TPTP Problem File: DAT218^1.p
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%------------------------------------------------------------------------------
% File : DAT218^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Splay tree analysis 380
% Version : [Bla16] axioms : Especial.
% English :
% Refs : [Nip14] Nipkow (2014), Amortized Complexity Verified
% : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% Source : [Bla16]
% Names : splay_tree_analysis__380.p [Bla16]
% Status : Theorem
% Rating : 1.00 v7.5.0, 0.33 v7.2.0, 0.50 v7.1.0
% Syntax : Number of formulae : 442 ( 205 unt; 77 typ; 0 def)
% Number of atoms : 763 ( 316 equ; 0 cnn)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 3439 ( 45 ~; 18 |; 19 &;3112 @)
% ( 0 <=>; 245 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 5 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 76 ( 76 >; 0 *; 0 +; 0 <<)
% Number of symbols : 74 ( 72 usr; 10 con; 0-4 aty)
% Number of variables : 689 ( 9 ^; 616 !; 7 ?; 689 :)
% ( 57 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:51:37.818
%------------------------------------------------------------------------------
%----Could-be-implicit typings (8)
thf(ty_t_Tree_Otree,type,
tree: $tType > $tType ).
thf(ty_t_Real_Oreal,type,
real: $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Num_Onum,type,
num: $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_Int_Oint,type,
int: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (69)
thf(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Oring,type,
ring:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Num_Onumeral,type,
numeral:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Osemiring,type,
semiring:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Num_Oneg__numeral,type,
neg_numeral:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Omult__zero,type,
mult_zero:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Omonoid__mult,type,
monoid_mult:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Oordered__ring,type,
ordered_ring:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Nat_Osemiring__char__0,type,
semiring_char_0:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Ocomm__semiring,type,
comm_semiring:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Num_Osemiring__numeral,type,
semiring_numeral:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Osemigroup__mult,type,
semigroup_mult:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__ring,type,
linordered_ring:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Oordered__semiring,type,
ordered_semiring:
!>[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_Oab__semigroup__mult,type,
ab_semigroup_mult:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Oordered__semiring__0,type,
ordered_semiring_0:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__semiring__1,type,
linord1278240602ring_1:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Oordered__comm__semiring,type,
ordere1490568538miring:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__ring__strict,type,
linord581940658strict:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1352612707id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Oring__1__no__zero__divisors,type,
ring_11004092258visors:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere779506340up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
semiri1193490041visors:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
linord1659791738miring:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere236663937imp_le:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors__cancel,type,
semiri1923998003cancel:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
ordere516151231imp_le:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
thf(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Otimes__class_Otimes,type,
times_times:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
semiring_1_of_nat:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OBit1,type,
bit1: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one2: num ).
thf(sy_c_Num_Onumeral__class_Onumeral,type,
numeral_numeral:
!>[A: $tType] : ( num > A ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Pure_Otype,type,
type:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Splay__Tree_Osplay,type,
splay_splay:
!>[A: $tType] : ( A > ( tree @ A ) > ( tree @ A ) ) ).
thf(sy_c_Splay__Tree__Analysis__Base_O_092_060Phi_062,type,
splay_1414789891is_Phi:
!>[A: $tType] : ( ( tree @ A ) > real ) ).
thf(sy_c_Splay__Tree__Analysis__Base_Ot__splay,type,
splay_914434265_splay:
!>[A: $tType] : ( A > ( tree @ A ) > nat ) ).
thf(sy_c_Transcendental_Olog,type,
log: real > real > real ).
thf(sy_c_Tree_Olinorder__class_Obst,type,
linorder_bst:
!>[A: $tType] : ( ( tree @ A ) > $o ) ).
thf(sy_c_Tree_Otree_OLeaf,type,
leaf:
!>[A: $tType] : ( tree @ A ) ).
thf(sy_c_Tree_Otree_ONode,type,
node:
!>[A: $tType] : ( ( tree @ A ) > A > ( tree @ A ) > ( tree @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_a____,type,
a2: a ).
thf(sy_v_e____,type,
e: a ).
thf(sy_v_l____,type,
l: tree @ a ).
thf(sy_v_ls____,type,
ls: tree @ a ).
thf(sy_v_r____,type,
r: tree @ a ).
thf(sy_v_rs____,type,
rs: tree @ a ).
thf(sy_v_s____,type,
s: tree @ a ).
thf(sy_v_x____,type,
x: a ).
%----Relevant facts (251)
thf(fact_0_Leaf,axiom,
( l
= ( leaf @ a ) ) ).
% Leaf
thf(fact_1__C1_C,axiom,
ord_less_eq @ real @ ( zero_zero @ real ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( plus_plus @ real @ ( semiring_1_of_nat @ real @ ( size_size @ ( tree @ a ) @ r ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% "1"
thf(fact_2__092_060open_0620_A_092_060le_062_Alog_A2_A_Ireal_A_Isize_Als_J_A_L_A_Ireal_A_Isize_Ars_J_A_L_A2_J_J_092_060close_062,axiom,
ord_less_eq @ real @ ( zero_zero @ real ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( plus_plus @ real @ ( semiring_1_of_nat @ real @ ( size_size @ ( tree @ a ) @ ls ) ) @ ( plus_plus @ real @ ( semiring_1_of_nat @ real @ ( size_size @ ( tree @ a ) @ rs ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% \<open>0 \<le> log 2 (real (size ls) + (real (size rs) + 2))\<close>
thf(fact_3_one__add__one,axiom,
! [A: $tType] :
( ( numeral @ A @ ( type @ A ) )
=> ( ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).
