TPTP Problem File: DAT136^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : DAT136^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Coinductive list 4627
% 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__4627.p [Bla16]
% Status : Theorem
% Rating : 0.00 v8.1.0, 0.25 v7.5.0, 0.00 v7.1.0
% Syntax : Number of formulae : 354 ( 148 unt; 62 typ; 0 def)
% Number of atoms : 662 ( 301 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 2885 ( 84 ~; 10 |; 27 &;2512 @)
% ( 0 <=>; 252 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 6 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 107 ( 107 >; 0 *; 0 +; 0 <<)
% Number of symbols : 60 ( 59 usr; 10 con; 0-4 aty)
% Number of variables : 756 ( 26 ^; 657 !; 30 ?; 756 :)
% ( 43 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 15:08:08.226
%------------------------------------------------------------------------------
%----Could-be-implicit typings (7)
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_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (55)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Ocomm__semiring__1,type,
comm_semiring_1:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1352612707id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
ordere216010020id_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_Ostrict__ordered__comm__monoid__add,type,
strict797366125id_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_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
semiri456707255roduct:
!>[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_Olconcat,type,
coindu441856546concat:
!>[A: $tType] : ( ( coindu1593790203_llist @ ( 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_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_Olnull,type,
coindu335574135_lnull:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist__of,type,
coindu1079202755ist_of:
!>[A: $tType] : ( ( list @ 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_Extended__Nat_Oenat,type,
extended_enat2: nat > extended_enat ).
thf(sy_c_Extended__Nat_Oenat_Osize__enat,type,
extended_size_enat: extended_enat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osetsum,type,
groups15040474setsum:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_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_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OlessThan,type,
set_ord_lessThan:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_N____,type,
n: nat ).
thf(sy_v_m____,type,
m: nat ).
thf(sy_v_n,type,
n2: nat ).
thf(sy_v_n_H____,type,
n3: nat ).
thf(sy_v_na____,type,
na: nat ).
thf(sy_v_x____,type,
x: a ).
thf(sy_v_xs____,type,
xs: coindu1593790203_llist @ a ).
thf(sy_v_xss,type,
xss: coindu1593790203_llist @ ( coindu1593790203_llist @ a ) ).
thf(sy_v_xss_H____,type,
xss2: list @ ( coindu1593790203_llist @ a ) ).
thf(sy_v_xssa____,type,
xssa: coindu1593790203_llist @ ( coindu1593790203_llist @ a ) ).
%----Relevant facts (250)
thf(fact_0__092_060open_062enat_A_Ilength_Axss_H_J_A_060_Allength_Axss_092_060close_062,axiom,
ord_less @ extended_enat @ ( extended_enat2 @ ( size_size @ ( list @ ( coindu1593790203_llist @ a ) ) @ xss2 ) ) @ ( coindu1018505716length @ ( coindu1593790203_llist @ a ) @ xssa ) ).
% \<open>enat (length xss') < llength xss\<close>
thf(fact_1_assms,axiom,
ord_less @ extended_enat @ ( extended_enat2 @ n2 ) @ ( coindu1018505716length @ a @ ( coindu441856546concat @ a @ xss ) ) ).
% assms
thf(fact_2__092_060open_062enat_A_ISuc_An_H_J_A_060_Allength_A_Ilnth_Axss_A_Ilength_Axss_H_J_J_092_060close_062,axiom,
ord_less @ extended_enat @ ( extended_enat2 @ ( suc @ n3 ) ) @ ( coindu1018505716length @ a @ ( coindu749330388e_lnth @ ( coindu1593790203_llist @ a ) @ xssa @ ( size_size @ ( list @ ( coindu1593790203_llist @ a ) ) @ xss2 ) ) ) ).
% \<open>enat (Suc n') < llength (lnth xss (length xss'))\<close>
thf(fact_3_Suc_Oprems,axiom,
ord_less @ extended_enat @ ( extended_enat2 @ ( suc @ na ) ) @ ( coindu1018505716length @ a @ ( coindu441856546concat @ a @ xssa ) ) ).
% Suc.prems
thf(fact_4__092_060open_062lnth_A_Ilconcat_Axss_J_A_ISuc_An_J_A_061_Alnth_A_Ilnth_Axss_A_Ilength_Axss_H_J_J_A_ISuc_An_H_J_092_060close_062,axiom,
( ( coindu749330388e_lnth @ a @ ( coindu441856546concat @ a @ xssa ) @ ( suc @ na ) )
= ( coindu749330388e_lnth @ a @ ( coindu749330388e_lnth @ ( coindu1593790203_llist @ a ) @ xssa @ ( size_size @ ( list @ ( coindu1593790203_llist @ a ) ) @ xss2 ) ) @ ( suc @ n3 ) ) ) ).
% \<open>lnth (lconcat xss) (Suc n) = lnth (lnth xss (length xss')) (Suc n')\<close>
thf(fact_5__092_060open_062enat_A_ISuc_An_J_A_061_A_I_092_060Sum_062i_060length_Axss_H_O_Allength_A_Ilnth_Axss_Ai_J_J_A_L_Aenat_A_ISuc_An_H_J_092_060close_062,axiom,
( ( extended_enat2 @ ( suc @ na ) )
= ( plus_plus @ extended_enat
@ ( groups15040474setsum @ nat @ extended_enat
@ ^ [I: nat] : ( coindu1018505716length @ a @ ( coindu749330388e_lnth @ ( coindu1593790203_llist @ a ) @ xssa @ I ) )
@ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ ( coindu1593790203_llist @ a ) ) @ xss2 ) ) )
@ ( extended_enat2 @ ( suc @ n3 ) ) ) ) ).
% \<open>enat (Suc n) = (\<Sum>i<length xss'. llength (lnth xss i)) + enat (Suc n')\<close>
thf(fact_6_Suc_Ohyps,axiom,
! [Xss: coindu1593790203_llist @ ( coindu1593790203_llist @ a )] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ na ) @ ( coindu1018505716length @ a @ ( coindu441856546concat @ a @ Xss ) ) )
=> ? [M: nat,N: nat] :
( ( ( coindu749330388e_lnth @ a @ ( coindu441856546concat @ a @ Xss ) @ na )
= ( coindu749330388e_lnth @ a @ ( coindu749330388e_lnth @ ( coindu1593790203_llist @ a ) @ Xss @ M ) @ N ) )
& ( ord_less @ extended_enat @ ( extended_enat2 @ N ) @ ( coindu1018505716length @ a @ ( coindu749330388e_lnth @ ( coindu1593790203_llist @ a ) @ Xss @ M ) ) )
& ( ord_less @ extended_enat @ ( extended_enat2 @ M ) @ ( coindu1018505716length @ ( coindu1593790203_llist @ a ) @ Xss ) )
& ( ( extended_enat2 @ na )
= ( plus_plus @ extended_enat
@ ( groups15040474setsum @ nat @ extended_enat
@ ^ [I: nat] : ( coindu1018505716length @ a @ ( coindu749330388e_lnth @ ( coindu1593790203_llist @ a ) @ Xss @ I ) )
@ ( set_ord_lessThan @ nat @ M ) )
@ ( extended_enat2 @ N ) ) ) ) ) ).
% Suc.hyps
thf(fact_7_setsum__lessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A @ ( type2 @ A ) )
=> ! [F: nat > A,N2: nat] :
( ( groups15040474setsum @ nat @ A @ F @ ( set_ord_lessThan @ nat @ ( suc @ N2 ) ) )
= ( plus_plus @ A @ ( groups15040474setsum @ nat @ A @ F @ ( set_ord_lessThan @ nat @ N2 ) ) @ ( F @ N2 ) ) ) ) ).
