TPTP Problem File: ITP043^2.p
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%------------------------------------------------------------------------------
% File : ITP043^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Descartes_Sign_Rule problem prob_378__5868812_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Descartes_Sign_Rule/prob_378__5868812_1 [Des21]
% Status : Theorem
% Rating : 0.00 v7.5.0
% Syntax : Number of formulae : 380 ( 123 unt; 50 typ; 0 def)
% Number of atoms : 858 ( 355 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 3700 ( 77 ~; 10 |; 44 &;3118 @)
% ( 0 <=>; 451 =>; 0 <=; 0 <~>)
% Maximal formula depth : 34 ( 7 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 114 ( 114 >; 0 *; 0 +; 0 <<)
% Number of symbols : 48 ( 47 usr; 6 con; 0-3 aty)
% Number of variables : 1050 ( 27 ^; 964 !; 20 ?;1050 :)
% ( 39 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:24:53.415
%------------------------------------------------------------------------------
% Could-be-implicit typings (4)
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_Int_Oint,type,
int: $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (46)
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Osize,type,
size:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oplus,type,
plus:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1352612707id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
ordere216010020id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere779506340up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__comm__monoid__add,type,
ordere1818651114id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ostrict__ordered__comm__monoid__add,type,
strict797366125id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere236663937imp_le:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
ordere223160158up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
strict2144017051up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
ordere516151231imp_le:
!>[A: $tType] : $o ).
thf(sy_c_Descartes__Sign__Rule__Mirabelle__vuqjybseel_Opsums,type,
descar1668888542_psums:
!>[A: $tType] : ( ( list @ A ) > ( list @ 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_List_Oappend,type,
append:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olast,type,
last:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : ( list @ A ) ).
thf(sy_c_List_Onth,type,
nth:
!>[A: $tType] : ( ( list @ A ) > nat > A ) ).
thf(sy_c_List_Oreplicate,type,
replicate:
!>[A: $tType] : ( nat > A > ( list @ A ) ) ).
thf(sy_c_List_Otake,type,
take:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
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_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_v_p____,type,
p: nat ).
thf(sy_v_v,type,
v: ( list @ a ) > int ).
thf(sy_v_xa____,type,
xa: a ).
thf(sy_v_xs,type,
xs: list @ a ).
thf(sy_v_xs1____,type,
xs1: list @ a ).
thf(sy_v_xs2____,type,
xs2: list @ a ).
thf(sy_v_xsa____,type,
xsa: list @ a ).
% Relevant facts (253)
thf(fact_0_nonneg_Ohyps_I1_J,axiom,
( xsa
!= ( nil @ a ) ) ).
% nonneg.hyps(1)
thf(fact_1_p__nz,axiom,
( ( nth @ a @ xsa @ p )
!= ( zero_zero @ a ) ) ).
% p_nz
thf(fact_2_nonneg_Ohyps_I2_J,axiom,
( ( last @ a @ xsa )
!= ( zero_zero @ a ) ) ).
% nonneg.hyps(2)
thf(fact_3_psums__0__Cons,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [Xs: list @ A] :
( ( descar1668888542_psums @ A @ ( cons @ A @ ( zero_zero @ A ) @ Xs ) )
= ( cons @ A @ ( zero_zero @ A ) @ ( descar1668888542_psums @ A @ Xs ) ) ) ) ).
% psums_0_Cons
thf(fact_4_xs1__def,axiom,
( xs1
= ( replicate @ a @ p @ ( zero_zero @ a ) ) ) ).
% xs1_def
thf(fact_5_xs__decompose,axiom,
( xsa
= ( append @ a @ xs1 @ ( cons @ a @ ( nth @ a @ xsa @ p ) @ xs2 ) ) ) ).
% xs_decompose
thf(fact_6_psums__replicate__0__append,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [N: nat,Xs: list @ A] :
( ( descar1668888542_psums @ A @ ( append @ A @ ( replicate @ A @ N @ ( zero_zero @ A ) ) @ Xs ) )
= ( append @ A @ ( replicate @ A @ N @ ( zero_zero @ A ) ) @ ( descar1668888542_psums @ A @ Xs ) ) ) ) ).
% psums_replicate_0_append
thf(fact_7_less__p__zero,axiom,
! [Q: nat] :
( ( ord_less @ nat @ Q @ p )
=> ( ( nth @ a @ xsa @ Q )
= ( zero_zero @ a ) ) ) ).
% less_p_zero
thf(fact_8__092_060open_062take_Ap_Axs_A_061_Areplicate_Ap_A_I0_058_058_Ha_J_092_060close_062,axiom,
( ( take @ a @ p @ xsa )
= ( replicate @ a @ p @ ( zero_zero @ a ) ) ) ).
% \<open>take p xs = replicate p (0::'a)\<close>
thf(fact_9_replicate__app__Cons__same,axiom,
! [A: $tType,N: nat,X: A,Xs: list @ A] :
( ( append @ A @ ( replicate @ A @ N @ X ) @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( append @ A @ ( replicate @ A @ N @ X ) @ Xs ) ) ) ).
% replicate_app_Cons_same
thf(fact_10_append_Oassoc,axiom,
! [A: $tType,A2: list @ A,B: list @ A,C: list @ A] :
( ( append @ A @ ( append @ A @ A2 @ B ) @ C )
= ( append @ A @ A2 @ ( append @ A @ B @ C ) ) ) ).
% append.assoc
thf(fact_11_append__assoc,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( append @ A @ ( append @ A @ Xs @ Ys ) @ Zs )
= ( append @ A @ Xs @ ( append @ A @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_12_append__same__eq,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A,Zs: list @ A] :
( ( ( append @ A @ Ys @ Xs )
= ( append @ A @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_13_same__append__eq,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_14_nonneg_Ohyps_I4_J,axiom,
ord_less_eq @ a @ ( zero_zero @ a ) @ xa ).
% nonneg.hyps(4)
thf(fact_15_list_Oinject,axiom,
! [A: $tType,X21: A,X22: list @ A,Y21: A,Y22: list @ A] :
( ( ( cons @ A @ X21 @ X22 )
= ( cons @ A @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_16_append__replicate__commute,axiom,
! [A: $tType,N: nat,X: A,K: nat] :
( ( append @ A @ ( replicate @ A @ N @ X ) @ ( replicate @ A @ K @ X ) )
= ( append @ A @ ( replicate @ A @ K @ X ) @ ( replicate @ A @ N @ X ) ) ) ).
% append_replicate_commute
thf(fact_17_Cons__eq__appendI,axiom,
! [A: $tType,X: A,Xs1: list @ A,Ys: list @ A,Xs: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append @ A @ Xs1 @ Zs ) )
=> ( ( cons @ A @ X @ Xs )
= ( append @ A @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_18_assms_I1_J,axiom,
( xs
!= ( nil @ a ) ) ).
% assms(1)
thf(fact_19_assms_I2_J,axiom,
( ( last @ a @ xs )
!= ( zero_zero @ a ) ) ).
