TPTP Problem File: DAT133^1.p
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%------------------------------------------------------------------------------
% File : DAT133^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Coinductive list 4159
% 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__4159.p [Bla16]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.25 v7.5.0, 0.00 v7.2.0, 0.25 v7.1.0
% Syntax : Number of formulae : 335 ( 110 unt; 54 typ; 0 def)
% Number of atoms : 699 ( 214 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 3529 ( 90 ~; 11 |; 54 &;3076 @)
% ( 0 <=>; 298 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 8 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 214 ( 214 >; 0 *; 0 +; 0 <<)
% Number of symbols : 53 ( 51 usr; 4 con; 0-4 aty)
% Number of variables : 987 ( 80 ^; 828 !; 36 ?; 987 :)
% ( 43 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 15:05:08.605
%------------------------------------------------------------------------------
%----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 (48)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oone,type,
one:
!>[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_Lattices_Olattice,type,
lattice:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Osemilattice__inf,type,
semilattice_inf:
!>[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_Oordered__ab__semigroup__add,type,
ordere779506340up_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_c_Coinductive__List__Mirabelle__kmikjhschf_Ogen__llength,type,
coindu493225792length:
!>[A: $tType] : ( nat > ( coindu1593790203_llist @ A ) > extended_enat ) ).
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_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_Olhd,type,
coindu1046438764le_lhd:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > A ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Olnull,type,
coindu335574135_lnull:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Olset,type,
coindu1112613586e_lset:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( set @ A ) ) ).
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_Olnth,type,
coindu749330388e_lnth:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > nat > 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_Oenat__unfold,type,
coindu1491768222unfold:
!>[A: $tType] : ( ( A > $o ) > ( A > A ) > A > extended_enat ) ).
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_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
thf(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Lattices_Oinf__class_Oinf,type,
inf_inf:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Oimage,type,
image:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_P,type,
p: a > $o ).
thf(sy_v_n____,type,
n: nat ).
thf(sy_v_xs,type,
xs: coindu1593790203_llist @ a ).
thf(sy_v_xsa____,type,
xsa: coindu1593790203_llist @ a ).
%----Relevant facts (253)
thf(fact_0__092_060open_062finite_A_I_123n_O_Aenat_An_A_060_Allength_A_Iltl_A_IldropWhile_A_I_092_060lambda_062x_O_A_092_060not_062_AP_Ax_J_Axs_J_J_125_A_092_060inter_062_A_123n_O_AP_A_Ilnth_A_Iltl_A_IldropWhile_A_I_092_060lambda_062x_O_A_092_060not_062_AP_Ax_J_Axs_J_J_An_J_125_J_092_060close_062,axiom,
( finite_finite2 @ nat
@ ( inf_inf @ ( set @ nat )
@ ( collect @ nat
@ ^ [N: nat] :
( ord_less @ extended_enat @ ( extended_enat2 @ N )
@ ( coindu1018505716length @ a
@ ( coindu1047225960le_ltl @ a
@ ( coindu438612276pWhile @ a
@ ^ [X: a] :
~ ( p @ X )
@ xsa ) ) ) ) )
@ ( collect @ nat
@ ^ [N: nat] :
( p
@ ( coindu749330388e_lnth @ a
@ ( coindu1047225960le_ltl @ a
@ ( coindu438612276pWhile @ a
@ ^ [X: a] :
~ ( p @ X )
@ xsa ) )
@ N ) ) ) ) ) ).
% \<open>finite ({n. enat n < llength (ltl (ldropWhile (\<lambda>x. \<not> P x) xs))} \<inter> {n. P (lnth (ltl (ldropWhile (\<lambda>x. \<not> P x) xs)) n)})\<close>
thf(fact_1_LCons_Oprems,axiom,
~ ( coindu1213758845finite @ a @ xsa ) ).
% LCons.prems
thf(fact_2_n,axiom,
( ( coindu1018505716length @ a
@ ( coindu721411036eWhile @ a
@ ^ [X: a] :
~ ( p @ X )
@ xsa ) )
= ( extended_enat2 @ n ) ) ).
% n
thf(fact_3_finite__Int,axiom,
! [A: $tType,F: set @ A,G: set @ A] :
( ( ( finite_finite2 @ A @ F )
| ( finite_finite2 @ A @ G ) )
=> ( finite_finite2 @ A @ ( inf_inf @ ( set @ A ) @ F @ G ) ) ) ).
% finite_Int
thf(fact_4_finite__imageI,axiom,
! [B: $tType,A: $tType,F: set @ A,H: A > B] :
( ( finite_finite2 @ A @ F )
=> ( finite_finite2 @ B @ ( image @ A @ B @ H @ F ) ) ) ).
% finite_imageI
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__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062n_O_Allength_A_IltakeWhile_A_I_092_060lambda_062x_O_A_092_060not_062_AP_Ax_J_Axs_J_A_061_Aenat_An_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [N2: nat] :
( ( coindu1018505716length @ a
@ ( coindu721411036eWhile @ a
@ ^ [X: a] :
~ ( p @ X )
@ xsa ) )
!= ( extended_enat2 @ N2 ) ) ).
% \<open>\<And>thesis. (\<And>n. llength (ltakeWhile (\<lambda>x. \<not> P x) xs) = enat n \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_8_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
@ ^ [X: A] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_9_finite__Collect__disjI,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
& ( finite_finite2 @ A @ ( collect @ A @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_10_image__ident,axiom,
! [A: $tType,Y: set @ A] :
( ( image @ A @ A
@ ^ [X: A] : X
@ Y )
= Y ) ).
% image_ident
thf(fact_11_less__add__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) ) ) ).
% less_add_one
thf(fact_12_exP,axiom,
? [X2: a] :
( ( member @ a @ X2 @ ( coindu1112613586e_lset @ a @ xsa ) )
& ( p @ X2 ) ) ).
% exP
thf(fact_13_enat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( extended_enat2 @ Nat )
= ( extended_enat2 @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% enat.inject
thf(fact_14_LCons_Ohyps_I3_J,axiom,
! [Xs: coindu1593790203_llist @ a] :
( ( ( coindu1047225960le_ltl @ a @ ( coindu1889960678filter @ a @ p @ xsa ) )
= ( coindu1889960678filter @ a @ p @ Xs ) )
=> ( ~ ( coindu1213758845finite @ a @ Xs )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N: nat] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ N ) @ ( coindu1018505716length @ a @ Xs ) )
& ( p @ ( coindu749330388e_lnth @ a @ Xs @ N ) ) ) ) ) ) ) ).
% LCons.hyps(3)
thf(fact_15__092_060open_062_092_060not_062_Alfinite_Axs_092_060close_062,axiom,
~ ( coindu1213758845finite @ a @ xs ) ).
% \<open>\<not> lfinite xs\<close>
thf(fact_16_LCons_Ohyps_I1_J,axiom,
coindu1213758845finite @ a @ ( coindu1889960678filter @ a @ p @ xsa ) ).
% LCons.hyps(1)
thf(fact_17__092_060open_062lfinite_A_Ilfilter_AP_Axs_J_092_060close_062,axiom,
coindu1213758845finite @ a @ ( coindu1889960678filter @ a @ p @ xs ) ).
% \<open>lfinite (lfilter P xs)\<close>
thf(fact_18_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_19_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_20_image__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F2: B > A,X3: B,A3: set @ B] :
( ( B2
= ( F2 @ X3 ) )
=> ( ( member @ B @ X3 @ A3 )
=> ( member @ A @ B2 @ ( image @ B @ A @ F2 @ A3 ) ) ) ) ).
% image_eqI
thf(fact_21_finite__code,axiom,
! [A: $tType] :
( ( finite_finite @ A @ ( type2 @ A ) )
=> ( ( finite_finite2 @ A )
= ( ^ [A4: set @ A] : $true ) ) ) ).
% finite_code
thf(fact_22_Int__iff,axiom,
! [A: $tType,C: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) )
= ( ( member @ A @ C @ A3 )
& ( member @ A @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_23_IntI,axiom,
! [A: $tType,C: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C @ A3 )
=> ( ( member @ A @ C @ B3 )
=> ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ) ) ).
% IntI
thf(fact_24_lfilter__idem,axiom,
! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
( ( coindu1889960678filter @ A @ P @ ( coindu1889960678filter @ A @ P @ Xs ) )
= ( coindu1889960678filter @ A @ P @ Xs ) ) ).
% lfilter_idem
thf(fact_25_ltakeWhile__K__True,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu721411036eWhile @ A
@ ^ [Uu: A] : $true
@ Xs )
= Xs ) ).
% ltakeWhile_K_True
thf(fact_26_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N: nat] : ( ord_less @ nat @ N @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_27_lfilter__K__True,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu1889960678filter @ A
@ ^ [Uu: A] : $true
@ Xs )
= Xs ) ).
% lfilter_K_True
thf(fact_28_plus__enat__simps_I1_J,axiom,
! [M: nat,N3: nat] :
( ( plus_plus @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N3 ) )
= ( extended_enat2 @ ( plus_plus @ nat @ M @ N3 ) ) ) ).
