TPTP Problem File: DAT129^1.p
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%------------------------------------------------------------------------------
% File : DAT129^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Coinductive list 3306
% 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__3306.p [Bla16]
% Status : Theorem
% Rating : 0.00 v7.2.0, 0.25 v7.1.0
% Syntax : Number of formulae : 337 ( 140 unt; 54 typ; 0 def)
% Number of atoms : 617 ( 313 equ; 0 cnn)
% Maximal formula atoms : 24 ( 2 avg)
% Number of connectives : 2740 ( 91 ~; 5 |; 19 &;2364 @)
% ( 0 <=>; 261 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 104 ( 104 >; 0 *; 0 +; 0 <<)
% Number of symbols : 52 ( 51 usr; 5 con; 0-4 aty)
% Number of variables : 736 ( 14 ^; 655 !; 27 ?; 736 :)
% ( 40 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 15:01:01.078
%------------------------------------------------------------------------------
%----Could-be-implicit typings (5)
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_a,type,
a: $tType ).
%----Explicit typings (49)
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_Groups_Ominus,type,
minus:
!>[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_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oab__group__add,type,
ab_group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocomm__monoid__diff,type,
comm_monoid_diff:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_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_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere779506340up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere236663937imp_le:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
ordere223160158up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
strict2144017051up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
ordere516151231imp_le:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_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_Oldropn,type,
coindu531130065ldropn:
!>[A: $tType] : ( nat > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollast,type,
coindu2000965700_llast:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > A ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollength,type,
coindu1018505716length:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > extended_enat ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_OLCons,type,
coindu1121789889_LCons:
!>[A: $tType] : ( A > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olnth,type,
coindu749330388e_lnth:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > nat > A ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olstrict__prefix,type,
coindu574146665prefix:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Extended__Nat_Oenat,type,
extended_enat2: nat > extended_enat ).
thf(sy_c_Extended__Nat_Oenat_Ocase__enat,type,
extended_case_enat:
!>[T: $tType] : ( ( nat > T ) > T > extended_enat > T ) ).
thf(sy_c_Extended__Nat_Oenat_Orec__enat,type,
extended_rec_enat:
!>[T: $tType] : ( ( nat > T ) > T > extended_enat > T ) ).
thf(sy_c_Extended__Nat_Oenat_Osize__enat,type,
extended_size_enat: extended_enat > nat ).
thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity,type,
extend1396239628finity:
!>[A: $tType] : A ).
thf(sy_c_Extended__Nat_Othe__enat,type,
extended_the_enat: extended_enat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_v_n,type,
n: extended_enat ).
thf(sy_v_n_H____,type,
n2: nat ).
thf(sy_v_n_Ha______,type,
n_a: nat ).
thf(sy_v_xs,type,
xs: coindu1593790203_llist @ a ).
thf(sy_v_xsa____,type,
xsa: coindu1593790203_llist @ a ).
%----Relevant facts (252)
thf(fact_0_n,axiom,
( n
= ( extended_enat2 @ n2 ) ) ).
% n
thf(fact_1_Suc_Ohyps,axiom,
! [Xs: coindu1593790203_llist @ a] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ n_a ) @ ( coindu1018505716length @ a @ Xs ) )
=> ( ( coindu2000965700_llast @ a @ ( coindu191418589_ldrop @ a @ ( extended_enat2 @ n_a ) @ Xs ) )
= ( coindu2000965700_llast @ a @ Xs ) ) ) ).
% Suc.hyps
thf(fact_2_Suc_Oprems,axiom,
ord_less @ extended_enat @ ( extended_enat2 @ ( suc @ n_a ) ) @ ( coindu1018505716length @ a @ xsa ) ).
% Suc.prems
thf(fact_3__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062n_H_O_An_A_061_Aenat_An_H_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [N: nat] :
( n
!= ( extended_enat2 @ N ) ) ).
% \<open>\<And>thesis. (\<And>n'. n = enat n' \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_4_enat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( extended_enat2 @ Nat )
= ( extended_enat2 @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% enat.inject
thf(fact_5_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_6_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_7_assms,axiom,
ord_less @ extended_enat @ n @ ( coindu1018505716length @ a @ xs ) ).
% assms
thf(fact_8_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_9_n__not__Suc__n,axiom,
! [N2: nat] :
( N2
!= ( suc @ N2 ) ) ).
% n_not_Suc_n
thf(fact_10_ldrop__enat,axiom,
! [A: $tType,N2: nat,Xs: coindu1593790203_llist @ A] :
( ( coindu191418589_ldrop @ A @ ( extended_enat2 @ N2 ) @ Xs )
= ( coindu531130065ldropn @ A @ N2 @ Xs ) ) ).
% ldrop_enat
thf(fact_11_enat_Osimps_I6_J,axiom,
! [T: $tType,F1: nat > T,F2: T,Nat: nat] :
( ( extended_rec_enat @ T @ F1 @ F2 @ ( extended_enat2 @ Nat ) )
= ( F1 @ Nat ) ) ).
% enat.simps(6)
thf(fact_12_chain__incr,axiom,
! [A: $tType,Y3: A > extended_enat,K: nat] :
( ! [I: A] :
? [J: A] : ( ord_less @ extended_enat @ ( Y3 @ I ) @ ( Y3 @ J ) )
=> ? [J2: A] : ( ord_less @ extended_enat @ ( extended_enat2 @ K ) @ ( Y3 @ J2 ) ) ) ).
% chain_incr
thf(fact_13_enat__iless,axiom,
! [N2: extended_enat,M: nat] :
( ( ord_less @ extended_enat @ N2 @ ( extended_enat2 @ M ) )
=> ? [K2: nat] :
( N2
= ( extended_enat2 @ K2 ) ) ) ).
% enat_iless
thf(fact_14_ldrop__0,axiom,
! [B: $tType,Xs: coindu1593790203_llist @ B] :
( ( coindu191418589_ldrop @ B @ ( zero_zero @ extended_enat ) @ Xs )
= Xs ) ).
% ldrop_0
thf(fact_15_i0__less,axiom,
! [N2: extended_enat] :
( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ N2 )
= ( N2
!= ( zero_zero @ extended_enat ) ) ) ).
% i0_less
thf(fact_16_not__iless0,axiom,
! [N2: extended_enat] :
~ ( ord_less @ extended_enat @ N2 @ ( zero_zero @ extended_enat ) ) ).
% not_iless0
thf(fact_17_enat__less__induct,axiom,
! [P: extended_enat > $o,N2: extended_enat] :
( ! [N3: extended_enat] :
( ! [M2: extended_enat] :
( ( ord_less @ extended_enat @ M2 @ N3 )
=> ( P @ M2 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% enat_less_induct
thf(fact_18_lift__Suc__mono__less__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [F: nat > A,N2: nat,M: nat] :
( ! [N3: nat] : ( ord_less @ A @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less @ A @ ( F @ N2 ) @ ( F @ M ) )
= ( ord_less @ nat @ N2 @ M ) ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_19_lift__Suc__mono__less,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [F: nat > A,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less @ A @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less @ nat @ N2 @ N4 )
=> ( ord_less @ A @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_20_llast__ldropn,axiom,
! [A: $tType,N2: nat,Xs: coindu1593790203_llist @ A] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ N2 ) @ ( coindu1018505716length @ A @ Xs ) )
=> ( ( coindu2000965700_llast @ A @ ( coindu531130065ldropn @ A @ N2 @ Xs ) )
= ( coindu2000965700_llast @ A @ Xs ) ) ) ).