% setsum_lessThan_Suc
thf(fact_8_enat__add__mono,axiom,
! [X: nat,Y: extended_enat,Z: extended_enat] :
( ( ord_less @ extended_enat @ ( plus_plus @ extended_enat @ ( extended_enat2 @ X ) @ Y ) @ ( plus_plus @ extended_enat @ ( extended_enat2 @ X ) @ Z ) )
= ( ord_less @ extended_enat @ Y @ Z ) ) ).
% enat_add_mono
thf(fact_9_enat__add1__eq,axiom,
! [X: nat,Y: extended_enat,Z: extended_enat] :
( ( ( plus_plus @ extended_enat @ ( extended_enat2 @ X ) @ Y )
= ( plus_plus @ extended_enat @ ( extended_enat2 @ X ) @ Z ) )
= ( Y = Z ) ) ).
% enat_add1_eq
thf(fact_10_enat__add2__eq,axiom,
! [Y: extended_enat,X: nat,Z: extended_enat] :
( ( ( plus_plus @ extended_enat @ Y @ ( extended_enat2 @ X ) )
= ( plus_plus @ extended_enat @ Z @ ( extended_enat2 @ X ) ) )
= ( Y = Z ) ) ).
% enat_add2_eq
thf(fact_11_lessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [I2: A,K: A] :
( ( member @ A @ I2 @ ( set_ord_lessThan @ A @ K ) )
= ( ord_less @ A @ I2 @ K ) ) ) ).
% lessThan_iff
thf(fact_12_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_13_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_14__092_060open_062_I_092_060Sum_062i_060length_Axss_H_O_Allength_A_Ilnth_Axss_Ai_J_J_A_061_A0_092_060close_062,axiom,
( ( groups15040474setsum @ nat @ extended_enat
@ ^ [I: nat] : ( coindu1018505716length @ a @ ( coindu749330388e_lnth @ ( coindu1593790203_llist @ a ) @ xssa @ I ) )
@ ( set_ord_lessThan @ nat @ ( size_size @ ( list @ ( coindu1593790203_llist @ a ) ) @ xss2 ) ) )
= ( zero_zero @ extended_enat ) ) ).
% \<open>(\<Sum>i<length xss'. llength (lnth xss i)) = 0\<close>
thf(fact_15_enat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( extended_enat2 @ Nat )
= ( extended_enat2 @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% enat.inject
thf(fact_16_lessThan__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ( set_ord_lessThan @ A @ X )
= ( set_ord_lessThan @ A @ Y ) )
= ( X = Y ) ) ) ).
% lessThan_eq_iff
thf(fact_17_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_18_n,axiom,
( na
= ( plus_plus @ nat @ n @ n3 ) ) ).
% n
thf(fact_19_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_20_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_21_add__Suc__right,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus @ nat @ M2 @ ( suc @ N2 ) )
= ( suc @ ( plus_plus @ nat @ M2 @ N2 ) ) ) ).
% add_Suc_right
thf(fact_22_Suc__less__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
= ( ord_less @ nat @ M2 @ N2 ) ) ).
% Suc_less_eq
thf(fact_23_Suc__mono,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( ord_less @ nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) ) ).
% Suc_mono
thf(fact_24_lessI,axiom,
! [N2: nat] : ( ord_less @ nat @ N2 @ ( suc @ N2 ) ) ).
% lessI
thf(fact_25_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_26_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_27_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_28_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_29_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_30_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_31_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_32_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_33_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_34_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_35_enat__ord__simps_I2_J,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ M2 ) @ ( extended_enat2 @ N2 ) )
= ( ord_less @ nat @ M2 @ N2 ) ) ).
% enat_ord_simps(2)
thf(fact_36_i0__less,axiom,
! [N2: extended_enat] :
( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ N2 )
= ( N2
!= ( zero_zero @ extended_enat ) ) ) ).
% i0_less
thf(fact_37_plus__enat__simps_I1_J,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus @ extended_enat @ ( extended_enat2 @ M2 ) @ ( extended_enat2 @ N2 ) )
= ( extended_enat2 @ ( plus_plus @ nat @ M2 @ N2 ) ) ) ).
% plus_enat_simps(1)
thf(fact_38_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_39_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_40_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_41_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_42_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_43_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_44_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_45_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X4: A] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_46_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X4: A] :
( ( F @ X4 )
= ( G @ X4 ) )
=> ( F = G ) ) ).
% ext
thf(fact_47_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_48_less__add__Suc1,axiom,
! [I2: nat,M2: nat] : ( ord_less @ nat @ I2 @ ( suc @ ( plus_plus @ nat @ I2 @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_49_less__add__Suc2,axiom,
! [I2: nat,M2: nat] : ( ord_less @ nat @ I2 @ ( suc @ ( plus_plus @ nat @ M2 @ I2 ) ) ) ).
% less_add_Suc2
thf(fact_50_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_51_less__iff__Suc__add,axiom,
( ( ord_less @ nat )
= ( ^ [M3: nat,N3: nat] :
? [K2: nat] :
( N3
= ( suc @ ( plus_plus @ nat @ M3 @ K2 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_52_less__imp__Suc__add,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ? [K3: nat] :
( N2
= ( suc @ ( plus_plus @ nat @ M2 @ K3 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_53_add__Suc,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus @ nat @ ( suc @ M2 ) @ N2 )
= ( suc @ ( plus_plus @ nat @ M2 @ N2 ) ) ) ).
% add_Suc
thf(fact_54_add__Suc__shift,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus @ nat @ ( suc @ M2 ) @ N2 )
= ( plus_plus @ nat @ M2 @ ( suc @ N2 ) ) ) ).
% add_Suc_shift
thf(fact_55_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_56_gr__implies__not__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [M2: A,N2: A] :
( ( ord_less @ A @ M2 @ N2 )
=> ( N2
!= ( zero_zero @ A ) ) ) ) ).
% gr_implies_not_zero
thf(fact_57_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_58_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_59_not__less__less__Suc__eq,axiom,
! [N2: nat,M2: nat] :
( ~ ( ord_less @ nat @ N2 @ M2 )
=> ( ( ord_less @ nat @ N2 @ ( suc @ M2 ) )
= ( N2 = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_60_strict__inc__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less @ nat @ I2 @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I2 ) ) ) ) ).
% strict_inc_induct
thf(fact_61_less__Suc__induct,axiom,
! [I2: nat,J: nat,P: nat > nat > $o] :
( ( ord_less @ nat @ I2 @ J )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J2: nat,K3: nat] :
( ( ord_less @ nat @ I3 @ J2 )
=> ( ( ord_less @ nat @ J2 @ K3 )
=> ( ( P @ I3 @ J2 )
=> ( ( P @ J2 @ K3 )
=> ( P @ I3 @ K3 ) ) ) ) )
=> ( P @ I2 @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_62_less__trans__Suc,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( ( ord_less @ nat @ J @ K )
=> ( ord_less @ nat @ ( suc @ I2 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_63_Suc__less__SucD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
=> ( ord_less @ nat @ M2 @ N2 ) ) ).
% Suc_less_SucD
thf(fact_64_less__antisym,axiom,
! [N2: nat,M2: nat] :
( ~ ( ord_less @ nat @ N2 @ M2 )
=> ( ( ord_less @ nat @ N2 @ ( suc @ M2 ) )
=> ( M2 = N2 ) ) ) ).
% less_antisym
thf(fact_65_Suc__less__eq2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less @ nat @ ( suc @ N2 ) @ M2 )
= ( ? [M4: nat] :
( ( M2
= ( suc @ M4 ) )
& ( ord_less @ nat @ N2 @ M4 ) ) ) ) ).
