TPTP Problem File: DAT135^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : DAT135^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Coinductive list 4499
% 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__4499.p [Bla16]
% Status : Theorem
% Rating : 1.00 v7.1.0
% Syntax : Number of formulae : 353 ( 135 unt; 60 typ; 0 def)
% Number of atoms : 697 ( 274 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 3212 ( 70 ~; 7 |; 35 &;2812 @)
% ( 0 <=>; 288 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 7 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 167 ( 167 >; 0 *; 0 +; 0 <<)
% Number of symbols : 59 ( 57 usr; 3 con; 0-4 aty)
% Number of variables : 898 ( 40 ^; 780 !; 27 ?; 898 :)
% ( 51 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 15:07:06.953
%------------------------------------------------------------------------------
%----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 (54)
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_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_Ono__bot,type,
no_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[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_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[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_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_Olinordered__ab__group__add,type,
linord219039673up_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_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__cancel__ab__semigroup__add,type,
ordere223160158up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
strict2144017051up_add:
!>[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_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_Olconcat,type,
coindu441856546concat:
!>[A: $tType] : ( ( coindu1593790203_llist @ ( coindu1593790203_llist @ A ) ) > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Oldistinct,type,
coindu6345450stinct:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > $o ) ).
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_Oldropn,type,
coindu531130065ldropn:
!>[A: $tType] : ( nat > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olfilter,type,
coindu1889960678filter:
!>[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_Ollast,type,
coindu2000965700_llast:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > A ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollength,type,
coindu1018505716length:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > extended_enat ) ).
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_Olhd,type,
coindu1046438764le_lhd:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > 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_Olnth,type,
coindu749330388e_lnth:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > nat > A ) ).
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_OltakeWhile,type,
coindu721411036eWhile:
!>[A: $tType] : ( ( A > $o ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Extended__Nat_Oenat,type,
extended_enat2: nat > extended_enat ).
thf(sy_c_Extended__Nat_Oenat_Orec__enat,type,
extended_rec_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_Finite__Set_Ofinite,type,
finite_finite2:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
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_Groups__Big_Ocomm__monoid__add__class_Osetsum,type,
groups15040474setsum:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate,type,
infini455366010merate:
!>[A: $tType] : ( ( set @ A ) > nat > A ) ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OlessThan,type,
set_ord_lessThan:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_xss,type,
xss: coindu1593790203_llist @ ( coindu1593790203_llist @ a ) ).
%----Relevant facts (253)
thf(fact_0_assms,axiom,
ord_less @ extended_enat @ ( extended_enat2 @ n ) @ ( coindu1018505716length @ a @ ( coindu441856546concat @ a @ xss ) ) ).
% assms
thf(fact_1_enat__add__mono,axiom,
! [X: nat,Y: extended_enat,Z: extended_enat] :
( ( ord_less @ extended_enat @ ( plus_plus @ extended_enat @ ( extended_enat2 @ X ) @ Y ) @ ( plus_plus @ extended_enat @ ( extended_enat2 @ X ) @ Z ) )
= ( ord_less @ extended_enat @ Y @ Z ) ) ).
% enat_add_mono
thf(fact_2_enat__add1__eq,axiom,
! [X: nat,Y: extended_enat,Z: extended_enat] :
( ( ( plus_plus @ extended_enat @ ( extended_enat2 @ X ) @ Y )
= ( plus_plus @ extended_enat @ ( extended_enat2 @ X ) @ Z ) )
= ( Y = Z ) ) ).
% enat_add1_eq
thf(fact_3_enat__add2__eq,axiom,
! [Y: extended_enat,X: nat,Z: extended_enat] :
( ( ( plus_plus @ extended_enat @ Y @ ( extended_enat2 @ X ) )
= ( plus_plus @ extended_enat @ Z @ ( extended_enat2 @ X ) ) )
= ( Y = Z ) ) ).
% enat_add2_eq
thf(fact_4_lessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [I: A,K: A] :
( ( member @ A @ I @ ( set_ord_lessThan @ A @ K ) )
= ( ord_less @ A @ I @ K ) ) ) ).
% lessThan_iff
thf(fact_5_add__less__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [C: A,A2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_cancel_left
thf(fact_6_add__less__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_cancel_right
thf(fact_7_llength__lconcat__lfinite__conv__sum,axiom,
! [A: $tType,Xss: coindu1593790203_llist @ ( coindu1593790203_llist @ A )] :
( ( coindu1213758845finite @ ( coindu1593790203_llist @ A ) @ Xss )
=> ( ( coindu1018505716length @ A @ ( coindu441856546concat @ A @ Xss ) )
= ( groups15040474setsum @ nat @ extended_enat
@ ^ [I2: nat] : ( coindu1018505716length @ A @ ( coindu749330388e_lnth @ ( coindu1593790203_llist @ A ) @ Xss @ I2 ) )
@ ( collect @ nat
@ ^ [I2: nat] : ( ord_less @ extended_enat @ ( extended_enat2 @ I2 ) @ ( coindu1018505716length @ ( coindu1593790203_llist @ A ) @ Xss ) ) ) ) ) ) ).
% llength_lconcat_lfinite_conv_sum
thf(fact_8_enat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( extended_enat2 @ Nat )
= ( extended_enat2 @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% enat.inject
thf(fact_9_lessThan__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ( set_ord_lessThan @ A @ X )
= ( set_ord_lessThan @ A @ Y ) )
= ( X = Y ) ) ) ).
% lessThan_eq_iff
thf(fact_10_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_11_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_12_lessThan__def,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ( ( set_ord_lessThan @ A )
= ( ^ [U: A] :
( collect @ A
@ ^ [X2: A] : ( ord_less @ A @ X2 @ U ) ) ) ) ) ).
% lessThan_def
thf(fact_13_enat__ord__simps_I2_J,axiom,
! [M: nat,N: nat] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
= ( ord_less @ nat @ M @ N ) ) ).
% enat_ord_simps(2)
thf(fact_14_plus__enat__simps_I1_J,axiom,
! [M: nat,N: nat] :
( ( plus_plus @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
= ( extended_enat2 @ ( plus_plus @ nat @ M @ N ) ) ) ).
% plus_enat_simps(1)
thf(fact_15_less__enatE,axiom,
! [N: extended_enat,M: nat] :
( ( ord_less @ extended_enat @ N @ ( extended_enat2 @ M ) )
=> ~ ! [K2: nat] :
( ( N
= ( extended_enat2 @ K2 ) )
=> ~ ( ord_less @ nat @ K2 @ M ) ) ) ).
% less_enatE
thf(fact_16_lfinite__conv__llength__enat,axiom,
! [A: $tType] :
( ( coindu1213758845finite @ A )
= ( ^ [Xs: coindu1593790203_llist @ A] :
? [N2: nat] :
( ( coindu1018505716length @ A @ Xs )
= ( extended_enat2 @ N2 ) ) ) ) ).
% lfinite_conv_llength_enat
thf(fact_17_llength__eq__enat__lfiniteD,axiom,
! [A: $tType,Xs2: coindu1593790203_llist @ A,N: nat] :
( ( ( coindu1018505716length @ A @ Xs2 )
= ( extended_enat2 @ N ) )
=> ( coindu1213758845finite @ A @ Xs2 ) ) ).
% llength_eq_enat_lfiniteD
thf(fact_18_lfinite__llength__enat,axiom,
! [A: $tType,Xs2: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ Xs2 )
=> ? [N3: nat] :
( ( coindu1018505716length @ A @ Xs2 )
= ( extended_enat2 @ N3 ) ) ) ).
% lfinite_llength_enat
thf(fact_19_enat__less__enat__plusI2,axiom,
! [Y: nat,Z: extended_enat,X: nat] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ Y ) @ Z )
=> ( ord_less @ extended_enat @ ( extended_enat2 @ ( plus_plus @ nat @ X @ Y ) ) @ ( plus_plus @ extended_enat @ ( extended_enat2 @ X ) @ Z ) ) ) ).
