TPTP Problem File: DAT121^1.p
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%------------------------------------------------------------------------------
% File : DAT121^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Coinductive list 1690
% 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__1690.p [Bla16]
% Status : Theorem
% Rating : 0.00 v7.2.0, 0.25 v7.1.0
% Syntax : Number of formulae : 416 ( 179 unt; 70 typ; 0 def)
% Number of atoms : 784 ( 267 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 3118 ( 67 ~; 20 |; 22 &;2713 @)
% ( 0 <=>; 296 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 6 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 151 ( 151 >; 0 *; 0 +; 0 <<)
% Number of symbols : 66 ( 63 usr; 4 con; 0-4 aty)
% Number of variables : 750 ( 23 ^; 666 !; 6 ?; 750 :)
% ( 55 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:51:55.110
%------------------------------------------------------------------------------
%----Could-be-implicit typings (9)
thf(ty_t_Coinductive__List__Mirabelle__kmikjhschf_Ollist,type,
coindu1593790203_llist: $tType > $tType ).
thf(ty_t_Code__Numeral_Onatural,type,
code_natural: $tType ).
thf(ty_t_Extended__Nat_Oenat,type,
extended_enat: $tType ).
thf(ty_t_Real_Oreal,type,
real: $tType ).
thf(ty_t_Num_Onum,type,
num: $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_Int_Oint,type,
int: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (61)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Num_Onumeral,type,
numeral:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ominus,type,
minus:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Oring__1,type,
ring_1:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Nat_Oring__char__0,type,
ring_char_0:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Num_Oneg__numeral,type,
neg_numeral:
!>[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_Rings_Osemiring__1,type,
semiring_1:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Nat_Osemiring__char__0,type,
semiring_char_0:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocomm__monoid__diff,type,
comm_monoid_diff:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1352612707id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel146912293up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Archimedean__Field_Ofloor__ceiling,type,
archim1727834104eiling:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__group__add__abs,type,
ordere142940540dd_abs:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
comple187826305attice:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Archimedean__Field_Oarchimedean__field,type,
archim1804426504_field:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Archimedean__Field_Oceiling,type,
archimedean_ceiling:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_Archimedean__Field_Oround,type,
archimedean_round:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_Code__Numeral_OSuc,type,
code_Suc: code_natural > code_natural ).
thf(sy_c_Code__Numeral_Onatural_Ocase__natural,type,
code_case_natural:
!>[T: $tType] : ( T > ( code_natural > T ) > code_natural > T ) ).
thf(sy_c_Code__Numeral_Onatural_Orec__natural,type,
code_rec_natural:
!>[T: $tType] : ( T > ( code_natural > T > T ) > code_natural > T ) ).
thf(sy_c_Code__Numeral_Onatural_Osize__natural,type,
code_size_natural: code_natural > nat ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ogen__llength,type,
coindu493225792length:
!>[A: $tType] : ( nat > ( coindu1593790203_llist @ A ) > extended_enat ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olappend,type,
coindu268472904append:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Oldrop,type,
coindu191418589_ldrop:
!>[A: $tType] : ( extended_enat > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ 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_OLNil,type,
coindu1598213697e_LNil:
!>[A: $tType] : ( coindu1593790203_llist @ A ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Olhd,type,
coindu1046438764le_lhd:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > A ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Olnull,type,
coindu335574135_lnull:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olmember,type,
coindu567634248member:
!>[A: $tType] : ( A > ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olprefix,type,
coindu1696667936prefix:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olstrict__prefix,type,
coindu574146665prefix:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Oltake,type,
coindu1802687541_ltake:
!>[A: $tType] : ( extended_enat > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_OltakeWhile,type,
coindu721411036eWhile:
!>[A: $tType] : ( ( A > $o ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Extended__Nat_Oenat_Ocase__enat,type,
extended_case_enat:
!>[T: $tType] : ( ( nat > T ) > T > extended_enat > T ) ).
thf(sy_c_Extended__Nat_Othe__enat,type,
extended_the_enat: extended_enat > nat ).
thf(sy_c_Groups_Oabs__class_Oabs,type,
abs_abs:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_Hilbert__Choice_OLeastM,type,
hilbert_LeastM:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( A > $o ) > A ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Int_Oring__1__class_Oof__int,type,
ring_1_of_int:
!>[A: $tType] : ( int > A ) ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
semiring_1_of_nat:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Nat__Transfer_Otsub,type,
nat_tsub: int > int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl,type,
neg_numeral_dbl:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Onumeral__class_Onumeral,type,
numeral_numeral:
!>[A: $tType] : ( num > A ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_v_m,type,
m: extended_enat ).
thf(sy_v_xs,type,
xs: coindu1593790203_llist @ a ).
%----Relevant facts (250)
thf(fact_0_ldrop__LNil,axiom,
! [A: $tType,N: extended_enat] :
( ( coindu191418589_ldrop @ A @ N @ ( coindu1598213697e_LNil @ A ) )
= ( coindu1598213697e_LNil @ A ) ) ).
% ldrop_LNil
thf(fact_1_ldrop__eq__LNil,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( ( coindu191418589_ldrop @ A @ N @ Xs )
= ( coindu1598213697e_LNil @ A ) )
= ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ N ) ) ).
% ldrop_eq_LNil
thf(fact_2_wlog__linorder__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,B2: A,A2: A] :
( ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% wlog_linorder_le
thf(fact_3_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_4_lnull__ldrop,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ ( coindu191418589_ldrop @ A @ N @ Xs ) )
= ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ N ) ) ).
% lnull_ldrop
thf(fact_5_lmember__code_I1_J,axiom,
! [A: $tType,X: A] :
~ ( coindu567634248member @ A @ X @ ( coindu1598213697e_LNil @ A ) ) ).
% lmember_code(1)
thf(fact_6_llength__LNil,axiom,
! [A: $tType] :
( ( coindu1018505716length @ A @ ( coindu1598213697e_LNil @ A ) )
= ( zero_zero @ extended_enat ) ) ).
% llength_LNil
thf(fact_7_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funD
thf(fact_8_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funE
thf(fact_9_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B] :
( ! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_10_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
! [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% le_fun_def
thf(fact_11_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B2: B,C: B] :
( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C )
=> ( ! [X2: B,Y: B] :
( ( ord_less_eq @ B @ X2 @ Y )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_subst1
thf(fact_12_order__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > C2,C: C2] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C2 @ ( F @ B2 ) @ C )
=> ( ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ord_less_eq @ C2 @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_subst2
thf(fact_13_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B2: B,C: B] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C )
=> ( ! [X2: B,Y: B] :
( ( ord_less_eq @ B @ X2 @ Y )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_14_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > B,C: B] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_15_ldrop__0,axiom,
! [B: $tType,Xs: coindu1593790203_llist @ B] :
( ( coindu191418589_ldrop @ B @ ( zero_zero @ extended_enat ) @ Xs )
= Xs ) ).
% ldrop_0
thf(fact_16_llength__lnull,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu1018505716length @ A @ Xs )
= ( zero_zero @ extended_enat ) ) ) ).
% llength_lnull
thf(fact_17_llength__eq__0,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( ( coindu1018505716length @ A @ Xs )
= ( zero_zero @ extended_enat ) )
= ( coindu335574135_lnull @ A @ Xs ) ) ).
% llength_eq_0
thf(fact_18_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_19_ltake_Oexhaust,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ~ ( ( N
= ( zero_zero @ extended_enat ) )
| ( coindu335574135_lnull @ A @ Xs ) )
=> ~ ( ( N
!= ( zero_zero @ extended_enat ) )
=> ( coindu335574135_lnull @ A @ Xs ) ) ) ).
