TPTP Problem File: DAT128^1.p
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%------------------------------------------------------------------------------
% File : DAT128^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Coinductive list 3119
% 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__3119.p [Bla16]
% Status : Theorem
% Rating : 0.67 v8.2.0, 0.33 v8.1.0, 0.75 v7.5.0, 1.00 v7.1.0
% Syntax : Number of formulae : 330 ( 111 unt; 44 typ; 0 def)
% Number of atoms : 797 ( 203 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 3427 ( 89 ~; 15 |; 47 &;2877 @)
% ( 0 <=>; 399 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 8 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 175 ( 175 >; 0 *; 0 +; 0 <<)
% Number of symbols : 41 ( 40 usr; 4 con; 0-5 aty)
% Number of variables : 1007 ( 36 ^; 907 !; 31 ?;1007 :)
% ( 33 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:59:56.275
%------------------------------------------------------------------------------
%----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_Nat_Onat,type,
nat: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_b,type,
b: $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (38)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Ono__top,type,
no_top:
!>[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_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere779506340up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere236663937imp_le:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
ordere223160158up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
strict2144017051up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ogen__llength,type,
coindu493225792length:
!>[A: $tType] : ( nat > ( coindu1593790203_llist @ A ) > extended_enat ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Oiterates,type,
coindu1529956631erates:
!>[A: $tType] : ( ( A > A ) > A > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olappend,type,
coindu268472904append:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olfinite,type,
coindu1213758845finite:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollength,type,
coindu1018505716length:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > extended_enat ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_OLCons,type,
coindu1121789889_LCons:
!>[A: $tType] : ( A > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_OLNil,type,
coindu1598213697e_LNil:
!>[A: $tType] : ( coindu1593790203_llist @ A ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Ollist__all2,type,
coindu987187967t_all2:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ B ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Olnull,type,
coindu335574135_lnull:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olnth,type,
coindu749330388e_lnth:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > nat > A ) ).
thf(sy_c_Extended__Nat_Oenat,type,
extended_enat2: nat > extended_enat ).
thf(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[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_v_P,type,
p: a > b > $o ).
thf(sy_v_n____,type,
n: nat ).
thf(sy_v_xs,type,
xs: coindu1593790203_llist @ a ).
thf(sy_v_xs_H,type,
xs2: coindu1593790203_llist @ a ).
thf(sy_v_ys,type,
ys: coindu1593790203_llist @ b ).
%----Relevant facts (254)
thf(fact_0_assms,axiom,
coindu987187967t_all2 @ a @ b @ p @ ( coindu268472904append @ a @ xs @ xs2 ) @ ys ).
% assms
thf(fact_1_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_2__092_060open_062llength_Axs_A_092_060le_062_Allength_A_Ilappend_Axs_Axs_H_J_092_060close_062,axiom,
ord_less_eq @ extended_enat @ ( coindu1018505716length @ a @ xs ) @ ( coindu1018505716length @ a @ ( coindu268472904append @ a @ xs @ xs2 ) ) ).
% \<open>llength xs \<le> llength (lappend xs xs')\<close>
thf(fact_3_n,axiom,
ord_less @ extended_enat @ ( extended_enat2 @ n ) @ ( coindu1018505716length @ a @ xs ) ).
% n
thf(fact_4_len,axiom,
( ( plus_plus @ extended_enat @ ( coindu1018505716length @ a @ xs ) @ ( coindu1018505716length @ a @ xs2 ) )
= ( coindu1018505716length @ b @ ys ) ) ).
% len
thf(fact_5_len__xs_H,axiom,
ord_less_eq @ extended_enat @ ( coindu1018505716length @ a @ xs2 ) @ ( coindu1018505716length @ b @ ys ) ).
% len_xs'
thf(fact_6_len__xs,axiom,
ord_less_eq @ extended_enat @ ( coindu1018505716length @ a @ xs ) @ ( coindu1018505716length @ b @ ys ) ).
% len_xs
thf(fact_7_lappend__code_I1_J,axiom,
! [A: $tType,Ys: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ ( coindu1598213697e_LNil @ A ) @ Ys )
= Ys ) ).
% lappend_code(1)
thf(fact_8_lappend__LNil2,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ Xs @ ( coindu1598213697e_LNil @ A ) )
= Xs ) ).
% lappend_LNil2
thf(fact_9_lappend__iterates,axiom,
! [A: $tType,F: A > A,X: A,Xs: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ ( coindu1529956631erates @ A @ F @ X ) @ Xs )
= ( coindu1529956631erates @ A @ F @ X ) ) ).
% lappend_iterates
thf(fact_10_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_11_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_12_lappend__code_I2_J,axiom,
! [A: $tType,Xa: A,X: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ ( coindu1121789889_LCons @ A @ Xa @ X ) @ Ys )
= ( coindu1121789889_LCons @ A @ Xa @ ( coindu268472904append @ A @ X @ Ys ) ) ) ).
% lappend_code(2)
thf(fact_13_lfinite__lappend,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( ( coindu1213758845finite @ A @ Xs )
& ( coindu1213758845finite @ A @ Ys ) ) ) ).
% lfinite_lappend
thf(fact_14_llist_Oinject,axiom,
! [A: $tType,X21: A,X22: coindu1593790203_llist @ A,Y21: A,Y22: coindu1593790203_llist @ A] :
( ( ( coindu1121789889_LCons @ A @ X21 @ X22 )
= ( coindu1121789889_LCons @ A @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% llist.inject
thf(fact_15_llist__all2__LCons__LCons,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X21: A,X22: coindu1593790203_llist @ A,Y21: B,Y22: coindu1593790203_llist @ B] :
( ( coindu987187967t_all2 @ A @ B @ R @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) @ ( coindu1121789889_LCons @ B @ Y21 @ Y22 ) )
= ( ( R @ X21 @ Y21 )
& ( coindu987187967t_all2 @ A @ B @ R @ X22 @ Y22 ) ) ) ).
% llist_all2_LCons_LCons
thf(fact_16_lfinite__code_I2_J,axiom,
! [B: $tType,X: B,Xs: coindu1593790203_llist @ B] :
( ( coindu1213758845finite @ B @ ( coindu1121789889_LCons @ B @ X @ Xs ) )
= ( coindu1213758845finite @ B @ Xs ) ) ).