% Suc_less_eq2
thf(fact_66_not__less__eq,axiom,
! [M2: nat,N2: nat] :
( ( ~ ( ord_less @ nat @ M2 @ N2 ) )
= ( ord_less @ nat @ N2 @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_67_less__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ ( suc @ N2 ) )
= ( ( ord_less @ nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ).
% less_Suc_eq
thf(fact_68_less__SucI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( ord_less @ nat @ M2 @ ( suc @ N2 ) ) ) ).
% less_SucI
thf(fact_69_less__SucE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ ( suc @ N2 ) )
=> ( ~ ( ord_less @ nat @ M2 @ N2 )
=> ( M2 = N2 ) ) ) ).
% less_SucE
thf(fact_70_Suc__lessI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( ( ( suc @ M2 )
!= N2 )
=> ( ord_less @ nat @ ( suc @ M2 ) @ N2 ) ) ) ).
% Suc_lessI
thf(fact_71_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_72_Suc__lessD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less @ nat @ M2 @ N2 ) ) ).
% Suc_lessD
thf(fact_73_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_74_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_75_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_76_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_77_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_78_not__iless0,axiom,
! [N2: extended_enat] :
~ ( ord_less @ extended_enat @ N2 @ ( zero_zero @ extended_enat ) ) ).
% not_iless0
thf(fact_79_plus__enat__eq__0__conv,axiom,
! [M2: extended_enat,N2: extended_enat] :
( ( ( plus_plus @ extended_enat @ M2 @ N2 )
= ( zero_zero @ extended_enat ) )
= ( ( M2
= ( zero_zero @ extended_enat ) )
& ( N2
= ( zero_zero @ extended_enat ) ) ) ) ).
% plus_enat_eq_0_conv
thf(fact_80_pos__add__strict,axiom,
! [A: $tType] :
( ( strict797366125id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ B2 @ C )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% pos_add_strict
thf(fact_81_add__pos__pos,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% add_pos_pos
thf(fact_82_add__neg__neg,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_neg
thf(fact_83_less__enatE,axiom,
! [N2: extended_enat,M2: nat] :
( ( ord_less @ extended_enat @ N2 @ ( extended_enat2 @ M2 ) )
=> ~ ! [K3: nat] :
( ( N2
= ( extended_enat2 @ K3 ) )
=> ~ ( ord_less @ nat @ K3 @ M2 ) ) ) ).
% less_enatE
thf(fact_84_enat__less__enat__plusI2,axiom,
! [Y: nat,Z: extended_enat,X: nat] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ Y ) @ Z )
=> ( ord_less @ extended_enat @ ( extended_enat2 @ ( plus_plus @ nat @ X @ Y ) ) @ ( plus_plus @ extended_enat @ ( extended_enat2 @ X ) @ Z ) ) ) ).
% enat_less_enat_plusI2
thf(fact_85_enat__less__enat__plusI,axiom,
! [X: nat,Y: nat,Z: extended_enat] :
( ( ord_less @ nat @ X @ Y )
=> ( ord_less @ extended_enat @ ( extended_enat2 @ X ) @ ( plus_plus @ extended_enat @ ( extended_enat2 @ Y ) @ Z ) ) ) ).
% enat_less_enat_plusI
thf(fact_86_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_87_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_88_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_89_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type2 @ A ) )
=> ( ( plus_plus @ A )
= ( ^ [A4: A,B3: A] : ( plus_plus @ A @ B3 @ A4 ) ) ) ) ).
% add.commute
thf(fact_90_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_91_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_92_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_93_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [I2: A,J: A,K: A,L: A] :
( ( ( I2 = J )
& ( K = L ) )
=> ( ( plus_plus @ A @ I2 @ K )
= ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_94_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_95_n__not__Suc__n,axiom,
! [N2: nat] :
( N2
!= ( suc @ N2 ) ) ).
% n_not_Suc_n
thf(fact_96_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_97_enat__less__induct,axiom,
! [P: extended_enat > $o,N2: extended_enat] :
( ! [N4: extended_enat] :
( ! [M5: extended_enat] :
( ( ord_less @ extended_enat @ M5 @ N4 )
=> ( P @ M5 ) )
=> ( P @ N4 ) )
=> ( P @ N2 ) ) ).
% enat_less_induct
thf(fact_98_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_99_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_100_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_101_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_102_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_103_add__mono__thms__linordered__field_I1_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [I2: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I2 @ J )
& ( K = L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_104_add__mono__thms__linordered__field_I2_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [I2: A,J: A,K: A,L: A] :
( ( ( I2 = J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_105_add__mono__thms__linordered__field_I5_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [I2: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I2 @ J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I2 @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_106_lift__Suc__mono__less__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [F: nat > A,N2: nat,M2: nat] :
( ! [N4: nat] : ( ord_less @ A @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less @ A @ ( F @ N2 ) @ ( F @ M2 ) )
= ( ord_less @ nat @ N2 @ M2 ) ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_107_lift__Suc__mono__less,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [F: nat > A,N2: nat,N5: nat] :
( ! [N4: nat] : ( ord_less @ A @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less @ nat @ N2 @ N5 )
=> ( ord_less @ A @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_108_lessThan__strict__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [M2: A,N2: A] :
( ( ord_less @ ( set @ A ) @ ( set_ord_lessThan @ A @ M2 ) @ ( set_ord_lessThan @ A @ N2 ) )
= ( ord_less @ A @ M2 @ N2 ) ) ) ).
% lessThan_strict_subset_iff
thf(fact_109_enat__iless,axiom,
! [N2: extended_enat,M2: nat] :
( ( ord_less @ extended_enat @ N2 @ ( extended_enat2 @ M2 ) )
=> ? [K3: nat] :
( N2
= ( extended_enat2 @ K3 ) ) ) ).
% enat_iless
thf(fact_110_chain__incr,axiom,
! [A: $tType,Y3: A > extended_enat,K: nat] :
( ! [I3: A] :
? [J3: A] : ( ord_less @ extended_enat @ ( Y3 @ I3 ) @ ( Y3 @ J3 ) )
=> ? [J2: A] : ( ord_less @ extended_enat @ ( extended_enat2 @ K ) @ ( Y3 @ J2 ) ) ) ).
% chain_incr
thf(fact_111_lessThan__def,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ( ( set_ord_lessThan @ A )
= ( ^ [U: A] :
( collect @ A
@ ^ [X3: A] : ( ord_less @ A @ X3 @ U ) ) ) ) ) ).
% lessThan_def
thf(fact_112_setsum_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A @ ( type2 @ A ) )
=> ! [A3: set @ B] :
( ( groups15040474setsum @ B @ A
@ ^ [Uu: B] : ( zero_zero @ A )
@ A3 )
= ( zero_zero @ A ) ) ) ).
% setsum.neutral_const
thf(fact_113_lnth__prefix,axiom,
! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ ( coindu1593790203_llist @ a ) ) @ xss2 ) )
=> ( ( coindu749330388e_lnth @ ( coindu1593790203_llist @ a ) @ xssa @ I4 )
= ( coindu1598213697e_LNil @ a ) ) ) ).