% one_add_one
thf(fact_4_numeral__plus__one,axiom,
! [A: $tType] :
( ( numeral @ A @ ( type @ A ) )
=> ! [N: num] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ N @ one2 ) ) ) ) ).
% numeral_plus_one
thf(fact_5_one__plus__numeral,axiom,
! [A: $tType] :
( ( numeral @ A @ ( type @ A ) )
=> ! [N: num] :
( ( plus_plus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ N ) ) ) ) ).
% one_plus_numeral
thf(fact_6_numeral__le__one__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type @ A ) )
=> ! [N: num] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( one_one @ A ) )
= ( ord_less_eq @ num @ N @ one2 ) ) ) ).
% numeral_le_one_iff
thf(fact_7__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062l_Ae_Ar_O_Asplay_Aa_A_092_060langle_062ls_M_Ax_M_Ars_092_060rangle_062_A_061_A_092_060langle_062l_M_Ae_M_Ar_092_060rangle_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [L: tree @ a,E: a,R: tree @ a] :
( ( splay_splay @ a @ a2 @ ( node @ a @ ls @ x @ rs ) )
!= ( node @ a @ L @ E @ R ) ) ).
% \<open>\<And>thesis. (\<And>l e r. splay a \<langle>ls, x, rs\<rangle> = \<langle>l, e, r\<rangle> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_8_numeral__eq__one__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A @ ( type @ A ) )
=> ! [N: num] :
( ( ( numeral_numeral @ A @ N )
= ( one_one @ A ) )
= ( N = one2 ) ) ) ).
% numeral_eq_one_iff
thf(fact_9_one__eq__numeral__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A @ ( type @ A ) )
=> ! [N: num] :
( ( ( one_one @ A )
= ( numeral_numeral @ A @ N ) )
= ( one2 = N ) ) ) ).
% one_eq_numeral_iff
thf(fact_10_distrib__left__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A @ ( type @ A ) )
& ( semiring @ A @ ( type @ A ) ) )
=> ! [V: num,B: A,C: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V ) @ ( plus_plus @ A @ B @ C ) )
= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ B ) @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ C ) ) ) ) ).
% distrib_left_numeral
thf(fact_11_distrib__right__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A @ ( type @ A ) )
& ( semiring @ A @ ( type @ A ) ) )
=> ! [A2: A,B: A,V: num] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B ) @ ( numeral_numeral @ A @ V ) )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ V ) ) @ ( times_times @ A @ B @ ( numeral_numeral @ A @ V ) ) ) ) ) ).
% distrib_right_numeral
thf(fact_12_Node,axiom,
( s
= ( node @ a @ ls @ x @ rs ) ) ).
% Node
thf(fact_13_of__nat__1,axiom,
! [A: $tType] :
( ( semiring_1 @ A @ ( type @ A ) )
=> ( ( semiring_1_of_nat @ A @ ( one_one @ nat ) )
= ( one_one @ A ) ) ) ).
% of_nat_1
thf(fact_14_of__nat__add,axiom,
! [A: $tType] :
( ( semiring_1 @ A @ ( type @ A ) )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M @ N ) )
= ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_add
thf(fact_15_num_Oinject_I1_J,axiom,
! [X2: num,Y2: num] :
( ( ( bit0 @ X2 )
= ( bit0 @ Y2 ) )
= ( X2 = Y2 ) ) ).
% num.inject(1)
thf(fact_16_numeral__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A @ ( type @ A ) )
=> ! [M: num,N: num] :
( ( ( numeral_numeral @ A @ M )
= ( numeral_numeral @ A @ N ) )
= ( M = N ) ) ) ).
% numeral_eq_iff
thf(fact_17_num_Oinject_I2_J,axiom,
! [X3: num,Y3: num] :
( ( ( bit1 @ X3 )
= ( bit1 @ Y3 ) )
= ( X3 = Y3 ) ) ).
% num.inject(2)
thf(fact_18_of__nat__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A @ ( type @ A ) )
=> ! [M: nat,N: nat] :
( ( ( semiring_1_of_nat @ A @ M )
= ( semiring_1_of_nat @ A @ N ) )
= ( M = N ) ) ) ).
% of_nat_eq_iff
thf(fact_19_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_20_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( one_one @ nat )
= ( times_times @ nat @ M @ N ) )
= ( ( M
= ( one_one @ nat ) )
& ( N
= ( one_one @ nat ) ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_21_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times @ nat @ M @ N )
= ( one_one @ nat ) )
= ( ( M
= ( one_one @ nat ) )
& ( N
= ( one_one @ nat ) ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_22__C5_C,axiom,
linorder_bst @ a @ s ).