% lfinite_code(2)
thf(fact_17_lfinite__LCons,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) )
= ( coindu1213758845finite @ A @ Xs ) ) ).
% lfinite_LCons
thf(fact_18_llist__all2__LNil1,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: coindu1593790203_llist @ B] :
( ( coindu987187967t_all2 @ A @ B @ P @ ( coindu1598213697e_LNil @ A ) @ Xs )
= ( Xs
= ( coindu1598213697e_LNil @ B ) ) ) ).
% llist_all2_LNil1
thf(fact_19_llist__all2__LNil2,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Xs: coindu1593790203_llist @ A] :
( ( coindu987187967t_all2 @ A @ B @ P @ Xs @ ( coindu1598213697e_LNil @ B ) )
= ( Xs
= ( coindu1598213697e_LNil @ A ) ) ) ).
% llist_all2_LNil2
thf(fact_20_lfinite__code_I1_J,axiom,
! [A: $tType] : ( coindu1213758845finite @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% lfinite_code(1)
thf(fact_21_llength__lappend,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1018505716length @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( plus_plus @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ ( coindu1018505716length @ A @ Ys ) ) ) ).
% llength_lappend
thf(fact_22_llist_Odisc_I2_J,axiom,
! [A: $tType,X21: A,X22: coindu1593790203_llist @ A] :
~ ( coindu335574135_lnull @ A @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) ) ).
% llist.disc(2)
thf(fact_23_llist_Odisc_I1_J,axiom,
! [A: $tType] : ( coindu335574135_lnull @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% llist.disc(1)
thf(fact_24_llist_OdiscI_I2_J,axiom,
! [A: $tType,Llist: coindu1593790203_llist @ A,X21: A,X22: coindu1593790203_llist @ A] :
( ( Llist
= ( coindu1121789889_LCons @ A @ X21 @ X22 ) )
=> ~ ( coindu335574135_lnull @ A @ Llist ) ) ).
% llist.discI(2)
thf(fact_25_llist_OdiscI_I1_J,axiom,
! [A: $tType,Llist: coindu1593790203_llist @ A] :
( ( Llist
= ( coindu1598213697e_LNil @ A ) )
=> ( coindu335574135_lnull @ A @ Llist ) ) ).
% llist.discI(1)
thf(fact_26_llist_Ocollapse_I1_J,axiom,
! [A: $tType,Llist: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Llist )
=> ( Llist
= ( coindu1598213697e_LNil @ A ) ) ) ).
% llist.collapse(1)
thf(fact_27_llist_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: coindu1593790203_llist @ A] :
( ( coindu1598213697e_LNil @ A )
!= ( coindu1121789889_LCons @ A @ X21 @ X22 ) ) ).
% llist.distinct(1)
thf(fact_28_llist_Orel__intros_I2_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X21: A,Y21: B,X22: coindu1593790203_llist @ A,Y22: coindu1593790203_llist @ B] :
( ( R @ X21 @ Y21 )
=> ( ( coindu987187967t_all2 @ A @ B @ R @ X22 @ Y22 )
=> ( coindu987187967t_all2 @ A @ B @ R @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) @ ( coindu1121789889_LCons @ B @ Y21 @ Y22 ) ) ) ) ).
% llist.rel_intros(2)
thf(fact_29_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_30_iterates_Ocode,axiom,
! [A: $tType] :
( ( coindu1529956631erates @ A )
= ( ^ [F2: A > A,X2: A] : ( coindu1121789889_LCons @ A @ X2 @ ( coindu1529956631erates @ A @ F2 @ ( F2 @ X2 ) ) ) ) ) ).
% iterates.code
thf(fact_31_lfinite_Ocases,axiom,
! [A: $tType,A2: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ A2 )
=> ( ( A2
!= ( coindu1598213697e_LNil @ A ) )
=> ~ ! [Xs2: coindu1593790203_llist @ A] :
( ? [X3: A] :
( A2
= ( coindu1121789889_LCons @ A @ X3 @ Xs2 ) )
=> ~ ( coindu1213758845finite @ A @ Xs2 ) ) ) ) ).
% lfinite.cases
thf(fact_32_lfinite_Osimps,axiom,
! [A: $tType] :
( ( coindu1213758845finite @ A )
= ( ^ [A3: coindu1593790203_llist @ A] :
( ( A3
= ( coindu1598213697e_LNil @ A ) )
| ? [Xs3: coindu1593790203_llist @ A,X2: A] :
( ( A3
= ( coindu1121789889_LCons @ A @ X2 @ Xs3 ) )
& ( coindu1213758845finite @ A @ Xs3 ) ) ) ) ) ).
% lfinite.simps
thf(fact_33_llist_Oexhaust,axiom,
! [A: $tType,Y: coindu1593790203_llist @ A] :
( ( Y
!= ( coindu1598213697e_LNil @ A ) )
=> ~ ! [X212: A,X222: coindu1593790203_llist @ A] :
( Y
!= ( coindu1121789889_LCons @ A @ X212 @ X222 ) ) ) ).
% llist.exhaust
thf(fact_34_neq__LNil__conv,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( Xs
!= ( coindu1598213697e_LNil @ A ) )
= ( ? [X2: A,Xs4: coindu1593790203_llist @ A] :
( Xs
= ( coindu1121789889_LCons @ A @ X2 @ Xs4 ) ) ) ) ).
% neq_LNil_conv
thf(fact_35_not__lnull__conv,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( ~ ( coindu335574135_lnull @ A @ Xs ) )
= ( ? [X2: A,Xs4: coindu1593790203_llist @ A] :
( Xs
= ( coindu1121789889_LCons @ A @ X2 @ Xs4 ) ) ) ) ).
% not_lnull_conv
thf(fact_36_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_37_lfinite_Oinducts,axiom,
! [A: $tType,X: coindu1593790203_llist @ A,P: ( coindu1593790203_llist @ A ) > $o] :
( ( coindu1213758845finite @ A @ X )
=> ( ( P @ ( coindu1598213697e_LNil @ A ) )
=> ( ! [Xs2: coindu1593790203_llist @ A,X3: A] :
( ( coindu1213758845finite @ A @ Xs2 )
=> ( ( P @ Xs2 )
=> ( P @ ( coindu1121789889_LCons @ A @ X3 @ Xs2 ) ) ) )
=> ( P @ X ) ) ) ) ).