% plus_enat_simps(1)
thf(fact_29_lfinite__ltl,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ ( coindu1047225960le_ltl @ A @ Xs ) )
= ( coindu1213758845finite @ A @ Xs ) ) ).
% lfinite_ltl
thf(fact_30_enat__ord__simps_I2_J,axiom,
! [M: nat,N3: nat] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N3 ) )
= ( ord_less @ nat @ M @ N3 ) ) ).
% enat_ord_simps(2)
thf(fact_31_lset__lfilter,axiom,
! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
( ( coindu1112613586e_lset @ A @ ( coindu1889960678filter @ A @ P @ Xs ) )
= ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
& ( P @ X ) ) ) ) ).
% lset_lfilter
thf(fact_32_LCons_Ohyps_I2_J,axiom,
~ ( coindu335574135_lnull @ a @ ( coindu1889960678filter @ a @ p @ xsa ) ) ).
% LCons.hyps(2)
thf(fact_33_lfilter__cong,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,P: A > $o,Q: A > $o] :
( ( Xs = Ys )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ ( coindu1112613586e_lset @ A @ Ys ) )
=> ( ( P @ X2 )
= ( Q @ X2 ) ) )
=> ( ( coindu1889960678filter @ A @ P @ Xs )
= ( coindu1889960678filter @ A @ Q @ Ys ) ) ) ) ).
% lfilter_cong
thf(fact_34_ltakeWhile__all,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
( ! [X2: A] :
( ( member @ A @ X2 @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ( P @ X2 ) )
=> ( ( coindu721411036eWhile @ A @ P @ Xs )
= Xs ) ) ).
% ltakeWhile_all
thf(fact_35_ltakeWhile__cong,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,P: A > $o,Q: A > $o] :
( ( Xs = Ys )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ ( coindu1112613586e_lset @ A @ Ys ) )
=> ( ( P @ X2 )
= ( Q @ X2 ) ) )
=> ( ( coindu721411036eWhile @ A @ P @ Xs )
= ( coindu721411036eWhile @ A @ Q @ Ys ) ) ) ) ).
% ltakeWhile_cong
thf(fact_36_lfinite__lfilterI,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
( ( coindu1213758845finite @ A @ Xs )
=> ( coindu1213758845finite @ A @ ( coindu1889960678filter @ A @ P @ Xs ) ) ) ).
% lfinite_lfilterI
thf(fact_37_lset__ltakeWhileD,axiom,
! [A: $tType,X3: A,P: A > $o,Xs: coindu1593790203_llist @ A] :
( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) )
=> ( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ Xs ) )
& ( P @ X3 ) ) ) ).
% lset_ltakeWhileD
thf(fact_38_lfinite__ltakeWhile,axiom,
! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) )
= ( ( coindu1213758845finite @ A @ Xs )
| ? [X: A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
& ~ ( P @ X ) ) ) ) ).
% lfinite_ltakeWhile
thf(fact_39_lfinite__imp__finite__lset,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ Xs )
=> ( finite_finite2 @ A @ ( coindu1112613586e_lset @ A @ Xs ) ) ) ).
% lfinite_imp_finite_lset
thf(fact_40_lfinite__ldropWhile,axiom,
! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ ( coindu438612276pWhile @ A @ P @ Xs ) )
= ( ? [X: A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
& ~ ( P @ X ) )
=> ( coindu1213758845finite @ A @ Xs ) ) ) ).
% lfinite_ldropWhile
thf(fact_41_finite__psubset__induct,axiom,
! [A: $tType,A3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [B4: set @ A] :
( ( ord_less @ ( set @ A ) @ B4 @ A5 )
=> ( P @ B4 ) )
=> ( P @ A5 ) ) )
=> ( P @ A3 ) ) ) ).
% finite_psubset_induct
thf(fact_42_lfilter__lfilter,axiom,
! [A: $tType,P: A > $o,Q: A > $o,Xs: coindu1593790203_llist @ A] :
( ( coindu1889960678filter @ A @ P @ ( coindu1889960678filter @ A @ Q @ Xs ) )
= ( coindu1889960678filter @ A
@ ^ [X: A] :
( ( P @ X )
& ( Q @ X ) )
@ Xs ) ) ).
% lfilter_lfilter
thf(fact_43_llength__ltakeWhile__lt__iff,axiom,
! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
( ( ord_less @ extended_enat @ ( coindu1018505716length @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) @ ( coindu1018505716length @ A @ Xs ) )
= ( ? [X: A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
& ~ ( P @ X ) ) ) ) ).
% llength_ltakeWhile_lt_iff
thf(fact_44_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X: A] : ( member @ A @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_46_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_47_ext,axiom,
! [B: $tType,A: $tType,F2: A > B,G2: A > B] :
( ! [X2: A] :
( ( F2 @ X2 )
= ( G2 @ X2 ) )
=> ( F2 = G2 ) ) ).
% ext
thf(fact_48_in__lset__ltlD,axiom,
! [A: $tType,X3: A,Xs: coindu1593790203_llist @ A] :
( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ ( coindu1047225960le_ltl @ A @ Xs ) ) )
=> ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ Xs ) ) ) ).
% in_lset_ltlD
thf(fact_49_llength__ltakeWhile__all,axiom,
! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
( ( ( coindu1018505716length @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) )
= ( coindu1018505716length @ A @ Xs ) )
= ( ( coindu721411036eWhile @ A @ P @ Xs )
= Xs ) ) ).
% llength_ltakeWhile_all
thf(fact_50_in__lset__ldropWhileD,axiom,
! [A: $tType,X3: A,P: A > $o,Xs: coindu1593790203_llist @ A] :
( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ ( coindu438612276pWhile @ A @ P @ Xs ) ) )
=> ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ Xs ) ) ) ).
% in_lset_ldropWhileD
thf(fact_51_ldropWhile__cong,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,P: A > $o,Q: A > $o] :
( ( Xs = Ys )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ ( coindu1112613586e_lset @ A @ Ys ) )
=> ( ( P @ X2 )
= ( Q @ X2 ) ) )
=> ( ( coindu438612276pWhile @ A @ P @ Xs )
= ( coindu438612276pWhile @ A @ Q @ Ys ) ) ) ) ).
% ldropWhile_cong
thf(fact_52_one__enat__def,axiom,
( ( one_one @ extended_enat )
= ( extended_enat2 @ ( one_one @ nat ) ) ) ).
% one_enat_def
thf(fact_53_enat__1__iff_I1_J,axiom,
! [X3: nat] :
( ( ( extended_enat2 @ X3 )
= ( one_one @ extended_enat ) )
= ( X3
= ( one_one @ nat ) ) ) ).
% enat_1_iff(1)
thf(fact_54_enat__1__iff_I2_J,axiom,
! [X3: nat] :
( ( ( one_one @ extended_enat )
= ( extended_enat2 @ X3 ) )
= ( X3
= ( one_one @ nat ) ) ) ).
% enat_1_iff(2)
thf(fact_55_less__enatE,axiom,
! [N3: extended_enat,M: nat] :
( ( ord_less @ extended_enat @ N3 @ ( extended_enat2 @ M ) )
=> ~ ! [K2: nat] :
( ( N3
= ( extended_enat2 @ K2 ) )
=> ~ ( ord_less @ nat @ K2 @ M ) ) ) ).
% less_enatE
thf(fact_56_finite__conv__nat__seg__image,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A4: set @ A] :
? [N: nat,F3: nat > A] :
( A4
= ( image @ nat @ A @ F3
@ ( collect @ nat
@ ^ [I: nat] : ( ord_less @ nat @ I @ N ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_57_nat__seg__image__imp__finite,axiom,
! [A: $tType,A3: set @ A,F2: nat > A,N3: nat] :
( ( A3
= ( image @ nat @ A @ F2
@ ( collect @ nat
@ ^ [I: nat] : ( ord_less @ nat @ I @ N3 ) ) ) )
=> ( finite_finite2 @ A @ A3 ) ) ).
% nat_seg_image_imp_finite
thf(fact_58_lfinite__conv__llength__enat,axiom,
! [A: $tType] :
( ( coindu1213758845finite @ A )
= ( ^ [Xs2: coindu1593790203_llist @ A] :
? [N: nat] :
( ( coindu1018505716length @ A @ Xs2 )
= ( extended_enat2 @ N ) ) ) ) ).
% lfinite_conv_llength_enat
thf(fact_59_llength__eq__enat__lfiniteD,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,N3: nat] :
( ( ( coindu1018505716length @ A @ Xs )
= ( extended_enat2 @ N3 ) )
=> ( coindu1213758845finite @ A @ Xs ) ) ).
% llength_eq_enat_lfiniteD
thf(fact_60_lfinite__llength__enat,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ Xs )
=> ? [N2: nat] :
( ( coindu1018505716length @ A @ Xs )
= ( extended_enat2 @ N2 ) ) ) ).
% lfinite_llength_enat
thf(fact_61_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( X3 != Y2 )
=> ( ~ ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less @ A @ Y2 @ X3 ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_62_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_63_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_64_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_65_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type2 @ A ) )
=> ( ( plus_plus @ A )
= ( ^ [A6: A,B5: A] : ( plus_plus @ A @ B5 @ A6 ) ) ) ) ).