% enat_less_enat_plusI2
thf(fact_20_enat__less__enat__plusI,axiom,
! [X: nat,Y: nat,Z: extended_enat] :
( ( ord_less @ nat @ X @ Y )
=> ( ord_less @ extended_enat @ ( extended_enat2 @ X ) @ ( plus_plus @ extended_enat @ ( extended_enat2 @ Y ) @ Z ) ) ) ).
% enat_less_enat_plusI
thf(fact_21_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_22_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_23_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_24_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type2 @ A ) )
=> ( ( plus_plus @ A )
= ( ^ [A3: A,B3: A] : ( plus_plus @ A @ B3 @ A3 ) ) ) ) ).
% add.commute
thf(fact_25_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_26_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_27_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_28_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_29_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_30_enat__less__induct,axiom,
! [P: extended_enat > $o,N: extended_enat] :
( ! [N3: extended_enat] :
( ! [M2: extended_enat] :
( ( ord_less @ extended_enat @ M2 @ N3 )
=> ( P @ M2 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% enat_less_induct
thf(fact_31_add__less__imp__less__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_imp_less_right
thf(fact_32_add__less__imp__less__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [C: A,A2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_imp_less_left
thf(fact_33_add__strict__right__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% add_strict_right_mono
thf(fact_34_add__strict__left__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) ) ) ) ).
% add_strict_left_mono
thf(fact_35_add__strict__mono,axiom,
! [A: $tType] :
( ( strict2144017051up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A,D: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C @ D )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ D ) ) ) ) ) ).
% add_strict_mono
thf(fact_36_add__mono__thms__linordered__field_I1_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( K = L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_37_add__mono__thms__linordered__field_I2_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_38_add__mono__thms__linordered__field_I5_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_39_lessThan__strict__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [M: A,N: A] :
( ( ord_less @ ( set @ A ) @ ( set_ord_lessThan @ A @ M ) @ ( set_ord_lessThan @ A @ N ) )
= ( ord_less @ A @ M @ N ) ) ) ).
% lessThan_strict_subset_iff
thf(fact_40_enat__iless,axiom,
! [N: extended_enat,M: nat] :
( ( ord_less @ extended_enat @ N @ ( extended_enat2 @ M ) )
=> ? [K2: nat] :
( N
= ( extended_enat2 @ K2 ) ) ) ).
% enat_iless
thf(fact_41_chain__incr,axiom,
! [A: $tType,Y2: A > extended_enat,K: nat] :
( ! [I3: A] :
? [J2: A] : ( ord_less @ extended_enat @ ( Y2 @ I3 ) @ ( Y2 @ J2 ) )
=> ? [J3: A] : ( ord_less @ extended_enat @ ( extended_enat2 @ K ) @ ( Y2 @ J3 ) ) ) ).
% chain_incr
thf(fact_42_setsum_Odistrib,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A @ ( type2 @ A ) )
=> ! [G: B > A,H: B > A,A4: set @ B] :
( ( groups15040474setsum @ B @ A
@ ^ [X2: B] : ( plus_plus @ A @ ( G @ X2 ) @ ( H @ X2 ) )
@ A4 )
= ( plus_plus @ A @ ( groups15040474setsum @ B @ A @ G @ A4 ) @ ( groups15040474setsum @ B @ A @ H @ A4 ) ) ) ) ).
% setsum.distrib
thf(fact_43_gen__llength__def,axiom,
! [A: $tType] :
( ( coindu493225792length @ A )
= ( ^ [N2: nat,Xs: coindu1593790203_llist @ A] : ( plus_plus @ extended_enat @ ( extended_enat2 @ N2 ) @ ( coindu1018505716length @ A @ Xs ) ) ) ) ).
% gen_llength_def
thf(fact_44_enat_Osimps_I6_J,axiom,
! [T: $tType,F1: nat > T,F2: T,Nat: nat] :
( ( extended_rec_enat @ T @ F1 @ F2 @ ( extended_enat2 @ Nat ) )
= ( F1 @ Nat ) ) ).
% enat.simps(6)
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] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_lnth__ldropn,axiom,
! [A: $tType,N: nat,M: nat,Xs2: coindu1593790203_llist @ A] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ ( plus_plus @ nat @ N @ M ) ) @ ( coindu1018505716length @ A @ Xs2 ) )
=> ( ( coindu749330388e_lnth @ A @ ( coindu531130065ldropn @ A @ N @ Xs2 ) @ M )
= ( coindu749330388e_lnth @ A @ Xs2 @ ( plus_plus @ nat @ M @ N ) ) ) ) ).
% lnth_ldropn
thf(fact_50_ldistinct__conv__lnth,axiom,
! [A: $tType] :
( ( coindu6345450stinct @ A )
= ( ^ [Xs: coindu1593790203_llist @ A] :
! [I2: nat,J4: nat] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ I2 ) @ ( coindu1018505716length @ A @ Xs ) )
=> ( ( ord_less @ extended_enat @ ( extended_enat2 @ J4 ) @ ( coindu1018505716length @ A @ Xs ) )
=> ( ( I2 != J4 )
=> ( ( coindu749330388e_lnth @ A @ Xs @ I2 )
!= ( coindu749330388e_lnth @ A @ Xs @ J4 ) ) ) ) ) ) ) ).
% ldistinct_conv_lnth
thf(fact_51_lfinite__finite__index,axiom,
! [A: $tType,Xs2: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ Xs2 )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N2: nat] : ( ord_less @ extended_enat @ ( extended_enat2 @ N2 ) @ ( coindu1018505716length @ A @ Xs2 ) ) ) ) ) ).
% lfinite_finite_index
thf(fact_52_setsum_Ocommute,axiom,
! [A: $tType,B: $tType,C2: $tType] :
( ( comm_monoid_add @ A @ ( type2 @ A ) )
=> ! [G: B > C2 > A,B4: set @ C2,A4: set @ B] :
( ( groups15040474setsum @ B @ A
@ ^ [I2: B] : ( groups15040474setsum @ C2 @ A @ ( G @ I2 ) @ B4 )
@ A4 )
= ( groups15040474setsum @ C2 @ A
@ ^ [J4: C2] :
( groups15040474setsum @ B @ A
@ ^ [I2: B] : ( G @ I2 @ J4 )
@ A4 )
@ B4 ) ) ) ).
% setsum.commute
thf(fact_53_lnth__lmap,axiom,
! [B: $tType,A: $tType,N: nat,Xs2: coindu1593790203_llist @ A,F: A > B] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ N ) @ ( coindu1018505716length @ A @ Xs2 ) )
=> ( ( coindu749330388e_lnth @ B @ ( coindu1062782156e_lmap @ A @ B @ F @ Xs2 ) @ N )
= ( F @ ( coindu749330388e_lnth @ A @ Xs2 @ N ) ) ) ) ).
% lnth_lmap
thf(fact_54_lprefix__lnthD,axiom,
! [A: $tType,Xs2: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,N: nat] :
( ( coindu1696667936prefix @ A @ Xs2 @ Ys )
=> ( ( ord_less @ extended_enat @ ( extended_enat2 @ N ) @ ( coindu1018505716length @ A @ Xs2 ) )
=> ( ( coindu749330388e_lnth @ A @ Xs2 @ N )
= ( coindu749330388e_lnth @ A @ Ys @ N ) ) ) ) ).
% lprefix_lnthD
thf(fact_55_setsum__lessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A @ ( type2 @ A ) )
=> ! [F: nat > A,N: nat] :
( ( groups15040474setsum @ nat @ A @ F @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups15040474setsum @ nat @ A @ F @ ( set_ord_lessThan @ nat @ N ) ) @ ( F @ N ) ) ) ) ).