% lfinite.inducts
thf(fact_38_lnull__def,axiom,
! [A: $tType] :
( ( coindu335574135_lnull @ A )
= ( ^ [Llist2: coindu1593790203_llist @ A] :
( Llist2
= ( coindu1598213697e_LNil @ A ) ) ) ) ).
% lnull_def
thf(fact_39_llist_Orel__cases,axiom,
! [A: $tType,B: $tType,R: A > B > $o,A2: coindu1593790203_llist @ A,B2: coindu1593790203_llist @ B] :
( ( coindu987187967t_all2 @ A @ B @ R @ A2 @ B2 )
=> ( ( ( A2
= ( coindu1598213697e_LNil @ A ) )
=> ( B2
!= ( coindu1598213697e_LNil @ B ) ) )
=> ~ ! [X1: A,X23: coindu1593790203_llist @ A] :
( ( A2
= ( coindu1121789889_LCons @ A @ X1 @ X23 ) )
=> ! [Y1: B,Y2: coindu1593790203_llist @ B] :
( ( B2
= ( coindu1121789889_LCons @ B @ Y1 @ Y2 ) )
=> ( ( R @ X1 @ Y1 )
=> ~ ( coindu987187967t_all2 @ A @ B @ R @ X23 @ Y2 ) ) ) ) ) ) ).
% llist.rel_cases
thf(fact_40_lfinite__iterates,axiom,
! [A: $tType,F: A > A,X: A] :
~ ( coindu1213758845finite @ A @ ( coindu1529956631erates @ A @ F @ X ) ) ).
% lfinite_iterates
thf(fact_41_llist__all2__cases,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B] :
( ( coindu987187967t_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ( ( Xs
= ( coindu1598213697e_LNil @ A ) )
=> ( Ys
!= ( coindu1598213697e_LNil @ B ) ) )
=> ~ ! [X3: A,Xs5: coindu1593790203_llist @ A] :
( ( Xs
= ( coindu1121789889_LCons @ A @ X3 @ Xs5 ) )
=> ! [Y3: B,Ys2: coindu1593790203_llist @ B] :
( ( Ys
= ( coindu1121789889_LCons @ B @ Y3 @ Ys2 ) )
=> ( ( P @ X3 @ Y3 )
=> ~ ( coindu987187967t_all2 @ A @ B @ P @ Xs5 @ Ys2 ) ) ) ) ) ) ).
% llist_all2_cases
thf(fact_42_llist__all2__lnthD,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B,N: nat] :
( ( coindu987187967t_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ( ord_less @ extended_enat @ ( extended_enat2 @ N ) @ ( coindu1018505716length @ A @ Xs ) )
=> ( P @ ( coindu749330388e_lnth @ A @ Xs @ N ) @ ( coindu749330388e_lnth @ B @ Ys @ N ) ) ) ) ).
% llist_all2_lnthD
thf(fact_43_wlog__linorder__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,B2: A,A2: A] :
( ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% wlog_linorder_le
thf(fact_44_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_45_iterates_Odisc__iff,axiom,
! [A: $tType,F: A > A,X: A] :
~ ( coindu335574135_lnull @ A @ ( coindu1529956631erates @ A @ F @ X ) ) ).
% iterates.disc_iff
thf(fact_46_llist__all2__LCons1,axiom,
! [A: $tType,B: $tType,P: A > B > $o,X: A,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B] :
( ( coindu987187967t_all2 @ A @ B @ P @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ Ys )
= ( ? [Y4: B,Ys3: coindu1593790203_llist @ B] :
( ( Ys
= ( coindu1121789889_LCons @ B @ Y4 @ Ys3 ) )
& ( P @ X @ Y4 )
& ( coindu987187967t_all2 @ A @ B @ P @ Xs @ Ys3 ) ) ) ) ).
% llist_all2_LCons1
thf(fact_47_llist__all2__LCons2,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Xs: coindu1593790203_llist @ A,Y: B,Ys: coindu1593790203_llist @ B] :
( ( coindu987187967t_all2 @ A @ B @ P @ Xs @ ( coindu1121789889_LCons @ B @ Y @ Ys ) )
= ( ? [X2: A,Xs4: coindu1593790203_llist @ A] :
( ( Xs
= ( coindu1121789889_LCons @ A @ X2 @ Xs4 ) )
& ( P @ X2 @ Y )
& ( coindu987187967t_all2 @ A @ B @ P @ Xs4 @ Ys ) ) ) ) ).
% llist_all2_LCons2
thf(fact_48_llist__all2__lnthD2,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B,N: nat] :
( ( coindu987187967t_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ( ord_less @ extended_enat @ ( extended_enat2 @ N ) @ ( coindu1018505716length @ B @ Ys ) )
=> ( P @ ( coindu749330388e_lnth @ A @ Xs @ N ) @ ( coindu749330388e_lnth @ B @ Ys @ N ) ) ) ) ).
% llist_all2_lnthD2
thf(fact_49_llist__all2__lnullD,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B] :
( ( coindu987187967t_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ( coindu335574135_lnull @ A @ Xs )
= ( coindu335574135_lnull @ B @ Ys ) ) ) ).
% llist_all2_lnullD
thf(fact_50_lnull__imp__lfinite,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( coindu1213758845finite @ A @ Xs ) ) ).
% lnull_imp_lfinite
thf(fact_51_lfinite__rev__induct,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: ( coindu1593790203_llist @ A ) > $o] :
( ( coindu1213758845finite @ A @ Xs )
=> ( ( P @ ( coindu1598213697e_LNil @ A ) )
=> ( ! [X3: A,Xs2: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ Xs2 )
=> ( ( P @ Xs2 )
=> ( P @ ( coindu268472904append @ A @ Xs2 @ ( coindu1121789889_LCons @ A @ X3 @ ( coindu1598213697e_LNil @ A ) ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lfinite_rev_induct
thf(fact_52_llist__all2__lfiniteD,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B] :
( ( coindu987187967t_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ( coindu1213758845finite @ A @ Xs )
= ( coindu1213758845finite @ B @ Ys ) ) ) ).
% llist_all2_lfiniteD
thf(fact_53_llist__all2__llengthD,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B] :
( ( coindu987187967t_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ( coindu1018505716length @ A @ Xs )
= ( coindu1018505716length @ B @ Ys ) ) ) ).