% ltake.exhaust
thf(fact_20_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_21_lnull__def,axiom,
! [A: $tType] :
( ( coindu335574135_lnull @ A )
= ( ^ [Llist: coindu1593790203_llist @ A] :
( Llist
= ( coindu1598213697e_LNil @ A ) ) ) ) ).
% lnull_def
thf(fact_22_llist_Ocollapse_I1_J,axiom,
! [A: $tType,Llist2: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Llist2 )
=> ( Llist2
= ( coindu1598213697e_LNil @ A ) ) ) ).
% llist.collapse(1)
thf(fact_23_llist_OdiscI_I1_J,axiom,
! [A: $tType,Llist2: coindu1593790203_llist @ A] :
( ( Llist2
= ( coindu1598213697e_LNil @ A ) )
=> ( coindu335574135_lnull @ A @ Llist2 ) ) ).
% llist.discI(1)
thf(fact_24_llist_Odisc_I1_J,axiom,
! [A: $tType] : ( coindu335574135_lnull @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% llist.disc(1)
thf(fact_25_ldrop__lnull,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,N: extended_enat] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu191418589_ldrop @ A @ N @ Xs )
= ( coindu1598213697e_LNil @ A ) ) ) ).
% ldrop_lnull
thf(fact_26_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% dual_order.antisym
thf(fact_27_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C @ B2 )
=> ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_28_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: A,B3: A] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_29_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_30_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ Z )
=> ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% order_trans
thf(fact_31_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ) ).
% order_class.order.antisym
thf(fact_32_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% ord_le_eq_trans
thf(fact_33_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% ord_eq_le_trans
thf(fact_34_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Y2: A,X: A] :
( ( ord_less_eq @ A @ Y2 @ X )
=> ( ( ord_less_eq @ A @ X @ Y2 )
= ( X = Y2 ) ) ) ) ).
% antisym_conv
thf(fact_35_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A,Z: A] :
( ( ( ord_less_eq @ A @ X @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ Z ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ X )
=> ~ ( ord_less_eq @ A @ X @ Z ) )
=> ( ( ( ord_less_eq @ A @ X @ Z )
=> ~ ( ord_less_eq @ A @ Z @ Y2 ) )
=> ( ( ( ord_less_eq @ A @ Z @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ X ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ Z )
=> ~ ( ord_less_eq @ A @ Z @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z @ X )
=> ~ ( ord_less_eq @ A @ X @ Y2 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_36_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% order.trans
thf(fact_37_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ~ ( ord_less_eq @ A @ X @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% le_cases
thf(fact_38_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( X = Y2 )
=> ( ord_less_eq @ A @ X @ Y2 ) ) ) ).
% eq_refl
thf(fact_39_linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
| ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% linear
thf(fact_40_antisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ X )
=> ( X = Y2 ) ) ) ) ).
% antisym
thf(fact_41_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y3: A,Z2: A] : Y3 = Z2 )
= ( ^ [X3: A,Y4: A] :
( ( ord_less_eq @ A @ X3 @ Y4 )
& ( ord_less_eq @ A @ Y4 @ X3 ) ) ) ) ) ).
% eq_iff
thf(fact_42_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_43_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X2: A] :
( ( F @ X2 )
= ( G @ X2 ) )
=> ( F = G ) ) ).
% ext
thf(fact_44_ile0__eq,axiom,
! [N: extended_enat] :
( ( ord_less_eq @ extended_enat @ N @ ( zero_zero @ extended_enat ) )
= ( N
= ( zero_zero @ extended_enat ) ) ) ).
% ile0_eq
thf(fact_45_i0__lb,axiom,
! [N: extended_enat] : ( ord_less_eq @ extended_enat @ ( zero_zero @ extended_enat ) @ N ) ).
% i0_lb
thf(fact_46_zero__le,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% zero_le
thf(fact_47_le__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% le_numeral_extra(3)
thf(fact_48_ltake_Octr_I1_J,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( ( N
= ( zero_zero @ extended_enat ) )
| ( coindu335574135_lnull @ A @ Xs ) )
=> ( ( coindu1802687541_ltake @ A @ N @ Xs )
= ( coindu1598213697e_LNil @ A ) ) ) ).
% ltake.ctr(1)
thf(fact_49_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_50_lfp_Oleq__antisym,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ X )
=> ( X = Y2 ) ) ) ) ).
% lfp.leq_antisym
thf(fact_51_ltake__LNil,axiom,
! [A: $tType,N: extended_enat] :
( ( coindu1802687541_ltake @ A @ N @ ( coindu1598213697e_LNil @ A ) )
= ( coindu1598213697e_LNil @ A ) ) ).
% ltake_LNil
thf(fact_52_lnull__ltake,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) )
= ( ( coindu335574135_lnull @ A @ Xs )
| ( N
= ( zero_zero @ extended_enat ) ) ) ) ).
% lnull_ltake
thf(fact_53_ltake_Odisc__iff_I1_J,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) )
= ( ( N
= ( zero_zero @ extended_enat ) )
| ( coindu335574135_lnull @ A @ Xs ) ) ) ).
% ltake.disc_iff(1)
thf(fact_54_ltake_Odisc__iff_I2_J,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( ~ ( coindu335574135_lnull @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) ) )
= ( ( N
!= ( zero_zero @ extended_enat ) )
& ~ ( coindu335574135_lnull @ A @ Xs ) ) ) ).
% ltake.disc_iff(2)
thf(fact_55_ltake__0,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu1802687541_ltake @ A @ ( zero_zero @ extended_enat ) @ Xs )
= ( coindu1598213697e_LNil @ A ) ) ).
% ltake_0
thf(fact_56_LNil__eq__ltake__iff,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( ( coindu1598213697e_LNil @ A )
= ( coindu1802687541_ltake @ A @ N @ Xs ) )
= ( ( Xs
= ( coindu1598213697e_LNil @ A ) )
| ( N
= ( zero_zero @ extended_enat ) ) ) ) ).
% LNil_eq_ltake_iff
thf(fact_57_ltake__eq__ltake__antimono,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,M: extended_enat] :
( ( ( coindu1802687541_ltake @ A @ N @ Xs )
= ( coindu1802687541_ltake @ A @ N @ Ys ) )
=> ( ( ord_less_eq @ extended_enat @ M @ N )
=> ( ( coindu1802687541_ltake @ A @ M @ Xs )
= ( coindu1802687541_ltake @ A @ M @ Ys ) ) ) ) ).
% ltake_eq_ltake_antimono
thf(fact_58_ltake_Odisc_I1_J,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( ( N
= ( zero_zero @ extended_enat ) )
| ( coindu335574135_lnull @ A @ Xs ) )
=> ( coindu335574135_lnull @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) ) ) ).
% ltake.disc(1)
thf(fact_59_ltake_Odisc_I2_J,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( N
!= ( zero_zero @ extended_enat ) )
=> ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ~ ( coindu335574135_lnull @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) ) ) ) ).
% ltake.disc(2)
thf(fact_60_ltake__eq__LNil__iff,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( ( coindu1802687541_ltake @ A @ N @ Xs )
= ( coindu1598213697e_LNil @ A ) )
= ( ( Xs
= ( coindu1598213697e_LNil @ A ) )
| ( N
= ( zero_zero @ extended_enat ) ) ) ) ).
% ltake_eq_LNil_iff
thf(fact_61_ltake__all,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,M: extended_enat] :
( ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ M )
=> ( ( coindu1802687541_ltake @ A @ M @ Xs )
= Xs ) ) ).