% add.commute
thf(fact_66_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_67_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_68_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_69_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [I2: A,J: A,K: A,L: A] :
( ( ( I2 = J )
& ( K = L ) )
=> ( ( plus_plus @ A @ I2 @ K )
= ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_70_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_71_one__reorient,axiom,
! [A: $tType] :
( ( one @ A @ ( type2 @ A ) )
=> ! [X3: A] :
( ( ( one_one @ A )
= X3 )
= ( X3
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_72_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X3: A,A3: set @ A,B2: B,F2: A > B] :
( ( member @ A @ X3 @ A3 )
=> ( ( B2
= ( F2 @ X3 ) )
=> ( member @ B @ B2 @ ( image @ A @ B @ F2 @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_73_ball__imageD,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B,P: A > $o] :
( ! [X2: A] :
( ( member @ A @ X2 @ ( image @ B @ A @ F2 @ A3 ) )
=> ( P @ X2 ) )
=> ! [X4: B] :
( ( member @ B @ X4 @ A3 )
=> ( P @ ( F2 @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_74_image__cong,axiom,
! [B: $tType,A: $tType,M2: set @ A,N4: set @ A,F2: A > B,G2: A > B] :
( ( M2 = N4 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ N4 )
=> ( ( F2 @ X2 )
= ( G2 @ X2 ) ) )
=> ( ( image @ A @ B @ F2 @ M2 )
= ( image @ A @ B @ G2 @ N4 ) ) ) ) ).
% image_cong
thf(fact_75_bex__imageD,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B,P: A > $o] :
( ? [X4: A] :
( ( member @ A @ X4 @ ( image @ B @ A @ F2 @ A3 ) )
& ( P @ X4 ) )
=> ? [X2: B] :
( ( member @ B @ X2 @ A3 )
& ( P @ ( F2 @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_76_image__iff,axiom,
! [A: $tType,B: $tType,Z: A,F2: B > A,A3: set @ B] :
( ( member @ A @ Z @ ( image @ B @ A @ F2 @ A3 ) )
= ( ? [X: B] :
( ( member @ B @ X @ A3 )
& ( Z
= ( F2 @ X ) ) ) ) ) ).
% image_iff
thf(fact_77_imageI,axiom,
! [B: $tType,A: $tType,X3: A,A3: set @ A,F2: A > B] :
( ( member @ A @ X3 @ A3 )
=> ( member @ B @ ( F2 @ X3 ) @ ( image @ A @ B @ F2 @ A3 ) ) ) ).
% imageI
thf(fact_78_finite,axiom,
! [A: $tType] :
( ( finite_finite @ A @ ( type2 @ A ) )
=> ! [A3: set @ A] : ( finite_finite2 @ A @ A3 ) ) ).
% finite
thf(fact_79_finite__set__choice,axiom,
! [B: $tType,A: $tType,A3: set @ A,P: A > B > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ? [X1: B] : ( P @ X2 @ X1 ) )
=> ? [F4: A > B] :
! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ( P @ X4 @ ( F4 @ X4 ) ) ) ) ) ).
% finite_set_choice
thf(fact_80_Int__left__commute,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C2: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B3 @ C2 ) )
= ( inf_inf @ ( set @ A ) @ B3 @ ( inf_inf @ ( set @ A ) @ A3 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_81_Int__left__absorb,axiom,
! [A: $tType,A3: set @ A,B3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) )
= ( inf_inf @ ( set @ A ) @ A3 @ B3 ) ) ).
% Int_left_absorb
thf(fact_82_Int__commute,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A4: set @ A,B6: set @ A] : ( inf_inf @ ( set @ A ) @ B6 @ A4 ) ) ) ).
% Int_commute
thf(fact_83_Int__absorb,axiom,
! [A: $tType,A3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ A3 )
= A3 ) ).
% Int_absorb
thf(fact_84_Int__assoc,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C2: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) @ C2 )
= ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B3 @ C2 ) ) ) ).
% Int_assoc
thf(fact_85_IntD2,axiom,
! [A: $tType,C: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) )
=> ( member @ A @ C @ B3 ) ) ).
% IntD2
thf(fact_86_IntD1,axiom,
! [A: $tType,C: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) )
=> ( member @ A @ C @ A3 ) ) ).
% IntD1
thf(fact_87_IntE,axiom,
! [A: $tType,C: A,A3: set @ A,B3: set @ A] :
( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A3 @ B3 ) )
=> ~ ( ( member @ A @ C @ A3 )
=> ~ ( member @ A @ C @ B3 ) ) ) ).
% IntE
thf(fact_88_enat__less__induct,axiom,
! [P: extended_enat > $o,N3: extended_enat] :
( ! [N2: extended_enat] :
( ! [M3: extended_enat] :
( ( ord_less @ extended_enat @ M3 @ N2 )
=> ( P @ M3 ) )
=> ( P @ N2 ) )
=> ( P @ N3 ) ) ).
% enat_less_induct
thf(fact_89_in__lset__conv__lnth,axiom,
! [A: $tType,X3: A,Xs: coindu1593790203_llist @ A] :
( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ Xs ) )
= ( ? [N: nat] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ N ) @ ( coindu1018505716length @ A @ Xs ) )
& ( ( coindu749330388e_lnth @ A @ Xs @ N )
= X3 ) ) ) ) ).
% in_lset_conv_lnth
thf(fact_90_ltakeWhile__nth,axiom,
! [A: $tType,I2: nat,P: A > $o,Xs: coindu1593790203_llist @ A] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ I2 ) @ ( coindu1018505716length @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) )
=> ( ( coindu749330388e_lnth @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) @ I2 )
= ( coindu749330388e_lnth @ A @ Xs @ I2 ) ) ) ).
% ltakeWhile_nth
thf(fact_91_lfinite__finite__index,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ Xs )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N: nat] : ( ord_less @ extended_enat @ ( extended_enat2 @ N ) @ ( coindu1018505716length @ A @ Xs ) ) ) ) ) ).
% lfinite_finite_index
thf(fact_92_Compr__image__eq,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B,P: A > $o] :
( ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ ( image @ B @ A @ F2 @ A3 ) )
& ( P @ X ) ) )
= ( image @ B @ A @ F2
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ A3 )
& ( P @ ( F2 @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_93_image__image,axiom,
! [A: $tType,B: $tType,C3: $tType,F2: B > A,G2: C3 > B,A3: set @ C3] :
( ( image @ B @ A @ F2 @ ( image @ C3 @ B @ G2 @ A3 ) )
= ( image @ C3 @ A
@ ^ [X: C3] : ( F2 @ ( G2 @ X ) )
@ A3 ) ) ).
% image_image
thf(fact_94_imageE,axiom,
! [A: $tType,B: $tType,B2: A,F2: B > A,A3: set @ B] :
( ( member @ A @ B2 @ ( image @ B @ A @ F2 @ A3 ) )
=> ~ ! [X2: B] :
( ( B2
= ( F2 @ X2 ) )
=> ~ ( member @ B @ X2 @ A3 ) ) ) ).
% imageE
thf(fact_95_pigeonhole__infinite__rel,axiom,
! [B: $tType,A: $tType,A3: set @ A,B3: set @ B,R: A > B > $o] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ B @ B3 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ? [Xa: B] :
( ( member @ B @ Xa @ B3 )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: B] :
( ( member @ B @ X2 @ B3 )
& ~ ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A6: A] :
( ( member @ A @ A6 @ A3 )
& ( R @ A6 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_96_not__finite__existsD,axiom,
! [A: $tType,P: A > $o] :
( ~ ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ? [X12: A] : ( P @ X12 ) ) ).
% not_finite_existsD
thf(fact_97_Collect__conj__eq,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X: A] :
( ( P @ X )
& ( Q @ X ) ) )
= ( inf_inf @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_98_Int__Collect,axiom,
! [A: $tType,X3: A,A3: set @ A,P: A > $o] :
( ( member @ A @ X3 @ ( inf_inf @ ( set @ A ) @ A3 @ ( collect @ A @ P ) ) )
= ( ( member @ A @ X3 @ A3 )
& ( P @ X3 ) ) ) ).
% Int_Collect
thf(fact_99_Int__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A4: set @ A,B6: set @ A] :
( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ A4 )
& ( member @ A @ X @ B6 ) ) ) ) ) ).
% Int_def
thf(fact_100_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_101_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_102_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_103_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_104_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_105_add__mono__thms__linordered__field_I1_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [I2: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I2 @ J )
& ( K = L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_106_add__mono__thms__linordered__field_I2_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [I2: A,J: A,K: A,L: A] :
( ( ( I2 = J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_107_add__mono__thms__linordered__field_I5_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [I2: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I2 @ J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_108_enat__iless,axiom,
! [N3: extended_enat,M: nat] :
( ( ord_less @ extended_enat @ N3 @ ( extended_enat2 @ M ) )
=> ? [K2: nat] :
( N3
= ( extended_enat2 @ K2 ) ) ) ).
% enat_iless
thf(fact_109_chain__incr,axiom,
! [A: $tType,Y: A > extended_enat,K: nat] :
( ! [I3: A] :
? [J2: A] : ( ord_less @ extended_enat @ ( Y @ I3 ) @ ( Y @ J2 ) )
=> ? [J3: A] : ( ord_less @ extended_enat @ ( extended_enat2 @ K ) @ ( Y @ J3 ) ) ) ).