% setsum_lessThan_Suc
thf(fact_56_ltakeWhile__nth,axiom,
! [A: $tType,I: nat,P: A > $o,Xs2: coindu1593790203_llist @ A] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ I ) @ ( coindu1018505716length @ A @ ( coindu721411036eWhile @ A @ P @ Xs2 ) ) )
=> ( ( coindu749330388e_lnth @ A @ ( coindu721411036eWhile @ A @ P @ Xs2 ) @ I )
= ( coindu749330388e_lnth @ A @ Xs2 @ I ) ) ) ).
% ltakeWhile_nth
thf(fact_57_llist_Oleq__refl,axiom,
! [A: $tType,X: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ X @ X ) ).
% llist.leq_refl
thf(fact_58_lprefix__refl,axiom,
! [A: $tType,Xs2: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ Xs2 @ Xs2 ) ).
% lprefix_refl
thf(fact_59_llist_Omap__ident,axiom,
! [A: $tType,T2: coindu1593790203_llist @ A] :
( ( coindu1062782156e_lmap @ A @ A
@ ^ [X2: A] : X2
@ T2 )
= T2 ) ).
% llist.map_ident
thf(fact_60_ltakeWhile__K__True,axiom,
! [A: $tType,Xs2: coindu1593790203_llist @ A] :
( ( coindu721411036eWhile @ A
@ ^ [Uu: A] : $true
@ Xs2 )
= Xs2 ) ).
% ltakeWhile_K_True
thf(fact_61_ldropn__ldropn,axiom,
! [A: $tType,N: nat,M: nat,Xs2: coindu1593790203_llist @ A] :
( ( coindu531130065ldropn @ A @ N @ ( coindu531130065ldropn @ A @ M @ Xs2 ) )
= ( coindu531130065ldropn @ A @ ( plus_plus @ nat @ N @ M ) @ Xs2 ) ) ).
% ldropn_ldropn
thf(fact_62_llength__lmap,axiom,
! [A: $tType,B: $tType,F: B > A,Xs2: coindu1593790203_llist @ B] :
( ( coindu1018505716length @ A @ ( coindu1062782156e_lmap @ B @ A @ F @ Xs2 ) )
= ( coindu1018505716length @ B @ Xs2 ) ) ).
% llength_lmap
thf(fact_63_lfinite__lmap,axiom,
! [A: $tType,B: $tType,F: B > A,Xs2: coindu1593790203_llist @ B] :
( ( coindu1213758845finite @ A @ ( coindu1062782156e_lmap @ B @ A @ F @ Xs2 ) )
= ( coindu1213758845finite @ B @ Xs2 ) ) ).
% lfinite_lmap
thf(fact_64_lfinite__ldropn,axiom,
! [A: $tType,N: nat,Xs2: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ ( coindu531130065ldropn @ A @ N @ Xs2 ) )
= ( coindu1213758845finite @ A @ Xs2 ) ) ).
% lfinite_ldropn
thf(fact_65_finite__lessThan,axiom,
! [K: nat] : ( finite_finite2 @ nat @ ( set_ord_lessThan @ nat @ K ) ) ).
% finite_lessThan
thf(fact_66_ldropn__lmap,axiom,
! [A: $tType,B: $tType,N: nat,F: B > A,Xs2: coindu1593790203_llist @ B] :
( ( coindu531130065ldropn @ A @ N @ ( coindu1062782156e_lmap @ B @ A @ F @ Xs2 ) )
= ( coindu1062782156e_lmap @ B @ A @ F @ ( coindu531130065ldropn @ B @ N @ Xs2 ) ) ) ).
% ldropn_lmap
thf(fact_67_bounded__nat__set__is__finite,axiom,
! [N4: set @ nat,N: nat] :
( ! [X3: nat] :
( ( member @ nat @ X3 @ N4 )
=> ( ord_less @ nat @ X3 @ N ) )
=> ( finite_finite2 @ nat @ N4 ) ) ).
% bounded_nat_set_is_finite
thf(fact_68_finite__nat__set__iff__bounded,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N5: set @ nat] :
? [M3: nat] :
! [X2: nat] :
( ( member @ nat @ X2 @ N5 )
=> ( ord_less @ nat @ X2 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_69_lprefix__down__linear,axiom,
! [A: $tType,Xs2: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs2 @ Zs )
=> ( ( coindu1696667936prefix @ A @ Ys @ Zs )
=> ( ( coindu1696667936prefix @ A @ Xs2 @ Ys )
| ( coindu1696667936prefix @ A @ Ys @ Xs2 ) ) ) ) ).
% lprefix_down_linear
thf(fact_70_lprefix__ltakeWhile,axiom,
! [A: $tType,P: A > $o,Xs2: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ ( coindu721411036eWhile @ A @ P @ Xs2 ) @ Xs2 ) ).
% lprefix_ltakeWhile
thf(fact_71_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_72_ldistinct__lprefix,axiom,
! [A: $tType,Xs2: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu6345450stinct @ A @ Xs2 )
=> ( ( coindu1696667936prefix @ A @ Ys @ Xs2 )
=> ( coindu6345450stinct @ A @ Ys ) ) ) ).
% ldistinct_lprefix
thf(fact_73_ldistinct__ldropn,axiom,
! [A: $tType,Xs2: coindu1593790203_llist @ A,N: nat] :
( ( coindu6345450stinct @ A @ Xs2 )
=> ( coindu6345450stinct @ A @ ( coindu531130065ldropn @ A @ N @ Xs2 ) ) ) ).
% ldistinct_ldropn
thf(fact_74_lprefix__antisym,axiom,
! [A: $tType,Xs2: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs2 @ Ys )
=> ( ( coindu1696667936prefix @ A @ Ys @ Xs2 )
=> ( Xs2 = Ys ) ) ) ).
% lprefix_antisym
thf(fact_75_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_76_lprefix__trans,axiom,
! [A: $tType,Xs2: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs2 @ Ys )
=> ( ( coindu1696667936prefix @ A @ Ys @ Zs )
=> ( coindu1696667936prefix @ A @ Xs2 @ Zs ) ) ) ).
% lprefix_trans
thf(fact_77_lmap__lprefix,axiom,
! [B: $tType,A: $tType,Xs2: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,F: A > B] :
( ( coindu1696667936prefix @ A @ Xs2 @ Ys )
=> ( coindu1696667936prefix @ B @ ( coindu1062782156e_lmap @ A @ B @ F @ Xs2 ) @ ( coindu1062782156e_lmap @ A @ B @ F @ Ys ) ) ) ).
% lmap_lprefix
thf(fact_78_lmap__lconcat,axiom,
! [A: $tType,B: $tType,F: B > A,Xss: coindu1593790203_llist @ ( coindu1593790203_llist @ B )] :
( ( coindu1062782156e_lmap @ B @ A @ F @ ( coindu441856546concat @ B @ Xss ) )
= ( coindu441856546concat @ A @ ( coindu1062782156e_lmap @ ( coindu1593790203_llist @ B ) @ ( coindu1593790203_llist @ A ) @ ( coindu1062782156e_lmap @ B @ A @ F ) @ Xss ) ) ) ).
% lmap_lconcat
thf(fact_79_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( P @ K3 )
& ( ord_less @ nat @ K3 @ I ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_80_setsum_Ocommute__restrict,axiom,
! [A: $tType,B: $tType,C2: $tType] :
( ( comm_monoid_add @ A @ ( type2 @ A ) )
=> ! [A4: set @ B,B4: set @ C2,G: B > C2 > A,R: B > C2 > $o] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ C2 @ B4 )
=> ( ( groups15040474setsum @ B @ A
@ ^ [X2: B] :
( groups15040474setsum @ C2 @ A @ ( G @ X2 )
@ ( collect @ C2
@ ^ [Y3: C2] :
( ( member @ C2 @ Y3 @ B4 )
& ( R @ X2 @ Y3 ) ) ) )
@ A4 )
= ( groups15040474setsum @ C2 @ A
@ ^ [Y3: C2] :
( groups15040474setsum @ B @ A
@ ^ [X2: B] : ( G @ X2 @ Y3 )
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ A4 )
& ( R @ X2 @ Y3 ) ) ) )
@ B4 ) ) ) ) ) ).