% ltake_all
thf(fact_62_lfp_Oleq__refl,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% lfp.leq_refl
thf(fact_63_gfp_Oleq__trans,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [Y2: A,X: A,Z: A] :
( ( ord_less_eq @ A @ Y2 @ X )
=> ( ( ord_less_eq @ A @ Z @ Y2 )
=> ( ord_less_eq @ A @ Z @ X ) ) ) ) ).
% gfp.leq_trans
thf(fact_64_lfp_Oleq__trans,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ Z )
=> ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% lfp.leq_trans
thf(fact_65_gfp_Oleq__antisym,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [Y2: A,X: A] :
( ( ord_less_eq @ A @ Y2 @ X )
=> ( ( ord_less_eq @ A @ X @ Y2 )
=> ( X = Y2 ) ) ) ) ).
% gfp.leq_antisym
thf(fact_66_the__enat__0,axiom,
( ( extended_the_enat @ ( zero_zero @ extended_enat ) )
= ( zero_zero @ nat ) ) ).
% the_enat_0
thf(fact_67_case__enat__0,axiom,
! [A: $tType,F: nat > A,I: A] :
( ( extended_case_enat @ A @ F @ I @ ( zero_zero @ extended_enat ) )
= ( F @ ( zero_zero @ nat ) ) ) ).
% case_enat_0
thf(fact_68_lprefix__ltake__same,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A,M: extended_enat] :
( ( coindu1696667936prefix @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) @ ( coindu1802687541_ltake @ A @ M @ Xs ) )
= ( ( ord_less_eq @ extended_enat @ N @ M )
| ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ M ) ) ) ).
% lprefix_ltake_same
thf(fact_69_ltake__lappend1,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ord_less_eq @ extended_enat @ N @ ( coindu1018505716length @ A @ Xs ) )
=> ( ( coindu1802687541_ltake @ A @ N @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( coindu1802687541_ltake @ A @ N @ Xs ) ) ) ).
% ltake_lappend1
thf(fact_70_llength__code,axiom,
! [A: $tType] :
( ( coindu1018505716length @ A )
= ( coindu493225792length @ A @ ( zero_zero @ nat ) ) ) ).
% llength_code
thf(fact_71_dbl__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A @ ( type2 @ A ) )
=> ( ( neg_numeral_dbl @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% dbl_simps(2)
thf(fact_72_ltake_Osimps_I3_J,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( N
!= ( zero_zero @ extended_enat ) )
=> ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu1046438764le_lhd @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) )
= ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% ltake.simps(3)
thf(fact_73_lprefix__refl,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ Xs @ Xs ) ).
% lprefix_refl
thf(fact_74_llist_Oleq__refl,axiom,
! [A: $tType,X: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ X @ X ) ).
% llist.leq_refl
thf(fact_75_lprefix__code_I1_J,axiom,
! [A: $tType,Ys: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ ( coindu1598213697e_LNil @ A ) @ Ys ) ).
% lprefix_code(1)
thf(fact_76_ltake__is__lprefix,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) @ Xs ) ).
% ltake_is_lprefix
thf(fact_77_lappend_Odisc__iff_I2_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ~ ( coindu335574135_lnull @ A @ ( coindu268472904append @ A @ Xs @ Ys ) ) )
= ( ~ ( coindu335574135_lnull @ A @ Xs )
| ~ ( coindu335574135_lnull @ A @ Ys ) ) ) ).
% lappend.disc_iff(2)
thf(fact_78_lnull__lappend,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( ( coindu335574135_lnull @ A @ Xs )
& ( coindu335574135_lnull @ A @ Ys ) ) ) ).
% lnull_lappend
thf(fact_79_lappend__LNil2,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ Xs @ ( coindu1598213697e_LNil @ A ) )
= Xs ) ).
% lappend_LNil2
thf(fact_80_lappend__code_I1_J,axiom,
! [A: $tType,Ys: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ ( coindu1598213697e_LNil @ A ) @ Ys )
= Ys ) ).
% lappend_code(1)
thf(fact_81_lprefix__LNil,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ ( coindu1598213697e_LNil @ A ) )
= ( coindu335574135_lnull @ A @ Xs ) ) ).
% lprefix_LNil
thf(fact_82_lhd__lappend,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu1046438764le_lhd @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( coindu1046438764le_lhd @ A @ Ys ) ) )
& ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu1046438764le_lhd @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% lhd_lappend
thf(fact_83_lhd__ltake,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( N
!= ( zero_zero @ extended_enat ) )
=> ( ( coindu1046438764le_lhd @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) )
= ( coindu1046438764le_lhd @ A @ Xs ) ) ) ).
% lhd_ltake
thf(fact_84_lprefix__lhdD,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu1046438764le_lhd @ A @ Xs )
= ( coindu1046438764le_lhd @ A @ Ys ) ) ) ) ).
% lprefix_lhdD
thf(fact_85_lappend__assoc,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ ( coindu268472904append @ A @ Xs @ Ys ) @ Zs )
= ( coindu268472904append @ A @ Xs @ ( coindu268472904append @ A @ Ys @ Zs ) ) ) ).
% lappend_assoc
thf(fact_86_lprefix__trans,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( ( coindu1696667936prefix @ A @ Ys @ Zs )
=> ( coindu1696667936prefix @ A @ Xs @ Zs ) ) ) ).
% lprefix_trans
thf(fact_87_llist_Oleq__trans,axiom,
! [A: $tType,X: coindu1593790203_llist @ A,Y2: coindu1593790203_llist @ A,Z: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ X @ Y2 )
=> ( ( coindu1696667936prefix @ A @ Y2 @ Z )
=> ( coindu1696667936prefix @ A @ X @ Z ) ) ) ).
% llist.leq_trans
thf(fact_88_lprefix__antisym,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( ( coindu1696667936prefix @ A @ Ys @ Xs )
=> ( Xs = Ys ) ) ) ).
% lprefix_antisym
thf(fact_89_lprefix__lappend,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ Xs @ ( coindu268472904append @ A @ Xs @ Ys ) ) ).
% lprefix_lappend
thf(fact_90_llist_Oleq__antisym,axiom,
! [A: $tType,X: coindu1593790203_llist @ A,Y2: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ X @ Y2 )
=> ( ( coindu1696667936prefix @ A @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% llist.leq_antisym
thf(fact_91_lprefix__down__linear,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Zs )
=> ( ( coindu1696667936prefix @ A @ Ys @ Zs )
=> ( ( coindu1696667936prefix @ A @ Xs @ Ys )
| ( coindu1696667936prefix @ A @ Ys @ Xs ) ) ) ) ).
% lprefix_down_linear
thf(fact_92_lprefix__lappend__sameI,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( coindu1696667936prefix @ A @ ( coindu268472904append @ A @ Zs @ Xs ) @ ( coindu268472904append @ A @ Zs @ Ys ) ) ) ).
% lprefix_lappend_sameI
thf(fact_93_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_94_lnull__lprefix,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( coindu1696667936prefix @ A @ Xs @ Ys ) ) ).
% lnull_lprefix
thf(fact_95_lprefix__lnull,axiom,
! [A: $tType,Ys: coindu1593790203_llist @ A,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Ys )
=> ( ( coindu1696667936prefix @ A @ Xs @ Ys )
= ( coindu335574135_lnull @ A @ Xs ) ) ) ).