% "5"
thf(fact_23_numeral__times__numeral,axiom,
! [A: $tType] :
( ( semiring_numeral @ A @ ( type @ A ) )
=> ! [M: num,N: num] :
( ( times_times @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ A @ ( times_times @ num @ M @ N ) ) ) ) ).
% numeral_times_numeral
thf(fact_24_of__nat__le__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type @ A ) )
=> ! [M: nat,N: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ) ).
% of_nat_le_iff
thf(fact_25_of__nat__eq__0__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A @ ( type @ A ) )
=> ! [M: nat] :
( ( ( semiring_1_of_nat @ A @ M )
= ( zero_zero @ A ) )
= ( M
= ( zero_zero @ nat ) ) ) ) ).
% of_nat_eq_0_iff
thf(fact_26_of__nat__0__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A @ ( type @ A ) )
=> ! [N: nat] :
( ( ( zero_zero @ A )
= ( semiring_1_of_nat @ A @ N ) )
= ( ( zero_zero @ nat )
= N ) ) ) ).
% of_nat_0_eq_iff
thf(fact_27_of__nat__0,axiom,
! [A: $tType] :
( ( semiring_1 @ A @ ( type @ A ) )
=> ( ( semiring_1_of_nat @ A @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) ) ) ).
% of_nat_0
thf(fact_28_of__nat__mult,axiom,
! [A: $tType] :
( ( semiring_1 @ A @ ( type @ A ) )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( times_times @ nat @ M @ N ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_mult
thf(fact_29_of__nat__numeral,axiom,
! [A: $tType] :
( ( semiring_1 @ A @ ( type @ A ) )
=> ! [N: num] :
( ( semiring_1_of_nat @ A @ ( numeral_numeral @ nat @ N ) )
= ( numeral_numeral @ A @ N ) ) ) ).
% of_nat_numeral
thf(fact_30_of__nat__le__0__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type @ A ) )
=> ! [M: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) )
= ( M
= ( zero_zero @ nat ) ) ) ) ).
% of_nat_le_0_iff
thf(fact_31_numeral__plus__numeral,axiom,
! [A: $tType] :
( ( numeral @ A @ ( type @ A ) )
=> ! [M: num,N: num] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ M @ N ) ) ) ) ).
% numeral_plus_numeral
thf(fact_32_add__numeral__left,axiom,
! [A: $tType] :
( ( numeral @ A @ ( type @ A ) )
=> ! [V: num,W: num,Z: A] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( plus_plus @ A @ ( numeral_numeral @ A @ W ) @ Z ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V @ W ) ) @ Z ) ) ) ).
% add_numeral_left
thf(fact_33_numeral__le__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type @ A ) )
=> ! [M: num,N: num] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less_eq @ num @ M @ N ) ) ) ).
% numeral_le_iff
thf(fact_34_numerals_I1_J,axiom,
( ( numeral_numeral @ nat @ one2 )
= ( one_one @ nat ) ) ).
% numerals(1)
thf(fact_35_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq @ nat @ M @ N )
=> ~ ( ord_less_eq @ nat @ K @ N ) ) ) ).
% add_leE
thf(fact_36_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ N @ M ) ) ).
% le_add1
thf(fact_37_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ M @ N ) ) ).
% le_add2
thf(fact_38_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% add_leD1
thf(fact_39_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
=> ( ord_less_eq @ nat @ K @ N ) ) ).
% add_leD2
thf(fact_40_le__Suc__ex,axiom,
! [K: nat,L2: nat] :
( ( ord_less_eq @ nat @ K @ L2 )
=> ? [N2: nat] :
( L2
= ( plus_plus @ nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_41_nat__mult__1,axiom,
! [N: nat] :
( ( times_times @ nat @ ( one_one @ nat ) @ N )
= N ) ).
% nat_mult_1
thf(fact_42_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L2: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ K @ L2 )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L2 ) ) ) ) ).
% add_le_mono
thf(fact_43_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_44_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X: A] : ( member @ A @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_46_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_47_zadd__int__left,axiom,
! [M: nat,N: nat,Z: int] :
( ( plus_plus @ int @ ( semiring_1_of_nat @ int @ M ) @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ Z ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ M @ N ) ) @ Z ) ) ).
% zadd_int_left
thf(fact_48_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_49_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_50_nat__le__iff__add,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M2: nat,N3: nat] :
? [K2: nat] :
( N3
= ( plus_plus @ nat @ M2 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_51_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times @ nat @ ( plus_plus @ nat @ M @ N ) @ K )
= ( plus_plus @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_52_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times @ nat @ N @ ( one_one @ nat ) )
= N ) ).
% nat_mult_1_right
thf(fact_53_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times @ nat @ K @ ( plus_plus @ nat @ M @ N ) )
= ( plus_plus @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_54_nat__add__left__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( plus_plus @ nat @ K @ M )
= ( plus_plus @ nat @ K @ N ) )
= ( M = N ) ) ).