% lnth_prefix
thf(fact_114_llength__lconcat__lfinite__conv__sum,axiom,
! [A: $tType,Xss: coindu1593790203_llist @ ( coindu1593790203_llist @ A )] :
( ( coindu1213758845finite @ ( coindu1593790203_llist @ A ) @ Xss )
=> ( ( coindu1018505716length @ A @ ( coindu441856546concat @ A @ Xss ) )
= ( groups15040474setsum @ nat @ extended_enat
@ ^ [I: nat] : ( coindu1018505716length @ A @ ( coindu749330388e_lnth @ ( coindu1593790203_llist @ A ) @ Xss @ I ) )
@ ( collect @ nat
@ ^ [I: nat] : ( ord_less @ extended_enat @ ( extended_enat2 @ I ) @ ( coindu1018505716length @ ( coindu1593790203_llist @ A ) @ Xss ) ) ) ) ) ) ).
% llength_lconcat_lfinite_conv_sum
thf(fact_115_setsum_Odistrib,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A @ ( type2 @ A ) )
=> ! [G: B > A,H: B > A,A3: set @ B] :
( ( groups15040474setsum @ B @ A
@ ^ [X3: B] : ( plus_plus @ A @ ( G @ X3 ) @ ( H @ X3 ) )
@ A3 )
= ( plus_plus @ A @ ( groups15040474setsum @ B @ A @ G @ A3 ) @ ( groups15040474setsum @ B @ A @ H @ A3 ) ) ) ) ).
% setsum.distrib
thf(fact_116__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062x_Axs_O_Alconcat_Axss_A_061_ALCons_Ax_Axs_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [X4: a,Xs: coindu1593790203_llist @ a] :
( ( coindu441856546concat @ a @ xssa )
!= ( coindu1121789889_LCons @ a @ X4 @ Xs ) ) ).
% \<open>\<And>thesis. (\<And>x xs. lconcat xss = LCons x xs \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_117__092_060open_062_092_060not_062_Alnull_A_Ilconcat_Axss_J_092_060close_062,axiom,
~ ( coindu335574135_lnull @ a @ ( coindu441856546concat @ a @ xssa ) ) ).
% \<open>\<not> lnull (lconcat xss)\<close>
thf(fact_118_gen__llength__def,axiom,
! [A: $tType] :
( ( coindu493225792length @ A )
= ( ^ [N3: nat,Xs2: coindu1593790203_llist @ A] : ( plus_plus @ extended_enat @ ( extended_enat2 @ N3 ) @ ( coindu1018505716length @ A @ Xs2 ) ) ) ) ).
% gen_llength_def
thf(fact_119_setsum_Onot__neutral__contains__not__neutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A @ ( type2 @ A ) )
=> ! [G: B > A,A3: set @ B] :
( ( ( groups15040474setsum @ B @ A @ G @ A3 )
!= ( zero_zero @ A ) )
=> ~ ! [A5: B] :
( ( member @ B @ A5 @ A3 )
=> ( ( G @ A5 )
= ( zero_zero @ A ) ) ) ) ) ).
% setsum.not_neutral_contains_not_neutral
thf(fact_120_setsum_Oneutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A @ ( type2 @ A ) )
=> ! [A3: set @ B,G: B > A] :
( ! [X4: B] :
( ( member @ B @ X4 @ A3 )
=> ( ( G @ X4 )
= ( zero_zero @ A ) ) )
=> ( ( groups15040474setsum @ B @ A @ G @ A3 )
= ( zero_zero @ A ) ) ) ) ).
% setsum.neutral
thf(fact_121_add__0__iff,axiom,
! [A: $tType] :
( ( semiri456707255roduct @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( B2
= ( plus_plus @ A @ B2 @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% add_0_iff
thf(fact_122_False,axiom,
~ ( ord_less @ nat @ ( zero_zero @ nat ) @ m ) ).
% False
thf(fact_123_neq0__conv,axiom,
! [N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).
% neq0_conv
thf(fact_124_add__is__0,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus @ nat @ M2 @ N2 )
= ( zero_zero @ nat ) )
= ( ( M2
= ( zero_zero @ nat ) )
& ( N2
= ( zero_zero @ nat ) ) ) ) ).
% add_is_0
thf(fact_125_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus @ nat @ M2 @ ( zero_zero @ nat ) )
= M2 ) ).
% Nat.add_0_right
thf(fact_126_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_127_zero__less__Suc,axiom,
! [N2: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) ).
% zero_less_Suc
thf(fact_128_less__Suc0,axiom,
! [N2: nat] :
( ( ord_less @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) )
= ( N2
= ( zero_zero @ nat ) ) ) ).
% less_Suc0
thf(fact_129_add__gr__0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ M2 @ N2 ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
| ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% add_gr_0
thf(fact_130_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M2 ) @ ( plus_plus @ nat @ K @ N2 ) )
= ( ord_less @ nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_131_lfinite__LCons,axiom,
! [A: $tType,X: A,Xs3: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ ( coindu1121789889_LCons @ A @ X @ Xs3 ) )
= ( coindu1213758845finite @ A @ Xs3 ) ) ).
% lfinite_LCons
thf(fact_132_lfinite__code_I2_J,axiom,
! [B: $tType,X: B,Xs3: coindu1593790203_llist @ B] :
( ( coindu1213758845finite @ B @ ( coindu1121789889_LCons @ B @ X @ Xs3 ) )
= ( coindu1213758845finite @ B @ Xs3 ) ) ).
% lfinite_code(2)
thf(fact_133_lfinite__code_I1_J,axiom,
! [A: $tType] : ( coindu1213758845finite @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% lfinite_code(1)
thf(fact_134_lnth__0,axiom,
! [A: $tType,X: A,Xs3: coindu1593790203_llist @ A] :
( ( coindu749330388e_lnth @ A @ ( coindu1121789889_LCons @ A @ X @ Xs3 ) @ ( zero_zero @ nat ) )
= X ) ).
% lnth_0
thf(fact_135_lconcat__LNil,axiom,
! [A: $tType] :
( ( coindu441856546concat @ A @ ( coindu1598213697e_LNil @ ( coindu1593790203_llist @ A ) ) )
= ( coindu1598213697e_LNil @ A ) ) ).
% lconcat_LNil
thf(fact_136_lnth__Suc__LCons,axiom,
! [A: $tType,X: A,Xs3: coindu1593790203_llist @ A,N2: nat] :
( ( coindu749330388e_lnth @ A @ ( coindu1121789889_LCons @ A @ X @ Xs3 ) @ ( suc @ N2 ) )
= ( coindu749330388e_lnth @ A @ Xs3 @ N2 ) ) ).
% lnth_Suc_LCons
thf(fact_137_llength__eq__0,axiom,
! [A: $tType,Xs3: coindu1593790203_llist @ A] :
( ( ( coindu1018505716length @ A @ Xs3 )
= ( zero_zero @ extended_enat ) )
= ( coindu335574135_lnull @ A @ Xs3 ) ) ).
% llength_eq_0
thf(fact_138_llength__lnull,axiom,
! [A: $tType,Xs3: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs3 )
=> ( ( coindu1018505716length @ A @ Xs3 )
= ( zero_zero @ extended_enat ) ) ) ).
% llength_lnull
thf(fact_139_llength__LNil,axiom,
! [A: $tType] :
( ( coindu1018505716length @ A @ ( coindu1598213697e_LNil @ A ) )
= ( zero_zero @ extended_enat ) ) ).
% llength_LNil
thf(fact_140_concat__xss,axiom,
( ( coindu441856546concat @ a @ xssa )
= ( coindu1121789889_LCons @ a @ x @ xs ) ) ).
% concat_xss
thf(fact_141_gr0I,axiom,
! [N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).
% gr0I
thf(fact_142_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) )
= ( N2
= ( zero_zero @ nat ) ) ) ).
% not_gr0
thf(fact_143_not__less0,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% not_less0
thf(fact_144_add__lessD1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ J ) @ K )
=> ( ord_less @ nat @ I2 @ K ) ) ).