% lprefix_lnull
thf(fact_96_lprefix__lnullD,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( ( coindu335574135_lnull @ A @ Ys )
=> ( coindu335574135_lnull @ A @ Xs ) ) ) ).
% lprefix_lnullD
thf(fact_97_lprefix__not__lnullD,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ~ ( coindu335574135_lnull @ A @ Ys ) ) ) ).
% lprefix_not_lnullD
thf(fact_98_lappend_Odisc_I2_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ~ ( coindu335574135_lnull @ A @ Xs )
| ~ ( coindu335574135_lnull @ A @ Ys ) )
=> ~ ( coindu335574135_lnull @ A @ ( coindu268472904append @ A @ Xs @ Ys ) ) ) ).
% lappend.disc(2)
thf(fact_99_lappend_Odisc_I1_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu335574135_lnull @ A @ Ys )
=> ( coindu335574135_lnull @ A @ ( coindu268472904append @ A @ Xs @ Ys ) ) ) ) ).
% lappend.disc(1)
thf(fact_100_lappend__lnull1,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu268472904append @ A @ Xs @ Ys )
= Ys ) ) ).
% lappend_lnull1
thf(fact_101_lappend__lnull2,axiom,
! [A: $tType,Ys: coindu1593790203_llist @ A,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Ys )
=> ( ( coindu268472904append @ A @ Xs @ Ys )
= Xs ) ) ).
% lappend_lnull2
thf(fact_102_LNil__lprefix,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ ( coindu1598213697e_LNil @ A ) @ Xs ) ).
% LNil_lprefix
thf(fact_103_lappend__eq__LNil__iff,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu268472904append @ A @ Xs @ Ys )
= ( coindu1598213697e_LNil @ A ) )
= ( ( Xs
= ( coindu1598213697e_LNil @ A ) )
& ( Ys
= ( coindu1598213697e_LNil @ A ) ) ) ) ).
% lappend_eq_LNil_iff
thf(fact_104_LNil__eq__lappend__iff,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu1598213697e_LNil @ A )
= ( coindu268472904append @ A @ Xs @ Ys ) )
= ( ( Xs
= ( coindu1598213697e_LNil @ A ) )
& ( Ys
= ( coindu1598213697e_LNil @ A ) ) ) ) ).
% LNil_eq_lappend_iff
thf(fact_105_lappend__LNil__LNil,axiom,
! [A: $tType] :
( ( coindu268472904append @ A @ ( coindu1598213697e_LNil @ A ) @ ( coindu1598213697e_LNil @ A ) )
= ( coindu1598213697e_LNil @ A ) ) ).
% lappend_LNil_LNil
thf(fact_106_lappend_Octr_I1_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu335574135_lnull @ A @ Ys )
=> ( ( coindu268472904append @ A @ Xs @ Ys )
= ( coindu1598213697e_LNil @ A ) ) ) ) ).
% lappend.ctr(1)
thf(fact_107_le0,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% le0
thf(fact_108_lstrict__prefix__def,axiom,
! [A: $tType] :
( ( coindu574146665prefix @ A )
= ( ^ [Xs2: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs2 @ Ys2 )
& ( Xs2 != Ys2 ) ) ) ) ).
% lstrict_prefix_def
thf(fact_109_ltakeWhile_Octr_I1_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
( ( ( coindu335574135_lnull @ A @ Xs )
| ~ ( P @ ( coindu1046438764le_lhd @ A @ Xs ) ) )
=> ( ( coindu721411036eWhile @ A @ P @ Xs )
= ( coindu1598213697e_LNil @ A ) ) ) ).
% ltakeWhile.ctr(1)
thf(fact_110_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_111_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_112_ltake__lappend2,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,N: extended_enat,Ys: coindu1593790203_llist @ A] :
( ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ N )
=> ( ( coindu1802687541_ltake @ A @ N @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( coindu268472904append @ A @ Xs @ ( coindu1802687541_ltake @ A @ ( minus_minus @ extended_enat @ N @ ( coindu1018505716length @ A @ Xs ) ) @ Ys ) ) ) ) ).
% ltake_lappend2
thf(fact_113_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_114_diff__zero,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% diff_zero
thf(fact_115_zero__diff,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% zero_diff
thf(fact_116_diff__0__right,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% diff_0_right
thf(fact_117_diff__self,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% diff_self
thf(fact_118_idiff__0__right,axiom,
! [N: extended_enat] :
( ( minus_minus @ extended_enat @ N @ ( zero_zero @ extended_enat ) )
= N ) ).
% idiff_0_right
thf(fact_119_idiff__0,axiom,
! [N: extended_enat] :
( ( minus_minus @ extended_enat @ ( zero_zero @ extended_enat ) @ N )
= ( zero_zero @ extended_enat ) ) ).
% idiff_0
thf(fact_120_diff__ge__0__iff__ge,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% diff_ge_0_iff_ge
thf(fact_121_le__refl,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% le_refl
thf(fact_122_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ J @ K )
=> ( ord_less_eq @ nat @ I @ K ) ) ) ).
% le_trans
thf(fact_123_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_124_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_125_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
| ( ord_less_eq @ nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_126_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A,D: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C @ D ) )
=> ( ( A2 = B2 )
= ( C = D ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_127_diff__right__commute,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,B2: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C ) @ B2 )
= ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C ) ) ) ).
% diff_right_commute
thf(fact_128_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,D: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ D @ C )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B2 @ D ) ) ) ) ) ).
% diff_mono
thf(fact_129_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C @ A2 ) @ ( minus_minus @ A @ C @ B2 ) ) ) ) ).
% diff_left_mono
thf(fact_130_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B2 @ C ) ) ) ) ).
% diff_right_mono
thf(fact_131_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A,D: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C @ D ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
= ( ord_less_eq @ A @ C @ D ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_132_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ( ( ^ [Y3: A,Z2: A] : Y3 = Z2 )
= ( ^ [A4: A,B4: A] :
( ( minus_minus @ A @ A4 @ B4 )
= ( zero_zero @ A ) ) ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_133_enat__minus__mono1,axiom,
! [X: extended_enat,Y2: extended_enat,M: extended_enat] :
( ( ord_less_eq @ extended_enat @ X @ Y2 )
=> ( ord_less_eq @ extended_enat @ ( minus_minus @ extended_enat @ X @ M ) @ ( minus_minus @ extended_enat @ Y2 @ M ) ) ) ).
% enat_minus_mono1
thf(fact_134_enat__diff__cancel__left,axiom,
! [M: extended_enat,X: extended_enat,Y2: extended_enat] :
( ( ord_less_eq @ extended_enat @ M @ X )
=> ( ( ord_less_eq @ extended_enat @ M @ Y2 )
=> ( ( ( minus_minus @ extended_enat @ X @ M )
= ( minus_minus @ extended_enat @ Y2 @ M ) )
= ( X = Y2 ) ) ) ) ).
% enat_diff_cancel_left
thf(fact_135_le__iff__diff__le__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B4: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ B4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% le_iff_diff_le_0
thf(fact_136_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% le_0_eq
thf(fact_137_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% less_eq_nat.simps(1)
thf(fact_138_ltake__lappend,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1802687541_ltake @ A @ N @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( coindu268472904append @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) @ ( coindu1802687541_ltake @ A @ ( minus_minus @ extended_enat @ N @ ( coindu1018505716length @ A @ Xs ) ) @ Ys ) ) ) ).
