Y )
= ( ? [Y5: extended_enat] :
( ( Y
= ( extended_eSuc @ Y5 ) )
& ( ord_less_eq @ extended_enat @ X @ Y5 ) ) ) ) ).
% eSuc_le_iff
thf(fact_234_monotone__eSuc,axiom,
comple1396247847notone @ extended_enat @ extended_enat @ ( ord_less_eq @ extended_enat ) @ ( ord_less_eq @ extended_enat ) @ extended_eSuc ).
% monotone_eSuc
thf(fact_235_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_236_ldrop__all,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,M: extended_enat] :
( ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ M )
=> ( ( coindu191418589_ldrop @ A @ M @ Xs )
= ( coindu1598213697e_LNil @ A ) ) ) ).
% ldrop_all
thf(fact_237_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_238_lfp_Omonotone__if__bot,axiom,
! [B: $tType,A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [Bound: A,G: A > B,Bot: B,F: A > B,Ord: B > B > $o] :
( ! [X2: A] :
( ( ( ord_less_eq @ A @ X2 @ Bound )
=> ( ( G @ X2 )
= Bot ) )
& ( ~ ( ord_less_eq @ A @ X2 @ Bound )
=> ( ( G @ X2 )
= ( F @ X2 ) ) ) )
=> ( ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ~ ( ord_less_eq @ A @ X2 @ Bound )
=> ( Ord @ ( F @ X2 ) @ ( F @ Y2 ) ) ) )
=> ( ! [X2: A] :
( ~ ( ord_less_eq @ A @ X2 @ Bound )
=> ( Ord @ Bot @ ( F @ X2 ) ) )
=> ( ( Ord @ Bot @ Bot )
=> ( comple1396247847notone @ A @ B @ ( ord_less_eq @ A ) @ Ord @ G ) ) ) ) ) ) ).
% lfp.monotone_if_bot
thf(fact_239_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_240_le0,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% le0
thf(fact_241_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% less_eq_nat.simps(1)
thf(fact_242_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% le_0_eq
thf(fact_243_is__num__normalize_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% is_num_normalize(1)
thf(fact_244_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_245_enat__set_Ocases,axiom,
! [A2: extended_enat] :
( ( member @ extended_enat @ A2 @ coinductive_enat_set )
=> ( ( A2
!= ( zero_zero @ extended_enat ) )
=> ~ ! [N4: extended_enat] :
( ( A2
= ( extended_eSuc @ N4 ) )
=> ~ ( member @ extended_enat @ N4 @ coinductive_enat_set ) ) ) ) ).
% enat_set.cases
thf(fact_246_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_247_diff__zero,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% diff_zero
thf(fact_248_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_249_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_250_diff__self,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% diff_self
thf(fact_251_add__diff__cancel__right_H,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% add_diff_cancel_right'
thf(fact_252_add__diff__cancel__right,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
= ( minus_minus @ A @ A2 @ B2 ) ) ) ).
% add_diff_cancel_right
%----Type constructors (33)
thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A6: $tType,A7: $tType] :
( ( comple187826305attice @ A7 @ ( type2 @ A7 ) )
=> ( comple187826305attice @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
semiri456707255roduct @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
ordere516151231imp_le @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add,axiom,
ordere216010020id_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___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_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___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1 @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add,axiom,
semigroup_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Omonoid__add,axiom,
monoid_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ozero,axiom,
zero @ nat @ ( type2 @ nat ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_1,axiom,
! [A6: $tType] : ( comple187826305attice @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_2,axiom,
comple187826305attice @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder_3,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_4,axiom,
canoni770627133id_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__lattice_5,axiom,
comple187826305attice @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_6,axiom,
ordere779506340up_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_7,axiom,
ordere216010020id_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_8,axiom,
ab_semigroup_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_9,axiom,
comm_monoid_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_10,axiom,
comm_semiring_1 @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_11,axiom,
semigroup_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_12,axiom,
linorder @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_13,axiom,
monoid_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Ozero_14,axiom,
zero @ extended_enat @ ( type2 @ extended_enat ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
comple1396247847notone @ ( coindu1593790203_llist @ a ) @ ( coindu1593790203_llist @ a ) @ ( coindu1696667936prefix @ a ) @ ( coindu1696667936prefix @ a ) @ ( coindu438612276pWhile @ a @ p ) ).
%------------------------------------------------------------------------------