% chain_incr
thf(fact_110_pigeonhole__infinite,axiom,
! [B: $tType,A: $tType,A3: set @ A,F2: A > B] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ B @ ( image @ A @ B @ F2 @ A3 ) )
=> ? [X2: A] :
( ( member @ A @ X2 @ A3 )
& ~ ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A6: A] :
( ( member @ A @ A6 @ A3 )
& ( ( F2 @ A6 )
= ( F2 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_111_enat__add__mono,axiom,
! [X3: nat,Y2: extended_enat,Z: extended_enat] :
( ( ord_less @ extended_enat @ ( plus_plus @ extended_enat @ ( extended_enat2 @ X3 ) @ Y2 ) @ ( plus_plus @ extended_enat @ ( extended_enat2 @ X3 ) @ Z ) )
= ( ord_less @ extended_enat @ Y2 @ Z ) ) ).
% enat_add_mono
thf(fact_112_enat__less__enat__plusI2,axiom,
! [Y2: nat,Z: extended_enat,X3: nat] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ Y2 ) @ Z )
=> ( ord_less @ extended_enat @ ( extended_enat2 @ ( plus_plus @ nat @ X3 @ Y2 ) ) @ ( plus_plus @ extended_enat @ ( extended_enat2 @ X3 ) @ Z ) ) ) ).
% enat_less_enat_plusI2
thf(fact_113_enat__add2__eq,axiom,
! [Y2: extended_enat,X3: nat,Z: extended_enat] :
( ( ( plus_plus @ extended_enat @ Y2 @ ( extended_enat2 @ X3 ) )
= ( plus_plus @ extended_enat @ Z @ ( extended_enat2 @ X3 ) ) )
= ( Y2 = Z ) ) ).
% enat_add2_eq
thf(fact_114_enat__add1__eq,axiom,
! [X3: nat,Y2: extended_enat,Z: extended_enat] :
( ( ( plus_plus @ extended_enat @ ( extended_enat2 @ X3 ) @ Y2 )
= ( plus_plus @ extended_enat @ ( extended_enat2 @ X3 ) @ Z ) )
= ( Y2 = Z ) ) ).
% enat_add1_eq
thf(fact_115_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N3: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N3 ) )
= ( ord_less @ nat @ M @ N3 ) ) ).
% nat_add_left_cancel_less
thf(fact_116_inf__right__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X3 @ Y2 ) @ Y2 )
= ( inf_inf @ A @ X3 @ Y2 ) ) ) ).
% inf_right_idem
thf(fact_117_inf_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B2 ) @ B2 )
= ( inf_inf @ A @ A2 @ B2 ) ) ) ).
% inf.right_idem
thf(fact_118_inf__left__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( inf_inf @ A @ X3 @ ( inf_inf @ A @ X3 @ Y2 ) )
= ( inf_inf @ A @ X3 @ Y2 ) ) ) ).
% inf_left_idem
thf(fact_119_inf_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( inf_inf @ A @ A2 @ ( inf_inf @ A @ A2 @ B2 ) )
= ( inf_inf @ A @ A2 @ B2 ) ) ) ).
% inf.left_idem
thf(fact_120_inf__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [X3: A] :
( ( inf_inf @ A @ X3 @ X3 )
= X3 ) ) ).
% inf_idem
thf(fact_121_inf__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf @ B @ ( type2 @ B ) )
=> ( ( inf_inf @ ( A > B ) )
= ( ^ [F3: A > B,G3: A > B,X: A] : ( inf_inf @ B @ ( F3 @ X ) @ ( G3 @ X ) ) ) ) ) ).
% inf_apply
thf(fact_122_inf_Oidem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( inf_inf @ A @ A2 @ A2 )
= A2 ) ) ).
% inf.idem
thf(fact_123_lnull__lfilter,axiom,
! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ ( coindu1889960678filter @ A @ P @ Xs ) )
= ( ! [X: A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ~ ( P @ X ) ) ) ) ).
% lnull_lfilter
thf(fact_124_lnull__ldropWhile,axiom,
! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ ( coindu438612276pWhile @ A @ P @ Xs ) )
= ( ! [X: A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ( P @ X ) ) ) ) ).
% lnull_ldropWhile
thf(fact_125_psubsetD,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C: A] :
( ( ord_less @ ( set @ A ) @ A3 @ B3 )
=> ( ( member @ A @ C @ A3 )
=> ( member @ A @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_126_nat__neq__iff,axiom,
! [M: nat,N3: nat] :
( ( M != N3 )
= ( ( ord_less @ nat @ M @ N3 )
| ( ord_less @ nat @ N3 @ M ) ) ) ).
% nat_neq_iff
thf(fact_127_less__set__def,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A4: set @ A,B6: set @ A] :
( ord_less @ ( A > $o )
@ ^ [X: A] : ( member @ A @ X @ A4 )
@ ^ [X: A] : ( member @ A @ X @ B6 ) ) ) ) ).
% less_set_def
thf(fact_128_less__not__refl,axiom,
! [N3: nat] :
~ ( ord_less @ nat @ N3 @ N3 ) ).
% less_not_refl
thf(fact_129_psubset__trans,axiom,
! [A: $tType,A3: set @ A,B3: set @ A,C2: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B3 )
=> ( ( ord_less @ ( set @ A ) @ B3 @ C2 )
=> ( ord_less @ ( set @ A ) @ A3 @ C2 ) ) ) ).
% psubset_trans
thf(fact_130_less__not__refl2,axiom,
! [N3: nat,M: nat] :
( ( ord_less @ nat @ N3 @ M )
=> ( M != N3 ) ) ).
% less_not_refl2
thf(fact_131_less__not__refl3,axiom,
! [S: nat,T2: nat] :
( ( ord_less @ nat @ S @ T2 )
=> ( S != T2 ) ) ).
% less_not_refl3
thf(fact_132_measure__induct,axiom,
! [A: $tType,F2: A > nat,P: A > $o,A2: A] :
( ! [X2: A] :
( ! [Y3: A] :
( ( ord_less @ nat @ ( F2 @ Y3 ) @ ( F2 @ X2 ) )
=> ( P @ Y3 ) )
=> ( P @ X2 ) )
=> ( P @ A2 ) ) ).
% measure_induct
thf(fact_133_less__irrefl__nat,axiom,
! [N3: nat] :
~ ( ord_less @ nat @ N3 @ N3 ) ).
% less_irrefl_nat
thf(fact_134_nat__less__induct,axiom,
! [P: nat > $o,N3: nat] :
( ! [N2: nat] :
( ! [M3: nat] :
( ( ord_less @ nat @ M3 @ N2 )
=> ( P @ M3 ) )
=> ( P @ N2 ) )
=> ( P @ N3 ) ) ).
% nat_less_induct
thf(fact_135_infinite__descent,axiom,
! [P: nat > $o,N3: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M3: nat] :
( ( ord_less @ nat @ M3 @ N2 )
& ~ ( P @ M3 ) ) )
=> ( P @ N3 ) ) ).
% infinite_descent
thf(fact_136_linorder__neqE__nat,axiom,
! [X3: nat,Y2: nat] :
( ( X3 != Y2 )
=> ( ~ ( ord_less @ nat @ X3 @ Y2 )
=> ( ord_less @ nat @ Y2 @ X3 ) ) ) ).
% linorder_neqE_nat
thf(fact_137_measure__induct__rule,axiom,
! [A: $tType,F2: A > nat,P: A > $o,A2: A] :
( ! [X2: A] :
( ! [Y3: A] :
( ( ord_less @ nat @ ( F2 @ Y3 ) @ ( F2 @ X2 ) )
=> ( P @ Y3 ) )
=> ( P @ X2 ) )
=> ( P @ A2 ) ) ).
% measure_induct_rule
thf(fact_138_infinite__descent__measure,axiom,
! [A: $tType,P: A > $o,V: A > nat,X3: A] :
( ! [X2: A] :
( ~ ( P @ X2 )
=> ? [Y3: A] :
( ( ord_less @ nat @ ( V @ Y3 ) @ ( V @ X2 ) )
& ~ ( P @ Y3 ) ) )
=> ( P @ X3 ) ) ).
% infinite_descent_measure
thf(fact_139_lappend_Oexhaust,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu335574135_lnull @ A @ Xs )
=> ~ ( coindu335574135_lnull @ A @ Ys ) )
=> ( ~ ( coindu335574135_lnull @ A @ Xs )
| ~ ( coindu335574135_lnull @ A @ Ys ) ) ) ).
% lappend.exhaust
thf(fact_140_lzip_Oexhaust,axiom,
! [A: $tType,B: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B] :
( ~ ( ( coindu335574135_lnull @ A @ Xs )
| ( coindu335574135_lnull @ B @ Ys ) )
=> ~ ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( coindu335574135_lnull @ B @ Ys ) ) ) ).
% lzip.exhaust
thf(fact_141_lnull__ltlI,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( coindu335574135_lnull @ A @ ( coindu1047225960le_ltl @ A @ Xs ) ) ) ).