% ltake_lappend
thf(fact_139_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_140_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_141_ltakeWhile_Osimps_I3_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( P @ ( coindu1046438764le_lhd @ A @ Xs ) )
=> ( ( coindu1046438764le_lhd @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) )
= ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% ltakeWhile.simps(3)
thf(fact_142_minus__apply,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B @ ( type2 @ B ) )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A5: A > B,B5: A > B,X3: A] : ( minus_minus @ B @ ( A5 @ X3 ) @ ( B5 @ X3 ) ) ) ) ) ).
% minus_apply
thf(fact_143_of__nat__le__0__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [M: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) )
= ( M
= ( zero_zero @ nat ) ) ) ) ).
% of_nat_le_0_iff
thf(fact_144_ex__has__least__nat,axiom,
! [A: $tType,P: A > $o,K: A,M: A > nat] :
( ( P @ K )
=> ? [X2: A] :
( ( P @ X2 )
& ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ ( M @ X2 ) @ ( M @ Y5 ) ) ) ) ) ).
% ex_has_least_nat
thf(fact_145_zero__natural_Orsp,axiom,
( ( zero_zero @ nat )
= ( zero_zero @ nat ) ) ).
% zero_natural.rsp
thf(fact_146_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus @ nat @ M @ M )
= ( zero_zero @ nat ) ) ).
% diff_self_eq_0
thf(fact_147_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% diff_0_eq_0
thf(fact_148_of__nat__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A @ ( type2 @ A ) )
=> ! [M: nat,N: nat] :
( ( ( semiring_1_of_nat @ A @ M )
= ( semiring_1_of_nat @ A @ N ) )
= ( M = N ) ) ) ).
% of_nat_eq_iff
thf(fact_149_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ N )
=> ( ( minus_minus @ nat @ N @ ( minus_minus @ nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_150_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( minus_minus @ nat @ M @ N )
= ( zero_zero @ nat ) ) ) ).
% diff_is_0_eq'
thf(fact_151_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus @ nat @ M @ N )
= ( zero_zero @ nat ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_152_of__nat__eq__0__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A @ ( type2 @ A ) )
=> ! [M: nat] :
( ( ( semiring_1_of_nat @ A @ M )
= ( zero_zero @ A ) )
= ( M
= ( zero_zero @ nat ) ) ) ) ).
% of_nat_eq_0_iff
thf(fact_153_of__nat__0__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A @ ( type2 @ A ) )
=> ! [N: nat] :
( ( ( zero_zero @ A )
= ( semiring_1_of_nat @ A @ N ) )
= ( ( zero_zero @ nat )
= N ) ) ) ).
% of_nat_0_eq_iff
thf(fact_154_of__nat__0,axiom,
! [A: $tType] :
( ( semiring_1 @ A @ ( type2 @ A ) )
=> ( ( semiring_1_of_nat @ A @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) ) ) ).
% of_nat_0
thf(fact_155_of__nat__le__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [M: nat,N: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ) ).
% of_nat_le_iff
thf(fact_156_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus @ nat @ M @ N )
= ( zero_zero @ nat ) )
=> ( ( ( minus_minus @ nat @ N @ M )
= ( zero_zero @ nat ) )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_157_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% minus_nat.diff_0
thf(fact_158_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( ( minus_minus @ nat @ M @ K )
= ( minus_minus @ nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_159_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
= ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_160_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( minus_minus @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
= ( minus_minus @ nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_161_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ L ) @ ( minus_minus @ nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_162_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_163_le__diff__iff_H,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq @ nat @ A2 @ C )
=> ( ( ord_less_eq @ nat @ B2 @ C )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C @ A2 ) @ ( minus_minus @ nat @ C @ B2 ) )
= ( ord_less_eq @ nat @ B2 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_164_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_165_of__nat__0__le__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N ) ) ) ).
% of_nat_0_le_iff
thf(fact_166_of__nat__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A @ ( type2 @ A ) )
=> ! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ M @ N ) )
= ( minus_minus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ) ).
% of_nat_diff
thf(fact_167_fun__diff__def,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B @ ( type2 @ B ) )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A5: A > B,B5: A > B,X3: A] : ( minus_minus @ B @ ( A5 @ X3 ) @ ( B5 @ X3 ) ) ) ) ) ).
% fun_diff_def
thf(fact_168_real__arch__simple,axiom,
! [A: $tType] :
( ( archim1804426504_field @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [N2: nat] : ( ord_less_eq @ A @ X @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% real_arch_simple
thf(fact_169_natural_Osize_I1_J,axiom,
( ( code_size_natural @ ( zero_zero @ code_natural ) )
= ( zero_zero @ nat ) ) ).
% natural.size(1)
thf(fact_170_natural_Osize_I3_J,axiom,
( ( size_size @ code_natural @ ( zero_zero @ code_natural ) )
= ( zero_zero @ nat ) ) ).
% natural.size(3)
thf(fact_171_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K )
= ( minus_minus @ nat @ ( minus_minus @ nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_172_natural_Osimps_I4_J,axiom,
! [T: $tType,F1: T,F22: code_natural > T] :
( ( code_case_natural @ T @ F1 @ F22 @ ( zero_zero @ code_natural ) )
= F1 ) ).
% natural.simps(4)
thf(fact_173_natural_Osimps_I6_J,axiom,
! [T: $tType,F1: T,F22: code_natural > T > T] :
( ( code_rec_natural @ T @ F1 @ F22 @ ( zero_zero @ code_natural ) )
= F1 ) ).
% natural.simps(6)
thf(fact_174_LeastMI2,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [P: A > $o,X: A,M: A > B,Q: A > $o] :
( ( P @ X )
=> ( ! [Y: A] :
( ( P @ Y )
=> ( ord_less_eq @ B @ ( M @ X ) @ ( M @ Y ) ) )
=> ( ! [X2: A] :
( ( P @ X2 )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ B @ ( M @ X2 ) @ ( M @ Y5 ) ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( hilbert_LeastM @ A @ B @ M @ P ) ) ) ) ) ) ).
% LeastMI2
thf(fact_175_LeastM__nat__le,axiom,
! [A: $tType,P: A > $o,X: A,M: A > nat] :
( ( P @ X )
=> ( ord_less_eq @ nat @ ( M @ ( hilbert_LeastM @ A @ nat @ M @ P ) ) @ ( M @ X ) ) ) ).
% LeastM_nat_le
thf(fact_176_LeastM__nat__lemma,axiom,
! [A: $tType,P: A > $o,K: A,M: A > nat] :
( ( P @ K )
=> ( ( P @ ( hilbert_LeastM @ A @ nat @ M @ P ) )
& ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ ( M @ ( hilbert_LeastM @ A @ nat @ M @ P ) ) @ ( M @ Y5 ) ) ) ) ) ).
% LeastM_nat_lemma
thf(fact_177_natural_Osimps_I5_J,axiom,
! [T: $tType,F1: T,F22: code_natural > T,Natural: code_natural] :
( ( code_case_natural @ T @ F1 @ F22 @ ( code_Suc @ Natural ) )
= ( F22 @ Natural ) ) ).
% natural.simps(5)
thf(fact_178_transfer__int__nat__relations_I3_J,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ X ) @ ( semiring_1_of_nat @ int @ Y2 ) )
= ( ord_less_eq @ nat @ X @ Y2 ) ) ).
% transfer_int_nat_relations(3)
thf(fact_179_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% zle_int
thf(fact_180_natural_Oinject,axiom,
! [Natural: code_natural,Natural2: code_natural] :
( ( ( code_Suc @ Natural )
= ( code_Suc @ Natural2 ) )
= ( Natural = Natural2 ) ) ).