% nat_add_left_cancel
thf(fact_55_nat__add__right__cancel,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( plus_plus @ nat @ M @ K )
= ( plus_plus @ nat @ N @ K ) )
= ( M = N ) ) ).
% nat_add_right_cancel
thf(fact_56_nat__1__add__1,axiom,
( ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% nat_1_add_1
thf(fact_57_add__One__commute,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ N )
= ( plus_plus @ num @ N @ one2 ) ) ).
% add_One_commute
thf(fact_58_le__num__One__iff,axiom,
! [X5: num] :
( ( ord_less_eq @ num @ X5 @ one2 )
= ( X5 = one2 ) ) ).
% le_num_One_iff
thf(fact_59_le__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% le_numeral_extra(3)
thf(fact_60_zero__neq__numeral,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A @ ( type @ A ) )
=> ! [N: num] :
( ( zero_zero @ A )
!= ( numeral_numeral @ A @ N ) ) ) ).
% zero_neq_numeral
thf(fact_61_is__num__normalize_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A @ ( type @ 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 ) ) ) ) ).
% is_num_normalize(1)
thf(fact_62_not__numeral__le__zero,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type @ A ) )
=> ! [N: num] :
~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( zero_zero @ A ) ) ) ).
% not_numeral_le_zero
thf(fact_63_zero__le__numeral,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type @ A ) )
=> ! [N: num] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ N ) ) ) ).
% zero_le_numeral
thf(fact_64_le__numeral__extra_I1_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% le_numeral_extra(1)
thf(fact_65_le__numeral__extra_I2_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type @ A ) )
=> ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% le_numeral_extra(2)
thf(fact_66_of__nat__0__le__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type @ A ) )
=> ! [N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N ) ) ) ).
% of_nat_0_le_iff
thf(fact_67_le__numeral__extra_I4_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% le_numeral_extra(4)
thf(fact_68_num_Odistinct_I5_J,axiom,
! [X2: num,X3: num] :
( ( bit0 @ X2 )
!= ( bit1 @ X3 ) ) ).
% num.distinct(5)
thf(fact_69_num_Odistinct_I1_J,axiom,
! [X2: num] :
( one2
!= ( bit0 @ X2 ) ) ).
% num.distinct(1)
thf(fact_70_num_Odistinct_I3_J,axiom,
! [X3: num] :
( one2
!= ( bit1 @ X3 ) ) ).
% num.distinct(3)
thf(fact_71_mult__of__nat__commute,axiom,
! [A: $tType] :
( ( semiring_1 @ A @ ( type @ A ) )
=> ! [X5: nat,Y: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ X5 ) @ Y )
= ( times_times @ A @ Y @ ( semiring_1_of_nat @ A @ X5 ) ) ) ) ).
% mult_of_nat_commute
thf(fact_72_one__le__numeral,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type @ A ) )
=> ! [N: num] : ( ord_less_eq @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) ) ) ).
% one_le_numeral
thf(fact_73_numeral__Bit0,axiom,
! [A: $tType] :
( ( numeral @ A @ ( type @ A ) )
=> ! [N: num] :
( ( numeral_numeral @ A @ ( bit0 @ N ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ N ) ) ) ) ).
% numeral_Bit0
thf(fact_74_num_Oexhaust,axiom,
! [Y: num] :
( ( Y != one2 )
=> ( ! [X22: num] :
( Y
!= ( bit0 @ X22 ) )
=> ~ ! [X32: num] :
( Y
!= ( bit1 @ X32 ) ) ) ) ).
% num.exhaust
thf(fact_75_num_Oinduct,axiom,
! [P: num > $o,Num: num] :
( ( P @ one2 )
=> ( ! [X4: num] :
( ( P @ X4 )
=> ( P @ ( bit0 @ X4 ) ) )
=> ( ! [X4: num] :
( ( P @ X4 )
=> ( P @ ( bit1 @ X4 ) ) )
=> ( P @ Num ) ) ) ) ).
% num.induct
thf(fact_76_mult__numeral__1__right,axiom,
! [A: $tType] :
( ( semiring_numeral @ A @ ( type @ A ) )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( numeral_numeral @ A @ one2 ) )
= A2 ) ) ).
% mult_numeral_1_right
thf(fact_77_mult__numeral__1,axiom,
! [A: $tType] :
( ( semiring_numeral @ A @ ( type @ A ) )
=> ! [A2: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ one2 ) @ A2 )
= A2 ) ) ).
% mult_numeral_1
thf(fact_78_one__plus__numeral__commute,axiom,
! [A: $tType] :
( ( numeral @ A @ ( type @ A ) )
=> ! [X5: num] :
( ( plus_plus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X5 ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ X5 ) @ ( one_one @ A ) ) ) ) ).
% one_plus_numeral_commute
thf(fact_79_numeral__One,axiom,
! [A: $tType] :
( ( numeral @ A @ ( type @ A ) )
=> ( ( numeral_numeral @ A @ one2 )
= ( one_one @ A ) ) ) ).