% setsum.commute_restrict
thf(fact_81_llength__ltakeWhile__all,axiom,
! [A: $tType,P: A > $o,Xs2: coindu1593790203_llist @ A] :
( ( ( coindu1018505716length @ A @ ( coindu721411036eWhile @ A @ P @ Xs2 ) )
= ( coindu1018505716length @ A @ Xs2 ) )
= ( ( coindu721411036eWhile @ A @ P @ Xs2 )
= Xs2 ) ) ).
% llength_ltakeWhile_all
thf(fact_82_lprefix__llength__eq__imp__eq,axiom,
! [A: $tType,Xs2: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs2 @ Ys )
=> ( ( ( coindu1018505716length @ A @ Xs2 )
= ( coindu1018505716length @ A @ Ys ) )
=> ( Xs2 = Ys ) ) ) ).
% lprefix_llength_eq_imp_eq
thf(fact_83_infinite__Iio,axiom,
! [A: $tType] :
( ( ( linorder @ A @ ( type2 @ A ) )
& ( no_bot @ A @ ( type2 @ A ) ) )
=> ! [A2: A] :
~ ( finite_finite2 @ A @ ( set_ord_lessThan @ A @ A2 ) ) ) ).
% infinite_Iio
thf(fact_84_not__lfinite__lprefix__conv__eq,axiom,
! [A: $tType,Xs2: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ~ ( coindu1213758845finite @ A @ Xs2 )
=> ( ( coindu1696667936prefix @ A @ Xs2 @ Ys )
= ( Xs2 = Ys ) ) ) ).
% not_lfinite_lprefix_conv_eq
thf(fact_85_lprefix__lfiniteD,axiom,
! [A: $tType,Xs2: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs2 @ Ys )
=> ( ( coindu1213758845finite @ A @ Ys )
=> ( coindu1213758845finite @ A @ Xs2 ) ) ) ).
% lprefix_lfiniteD
thf(fact_86_setsum_Ocong,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A @ ( type2 @ A ) )
=> ! [A4: set @ B,B4: set @ B,G: B > A,H: B > A] :
( ( A4 = B4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B4 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups15040474setsum @ B @ A @ G @ A4 )
= ( groups15040474setsum @ B @ A @ H @ B4 ) ) ) ) ) ).
% setsum.cong
thf(fact_87_setsum_Oreindex__bij__witness,axiom,
! [B: $tType,A: $tType,C2: $tType] :
( ( comm_monoid_add @ A @ ( type2 @ A ) )
=> ! [S: set @ B,I: C2 > B,J: B > C2,T3: set @ C2,H: C2 > A,G: B > A] :
( ! [A5: B] :
( ( member @ B @ A5 @ S )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ S )
=> ( member @ C2 @ ( J @ A5 ) @ T3 ) )
=> ( ! [B5: C2] :
( ( member @ C2 @ B5 @ T3 )
=> ( ( J @ ( I @ B5 ) )
= B5 ) )
=> ( ! [B5: C2] :
( ( member @ C2 @ B5 @ T3 )
=> ( member @ B @ ( I @ B5 ) @ S ) )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ S )
=> ( ( H @ ( J @ A5 ) )
= ( G @ A5 ) ) )
=> ( ( groups15040474setsum @ B @ A @ G @ S )
= ( groups15040474setsum @ C2 @ A @ H @ T3 ) ) ) ) ) ) ) ) ).
% setsum.reindex_bij_witness
thf(fact_88_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N2: nat] : ( ord_less @ nat @ N2 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_89_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
= ( ord_less @ nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_90_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus @ nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus @ nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_91_lessI,axiom,
! [N: nat] : ( ord_less @ nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_92_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_93_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less @ nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_94_finite__Collect__disjI,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
& ( finite_finite2 @ A @ ( collect @ A @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_95_finite__code,axiom,
! [A: $tType] :
( ( finite_finite @ A @ ( type2 @ A ) )
=> ( ( finite_finite2 @ A )
= ( ^ [A6: set @ A] : $true ) ) ) ).
% finite_code
thf(fact_96_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_97_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_98_finite__Collect__conjI,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
| ( finite_finite2 @ A @ ( collect @ A @ Q ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_99_finite__set__choice,axiom,
! [B: $tType,A: $tType,A4: set @ A,P: A > B > $o] :
( ( finite_finite2 @ A @ A4 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ? [X1: B] : ( P @ X3 @ X1 ) )
=> ? [F3: A > B] :
! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( P @ X4 @ ( F3 @ X4 ) ) ) ) ) ).
% finite_set_choice
thf(fact_100_finite,axiom,
! [A: $tType] :
( ( finite_finite @ A @ ( type2 @ A ) )
=> ! [A4: set @ A] : ( finite_finite2 @ A @ A4 ) ) ).
% finite
thf(fact_101_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_102_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_103_infinite__descent__measure,axiom,
! [A: $tType,P: A > $o,V: A > nat,X: A] :
( ! [X3: A] :
( ~ ( P @ X3 )
=> ? [Y4: A] :
( ( ord_less @ nat @ ( V @ Y4 ) @ ( V @ X3 ) )
& ~ ( P @ Y4 ) ) )
=> ( P @ X ) ) ).
% infinite_descent_measure
thf(fact_104_measure__induct__rule,axiom,
! [A: $tType,F: A > nat,P: A > $o,A2: A] :
( ! [X3: A] :
( ! [Y4: A] :
( ( ord_less @ nat @ ( F @ Y4 ) @ ( F @ X3 ) )
=> ( P @ Y4 ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ).
% measure_induct_rule
thf(fact_105_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less @ nat @ X @ Y )
=> ( ord_less @ nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_106_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M2: nat] :
( ( ord_less @ nat @ M2 @ N3 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_107_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M2: nat] :
( ( ord_less @ nat @ M2 @ N3 )
=> ( P @ M2 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_108_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_109_measure__induct,axiom,
! [A: $tType,F: A > nat,P: A > $o,A2: A] :
( ! [X3: A] :
( ! [Y4: A] :
( ( ord_less @ nat @ ( F @ Y4 ) @ ( F @ X3 ) )
=> ( P @ Y4 ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ).
% measure_induct
thf(fact_110_less__not__refl3,axiom,
! [S2: nat,T2: nat] :
( ( ord_less @ nat @ S2 @ T2 )
=> ( S2 != T2 ) ) ).
% less_not_refl3
thf(fact_111_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_112_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_not_refl
thf(fact_113_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less @ nat @ M @ N )
| ( ord_less @ nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_114_nat__add__right__cancel,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( plus_plus @ nat @ M @ K )
= ( plus_plus @ nat @ N @ K ) )
= ( M = N ) ) ).
% nat_add_right_cancel
thf(fact_115_nat__add__left__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( plus_plus @ nat @ K @ M )
= ( plus_plus @ nat @ K @ N ) )
= ( M = N ) ) ).
% nat_add_left_cancel
thf(fact_116_not__finite__existsD,axiom,
! [A: $tType,P: A > $o] :
( ~ ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ? [X12: A] : ( P @ X12 ) ) ).
% not_finite_existsD
thf(fact_117_pigeonhole__infinite__rel,axiom,
! [B: $tType,A: $tType,A4: set @ A,B4: set @ B,R: A > B > $o] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ? [Xa: B] :
( ( member @ B @ Xa @ B4 )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: B] :
( ( member @ B @ X3 @ B4 )
& ~ ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A3: A] :
( ( member @ A @ A3 @ A4 )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_118_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less @ nat @ N @ M )
=> ( ( ord_less @ nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_119_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less @ nat @ I @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_120_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less @ nat @ I @ J )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J3: nat,K2: nat] :
( ( ord_less @ nat @ I3 @ J3 )
=> ( ( ord_less @ nat @ J3 @ K2 )
=> ( ( P @ I3 @ J3 )
=> ( ( P @ J3 @ K2 )
=> ( P @ I3 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_121_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ K )
=> ( ord_less @ nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_122_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_123_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less @ nat @ N @ M )
=> ( ( ord_less @ nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_124_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( suc @ N ) @ M )
= ( ? [M4: nat] :
( ( M
= ( suc @ M4 ) )
& ( ord_less @ nat @ N @ M4 ) ) ) ) ).