% lnull_ltlI
thf(fact_142_lnull__imp__lfinite,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( coindu1213758845finite @ A @ Xs ) ) ).
% lnull_imp_lfinite
thf(fact_143_inf__set__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A4: set @ A,B6: set @ A] :
( collect @ A
@ ( inf_inf @ ( A > $o )
@ ^ [X: A] : ( member @ A @ X @ A4 )
@ ^ [X: A] : ( member @ A @ X @ B6 ) ) ) ) ) ).
% inf_set_def
thf(fact_144_llist_Oset__sel_I2_J,axiom,
! [A: $tType,A2: coindu1593790203_llist @ A,X3: A] :
( ~ ( coindu335574135_lnull @ A @ A2 )
=> ( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ ( coindu1047225960le_ltl @ A @ A2 ) ) )
=> ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ A2 ) ) ) ) ).
% llist.set_sel(2)
thf(fact_145_lfinite__induct,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: ( coindu1593790203_llist @ A ) > $o] :
( ( coindu1213758845finite @ A @ Xs )
=> ( ! [Xs3: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs3 )
=> ( P @ Xs3 ) )
=> ( ! [Xs3: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ Xs3 )
=> ( ~ ( coindu335574135_lnull @ A @ Xs3 )
=> ( ( P @ ( coindu1047225960le_ltl @ A @ Xs3 ) )
=> ( P @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lfinite_induct
thf(fact_146_inf__sup__aci_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] :
( ( inf_inf @ A @ X3 @ ( inf_inf @ A @ X3 @ Y2 ) )
= ( inf_inf @ A @ X3 @ Y2 ) ) ) ).
% inf_sup_aci(4)
thf(fact_147_inf__sup__aci_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A,Z: A] :
( ( inf_inf @ A @ X3 @ ( inf_inf @ A @ Y2 @ Z ) )
= ( inf_inf @ A @ Y2 @ ( inf_inf @ A @ X3 @ Z ) ) ) ) ).
% inf_sup_aci(3)
thf(fact_148_inf__sup__aci_I2_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A,Z: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X3 @ Y2 ) @ Z )
= ( inf_inf @ A @ X3 @ ( inf_inf @ A @ Y2 @ Z ) ) ) ) ).
% inf_sup_aci(2)
thf(fact_149_inf__sup__aci_I1_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ( ( inf_inf @ A )
= ( ^ [X: A,Y4: A] : ( inf_inf @ A @ Y4 @ X ) ) ) ) ).
% inf_sup_aci(1)
thf(fact_150_inf__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf @ B @ ( type2 @ B ) )
=> ( ( inf_inf @ ( A > B ) )
= ( ^ [F3: A > B,G3: A > B,X: A] : ( inf_inf @ B @ ( F3 @ X ) @ ( G3 @ X ) ) ) ) ) ).
% inf_fun_def
thf(fact_151_inf_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C )
= ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B2 @ C ) ) ) ) ).
% inf.assoc
thf(fact_152_inf__assoc,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A,Z: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X3 @ Y2 ) @ Z )
= ( inf_inf @ A @ X3 @ ( inf_inf @ A @ Y2 @ Z ) ) ) ) ).
% inf_assoc
thf(fact_153_inf_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ( ( inf_inf @ A )
= ( ^ [A6: A,B5: A] : ( inf_inf @ A @ B5 @ A6 ) ) ) ) ).
% inf.commute
thf(fact_154_inf__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ( ( inf_inf @ A )
= ( ^ [X: A,Y4: A] : ( inf_inf @ A @ Y4 @ X ) ) ) ) ).
% inf_commute
thf(fact_155_inf_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( inf_inf @ A @ B2 @ ( inf_inf @ A @ A2 @ C ) )
= ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B2 @ C ) ) ) ) ).
% inf.left_commute
thf(fact_156_inf__left__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A,Z: A] :
( ( inf_inf @ A @ X3 @ ( inf_inf @ A @ Y2 @ Z ) )
= ( inf_inf @ A @ Y2 @ ( inf_inf @ A @ X3 @ Z ) ) ) ) ).
% inf_left_commute
thf(fact_157_nat__add__left__cancel,axiom,
! [K: nat,M: nat,N3: nat] :
( ( ( plus_plus @ nat @ K @ M )
= ( plus_plus @ nat @ K @ N3 ) )
= ( M = N3 ) ) ).
% nat_add_left_cancel
thf(fact_158_nat__add__right__cancel,axiom,
! [M: nat,K: nat,N3: nat] :
( ( ( plus_plus @ nat @ M @ K )
= ( plus_plus @ nat @ N3 @ K ) )
= ( M = N3 ) ) ).
% nat_add_right_cancel
thf(fact_159_add__lessD1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ J ) @ K )
=> ( ord_less @ nat @ I2 @ K ) ) ).
% add_lessD1
thf(fact_160_add__less__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( ( ord_less @ nat @ K @ L )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_161_not__add__less1,axiom,
! [I2: nat,J: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ J ) @ I2 ) ).
% not_add_less1
thf(fact_162_not__add__less2,axiom,
! [J: nat,I2: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_163_add__less__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_164_trans__less__add1,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( ord_less @ nat @ I2 @ ( plus_plus @ nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_165_trans__less__add2,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( ord_less @ nat @ I2 @ ( plus_plus @ nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_166_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N3: nat] :
( ( ord_less @ nat @ K @ L )
=> ( ( ( plus_plus @ nat @ M @ L )
= ( plus_plus @ nat @ K @ N3 ) )
=> ( ord_less @ nat @ M @ N3 ) ) ) ).
% less_add_eq_less
thf(fact_167_less__infI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [A2: A,X3: A,B2: A] :
( ( ord_less @ A @ A2 @ X3 )
=> ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X3 ) ) ) ).
% less_infI1
thf(fact_168_less__infI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [B2: A,X3: A,A2: A] :
( ( ord_less @ A @ B2 @ X3 )
=> ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X3 ) ) ) ).
% less_infI2
thf(fact_169_inf_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less @ A @ A2 @ ( inf_inf @ A @ B2 @ C ) )
=> ~ ( ( ord_less @ A @ A2 @ B2 )
=> ~ ( ord_less @ A @ A2 @ C ) ) ) ) ).
% inf.strict_boundedE
thf(fact_170_inf_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [A6: A,B5: A] :
( ( A6
= ( inf_inf @ A @ A6 @ B5 ) )
& ( A6 != B5 ) ) ) ) ) ).
% inf.strict_order_iff
thf(fact_171_inf_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,B2: A] :
( ( ord_less @ A @ A2 @ C )
=> ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C ) ) ) ).
% inf.strict_coboundedI1
thf(fact_172_inf_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [B2: A,C: A,A2: A] :
( ( ord_less @ A @ B2 @ C )
=> ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C ) ) ) ).
% inf.strict_coboundedI2
thf(fact_173_enat__less__enat__plusI,axiom,
! [X3: nat,Y2: nat,Z: extended_enat] :
( ( ord_less @ nat @ X3 @ Y2 )
=> ( ord_less @ extended_enat @ ( extended_enat2 @ X3 ) @ ( plus_plus @ extended_enat @ ( extended_enat2 @ Y2 ) @ Z ) ) ) ).
% enat_less_enat_plusI
thf(fact_174_fold__atLeastAtMost__nat_Oinduct,axiom,
! [A: $tType,P: ( nat > A > A ) > nat > nat > A > $o,A0: nat > A > A,A1: nat,A22: nat,A32: A] :
( ! [F4: nat > A > A,A7: nat,B7: nat,Acc: A] :
( ( ~ ( ord_less @ nat @ B7 @ A7 )
=> ( P @ F4 @ ( plus_plus @ nat @ A7 @ ( one_one @ nat ) ) @ B7 @ ( F4 @ A7 @ Acc ) ) )
=> ( P @ F4 @ A7 @ B7 @ Acc ) )
=> ( P @ A0 @ A1 @ A22 @ A32 ) ) ).
% fold_atLeastAtMost_nat.induct
thf(fact_175_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I2: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( P @ K3 )
& ( ord_less @ nat @ K3 @ I2 ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_176_gen__llength__def,axiom,
! [A: $tType] :
( ( coindu493225792length @ A )
= ( ^ [N: nat,Xs2: coindu1593790203_llist @ A] : ( plus_plus @ extended_enat @ ( extended_enat2 @ N ) @ ( coindu1018505716length @ A @ Xs2 ) ) ) ) ).
% gen_llength_def
thf(fact_177_unbounded__k__infinite,axiom,
! [K: nat,S2: set @ nat] :
( ! [M4: nat] :
( ( ord_less @ nat @ K @ M4 )
=> ? [N5: nat] :
( ( ord_less @ nat @ M4 @ N5 )
& ( member @ nat @ N5 @ S2 ) ) )
=> ~ ( finite_finite2 @ nat @ S2 ) ) ).
% unbounded_k_infinite
thf(fact_178_inf1I,axiom,
! [A: $tType,A3: A > $o,X3: A,B3: A > $o] :
( ( A3 @ X3 )
=> ( ( B3 @ X3 )
=> ( inf_inf @ ( A > $o ) @ A3 @ B3 @ X3 ) ) ) ).