% numeral_One
thf(fact_80_numeral__Bit1,axiom,
! [A: $tType] :
( ( numeral @ A @ ( type @ A ) )
=> ! [N: num] :
( ( numeral_numeral @ A @ ( bit1 @ N ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ N ) ) @ ( one_one @ A ) ) ) ) ).
% numeral_Bit1
thf(fact_81_mult__2__right,axiom,
! [A: $tType] :
( ( semiring_numeral @ A @ ( type @ A ) )
=> ! [Z: A] :
( ( times_times @ A @ Z @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ Z @ Z ) ) ) ).
% mult_2_right
thf(fact_82_mult__2,axiom,
! [A: $tType] :
( ( semiring_numeral @ A @ ( type @ A ) )
=> ! [Z: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z )
= ( plus_plus @ A @ Z @ Z ) ) ) ).
% mult_2
thf(fact_83_sp,axiom,
( ( splay_splay @ a @ a2 @ ( node @ a @ ls @ x @ rs ) )
= ( node @ a @ l @ e @ r ) ) ).
% sp
thf(fact_84_semiring__norm_I3_J,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ ( bit0 @ N ) )
= ( bit1 @ N ) ) ).
% semiring_norm(3)
thf(fact_85_semiring__norm_I4_J,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus @ num @ N @ one2 ) ) ) ).
% semiring_norm(4)
thf(fact_86_semiring__norm_I5_J,axiom,
! [M: num] :
( ( plus_plus @ num @ ( bit0 @ M ) @ one2 )
= ( bit1 @ M ) ) ).
% semiring_norm(5)
thf(fact_87_semiring__norm_I8_J,axiom,
! [M: num] :
( ( plus_plus @ num @ ( bit1 @ M ) @ one2 )
= ( bit0 @ ( plus_plus @ num @ M @ one2 ) ) ) ).
% semiring_norm(8)
thf(fact_88_semiring__norm_I10_J,axiom,
! [M: num,N: num] :
( ( plus_plus @ num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus @ num @ ( plus_plus @ num @ M @ N ) @ one2 ) ) ) ).
% semiring_norm(10)
thf(fact_89_t__splay__simps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type @ A ) )
=> ! [A2: A,L2: tree @ A,R2: tree @ A] :
( ( splay_914434265_splay @ A @ A2 @ ( node @ A @ L2 @ A2 @ R2 ) )
= ( one_one @ nat ) ) ) ).
% t_splay_simps(1)
thf(fact_90_semiring__norm_I70_J,axiom,
! [M: num] :
~ ( ord_less_eq @ num @ ( bit1 @ M ) @ one2 ) ).
% semiring_norm(70)
thf(fact_91_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq @ num @ ( bit0 @ M ) @ one2 ) ).
% semiring_norm(69)
thf(fact_92_semiring__norm_I72_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq @ num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq @ num @ M @ N ) ) ).
% semiring_norm(72)
thf(fact_93_semiring__norm_I87_J,axiom,
! [M: num,N: num] :
( ( ( bit0 @ M )
= ( bit0 @ N ) )
= ( M = N ) ) ).
% semiring_norm(87)
thf(fact_94_semiring__norm_I90_J,axiom,
! [M: num,N: num] :
( ( ( bit1 @ M )
= ( bit1 @ N ) )
= ( M = N ) ) ).
% semiring_norm(90)
thf(fact_95_semiring__norm_I89_J,axiom,
! [M: num,N: num] :
( ( bit1 @ M )
!= ( bit0 @ N ) ) ).
% semiring_norm(89)
thf(fact_96_semiring__norm_I88_J,axiom,
! [M: num,N: num] :
( ( bit0 @ M )
!= ( bit1 @ N ) ) ).
% semiring_norm(88)
thf(fact_97_semiring__norm_I85_J,axiom,
! [M: num] :
( ( bit0 @ M )
!= one2 ) ).
% semiring_norm(85)
thf(fact_98_semiring__norm_I83_J,axiom,
! [N: num] :
( one2
!= ( bit0 @ N ) ) ).
% semiring_norm(83)
thf(fact_99_semiring__norm_I86_J,axiom,
! [M: num] :
( ( bit1 @ M )
!= one2 ) ).
% semiring_norm(86)
thf(fact_100_semiring__norm_I84_J,axiom,
! [N: num] :
( one2
!= ( bit1 @ N ) ) ).
% semiring_norm(84)
thf(fact_101_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% Nat.add_0_right
thf(fact_102_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus @ nat @ M @ N )
= ( zero_zero @ nat ) )
= ( ( M
= ( zero_zero @ nat ) )
& ( N
= ( zero_zero @ nat ) ) ) ) ).
% add_is_0
thf(fact_103_le0,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% le0
thf(fact_104_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times @ nat @ M @ N )
= ( zero_zero @ nat ) )
= ( ( M
= ( zero_zero @ nat ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% mult_is_0
thf(fact_105_mult__0__right,axiom,
! [M: nat] :
( ( times_times @ nat @ M @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% mult_0_right
thf(fact_106_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times @ nat @ K @ M )
= ( times_times @ nat @ K @ N ) )
= ( ( M = N )
| ( K
= ( zero_zero @ nat ) ) ) ) ).