% llast_ldropn
thf(fact_21_not__gr__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N2: A] :
( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N2 ) )
= ( N2
= ( zero_zero @ A ) ) ) ) ).
% not_gr_zero
thf(fact_22_less__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% less_numeral_extra(3)
thf(fact_23_gr__zeroI,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N2: A] :
( ( N2
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ N2 ) ) ) ).
% gr_zeroI
thf(fact_24_not__less__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N2: A] :
~ ( ord_less @ A @ N2 @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_25_gr__implies__not__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [M: A,N2: A] :
( ( ord_less @ A @ M @ N2 )
=> ( N2
!= ( zero_zero @ A ) ) ) ) ).
% gr_implies_not_zero
thf(fact_26_zero__less__iff__neq__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ N2 )
= ( N2
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_iff_neq_zero
thf(fact_27_ldropn__eq__LConsD,axiom,
! [A: $tType,N2: nat,Xs: coindu1593790203_llist @ A,Y: A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu531130065ldropn @ A @ N2 @ Xs )
= ( coindu1121789889_LCons @ A @ Y @ Ys ) )
=> ( ord_less @ extended_enat @ ( extended_enat2 @ N2 ) @ ( coindu1018505716length @ A @ Xs ) ) ) ).
% ldropn_eq_LConsD
thf(fact_28_enat_Osimps_I4_J,axiom,
! [T: $tType,F1: nat > T,F2: T,Nat: nat] :
( ( extended_case_enat @ T @ F1 @ F2 @ ( extended_enat2 @ Nat ) )
= ( F1 @ Nat ) ) ).
% enat.simps(4)
thf(fact_29_llength__ldropn,axiom,
! [A: $tType,N2: nat,Xs: coindu1593790203_llist @ A] :
( ( coindu1018505716length @ A @ ( coindu531130065ldropn @ A @ N2 @ Xs ) )
= ( minus_minus @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ ( extended_enat2 @ N2 ) ) ) ).
% llength_ldropn
thf(fact_30_neq0__conv,axiom,
! [N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).
% neq0_conv
thf(fact_31_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_32_ldropn__0,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu531130065ldropn @ A @ ( zero_zero @ nat ) @ Xs )
= Xs ) ).
% ldropn_0
thf(fact_33_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_34_diff__zero,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% diff_zero
thf(fact_35_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_36_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_37_diff__self,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% diff_self
thf(fact_38_zero__less__Suc,axiom,
! [N2: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) ).
% zero_less_Suc
thf(fact_39_less__Suc0,axiom,
! [N2: nat] :
( ( ord_less @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) )
= ( N2
= ( zero_zero @ nat ) ) ) ).
% less_Suc0
thf(fact_40_lessI,axiom,
! [N2: nat] : ( ord_less @ nat @ N2 @ ( suc @ N2 ) ) ).
% lessI
thf(fact_41_Suc__mono,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ N2 )
=> ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) ).
% Suc_mono
thf(fact_42_Suc__less__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N2 ) )
= ( ord_less @ nat @ M @ N2 ) ) ).
% Suc_less_eq
thf(fact_43_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_44_idiff__enat__0__right,axiom,
! [N2: extended_enat] :
( ( minus_minus @ extended_enat @ N2 @ ( extended_enat2 @ ( zero_zero @ nat ) ) )
= N2 ) ).
% idiff_enat_0_right
thf(fact_45_idiff__enat__0,axiom,
! [N2: extended_enat] :
( ( minus_minus @ extended_enat @ ( extended_enat2 @ ( zero_zero @ nat ) ) @ N2 )
= ( extended_enat2 @ ( zero_zero @ nat ) ) ) ).
% idiff_enat_0
thf(fact_46_idiff__0__right,axiom,
! [N2: extended_enat] :
( ( minus_minus @ extended_enat @ N2 @ ( zero_zero @ extended_enat ) )
= N2 ) ).
% idiff_0_right
thf(fact_47_idiff__0,axiom,
! [N2: extended_enat] :
( ( minus_minus @ extended_enat @ ( zero_zero @ extended_enat ) @ N2 )
= ( zero_zero @ extended_enat ) ) ).
% idiff_0
thf(fact_48_llast__LCons2,axiom,
! [A: $tType,X: A,Y: A,Xs: coindu1593790203_llist @ A] :
( ( coindu2000965700_llast @ A @ ( coindu1121789889_LCons @ A @ X @ ( coindu1121789889_LCons @ A @ Y @ Xs ) ) )
= ( coindu2000965700_llast @ A @ ( coindu1121789889_LCons @ A @ Y @ Xs ) ) ) ).
% llast_LCons2
thf(fact_49_diff__gt__0__iff__gt,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ).
% diff_gt_0_iff_gt
thf(fact_50_enat__ord__simps_I2_J,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N2 ) )
= ( ord_less @ nat @ M @ N2 ) ) ).
% enat_ord_simps(2)
thf(fact_51_ldropn__Suc__LCons,axiom,
! [A: $tType,N2: nat,X: A,Xs: coindu1593790203_llist @ A] :
( ( coindu531130065ldropn @ A @ ( suc @ N2 ) @ ( coindu1121789889_LCons @ A @ X @ Xs ) )
= ( coindu531130065ldropn @ A @ N2 @ Xs ) ) ).
% ldropn_Suc_LCons
thf(fact_52_gr0I,axiom,
! [N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).
% gr0I
thf(fact_53_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) )
= ( N2
= ( zero_zero @ nat ) ) ) ).
% not_gr0
thf(fact_54_not__less0,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% not_less0
thf(fact_55_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% less_zeroE
thf(fact_56_nat__neq__iff,axiom,
! [M: nat,N2: nat] :
( ( M != N2 )
= ( ( ord_less @ nat @ M @ N2 )
| ( ord_less @ nat @ N2 @ M ) ) ) ).
% nat_neq_iff
thf(fact_57_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_58_less__not__refl2,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ N2 @ M )
=> ( M != N2 ) ) ).
% less_not_refl2
thf(fact_59_less__not__refl3,axiom,
! [S: nat,T2: nat] :
( ( ord_less @ nat @ S @ T2 )
=> ( S != T2 ) ) ).
% less_not_refl3
thf(fact_60_measure__induct,axiom,
! [A: $tType,F: A > nat,P: A > $o,A2: A] :
( ! [X3: A] :
( ! [Y4: A] :
( ( ord_less @ nat @ ( F @ Y4 ) @ ( F @ X3 ) )
=> ( P @ Y4 ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ).
% measure_induct
thf(fact_61_gr__implies__not0,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ N2 )
=> ( N2
!= ( zero_zero @ nat ) ) ) ).