% inf1I
thf(fact_179_inf1E,axiom,
! [A: $tType,A3: A > $o,B3: A > $o,X3: A] :
( ( inf_inf @ ( A > $o ) @ A3 @ B3 @ X3 )
=> ~ ( ( A3 @ X3 )
=> ~ ( B3 @ X3 ) ) ) ).
% inf1E
thf(fact_180_inf1D1,axiom,
! [A: $tType,A3: A > $o,B3: A > $o,X3: A] :
( ( inf_inf @ ( A > $o ) @ A3 @ B3 @ X3 )
=> ( A3 @ X3 ) ) ).
% inf1D1
thf(fact_181_inf1D2,axiom,
! [A: $tType,A3: A > $o,B3: A > $o,X3: A] :
( ( inf_inf @ ( A > $o ) @ A3 @ B3 @ X3 )
=> ( B3 @ X3 ) ) ).
% inf1D2
thf(fact_182_finite__nat__set__iff__bounded,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N6: set @ nat] :
? [M5: nat] :
! [X: nat] :
( ( member @ nat @ X @ N6 )
=> ( ord_less @ nat @ X @ M5 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_183_infinite__nat__iff__unbounded,axiom,
! [S2: set @ nat] :
( ( ~ ( finite_finite2 @ nat @ S2 ) )
= ( ! [M5: nat] :
? [N: nat] :
( ( ord_less @ nat @ M5 @ N )
& ( member @ nat @ N @ S2 ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_184_bounded__nat__set__is__finite,axiom,
! [N4: set @ nat,N3: nat] :
( ! [X2: nat] :
( ( member @ nat @ X2 @ N4 )
=> ( ord_less @ nat @ X2 @ N3 ) )
=> ( finite_finite2 @ nat @ N4 ) ) ).
% bounded_nat_set_is_finite
thf(fact_185_inf__Int__eq,axiom,
! [A: $tType,R: set @ A,S2: set @ A] :
( ( inf_inf @ ( A > $o )
@ ^ [X: A] : ( member @ A @ X @ R )
@ ^ [X: A] : ( member @ A @ X @ S2 ) )
= ( ^ [X: A] : ( member @ A @ X @ ( inf_inf @ ( set @ A ) @ R @ S2 ) ) ) ) ).
% inf_Int_eq
thf(fact_186_lnth__llength__ltakeWhile,axiom,
! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
( ( ord_less @ extended_enat @ ( coindu1018505716length @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) @ ( coindu1018505716length @ A @ Xs ) )
=> ~ ( P @ ( coindu749330388e_lnth @ A @ Xs @ ( extended_the_enat @ ( coindu1018505716length @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) ) ) ) ) ).
% lnth_llength_ltakeWhile
thf(fact_187_lnth__ldropn,axiom,
! [A: $tType,N3: nat,M: nat,Xs: coindu1593790203_llist @ A] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ ( plus_plus @ nat @ N3 @ M ) ) @ ( coindu1018505716length @ A @ Xs ) )
=> ( ( coindu749330388e_lnth @ A @ ( coindu531130065ldropn @ A @ N3 @ Xs ) @ M )
= ( coindu749330388e_lnth @ A @ Xs @ ( plus_plus @ nat @ M @ N3 ) ) ) ) ).
% lnth_ldropn
thf(fact_188_ldropn__ldropn,axiom,
! [A: $tType,N3: nat,M: nat,Xs: coindu1593790203_llist @ A] :
( ( coindu531130065ldropn @ A @ N3 @ ( coindu531130065ldropn @ A @ M @ Xs ) )
= ( coindu531130065ldropn @ A @ ( plus_plus @ nat @ N3 @ M ) @ Xs ) ) ).
% ldropn_ldropn
thf(fact_189_lfinite__ldropn,axiom,
! [A: $tType,N3: nat,Xs: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ ( coindu531130065ldropn @ A @ N3 @ Xs ) )
= ( coindu1213758845finite @ A @ Xs ) ) ).
% lfinite_ldropn
thf(fact_190_the__enat_Osimps,axiom,
! [N3: nat] :
( ( extended_the_enat @ ( extended_enat2 @ N3 ) )
= N3 ) ).
% the_enat.simps
thf(fact_191_ltl__ldropn,axiom,
! [A: $tType,N3: nat,Xs: coindu1593790203_llist @ A] :
( ( coindu1047225960le_ltl @ A @ ( coindu531130065ldropn @ A @ N3 @ Xs ) )
= ( coindu531130065ldropn @ A @ N3 @ ( coindu1047225960le_ltl @ A @ Xs ) ) ) ).
% ltl_ldropn
thf(fact_192_in__lset__ldropnD,axiom,
! [A: $tType,X3: A,N3: nat,Xs: coindu1593790203_llist @ A] :
( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ ( coindu531130065ldropn @ A @ N3 @ Xs ) ) )
=> ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ Xs ) ) ) ).
% in_lset_ldropnD
thf(fact_193_lnth__ldrop,axiom,
! [A: $tType,N3: extended_enat,M: nat,Xs: coindu1593790203_llist @ A] :
( ( ord_less @ extended_enat @ ( plus_plus @ extended_enat @ N3 @ ( extended_enat2 @ M ) ) @ ( coindu1018505716length @ A @ Xs ) )
=> ( ( coindu749330388e_lnth @ A @ ( coindu191418589_ldrop @ A @ N3 @ Xs ) @ M )
= ( coindu749330388e_lnth @ A @ Xs @ ( plus_plus @ nat @ M @ ( extended_the_enat @ N3 ) ) ) ) ) ).
% lnth_ldrop
thf(fact_194_less__numeral__extra_I4_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% less_numeral_extra(4)
thf(fact_195_ldrop__ldrop,axiom,
! [A: $tType,N3: extended_enat,M: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( coindu191418589_ldrop @ A @ N3 @ ( coindu191418589_ldrop @ A @ M @ Xs ) )
= ( coindu191418589_ldrop @ A @ ( plus_plus @ extended_enat @ N3 @ M ) @ Xs ) ) ).
% ldrop_ldrop
thf(fact_196_in__lset__ldropD,axiom,
! [A: $tType,X3: A,N3: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ ( coindu191418589_ldrop @ A @ N3 @ Xs ) ) )
=> ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ Xs ) ) ) ).
% in_lset_ldropD
thf(fact_197_ltl__ldrop,axiom,
! [A: $tType,N3: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( coindu1047225960le_ltl @ A @ ( coindu191418589_ldrop @ A @ N3 @ Xs ) )
= ( coindu191418589_ldrop @ A @ N3 @ ( coindu1047225960le_ltl @ A @ Xs ) ) ) ).
% ltl_ldrop
thf(fact_198_ldrop__enat,axiom,
! [A: $tType,N3: nat,Xs: coindu1593790203_llist @ A] :
( ( coindu191418589_ldrop @ A @ ( extended_enat2 @ N3 ) @ Xs )
= ( coindu531130065ldropn @ A @ N3 @ Xs ) ) ).
% ldrop_enat
thf(fact_199_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_200_ldropWhile__eq__ldrop,axiom,
! [A: $tType] :
( ( coindu438612276pWhile @ A )
= ( ^ [P2: A > $o,Xs2: coindu1593790203_llist @ A] : ( coindu191418589_ldrop @ A @ ( coindu1018505716length @ A @ ( coindu721411036eWhile @ A @ P2 @ Xs2 ) ) @ Xs2 ) ) ) ).
% ldropWhile_eq_ldrop
thf(fact_201_lhd__ldropWhile__conv__lnth,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
( ? [X4: A] :
( ( member @ A @ X4 @ ( coindu1112613586e_lset @ A @ Xs ) )
& ~ ( P @ X4 ) )
=> ( ( coindu1046438764le_lhd @ A @ ( coindu438612276pWhile @ A @ P @ Xs ) )
= ( coindu749330388e_lnth @ A @ Xs @ ( extended_the_enat @ ( coindu1018505716length @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) ) ) ) ) ).
% lhd_ldropWhile_conv_lnth
thf(fact_202_llast__ldropn,axiom,
! [A: $tType,N3: nat,Xs: coindu1593790203_llist @ A] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ N3 ) @ ( coindu1018505716length @ A @ Xs ) )
=> ( ( coindu2000965700_llast @ A @ ( coindu531130065ldropn @ A @ N3 @ Xs ) )
= ( coindu2000965700_llast @ A @ Xs ) ) ) ).
% llast_ldropn
thf(fact_203_lnull__ltakeWhile,axiom,
! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) )
= ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ~ ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% lnull_ltakeWhile
thf(fact_204_ltakeWhile_Odisc__iff_I1_J,axiom,
! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) )
= ( ( coindu335574135_lnull @ A @ Xs )
| ~ ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% ltakeWhile.disc_iff(1)
thf(fact_205_ltakeWhile_Odisc__iff_I2_J,axiom,
! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
( ( ~ ( coindu335574135_lnull @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) )
= ( ~ ( coindu335574135_lnull @ A @ Xs )
& ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% ltakeWhile.disc_iff(2)
thf(fact_206_ltakeWhile_Oexhaust,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
( ~ ( ( coindu335574135_lnull @ A @ Xs )
| ~ ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) )
=> ~ ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ~ ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% ltakeWhile.exhaust
thf(fact_207_llist_Oset__sel_I1_J,axiom,
! [A: $tType,A2: coindu1593790203_llist @ A] :
( ~ ( coindu335574135_lnull @ A @ A2 )
=> ( member @ A @ ( coindu1046438764le_lhd @ A @ A2 ) @ ( coindu1112613586e_lset @ A @ A2 ) ) ) ).