% assms(2)
thf(fact_20_replicate__eq__replicate,axiom,
! [A: $tType,M: nat,X: A,N: nat,Y: A] :
( ( ( replicate @ A @ M @ X )
= ( replicate @ A @ N @ Y ) )
= ( ( M = N )
& ( ( M
!= ( zero_zero @ nat ) )
=> ( X = Y ) ) ) ) ).
% replicate_eq_replicate
thf(fact_21_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_22_not__gr__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% not_gr_zero
thf(fact_23_append_Oright__neutral,axiom,
! [A: $tType,A2: list @ A] :
( ( append @ A @ A2 @ ( nil @ A ) )
= A2 ) ).
% append.right_neutral
thf(fact_24_append__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( nil @ A ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys
= ( nil @ A ) ) ) ) ).
% append_is_Nil_conv
thf(fact_25_Nil__is__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( nil @ A )
= ( append @ A @ Xs @ Ys ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys
= ( nil @ A ) ) ) ) ).
% Nil_is_append_conv
thf(fact_26_self__append__conv2,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( Ys
= ( append @ A @ Xs @ Ys ) )
= ( Xs
= ( nil @ A ) ) ) ).
% self_append_conv2
thf(fact_27_append__self__conv2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= Ys )
= ( Xs
= ( nil @ A ) ) ) ).
% append_self_conv2
thf(fact_28_self__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
= ( append @ A @ Xs @ Ys ) )
= ( Ys
= ( nil @ A ) ) ) ).
% self_append_conv
thf(fact_29_append__self__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= Xs )
= ( Ys
= ( nil @ A ) ) ) ).
% append_self_conv
thf(fact_30_append__Nil2,axiom,
! [A: $tType,Xs: list @ A] :
( ( append @ A @ Xs @ ( nil @ A ) )
= Xs ) ).
% append_Nil2
thf(fact_31_nth__Cons__0,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( nth @ A @ ( cons @ A @ X @ Xs ) @ ( zero_zero @ nat ) )
= X ) ).
% nth_Cons_0
thf(fact_32_take0,axiom,
! [A: $tType] :
( ( take @ A @ ( zero_zero @ nat ) )
= ( ^ [Xs2: list @ A] : ( nil @ A ) ) ) ).
% take0
thf(fact_33_take__eq__Nil,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( take @ A @ N @ Xs )
= ( nil @ A ) )
= ( ( N
= ( zero_zero @ nat ) )
| ( Xs
= ( nil @ A ) ) ) ) ).
% take_eq_Nil
thf(fact_34_empty__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( ( nil @ A )
= ( replicate @ A @ N @ X ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% empty_replicate
thf(fact_35_replicate__empty,axiom,
! [A: $tType,N: nat,X: A] :
( ( ( replicate @ A @ N @ X )
= ( nil @ A ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% replicate_empty
thf(fact_36_last__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( last @ A @ ( replicate @ A @ N @ X ) )
= X ) ) ).
% last_replicate
thf(fact_37_append1__eq__conv,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys: list @ A,Y: A] :
( ( ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) )
= ( append @ A @ Ys @ ( cons @ A @ Y @ ( nil @ A ) ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_38_nth__take,axiom,
! [A: $tType,I: nat,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ I @ N )
=> ( ( nth @ A @ ( take @ A @ N @ Xs ) @ I )
= ( nth @ A @ Xs @ I ) ) ) ).
% nth_take
thf(fact_39_last__appendL,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( Ys
= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
= ( last @ A @ Xs ) ) ) ).
% last_appendL
thf(fact_40_last__appendR,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( Ys
!= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
= ( last @ A @ Ys ) ) ) ).
% last_appendR
thf(fact_41_nth__replicate,axiom,
! [A: $tType,I: nat,N: nat,X: A] :
( ( ord_less @ nat @ I @ N )
=> ( ( nth @ A @ ( replicate @ A @ N @ X ) @ I )
= X ) ) ).
% nth_replicate
thf(fact_42_last__snoc,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( last @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= X ) ).
% last_snoc
thf(fact_43_take__0,axiom,
! [A: $tType,Xs: list @ A] :
( ( take @ A @ ( zero_zero @ nat ) @ Xs )
= ( nil @ A ) ) ).
% take_0
thf(fact_44_ext,axiom,
! [B2: $tType,A: $tType,F: A > B2,G: A > B2] :
( ! [X2: A] :
( ( F @ X2 )
= ( G @ X2 ) )
=> ( F = G ) ) ).
% ext
thf(fact_45_last_Osimps,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( ( Xs
= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X @ Xs ) )
= X ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X @ Xs ) )
= ( last @ A @ Xs ) ) ) ) ).
% last.simps
thf(fact_46_last__ConsL,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( Xs
= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_47_last__ConsR,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X @ Xs ) )
= ( last @ A @ Xs ) ) ) ).
% last_ConsR
thf(fact_48_last__append,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( ( Ys
= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
= ( last @ A @ Xs ) ) )
& ( ( Ys
!= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
= ( last @ A @ Ys ) ) ) ) ).
% last_append
thf(fact_49_take__Nil,axiom,
! [A: $tType,N: nat] :
( ( take @ A @ N @ ( nil @ A ) )
= ( nil @ A ) ) ).
% take_Nil
thf(fact_50_take__equalityI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ! [I2: nat] :
( ( take @ A @ I2 @ Xs )
= ( take @ A @ I2 @ Ys ) )
=> ( Xs = Ys ) ) ).
% take_equalityI
thf(fact_51_transpose_Ocases,axiom,
! [A: $tType,X: list @ ( list @ A )] :
( ( X
!= ( nil @ ( list @ A ) ) )
=> ( ! [Xss: list @ ( list @ A )] :
( X
!= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
=> ~ ! [X2: A,Xs3: list @ A,Xss: list @ ( list @ A )] :
( X
!= ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_52_longest__common__suffix,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
? [Ss: list @ A,Xs4: list @ A,Ys2: list @ A] :
( ( Xs
= ( append @ A @ Xs4 @ Ss ) )
& ( Ys
= ( append @ A @ Ys2 @ Ss ) )
& ( ( Xs4
= ( nil @ A ) )
| ( Ys2
= ( nil @ A ) )
| ( ( last @ A @ Xs4 )
!= ( last @ A @ Ys2 ) ) ) ) ).
% longest_common_suffix
thf(fact_53_replicate__0,axiom,
! [A: $tType,X: A] :
( ( replicate @ A @ ( zero_zero @ nat ) @ X )
= ( nil @ A ) ) ).
% replicate_0
thf(fact_54_zero__less__iff__neq__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ N )
= ( N
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_iff_neq_zero
thf(fact_55_gr__implies__not__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [M: A,N: A] :
( ( ord_less @ A @ M @ N )
=> ( N
!= ( zero_zero @ A ) ) ) ) ).
% gr_implies_not_zero
thf(fact_56_not__less__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_57_gr__zeroI,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
( ( N
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% gr_zeroI
thf(fact_58_zero__le,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% zero_le
thf(fact_59_strict__sorted_Oinduct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: ( list @ A ) > $o,A0: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X2: A,Ys3: list @ A] :
( ( P @ Ys3 )
=> ( P @ ( cons @ A @ X2 @ Ys3 ) ) )
=> ( P @ A0 ) ) ) ) ).