% mult_cancel1
thf(fact_107_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times @ nat @ M @ K )
= ( times_times @ nat @ N @ K ) )
= ( ( M = N )
| ( K
= ( zero_zero @ nat ) ) ) ) ).
% mult_cancel2
thf(fact_108_semiring__norm_I6_J,axiom,
! [M: num,N: num] :
( ( plus_plus @ num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( plus_plus @ num @ M @ N ) ) ) ).
% semiring_norm(6)
thf(fact_109_semiring__norm_I13_J,axiom,
! [M: num,N: num] :
( ( times_times @ num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( bit0 @ ( times_times @ num @ M @ N ) ) ) ) ).
% semiring_norm(13)
thf(fact_110_semiring__norm_I12_J,axiom,
! [N: num] :
( ( times_times @ num @ one2 @ N )
= N ) ).
% semiring_norm(12)
thf(fact_111_semiring__norm_I11_J,axiom,
! [M: num] :
( ( times_times @ num @ M @ one2 )
= M ) ).
% semiring_norm(11)
thf(fact_112_semiring__norm_I71_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq @ num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_eq @ num @ M @ N ) ) ).
% semiring_norm(71)
thf(fact_113_semiring__norm_I73_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq @ num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq @ num @ M @ N ) ) ).
% semiring_norm(73)
thf(fact_114_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq @ num @ one2 @ N ) ).
% semiring_norm(68)
thf(fact_115_semiring__norm_I9_J,axiom,
! [M: num,N: num] :
( ( plus_plus @ num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( bit1 @ ( plus_plus @ num @ M @ N ) ) ) ).
% semiring_norm(9)
thf(fact_116_semiring__norm_I7_J,axiom,
! [M: num,N: num] :
( ( plus_plus @ num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( bit1 @ ( plus_plus @ num @ M @ N ) ) ) ).
% semiring_norm(7)
thf(fact_117_semiring__norm_I2_J,axiom,
( ( plus_plus @ num @ one2 @ one2 )
= ( bit0 @ one2 ) ) ).
% semiring_norm(2)
thf(fact_118_semiring__norm_I15_J,axiom,
! [M: num,N: num] :
( ( times_times @ num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( times_times @ num @ ( bit1 @ M ) @ N ) ) ) ).
% semiring_norm(15)
thf(fact_119_semiring__norm_I14_J,axiom,
! [M: num,N: num] :
( ( times_times @ num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( bit0 @ ( times_times @ num @ M @ ( bit1 @ N ) ) ) ) ).
% semiring_norm(14)
thf(fact_120_semiring__norm_I169_J,axiom,
! [A: $tType] :
( ( semiring_numeral @ A @ ( type @ A ) )
=> ! [V: num,W: num,Z: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V ) @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ Z ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V @ W ) ) @ Z ) ) ) ).
% semiring_norm(169)
thf(fact_121_semiring__norm_I16_J,axiom,
! [M: num,N: num] :
( ( times_times @ num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( bit1 @ ( plus_plus @ num @ ( plus_plus @ num @ M @ N ) @ ( bit0 @ ( times_times @ num @ M @ N ) ) ) ) ) ).
% semiring_norm(16)
thf(fact_122_True,axiom,
e = a2 ).
% True
thf(fact_123_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z2: int] :
( ( times_times @ int @ W @ ( plus_plus @ int @ Z1 @ Z2 ) )
= ( plus_plus @ int @ ( times_times @ int @ W @ Z1 ) @ ( times_times @ int @ W @ Z2 ) ) ) ).
% int_distrib(2)
thf(fact_124_int__distrib_I1_J,axiom,
! [Z1: int,Z2: int,W: int] :
( ( times_times @ int @ ( plus_plus @ int @ Z1 @ Z2 ) @ W )
= ( plus_plus @ int @ ( times_times @ int @ Z1 @ W ) @ ( times_times @ int @ Z2 @ W ) ) ) ).
% int_distrib(1)
thf(fact_125_plus__int__code_I2_J,axiom,
! [L2: int] :
( ( plus_plus @ int @ ( zero_zero @ int ) @ L2 )
= L2 ) ).
% plus_int_code(2)
thf(fact_126_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus @ int @ K @ ( zero_zero @ int ) )
= K ) ).
% plus_int_code(1)
thf(fact_127_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% less_eq_nat.simps(1)
thf(fact_128_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% zle_int
thf(fact_129_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% le_0_eq
thf(fact_130_le__cube,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ ( times_times @ nat @ M @ M ) ) ) ).
% le_cube
thf(fact_131_le__refl,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% le_refl
thf(fact_132_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ J @ K )
=> ( ord_less_eq @ nat @ I @ K ) ) ) ).
% le_trans
thf(fact_133_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_134_le__square,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ M ) ) ).
% le_square
thf(fact_135_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiring_1_of_nat @ int @ M )
= ( semiring_1_of_nat @ int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_136_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_137_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ Z ) @ Z )
!= ( zero_zero @ int ) ) ).