% add_lessD1
thf(fact_145_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% less_zeroE
thf(fact_146_nat__neq__iff,axiom,
! [M2: nat,N2: nat] :
( ( M2 != N2 )
= ( ( ord_less @ nat @ M2 @ N2 )
| ( ord_less @ nat @ N2 @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_147_add__less__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( ( ord_less @ nat @ K @ L )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_148_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_149_not__add__less1,axiom,
! [I2: nat,J: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ J ) @ I2 ) ).
% not_add_less1
thf(fact_150_not__add__less2,axiom,
! [J: nat,I2: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_151_add__less__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_152_less__not__refl2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less @ nat @ N2 @ M2 )
=> ( M2 != N2 ) ) ).
% less_not_refl2
thf(fact_153_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less @ nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_154_measure__induct,axiom,
! [A: $tType,F: A > nat,P: A > $o,A2: A] :
( ! [X4: A] :
( ! [Y4: A] :
( ( ord_less @ nat @ ( F @ Y4 ) @ ( F @ X4 ) )
=> ( P @ Y4 ) )
=> ( P @ X4 ) )
=> ( P @ A2 ) ) ).
% measure_induct
thf(fact_155_plus__nat_Oadd__0,axiom,
! [N2: nat] :
( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N2 )
= N2 ) ).
% plus_nat.add_0
thf(fact_156_gr__implies__not0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( N2
!= ( zero_zero @ nat ) ) ) ).
% gr_implies_not0
thf(fact_157_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_158_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N4: nat] :
( ! [M5: nat] :
( ( ord_less @ nat @ M5 @ N4 )
=> ( P @ M5 ) )
=> ( P @ N4 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_159_trans__less__add1,axiom,
! [I2: nat,J: nat,M2: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( ord_less @ nat @ I2 @ ( plus_plus @ nat @ J @ M2 ) ) ) ).
% trans_less_add1
thf(fact_160_trans__less__add2,axiom,
! [I2: nat,J: nat,M2: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( ord_less @ nat @ I2 @ ( plus_plus @ nat @ M2 @ J ) ) ) ).
% trans_less_add2
thf(fact_161_add__eq__self__zero,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus @ nat @ M2 @ N2 )
= M2 )
=> ( N2
= ( zero_zero @ nat ) ) ) ).
% add_eq_self_zero
thf(fact_162_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N4: nat] :
( ~ ( P @ N4 )
=> ? [M5: nat] :
( ( ord_less @ nat @ M5 @ N4 )
& ~ ( P @ M5 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_163_less__add__eq__less,axiom,
! [K: nat,L: nat,M2: nat,N2: nat] :
( ( ord_less @ nat @ K @ L )
=> ( ( ( plus_plus @ nat @ M2 @ L )
= ( plus_plus @ nat @ K @ N2 ) )
=> ( ord_less @ nat @ M2 @ N2 ) ) ) ).
% less_add_eq_less
thf(fact_164_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N4: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
=> ( ~ ( P @ N4 )
=> ? [M5: nat] :
( ( ord_less @ nat @ M5 @ N4 )
& ~ ( P @ M5 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_165_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_166_setsum__SucD,axiom,
! [A: $tType,F: A > nat,A3: set @ A,N2: nat] :
( ( ( groups15040474setsum @ A @ nat @ F @ A3 )
= ( suc @ N2 ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ A3 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X4 ) ) ) ) ).
% setsum_SucD
thf(fact_167_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% less_nat_zero_code
thf(fact_168_measure__induct__rule,axiom,
! [A: $tType,F: A > nat,P: A > $o,A2: A] :
( ! [X4: A] :
( ! [Y4: A] :
( ( ord_less @ nat @ ( F @ Y4 ) @ ( F @ X4 ) )
=> ( P @ Y4 ) )
=> ( P @ X4 ) )
=> ( P @ A2 ) ) ).
% measure_induct_rule
thf(fact_169_nat__add__left__cancel,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ( plus_plus @ nat @ K @ M2 )
= ( plus_plus @ nat @ K @ N2 ) )
= ( M2 = N2 ) ) ).
% nat_add_left_cancel
thf(fact_170_nat__add__right__cancel,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ( plus_plus @ nat @ M2 @ K )
= ( plus_plus @ nat @ N2 @ K ) )
= ( M2 = N2 ) ) ).
% nat_add_right_cancel
thf(fact_171_less__imp__add__positive,axiom,
! [I2: nat,J: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ? [K3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K3 )
& ( ( plus_plus @ nat @ I2 @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_172_infinite__descent__measure,axiom,
! [A: $tType,P: A > $o,V: A > nat,X: A] :
( ! [X4: A] :
( ~ ( P @ X4 )
=> ? [Y4: A] :
( ( ord_less @ nat @ ( V @ Y4 ) @ ( V @ X4 ) )
& ~ ( P @ Y4 ) ) )
=> ( P @ X ) ) ).
% infinite_descent_measure
thf(fact_173_infinite__descent0__measure,axiom,
! [A: $tType,V: A > nat,P: A > $o,X: A] :
( ! [X4: A] :
( ( ( V @ X4 )
= ( zero_zero @ nat ) )
=> ( P @ X4 ) )
=> ( ! [X4: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X4 ) )
=> ( ~ ( P @ X4 )
=> ? [Y4: A] :
( ( ord_less @ nat @ ( V @ Y4 ) @ ( V @ X4 ) )
& ~ ( P @ Y4 ) ) ) )
=> ( P @ X ) ) ) ).
% infinite_descent0_measure
thf(fact_174_lfinite__LConsI,axiom,
! [A: $tType,Xs3: coindu1593790203_llist @ A,X: A] :
( ( coindu1213758845finite @ A @ Xs3 )
=> ( coindu1213758845finite @ A @ ( coindu1121789889_LCons @ A @ X @ Xs3 ) ) ) ).
% lfinite_LConsI
thf(fact_175_lfinite__LNil,axiom,
! [A: $tType] : ( coindu1213758845finite @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% lfinite_LNil
thf(fact_176_lnull__imp__lfinite,axiom,
! [A: $tType,Xs3: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs3 )
=> ( coindu1213758845finite @ A @ Xs3 ) ) ).
% lnull_imp_lfinite
thf(fact_177_lnull__def,axiom,
! [A: $tType] :
( ( coindu335574135_lnull @ A )
= ( ^ [Llist: coindu1593790203_llist @ A] :
( Llist
= ( coindu1598213697e_LNil @ A ) ) ) ) ).
% lnull_def
thf(fact_178_lfinite_Oinducts,axiom,
! [A: $tType,X: coindu1593790203_llist @ A,P: ( coindu1593790203_llist @ A ) > $o] :
( ( coindu1213758845finite @ A @ X )
=> ( ( P @ ( coindu1598213697e_LNil @ A ) )
=> ( ! [Xs: coindu1593790203_llist @ A,X4: A] :
( ( coindu1213758845finite @ A @ Xs )
=> ( ( P @ Xs )
=> ( P @ ( coindu1121789889_LCons @ A @ X4 @ Xs ) ) ) )
=> ( P @ X ) ) ) ) ).
% lfinite.inducts
thf(fact_179_lappend_Oexhaust,axiom,
! [A: $tType,Xs3: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu335574135_lnull @ A @ Xs3 )
=> ~ ( coindu335574135_lnull @ A @ Ys ) )
=> ( ~ ( coindu335574135_lnull @ A @ Xs3 )
| ~ ( coindu335574135_lnull @ A @ Ys ) ) ) ).