% strict_sorted.induct
thf(fact_60_strict__sorted_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: list @ A] :
( ( X
!= ( nil @ A ) )
=> ~ ! [X2: A,Ys3: list @ A] :
( X
!= ( cons @ A @ X2 @ Ys3 ) ) ) ) ).
% strict_sorted.cases
thf(fact_61_map__tailrec__rev_Oinduct,axiom,
! [A: $tType,B2: $tType,P: ( A > B2 ) > ( list @ A ) > ( list @ B2 ) > $o,A0: A > B2,A1: list @ A,A22: list @ B2] :
( ! [F2: A > B2,X_1: list @ B2] : ( P @ F2 @ ( nil @ A ) @ X_1 )
=> ( ! [F2: A > B2,A3: A,As: list @ A,Bs: list @ B2] :
( ( P @ F2 @ As @ ( cons @ B2 @ ( F2 @ A3 ) @ Bs ) )
=> ( P @ F2 @ ( cons @ A @ A3 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_62_list__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ! [X2: A,Xs3: list @ A] :
( ( Xs3
!= ( nil @ A ) )
=> ( ( P @ Xs3 )
=> ( P @ ( cons @ A @ X2 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_63_successively_Oinduct,axiom,
! [A: $tType,P: ( A > A > $o ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A] :
( ! [P2: A > A > $o] : ( P @ P2 @ ( nil @ A ) )
=> ( ! [P2: A > A > $o,X2: A] : ( P @ P2 @ ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ! [P2: A > A > $o,X2: A,Y2: A,Xs3: list @ A] :
( ( P @ P2 @ ( cons @ A @ Y2 @ Xs3 ) )
=> ( P @ P2 @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs3 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% successively.induct
thf(fact_64_arg__min__list_Oinduct,axiom,
! [B2: $tType,A: $tType] :
( ( linorder @ B2 )
=> ! [P: ( A > B2 ) > ( list @ A ) > $o,A0: A > B2,A1: list @ A] :
( ! [F2: A > B2,X2: A] : ( P @ F2 @ ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ! [F2: A > B2,X2: A,Y2: A,Zs2: list @ A] :
( ( P @ F2 @ ( cons @ A @ Y2 @ Zs2 ) )
=> ( P @ F2 @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Zs2 ) ) ) )
=> ( ! [A3: A > B2] : ( P @ A3 @ ( nil @ A ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ).
% arg_min_list.induct
thf(fact_65_remdups__adj_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > $o,A0: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ! [X2: A,Y2: A,Xs3: list @ A] :
( ( ( X2 = Y2 )
=> ( P @ ( cons @ A @ X2 @ Xs3 ) ) )
=> ( ( ( X2 != Y2 )
=> ( P @ ( cons @ A @ Y2 @ Xs3 ) ) )
=> ( P @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs3 ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_66_sorted__wrt_Oinduct,axiom,
! [A: $tType,P: ( A > A > $o ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A] :
( ! [P2: A > A > $o] : ( P @ P2 @ ( nil @ A ) )
=> ( ! [P2: A > A > $o,X2: A,Ys3: list @ A] :
( ( P @ P2 @ Ys3 )
=> ( P @ P2 @ ( cons @ A @ X2 @ Ys3 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% sorted_wrt.induct
thf(fact_67_remdups__adj_Ocases,axiom,
! [A: $tType,X: list @ A] :
( ( X
!= ( nil @ A ) )
=> ( ! [X2: A] :
( X
!= ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ~ ! [X2: A,Y2: A,Xs3: list @ A] :
( X
!= ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_68_shuffles_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X_1: list @ A] : ( P @ ( nil @ A ) @ X_1 )
=> ( ! [Xs3: list @ A] : ( P @ Xs3 @ ( nil @ A ) )
=> ( ! [X2: A,Xs3: list @ A,Y2: A,Ys3: list @ A] :
( ( P @ Xs3 @ ( cons @ A @ Y2 @ Ys3 ) )
=> ( ( P @ ( cons @ A @ X2 @ Xs3 ) @ Ys3 )
=> ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ A @ Y2 @ Ys3 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% shuffles.induct
thf(fact_69_min__list_Oinduct,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [P: ( list @ A ) > $o,A0: list @ A] :
( ! [X2: A,Xs3: list @ A] :
( ! [X212: A,X222: list @ A] :
( ( Xs3
= ( cons @ A @ X212 @ X222 ) )
=> ( P @ Xs3 ) )
=> ( P @ ( cons @ A @ X2 @ Xs3 ) ) )
=> ( ( P @ ( nil @ A ) )
=> ( P @ A0 ) ) ) ) ).
% min_list.induct
thf(fact_70_min__list_Ocases,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: list @ A] :
( ! [X2: A,Xs3: list @ A] :
( X
!= ( cons @ A @ X2 @ Xs3 ) )
=> ( X
= ( nil @ A ) ) ) ) ).
% min_list.cases
thf(fact_71_induct__list012,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ! [X2: A,Y2: A,Zs2: list @ A] :
( ( P @ Zs2 )
=> ( ( P @ ( cons @ A @ Y2 @ Zs2 ) )
=> ( P @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Zs2 ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% induct_list012
thf(fact_72_splice_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X_1: list @ A] : ( P @ ( nil @ A ) @ X_1 )
=> ( ! [X2: A,Xs3: list @ A,Ys3: list @ A] :
( ( P @ Ys3 @ Xs3 )
=> ( P @ ( cons @ A @ X2 @ Xs3 ) @ Ys3 ) )
=> ( P @ A0 @ A1 ) ) ) ).
% splice.induct
thf(fact_73_list__induct2_H,axiom,
! [A: $tType,B2: $tType,P: ( list @ A ) > ( list @ B2 ) > $o,Xs: list @ A,Ys: list @ B2] :
( ( P @ ( nil @ A ) @ ( nil @ B2 ) )
=> ( ! [X2: A,Xs3: list @ A] : ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( nil @ B2 ) )
=> ( ! [Y2: B2,Ys3: list @ B2] : ( P @ ( nil @ A ) @ ( cons @ B2 @ Y2 @ Ys3 ) )
=> ( ! [X2: A,Xs3: list @ A,Y2: B2,Ys3: list @ B2] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ B2 @ Y2 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_74_neq__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
= ( ? [Y3: A,Ys4: list @ A] :
( Xs
= ( cons @ A @ Y3 @ Ys4 ) ) ) ) ).
% neq_Nil_conv
thf(fact_75_list_Oinducts,axiom,
! [A: $tType,P: ( list @ A ) > $o,List: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X1: A,X23: list @ A] :
( ( P @ X23 )
=> ( P @ ( cons @ A @ X1 @ X23 ) ) )
=> ( P @ List ) ) ) ).
% list.inducts
thf(fact_76_list_Oexhaust,axiom,
! [A: $tType,Y: list @ A] :
( ( Y
!= ( nil @ A ) )
=> ~ ! [X213: A,X223: list @ A] :
( Y
!= ( cons @ A @ X213 @ X223 ) ) ) ).