% odd_nonzero
thf(fact_138_zle__iff__zadd,axiom,
( ( ord_less_eq @ int )
= ( ^ [W2: int,Z3: int] :
? [N3: nat] :
( Z3
= ( plus_plus @ int @ W2 @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_139_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L2: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ K @ L2 )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J @ L2 ) ) ) ) ).
% mult_le_mono
thf(fact_140_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq @ int @ K @ I )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq @ int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_141_nonneg__eq__int,axiom,
! [Z: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ~ ! [M3: nat] :
( Z
!= ( semiring_1_of_nat @ int @ M3 ) ) ) ).
% nonneg_eq_int
thf(fact_142_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_143_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ K @ I ) @ ( times_times @ nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_144_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
| ( ord_less_eq @ nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_145_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ~ ! [N2: nat] :
( K
!= ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% nonneg_int_cases
thf(fact_146_mult__0,axiom,
! [N: nat] :
( ( times_times @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% mult_0
thf(fact_147_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ? [N2: nat] :
( K
= ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_148_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times @ nat @ K @ M )
= ( times_times @ nat @ K @ N ) )
= ( ( K
= ( zero_zero @ nat ) )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_149_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus @ nat @ M @ N )
= M )
=> ( N
= ( zero_zero @ nat ) ) ) ).
% add_eq_self_zero
thf(fact_150_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% plus_nat.add_0
thf(fact_151_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times @ nat @ M @ N ) )
=> ( ( N
= ( one_one @ nat ) )
| ( M
= ( zero_zero @ nat ) ) ) ) ).
% mult_eq_self_implies_10
thf(fact_152_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ K ) )
= ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ I @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_153__092_060Phi_062_Ocases,axiom,
! [A: $tType,X5: tree @ A] :
( ( X5
!= ( leaf @ A ) )
=> ~ ! [L: tree @ A,A4: A,R: tree @ A] :
( X5
!= ( node @ A @ L @ A4 @ R ) ) ) ).
% \<Phi>.cases
thf(fact_154__092_060Phi_062_Oinduct,axiom,
! [A: $tType,P: ( tree @ A ) > $o,A0: tree @ A] :
( ( P @ ( leaf @ A ) )
=> ( ! [L: tree @ A,A4: A,R: tree @ A] :
( ( P @ L )
=> ( ( P @ R )
=> ( P @ ( node @ A @ L @ A4 @ R ) ) ) )
=> ( P @ A0 ) ) ) ).
% \<Phi>.induct
thf(fact_155_t__splay__max_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type @ A ) )
=> ! [X5: tree @ A] :
( ( X5
!= ( leaf @ A ) )
=> ( ! [L: tree @ A,B2: A] :
( X5
!= ( node @ A @ L @ B2 @ ( leaf @ A ) ) )
=> ~ ! [L: tree @ A,B2: A,Rl: tree @ A,C2: A,Rr: tree @ A] :
( X5
!= ( node @ A @ L @ B2 @ ( node @ A @ Rl @ C2 @ Rr ) ) ) ) ) ) ).
% t_splay_max.cases
thf(fact_156_t__splay__max_Oinduct,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type @ A ) )
=> ! [P: ( tree @ A ) > $o,A0: tree @ A] :
( ( P @ ( leaf @ A ) )
=> ( ! [L: tree @ A,B2: A] : ( P @ ( node @ A @ L @ B2 @ ( leaf @ A ) ) )
=> ( ! [L: tree @ A,B2: A,Rl: tree @ A,C2: A,Rr: tree @ A] :
( ( ( Rr
!= ( leaf @ A ) )
=> ( P @ Rr ) )
=> ( P @ ( node @ A @ L @ B2 @ ( node @ A @ Rl @ C2 @ Rr ) ) ) )
=> ( P @ A0 ) ) ) ) ) ).
% t_splay_max.induct
thf(fact_157_t__splay_Osimps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type @ A ) )
=> ! [A2: A] :
( ( splay_914434265_splay @ A @ A2 @ ( leaf @ A ) )
= ( one_one @ nat ) ) ) ).
% t_splay.simps(1)
thf(fact_158__092_060Phi_062_Osimps_I1_J,axiom,
! [A: $tType] :
( ( splay_1414789891is_Phi @ A @ ( leaf @ A ) )
= ( zero_zero @ real ) ) ).
% \<Phi>.simps(1)
thf(fact_159_numeral__le__real__of__nat__iff,axiom,
! [N: num,M: nat] :
( ( ord_less_eq @ real @ ( numeral_numeral @ real @ N ) @ ( semiring_1_of_nat @ real @ M ) )
= ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ N ) @ M ) ) ).
% numeral_le_real_of_nat_iff
thf(fact_160_log__one,axiom,
! [A2: real] :
( ( log @ A2 @ ( one_one @ real ) )
= ( zero_zero @ real ) ) ).