% lappend.exhaust
thf(fact_180_not__lnull__conv,axiom,
! [A: $tType,Xs3: coindu1593790203_llist @ A] :
( ( ~ ( coindu335574135_lnull @ A @ Xs3 ) )
= ( ? [X3: A,Xs4: coindu1593790203_llist @ A] :
( Xs3
= ( coindu1121789889_LCons @ A @ X3 @ Xs4 ) ) ) ) ).
% not_lnull_conv
thf(fact_181_neq__LNil__conv,axiom,
! [A: $tType,Xs3: coindu1593790203_llist @ A] :
( ( Xs3
!= ( coindu1598213697e_LNil @ A ) )
= ( ? [X3: A,Xs4: coindu1593790203_llist @ A] :
( Xs3
= ( coindu1121789889_LCons @ A @ X3 @ Xs4 ) ) ) ) ).
% neq_LNil_conv
thf(fact_182_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_183_lfinite_Osimps,axiom,
! [A: $tType] :
( ( coindu1213758845finite @ A )
= ( ^ [A4: coindu1593790203_llist @ A] :
( ( A4
= ( coindu1598213697e_LNil @ A ) )
| ? [Xs2: coindu1593790203_llist @ A,X3: A] :
( ( A4
= ( coindu1121789889_LCons @ A @ X3 @ Xs2 ) )
& ( coindu1213758845finite @ A @ Xs2 ) ) ) ) ) ).
% lfinite.simps
thf(fact_184_lfinite_Ocases,axiom,
! [A: $tType,A2: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ A2 )
=> ( ( A2
!= ( coindu1598213697e_LNil @ A ) )
=> ~ ! [Xs: coindu1593790203_llist @ A] :
( ? [X4: A] :
( A2
= ( coindu1121789889_LCons @ A @ X4 @ Xs ) )
=> ~ ( coindu1213758845finite @ A @ Xs ) ) ) ) ).
% lfinite.cases
thf(fact_185_lzip_Oexhaust,axiom,
! [A: $tType,B: $tType,Xs3: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B] :
( ~ ( ( coindu335574135_lnull @ A @ Xs3 )
| ( coindu335574135_lnull @ B @ Ys ) )
=> ~ ( ~ ( coindu335574135_lnull @ A @ Xs3 )
=> ( coindu335574135_lnull @ B @ Ys ) ) ) ).
% lzip.exhaust
thf(fact_186_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_187_llist_Ocollapse_I1_J,axiom,
! [A: $tType,Llist2: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Llist2 )
=> ( Llist2
= ( coindu1598213697e_LNil @ A ) ) ) ).
% llist.collapse(1)
thf(fact_188_llist_OdiscI_I1_J,axiom,
! [A: $tType,Llist2: coindu1593790203_llist @ A] :
( ( Llist2
= ( coindu1598213697e_LNil @ A ) )
=> ( coindu335574135_lnull @ A @ Llist2 ) ) ).
% llist.discI(1)
thf(fact_189_llist_OdiscI_I2_J,axiom,
! [A: $tType,Llist2: coindu1593790203_llist @ A,X21: A,X22: coindu1593790203_llist @ A] :
( ( Llist2
= ( coindu1121789889_LCons @ A @ X21 @ X22 ) )
=> ~ ( coindu335574135_lnull @ A @ Llist2 ) ) ).
% llist.discI(2)
thf(fact_190_llist_Odisc_I1_J,axiom,
! [A: $tType] : ( coindu335574135_lnull @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% llist.disc(1)
thf(fact_191_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_192_gen__llength__code_I1_J,axiom,
! [A: $tType,N2: nat] :
( ( coindu493225792length @ A @ N2 @ ( coindu1598213697e_LNil @ A ) )
= ( extended_enat2 @ N2 ) ) ).
% gen_llength_code(1)
thf(fact_193_llength__code,axiom,
! [A: $tType] :
( ( coindu1018505716length @ A )
= ( coindu493225792length @ A @ ( zero_zero @ nat ) ) ) ).
% llength_code
thf(fact_194_ltake_Oexhaust,axiom,
! [A: $tType,N2: extended_enat,Xs3: coindu1593790203_llist @ A] :
( ~ ( ( N2
= ( zero_zero @ extended_enat ) )
| ( coindu335574135_lnull @ A @ Xs3 ) )
=> ~ ( ( N2
!= ( zero_zero @ extended_enat ) )
=> ( coindu335574135_lnull @ A @ Xs3 ) ) ) ).
% ltake.exhaust
thf(fact_195_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( ( zero_zero @ nat )
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_196_old_Onat_Odistinct_I2_J,axiom,
! [Nat3: nat] :
( ( suc @ Nat3 )
!= ( zero_zero @ nat ) ) ).
% old.nat.distinct(2)
thf(fact_197_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( ( zero_zero @ nat )
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_198_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat
!= ( zero_zero @ nat ) ) ) ).
% nat.discI
thf(fact_199_nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N4: nat] :
( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) )
=> ( P @ N2 ) ) ) ).
% nat_induct
thf(fact_200_diff__induct,axiom,
! [P: nat > nat > $o,M2: nat,N2: nat] :
( ! [X4: nat] : ( P @ X4 @ ( zero_zero @ nat ) )
=> ( ! [Y5: nat] : ( P @ ( zero_zero @ nat ) @ ( suc @ Y5 ) )
=> ( ! [X4: nat,Y5: nat] :
( ( P @ X4 @ Y5 )
=> ( P @ ( suc @ X4 ) @ ( suc @ Y5 ) ) )
=> ( P @ M2 @ N2 ) ) ) ) ).
% diff_induct
thf(fact_201_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N4: nat] :
( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) )
=> ( P @ ( zero_zero @ nat ) ) ) ) ).
% zero_induct
thf(fact_202_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= ( zero_zero @ nat ) ) ).
% Suc_neq_Zero
thf(fact_203_Zero__neq__Suc,axiom,
! [M2: nat] :
( ( zero_zero @ nat )
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_204_Zero__not__Suc,axiom,
! [M2: nat] :
( ( zero_zero @ nat )
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_205_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y
!= ( zero_zero @ nat ) )
=> ~ ! [Nat4: nat] :
( Y
!= ( suc @ Nat4 ) ) ) ).
% old.nat.exhaust
thf(fact_206_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_207_not0__implies__Suc,axiom,
! [N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
=> ? [M: nat] :
( N2
= ( suc @ M ) ) ) ).
% not0_implies_Suc
thf(fact_208_less__Suc__eq__0__disj,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ ( suc @ N2 ) )
= ( ( M2
= ( zero_zero @ nat ) )
| ? [J4: nat] :
( ( M2
= ( suc @ J4 ) )
& ( ord_less @ nat @ J4 @ N2 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_209_gr0__implies__Suc,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ? [M: nat] :
( N2
= ( suc @ M ) ) ) ).
% gr0_implies_Suc
thf(fact_210_gr0__conv__Suc,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
= ( ? [M3: nat] :
( N2
= ( suc @ M3 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_211_one__is__add,axiom,
! [M2: nat,N2: nat] :
( ( ( suc @ ( zero_zero @ nat ) )
= ( plus_plus @ nat @ M2 @ N2 ) )
= ( ( ( M2
= ( suc @ ( zero_zero @ nat ) ) )
& ( N2
= ( zero_zero @ nat ) ) )
| ( ( M2
= ( zero_zero @ nat ) )
& ( N2
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% one_is_add
thf(fact_212_add__is__1,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus @ nat @ M2 @ N2 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( ( M2
= ( suc @ ( zero_zero @ nat ) ) )
& ( N2
= ( zero_zero @ nat ) ) )
| ( ( M2
= ( zero_zero @ nat ) )
& ( N2
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% add_is_1
thf(fact_213_setsum__cong__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A @ ( type2 @ A ) )
=> ! [A3: set @ nat,F: nat > A,G: nat > A] :
( ~ ( member @ nat @ ( zero_zero @ nat ) @ A3 )
=> ( ! [X4: nat] :
( ( member @ nat @ ( suc @ X4 ) @ A3 )
=> ( ( F @ ( suc @ X4 ) )
= ( G @ ( suc @ X4 ) ) ) )
=> ( ( groups15040474setsum @ nat @ A @ F @ A3 )
= ( groups15040474setsum @ nat @ A @ G @ A3 ) ) ) ) ) ).