% list.exhaust
thf(fact_77_list_OdiscI,axiom,
! [A: $tType,List: list @ A,X21: A,X22: list @ A] :
( ( List
= ( cons @ A @ X21 @ X22 ) )
=> ( List
!= ( nil @ A ) ) ) ).
% list.discI
thf(fact_78_list_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( nil @ A )
!= ( cons @ A @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_79_append_Oleft__neutral,axiom,
! [A: $tType,A2: list @ A] :
( ( append @ A @ ( nil @ A ) @ A2 )
= A2 ) ).
% append.left_neutral
thf(fact_80_append__Nil,axiom,
! [A: $tType,Ys: list @ A] :
( ( append @ A @ ( nil @ A ) @ Ys )
= Ys ) ).
% append_Nil
thf(fact_81_eq__Nil__appendI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs = Ys )
=> ( Xs
= ( append @ A @ ( nil @ A ) @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_82_psums_Ocases,axiom,
! [A: $tType] :
( ( plus @ A )
=> ! [X: list @ A] :
( ( X
!= ( nil @ A ) )
=> ( ! [X2: A] :
( X
!= ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ~ ! [X2: A,Y2: A,Xs3: list @ A] :
( X
!= ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs3 ) ) ) ) ) ) ).
% psums.cases
thf(fact_83_rev__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ! [X2: A,Xs3: list @ A] :
( ( Xs3
!= ( nil @ A ) )
=> ( ( P @ Xs3 )
=> ( P @ ( append @ A @ Xs3 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_84_append__eq__Cons__conv,axiom,
! [A: $tType,Ys: list @ A,Zs: list @ A,X: A,Xs: list @ A] :
( ( ( append @ A @ Ys @ Zs )
= ( cons @ A @ X @ Xs ) )
= ( ( ( Ys
= ( nil @ A ) )
& ( Zs
= ( cons @ A @ X @ Xs ) ) )
| ? [Ys5: list @ A] :
( ( Ys
= ( cons @ A @ X @ Ys5 ) )
& ( ( append @ A @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_85_Cons__eq__append__conv,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X @ Xs )
= ( append @ A @ Ys @ Zs ) )
= ( ( ( Ys
= ( nil @ A ) )
& ( ( cons @ A @ X @ Xs )
= Zs ) )
| ? [Ys5: list @ A] :
( ( ( cons @ A @ X @ Ys5 )
= Ys )
& ( Xs
= ( append @ A @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_86_rev__exhaust,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ~ ! [Ys3: list @ A,Y2: A] :
( Xs
!= ( append @ A @ Ys3 @ ( cons @ A @ Y2 @ ( nil @ A ) ) ) ) ) ).
% rev_exhaust
thf(fact_87_rev__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X2: A,Xs3: list @ A] :
( ( P @ Xs3 )
=> ( P @ ( append @ A @ Xs3 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_88_psums_Osimps_I1_J,axiom,
! [A: $tType] :
( ( plus @ A )
=> ( ( descar1668888542_psums @ A @ ( nil @ A ) )
= ( nil @ A ) ) ) ).
% psums.simps(1)
thf(fact_89_replicate__append__same,axiom,
! [A: $tType,I: nat,X: A] :
( ( append @ A @ ( replicate @ A @ I @ X ) @ ( cons @ A @ X @ ( nil @ A ) ) )
= ( cons @ A @ X @ ( replicate @ A @ I @ X ) ) ) ).
% replicate_append_same
thf(fact_90_psums_Osimps_I2_J,axiom,
! [A: $tType] :
( ( plus @ A )
=> ! [X: A] :
( ( descar1668888542_psums @ A @ ( cons @ A @ X @ ( nil @ A ) ) )
= ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% psums.simps(2)
thf(fact_91_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_92_not__Cons__self2,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( cons @ A @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_93_append__eq__append__conv2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A,Ts: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Zs @ Ts ) )
= ( ? [Us: list @ A] :
( ( ( Xs
= ( append @ A @ Zs @ Us ) )
& ( ( append @ A @ Us @ Ys )
= Ts ) )
| ( ( ( append @ A @ Xs @ Us )
= Zs )
& ( Ys
= ( append @ A @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_94_append__eq__appendI,axiom,
! [A: $tType,Xs: list @ A,Xs1: list @ A,Zs: list @ A,Ys: list @ A,Us2: list @ A] :
( ( ( append @ A @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append @ A @ Xs1 @ Us2 ) )
=> ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_95_append__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
( ( append @ A @ ( cons @ A @ X @ Xs ) @ Ys )
= ( cons @ A @ X @ ( append @ A @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_96_v__decompose,axiom,
! [Xs5: list @ a] :
( ( v @ ( append @ a @ Xs5 @ xsa ) )
= ( plus_plus @ int @ ( v @ ( append @ a @ Xs5 @ ( cons @ a @ ( nth @ a @ xsa @ p ) @ ( nil @ a ) ) ) ) @ ( v @ ( cons @ a @ ( nth @ a @ xsa @ p ) @ xs2 ) ) ) ) ).
% v_decompose
thf(fact_97_p__less__length,axiom,
ord_less @ nat @ p @ ( size_size @ ( list @ a ) @ xsa ) ).
% p_less_length
thf(fact_98_neq0__conv,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% neq0_conv
thf(fact_99_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_nat_zero_code
thf(fact_100_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_101_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_102_minus__poly__rev__list_Oinduct,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X2: A,Xs3: list @ A,Y2: A,Ys3: list @ A] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ A @ Y2 @ Ys3 ) ) )
=> ( ! [Xs3: list @ A] : ( P @ Xs3 @ ( nil @ A ) )
=> ( ! [Y2: A,Ys3: list @ A] : ( P @ ( nil @ A ) @ ( cons @ A @ Y2 @ Ys3 ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ).
% minus_poly_rev_list.induct
thf(fact_103_plus__coeffs_Oinduct,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [Xs3: list @ A] : ( P @ Xs3 @ ( nil @ A ) )
=> ( ! [V: A,Va: list @ A] : ( P @ ( nil @ A ) @ ( cons @ A @ V @ Va ) )
=> ( ! [X2: A,Xs3: list @ A,Y2: A,Ys3: list @ A] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ A @ Y2 @ Ys3 ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ).
% plus_coeffs.induct
thf(fact_104_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z )
=> ~ ( ord_less_eq @ A @ T @ X3 ) ) ) ).
% minf(8)
thf(fact_105_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z )
=> ( ord_less_eq @ A @ X3 @ T ) ) ) ).
% minf(6)
thf(fact_106_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,B: A,C: A] :
( ( ( plus_plus @ A @ A2 @ B )
= ( plus_plus @ A @ A2 @ C ) )
= ( B = C ) ) ) ).
% add_left_cancel
thf(fact_107_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B: A,A2: A,C: A] :
( ( ( plus_plus @ A @ B @ A2 )
= ( plus_plus @ A @ C @ A2 ) )
= ( B = C ) ) ) ).