% gr_implies_not0
thf(fact_62_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_63_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M2: nat] :
( ( ord_less @ nat @ M2 @ N3 )
=> ( P @ M2 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_64_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M2: nat] :
( ( ord_less @ nat @ M2 @ N3 )
& ~ ( P @ M2 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_65_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M2: nat] :
( ( ord_less @ nat @ M2 @ N3 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_66_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_67_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% less_nat_zero_code
thf(fact_68_measure__induct__rule,axiom,
! [A: $tType,F: A > nat,P: A > $o,A2: A] :
( ! [X3: A] :
( ! [Y4: A] :
( ( ord_less @ nat @ ( F @ Y4 ) @ ( F @ X3 ) )
=> ( P @ Y4 ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ).
% measure_induct_rule
thf(fact_69_infinite__descent__measure,axiom,
! [A: $tType,P: A > $o,V: A > nat,X: A] :
( ! [X3: A] :
( ~ ( P @ X3 )
=> ? [Y4: A] :
( ( ord_less @ nat @ ( V @ Y4 ) @ ( V @ X3 ) )
& ~ ( P @ Y4 ) ) )
=> ( P @ X ) ) ).
% infinite_descent_measure
thf(fact_70_infinite__descent0__measure,axiom,
! [A: $tType,V: A > nat,P: A > $o,X: A] :
( ! [X3: A] :
( ( ( V @ X3 )
= ( zero_zero @ nat ) )
=> ( P @ X3 ) )
=> ( ! [X3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X3 ) )
=> ( ~ ( P @ X3 )
=> ? [Y4: A] :
( ( ord_less @ nat @ ( V @ Y4 ) @ ( V @ X3 ) )
& ~ ( P @ Y4 ) ) ) )
=> ( P @ X ) ) ) ).
% infinite_descent0_measure
thf(fact_71_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_72_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_73_less__Suc__eq__0__disj,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ ( suc @ N2 ) )
= ( ( M
= ( zero_zero @ nat ) )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less @ nat @ J3 @ N2 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_74_gr0__implies__Suc,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ? [M3: nat] :
( N2
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_75_gr0__conv__Suc,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
= ( ? [M4: nat] :
( N2
= ( suc @ M4 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_76_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ( ( ^ [Y5: A,Z: A] : Y5 = Z )
= ( ^ [A3: A,B3: A] :
( ( minus_minus @ A @ A3 @ B3 )
= ( zero_zero @ A ) ) ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_77_diff__strict__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B2 @ C ) ) ) ) ).
% diff_strict_right_mono
thf(fact_78_diff__strict__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ord_less @ A @ ( minus_minus @ A @ C @ A2 ) @ ( minus_minus @ A @ C @ B2 ) ) ) ) ).
% diff_strict_left_mono
thf(fact_79_diff__eq__diff__less,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 @ A @ A2 @ B2 )
= ( ord_less @ A @ C @ D ) ) ) ) ).
% diff_eq_diff_less
thf(fact_80_diff__strict__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,D: A,C: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ D @ C )
=> ( ord_less @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B2 @ D ) ) ) ) ) ).
% diff_strict_mono
thf(fact_81_lessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less @ nat @ I2 @ K )
=> ( ( K
!= ( suc @ I2 ) )
=> ~ ! [J2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% lessE
thf(fact_82_Suc__lessD,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ N2 )
=> ( ord_less @ nat @ M @ N2 ) ) ).
% Suc_lessD
thf(fact_83_Suc__lessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less @ nat @ ( suc @ I2 ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_84_Suc__lessI,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ N2 )
=> ( ( ( suc @ M )
!= N2 )
=> ( ord_less @ nat @ ( suc @ M ) @ N2 ) ) ) ).
% Suc_lessI
thf(fact_85_less__SucE,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ ( suc @ N2 ) )
=> ( ~ ( ord_less @ nat @ M @ N2 )
=> ( M = N2 ) ) ) ).
% less_SucE
thf(fact_86_less__SucI,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ N2 )
=> ( ord_less @ nat @ M @ ( suc @ N2 ) ) ) ).
% less_SucI
thf(fact_87_less__Suc__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ ( suc @ N2 ) )
= ( ( ord_less @ nat @ M @ N2 )
| ( M = N2 ) ) ) ).
% less_Suc_eq
thf(fact_88_not__less__eq,axiom,
! [M: nat,N2: nat] :
( ( ~ ( ord_less @ nat @ M @ N2 ) )
= ( ord_less @ nat @ N2 @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_89_Suc__less__eq2,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( suc @ N2 ) @ M )
= ( ? [M5: nat] :
( ( M
= ( suc @ M5 ) )
& ( ord_less @ nat @ N2 @ M5 ) ) ) ) ).
% Suc_less_eq2
thf(fact_90_less__antisym,axiom,
! [N2: nat,M: nat] :
( ~ ( ord_less @ nat @ N2 @ M )
=> ( ( ord_less @ nat @ N2 @ ( suc @ M ) )
=> ( M = N2 ) ) ) ).
% less_antisym
thf(fact_91_Suc__less__SucD,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N2 ) )
=> ( ord_less @ nat @ M @ N2 ) ) ).
% Suc_less_SucD
thf(fact_92_less__trans__Suc,axiom,
! [I2: nat,J4: nat,K: nat] :
( ( ord_less @ nat @ I2 @ J4 )
=> ( ( ord_less @ nat @ J4 @ K )
=> ( ord_less @ nat @ ( suc @ I2 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_93_less__Suc__induct,axiom,
! [I2: nat,J4: nat,P: nat > nat > $o] :
( ( ord_less @ nat @ I2 @ J4 )
=> ( ! [I: nat] : ( P @ I @ ( suc @ I ) )
=> ( ! [I: nat,J2: nat,K2: nat] :
( ( ord_less @ nat @ I @ J2 )
=> ( ( ord_less @ nat @ J2 @ K2 )
=> ( ( P @ I @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I @ K2 ) ) ) ) )
=> ( P @ I2 @ J4 ) ) ) ) ).
% less_Suc_induct
thf(fact_94_strict__inc__induct,axiom,
! [I2: nat,J4: nat,P: nat > $o] :
( ( ord_less @ nat @ I2 @ J4 )
=> ( ! [I: nat] :
( ( J4
= ( suc @ I ) )
=> ( P @ I ) )
=> ( ! [I: nat] :
( ( ord_less @ nat @ I @ J4 )
=> ( ( P @ ( suc @ I ) )
=> ( P @ I ) ) )
=> ( P @ I2 ) ) ) ) ).
% strict_inc_induct
thf(fact_95_not__less__less__Suc__eq,axiom,
! [N2: nat,M: nat] :
( ~ ( ord_less @ nat @ N2 @ M )
=> ( ( ord_less @ nat @ N2 @ ( suc @ M ) )
= ( N2 = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_96_less__iff__diff__less__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [A3: A,B3: A] : ( ord_less @ A @ ( minus_minus @ A @ A3 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% less_iff_diff_less_0
thf(fact_97_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( ( zero_zero @ nat )
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_98_old_Onat_Odistinct_I2_J,axiom,
! [Nat3: nat] :
( ( suc @ Nat3 )
!= ( zero_zero @ nat ) ) ).
% old.nat.distinct(2)
thf(fact_99_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( ( zero_zero @ nat )
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_100_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat
!= ( zero_zero @ nat ) ) ) ).
% nat.discI
thf(fact_101_nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N2 ) ) ) ).