% llist.set_sel(1)
thf(fact_208_llist_Ocoinduct__strong,axiom,
! [A: $tType,R: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o,Llist: coindu1593790203_llist @ A,Llist2: coindu1593790203_llist @ A] :
( ( R @ Llist @ Llist2 )
=> ( ! [Llist3: coindu1593790203_llist @ A,Llist4: coindu1593790203_llist @ A] :
( ( R @ Llist3 @ Llist4 )
=> ( ( ( coindu335574135_lnull @ A @ Llist3 )
= ( coindu335574135_lnull @ A @ Llist4 ) )
& ( ~ ( coindu335574135_lnull @ A @ Llist3 )
=> ( ~ ( coindu335574135_lnull @ A @ Llist4 )
=> ( ( ( coindu1046438764le_lhd @ A @ Llist3 )
= ( coindu1046438764le_lhd @ A @ Llist4 ) )
& ( ( R @ ( coindu1047225960le_ltl @ A @ Llist3 ) @ ( coindu1047225960le_ltl @ A @ Llist4 ) )
| ( ( coindu1047225960le_ltl @ A @ Llist3 )
= ( coindu1047225960le_ltl @ A @ Llist4 ) ) ) ) ) ) ) )
=> ( Llist = Llist2 ) ) ) ).
% llist.coinduct_strong
thf(fact_209_llist_Ocoinduct,axiom,
! [A: $tType,R: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o,Llist: coindu1593790203_llist @ A,Llist2: coindu1593790203_llist @ A] :
( ( R @ Llist @ Llist2 )
=> ( ! [Llist3: coindu1593790203_llist @ A,Llist4: coindu1593790203_llist @ A] :
( ( R @ Llist3 @ Llist4 )
=> ( ( ( coindu335574135_lnull @ A @ Llist3 )
= ( coindu335574135_lnull @ A @ Llist4 ) )
& ( ~ ( coindu335574135_lnull @ A @ Llist3 )
=> ( ~ ( coindu335574135_lnull @ A @ Llist4 )
=> ( ( ( coindu1046438764le_lhd @ A @ Llist3 )
= ( coindu1046438764le_lhd @ A @ Llist4 ) )
& ( R @ ( coindu1047225960le_ltl @ A @ Llist3 ) @ ( coindu1047225960le_ltl @ A @ Llist4 ) ) ) ) ) ) )
=> ( Llist = Llist2 ) ) ) ).
% llist.coinduct
thf(fact_210_llist_Oexpand,axiom,
! [A: $tType,Llist: coindu1593790203_llist @ A,Llist2: coindu1593790203_llist @ A] :
( ( ( coindu335574135_lnull @ A @ Llist )
= ( coindu335574135_lnull @ A @ Llist2 ) )
=> ( ( ~ ( coindu335574135_lnull @ A @ Llist )
=> ( ~ ( coindu335574135_lnull @ A @ Llist2 )
=> ( ( ( coindu1046438764le_lhd @ A @ Llist )
= ( coindu1046438764le_lhd @ A @ Llist2 ) )
& ( ( coindu1047225960le_ltl @ A @ Llist )
= ( coindu1047225960le_ltl @ A @ Llist2 ) ) ) ) )
=> ( Llist = Llist2 ) ) ) ).
% llist.expand
thf(fact_211_ltakeWhile_Odisc_I2_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( P @ ( coindu1046438764le_lhd @ A @ Xs ) )
=> ~ ( coindu335574135_lnull @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) ) ) ).
% ltakeWhile.disc(2)
thf(fact_212_ltakeWhile_Odisc_I1_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
( ( ( coindu335574135_lnull @ A @ Xs )
| ~ ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) )
=> ( coindu335574135_lnull @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) ) ).
% ltakeWhile.disc(1)
thf(fact_213_lhd__ltakeWhile,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( P @ ( coindu1046438764le_lhd @ A @ Xs ) )
=> ( ( coindu1046438764le_lhd @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) )
= ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% lhd_ltakeWhile
thf(fact_214_lhd__ldropWhile__in__lset,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
( ? [X4: A] :
( ( member @ A @ X4 @ ( coindu1112613586e_lset @ A @ Xs ) )
& ~ ( P @ X4 ) )
=> ( member @ A @ ( coindu1046438764le_lhd @ A @ ( coindu438612276pWhile @ A @ P @ Xs ) ) @ ( coindu1112613586e_lset @ A @ Xs ) ) ) ).
% lhd_ldropWhile_in_lset
thf(fact_215_lhd__ldropWhile,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
( ? [X4: A] :
( ( member @ A @ X4 @ ( coindu1112613586e_lset @ A @ Xs ) )
& ~ ( P @ X4 ) )
=> ~ ( P @ ( coindu1046438764le_lhd @ A @ ( coindu438612276pWhile @ A @ P @ Xs ) ) ) ) ).
% lhd_ldropWhile
thf(fact_216_llist__set__induct,axiom,
! [A: $tType,X3: A,Xs: coindu1593790203_llist @ A,P: A > ( coindu1593790203_llist @ A ) > $o] :
( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ( ! [Xs3: coindu1593790203_llist @ A] :
( ~ ( coindu335574135_lnull @ A @ Xs3 )
=> ( P @ ( coindu1046438764le_lhd @ A @ Xs3 ) @ Xs3 ) )
=> ( ! [Xs3: coindu1593790203_llist @ A,Y5: A] :
( ~ ( coindu335574135_lnull @ A @ Xs3 )
=> ( ( member @ A @ Y5 @ ( coindu1112613586e_lset @ A @ ( coindu1047225960le_ltl @ A @ Xs3 ) ) )
=> ( ( P @ Y5 @ ( coindu1047225960le_ltl @ A @ Xs3 ) )
=> ( P @ Y5 @ Xs3 ) ) ) )
=> ( P @ X3 @ Xs ) ) ) ) ).
% llist_set_induct
thf(fact_217_ltakeWhile_Osimps_I4_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( P @ ( coindu1046438764le_lhd @ A @ Xs ) )
=> ( ( coindu1047225960le_ltl @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) )
= ( coindu721411036eWhile @ A @ P @ ( coindu1047225960le_ltl @ A @ Xs ) ) ) ) ) ).
% ltakeWhile.simps(4)
thf(fact_218_llast__ldrop,axiom,
! [A: $tType,N3: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( ord_less @ extended_enat @ N3 @ ( coindu1018505716length @ A @ Xs ) )
=> ( ( coindu2000965700_llast @ A @ ( coindu191418589_ldrop @ A @ N3 @ Xs ) )
= ( coindu2000965700_llast @ A @ Xs ) ) ) ).
% llast_ldrop
thf(fact_219_lhd__ldropn,axiom,
! [A: $tType,N3: nat,Xs: coindu1593790203_llist @ A] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ N3 ) @ ( coindu1018505716length @ A @ Xs ) )
=> ( ( coindu1046438764le_lhd @ A @ ( coindu531130065ldropn @ A @ N3 @ Xs ) )
= ( coindu749330388e_lnth @ A @ Xs @ N3 ) ) ) ).
% lhd_ldropn
thf(fact_220_lhd__ldrop,axiom,
! [A: $tType,N3: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( ord_less @ extended_enat @ N3 @ ( coindu1018505716length @ A @ Xs ) )
=> ( ( coindu1046438764le_lhd @ A @ ( coindu191418589_ldrop @ A @ N3 @ Xs ) )
= ( coindu749330388e_lnth @ A @ Xs @ ( extended_the_enat @ N3 ) ) ) ) ).
% lhd_ldrop
thf(fact_221_enat_Osimps_I6_J,axiom,
! [T: $tType,F1: nat > T,F22: T,Nat: nat] :
( ( extended_rec_enat @ T @ F1 @ F22 @ ( extended_enat2 @ Nat ) )
= ( F1 @ Nat ) ) ).
% enat.simps(6)
thf(fact_222_lset__conv__lnth,axiom,
! [A: $tType] :
( ( coindu1112613586e_lset @ A )
= ( ^ [Xs2: coindu1593790203_llist @ A] :
( collect @ A
@ ^ [Uu: A] :
? [N: nat] :
( ( Uu
= ( coindu749330388e_lnth @ A @ Xs2 @ N ) )
& ( ord_less @ extended_enat @ ( extended_enat2 @ N ) @ ( coindu1018505716length @ A @ Xs2 ) ) ) ) ) ) ).