% llist_all2_llengthD
thf(fact_54_lfinite__LNil,axiom,
! [A: $tType] : ( coindu1213758845finite @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% lfinite_LNil
thf(fact_55_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_56_llist__all2__LNil__LNil,axiom,
! [A: $tType,B: $tType,R: A > B > $o] : ( coindu987187967t_all2 @ A @ B @ R @ ( coindu1598213697e_LNil @ A ) @ ( coindu1598213697e_LNil @ B ) ) ).
% llist_all2_LNil_LNil
thf(fact_57_llist__all2__all__lnthI,axiom,
! [A: $tType,B: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B,P: A > B > $o] :
( ( ( coindu1018505716length @ A @ Xs )
= ( coindu1018505716length @ B @ Ys ) )
=> ( ! [N2: nat] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ N2 ) @ ( coindu1018505716length @ A @ Xs ) )
=> ( P @ ( coindu749330388e_lnth @ A @ Xs @ N2 ) @ ( coindu749330388e_lnth @ B @ Ys @ N2 ) ) )
=> ( coindu987187967t_all2 @ A @ B @ P @ Xs @ Ys ) ) ) ).
% llist_all2_all_lnthI
thf(fact_58_llist__all2__LCons__LNil,axiom,
! [A: $tType,B: $tType,R: A > B > $o,Y21: A,Y22: coindu1593790203_llist @ A] :
~ ( coindu987187967t_all2 @ A @ B @ R @ ( coindu1121789889_LCons @ A @ Y21 @ Y22 ) @ ( coindu1598213697e_LNil @ B ) ) ).
% llist_all2_LCons_LNil
thf(fact_59_llist__all2__LNil__LCons,axiom,
! [A: $tType,B: $tType,R: A > B > $o,Y21: B,Y22: coindu1593790203_llist @ B] :
~ ( coindu987187967t_all2 @ A @ B @ R @ ( coindu1598213697e_LNil @ A ) @ ( coindu1121789889_LCons @ B @ Y21 @ Y22 ) ) ).
% llist_all2_LNil_LCons
thf(fact_60_lfinite__LConsI,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,X: A] :
( ( coindu1213758845finite @ A @ Xs )
=> ( coindu1213758845finite @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) ) ) ).
% lfinite_LConsI
thf(fact_61_llength__eq__enat__lfiniteD,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,N: nat] :
( ( ( coindu1018505716length @ A @ Xs )
= ( extended_enat2 @ N ) )
=> ( coindu1213758845finite @ A @ Xs ) ) ).
% llength_eq_enat_lfiniteD
thf(fact_62_llist__all2__conv__all__lnth,axiom,
! [B: $tType,A: $tType] :
( ( coindu987187967t_all2 @ A @ B )
= ( ^ [P2: A > B > $o,Xs3: coindu1593790203_llist @ A,Ys4: coindu1593790203_llist @ B] :
( ( ( coindu1018505716length @ A @ Xs3 )
= ( coindu1018505716length @ B @ Ys4 ) )
& ! [N3: nat] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ N3 ) @ ( coindu1018505716length @ B @ Ys4 ) )
=> ( P2 @ ( coindu749330388e_lnth @ A @ Xs3 @ N3 ) @ ( coindu749330388e_lnth @ B @ Ys4 @ N3 ) ) ) ) ) ) ).
% llist_all2_conv_all_lnth
thf(fact_63_lfinite__conv__llength__enat,axiom,
! [A: $tType] :
( ( coindu1213758845finite @ A )
= ( ^ [Xs3: coindu1593790203_llist @ A] :
? [N3: nat] :
( ( coindu1018505716length @ A @ Xs3 )
= ( extended_enat2 @ N3 ) ) ) ) ).
% lfinite_conv_llength_enat
thf(fact_64_lappend__eq__lappend__conv,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Us: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Vs: coindu1593790203_llist @ A] :
( ( ( coindu1018505716length @ A @ Xs )
= ( coindu1018505716length @ A @ Us ) )
=> ( ( ( coindu268472904append @ A @ Xs @ Ys )
= ( coindu268472904append @ A @ Us @ Vs ) )
= ( ( Xs = Us )
& ( ( coindu1213758845finite @ A @ Xs )
=> ( Ys = Vs ) ) ) ) ) ).
% lappend_eq_lappend_conv
thf(fact_65_lappend__snocL1__conv__LCons2,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Y: A,Ys: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ ( coindu268472904append @ A @ Xs @ ( coindu1121789889_LCons @ A @ Y @ ( coindu1598213697e_LNil @ A ) ) ) @ Ys )
= ( coindu268472904append @ A @ Xs @ ( coindu1121789889_LCons @ A @ Y @ Ys ) ) ) ).
% lappend_snocL1_conv_LCons2
thf(fact_66_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_67_lnth__lappend1,axiom,
! [A: $tType,N: nat,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ N ) @ ( coindu1018505716length @ A @ Xs ) )
=> ( ( coindu749330388e_lnth @ A @ ( coindu268472904append @ A @ Xs @ Ys ) @ N )
= ( coindu749330388e_lnth @ A @ Xs @ N ) ) ) ).
% lnth_lappend1
thf(fact_68_llist__all2__lappendI,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B,Xs6: coindu1593790203_llist @ A,Ys5: coindu1593790203_llist @ B] :
( ( coindu987187967t_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ( ( coindu1213758845finite @ A @ Xs )
=> ( ( coindu1213758845finite @ B @ Ys )
=> ( coindu987187967t_all2 @ A @ B @ P @ Xs6 @ Ys5 ) ) )
=> ( coindu987187967t_all2 @ A @ B @ P @ ( coindu268472904append @ A @ Xs @ Xs6 ) @ ( coindu268472904append @ B @ Ys @ Ys5 ) ) ) ) ).
% llist_all2_lappendI
thf(fact_69_llist_Orel__refl,axiom,
! [B: $tType,Ra: B > B > $o,X: coindu1593790203_llist @ B] :
( ! [X3: B] : ( Ra @ X3 @ X3 )
=> ( coindu987187967t_all2 @ B @ B @ Ra @ X @ X ) ) ).