% add_right_cancel
thf(fact_108_le0,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% le0
thf(fact_109_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% bot_nat_0.extremum
thf(fact_110_add_Oleft__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add.left_neutral
thf(fact_111_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.right_neutral
thf(fact_112_double__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ( plus_plus @ A @ A2 @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% double_zero
thf(fact_113_double__zero__sym,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ A2 @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% double_zero_sym
thf(fact_114_add__cancel__left__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [B: A,A2: A] :
( ( ( plus_plus @ A @ B @ A2 )
= A2 )
= ( B
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_left
thf(fact_115_add__cancel__left__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A2: A,B: A] :
( ( ( plus_plus @ A @ A2 @ B )
= A2 )
= ( B
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_right
thf(fact_116_add__cancel__right__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A2: A,B: A] :
( ( A2
= ( plus_plus @ A @ B @ A2 ) )
= ( B
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_left
thf(fact_117_add__cancel__right__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A2: A,B: A] :
( ( A2
= ( plus_plus @ A @ A2 @ B ) )
= ( B
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_right
thf(fact_118_add__eq__0__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ 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_119_zero__eq__add__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [X: A,Y: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ X @ Y ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_120_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A2: A,C: A,B: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
= ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% add_le_cancel_right
thf(fact_121_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C: A,A2: A,B: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
= ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% add_le_cancel_left
thf(fact_122_add__less__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A2: A,C: A,B: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
= ( ord_less @ A @ A2 @ B ) ) ) ).
% add_less_cancel_right
thf(fact_123_add__less__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C: A,A2: A,B: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
= ( ord_less @ A @ A2 @ B ) ) ) ).
% add_less_cancel_left
thf(fact_124_append__eq__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Us2: list @ A,Vs: list @ A] :
( ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
| ( ( size_size @ ( list @ A ) @ Us2 )
= ( size_size @ ( list @ A ) @ Vs ) ) )
=> ( ( ( append @ A @ Xs @ Us2 )
= ( append @ A @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us2 = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_125_take__all__iff,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( take @ A @ N @ Xs )
= Xs )
= ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) ) ).
% take_all_iff
thf(fact_126_take__all,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N )
=> ( ( take @ A @ N @ Xs )
= Xs ) ) ).
% take_all
thf(fact_127_length__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( size_size @ ( list @ A ) @ ( replicate @ A @ N @ X ) )
= N ) ).
% length_replicate
thf(fact_128_length__psums,axiom,
! [A: $tType] :
( ( plus @ A )
=> ! [Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( descar1668888542_psums @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_psums
thf(fact_129_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_130_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_131_le__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ B @ A2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% le_add_same_cancel2
thf(fact_132_le__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ A2 @ B ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% le_add_same_cancel1
thf(fact_133_add__le__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B ) @ B )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel2
thf(fact_134_add__le__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [B: A,A2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ B @ A2 ) @ B )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel1
thf(fact_135_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: $tType] :
( ( linord219039673up_add @ 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_136_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ 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_137_less__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ A2 @ ( plus_plus @ A @ B @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% less_add_same_cancel2
thf(fact_138_less__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ B ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% less_add_same_cancel1
thf(fact_139_add__less__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ B ) @ B )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel2
thf(fact_140_add__less__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [B: A,A2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ B @ A2 ) @ B )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel1
thf(fact_141_length__0__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( zero_zero @ nat ) )
= ( Xs
= ( nil @ A ) ) ) ).
% length_0_conv
thf(fact_142_length__greater__0__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) )
= ( Xs
!= ( nil @ A ) ) ) ).
% length_greater_0_conv
thf(fact_143_nth__append__length,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys: list @ A] :
( ( nth @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_144_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% less_eq_nat.simps(1)
thf(fact_145_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [A2: A,B: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_146_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus @ A @ I @ K )
= ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_147_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% le_0_eq
thf(fact_148_impossible__Cons,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,X: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) )
=> ( Xs
!= ( cons @ A @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_149_Ex__list__of__length,axiom,
! [A: $tType,N: nat] :
? [Xs3: list @ A] :
( ( size_size @ ( list @ A ) @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_150_neq__if__length__neq,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
!= ( size_size @ ( list @ A ) @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_151_group__cancel_Oadd1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: A,K: A,A2: A,B: A] :
( ( A4
= ( plus_plus @ A @ K @ A2 ) )
=> ( ( plus_plus @ A @ A4 @ B )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B ) ) ) ) ) ).
% group_cancel.add1
thf(fact_152_group__cancel_Oadd2,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [B3: A,K: A,B: A,A2: A] :
( ( B3
= ( plus_plus @ A @ K @ B ) )
=> ( ( plus_plus @ A @ A2 @ B3 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B ) ) ) ) ) ).
% group_cancel.add2
thf(fact_153_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
= ( A2
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_unique
thf(fact_154_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
=> ( A2
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_155_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ! [A2: A,B: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% add.assoc
thf(fact_156_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B: A,C: A] :
( ( ( plus_plus @ A @ A2 @ B )
= ( plus_plus @ A @ A2 @ C ) )
= ( B = C ) ) ) ).
% add.left_cancel
thf(fact_157_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [B: A,A2: A,C: A] :
( ( ( plus_plus @ A @ B @ A2 )
= ( plus_plus @ A @ C @ A2 ) )
= ( B = C ) ) ) ).
% add.right_cancel
thf(fact_158_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( ( plus_plus @ A )
= ( ^ [A5: A,B4: A] : ( plus_plus @ A @ B4 @ A5 ) ) ) ) ).
% add.commute
thf(fact_159_add_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [B: A,A2: A,C: A] :
( ( plus_plus @ A @ B @ ( plus_plus @ A @ A2 @ C ) )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% add.left_commute
thf(fact_160_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,B: A,C: A] :
( ( ( plus_plus @ A @ A2 @ B )
= ( plus_plus @ A @ A2 @ C ) )
=> ( B = C ) ) ) ).
% add_left_imp_eq
thf(fact_161_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B: A,A2: A,C: A] :
( ( ( plus_plus @ A @ B @ A2 )
= ( plus_plus @ A @ C @ A2 ) )
=> ( B = C ) ) ) ).
% add_right_imp_eq
thf(fact_162_size__neq__size__imp__neq,axiom,
! [A: $tType] :
( ( size @ A )
=> ! [X: A,Y: A] :
( ( ( size_size @ A @ X )
!= ( size_size @ A @ Y ) )
=> ( X != Y ) ) ) ).
% size_neq_size_imp_neq
thf(fact_163_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_164_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( M != N )
=> ( ord_less @ nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_165_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less @ nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_166_le__eq__less__or__eq,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_167_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_168_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ? [K2: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_169_nat__less__le,axiom,
( ( ord_less @ nat )
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_170_comm__monoid__add__class_Oadd__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% comm_monoid_add_class.add_0
thf(fact_171_add_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.comm_neutral
thf(fact_172_add_Ogroup__left__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add.group_left_neutral
thf(fact_173_add__le__imp__le__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A2: A,C: A,B: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
=> ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% add_le_imp_le_right
thf(fact_174_add__le__imp__le__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C: A,A2: A,B: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
=> ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% add_le_imp_le_left
thf(fact_175_le__iff__add,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
? [C2: A] :
( B4
= ( plus_plus @ A @ A5 @ C2 ) ) ) ) ) ).