% Suc_less_eq2
thf(fact_125_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less @ nat @ M @ N ) )
= ( ord_less @ nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_126_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( suc @ N ) )
= ( ( ord_less @ nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_127_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_128_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less @ nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_129_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less @ nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_130_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less @ nat @ ( suc @ I ) @ K )
=> ~ ! [J3: nat] :
( ( ord_less @ nat @ I @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_131_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ N )
=> ( ord_less @ nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_132_lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less @ nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J3: nat] :
( ( ord_less @ nat @ I @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ) ).
% lessE
thf(fact_133_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus @ nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus @ nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_134_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus @ nat @ ( suc @ M ) @ N )
= ( plus_plus @ nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_135_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less @ nat @ K @ L )
=> ( ( ( plus_plus @ nat @ M @ L )
= ( plus_plus @ nat @ K @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_136_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ord_less @ nat @ I @ ( plus_plus @ nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_137_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ord_less @ nat @ I @ ( plus_plus @ nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_138_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_139_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_140_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_141_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ K @ L )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_142_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ K )
=> ( ord_less @ nat @ I @ K ) ) ).
% add_lessD1
thf(fact_143_finite__psubset__induct,axiom,
! [A: $tType,A4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A4 )
=> ( ! [A7: set @ A] :
( ( finite_finite2 @ A @ A7 )
=> ( ! [B6: set @ A] :
( ( ord_less @ ( set @ A ) @ B6 @ A7 )
=> ( P @ B6 ) )
=> ( P @ A7 ) ) )
=> ( P @ A4 ) ) ) ).
% finite_psubset_induct
thf(fact_144_lift__Suc__mono__less__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [F: nat > A,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less @ A @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less @ A @ ( F @ N ) @ ( F @ M ) )
= ( ord_less @ nat @ N @ M ) ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_145_lift__Suc__mono__less,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [F: nat > A,N: nat,N6: nat] :
( ! [N3: nat] : ( ord_less @ A @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less @ nat @ N @ N6 )
=> ( ord_less @ A @ ( F @ N ) @ ( F @ N6 ) ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_146_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus @ nat @ M @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_147_less__iff__Suc__add,axiom,
( ( ord_less @ nat )
= ( ^ [M3: nat,N2: nat] :
? [K3: nat] :
( N2
= ( suc @ ( plus_plus @ nat @ M3 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_148_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_149_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_150_infinite__nat__iff__unbounded,axiom,
! [S: set @ nat] :
( ( ~ ( finite_finite2 @ nat @ S ) )
= ( ! [M3: nat] :
? [N2: nat] :
( ( ord_less @ nat @ M3 @ N2 )
& ( member @ nat @ N2 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_151_unbounded__k__infinite,axiom,
! [K: nat,S: set @ nat] :
( ! [M5: nat] :
( ( ord_less @ nat @ K @ M5 )
=> ? [N7: nat] :
( ( ord_less @ nat @ M5 @ N7 )
& ( member @ nat @ N7 @ S ) ) )
=> ~ ( finite_finite2 @ nat @ S ) ) ).
% unbounded_k_infinite
thf(fact_152_lnth__llength__ltakeWhile,axiom,
! [A: $tType,P: A > $o,Xs2: coindu1593790203_llist @ A] :
( ( ord_less @ extended_enat @ ( coindu1018505716length @ A @ ( coindu721411036eWhile @ A @ P @ Xs2 ) ) @ ( coindu1018505716length @ A @ Xs2 ) )
=> ~ ( P @ ( coindu749330388e_lnth @ A @ Xs2 @ ( extended_the_enat @ ( coindu1018505716length @ A @ ( coindu721411036eWhile @ A @ P @ Xs2 ) ) ) ) ) ) ).
% lnth_llength_ltakeWhile
thf(fact_153_lfinite__lfilter,axiom,
! [A: $tType,P: A > $o,Xs2: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ ( coindu1889960678filter @ A @ P @ Xs2 ) )
= ( ( coindu1213758845finite @ A @ Xs2 )
| ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N2: nat] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ N2 ) @ ( coindu1018505716length @ A @ Xs2 ) )
& ( P @ ( coindu749330388e_lnth @ A @ Xs2 @ N2 ) ) ) ) ) ) ) ).
% lfinite_lfilter
thf(fact_154_ldropn__Suc__conv__ldropn,axiom,
! [A: $tType,N: nat,Xs2: coindu1593790203_llist @ A] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ N ) @ ( coindu1018505716length @ A @ Xs2 ) )
=> ( ( coindu1121789889_LCons @ A @ ( coindu749330388e_lnth @ A @ Xs2 @ N ) @ ( coindu531130065ldropn @ A @ ( suc @ N ) @ Xs2 ) )
= ( coindu531130065ldropn @ A @ N @ Xs2 ) ) ) ).
% ldropn_Suc_conv_ldropn
thf(fact_155_llist_Oinject,axiom,
! [A: $tType,X21: A,X222: coindu1593790203_llist @ A,Y21: A,Y222: coindu1593790203_llist @ A] :
( ( ( coindu1121789889_LCons @ A @ X21 @ X222 )
= ( coindu1121789889_LCons @ A @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% llist.inject
thf(fact_156_lfilter__idem,axiom,
! [A: $tType,P: A > $o,Xs2: coindu1593790203_llist @ A] :
( ( coindu1889960678filter @ A @ P @ ( coindu1889960678filter @ A @ P @ Xs2 ) )
= ( coindu1889960678filter @ A @ P @ Xs2 ) ) ).
% lfilter_idem
thf(fact_157_lfilter__K__True,axiom,
! [A: $tType,Xs2: coindu1593790203_llist @ A] :
( ( coindu1889960678filter @ A
@ ^ [Uu: A] : $true
@ Xs2 )
= Xs2 ) ).
% lfilter_K_True
thf(fact_158_LCons__lprefix__LCons,axiom,
! [A: $tType,X: A,Xs2: coindu1593790203_llist @ A,Y: A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ ( coindu1121789889_LCons @ A @ X @ Xs2 ) @ ( coindu1121789889_LCons @ A @ Y @ Ys ) )
= ( ( X = Y )
& ( coindu1696667936prefix @ A @ Xs2 @ Ys ) ) ) ).
% LCons_lprefix_LCons
thf(fact_159_lfinite__LCons,axiom,
! [A: $tType,X: A,Xs2: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ ( coindu1121789889_LCons @ A @ X @ Xs2 ) )
= ( coindu1213758845finite @ A @ Xs2 ) ) ).
% lfinite_LCons
thf(fact_160_lfinite__code_I2_J,axiom,
! [B: $tType,X: B,Xs2: coindu1593790203_llist @ B] :
( ( coindu1213758845finite @ B @ ( coindu1121789889_LCons @ B @ X @ Xs2 ) )
= ( coindu1213758845finite @ B @ Xs2 ) ) ).
% lfinite_code(2)
thf(fact_161_lfilter__LCons,axiom,
! [A: $tType,P: A > $o,X: A,Xs2: coindu1593790203_llist @ A] :
( ( ( P @ X )
=> ( ( coindu1889960678filter @ A @ P @ ( coindu1121789889_LCons @ A @ X @ Xs2 ) )
= ( coindu1121789889_LCons @ A @ X @ ( coindu1889960678filter @ A @ P @ Xs2 ) ) ) )
& ( ~ ( P @ X )
=> ( ( coindu1889960678filter @ A @ P @ ( coindu1121789889_LCons @ A @ X @ Xs2 ) )
= ( coindu1889960678filter @ A @ P @ Xs2 ) ) ) ) ).