% nat_induct
thf(fact_102_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N2: nat] :
( ! [X3: nat] : ( P @ X3 @ ( zero_zero @ nat ) )
=> ( ! [Y6: nat] : ( P @ ( zero_zero @ nat ) @ ( suc @ Y6 ) )
=> ( ! [X3: nat,Y6: nat] :
( ( P @ X3 @ Y6 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y6 ) ) )
=> ( P @ M @ N2 ) ) ) ) ).
% diff_induct
thf(fact_103_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( zero_zero @ nat ) ) ) ) ).
% zero_induct
thf(fact_104_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= ( zero_zero @ nat ) ) ).
% Suc_neq_Zero
thf(fact_105_Zero__neq__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_106_Zero__not__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_107_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y
!= ( zero_zero @ nat ) )
=> ~ ! [Nat4: nat] :
( Y
!= ( suc @ Nat4 ) ) ) ).
% old.nat.exhaust
thf(fact_108_old_Onat_Oinducts,axiom,
! [P: nat > $o,Nat: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [Nat4: nat] :
( ( P @ Nat4 )
=> ( P @ ( suc @ Nat4 ) ) )
=> ( P @ Nat ) ) ) ).
% old.nat.inducts
thf(fact_109_not0__implies__Suc,axiom,
! [N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
=> ? [M3: nat] :
( N2
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_110_less__enatE,axiom,
! [N2: extended_enat,M: nat] :
( ( ord_less @ extended_enat @ N2 @ ( extended_enat2 @ M ) )
=> ~ ! [K2: nat] :
( ( N2
= ( extended_enat2 @ K2 ) )
=> ~ ( ord_less @ nat @ K2 @ M ) ) ) ).
% less_enatE
thf(fact_111_zero__enat__def,axiom,
( ( zero_zero @ extended_enat )
= ( extended_enat2 @ ( zero_zero @ nat ) ) ) ).
% zero_enat_def
thf(fact_112_enat__0__iff_I1_J,axiom,
! [X: nat] :
( ( ( extended_enat2 @ X )
= ( zero_zero @ extended_enat ) )
= ( X
= ( zero_zero @ nat ) ) ) ).
% enat_0_iff(1)
thf(fact_113_enat__0__iff_I2_J,axiom,
! [X: nat] :
( ( ( zero_zero @ extended_enat )
= ( extended_enat2 @ X ) )
= ( X
= ( zero_zero @ nat ) ) ) ).
% enat_0_iff(2)
thf(fact_114_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_115_ldrop__eq__LConsD,axiom,
! [A: $tType,N2: extended_enat,Xs: coindu1593790203_llist @ A,Y: A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu191418589_ldrop @ A @ N2 @ Xs )
= ( coindu1121789889_LCons @ A @ Y @ Ys ) )
=> ( ord_less @ extended_enat @ N2 @ ( coindu1018505716length @ A @ Xs ) ) ) ).
% ldrop_eq_LConsD
thf(fact_116_case__enat__0,axiom,
! [A: $tType,F: nat > A,I2: A] :
( ( extended_case_enat @ A @ F @ I2 @ ( zero_zero @ extended_enat ) )
= ( F @ ( zero_zero @ nat ) ) ) ).
% case_enat_0
thf(fact_117_minus__apply,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B @ ( type2 @ B ) )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A4: A > B,B4: A > B,X4: A] : ( minus_minus @ B @ ( A4 @ X4 ) @ ( B4 @ X4 ) ) ) ) ) ).
% minus_apply
thf(fact_118_enat_Osize_I1_J,axiom,
! [Nat: nat] :
( ( extended_size_enat @ ( extended_enat2 @ Nat ) )
= ( zero_zero @ nat ) ) ).
% enat.size(1)
thf(fact_119_ldropn__Suc__conv__ldropn,axiom,
! [A: $tType,N2: nat,Xs: coindu1593790203_llist @ A] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ N2 ) @ ( coindu1018505716length @ A @ Xs ) )
=> ( ( coindu1121789889_LCons @ A @ ( coindu749330388e_lnth @ A @ Xs @ N2 ) @ ( coindu531130065ldropn @ A @ ( suc @ N2 ) @ Xs ) )
= ( coindu531130065ldropn @ A @ N2 @ Xs ) ) ) ).
% ldropn_Suc_conv_ldropn
thf(fact_120_enat_Osize_I3_J,axiom,
! [Nat: nat] :
( ( size_size @ extended_enat @ ( extended_enat2 @ Nat ) )
= ( zero_zero @ nat ) ) ).
% enat.size(3)
thf(fact_121_llength__ldrop,axiom,
! [A: $tType,N2: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( ( N2
= ( extend1396239628finity @ extended_enat ) )
=> ( ( coindu1018505716length @ A @ ( coindu191418589_ldrop @ A @ N2 @ Xs ) )
= ( zero_zero @ extended_enat ) ) )
& ( ( N2
!= ( extend1396239628finity @ extended_enat ) )
=> ( ( coindu1018505716length @ A @ ( coindu191418589_ldrop @ A @ N2 @ Xs ) )
= ( minus_minus @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ N2 ) ) ) ) ).
% llength_ldrop
thf(fact_122_ldrop__lappend,axiom,
! [A: $tType,N2: extended_enat,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ( ord_less @ extended_enat @ N2 @ ( coindu1018505716length @ A @ Xs ) )
=> ( ( coindu191418589_ldrop @ A @ N2 @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( coindu268472904append @ A @ ( coindu191418589_ldrop @ A @ N2 @ Xs ) @ Ys ) ) )
& ( ~ ( ord_less @ extended_enat @ N2 @ ( coindu1018505716length @ A @ Xs ) )
=> ( ( coindu191418589_ldrop @ A @ N2 @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( coindu191418589_ldrop @ A @ ( minus_minus @ extended_enat @ N2 @ ( coindu1018505716length @ A @ Xs ) ) @ Ys ) ) ) ) ).
% ldrop_lappend
thf(fact_123_idiff__infinity__right,axiom,
! [A2: nat] :
( ( minus_minus @ extended_enat @ ( extended_enat2 @ A2 ) @ ( extend1396239628finity @ extended_enat ) )
= ( zero_zero @ extended_enat ) ) ).
% idiff_infinity_right
thf(fact_124_lstrict__prefix__llength__less,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu574146665prefix @ A @ Xs @ Ys )
=> ( ord_less @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ ( coindu1018505716length @ A @ Ys ) ) ) ).
% lstrict_prefix_llength_less
thf(fact_125_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N2 )
= ( zero_zero @ nat ) ) ).
% diff_0_eq_0
thf(fact_126_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus @ nat @ M @ M )
= ( zero_zero @ nat ) ) ).
% diff_self_eq_0
thf(fact_127_Suc__diff__diff,axiom,
! [M: nat,N2: nat,K: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ ( suc @ M ) @ N2 ) @ ( suc @ K ) )
= ( minus_minus @ nat @ ( minus_minus @ nat @ M @ N2 ) @ K ) ) ).
% Suc_diff_diff
thf(fact_128_diff__Suc__Suc,axiom,
! [M: nat,N2: nat] :
( ( minus_minus @ nat @ ( suc @ M ) @ ( suc @ N2 ) )
= ( minus_minus @ nat @ M @ N2 ) ) ).