% llist.rel_refl
thf(fact_70_llist_Orel__eq,axiom,
! [A: $tType] :
( ( coindu987187967t_all2 @ A @ A
@ ^ [Y5: A,Z: A] : Y5 = Z )
= ( ^ [Y5: coindu1593790203_llist @ A,Z: coindu1593790203_llist @ A] : Y5 = Z ) ) ).
% llist.rel_eq
thf(fact_71_llist__all2__mono,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B,P3: A > B > $o] :
( ( coindu987187967t_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ! [X3: A,Y3: B] :
( ( P @ X3 @ Y3 )
=> ( P3 @ X3 @ Y3 ) )
=> ( coindu987187967t_all2 @ A @ B @ P3 @ Xs @ Ys ) ) ) ).
% llist_all2_mono
thf(fact_72_lappend__inf,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ~ ( coindu1213758845finite @ A @ Xs )
=> ( ( coindu268472904append @ A @ Xs @ Ys )
= Xs ) ) ).
% lappend_inf
thf(fact_73_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_74_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_75_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_76_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_77_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_78_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_79_lappend__LNil__LNil,axiom,
! [A: $tType] :
( ( coindu268472904append @ A @ ( coindu1598213697e_LNil @ A ) @ ( coindu1598213697e_LNil @ A ) )
= ( coindu1598213697e_LNil @ A ) ) ).
% lappend_LNil_LNil
thf(fact_80_enat__add__mono,axiom,
! [X: nat,Y: extended_enat,Z2: extended_enat] :
( ( ord_less @ extended_enat @ ( plus_plus @ extended_enat @ ( extended_enat2 @ X ) @ Y ) @ ( plus_plus @ extended_enat @ ( extended_enat2 @ X ) @ Z2 ) )
= ( ord_less @ extended_enat @ Y @ Z2 ) ) ).
% enat_add_mono
thf(fact_81_enat__add1__eq,axiom,
! [X: nat,Y: extended_enat,Z2: extended_enat] :
( ( ( plus_plus @ extended_enat @ ( extended_enat2 @ X ) @ Y )
= ( plus_plus @ extended_enat @ ( extended_enat2 @ X ) @ Z2 ) )
= ( Y = Z2 ) ) ).
% enat_add1_eq
thf(fact_82_enat__add2__eq,axiom,
! [Y: extended_enat,X: nat,Z2: extended_enat] :
( ( ( plus_plus @ extended_enat @ Y @ ( extended_enat2 @ X ) )
= ( plus_plus @ extended_enat @ Z2 @ ( extended_enat2 @ X ) ) )
= ( Y = Z2 ) ) ).
% enat_add2_eq
thf(fact_83_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_84_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_85_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_86_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_87_enat__less__imp__le,axiom,
! [N: extended_enat,M: extended_enat] :
( ! [K: nat] :
( ( ord_less @ extended_enat @ N @ ( extended_enat2 @ K ) )
=> ( ord_less @ extended_enat @ M @ ( extended_enat2 @ K ) ) )
=> ( ord_less_eq @ extended_enat @ M @ N ) ) ).
% enat_less_imp_le
thf(fact_88_add__mono__thms__linordered__field_I4_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K2: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less @ A @ K2 @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K2 ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_89_add__mono__thms__linordered__field_I3_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K2: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less_eq @ A @ K2 @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K2 ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_90_add__le__less__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A,D: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C @ D )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ D ) ) ) ) ) ).
% add_le_less_mono
thf(fact_91_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_92_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_93_llist_Orel__mono,axiom,
! [B: $tType,A: $tType,R: A > B > $o,Ra: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ R @ Ra )
=> ( ord_less_eq @ ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ B ) > $o ) @ ( coindu987187967t_all2 @ A @ B @ R ) @ ( coindu987187967t_all2 @ A @ B @ Ra ) ) ) ).
% llist.rel_mono
thf(fact_94_enat__less__enat__plusI2,axiom,
! [Y: nat,Z2: extended_enat,X: nat] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ Y ) @ Z2 )
=> ( ord_less @ extended_enat @ ( extended_enat2 @ ( plus_plus @ nat @ X @ Y ) ) @ ( plus_plus @ extended_enat @ ( extended_enat2 @ X ) @ Z2 ) ) ) ).
% enat_less_enat_plusI2
thf(fact_95_enat__less__enat__plusI,axiom,
! [X: nat,Y: nat,Z2: extended_enat] :
( ( ord_less @ nat @ X @ Y )
=> ( ord_less @ extended_enat @ ( extended_enat2 @ X ) @ ( plus_plus @ extended_enat @ ( extended_enat2 @ Y ) @ Z2 ) ) ) ).
% enat_less_enat_plusI
thf(fact_96_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_97_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_98_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_99_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type2 @ A ) )
=> ( ( plus_plus @ A )
= ( ^ [A3: A,B4: A] : ( plus_plus @ A @ B4 @ A3 ) ) ) ) ).
% add.commute
thf(fact_100_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_101_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_102_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_103_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K2: A,L: A] :
( ( ( I = J )
& ( K2 = L ) )
=> ( ( plus_plus @ A @ I @ K2 )
= ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_104_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_105_add__le__imp__le__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_imp_le_right
thf(fact_106_add__le__imp__le__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [C: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_imp_le_left
thf(fact_107_le__iff__add,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [A3: A,B4: A] :
? [C2: A] :
( B4
= ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% le_iff_add
thf(fact_108_add__right__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% add_right_mono
thf(fact_109_add__left__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) ) ) ) ).
% add_left_mono
thf(fact_110_add__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A,D: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C @ D )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ D ) ) ) ) ) ).
% add_mono
thf(fact_111_add__mono__thms__linordered__semiring_I1_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K2: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less_eq @ A @ K2 @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K2 ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_112_add__mono__thms__linordered__semiring_I2_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K2: A,L: A] :
( ( ( I = J )
& ( ord_less_eq @ A @ K2 @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K2 ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_113_add__mono__thms__linordered__semiring_I3_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K2: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( K2 = L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K2 ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_114_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_115_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_116_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_117_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_118_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_119_add__mono__thms__linordered__field_I1_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K2: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( K2 = L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K2 ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_120_add__mono__thms__linordered__field_I2_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K2: A,L: A] :
( ( ( I = J )
& ( ord_less @ A @ K2 @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K2 ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_121_add__mono__thms__linordered__field_I5_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K2: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less @ A @ K2 @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K2 ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_122_enat__le__plus__same_I1_J,axiom,
! [X: extended_enat,Y: extended_enat] : ( ord_less_eq @ extended_enat @ X @ ( plus_plus @ extended_enat @ X @ Y ) ) ).