% natural.inject
thf(fact_181_natural_Osimps_I7_J,axiom,
! [T: $tType,F1: T,F22: code_natural > T > T,Natural: code_natural] :
( ( code_rec_natural @ T @ F1 @ F22 @ ( code_Suc @ Natural ) )
= ( F22 @ Natural @ ( code_rec_natural @ T @ F1 @ F22 @ Natural ) ) ) ).
% natural.simps(7)
thf(fact_182_LeastM__natI,axiom,
! [A: $tType,P: A > $o,K: A,M: A > nat] :
( ( P @ K )
=> ( P @ ( hilbert_LeastM @ A @ nat @ M @ P ) ) ) ).
% LeastM_natI
thf(fact_183_transfer__int__nat__relations_I1_J,axiom,
! [X: nat,Y2: nat] :
( ( ( semiring_1_of_nat @ int @ X )
= ( semiring_1_of_nat @ int @ Y2 ) )
= ( X = Y2 ) ) ).
% transfer_int_nat_relations(1)
thf(fact_184_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiring_1_of_nat @ int @ M )
= ( semiring_1_of_nat @ int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_185_int__if__cong,axiom,
! [P: $o,X: nat,Y2: nat] :
( ( P
=> ( ( semiring_1_of_nat @ int @ X )
= ( semiring_1_of_nat @ int @ ( if @ nat @ P @ X @ Y2 ) ) ) )
& ( ~ P
=> ( ( semiring_1_of_nat @ int @ Y2 )
= ( semiring_1_of_nat @ int @ ( if @ nat @ P @ X @ Y2 ) ) ) ) ) ).
% int_if_cong
thf(fact_186_natural_Odistinct_I2_J,axiom,
! [Natural3: code_natural] :
( ( code_Suc @ Natural3 )
!= ( zero_zero @ code_natural ) ) ).
% natural.distinct(2)
thf(fact_187_natural_Odistinct_I1_J,axiom,
! [Natural2: code_natural] :
( ( zero_zero @ code_natural )
!= ( code_Suc @ Natural2 ) ) ).
% natural.distinct(1)
thf(fact_188_natural_Oexhaust,axiom,
! [Y2: code_natural] :
( ( Y2
!= ( zero_zero @ code_natural ) )
=> ~ ! [Natural4: code_natural] :
( Y2
!= ( code_Suc @ Natural4 ) ) ) ).
% natural.exhaust
thf(fact_189_natural_Oinducts,axiom,
! [P: code_natural > $o,Natural: code_natural] :
( ( P @ ( zero_zero @ code_natural ) )
=> ( ! [Natural4: code_natural] :
( ( P @ Natural4 )
=> ( P @ ( code_Suc @ Natural4 ) ) )
=> ( P @ Natural ) ) ) ).
% natural.inducts
thf(fact_190_int__diff__cases,axiom,
! [Z: int] :
~ ! [M2: nat,N2: nat] :
( Z
!= ( minus_minus @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% int_diff_cases
thf(fact_191_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ? [N2: nat] :
( K
= ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_192_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ~ ! [N2: nat] :
( K
!= ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% nonneg_int_cases
thf(fact_193_nonneg__eq__int,axiom,
! [Z: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ~ ! [M2: nat] :
( Z
!= ( semiring_1_of_nat @ int @ M2 ) ) ) ).
% nonneg_eq_int
thf(fact_194_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J,axiom,
! [Z: nat] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( semiring_1_of_nat @ int @ Z ) ) ).
% Nat_Transfer.transfer_nat_int_function_closures(9)
thf(fact_195_transfer__int__nat__quantifiers_I1_J,axiom,
! [P: int > $o] :
( ( ! [X3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X3 )
=> ( P @ X3 ) ) )
= ( ! [X3: nat] : ( P @ ( semiring_1_of_nat @ int @ X3 ) ) ) ) ).
% transfer_int_nat_quantifiers(1)
thf(fact_196_transfer__int__nat__quantifiers_I2_J,axiom,
! [P: int > $o] :
( ( ? [X3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X3 )
& ( P @ X3 ) ) )
= ( ? [X3: nat] : ( P @ ( semiring_1_of_nat @ int @ X3 ) ) ) ) ).
% transfer_int_nat_quantifiers(2)
thf(fact_197_transfer__int__nat__numerals_I1_J,axiom,
( ( zero_zero @ int )
= ( semiring_1_of_nat @ int @ ( zero_zero @ nat ) ) ) ).
% transfer_int_nat_numerals(1)
thf(fact_198_Nat__Transfer_Otransfer__int__nat__functions_I3_J,axiom,
! [X: nat,Y2: nat] :
( ( nat_tsub @ ( semiring_1_of_nat @ int @ X ) @ ( semiring_1_of_nat @ int @ Y2 ) )
= ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ X @ Y2 ) ) ) ).
% Nat_Transfer.transfer_int_nat_functions(3)
thf(fact_199_round__of__nat,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A @ ( type2 @ A ) )
=> ! [N: nat] :
( ( archimedean_round @ A @ ( semiring_1_of_nat @ A @ N ) )
= ( semiring_1_of_nat @ int @ N ) ) ) ).
% round_of_nat
thf(fact_200_round__0,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A @ ( type2 @ A ) )
=> ( ( archimedean_round @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% round_0
thf(fact_201_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus @ int @ K @ ( zero_zero @ int ) )
= K ) ).
% minus_int_code(1)
thf(fact_202_zero__integer_Orsp,axiom,
( ( zero_zero @ int )
= ( zero_zero @ int ) ) ).
% zero_integer.rsp
thf(fact_203_tsub__def,axiom,
( nat_tsub
= ( ^ [X3: int,Y4: int] : ( if @ int @ ( ord_less_eq @ int @ Y4 @ X3 ) @ ( minus_minus @ int @ X3 @ Y4 ) @ ( zero_zero @ int ) ) ) ) ).
% tsub_def
thf(fact_204_tsub__eq,axiom,
! [Y2: int,X: int] :
( ( ord_less_eq @ int @ Y2 @ X )
=> ( ( nat_tsub @ X @ Y2 )
= ( minus_minus @ int @ X @ Y2 ) ) ) ).
% tsub_eq
thf(fact_205_round__mono,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ord_less_eq @ int @ ( archimedean_round @ A @ X ) @ ( archimedean_round @ A @ Y2 ) ) ) ) ).
% round_mono
thf(fact_206_ceiling__le__zero,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( zero_zero @ int ) )
= ( ord_less_eq @ A @ X @ ( zero_zero @ A ) ) ) ) ).
% ceiling_le_zero
thf(fact_207_of__int__0__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [Z: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z ) ) ) ).
% of_int_0_le_iff
thf(fact_208_of__int__0,axiom,
! [A: $tType] :
( ( ring_1 @ A @ ( type2 @ A ) )
=> ( ( ring_1_of_int @ A @ ( zero_zero @ int ) )
= ( zero_zero @ A ) ) ) ).
% of_int_0
thf(fact_209_of__int__0__eq__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A @ ( type2 @ A ) )
=> ! [Z: int] :
( ( ( zero_zero @ A )
= ( ring_1_of_int @ A @ Z ) )
= ( Z
= ( zero_zero @ int ) ) ) ) ).
% of_int_0_eq_iff
thf(fact_210_of__int__eq__0__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A @ ( type2 @ A ) )
=> ! [Z: int] :
( ( ( ring_1_of_int @ A @ Z )
= ( zero_zero @ A ) )
= ( Z
= ( zero_zero @ int ) ) ) ) ).
% of_int_eq_0_iff
thf(fact_211_of__int__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [W: int,Z: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less_eq @ int @ W @ Z ) ) ) ).