% diff_Suc_Suc
thf(fact_129_zero__less__diff,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N2 @ M ) )
= ( ord_less @ nat @ M @ N2 ) ) ).
% zero_less_diff
thf(fact_130_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_131_not__infinity__eq,axiom,
! [X: extended_enat] :
( ( X
!= ( extend1396239628finity @ extended_enat ) )
= ( ? [I3: nat] :
( X
= ( extended_enat2 @ I3 ) ) ) ) ).
% not_infinity_eq
thf(fact_132_not__enat__eq,axiom,
! [X: extended_enat] :
( ( ! [Y7: nat] :
( X
!= ( extended_enat2 @ Y7 ) ) )
= ( X
= ( extend1396239628finity @ extended_enat ) ) ) ).
% not_enat_eq
thf(fact_133_enat__ord__simps_I6_J,axiom,
! [Q: extended_enat] :
~ ( ord_less @ extended_enat @ ( extend1396239628finity @ extended_enat ) @ Q ) ).
% enat_ord_simps(6)
thf(fact_134_enat__ord__simps_I4_J,axiom,
! [Q: extended_enat] :
( ( ord_less @ extended_enat @ Q @ ( extend1396239628finity @ extended_enat ) )
= ( Q
!= ( extend1396239628finity @ extended_enat ) ) ) ).
% enat_ord_simps(4)
thf(fact_135_idiff__enat__enat,axiom,
! [A2: nat,B2: nat] :
( ( minus_minus @ extended_enat @ ( extended_enat2 @ A2 ) @ ( extended_enat2 @ B2 ) )
= ( extended_enat2 @ ( minus_minus @ nat @ A2 @ B2 ) ) ) ).
% idiff_enat_enat
thf(fact_136_idiff__infinity,axiom,
! [N2: extended_enat] :
( ( minus_minus @ extended_enat @ ( extend1396239628finity @ extended_enat ) @ N2 )
= ( extend1396239628finity @ extended_enat ) ) ).
% idiff_infinity
thf(fact_137_enat_Osimps_I5_J,axiom,
! [T: $tType,F1: nat > T,F2: T] :
( ( extended_case_enat @ T @ F1 @ F2 @ ( extend1396239628finity @ extended_enat ) )
= F2 ) ).
% enat.simps(5)
thf(fact_138_lstrict__prefix__code_I4_J,axiom,
! [B: $tType,X: B,Xs: coindu1593790203_llist @ B,Y: B,Ys: coindu1593790203_llist @ B] :
( ( coindu574146665prefix @ B @ ( coindu1121789889_LCons @ B @ X @ Xs ) @ ( coindu1121789889_LCons @ B @ Y @ Ys ) )
= ( ( X = Y )
& ( coindu574146665prefix @ B @ Xs @ Ys ) ) ) ).
% lstrict_prefix_code(4)
thf(fact_139_enat_Osimps_I7_J,axiom,
! [T: $tType,F1: nat > T,F2: T] :
( ( extended_rec_enat @ T @ F1 @ F2 @ ( extend1396239628finity @ extended_enat ) )
= F2 ) ).
% enat.simps(7)
thf(fact_140_Suc__pred,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( suc @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) )
= N2 ) ) ).
% Suc_pred
thf(fact_141_lnth__0,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] :
( ( coindu749330388e_lnth @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ ( zero_zero @ nat ) )
= X ) ).
% lnth_0
thf(fact_142_lnth__Suc__LCons,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A,N2: nat] :
( ( coindu749330388e_lnth @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ ( suc @ N2 ) )
= ( coindu749330388e_lnth @ A @ Xs @ N2 ) ) ).
% lnth_Suc_LCons
thf(fact_143_idiff__self,axiom,
! [N2: extended_enat] :
( ( N2
!= ( extend1396239628finity @ extended_enat ) )
=> ( ( minus_minus @ extended_enat @ N2 @ N2 )
= ( zero_zero @ extended_enat ) ) ) ).
% idiff_self
thf(fact_144_enat_Osize_I4_J,axiom,
( ( size_size @ extended_enat @ ( extend1396239628finity @ extended_enat ) )
= ( zero_zero @ nat ) ) ).
% enat.size(4)
thf(fact_145_enat_Osize_I2_J,axiom,
( ( extended_size_enat @ ( extend1396239628finity @ extended_enat ) )
= ( zero_zero @ nat ) ) ).
% enat.size(2)
thf(fact_146_diffs0__imp__equal,axiom,
! [M: nat,N2: nat] :
( ( ( minus_minus @ nat @ M @ N2 )
= ( zero_zero @ nat ) )
=> ( ( ( minus_minus @ nat @ N2 @ M )
= ( zero_zero @ nat ) )
=> ( M = N2 ) ) ) ).
% diffs0_imp_equal
thf(fact_147_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% minus_nat.diff_0
thf(fact_148_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_149_llist__less__induct,axiom,
! [A: $tType,P: ( coindu1593790203_llist @ A ) > $o,Xs: coindu1593790203_llist @ A] :
( ! [Xs2: coindu1593790203_llist @ A] :
( ! [Ys2: coindu1593790203_llist @ A] :
( ( coindu574146665prefix @ A @ Ys2 @ Xs2 )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% llist_less_induct
thf(fact_150_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I2: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus @ nat @ K @ I2 ) ) ) ) ).
% zero_induct_lemma
thf(fact_151_enat__ex__split,axiom,
( ( ^ [P2: extended_enat > $o] :
? [X5: extended_enat] : ( P2 @ X5 ) )
= ( ^ [P3: extended_enat > $o] :
( ( P3 @ ( extend1396239628finity @ extended_enat ) )
| ? [X4: nat] : ( P3 @ ( extended_enat2 @ X4 ) ) ) ) ) ).
% enat_ex_split
thf(fact_152_enat_Oinducts,axiom,
! [P: extended_enat > $o,Enat: extended_enat] :
( ! [Nat4: nat] : ( P @ ( extended_enat2 @ Nat4 ) )
=> ( ( P @ ( extend1396239628finity @ extended_enat ) )
=> ( P @ Enat ) ) ) ).
% enat.inducts
thf(fact_153_enat_Oexhaust,axiom,
! [Y: extended_enat] :
( ! [Nat4: nat] :
( Y
!= ( extended_enat2 @ Nat4 ) )
=> ( Y
= ( extend1396239628finity @ extended_enat ) ) ) ).