% lfilter_LCons
thf(fact_162_lnth__Suc__LCons,axiom,
! [A: $tType,X: A,Xs2: coindu1593790203_llist @ A,N: nat] :
( ( coindu749330388e_lnth @ A @ ( coindu1121789889_LCons @ A @ X @ Xs2 ) @ ( suc @ N ) )
= ( coindu749330388e_lnth @ A @ Xs2 @ N ) ) ).
% lnth_Suc_LCons
thf(fact_163_ldropn__Suc__LCons,axiom,
! [A: $tType,N: nat,X: A,Xs2: coindu1593790203_llist @ A] :
( ( coindu531130065ldropn @ A @ ( suc @ N ) @ ( coindu1121789889_LCons @ A @ X @ Xs2 ) )
= ( coindu531130065ldropn @ A @ N @ Xs2 ) ) ).
% ldropn_Suc_LCons
thf(fact_164_lfilter__LCons__found,axiom,
! [A: $tType,P: A > $o,X: A,Xs2: coindu1593790203_llist @ A] :
( ( P @ X )
=> ( ( coindu1889960678filter @ A @ P @ ( coindu1121789889_LCons @ A @ X @ Xs2 ) )
= ( coindu1121789889_LCons @ A @ X @ ( coindu1889960678filter @ A @ P @ Xs2 ) ) ) ) ).
% lfilter_LCons_found
thf(fact_165_lfilter__LCons__seek,axiom,
! [A: $tType,P2: A > $o,X: A,L: coindu1593790203_llist @ A] :
( ~ ( P2 @ X )
=> ( ( coindu1889960678filter @ A @ P2 @ ( coindu1121789889_LCons @ A @ X @ L ) )
= ( coindu1889960678filter @ A @ P2 @ L ) ) ) ).
% lfilter_LCons_seek
thf(fact_166_lfilter__lfilter,axiom,
! [A: $tType,P: A > $o,Q: A > $o,Xs2: coindu1593790203_llist @ A] :
( ( coindu1889960678filter @ A @ P @ ( coindu1889960678filter @ A @ Q @ Xs2 ) )
= ( coindu1889960678filter @ A
@ ^ [X2: A] :
( ( P @ X2 )
& ( Q @ X2 ) )
@ Xs2 ) ) ).
% lfilter_lfilter
thf(fact_167_lprefix__lfilterI,axiom,
! [A: $tType,Xs2: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,P: A > $o] :
( ( coindu1696667936prefix @ A @ Xs2 @ Ys )
=> ( coindu1696667936prefix @ A @ ( coindu1889960678filter @ A @ P @ Xs2 ) @ ( coindu1889960678filter @ A @ P @ Ys ) ) ) ).
% lprefix_lfilterI
thf(fact_168_LCons__lprefix__conv,axiom,
! [A: $tType,X: A,Xs2: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ ( coindu1121789889_LCons @ A @ X @ Xs2 ) @ Ys )
= ( ? [Ys2: coindu1593790203_llist @ A] :
( ( Ys
= ( coindu1121789889_LCons @ A @ X @ Ys2 ) )
& ( coindu1696667936prefix @ A @ Xs2 @ Ys2 ) ) ) ) ).
% LCons_lprefix_conv
thf(fact_169_Le__LCons,axiom,
! [A: $tType,Xs2: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,X: A] :
( ( coindu1696667936prefix @ A @ Xs2 @ Ys )
=> ( coindu1696667936prefix @ A @ ( coindu1121789889_LCons @ A @ X @ Xs2 ) @ ( coindu1121789889_LCons @ A @ X @ Ys ) ) ) ).
% Le_LCons
thf(fact_170_lfinite__lfilterI,axiom,
! [A: $tType,Xs2: coindu1593790203_llist @ A,P: A > $o] :
( ( coindu1213758845finite @ A @ Xs2 )
=> ( coindu1213758845finite @ A @ ( coindu1889960678filter @ A @ P @ Xs2 ) ) ) ).
% lfinite_lfilterI
thf(fact_171_lfinite__LConsI,axiom,
! [A: $tType,Xs2: coindu1593790203_llist @ A,X: A] :
( ( coindu1213758845finite @ A @ Xs2 )
=> ( coindu1213758845finite @ A @ ( coindu1121789889_LCons @ A @ X @ Xs2 ) ) ) ).
% lfinite_LConsI
thf(fact_172_lmap__eq__LCons__conv,axiom,
! [A: $tType,B: $tType,F: B > A,Xs2: coindu1593790203_llist @ B,Y: A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu1062782156e_lmap @ B @ A @ F @ Xs2 )
= ( coindu1121789889_LCons @ A @ Y @ Ys ) )
= ( ? [X2: B,Xs3: coindu1593790203_llist @ B] :
( ( Xs2
= ( coindu1121789889_LCons @ B @ X2 @ Xs3 ) )
& ( Y
= ( F @ X2 ) )
& ( Ys
= ( coindu1062782156e_lmap @ B @ A @ F @ Xs3 ) ) ) ) ) ).
% lmap_eq_LCons_conv
thf(fact_173_llist_Osimps_I13_J,axiom,
! [B: $tType,A: $tType,F: A > B,X21: A,X222: coindu1593790203_llist @ A] :
( ( coindu1062782156e_lmap @ A @ B @ F @ ( coindu1121789889_LCons @ A @ X21 @ X222 ) )
= ( coindu1121789889_LCons @ B @ ( F @ X21 ) @ ( coindu1062782156e_lmap @ A @ B @ F @ X222 ) ) ) ).
% llist.simps(13)
thf(fact_174_ldistinct__lfilterI,axiom,
! [A: $tType,Xs2: coindu1593790203_llist @ A,P: A > $o] :
( ( coindu6345450stinct @ A @ Xs2 )
=> ( coindu6345450stinct @ A @ ( coindu1889960678filter @ A @ P @ Xs2 ) ) ) ).
% ldistinct_lfilterI
thf(fact_175_the__enat_Osimps,axiom,
! [N: nat] :
( ( extended_the_enat @ ( extended_enat2 @ N ) )
= N ) ).
% the_enat.simps
thf(fact_176_ldropn__eq__LConsD,axiom,
! [A: $tType,N: nat,Xs2: coindu1593790203_llist @ A,Y: A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu531130065ldropn @ A @ N @ Xs2 )
= ( coindu1121789889_LCons @ A @ Y @ Ys ) )
=> ( ord_less @ extended_enat @ ( extended_enat2 @ N ) @ ( coindu1018505716length @ A @ Xs2 ) ) ) ).
% ldropn_eq_LConsD
thf(fact_177_ldistinct__lfilterD,axiom,
! [A: $tType,P: A > $o,Xs2: coindu1593790203_llist @ A,N: nat,M: nat,A2: A] :
( ( coindu6345450stinct @ A @ ( coindu1889960678filter @ A @ P @ Xs2 ) )
=> ( ( ord_less @ extended_enat @ ( extended_enat2 @ N ) @ ( coindu1018505716length @ A @ Xs2 ) )
=> ( ( ord_less @ extended_enat @ ( extended_enat2 @ M ) @ ( coindu1018505716length @ A @ Xs2 ) )
=> ( ( P @ A2 )
=> ( ( ( coindu749330388e_lnth @ A @ Xs2 @ N )
= A2 )
=> ( ( ( coindu749330388e_lnth @ A @ Xs2 @ M )
= A2 )
=> ( M = N ) ) ) ) ) ) ) ).
% ldistinct_lfilterD
thf(fact_178_lnth__ldrop,axiom,
! [A: $tType,N: extended_enat,M: nat,Xs2: coindu1593790203_llist @ A] :
( ( ord_less @ extended_enat @ ( plus_plus @ extended_enat @ N @ ( extended_enat2 @ M ) ) @ ( coindu1018505716length @ A @ Xs2 ) )
=> ( ( coindu749330388e_lnth @ A @ ( coindu191418589_ldrop @ A @ N @ Xs2 ) @ M )
= ( coindu749330388e_lnth @ A @ Xs2 @ ( plus_plus @ nat @ M @ ( extended_the_enat @ N ) ) ) ) ) ).