% enat_le_plus_same(1)
thf(fact_123_enat__le__plus__same_I2_J,axiom,
! [X: extended_enat,Y: extended_enat] : ( ord_less_eq @ extended_enat @ X @ ( plus_plus @ extended_enat @ Y @ X ) ) ).
% enat_le_plus_same(2)
thf(fact_124_add__less__le__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A,D: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C @ D )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ D ) ) ) ) ) ).
% add_less_le_mono
thf(fact_125_plus__enat__simps_I1_J,axiom,
! [M: nat,N: nat] :
( ( plus_plus @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
= ( extended_enat2 @ ( plus_plus @ nat @ M @ N ) ) ) ).
% plus_enat_simps(1)
thf(fact_126_enat__ord__simps_I2_J,axiom,
! [M: nat,N: nat] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
= ( ord_less @ nat @ M @ N ) ) ).
% enat_ord_simps(2)
thf(fact_127_enat__ord__simps_I1_J,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% enat_ord_simps(1)
thf(fact_128_iadd__le__enat__iff,axiom,
! [X: extended_enat,Y: extended_enat,N: nat] :
( ( ord_less_eq @ extended_enat @ ( plus_plus @ extended_enat @ X @ Y ) @ ( extended_enat2 @ N ) )
= ( ? [Y6: nat,X4: nat] :
( ( X
= ( extended_enat2 @ X4 ) )
& ( Y
= ( extended_enat2 @ Y6 ) )
& ( ord_less_eq @ nat @ ( plus_plus @ nat @ X4 @ Y6 ) @ N ) ) ) ) ).
% iadd_le_enat_iff
thf(fact_129_enat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( extended_enat2 @ Nat )
= ( extended_enat2 @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% enat.inject
thf(fact_130_enat__less__induct,axiom,
! [P: extended_enat > $o,N: extended_enat] :
( ! [N2: extended_enat] :
( ! [M2: extended_enat] :
( ( ord_less @ extended_enat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% enat_less_induct
thf(fact_131_enat__ile,axiom,
! [N: extended_enat,M: nat] :
( ( ord_less_eq @ extended_enat @ N @ ( extended_enat2 @ M ) )
=> ? [K: nat] :
( N
= ( extended_enat2 @ K ) ) ) ).
% enat_ile
thf(fact_132_chain__incr,axiom,
! [A: $tType,Y7: A > extended_enat,K2: nat] :
( ! [I2: A] :
? [J2: A] : ( ord_less @ extended_enat @ ( Y7 @ I2 ) @ ( Y7 @ J2 ) )
=> ? [J3: A] : ( ord_less @ extended_enat @ ( extended_enat2 @ K2 ) @ ( Y7 @ J3 ) ) ) ).
% chain_incr
thf(fact_133_enat__iless,axiom,
! [N: extended_enat,M: nat] :
( ( ord_less @ extended_enat @ N @ ( extended_enat2 @ M ) )
=> ? [K: nat] :
( N
= ( extended_enat2 @ K ) ) ) ).
% enat_iless
thf(fact_134_less__enatE,axiom,
! [N: extended_enat,M: nat] :
( ( ord_less @ extended_enat @ N @ ( extended_enat2 @ M ) )
=> ~ ! [K: nat] :
( ( N
= ( extended_enat2 @ K ) )
=> ~ ( ord_less @ nat @ K @ M ) ) ) ).
% less_enatE
thf(fact_135_nat__add__left__cancel__le,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K2 @ M ) @ ( plus_plus @ nat @ K2 @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_136_nat__add__left__cancel__less,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ K2 @ M ) @ ( plus_plus @ nat @ K2 @ N ) )
= ( ord_less @ nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_137_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K2: nat] :
( ! [M3: nat,N2: nat] :
( ( ord_less @ nat @ M3 @ N2 )
=> ( ord_less @ nat @ ( F @ M3 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( F @ M ) @ K2 ) @ ( F @ ( plus_plus @ nat @ M @ K2 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_138_ex__has__greatest__nat__lemma,axiom,
! [A: $tType,P: A > $o,K2: A,M: A > nat,N: nat] :
( ( P @ K2 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ? [Y8: A] :
( ( P @ Y8 )
& ~ ( ord_less_eq @ nat @ ( M @ Y8 ) @ ( M @ X3 ) ) ) )
=> ? [Y3: A] :
( ( P @ Y3 )
& ~ ( ord_less @ nat @ ( M @ Y3 ) @ ( plus_plus @ nat @ ( M @ K2 ) @ N ) ) ) ) ) ).
% ex_has_greatest_nat_lemma
thf(fact_139_infinite__descent__measure,axiom,
! [A: $tType,P: A > $o,V: A > nat,X: A] :
( ! [X3: A] :
( ~ ( P @ X3 )
=> ? [Y8: A] :
( ( ord_less @ nat @ ( V @ Y8 ) @ ( V @ X3 ) )
& ~ ( P @ Y8 ) ) )
=> ( P @ X ) ) ).
% infinite_descent_measure
thf(fact_140_measure__induct__rule,axiom,
! [A: $tType,F: A > nat,P: A > $o,A2: A] :
( ! [X3: A] :
( ! [Y8: A] :
( ( ord_less @ nat @ ( F @ Y8 ) @ ( F @ X3 ) )
=> ( P @ Y8 ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ).
% measure_induct_rule
thf(fact_141_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less @ nat @ X @ Y )
=> ( ord_less @ nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_142_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_143_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_144_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_145_measure__induct,axiom,
! [A: $tType,F: A > nat,P: A > $o,A2: A] :
( ! [X3: A] :
( ! [Y8: A] :
( ( ord_less @ nat @ ( F @ Y8 ) @ ( F @ X3 ) )
=> ( P @ Y8 ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ).
% measure_induct
thf(fact_146_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less @ nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_147_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_148_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_not_refl
thf(fact_149_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less @ nat @ M @ N )
| ( ord_less @ nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_150_ex__has__least__nat,axiom,
! [A: $tType,P: A > $o,K2: A,M: A > nat] :
( ( P @ K2 )
=> ? [X3: A] :
( ( P @ X3 )
& ! [Y8: A] :
( ( P @ Y8 )
=> ( ord_less_eq @ nat @ ( M @ X3 ) @ ( M @ Y8 ) ) ) ) ) ).