% le_iff_add
thf(fact_176_add__right__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [A2: A,B: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% add_right_mono
thf(fact_177_less__eqE,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ~ ! [C3: A] :
( B
!= ( plus_plus @ A @ A2 @ C3 ) ) ) ) ).
% less_eqE
thf(fact_178_add__left__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [A2: A,B: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) ) ) ) ).
% add_left_mono
thf(fact_179_add__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [A2: A,B: A,C: A,D: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less_eq @ A @ C @ D )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ D ) ) ) ) ) ).
% add_mono
thf(fact_180_add__mono__thms__linordered__semiring_I1_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_181_add__mono__thms__linordered__semiring_I2_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_182_add__mono__thms__linordered__semiring_I3_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( K = L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_183_add__less__imp__less__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A2: A,C: A,B: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
=> ( ord_less @ A @ A2 @ B ) ) ) ).
% add_less_imp_less_right
thf(fact_184_add__less__imp__less__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C: A,A2: A,B: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
=> ( ord_less @ A @ A2 @ B ) ) ) ).
% add_less_imp_less_left
thf(fact_185_add__strict__right__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [A2: A,B: A,C: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% add_strict_right_mono
thf(fact_186_add__strict__left__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [A2: A,B: A,C: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) ) ) ) ).
% add_strict_left_mono
thf(fact_187_add__strict__mono,axiom,
! [A: $tType] :
( ( strict2144017051up_add @ A )
=> ! [A2: A,B: A,C: A,D: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ord_less @ A @ C @ D )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ D ) ) ) ) ) ).
% add_strict_mono
thf(fact_188_add__mono__thms__linordered__field_I1_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( K = L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_189_add__mono__thms__linordered__field_I2_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_190_add__mono__thms__linordered__field_I5_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_191_list_Osize_I3_J,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) @ ( nil @ A ) )
= ( zero_zero @ nat ) ) ).
% list.size(3)
thf(fact_192_length__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ! [Xs3: list @ A] :
( ! [Ys6: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys6 ) @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( P @ Ys6 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_193_nth__take__lemma,axiom,
! [A: $tType,K: nat,Xs: list @ A,Ys: list @ A] :
( ( ord_less_eq @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less_eq @ nat @ K @ ( size_size @ ( list @ A ) @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ K )
=> ( ( nth @ A @ Xs @ I2 )
= ( nth @ A @ Ys @ I2 ) ) )
=> ( ( take @ A @ K @ Xs )
= ( take @ A @ K @ Ys ) ) ) ) ) ).
% nth_take_lemma
thf(fact_194_add__nonpos__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
=> ( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_195_add__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_196_add__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_nonpos
thf(fact_197_add__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B ) ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_198_add__increasing2,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [C: A,B: A,A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C )
=> ( ( ord_less_eq @ A @ B @ A2 )
=> ( ord_less_eq @ A @ B @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% add_increasing2
thf(fact_199_add__decreasing2,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [C: A,A2: A,B: A] :
( ( ord_less_eq @ A @ C @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A2 @ B )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ B ) ) ) ) ).
% add_decreasing2
thf(fact_200_add__increasing,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,B: A,C: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B @ C )
=> ( ord_less_eq @ A @ B @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% add_increasing
thf(fact_201_add__decreasing,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,C: A,B: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ C @ B )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ B ) ) ) ) ).
% add_decreasing
thf(fact_202_pos__add__strict,axiom,
! [A: $tType] :
( ( strict797366125id_add @ A )
=> ! [A2: A,B: A,C: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ B @ C )
=> ( ord_less @ A @ B @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% pos_add_strict
thf(fact_203_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ A2 @ B )
=> ~ ! [C3: A] :
( ( B
= ( plus_plus @ A @ A2 @ C3 ) )
=> ( C3
= ( zero_zero @ A ) ) ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_204_add__pos__pos,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B ) ) ) ) ) ).
% add_pos_pos
thf(fact_205_add__neg__neg,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ B ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_neg
thf(fact_206_add__less__le__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [A2: A,B: A,C: A,D: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ord_less_eq @ A @ C @ D )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ D ) ) ) ) ) ).
% add_less_le_mono
thf(fact_207_add__le__less__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [A2: A,B: A,C: A,D: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less @ A @ C @ D )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ D ) ) ) ) ) ).
% add_le_less_mono
thf(fact_208_add__mono__thms__linordered__field_I3_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_209_add__mono__thms__linordered__field_I4_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_210_list__induct4,axiom,
! [C4: $tType,A: $tType,B2: $tType,D2: $tType,Xs: list @ A,Ys: list @ B2,Zs: list @ C4,Ws: list @ D2,P: ( list @ A ) > ( list @ B2 ) > ( list @ C4 ) > ( list @ D2 ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B2 ) @ Ys ) )
=> ( ( ( size_size @ ( list @ B2 ) @ Ys )
= ( size_size @ ( list @ C4 ) @ Zs ) )
=> ( ( ( size_size @ ( list @ C4 ) @ Zs )
= ( size_size @ ( list @ D2 ) @ Ws ) )
=> ( ( P @ ( nil @ A ) @ ( nil @ B2 ) @ ( nil @ C4 ) @ ( nil @ D2 ) )
=> ( ! [X2: A,Xs3: list @ A,Y2: B2,Ys3: list @ B2,Z: C4,Zs2: list @ C4,W: D2,Ws2: list @ D2] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ B2 ) @ Ys3 ) )
=> ( ( ( size_size @ ( list @ B2 ) @ Ys3 )
= ( size_size @ ( list @ C4 ) @ Zs2 ) )
=> ( ( ( size_size @ ( list @ C4 ) @ Zs2 )
= ( size_size @ ( list @ D2 ) @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ B2 @ Y2 @ Ys3 ) @ ( cons @ C4 @ Z @ Zs2 ) @ ( cons @ D2 @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_211_list__induct3,axiom,
! [B2: $tType,A: $tType,C4: $tType,Xs: list @ A,Ys: list @ B2,Zs: list @ C4,P: ( list @ A ) > ( list @ B2 ) > ( list @ C4 ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B2 ) @ Ys ) )
=> ( ( ( size_size @ ( list @ B2 ) @ Ys )
= ( size_size @ ( list @ C4 ) @ Zs ) )
=> ( ( P @ ( nil @ A ) @ ( nil @ B2 ) @ ( nil @ C4 ) )
=> ( ! [X2: A,Xs3: list @ A,Y2: B2,Ys3: list @ B2,Z: C4,Zs2: list @ C4] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ B2 ) @ Ys3 ) )
=> ( ( ( size_size @ ( list @ B2 ) @ Ys3 )
= ( size_size @ ( list @ C4 ) @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys3 @ Zs2 )
=> ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ B2 @ Y2 @ Ys3 ) @ ( cons @ C4 @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_212_list__induct2,axiom,
! [A: $tType,B2: $tType,Xs: list @ A,Ys: list @ B2,P: ( list @ A ) > ( list @ B2 ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B2 ) @ Ys ) )
=> ( ( P @ ( nil @ A ) @ ( nil @ B2 ) )
=> ( ! [X2: A,Xs3: list @ A,Y2: B2,Ys3: list @ B2] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ B2 ) @ Ys3 ) )
=> ( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons @ A @ X2 @ Xs3 ) @ ( cons @ B2 @ Y2 @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_213_psums_Oinduct,axiom,
! [A: $tType] :
( ( plus @ A )
=> ! [P: ( list @ A ) > $o,A0: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ! [X2: A,Y2: A,Xs3: list @ A] :
( ( P @ ( cons @ A @ ( plus_plus @ A @ X2 @ Y2 ) @ Xs3 ) )
=> ( P @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs3 ) ) ) )
=> ( P @ A0 ) ) ) ) ) ).