% setsum_cong_Suc
thf(fact_214_lfinite__llength__enat,axiom,
! [A: $tType,Xs3: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ Xs3 )
=> ? [N4: nat] :
( ( coindu1018505716length @ A @ Xs3 )
= ( extended_enat2 @ N4 ) ) ) ).
% lfinite_llength_enat
thf(fact_215_llength__eq__enat__lfiniteD,axiom,
! [A: $tType,Xs3: coindu1593790203_llist @ A,N2: nat] :
( ( ( coindu1018505716length @ A @ Xs3 )
= ( extended_enat2 @ N2 ) )
=> ( coindu1213758845finite @ A @ Xs3 ) ) ).
% llength_eq_enat_lfiniteD
thf(fact_216_lfinite__conv__llength__enat,axiom,
! [A: $tType] :
( ( coindu1213758845finite @ A )
= ( ^ [Xs2: coindu1593790203_llist @ A] :
? [N3: nat] :
( ( coindu1018505716length @ A @ Xs2 )
= ( extended_enat2 @ N3 ) ) ) ) ).
% lfinite_conv_llength_enat
thf(fact_217_zero__enat__def,axiom,
( ( zero_zero @ extended_enat )
= ( extended_enat2 @ ( zero_zero @ nat ) ) ) ).
% zero_enat_def
thf(fact_218_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_219_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_220_semiring__normalization__rules_I25_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,D: A] :
( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ C @ D ) )
= ( plus_plus @ A @ ( plus_plus @ A @ A2 @ C ) @ D ) ) ) ).
% semiring_normalization_rules(25)
thf(fact_221_semiring__normalization__rules_I24_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ( ( plus_plus @ A )
= ( ^ [A4: A,C2: A] : ( plus_plus @ A @ C2 @ A4 ) ) ) ) ).
% semiring_normalization_rules(24)
thf(fact_222_semiring__normalization__rules_I23_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
= ( plus_plus @ A @ ( plus_plus @ A @ A2 @ C ) @ B2 ) ) ) ).
% semiring_normalization_rules(23)
thf(fact_223_semiring__normalization__rules_I22_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,D: A] :
( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ C @ D ) )
= ( plus_plus @ A @ C @ ( plus_plus @ A @ A2 @ D ) ) ) ) ).
% semiring_normalization_rules(22)
thf(fact_224_semiring__normalization__rules_I21_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ 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 ) ) ) ) ).
% semiring_normalization_rules(21)
thf(fact_225_semiring__normalization__rules_I20_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A,D: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ C @ D ) )
= ( plus_plus @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ D ) ) ) ) ).
% semiring_normalization_rules(20)
thf(fact_226_setsum_Ocong,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A @ ( type2 @ A ) )
=> ! [A3: set @ B,B4: set @ B,G: B > A,H: B > A] :
( ( A3 = B4 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ B4 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups15040474setsum @ B @ A @ G @ A3 )
= ( groups15040474setsum @ B @ A @ H @ B4 ) ) ) ) ) ).
% setsum.cong
thf(fact_227_setsum_Oreindex__bij__witness,axiom,
! [B: $tType,A: $tType,C3: $tType] :
( ( comm_monoid_add @ A @ ( type2 @ A ) )
=> ! [S2: set @ B,I2: C3 > B,J: B > C3,T2: set @ C3,H: C3 > A,G: B > A] :
( ! [A5: B] :
( ( member @ B @ A5 @ S2 )
=> ( ( I2 @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ S2 )
=> ( member @ C3 @ ( J @ A5 ) @ T2 ) )
=> ( ! [B5: C3] :
( ( member @ C3 @ B5 @ T2 )
=> ( ( J @ ( I2 @ B5 ) )
= B5 ) )
=> ( ! [B5: C3] :
( ( member @ C3 @ B5 @ T2 )
=> ( member @ B @ ( I2 @ B5 ) @ S2 ) )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ S2 )
=> ( ( H @ ( J @ A5 ) )
= ( G @ A5 ) ) )
=> ( ( groups15040474setsum @ B @ A @ G @ S2 )
= ( groups15040474setsum @ C3 @ A @ H @ T2 ) ) ) ) ) ) ) ) ).
% setsum.reindex_bij_witness
thf(fact_228_setsum_Ocommute,axiom,
! [A: $tType,B: $tType,C3: $tType] :
( ( comm_monoid_add @ A @ ( type2 @ A ) )
=> ! [G: B > C3 > A,B4: set @ C3,A3: set @ B] :
( ( groups15040474setsum @ B @ A
@ ^ [I: B] : ( groups15040474setsum @ C3 @ A @ ( G @ I ) @ B4 )
@ A3 )
= ( groups15040474setsum @ C3 @ A
@ ^ [J4: C3] :
( groups15040474setsum @ B @ A
@ ^ [I: B] : ( G @ I @ J4 )
@ A3 )
@ B4 ) ) ) ).
% setsum.commute
thf(fact_229_setsum__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A @ ( type2 @ A ) )
=> ! [F: nat > A,N2: nat] :
( ( groups15040474setsum @ nat @ A @ F @ ( set_ord_lessThan @ nat @ ( suc @ N2 ) ) )
= ( plus_plus @ A @ ( F @ ( zero_zero @ nat ) )
@ ( groups15040474setsum @ nat @ A
@ ^ [I: nat] : ( F @ ( suc @ I ) )
@ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ).
% setsum_lessThan_Suc_shift
thf(fact_230_semiring__normalization__rules_I6_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% semiring_normalization_rules(6)
thf(fact_231_semiring__normalization__rules_I5_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% semiring_normalization_rules(5)
thf(fact_232_concat__xss_H,axiom,
coindu335574135_lnull @ a @ ( coindu441856546concat @ a @ ( coindu1079202755ist_of @ ( coindu1593790203_llist @ a ) @ xss2 ) ) ).
% concat_xss'
thf(fact_233_enat_Osize_I1_J,axiom,
! [Nat: nat] :
( ( extended_size_enat @ ( extended_enat2 @ Nat ) )
= ( zero_zero @ nat ) ) ).
% enat.size(1)
thf(fact_234_length__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs3: list @ A] :
( ! [Xs: list @ A] :
( ! [Ys2: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys2 ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs ) )
=> ( P @ Xs3 ) ) ).
% length_induct
thf(fact_235_lnth__ldropn,axiom,
! [A: $tType,N2: nat,M2: nat,Xs3: coindu1593790203_llist @ A] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ ( plus_plus @ nat @ N2 @ M2 ) ) @ ( coindu1018505716length @ A @ Xs3 ) )
=> ( ( coindu749330388e_lnth @ A @ ( coindu531130065ldropn @ A @ N2 @ Xs3 ) @ M2 )
= ( coindu749330388e_lnth @ A @ Xs3 @ ( plus_plus @ nat @ M2 @ N2 ) ) ) ) ).