% of_int_le_iff
thf(fact_212_ceiling__zero,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A @ ( type2 @ A ) )
=> ( ( archimedean_ceiling @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% ceiling_zero
thf(fact_213_of__int__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A @ ( type2 @ A ) )
=> ! [W: int,Z: int] :
( ( ring_1_of_int @ A @ ( minus_minus @ int @ W @ Z ) )
= ( minus_minus @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% of_int_diff
thf(fact_214_of__int__of__nat__eq,axiom,
! [A: $tType] :
( ( ring_1 @ A @ ( type2 @ A ) )
=> ! [N: nat] :
( ( ring_1_of_int @ A @ ( semiring_1_of_nat @ int @ N ) )
= ( semiring_1_of_nat @ A @ N ) ) ) ).
% of_int_of_nat_eq
thf(fact_215_ceiling__of__nat,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A @ ( type2 @ A ) )
=> ! [N: nat] :
( ( archimedean_ceiling @ A @ ( semiring_1_of_nat @ A @ N ) )
= ( semiring_1_of_nat @ int @ N ) ) ) ).
% ceiling_of_nat
thf(fact_216_ceiling__diff__of__int,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A @ ( type2 @ A ) )
=> ! [X: A,Z: int] :
( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X @ ( ring_1_of_int @ A @ Z ) ) )
= ( minus_minus @ int @ ( archimedean_ceiling @ A @ X ) @ Z ) ) ) ).
% ceiling_diff_of_int
thf(fact_217_of__int__le__0__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [Z: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ int @ Z @ ( zero_zero @ int ) ) ) ) ).
% of_int_le_0_iff
thf(fact_218_ex__le__of__int,axiom,
! [A: $tType] :
( ( archim1804426504_field @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [Z3: int] : ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z3 ) ) ) ).
% ex_le_of_int
thf(fact_219_ceiling__mono,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A @ ( type2 @ A ) )
=> ! [Y2: A,X: A] :
( ( ord_less_eq @ A @ Y2 @ X )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ Y2 ) @ ( archimedean_ceiling @ A @ X ) ) ) ) ).
% ceiling_mono
thf(fact_220_le__of__int__ceiling,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) ) ) ) ).
% le_of_int_ceiling
thf(fact_221_ceiling__le__iff,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A @ ( type2 @ A ) )
=> ! [X: A,Z: int] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ Z )
= ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% ceiling_le_iff
thf(fact_222_of__int__nonneg,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [Z: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% of_int_nonneg
thf(fact_223_ceiling__le,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A @ ( type2 @ A ) )
=> ! [X: A,A2: int] :
( ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ A2 ) )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ A2 ) ) ) ).
% ceiling_le
thf(fact_224_of__nat__nat,axiom,
! [A: $tType] :
( ( ring_1 @ A @ ( type2 @ A ) )
=> ! [Z: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( semiring_1_of_nat @ A @ ( nat2 @ Z ) )
= ( ring_1_of_int @ A @ Z ) ) ) ) ).
% of_nat_nat
thf(fact_225_nat__ceiling__le__eq,axiom,
! [X: real,A2: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ ( archimedean_ceiling @ real @ X ) ) @ A2 )
= ( ord_less_eq @ real @ X @ ( semiring_1_of_nat @ real @ A2 ) ) ) ).
% nat_ceiling_le_eq
thf(fact_226_nat__int,axiom,
! [N: nat] :
( ( nat2 @ ( semiring_1_of_nat @ int @ N ) )
= N ) ).
% nat_int
thf(fact_227_nat__0,axiom,
( ( nat2 @ ( zero_zero @ int ) )
= ( zero_zero @ nat ) ) ).
% nat_0
thf(fact_228_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= ( zero_zero @ nat ) )
= ( ord_less_eq @ int @ I @ ( zero_zero @ int ) ) ) ).
% nat_0_iff
thf(fact_229_nat__le__0,axiom,
! [Z: int] :
( ( ord_less_eq @ int @ Z @ ( zero_zero @ int ) )
=> ( ( nat2 @ Z )
= ( zero_zero @ nat ) ) ) ).
% nat_le_0
thf(fact_230_int__nat__eq,axiom,
! [Z: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( semiring_1_of_nat @ int @ ( nat2 @ Z ) )
= Z ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( semiring_1_of_nat @ int @ ( nat2 @ Z ) )
= ( zero_zero @ int ) ) ) ) ).
% int_nat_eq
thf(fact_231_real__nat__ceiling__ge,axiom,
! [X: real] : ( ord_less_eq @ real @ X @ ( semiring_1_of_nat @ real @ ( nat2 @ ( archimedean_ceiling @ real @ X ) ) ) ) ).
% real_nat_ceiling_ge
thf(fact_232_nat__mono,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq @ int @ X @ Y2 )
=> ( ord_less_eq @ nat @ ( nat2 @ X ) @ ( nat2 @ Y2 ) ) ) ).
% nat_mono
thf(fact_233_transfer__nat__int__relations_I3_J,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
=> ( ( ord_less_eq @ nat @ ( nat2 @ X ) @ ( nat2 @ Y2 ) )
= ( ord_less_eq @ int @ X @ Y2 ) ) ) ) ).
% transfer_nat_int_relations(3)
thf(fact_234_nat__le__iff,axiom,
! [X: int,N: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ X ) @ N )
= ( ord_less_eq @ int @ X @ ( semiring_1_of_nat @ int @ N ) ) ) ).
% nat_le_iff
thf(fact_235_int__eq__iff,axiom,
! [M: nat,Z: int] :
( ( ( semiring_1_of_nat @ int @ M )
= Z )
= ( ( M
= ( nat2 @ Z ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z ) ) ) ).
% int_eq_iff
thf(fact_236_nat__0__le,axiom,
! [Z: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( semiring_1_of_nat @ int @ ( nat2 @ Z ) )
= Z ) ) ).
% nat_0_le
thf(fact_237_nat__eq__iff,axiom,
! [W: int,M: nat] :
( ( ( nat2 @ W )
= M )
= ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
=> ( W
= ( semiring_1_of_nat @ int @ M ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
=> ( M
= ( zero_zero @ nat ) ) ) ) ) ).
% nat_eq_iff
thf(fact_238_nat__eq__iff2,axiom,
! [M: nat,W: int] :
( ( M
= ( nat2 @ W ) )
= ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
=> ( W
= ( semiring_1_of_nat @ int @ M ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
=> ( M
= ( zero_zero @ nat ) ) ) ) ) ).
% nat_eq_iff2
thf(fact_239_le__nat__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less_eq @ nat @ N @ ( nat2 @ K ) )
= ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N ) @ K ) ) ) ).
% le_nat_iff
thf(fact_240_nat__diff__distrib_H,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
=> ( ( nat2 @ ( minus_minus @ int @ X @ Y2 ) )
= ( minus_minus @ nat @ ( nat2 @ X ) @ ( nat2 @ Y2 ) ) ) ) ) ).
% nat_diff_distrib'
thf(fact_241_nat__diff__distrib,axiom,
! [Z4: int,Z: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z4 )
=> ( ( ord_less_eq @ int @ Z4 @ Z )
=> ( ( nat2 @ ( minus_minus @ int @ Z @ Z4 ) )
= ( minus_minus @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z4 ) ) ) ) ) ).
% nat_diff_distrib
thf(fact_242_Nat__Transfer_Otransfer__nat__int__functions_I3_J,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
=> ( ( minus_minus @ nat @ ( nat2 @ X ) @ ( nat2 @ Y2 ) )
= ( nat2 @ ( nat_tsub @ X @ Y2 ) ) ) ) ) ).