% lnth_ldrop
thf(fact_179_llast__ldropn,axiom,
! [A: $tType,N: nat,Xs2: coindu1593790203_llist @ A] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ N ) @ ( coindu1018505716length @ A @ Xs2 ) )
=> ( ( coindu2000965700_llast @ A @ ( coindu531130065ldropn @ A @ N @ Xs2 ) )
= ( coindu2000965700_llast @ A @ Xs2 ) ) ) ).
% llast_ldropn
thf(fact_180_lhd__ldropn,axiom,
! [A: $tType,N: nat,Xs2: coindu1593790203_llist @ A] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ N ) @ ( coindu1018505716length @ A @ Xs2 ) )
=> ( ( coindu1046438764le_lhd @ A @ ( coindu531130065ldropn @ A @ N @ Xs2 ) )
= ( coindu749330388e_lnth @ A @ Xs2 @ N ) ) ) ).
% lhd_ldropn
thf(fact_181_ldrop__lmap,axiom,
! [A: $tType,B: $tType,N: extended_enat,F: B > A,Xs2: coindu1593790203_llist @ B] :
( ( coindu191418589_ldrop @ A @ N @ ( coindu1062782156e_lmap @ B @ A @ F @ Xs2 ) )
= ( coindu1062782156e_lmap @ B @ A @ F @ ( coindu191418589_ldrop @ B @ N @ Xs2 ) ) ) ).
% ldrop_lmap
thf(fact_182_ldrop__ldrop,axiom,
! [A: $tType,N: extended_enat,M: extended_enat,Xs2: coindu1593790203_llist @ A] :
( ( coindu191418589_ldrop @ A @ N @ ( coindu191418589_ldrop @ A @ M @ Xs2 ) )
= ( coindu191418589_ldrop @ A @ ( plus_plus @ extended_enat @ N @ M ) @ Xs2 ) ) ).
% ldrop_ldrop
thf(fact_183_llast__LCons2,axiom,
! [A: $tType,X: A,Y: A,Xs2: coindu1593790203_llist @ A] :
( ( coindu2000965700_llast @ A @ ( coindu1121789889_LCons @ A @ X @ ( coindu1121789889_LCons @ A @ Y @ Xs2 ) ) )
= ( coindu2000965700_llast @ A @ ( coindu1121789889_LCons @ A @ Y @ Xs2 ) ) ) ).
% llast_LCons2
thf(fact_184_lhd__LCons,axiom,
! [A: $tType,X21: A,X222: coindu1593790203_llist @ A] :
( ( coindu1046438764le_lhd @ A @ ( coindu1121789889_LCons @ A @ X21 @ X222 ) )
= X21 ) ).
% lhd_LCons
thf(fact_185_llast__ldrop,axiom,
! [A: $tType,N: extended_enat,Xs2: coindu1593790203_llist @ A] :
( ( ord_less @ extended_enat @ N @ ( coindu1018505716length @ A @ Xs2 ) )
=> ( ( coindu2000965700_llast @ A @ ( coindu191418589_ldrop @ A @ N @ Xs2 ) )
= ( coindu2000965700_llast @ A @ Xs2 ) ) ) ).
% llast_ldrop
thf(fact_186_ldistinct__ldrop,axiom,
! [A: $tType,Xs2: coindu1593790203_llist @ A,N: extended_enat] :
( ( coindu6345450stinct @ A @ Xs2 )
=> ( coindu6345450stinct @ A @ ( coindu191418589_ldrop @ A @ N @ Xs2 ) ) ) ).
% ldistinct_ldrop
thf(fact_187_lhd__ldrop,axiom,
! [A: $tType,N: extended_enat,Xs2: coindu1593790203_llist @ A] :
( ( ord_less @ extended_enat @ N @ ( coindu1018505716length @ A @ Xs2 ) )
=> ( ( coindu1046438764le_lhd @ A @ ( coindu191418589_ldrop @ A @ N @ Xs2 ) )
= ( coindu749330388e_lnth @ A @ Xs2 @ ( extended_the_enat @ N ) ) ) ) ).
% lhd_ldrop
thf(fact_188_ldrop__enat,axiom,
! [A: $tType,N: nat,Xs2: coindu1593790203_llist @ A] :
( ( coindu191418589_ldrop @ A @ ( extended_enat2 @ N ) @ Xs2 )
= ( coindu531130065ldropn @ A @ N @ Xs2 ) ) ).
% ldrop_enat
thf(fact_189_ldrop__eq__LConsD,axiom,
! [A: $tType,N: extended_enat,Xs2: coindu1593790203_llist @ A,Y: A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu191418589_ldrop @ A @ N @ Xs2 )
= ( coindu1121789889_LCons @ A @ Y @ Ys ) )
=> ( ord_less @ extended_enat @ N @ ( coindu1018505716length @ A @ Xs2 ) ) ) ).
% ldrop_eq_LConsD
thf(fact_190_Coinductive__List__Mirabelle__kmikjhschf_Olprefix__nitpick__simps,axiom,
! [A: $tType] :
( ( coindu1696667936prefix @ A )
= ( ^ [Xs: coindu1593790203_llist @ A,Ys3: coindu1593790203_llist @ A] :
( ( ( coindu1213758845finite @ A @ Xs )
=> ( coindu1571841457prefix @ A @ Xs @ Ys3 ) )
& ( ~ ( coindu1213758845finite @ A @ Xs )
=> ( Xs = Ys3 ) ) ) ) ) ).
% Coinductive_List_Mirabelle_kmikjhschf.lprefix_nitpick_simps
thf(fact_191_enumerate__mono,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [M: nat,N: nat,S: set @ A] :
( ( ord_less @ nat @ M @ N )
=> ( ~ ( finite_finite2 @ A @ S )
=> ( ord_less @ A @ ( infini455366010merate @ A @ S @ M ) @ ( infini455366010merate @ A @ S @ N ) ) ) ) ) ).
% enumerate_mono
thf(fact_192_setsum__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A @ ( type2 @ A ) )
=> ! [F: nat > A,N: nat] :
( ( groups15040474setsum @ nat @ A @ F @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( F @ ( zero_zero @ nat ) )
@ ( groups15040474setsum @ nat @ A
@ ^ [I2: nat] : ( F @ ( suc @ I2 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% setsum_lessThan_Suc_shift
thf(fact_193_not__gr__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% not_gr_zero
thf(fact_194_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_195_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_196_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_197_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_198_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_199_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_200_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_201_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_202_neq0__conv,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% neq0_conv
thf(fact_203_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% Nat.add_0_right
thf(fact_204_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_205_ldropn__0,axiom,
! [A: $tType,Xs2: coindu1593790203_llist @ A] :
( ( coindu531130065ldropn @ A @ ( zero_zero @ nat ) @ Xs2 )
= Xs2 ) ).
% ldropn_0
thf(fact_206_setsum_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A @ ( type2 @ A ) )
=> ! [A4: set @ B] :
( ( groups15040474setsum @ B @ A
@ ^ [Uu: B] : ( zero_zero @ A )
@ A4 )
= ( zero_zero @ A ) ) ) ).
% setsum.neutral_const
thf(fact_207_add__less__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel1
thf(fact_208_add__less__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel2
thf(fact_209_less__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% less_add_same_cancel1
thf(fact_210_less__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% less_add_same_cancel2
thf(fact_211_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_212_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_213_setsum__eq__0__iff,axiom,
! [A: $tType,B: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [F4: set @ B,F: B > A] :
( ( finite_finite2 @ B @ F4 )
=> ( ( ( groups15040474setsum @ B @ A @ F @ F4 )
= ( zero_zero @ A ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ F4 )
=> ( ( F @ X2 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% setsum_eq_0_iff
thf(fact_214_setsum_Oinfinite,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A @ ( type2 @ A ) )
=> ! [A4: set @ B,G: B > A] :
( ~ ( finite_finite2 @ B @ A4 )
=> ( ( groups15040474setsum @ B @ A @ G @ A4 )
= ( zero_zero @ A ) ) ) ) ).