% ex_has_least_nat
thf(fact_151_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_152_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_153_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_154_le__trans,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ J @ K2 )
=> ( ord_less_eq @ nat @ I @ K2 ) ) ) ).
% le_trans
thf(fact_155_le__refl,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% le_refl
thf(fact_156_nat__add__right__cancel,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ( plus_plus @ nat @ M @ K2 )
= ( plus_plus @ nat @ N @ K2 ) )
= ( M = N ) ) ).
% nat_add_right_cancel
thf(fact_157_nat__add__left__cancel,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ( plus_plus @ nat @ K2 @ M )
= ( plus_plus @ nat @ K2 @ N ) )
= ( M = N ) ) ).
% nat_add_left_cancel
thf(fact_158_ex__has__greatest__nat,axiom,
! [A: $tType,P: A > $o,K2: A,M: A > nat,B2: nat] :
( ( P @ K2 )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less @ nat @ ( M @ Y3 ) @ B2 ) )
=> ? [X3: A] :
( ( P @ X3 )
& ! [Y8: A] :
( ( P @ Y8 )
=> ( ord_less_eq @ nat @ ( M @ Y8 ) @ ( M @ X3 ) ) ) ) ) ) ).
% ex_has_greatest_nat
thf(fact_159_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J3: nat] :
( ( ord_less @ nat @ I2 @ J3 )
=> ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_160_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( M != N )
=> ( ord_less @ nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_161_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less @ nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_162_le__eq__less__or__eq,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M4: nat,N3: nat] :
( ( ord_less @ nat @ M4 @ N3 )
| ( M4 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_163_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_164_nat__less__le,axiom,
( ( ord_less @ nat )
= ( ^ [M4: nat,N3: nat] :
( ( ord_less_eq @ nat @ M4 @ N3 )
& ( M4 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_165_less__add__eq__less,axiom,
! [K2: nat,L: nat,M: nat,N: nat] :
( ( ord_less @ nat @ K2 @ L )
=> ( ( ( plus_plus @ nat @ M @ L )
= ( plus_plus @ nat @ K2 @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_166_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ord_less @ nat @ I @ ( plus_plus @ nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_167_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ord_less @ nat @ I @ ( plus_plus @ nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_168_add__less__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I @ K2 ) @ ( plus_plus @ nat @ J @ K2 ) ) ) ).
% add_less_mono1
thf(fact_169_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_170_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_171_add__less__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ K2 @ L )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I @ K2 ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_172_add__lessD1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ K2 )
=> ( ord_less @ nat @ I @ K2 ) ) ).
% add_lessD1
thf(fact_173_nat__le__iff__add,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M4: nat,N3: nat] :
? [K3: nat] :
( N3
= ( plus_plus @ nat @ M4 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_174_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_175_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_176_add__le__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K2 ) @ ( plus_plus @ nat @ J @ K2 ) ) ) ).
% add_le_mono1
thf(fact_177_add__le__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ K2 @ L )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K2 ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_178_le__Suc__ex,axiom,
! [K2: nat,L: nat] :
( ( ord_less_eq @ nat @ K2 @ L )
=> ? [N2: nat] :
( L
= ( plus_plus @ nat @ K2 @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_179_add__leD2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K2 ) @ N )
=> ( ord_less_eq @ nat @ K2 @ N ) ) ).
% add_leD2
thf(fact_180_add__leD1,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K2 ) @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% add_leD1
thf(fact_181_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ M @ N ) ) ).
% le_add2
thf(fact_182_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ N @ M ) ) ).
% le_add1
thf(fact_183_add__leE,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K2 ) @ N )
=> ~ ( ( ord_less_eq @ nat @ M @ N )
=> ~ ( ord_less_eq @ nat @ K2 @ N ) ) ) ).
% add_leE
thf(fact_184_predicate2I,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
( ! [X3: A,Y3: B] :
( ( P @ X3 @ Y3 )
=> ( Q @ X3 @ Y3 ) )
=> ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).
% predicate2I
thf(fact_185_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_186_gen__llength__def,axiom,
! [A: $tType] :
( ( coindu493225792length @ A )
= ( ^ [N3: nat,Xs3: coindu1593790203_llist @ A] : ( plus_plus @ extended_enat @ ( extended_enat2 @ N3 ) @ ( coindu1018505716length @ A @ Xs3 ) ) ) ) ).
% gen_llength_def
thf(fact_187_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
! [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% le_fun_def
thf(fact_188_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 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less_eq @ B @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_subst1
thf(fact_189_order__subst2,axiom,
! [A: $tType,C3: $tType] :
( ( ( order @ C3 @ ( type2 @ C3 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > C3,C: C3] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C3 @ ( F @ B2 ) @ C )
=> ( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ C3 @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ C3 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_subst2
thf(fact_190_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 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less_eq @ B @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_191_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 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_192_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y5: A,Z: A] : Y5 = Z )
= ( ^ [X2: A,Y4: A] :
( ( ord_less_eq @ A @ X2 @ Y4 )
& ( ord_less_eq @ A @ Y4 @ X2 ) ) ) ) ) ).
% eq_iff
thf(fact_193_antisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( X = Y ) ) ) ) ).
% antisym
thf(fact_194_linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linear
thf(fact_195_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( X = Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% eq_refl
thf(fact_196_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% le_cases
thf(fact_197_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_198_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z2: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less_eq @ A @ X @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X )
=> ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_199_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv
thf(fact_200_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_201_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_202_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_203_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z2 )
=> ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% order_trans
thf(fact_204_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_205_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: A,B3: A] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_206_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_207_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_208_ord__eq__less__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 @ B @ B2 @ C )
=> ( ! [X3: B,Y3: B] :
( ( ord_less @ B @ X3 @ Y3 )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_209_ord__less__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 @ A @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less @ B @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_210_order__less__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 @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C )
=> ( ! [X3: B,Y3: B] :
( ( ord_less @ B @ X3 @ Y3 )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_211_order__less__subst2,axiom,
! [A: $tType,C3: $tType] :
( ( ( order @ C3 @ ( type2 @ C3 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > C3,C: C3] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ C3 @ ( F @ B2 ) @ C )
=> ( ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less @ C3 @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ C3 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_less_subst2
thf(fact_212_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [Y3: A] : ( ord_less @ A @ Y3 @ X ) ) ).