% enat.exhaust
thf(fact_154_enat3__cases,axiom,
! [Y: extended_enat,Ya: extended_enat,Yb: extended_enat] :
( ( ? [Nat4: nat] :
( Y
= ( extended_enat2 @ Nat4 ) )
=> ( ? [Nata: nat] :
( Ya
= ( extended_enat2 @ Nata ) )
=> ! [Natb: nat] :
( Yb
!= ( extended_enat2 @ Natb ) ) ) )
=> ( ( ? [Nat4: nat] :
( Y
= ( extended_enat2 @ Nat4 ) )
=> ( ? [Nata: nat] :
( Ya
= ( extended_enat2 @ Nata ) )
=> ( Yb
!= ( extend1396239628finity @ extended_enat ) ) ) )
=> ( ( ? [Nat4: nat] :
( Y
= ( extended_enat2 @ Nat4 ) )
=> ( ( Ya
= ( extend1396239628finity @ extended_enat ) )
=> ! [Nata: nat] :
( Yb
!= ( extended_enat2 @ Nata ) ) ) )
=> ( ( ? [Nat4: nat] :
( Y
= ( extended_enat2 @ Nat4 ) )
=> ( ( Ya
= ( extend1396239628finity @ extended_enat ) )
=> ( Yb
!= ( extend1396239628finity @ extended_enat ) ) ) )
=> ( ( ( Y
= ( extend1396239628finity @ extended_enat ) )
=> ( ? [Nat4: nat] :
( Ya
= ( extended_enat2 @ Nat4 ) )
=> ! [Nata: nat] :
( Yb
!= ( extended_enat2 @ Nata ) ) ) )
=> ( ( ( Y
= ( extend1396239628finity @ extended_enat ) )
=> ( ? [Nat4: nat] :
( Ya
= ( extended_enat2 @ Nat4 ) )
=> ( Yb
!= ( extend1396239628finity @ extended_enat ) ) ) )
=> ( ( ( Y
= ( extend1396239628finity @ extended_enat ) )
=> ( ( Ya
= ( extend1396239628finity @ extended_enat ) )
=> ! [Nat4: nat] :
( Yb
!= ( extended_enat2 @ Nat4 ) ) ) )
=> ~ ( ( Y
= ( extend1396239628finity @ extended_enat ) )
=> ( ( Ya
= ( extend1396239628finity @ extended_enat ) )
=> ( Yb
!= ( extend1396239628finity @ extended_enat ) ) ) ) ) ) ) ) ) ) ) ).
% enat3_cases
thf(fact_155_enat2__cases,axiom,
! [Y: extended_enat,Ya: extended_enat] :
( ( ? [Nat4: nat] :
( Y
= ( extended_enat2 @ Nat4 ) )
=> ! [Nata: nat] :
( Ya
!= ( extended_enat2 @ Nata ) ) )
=> ( ( ? [Nat4: nat] :
( Y
= ( extended_enat2 @ Nat4 ) )
=> ( Ya
!= ( extend1396239628finity @ extended_enat ) ) )
=> ( ( ( Y
= ( extend1396239628finity @ extended_enat ) )
=> ! [Nat4: nat] :
( Ya
!= ( extended_enat2 @ Nat4 ) ) )
=> ~ ( ( Y
= ( extend1396239628finity @ extended_enat ) )
=> ( Ya
!= ( extend1396239628finity @ extended_enat ) ) ) ) ) ) ).
% enat2_cases
thf(fact_156_enat_Odistinct_I1_J,axiom,
! [Nat: nat] :
( ( extended_enat2 @ Nat )
!= ( extend1396239628finity @ extended_enat ) ) ).
% enat.distinct(1)
thf(fact_157_enat_Odistinct_I2_J,axiom,
! [Nat5: nat] :
( ( extend1396239628finity @ extended_enat )
!= ( extended_enat2 @ Nat5 ) ) ).
% enat.distinct(2)
thf(fact_158_infinity__ne__i0,axiom,
( ( extend1396239628finity @ extended_enat )
!= ( zero_zero @ extended_enat ) ) ).
% infinity_ne_i0
thf(fact_159_less__imp__diff__less,axiom,
! [J4: nat,K: nat,N2: nat] :
( ( ord_less @ nat @ J4 @ K )
=> ( ord_less @ nat @ ( minus_minus @ nat @ J4 @ N2 ) @ K ) ) ).
% less_imp_diff_less
thf(fact_160_diff__less__mono2,axiom,
! [M: nat,N2: nat,L: nat] :
( ( ord_less @ nat @ M @ N2 )
=> ( ( ord_less @ nat @ M @ L )
=> ( ord_less @ nat @ ( minus_minus @ nat @ L @ N2 ) @ ( minus_minus @ nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_161_diff__less,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ nat @ ( minus_minus @ nat @ M @ N2 ) @ M ) ) ) ).
% diff_less
thf(fact_162_lnth__lappend1,axiom,
! [A: $tType,N2: nat,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ N2 ) @ ( coindu1018505716length @ A @ Xs ) )
=> ( ( coindu749330388e_lnth @ A @ ( coindu268472904append @ A @ Xs @ Ys ) @ N2 )
= ( coindu749330388e_lnth @ A @ Xs @ N2 ) ) ) ).
% lnth_lappend1
thf(fact_163_Suc__diff__Suc,axiom,
! [N2: nat,M: nat] :
( ( ord_less @ nat @ N2 @ M )
=> ( ( suc @ ( minus_minus @ nat @ M @ ( suc @ N2 ) ) )
= ( minus_minus @ nat @ M @ N2 ) ) ) ).
% Suc_diff_Suc
thf(fact_164_diff__less__Suc,axiom,
! [M: nat,N2: nat] : ( ord_less @ nat @ ( minus_minus @ nat @ M @ N2 ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_165_diff__Suc__less,axiom,
! [N2: nat,I2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ N2 @ ( suc @ I2 ) ) @ N2 ) ) ).
% diff_Suc_less
thf(fact_166_infinity__ilessE,axiom,
! [M: nat] :
~ ( ord_less @ extended_enat @ ( extend1396239628finity @ extended_enat ) @ ( extended_enat2 @ M ) ) ).
% infinity_ilessE
thf(fact_167_less__infinityE,axiom,
! [N2: extended_enat] :
( ( ord_less @ extended_enat @ N2 @ ( extend1396239628finity @ extended_enat ) )
=> ~ ! [K2: nat] :
( N2
!= ( extended_enat2 @ K2 ) ) ) ).
% less_infinityE
thf(fact_168_enat__ord__code_I4_J,axiom,
! [M: nat] : ( ord_less @ extended_enat @ ( extended_enat2 @ M ) @ ( extend1396239628finity @ extended_enat ) ) ).
% enat_ord_code(4)
thf(fact_169_fun__diff__def,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B @ ( type2 @ B ) )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A4: A > B,B4: A > B,X4: A] : ( minus_minus @ B @ ( A4 @ X4 ) @ ( B4 @ X4 ) ) ) ) ) ).
% fun_diff_def
thf(fact_170_ldropn__lappend,axiom,
! [A: $tType,N2: nat,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ( ord_less @ extended_enat @ ( extended_enat2 @ N2 ) @ ( coindu1018505716length @ A @ Xs ) )
=> ( ( coindu531130065ldropn @ A @ N2 @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( coindu268472904append @ A @ ( coindu531130065ldropn @ A @ N2 @ Xs ) @ Ys ) ) )
& ( ~ ( ord_less @ extended_enat @ ( extended_enat2 @ N2 ) @ ( coindu1018505716length @ A @ Xs ) )
=> ( ( coindu531130065ldropn @ A @ N2 @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( coindu531130065ldropn @ A @ ( minus_minus @ nat @ N2 @ ( extended_the_enat @ ( coindu1018505716length @ A @ Xs ) ) ) @ Ys ) ) ) ) ).