% Nat_Transfer.transfer_nat_int_functions(3)
thf(fact_243_int__eq__iff__numeral,axiom,
! [M: nat,V: num] :
( ( ( semiring_1_of_nat @ int @ M )
= ( numeral_numeral @ int @ V ) )
= ( ( M
= ( nat2 @ ( numeral_numeral @ int @ V ) ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( numeral_numeral @ int @ V ) ) ) ) ).
% int_eq_iff_numeral
thf(fact_244_nat__abs__int__diff,axiom,
! [A2: nat,B2: nat] :
( ( ( ord_less_eq @ nat @ A2 @ B2 )
=> ( ( nat2 @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) )
= ( minus_minus @ nat @ B2 @ A2 ) ) )
& ( ~ ( ord_less_eq @ nat @ A2 @ B2 )
=> ( ( nat2 @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) )
= ( minus_minus @ nat @ A2 @ B2 ) ) ) ) ).
% nat_abs_int_diff
thf(fact_245_numeral__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A @ ( type2 @ A ) )
=> ! [M: num,N: num] :
( ( ( numeral_numeral @ A @ M )
= ( numeral_numeral @ A @ N ) )
= ( M = N ) ) ) ).
% numeral_eq_iff
thf(fact_246_abs__idempotent,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( abs_abs @ A @ ( abs_abs @ A @ A2 ) )
= ( abs_abs @ A @ A2 ) ) ) ).
% abs_idempotent
thf(fact_247_numeral__le__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [M: num,N: num] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less_eq @ num @ M @ N ) ) ) ).
% numeral_le_iff
thf(fact_248_abs__0__eq,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( abs_abs @ A @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% abs_0_eq
thf(fact_249_abs__eq__0,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ( abs_abs @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% abs_eq_0
%----Type constructors (91)
thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A6: $tType,A7: $tType] :
( ( comple187826305attice @ A7 @ ( type2 @ A7 ) )
=> ( comple187826305attice @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A6: $tType,A7: $tType] :
( ( preorder @ A7 @ ( type2 @ A7 ) )
=> ( preorder @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A6: $tType,A7: $tType] :
( ( order @ A7 @ ( type2 @ A7 ) )
=> ( order @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A6: $tType,A7: $tType] :
( ( ord @ A7 @ ( type2 @ A7 ) )
=> ( ord @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Groups_Ominus,axiom,
! [A6: $tType,A7: $tType] :
( ( minus @ A7 @ ( type2 @ A7 ) )
=> ( minus @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_Int_Oint___Groups_Oordered__ab__group__add__abs,axiom,
ordere142940540dd_abs @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
cancel1352612707id_add @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
linordered_semidom @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
semiring_char_0 @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Orderings_Opreorder_1,axiom,
preorder @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
linorder @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1 @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
group_add @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Orderings_Oorder_2,axiom,
order @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Num_Oneg__numeral,axiom,
neg_numeral @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Nat_Oring__char__0,axiom,
ring_char_0 @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Orderings_Oord_3,axiom,
ord @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Rings_Oring__1,axiom,
ring_1 @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Groups_Ominus_4,axiom,
minus @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Num_Onumeral,axiom,
numeral @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Groups_Ozero,axiom,
zero @ int @ ( type2 @ int ) ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_5,axiom,
cancel146912293up_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_6,axiom,
cancel1352612707id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom_7,axiom,
linordered_semidom @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
comm_monoid_diff @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Nat_Osemiring__char__0_8,axiom,
semiring_char_0 @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Opreorder_9,axiom,
preorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Olinorder_10,axiom,
linorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Osemiring__1_11,axiom,
semiring_1 @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oorder_12,axiom,
order @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oord_13,axiom,
ord @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ominus_14,axiom,
minus @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Num_Onumeral_15,axiom,
numeral @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ozero_16,axiom,
zero @ nat @ ( type2 @ nat ) ).
thf(tcon_Num_Onum___Orderings_Opreorder_17,axiom,
preorder @ num @ ( type2 @ num ) ).
thf(tcon_Num_Onum___Orderings_Olinorder_18,axiom,
linorder @ num @ ( type2 @ num ) ).
thf(tcon_Num_Onum___Orderings_Oorder_19,axiom,
order @ num @ ( type2 @ num ) ).
thf(tcon_Num_Onum___Orderings_Oord_20,axiom,
ord @ num @ ( type2 @ num ) ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_21,axiom,
comple187826305attice @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_22,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder_23,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_24,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_25,axiom,
ord @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Groups_Ominus_26,axiom,
minus @ $o @ ( type2 @ $o ) ).
thf(tcon_Real_Oreal___Archimedean__Field_Oarchimedean__field,axiom,
archim1804426504_field @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_27,axiom,
ordere142940540dd_abs @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling,axiom,
archim1727834104eiling @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_28,axiom,
cancel146912293up_add @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_29,axiom,
cancel1352612707id_add @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add_30,axiom,
ordered_ab_group_add @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Rings_Olinordered__semidom_31,axiom,
linordered_semidom @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Rings_Olinordered__idom_32,axiom,
linordered_idom @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Nat_Osemiring__char__0_33,axiom,
semiring_char_0 @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Orderings_Opreorder_34,axiom,
preorder @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Orderings_Olinorder_35,axiom,
linorder @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Rings_Osemiring__1_36,axiom,
semiring_1 @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Ogroup__add_37,axiom,
group_add @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Orderings_Oorder_38,axiom,
order @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Num_Oneg__numeral_39,axiom,
neg_numeral @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Nat_Oring__char__0_40,axiom,
ring_char_0 @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Orderings_Oord_41,axiom,
ord @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Rings_Oring__1_42,axiom,
ring_1 @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Ominus_43,axiom,
minus @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Num_Onumeral_44,axiom,
numeral @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Ozero_45,axiom,
zero @ real @ ( type2 @ real ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_46,axiom,
canoni770627133id_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__lattice_47,axiom,
comple187826305attice @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_48,axiom,
semiring_char_0 @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Opreorder_49,axiom,
preorder @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_50,axiom,
linorder @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_51,axiom,
semiring_1 @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_52,axiom,
order @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oord_53,axiom,
ord @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Ominus_54,axiom,
minus @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Num_Onumeral_55,axiom,
numeral @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Ozero_56,axiom,
zero @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__ab__semigroup__add_57,axiom,
cancel146912293up_add @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__comm__monoid__add_58,axiom,
cancel1352612707id_add @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__diff_59,axiom,
comm_monoid_diff @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Orderings_Opreorder_60,axiom,
preorder @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Orderings_Olinorder_61,axiom,
linorder @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_62,axiom,
semiring_1 @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Orderings_Oorder_63,axiom,
order @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Orderings_Oord_64,axiom,
ord @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Ominus_65,axiom,
minus @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Num_Onumeral_66,axiom,
numeral @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Ozero_67,axiom,
zero @ code_natural @ ( type2 @ code_natural ) ).
%----Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y2: A] :
( ( if @ A @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y2: A] :
( ( if @ A @ $true @ X @ Y2 )
= X ) ).
%----Conjectures (2)
thf(conj_0,hypothesis,
ord_less_eq @ extended_enat @ ( coindu1018505716length @ a @ xs ) @ m ).
thf(conj_1,conjecture,
( ( coindu191418589_ldrop @ a @ m @ xs )
= ( coindu1598213697e_LNil @ a ) ) ).
%------------------------------------------------------------------------------