% psums.induct
thf(fact_214_list__eq__iff__nth__eq,axiom,
! [A: $tType] :
( ( ^ [Y4: list @ A,Z2: list @ A] : Y4 = Z2 )
= ( ^ [Xs2: list @ A,Ys4: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys4 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ Xs2 @ I4 )
= ( nth @ A @ Ys4 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_215_Skolem__list__nth,axiom,
! [A: $tType,K: nat,P: nat > A > $o] :
( ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ K )
=> ? [X4: A] : ( P @ I4 @ X4 ) ) )
= ( ? [Xs2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= K )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ K )
=> ( P @ I4 @ ( nth @ A @ Xs2 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_216_nth__equalityI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I2 )
= ( nth @ A @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_217_psums_Osimps_I3_J,axiom,
! [A: $tType] :
( ( plus @ A )
=> ! [X: A,Y: A,Xs: list @ A] :
( ( descar1668888542_psums @ A @ ( cons @ A @ X @ ( cons @ A @ Y @ Xs ) ) )
= ( cons @ A @ X @ ( descar1668888542_psums @ A @ ( cons @ A @ ( plus_plus @ A @ X @ Y ) @ Xs ) ) ) ) ) ).
% psums.simps(3)
thf(fact_218_add__strict__increasing2,axiom,
! [A: $tType] :
( ( ordere1818651114id_add @ A )
=> ! [A2: A,B: A,C: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ B @ C )
=> ( ord_less @ A @ B @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% add_strict_increasing2
thf(fact_219_add__strict__increasing,axiom,
! [A: $tType] :
( ( ordere1818651114id_add @ A )
=> ! [A2: A,B: A,C: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B @ C )
=> ( ord_less @ A @ B @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% add_strict_increasing
thf(fact_220_add__pos__nonneg,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B ) ) ) ) ) ).
% add_pos_nonneg
thf(fact_221_add__nonpos__neg,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ B ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_neg
thf(fact_222_add__nonneg__pos,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B ) ) ) ) ) ).
% add_nonneg_pos
thf(fact_223_add__neg__nonpos,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ B ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_nonpos
thf(fact_224_sign__changes__induct,axiom,
! [A: $tType] :
( ( ( plus @ A )
& ( zero @ A ) )
=> ! [P: ( list @ A ) > $o,Xs: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ! [Xs3: list @ A] :
( ( P @ Xs3 )
=> ( P @ ( cons @ A @ ( zero_zero @ A ) @ Xs3 ) ) )
=> ( ! [X2: A,Y2: A,Xs3: list @ A] :
( ( X2
!= ( zero_zero @ A ) )
=> ( ( P @ ( cons @ A @ ( plus_plus @ A @ X2 @ Y2 ) @ Xs3 ) )
=> ( P @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs3 ) ) ) ) )
=> ( P @ Xs ) ) ) ) ) ) ).
% sign_changes_induct
thf(fact_225_same__length__different,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs != Ys )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ? [Pre: list @ A,X2: A,Xs4: list @ A,Y2: A,Ys2: list @ A] :
( ( X2 != Y2 )
& ( Xs
= ( append @ A @ Pre @ ( append @ A @ ( cons @ A @ X2 @ ( nil @ A ) ) @ Xs4 ) ) )
& ( Ys
= ( append @ A @ Pre @ ( append @ A @ ( cons @ A @ Y2 @ ( nil @ A ) ) @ Ys2 ) ) ) ) ) ) ).
% same_length_different
thf(fact_226_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B: A,A2: A] :
( ( ord_less_eq @ A @ B @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B )
=> ( A2 = B ) ) ) ) ).
% dual_order.antisym
thf(fact_227_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z2: A] : Y4 = Z2 )
= ( ^ [A5: A,B4: A] :
( ( ord_less_eq @ A @ B4 @ A5 )
& ( ord_less_eq @ A @ A5 @ B4 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_228_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B: A,A2: A,C: A] :
( ( ord_less_eq @ A @ B @ A2 )
=> ( ( ord_less_eq @ A @ C @ B )
=> ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_229_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A2: A,B: A] :
( ! [A3: A,B5: A] :
( ( ord_less_eq @ A @ A3 @ B5 )
=> ( P @ A3 @ B5 ) )
=> ( ! [A3: A,B5: A] :
( ( P @ B5 @ A3 )
=> ( P @ A3 @ B5 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_wlog
thf(fact_230_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_231_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z3 )
=> ( ord_less_eq @ A @ X @ Z3 ) ) ) ) ).
% order_trans
thf(fact_232_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less_eq @ A @ B @ A2 )
=> ( A2 = B ) ) ) ) ).
% order_class.order.antisym
thf(fact_233_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% ord_le_eq_trans
thf(fact_234_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B: A,C: A] :
( ( A2 = B )
=> ( ( ord_less_eq @ A @ B @ C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% ord_eq_le_trans
thf(fact_235_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z2: A] : Y4 = Z2 )
= ( ^ [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
& ( ord_less_eq @ A @ B4 @ A5 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_236_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv
thf(fact_237_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z3: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z3 ) )
=> ( ( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less_eq @ A @ X @ Z3 ) )
=> ( ( ( ord_less_eq @ A @ X @ Z3 )
=> ~ ( ord_less_eq @ A @ Z3 @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z3 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z3 )
=> ~ ( ord_less_eq @ A @ Z3 @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z3 @ X )
=> ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_238_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less_eq @ A @ B @ C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% order.trans
thf(fact_239_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% le_cases
thf(fact_240_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( X = Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% eq_refl
thf(fact_241_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linear
thf(fact_242_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( X = Y ) ) ) ) ).
% antisym
thf(fact_243_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z2: A] : Y4 = Z2 )
= ( ^ [X5: A,Y3: A] :
( ( ord_less_eq @ A @ X5 @ Y3 )
& ( ord_less_eq @ A @ Y3 @ X5 ) ) ) ) ) ).