% lt_ex
thf(fact_213_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [X1: A] : ( ord_less @ A @ X @ X1 ) ) ).
% gt_ex
thf(fact_214_neqE,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% neqE
thf(fact_215_neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( X != Y )
= ( ( ord_less @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ) ).
% neq_iff
thf(fact_216_order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% order.asym
thf(fact_217_dense,axiom,
! [A: $tType] :
( ( dense_order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ? [Z3: A] :
( ( ord_less @ A @ X @ Z3 )
& ( ord_less @ A @ Z3 @ Y ) ) ) ) ).
% dense
thf(fact_218_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% less_imp_neq
thf(fact_219_less__asym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% less_asym
thf(fact_220_less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% less_asym'
thf(fact_221_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z2 )
=> ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% less_trans
thf(fact_222_less__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
| ( X = Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% less_linear
thf(fact_223_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] :
~ ( ord_less @ A @ X @ X ) ) ).
% less_irrefl
thf(fact_224_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( A2 = B2 )
=> ( ( ord_less @ A @ B2 @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% ord_eq_less_trans
thf(fact_225_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% ord_less_eq_trans
thf(fact_226_dual__order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ~ ( ord_less @ A @ A2 @ B2 ) ) ) ).
% dual_order.asym
thf(fact_227_less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% less_imp_not_eq
thf(fact_228_less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% less_not_sym
thf(fact_229_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,A2: A] :
( ! [X3: A] :
( ! [Y8: A] :
( ( ord_less @ A @ Y8 @ X3 )
=> ( P @ Y8 ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ) ).
% less_induct
thf(fact_230_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ~ ( ord_less @ A @ Y @ X )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv3
thf(fact_231_less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( Y != X ) ) ) ).
% less_imp_not_eq2
thf(fact_232_less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,P: $o] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ X )
=> P ) ) ) ).
% less_imp_triv
thf(fact_233_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( X != Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_cases
thf(fact_234_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% dual_order.irrefl
thf(fact_235_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ B2 @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% order.strict_trans
thf(fact_236_less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% less_imp_not_less
thf(fact_237_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C @ B2 )
=> ( ord_less @ A @ C @ A2 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_238_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less @ A @ X @ Y ) )
= ( ( ord_less @ A @ Y @ X )
| ( X = Y ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_239_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( A2 != B2 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_240_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( A2 != B2 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_241_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
& ~ ( ord_less_eq @ ( A > B ) @ G2 @ F2 ) ) ) ) ) ).
% less_fun_def
thf(fact_242_predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,X: A,Y: B] :
( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( ( P @ X @ Y )
=> ( Q @ X @ Y ) ) ) ).
% predicate2D
thf(fact_243_rev__predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,X: A,Y: B,Q: A > B > $o] :
( ( P @ X @ Y )
=> ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( Q @ X @ Y ) ) ) ).
% rev_predicate2D
thf(fact_244_gen__llength__code_I1_J,axiom,
! [A: $tType,N: nat] :
( ( coindu493225792length @ A @ N @ ( coindu1598213697e_LNil @ A ) )
= ( extended_enat2 @ N ) ) ).
% gen_llength_code(1)
thf(fact_245_leD,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less @ A @ X @ Y ) ) ) ).
% leD
thf(fact_246_leI,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% leI
thf(fact_247_le__less,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [X2: A,Y4: A] :
( ( ord_less @ A @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ) ) ).
% le_less
thf(fact_248_less__le,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y4: A] :
( ( ord_less_eq @ A @ X2 @ Y4 )
& ( X2 != Y4 ) ) ) ) ) ).
% less_le
thf(fact_249_order__le__less__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 @ B @ B2 @ C )
=> ( ! [X3: B,Y3: B] :
( ( ord_less @ B @ X3 @ Y3 )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_250_order__le__less__subst2,axiom,
! [A: $tType,C3: $tType] :
( ( ( order @ C3 @ ( type2 @ C3 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > C3,C: C3] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ C3 @ ( F @ B2 ) @ C )
=> ( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ C3 @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ C3 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_251_order__less__le__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 @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C )
=> ( ! [X3: B,Y3: B] :
( ( ord_less_eq @ B @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_252_order__less__le__subst2,axiom,
! [A: $tType,C3: $tType] :
( ( ( order @ C3 @ ( type2 @ C3 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > C3,C: C3] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C3 @ ( F @ B2 ) @ C )
=> ( ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less @ C3 @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ C3 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_253_not__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less_eq @ A @ X @ Y ) )
= ( ord_less @ A @ Y @ X ) ) ) ).
% not_le
%----Type constructors (31)
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A5: $tType,A6: $tType] :
( ( preorder @ A6 @ ( type2 @ A6 ) )
=> ( preorder @ ( A5 > A6 ) @ ( type2 @ ( A5 > A6 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A5: $tType,A6: $tType] :
( ( order @ A6 @ ( type2 @ A6 ) )
=> ( order @ ( A5 > A6 ) @ ( type2 @ ( A5 > A6 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A5: $tType,A6: $tType] :
( ( ord @ A6 @ ( type2 @ A6 ) )
=> ( ord @ ( A5 > A6 ) @ ( type2 @ ( A5 > A6 ) ) ) ) ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
strict2144017051up_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere223160158up_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add,axiom,
semigroup_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
preorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
no_top @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
order @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
ord @ nat @ ( type2 @ nat ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_4,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder_5,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_6,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_7,axiom,
ord @ $o @ ( type2 @ $o ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_8,axiom,
strict2144017051up_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_9,axiom,
canoni770627133id_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___Groups_Oab__semigroup__add_11,axiom,
ab_semigroup_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_12,axiom,
semigroup_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Owellorder_13,axiom,
wellorder @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Opreorder_14,axiom,
preorder @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_15,axiom,
linorder @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_16,axiom,
order @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oord_17,axiom,
ord @ extended_enat @ ( type2 @ extended_enat ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
p @ ( coindu749330388e_lnth @ a @ ( coindu268472904append @ a @ xs @ xs2 ) @ n ) @ ( coindu749330388e_lnth @ b @ ys @ n ) ).
%------------------------------------------------------------------------------