% setsum.infinite
thf(fact_215_less__Suc0,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_Suc0
thf(fact_216_zero__less__Suc,axiom,
! [N: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_217_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ M @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
| ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% add_gr_0
thf(fact_218_lnth__0,axiom,
! [A: $tType,X: A,Xs2: coindu1593790203_llist @ A] :
( ( coindu749330388e_lnth @ A @ ( coindu1121789889_LCons @ A @ X @ Xs2 ) @ ( zero_zero @ nat ) )
= X ) ).
% lnth_0
thf(fact_219_enumerate__in__set,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [S: set @ A,N: nat] :
( ~ ( finite_finite2 @ A @ S )
=> ( member @ A @ ( infini455366010merate @ A @ S @ N ) @ S ) ) ) ).
% enumerate_in_set
thf(fact_220_enumerate__Ex,axiom,
! [S: set @ nat,S2: nat] :
( ~ ( finite_finite2 @ nat @ S )
=> ( ( member @ nat @ S2 @ S )
=> ? [N3: nat] :
( ( infini455366010merate @ nat @ S @ N3 )
= S2 ) ) ) ).
% enumerate_Ex
thf(fact_221_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_222_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% plus_nat.add_0
thf(fact_223_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_224_gr0I,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% gr0I
thf(fact_225_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% not_gr0
thf(fact_226_not__less0,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% not_less0
thf(fact_227_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_zeroE
thf(fact_228_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( N
!= ( zero_zero @ nat ) ) ) ).
% gr_implies_not0
thf(fact_229_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M2: nat] :
( ( ord_less @ nat @ M2 @ N3 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_230_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_nat_zero_code
thf(fact_231_infinite__descent0__measure,axiom,
! [A: $tType,V: A > nat,P: A > $o,X: A] :
( ! [X3: A] :
( ( ( V @ X3 )
= ( zero_zero @ nat ) )
=> ( P @ X3 ) )
=> ( ! [X3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X3 ) )
=> ( ~ ( P @ X3 )
=> ? [Y4: A] :
( ( ord_less @ nat @ ( V @ Y4 ) @ ( V @ X3 ) )
& ~ ( P @ Y4 ) ) ) )
=> ( P @ X ) ) ) ).
% infinite_descent0_measure
thf(fact_232_not0__implies__Suc,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% not0_implies_Suc
thf(fact_233_old_Onat_Oinducts,axiom,
! [P: nat > $o,Nat: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [Nat3: nat] :
( ( P @ Nat3 )
=> ( P @ ( suc @ Nat3 ) ) )
=> ( P @ Nat ) ) ) ).
% old.nat.inducts
thf(fact_234_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y
!= ( zero_zero @ nat ) )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_235_Zero__not__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_236_Zero__neq__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_237_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= ( zero_zero @ nat ) ) ).
% Suc_neq_Zero
thf(fact_238_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( zero_zero @ nat ) ) ) ) ).
% zero_induct
thf(fact_239_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ ( zero_zero @ nat ) )
=> ( ! [Y5: nat] : ( P @ ( zero_zero @ nat ) @ ( suc @ Y5 ) )
=> ( ! [X3: nat,Y5: nat] :
( ( P @ X3 @ Y5 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y5 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_240_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_241_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat
!= ( zero_zero @ nat ) ) ) ).
% nat.discI
thf(fact_242_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( ( zero_zero @ nat )
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_243_old_Onat_Odistinct_I2_J,axiom,
! [Nat4: nat] :
( ( suc @ Nat4 )
!= ( zero_zero @ nat ) ) ).
% old.nat.distinct(2)
thf(fact_244_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( ( zero_zero @ nat )
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_245_setsum_Onot__neutral__contains__not__neutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A @ ( type2 @ A ) )
=> ! [G: B > A,A4: set @ B] :
( ( ( groups15040474setsum @ B @ A @ G @ A4 )
!= ( zero_zero @ A ) )
=> ~ ! [A5: B] :
( ( member @ B @ A5 @ A4 )
=> ( ( G @ A5 )
= ( zero_zero @ A ) ) ) ) ) ).
% setsum.not_neutral_contains_not_neutral
thf(fact_246_setsum_Oneutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A @ ( type2 @ A ) )
=> ! [A4: set @ B,G: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups15040474setsum @ B @ A @ G @ A4 )
= ( zero_zero @ A ) ) ) ) ).
% setsum.neutral
thf(fact_247_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_248_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_249_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_250_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_251_zero__less__iff__neq__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ N )
= ( N
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_iff_neq_zero
thf(fact_252_gr__implies__not__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [M: A,N: A] :
( ( ord_less @ A @ M @ N )
=> ( N
!= ( zero_zero @ A ) ) ) ) ).
% gr_implies_not_zero
%----Type constructors (39)
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A8: $tType,A9: $tType] :
( ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
& ( finite_finite @ A9 @ ( type2 @ A9 ) ) )
=> ( finite_finite @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A8: $tType,A9: $tType] :
( ( order @ A9 @ ( type2 @ A9 ) )
=> ( order @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A8: $tType,A9: $tType] :
( ( ord @ A9 @ ( type2 @ A9 ) )
=> ( ord @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
ordere516151231imp_le @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
strict2144017051up_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere223160158up_add @ 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_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___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___Groups_Osemigroup__add,axiom,
semigroup_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ 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___Orderings_Oorder_1,axiom,
order @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oord_2,axiom,
ord @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ozero,axiom,
zero @ nat @ ( type2 @ nat ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_3,axiom,
! [A8: $tType] :
( ( finite_finite @ A8 @ ( type2 @ A8 ) )
=> ( finite_finite @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_4,axiom,
! [A8: $tType] : ( order @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_5,axiom,
! [A8: $tType] : ( ord @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder_6,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_7,axiom,
finite_finite @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_8,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_9,axiom,
ord @ $o @ ( type2 @ $o ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_10,axiom,
strict2144017051up_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_11,axiom,
canoni770627133id_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_12,axiom,
ordere779506340up_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_13,axiom,
ab_semigroup_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_14,axiom,
comm_monoid_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_15,axiom,
semigroup_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Owellorder_16,axiom,
wellorder @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_17,axiom,
linorder @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_18,axiom,
monoid_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_19,axiom,
order @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oord_20,axiom,
ord @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Ozero_21,axiom,
zero @ extended_enat @ ( type2 @ extended_enat ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
? [M2: nat,N8: nat] :
( ( ( coindu749330388e_lnth @ a @ ( coindu441856546concat @ a @ xss ) @ n )
= ( coindu749330388e_lnth @ a @ ( coindu749330388e_lnth @ ( coindu1593790203_llist @ a ) @ xss @ M2 ) @ N8 ) )
& ( ord_less @ extended_enat @ ( extended_enat2 @ N8 ) @ ( coindu1018505716length @ a @ ( coindu749330388e_lnth @ ( coindu1593790203_llist @ a ) @ xss @ M2 ) ) )
& ( ord_less @ extended_enat @ ( extended_enat2 @ M2 ) @ ( coindu1018505716length @ ( coindu1593790203_llist @ a ) @ xss ) )
& ( ( extended_enat2 @ n )
= ( plus_plus @ extended_enat
@ ( groups15040474setsum @ nat @ extended_enat
@ ^ [I2: nat] : ( coindu1018505716length @ a @ ( coindu749330388e_lnth @ ( coindu1593790203_llist @ a ) @ xss @ I2 ) )
@ ( set_ord_lessThan @ nat @ M2 ) )
@ ( extended_enat2 @ N8 ) ) ) ) ).
%------------------------------------------------------------------------------