% eq_iff
thf(fact_244_ord__le__eq__subst,axiom,
! [A: $tType,B2: $tType] :
( ( ( ord @ B2 )
& ( ord @ A ) )
=> ! [A2: A,B: A,F: A > B2,C: B2] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ord_less_eq @ B2 @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ B2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_245_ord__eq__le__subst,axiom,
! [A: $tType,B2: $tType] :
( ( ( ord @ B2 )
& ( ord @ A ) )
=> ! [A2: A,F: B2 > A,B: B2,C: B2] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq @ B2 @ B @ C )
=> ( ! [X2: B2,Y2: B2] :
( ( ord_less_eq @ B2 @ X2 @ Y2 )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_246_order__subst2,axiom,
! [A: $tType,C4: $tType] :
( ( ( order @ C4 )
& ( order @ A ) )
=> ! [A2: A,B: A,F: A > C4,C: C4] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less_eq @ C4 @ ( F @ B ) @ C )
=> ( ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ord_less_eq @ C4 @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ C4 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_subst2
thf(fact_247_order__subst1,axiom,
! [A: $tType,B2: $tType] :
( ( ( order @ B2 )
& ( order @ A ) )
=> ! [A2: A,F: B2 > A,B: B2,C: B2] :
( ( ord_less_eq @ A @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq @ B2 @ B @ C )
=> ( ! [X2: B2,Y2: B2] :
( ( ord_less_eq @ B2 @ X2 @ Y2 )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_subst1
thf(fact_248_le__fun__def,axiom,
! [B2: $tType,A: $tType] :
( ( ord @ B2 )
=> ( ( ord_less_eq @ ( A > B2 ) )
= ( ^ [F3: A > B2,G2: A > B2] :
! [X5: A] : ( ord_less_eq @ B2 @ ( F3 @ X5 ) @ ( G2 @ X5 ) ) ) ) ) ).
% le_fun_def
thf(fact_249_le__funI,axiom,
! [B2: $tType,A: $tType] :
( ( ord @ B2 )
=> ! [F: A > B2,G: A > B2] :
( ! [X2: A] : ( ord_less_eq @ B2 @ ( F @ X2 ) @ ( G @ X2 ) )
=> ( ord_less_eq @ ( A > B2 ) @ F @ G ) ) ) ).
% le_funI
thf(fact_250_le__funE,axiom,
! [B2: $tType,A: $tType] :
( ( ord @ B2 )
=> ! [F: A > B2,G: A > B2,X: A] :
( ( ord_less_eq @ ( A > B2 ) @ F @ G )
=> ( ord_less_eq @ B2 @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funE
thf(fact_251_le__funD,axiom,
! [B2: $tType,A: $tType] :
( ( ord @ B2 )
=> ! [F: A > B2,G: A > B2,X: A] :
( ( ord_less_eq @ ( A > B2 ) @ F @ G )
=> ( ord_less_eq @ B2 @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funD
thf(fact_252_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B: A,A2: A] :
( ( ord_less @ A @ B @ A2 )
=> ( A2 != B ) ) ) ).
% dual_order.strict_implies_not_eq
% Subclasses (23)
thf(subcl_Rings_Olinordered__idom___HOL_Otype,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( type @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Groups_Oplus,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( plus @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Groups_Ozero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( zero @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Orderings_Oord,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ord @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Orderings_Oorder,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( order @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Groups_Ogroup__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( group_add @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Groups_Omonoid__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( monoid_add @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Orderings_Olinorder,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( linorder @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Orderings_Opreorder,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( preorder @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Groups_Osemigroup__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( semigroup_add @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Groups_Ocomm__monoid__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( comm_monoid_add @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Groups_Oab__semigroup__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ab_semigroup_add @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Groups_Ocancel__semigroup__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( cancel_semigroup_add @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Groups_Ocancel__comm__monoid__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( cancel1352612707id_add @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Groups_Olinordered__ab__group__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( linord219039673up_add @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Groups_Oordered__comm__monoid__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ordere216010020id_add @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Groups_Oordered__ab__semigroup__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ordere779506340up_add @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Groups_Oordered__cancel__comm__monoid__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ordere1818651114id_add @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Groups_Ostrict__ordered__comm__monoid__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( strict797366125id_add @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ordere236663937imp_le @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Groups_Oordered__cancel__ab__semigroup__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ordere223160158up_add @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( strict2144017051up_add @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ordere516151231imp_le @ A ) ) ).
% Type constructors (52)
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A6: $tType,A7: $tType] :
( ( preorder @ A7 )
=> ( preorder @ ( A6 > A7 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A6: $tType,A7: $tType] :
( ( order @ A7 )
=> ( order @ ( A6 > A7 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A6: $tType,A7: $tType] :
( ( ord @ A7 )
=> ( ord @ ( A6 > A7 ) ) ) ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
ordere516151231imp_le @ int ).
thf(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
strict2144017051up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere223160158up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le @ int ).
thf(tcon_Int_Oint___Groups_Ostrict__ordered__comm__monoid__add,axiom,
strict797366125id_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__cancel__comm__monoid__add,axiom,
ordere1818651114id_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__comm__monoid__add,axiom,
ordere216010020id_add @ int ).
thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
linord219039673up_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
cancel1352612707id_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add @ int ).
thf(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add @ int ).
thf(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add @ int ).
thf(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
semigroup_add @ int ).
thf(tcon_Int_Oint___Orderings_Opreorder_1,axiom,
preorder @ int ).
thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
linorder @ int ).
thf(tcon_Int_Oint___Groups_Omonoid__add,axiom,
monoid_add @ int ).
thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
group_add @ int ).
thf(tcon_Int_Oint___Orderings_Oorder_2,axiom,
order @ int ).
thf(tcon_Int_Oint___Orderings_Oord_3,axiom,
ord @ int ).
thf(tcon_Int_Oint___Groups_Ozero,axiom,
zero @ int ).
thf(tcon_Int_Oint___Groups_Oplus,axiom,
plus @ int ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_4,axiom,
ordere516151231imp_le @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_5,axiom,
strict2144017051up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add_6,axiom,
ordere223160158up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_7,axiom,
ordere236663937imp_le @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_8,axiom,
strict797366125id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_9,axiom,
ordere1818651114id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_10,axiom,
ordere779506340up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_11,axiom,
ordere216010020id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_12,axiom,
cancel1352612707id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_13,axiom,
cancel_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__add_14,axiom,
ab_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_15,axiom,
comm_monoid_add @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add_16,axiom,
semigroup_add @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_17,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder_18,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__add_19,axiom,
monoid_add @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_20,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_21,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Groups_Ozero_22,axiom,
zero @ nat ).
thf(tcon_Nat_Onat___Groups_Oplus_23,axiom,
plus @ nat ).
thf(tcon_Nat_Onat___Nat_Osize,axiom,
size @ nat ).
thf(tcon_HOL_Obool___Orderings_Opreorder_24,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_25,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_26,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_27,axiom,
ord @ $o ).
thf(tcon_List_Olist___Nat_Osize_28,axiom,
! [A6: $tType] : ( size @ ( list @ A6 ) ) ).
% Free types (1)
thf(tfree_0,hypothesis,
linordered_idom @ a ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( descar1668888542_psums @ a @ xsa )
= ( append @ a @ ( replicate @ a @ p @ ( zero_zero @ a ) ) @ ( descar1668888542_psums @ a @ ( cons @ a @ ( nth @ a @ xsa @ p ) @ xs2 ) ) ) ) ).
%------------------------------------------------------------------------------