% ldropn_lappend
thf(fact_171_lnth__lappend,axiom,
! [A: $tType,N2: nat,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ( ord_less @ extended_enat @ ( extended_enat2 @ N2 ) @ ( coindu1018505716length @ A @ Xs ) )
=> ( ( coindu749330388e_lnth @ A @ ( coindu268472904append @ A @ Xs @ Ys ) @ N2 )
= ( coindu749330388e_lnth @ A @ Xs @ N2 ) ) )
& ( ~ ( ord_less @ extended_enat @ ( extended_enat2 @ N2 ) @ ( coindu1018505716length @ A @ Xs ) )
=> ( ( coindu749330388e_lnth @ A @ ( coindu268472904append @ A @ Xs @ Ys ) @ N2 )
= ( coindu749330388e_lnth @ A @ Ys @ ( minus_minus @ nat @ N2 @ ( extended_the_enat @ ( coindu1018505716length @ A @ Xs ) ) ) ) ) ) ) ).
% lnth_lappend
thf(fact_172_lnth__ldropn,axiom,
! [A: $tType,N2: nat,M: nat,Xs: coindu1593790203_llist @ A] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ ( plus_plus @ nat @ N2 @ M ) ) @ ( coindu1018505716length @ A @ Xs ) )
=> ( ( coindu749330388e_lnth @ A @ ( coindu531130065ldropn @ A @ N2 @ Xs ) @ M )
= ( coindu749330388e_lnth @ A @ Xs @ ( plus_plus @ nat @ M @ N2 ) ) ) ) ).
% lnth_ldropn
thf(fact_173_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ ( zero_zero @ nat ) )
=> ( ? [X1: nat] : ( P @ X1 )
=> ? [N3: nat] :
( ~ ( P @ N3 )
& ( P @ ( suc @ N3 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_174_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_175_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_176_add_Oleft__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add.left_neutral
thf(fact_177_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.right_neutral
thf(fact_178_double__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ( plus_plus @ A @ A2 @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% double_zero
thf(fact_179_double__zero__sym,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ A2 @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% double_zero_sym
thf(fact_180_add__cancel__left__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= A2 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_left
thf(fact_181_add__cancel__left__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= A2 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_right
thf(fact_182_add__cancel__right__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( A2
= ( plus_plus @ A @ B2 @ A2 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_left
thf(fact_183_add__cancel__right__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( A2
= ( plus_plus @ A @ A2 @ B2 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_right
thf(fact_184_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_185_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_186_add__diff__cancel,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% add_diff_cancel
thf(fact_187_diff__add__cancel,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% diff_add_cancel
thf(fact_188_add__diff__cancel__left,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A @ ( type2 @ A ) )
=> ! [C: A,A2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
= ( minus_minus @ A @ A2 @ B2 ) ) ) ).
% add_diff_cancel_left
thf(fact_189_add__diff__cancel__left_H,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ A2 )
= B2 ) ) ).
% add_diff_cancel_left'
thf(fact_190_add__diff__cancel__right,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
= ( minus_minus @ A @ A2 @ B2 ) ) ) ).
% add_diff_cancel_right
thf(fact_191_add__diff__cancel__right_H,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% add_diff_cancel_right'
thf(fact_192_add__is__0,axiom,
! [M: nat,N2: nat] :
( ( ( plus_plus @ nat @ M @ N2 )
= ( zero_zero @ nat ) )
= ( ( M
= ( zero_zero @ nat ) )
& ( N2
= ( zero_zero @ nat ) ) ) ) ).
% add_is_0
thf(fact_193_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% Nat.add_0_right
thf(fact_194_add__Suc__right,axiom,
! [M: nat,N2: nat] :
( ( plus_plus @ nat @ M @ ( suc @ N2 ) )
= ( suc @ ( plus_plus @ nat @ M @ N2 ) ) ) ).
% add_Suc_right
thf(fact_195_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N2 ) )
= ( ord_less @ nat @ M @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_196_diff__diff__left,axiom,
! [I2: nat,J4: nat,K: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ I2 @ J4 ) @ K )
= ( minus_minus @ nat @ I2 @ ( plus_plus @ nat @ J4 @ K ) ) ) ).
% diff_diff_left
thf(fact_197_ldropn__ldropn,axiom,
! [A: $tType,N2: nat,M: nat,Xs: coindu1593790203_llist @ A] :
( ( coindu531130065ldropn @ A @ N2 @ ( coindu531130065ldropn @ A @ M @ Xs ) )
= ( coindu531130065ldropn @ A @ ( plus_plus @ nat @ N2 @ M ) @ Xs ) ) ).
% ldropn_ldropn
thf(fact_198_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_199_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_200_less__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% less_add_same_cancel2
thf(fact_201_less__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% less_add_same_cancel1
thf(fact_202_add__less__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel2
thf(fact_203_add__less__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel1
thf(fact_204_diff__add__zero,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( zero_zero @ A ) ) ) ).
% diff_add_zero
thf(fact_205_add__gr__0,axiom,
! [M: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ M @ N2 ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
| ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% add_gr_0
thf(fact_206_the__enat__0,axiom,
( ( extended_the_enat @ ( zero_zero @ extended_enat ) )
= ( zero_zero @ nat ) ) ).
% the_enat_0
thf(fact_207_diff__add__0,axiom,
! [N2: nat,M: nat] :
( ( minus_minus @ nat @ N2 @ ( plus_plus @ nat @ N2 @ M ) )
= ( zero_zero @ nat ) ) ).
% diff_add_0
thf(fact_208_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N2: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N2 ) )
= ( minus_minus @ nat @ M @ N2 ) ) ).
% Nat.diff_cancel
thf(fact_209_diff__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N2 @ K ) )
= ( minus_minus @ nat @ M @ N2 ) ) ).
% diff_cancel2
thf(fact_210_diff__commute,axiom,
! [I2: nat,J4: nat,K: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ I2 @ J4 ) @ K )
= ( minus_minus @ nat @ ( minus_minus @ nat @ I2 @ K ) @ J4 ) ) ).
% diff_commute
thf(fact_211_less__diff__conv,axiom,
! [I2: nat,J4: nat,K: nat] :
( ( ord_less @ nat @ I2 @ ( minus_minus @ nat @ J4 @ K ) )
= ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ K ) @ J4 ) ) ).
% less_diff_conv
thf(fact_212_plus__nat_Oadd__0,axiom,
! [N2: nat] :
( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N2 )
= N2 ) ).
% plus_nat.add_0
thf(fact_213_add__eq__self__zero,axiom,
! [M: nat,N2: nat] :
( ( ( plus_plus @ nat @ M @ N2 )
= M )
=> ( N2
= ( zero_zero @ nat ) ) ) ).
% add_eq_self_zero
thf(fact_214_diff__add__inverse,axiom,
! [N2: nat,M: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ N2 @ M ) @ N2 )
= M ) ).
% diff_add_inverse
thf(fact_215_diff__add__inverse2,axiom,
! [M: nat,N2: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N2 ) @ N2 )
= M ) ).