% lset_conv_lnth
thf(fact_223_finite__Collect__bounded__ex,axiom,
! [B: $tType,A: $tType,P: A > $o,Q: B > A > $o] :
( ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X: B] :
? [Y4: A] :
( ( P @ Y4 )
& ( Q @ X @ Y4 ) ) ) )
= ( ! [Y4: A] :
( ( P @ Y4 )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X: B] : ( Q @ X @ Y4 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_224_finite__image__set,axiom,
! [A: $tType,B: $tType,P: A > $o,F2: A > B] :
( ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [Uu: B] :
? [X: A] :
( ( Uu
= ( F2 @ X ) )
& ( P @ X ) ) ) ) ) ).
% finite_image_set
thf(fact_225_finite__image__set2,axiom,
! [A: $tType,B: $tType,C3: $tType,P: A > $o,Q: B > $o,F2: A > B > C3] :
( ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ( ( finite_finite2 @ B @ ( collect @ B @ Q ) )
=> ( finite_finite2 @ C3
@ ( collect @ C3
@ ^ [Uu: C3] :
? [X: A,Y4: B] :
( ( Uu
= ( F2 @ X @ Y4 ) )
& ( P @ X )
& ( Q @ Y4 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_226_Setcompr__eq__image,axiom,
! [A: $tType,B: $tType,F2: B > A,A3: set @ B] :
( ( collect @ A
@ ^ [Uu: A] :
? [X: B] :
( ( Uu
= ( F2 @ X ) )
& ( member @ B @ X @ A3 ) ) )
= ( image @ B @ A @ F2 @ A3 ) ) ).
% Setcompr_eq_image
thf(fact_227_setcompr__eq__image,axiom,
! [A: $tType,B: $tType,F2: B > A,P: B > $o] :
( ( collect @ A
@ ^ [Uu: A] :
? [X: B] :
( ( Uu
= ( F2 @ X ) )
& ( P @ X ) ) )
= ( image @ B @ A @ F2 @ ( collect @ B @ P ) ) ) ).
% setcompr_eq_image
thf(fact_228_lnull__ldropn,axiom,
! [A: $tType,N3: nat,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ ( coindu531130065ldropn @ A @ N3 @ Xs ) )
= ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ ( extended_enat2 @ N3 ) ) ) ).
% lnull_ldropn
thf(fact_229_llength__def,axiom,
! [A: $tType] :
( ( coindu1018505716length @ A )
= ( coindu1491768222unfold @ ( coindu1593790203_llist @ A ) @ ( coindu335574135_lnull @ A ) @ ( coindu1047225960le_ltl @ A ) ) ) ).
% llength_def
thf(fact_230_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_231_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_232_inf_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B2 @ C ) )
= ( ( ord_less_eq @ A @ A2 @ B2 )
& ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% inf.bounded_iff
thf(fact_233_le__inf__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A,Z: A] :
( ( ord_less_eq @ A @ X3 @ ( inf_inf @ A @ Y2 @ Z ) )
= ( ( ord_less_eq @ A @ X3 @ Y2 )
& ( ord_less_eq @ A @ X3 @ Z ) ) ) ) ).
% le_inf_iff
thf(fact_234_lnull__ldrop,axiom,
! [A: $tType,N3: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ ( coindu191418589_ldrop @ A @ N3 @ Xs ) )
= ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ N3 ) ) ).
% lnull_ldrop
thf(fact_235_le__numeral__extra_I4_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% le_numeral_extra(4)
thf(fact_236_enat__less__imp__le,axiom,
! [N3: extended_enat,M: extended_enat] :
( ! [K2: nat] :
( ( ord_less @ extended_enat @ N3 @ ( extended_enat2 @ K2 ) )
=> ( ord_less @ extended_enat @ M @ ( extended_enat2 @ K2 ) ) )
=> ( ord_less_eq @ extended_enat @ M @ N3 ) ) ).
% enat_less_imp_le
thf(fact_237_llength__ltakeWhile__le,axiom,
! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] : ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) @ ( coindu1018505716length @ A @ Xs ) ) ).
% llength_ltakeWhile_le
thf(fact_238_inf__sup__ord_I2_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ Y2 ) @ Y2 ) ) ).
% inf_sup_ord(2)
thf(fact_239_inf__sup__ord_I1_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ Y2 ) @ X3 ) ) ).
% inf_sup_ord(1)
thf(fact_240_inf__le1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ Y2 ) @ X3 ) ) ).
% inf_le1
thf(fact_241_inf__le2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [X3: A,Y2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ Y2 ) @ Y2 ) ) ).
% inf_le2
thf(fact_242_le__infE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [X3: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ X3 @ ( inf_inf @ A @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq @ A @ X3 @ A2 )
=> ~ ( ord_less_eq @ A @ X3 @ B2 ) ) ) ) ).
% le_infE
thf(fact_243_le__infI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [X3: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ X3 @ A2 )
=> ( ( ord_less_eq @ A @ X3 @ B2 )
=> ( ord_less_eq @ A @ X3 @ ( inf_inf @ A @ A2 @ B2 ) ) ) ) ) ).
% le_infI
thf(fact_244_inf__mono,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,B2: A,D: A] :
( ( ord_less_eq @ A @ A2 @ C )
=> ( ( ord_less_eq @ A @ B2 @ D )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ ( inf_inf @ A @ C @ D ) ) ) ) ) ).
% inf_mono
thf(fact_245_le__infI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [A2: A,X3: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ X3 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X3 ) ) ) ).
% le_infI1
thf(fact_246_le__infI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [B2: A,X3: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ X3 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X3 ) ) ) ).
% le_infI2
thf(fact_247_inf_OorderE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2
= ( inf_inf @ A @ A2 @ B2 ) ) ) ) ).
% inf.orderE
thf(fact_248_inf_OorderI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( A2
= ( inf_inf @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% inf.orderI
thf(fact_249_inf__unique,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [F2: A > A > A,X3: A,Y2: A] :
( ! [X2: A,Y5: A] : ( ord_less_eq @ A @ ( F2 @ X2 @ Y5 ) @ X2 )
=> ( ! [X2: A,Y5: A] : ( ord_less_eq @ A @ ( F2 @ X2 @ Y5 ) @ Y5 )
=> ( ! [X2: A,Y5: A,Z2: A] :
( ( ord_less_eq @ A @ X2 @ Y5 )
=> ( ( ord_less_eq @ A @ X2 @ Z2 )
=> ( ord_less_eq @ A @ X2 @ ( F2 @ Y5 @ Z2 ) ) ) )
=> ( ( inf_inf @ A @ X3 @ Y2 )
= ( F2 @ X3 @ Y2 ) ) ) ) ) ) ).
% inf_unique
thf(fact_250_le__iff__inf,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [X: A,Y4: A] :
( ( inf_inf @ A @ X @ Y4 )
= X ) ) ) ) ).
% le_iff_inf
thf(fact_251_inf_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( inf_inf @ A @ A2 @ B2 )
= A2 ) ) ) ).
% inf.absorb1
thf(fact_252_inf_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( inf_inf @ A @ A2 @ B2 )
= B2 ) ) ) ).
% inf.absorb2
%----Type constructors (27)
thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
! [A8: $tType,A9: $tType] :
( ( semilattice_inf @ A9 @ ( type2 @ A9 ) )
=> ( semilattice_inf @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
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___Lattices_Olattice,axiom,
! [A8: $tType,A9: $tType] :
( ( lattice @ A9 @ ( type2 @ A9 ) )
=> ( lattice @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
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_Oordered__ab__semigroup__add,axiom,
ordere779506340up_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___Lattices_Osemilattice__inf_1,axiom,
semilattice_inf @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add,axiom,
semigroup_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Lattices_Olattice_2,axiom,
lattice @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oone,axiom,
one @ nat @ ( type2 @ nat ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__inf_3,axiom,
! [A8: $tType] : ( semilattice_inf @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_4,axiom,
! [A8: $tType] :
( ( finite_finite @ A8 @ ( type2 @ A8 ) )
=> ( finite_finite @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_5,axiom,
! [A8: $tType] : ( lattice @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_6,axiom,
semilattice_inf @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_7,axiom,
finite_finite @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Olattice_8,axiom,
lattice @ $o @ ( type2 @ $o ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_9,axiom,
strict2144017051up_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_10,axiom,
ordere779506340up_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__inf_11,axiom,
semilattice_inf @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_12,axiom,
ab_semigroup_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_13,axiom,
semigroup_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Lattices_Olattice_14,axiom,
lattice @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Oone_15,axiom,
one @ extended_enat @ ( type2 @ extended_enat ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( finite_finite2 @ nat
@ ( image @ nat @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ n @ ( one_one @ nat ) ) )
@ ( inf_inf @ ( set @ nat )
@ ( collect @ nat
@ ^ [N: nat] :
( ord_less @ extended_enat @ ( extended_enat2 @ N )
@ ( coindu1018505716length @ a
@ ( coindu1047225960le_ltl @ a
@ ( coindu438612276pWhile @ a
@ ^ [X: a] :
~ ( p @ X )
@ xsa ) ) ) ) )
@ ( collect @ nat
@ ^ [N: nat] :
( p
@ ( coindu749330388e_lnth @ a
@ ( coindu1047225960le_ltl @ a
@ ( coindu438612276pWhile @ a
@ ^ [X: a] :
~ ( p @ X )
@ xsa ) )
@ N ) ) ) ) ) ) ).
%------------------------------------------------------------------------------