% lnth_ldropn
thf(fact_236_llist__of__inject,axiom,
! [A: $tType,Xs3: list @ A,Ys: list @ A] :
( ( ( coindu1079202755ist_of @ A @ Xs3 )
= ( coindu1079202755ist_of @ A @ Ys ) )
= ( Xs3 = Ys ) ) ).
% llist_of_inject
thf(fact_237_ldropn__0,axiom,
! [A: $tType,Xs3: coindu1593790203_llist @ A] :
( ( coindu531130065ldropn @ A @ ( zero_zero @ nat ) @ Xs3 )
= Xs3 ) ).
% ldropn_0
thf(fact_238_ldropn__ldropn,axiom,
! [A: $tType,N2: nat,M2: nat,Xs3: coindu1593790203_llist @ A] :
( ( coindu531130065ldropn @ A @ N2 @ ( coindu531130065ldropn @ A @ M2 @ Xs3 ) )
= ( coindu531130065ldropn @ A @ ( plus_plus @ nat @ N2 @ M2 ) @ Xs3 ) ) ).
% ldropn_ldropn
thf(fact_239_ldropn__LNil,axiom,
! [A: $tType,N2: nat] :
( ( coindu531130065ldropn @ A @ N2 @ ( coindu1598213697e_LNil @ A ) )
= ( coindu1598213697e_LNil @ A ) ) ).
% ldropn_LNil
thf(fact_240_lfinite__ldropn,axiom,
! [A: $tType,N2: nat,Xs3: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ ( coindu531130065ldropn @ A @ N2 @ Xs3 ) )
= ( coindu1213758845finite @ A @ Xs3 ) ) ).
% lfinite_ldropn
thf(fact_241_ldropn__Suc__LCons,axiom,
! [A: $tType,N2: nat,X: A,Xs3: coindu1593790203_llist @ A] :
( ( coindu531130065ldropn @ A @ ( suc @ N2 ) @ ( coindu1121789889_LCons @ A @ X @ Xs3 ) )
= ( coindu531130065ldropn @ A @ N2 @ Xs3 ) ) ).
% ldropn_Suc_LCons
thf(fact_242__092_060open_062m_A_061_A0_092_060close_062,axiom,
( m
= ( zero_zero @ nat ) ) ).
% \<open>m = 0\<close>
thf(fact_243_llength__llist__of,axiom,
! [A: $tType,Xs3: list @ A] :
( ( coindu1018505716length @ A @ ( coindu1079202755ist_of @ A @ Xs3 ) )
= ( extended_enat2 @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) ).
% llength_llist_of
thf(fact_244_lfinite__llist__of,axiom,
! [A: $tType,Xs3: list @ A] : ( coindu1213758845finite @ A @ ( coindu1079202755ist_of @ A @ Xs3 ) ) ).
% lfinite_llist_of
thf(fact_245_ldropn__lnull,axiom,
! [A: $tType,Xs3: coindu1593790203_llist @ A,N2: nat] :
( ( coindu335574135_lnull @ A @ Xs3 )
=> ( ( coindu531130065ldropn @ A @ N2 @ Xs3 )
= ( coindu1598213697e_LNil @ A ) ) ) ).
% ldropn_lnull
thf(fact_246_Ex__list__of__length,axiom,
! [A: $tType,N2: nat] :
? [Xs: list @ A] :
( ( size_size @ ( list @ A ) @ Xs )
= N2 ) ).
% Ex_list_of_length
thf(fact_247_neq__if__length__neq,axiom,
! [A: $tType,Xs3: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
!= ( size_size @ ( list @ A ) @ Ys ) )
=> ( Xs3 != Ys ) ) ).
% neq_if_length_neq
thf(fact_248_ldropn__eq__LConsD,axiom,
! [A: $tType,N2: nat,Xs3: coindu1593790203_llist @ A,Y: A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu531130065ldropn @ A @ N2 @ Xs3 )
= ( coindu1121789889_LCons @ A @ Y @ Ys ) )
=> ( ord_less @ extended_enat @ ( extended_enat2 @ N2 ) @ ( coindu1018505716length @ A @ Xs3 ) ) ) ).
% ldropn_eq_LConsD
thf(fact_249_ldropn__Suc__conv__ldropn,axiom,
! [A: $tType,N2: nat,Xs3: coindu1593790203_llist @ A] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ N2 ) @ ( coindu1018505716length @ A @ Xs3 ) )
=> ( ( coindu1121789889_LCons @ A @ ( coindu749330388e_lnth @ A @ Xs3 @ N2 ) @ ( coindu531130065ldropn @ A @ ( suc @ N2 ) @ Xs3 ) )
= ( coindu531130065ldropn @ A @ N2 @ Xs3 ) ) ) ).
% ldropn_Suc_conv_ldropn
%----Type constructors (41)
thf(tcon_fun___Orderings_Oorder,axiom,
! [A6: $tType,A7: $tType] :
( ( order @ A7 @ ( type2 @ A7 ) )
=> ( order @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A6: $tType,A7: $tType] :
( ( ord @ A7 @ ( type2 @ A7 ) )
=> ( ord @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
semiri456707255roduct @ nat @ ( type2 @ nat ) ).
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_Ostrict__ordered__comm__monoid__add,axiom,
strict797366125id_add @ 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_Oordered__comm__monoid__add,axiom,
ordere216010020id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
cancel1352612707id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1 @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add,axiom,
semigroup_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Omonoid__add,axiom,
monoid_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oorder_1,axiom,
order @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oord_2,axiom,
ord @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ozero,axiom,
zero @ nat @ ( type2 @ nat ) ).
thf(tcon_Set_Oset___Orderings_Oorder_3,axiom,
! [A6: $tType] : ( order @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_4,axiom,
! [A6: $tType] : ( ord @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
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_Ostrict__ordered__comm__monoid__add_9,axiom,
strict797366125id_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_10,axiom,
canoni770627133id_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_11,axiom,
ordere779506340up_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_12,axiom,
ordere216010020id_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_13,axiom,
ab_semigroup_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_14,axiom,
comm_monoid_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_15,axiom,
comm_semiring_1 @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_16,axiom,
semigroup_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_17,axiom,
linorder @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_18,axiom,
monoid_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_19,axiom,
order @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oord_20,axiom,
ord @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Ozero_21,axiom,
zero @ extended_enat @ ( type2 @ extended_enat ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
? [M5: nat,N6: nat] :
( ( ( coindu749330388e_lnth @ a @ ( coindu441856546concat @ a @ xssa ) @ ( suc @ na ) )
= ( coindu749330388e_lnth @ a @ ( coindu749330388e_lnth @ ( coindu1593790203_llist @ a ) @ xssa @ M5 ) @ N6 ) )
& ( ord_less @ extended_enat @ ( extended_enat2 @ N6 ) @ ( coindu1018505716length @ a @ ( coindu749330388e_lnth @ ( coindu1593790203_llist @ a ) @ xssa @ M5 ) ) )
& ( ord_less @ extended_enat @ ( extended_enat2 @ M5 ) @ ( coindu1018505716length @ ( coindu1593790203_llist @ a ) @ xssa ) )
& ( ( extended_enat2 @ ( suc @ na ) )
= ( plus_plus @ extended_enat
@ ( groups15040474setsum @ nat @ extended_enat
@ ^ [I: nat] : ( coindu1018505716length @ a @ ( coindu749330388e_lnth @ ( coindu1593790203_llist @ a ) @ xssa @ I ) )
@ ( set_ord_lessThan @ nat @ M5 ) )
@ ( extended_enat2 @ N6 ) ) ) ) ).
%------------------------------------------------------------------------------