% diff_add_inverse2
thf(fact_216_add__diff__inverse__nat,axiom,
! [M: nat,N2: nat] :
( ~ ( ord_less @ nat @ M @ N2 )
=> ( ( plus_plus @ nat @ N2 @ ( minus_minus @ nat @ M @ N2 ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_217_comm__monoid__add__class_Oadd__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% comm_monoid_add_class.add_0
thf(fact_218_add_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.comm_neutral
thf(fact_219_add_Ogroup__left__neutral,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add.group_left_neutral
thf(fact_220_add__eq__0__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_221_add__mono__thms__linordered__field_I5_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [I2: A,J4: A,K: A,L: A] :
( ( ( ord_less @ A @ I2 @ J4 )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J4 @ L ) ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_222_add__mono__thms__linordered__field_I2_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [I2: A,J4: A,K: A,L: A] :
( ( ( I2 = J4 )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J4 @ L ) ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_223_add__mono__thms__linordered__field_I1_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [I2: A,J4: A,K: A,L: A] :
( ( ( ord_less @ A @ I2 @ J4 )
& ( K = L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J4 @ L ) ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_224_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_225_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_226_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_227_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_228_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_229_diff__eq__eq,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= C )
= ( A2
= ( plus_plus @ A @ C @ B2 ) ) ) ) ).
% diff_eq_eq
thf(fact_230_eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,B2: A] :
( ( A2
= ( minus_minus @ A @ C @ B2 ) )
= ( ( plus_plus @ A @ A2 @ B2 )
= C ) ) ) ).
% eq_diff_eq
thf(fact_231_add__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B2 @ C ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C ) ) ) ).
% add_diff_eq
thf(fact_232_diff__diff__eq2,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( minus_minus @ A @ A2 @ ( minus_minus @ A @ B2 @ C ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C ) @ B2 ) ) ) ).
% diff_diff_eq2
thf(fact_233_diff__add__eq,axiom,
! [A: $tType] :
( ( ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C ) @ B2 ) ) ) ).
% diff_add_eq
thf(fact_234_diff__add__eq__diff__diff__swap,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) )
= ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C ) @ B2 ) ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_235_diff__diff__add,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C )
= ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% diff_diff_add
thf(fact_236_add__implies__diff,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [C: A,B2: A,A2: A] :
( ( ( plus_plus @ A @ C @ B2 )
= A2 )
=> ( C
= ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).
% add_implies_diff
thf(fact_237_add__Suc,axiom,
! [M: nat,N2: nat] :
( ( plus_plus @ nat @ ( suc @ M ) @ N2 )
= ( suc @ ( plus_plus @ nat @ M @ N2 ) ) ) ).
% add_Suc
thf(fact_238_add__Suc__shift,axiom,
! [M: nat,N2: nat] :
( ( plus_plus @ nat @ ( suc @ M ) @ N2 )
= ( plus_plus @ nat @ M @ ( suc @ N2 ) ) ) ).
% add_Suc_shift
thf(fact_239_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N2: nat] :
( ( ord_less @ nat @ K @ L )
=> ( ( ( plus_plus @ nat @ M @ L )
= ( plus_plus @ nat @ K @ N2 ) )
=> ( ord_less @ nat @ M @ N2 ) ) ) ).
% less_add_eq_less
thf(fact_240_trans__less__add2,axiom,
! [I2: nat,J4: nat,M: nat] :
( ( ord_less @ nat @ I2 @ J4 )
=> ( ord_less @ nat @ I2 @ ( plus_plus @ nat @ M @ J4 ) ) ) ).
% trans_less_add2
thf(fact_241_trans__less__add1,axiom,
! [I2: nat,J4: nat,M: nat] :
( ( ord_less @ nat @ I2 @ J4 )
=> ( ord_less @ nat @ I2 @ ( plus_plus @ nat @ J4 @ M ) ) ) ).
% trans_less_add1
thf(fact_242_add__less__mono1,axiom,
! [I2: nat,J4: nat,K: nat] :
( ( ord_less @ nat @ I2 @ J4 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ K ) @ ( plus_plus @ nat @ J4 @ K ) ) ) ).
% add_less_mono1
thf(fact_243_not__add__less2,axiom,
! [J4: nat,I2: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ J4 @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_244_not__add__less1,axiom,
! [I2: nat,J4: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ J4 ) @ I2 ) ).
% not_add_less1
thf(fact_245_add__less__mono,axiom,
! [I2: nat,J4: nat,K: nat,L: nat] :
( ( ord_less @ nat @ I2 @ J4 )
=> ( ( ord_less @ nat @ K @ L )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ K ) @ ( plus_plus @ nat @ J4 @ L ) ) ) ) ).
% add_less_mono
thf(fact_246_add__lessD1,axiom,
! [I2: nat,J4: nat,K: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ J4 ) @ K )
=> ( ord_less @ nat @ I2 @ K ) ) ).
% add_lessD1
thf(fact_247_nat__add__left__cancel,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( plus_plus @ nat @ K @ M )
= ( plus_plus @ nat @ K @ N2 ) )
= ( M = N2 ) ) ).
% nat_add_left_cancel
thf(fact_248_nat__add__right__cancel,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ( plus_plus @ nat @ M @ K )
= ( plus_plus @ nat @ N2 @ K ) )
= ( M = N2 ) ) ).
% nat_add_right_cancel
thf(fact_249_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_250_is__num__normalize_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% is_num_normalize(1)
thf(fact_251_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [I2: A,J4: A,K: A,L: A] :
( ( ( I2 = J4 )
& ( K = L ) )
=> ( ( plus_plus @ A @ I2 @ K )
= ( plus_plus @ A @ J4 @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
%----Type constructors (30)
thf(tcon_fun___Orderings_Oorder,axiom,
! [A5: $tType,A6: $tType] :
( ( order @ A6 @ ( type2 @ A6 ) )
=> ( order @ ( A5 > A6 ) @ ( type2 @ ( A5 > A6 ) ) ) ) ).
thf(tcon_fun___Groups_Ominus,axiom,
! [A5: $tType,A6: $tType] :
( ( minus @ A6 @ ( type2 @ A6 ) )
=> ( minus @ ( A5 > A6 ) @ ( type2 @ ( A5 > A6 ) ) ) ) ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
ordere516151231imp_le @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
strict2144017051up_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere223160158up_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
cancel1352612707id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom,axiom,
linordered_semidom @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
comm_monoid_diff @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Omonoid__add,axiom,
monoid_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oorder_1,axiom,
order @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ominus_2,axiom,
minus @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ozero,axiom,
zero @ nat @ ( type2 @ nat ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_3,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Groups_Ominus_4,axiom,
minus @ $o @ ( type2 @ $o ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_5,axiom,
strict2144017051up_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_6,axiom,
canoni770627133id_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_7,axiom,
ordere779506340up_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_8,axiom,
ab_semigroup_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_9,axiom,
comm_monoid_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_10,axiom,
monoid_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_11,axiom,
order @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Ominus_12,axiom,
minus @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Ozero_13,axiom,
zero @ extended_enat @ ( type2 @ extended_enat ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( ( coindu2000965700_llast @ a @ ( coindu191418589_ldrop @ a @ ( extended_enat2 @ ( suc @ n_a ) ) @ xsa ) )
= ( coindu2000965700_llast @ a @ xsa ) ) ).
%------------------------------------------------------------------------------