% log_one
thf(fact_161_mult__cancel__left1,axiom,
! [A: $tType] :
( ( ring_11004092258visors @ A @ ( type @ A ) )
=> ! [C: A,B: A] :
( ( C
= ( times_times @ A @ C @ B ) )
= ( ( C
= ( zero_zero @ A ) )
| ( B
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left1
thf(fact_162_mult__cancel__left2,axiom,
! [A: $tType] :
( ( ring_11004092258visors @ A @ ( type @ A ) )
=> ! [C: A,A2: A] :
( ( ( times_times @ A @ C @ A2 )
= C )
= ( ( C
= ( zero_zero @ A ) )
| ( A2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left2
thf(fact_163_mult__cancel__right1,axiom,
! [A: $tType] :
( ( ring_11004092258visors @ A @ ( type @ A ) )
=> ! [C: A,B: A] :
( ( C
= ( times_times @ A @ B @ C ) )
= ( ( C
= ( zero_zero @ A ) )
| ( B
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right1
thf(fact_164_mult__cancel__right,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A @ ( type @ A ) )
=> ! [A2: A,C: A,B: A] :
( ( ( times_times @ A @ A2 @ C )
= ( times_times @ A @ B @ C ) )
= ( ( C
= ( zero_zero @ A ) )
| ( A2 = B ) ) ) ) ).
% mult_cancel_right
thf(fact_165_mult__cancel__left,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A @ ( type @ A ) )
=> ! [C: A,A2: A,B: A] :
( ( ( times_times @ A @ C @ A2 )
= ( times_times @ A @ C @ B ) )
= ( ( C
= ( zero_zero @ A ) )
| ( A2 = B ) ) ) ) ).
% mult_cancel_left
thf(fact_166_mult__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri1193490041visors @ A @ ( type @ A ) )
=> ! [A2: A,B: A] :
( ( ( times_times @ A @ A2 @ B )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ A ) )
| ( B
= ( zero_zero @ A ) ) ) ) ) ).
% mult_eq_0_iff
thf(fact_167_mult__zero__right,axiom,
! [A: $tType] :
( ( mult_zero @ A @ ( type @ A ) )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% mult_zero_right
thf(fact_168_mult__zero__left,axiom,
! [A: $tType] :
( ( mult_zero @ A @ ( type @ A ) )
=> ! [A2: A] :
( ( times_times @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% mult_zero_left
thf(fact_169_mult__cancel__right2,axiom,
! [A: $tType] :
( ( ring_11004092258visors @ A @ ( type @ A ) )
=> ! [A2: A,C: A] :
( ( ( times_times @ A @ A2 @ C )
= C )
= ( ( C
= ( zero_zero @ A ) )
| ( A2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right2
thf(fact_170_times__int__code_I2_J,axiom,
! [L2: int] :
( ( times_times @ int @ ( zero_zero @ int ) @ L2 )
= ( zero_zero @ int ) ) ).
% times_int_code(2)
thf(fact_171_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times @ int @ K @ ( zero_zero @ int ) )
= ( zero_zero @ int ) ) ).
% times_int_code(1)
thf(fact_172_less__eq__int__code_I1_J,axiom,
ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% less_eq_int_code(1)
thf(fact_173_complete__real,axiom,
! [S: set @ real] :
( ? [X6: real] : ( member @ real @ X6 @ S )
=> ( ? [Z4: real] :
! [X4: real] :
( ( member @ real @ X4 @ S )
=> ( ord_less_eq @ real @ X4 @ Z4 ) )
=> ? [Y4: real] :
( ! [X6: real] :
( ( member @ real @ X6 @ S )
=> ( ord_less_eq @ real @ X6 @ Y4 ) )
& ! [Z4: real] :
( ! [X4: real] :
( ( member @ real @ X4 @ S )
=> ( ord_less_eq @ real @ X4 @ Z4 ) )
=> ( ord_less_eq @ real @ Y4 @ Z4 ) ) ) ) ) ).
% complete_real
thf(fact_174_mult__right__cancel,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A @ ( type @ A ) )
=> ! [C: A,A2: A,B: A] :
( ( C
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A2 @ C )
= ( times_times @ A @ B @ C ) )
= ( A2 = B ) ) ) ) ).
% mult_right_cancel
thf(fact_175_mult__left__cancel,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A @ ( type @ A ) )
=> ! [C: A,A2: A,B: A] :
( ( C
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ C @ A2 )
= ( times_times @ A @ C @ B ) )
= ( A2 = B ) ) ) ) ).
% mult_left_cancel
thf(fact_176_no__zero__divisors,axiom,
! [A: $tType] :
( ( semiri1193490041visors @ A @ ( type @ A ) )
=> ! [A2: A,B: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( B
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ B )
!= ( zero_zero @ A ) ) ) ) ) ).
% no_zero_divisors
thf(fact_177_divisors__zero,axiom,
! [A: $tType] :
( ( semiri1193490041visors @ A @ ( type @ A ) )
=> ! [A2: A,B: A] :
( ( ( times_times @ A @ A2 @ B )
= ( zero_zero @ A ) )
=> ( ( A2
= ( zero_zero @ A ) )
| ( B
= ( zero_zero @ A ) ) ) ) ) ).
% divisors_zero
thf(fact_178_mult__not__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A @ ( type @ A ) )
=> ! [A2: A,B: A] :
( ( ( times_times @ A @ A2 @ B )
!= ( zero_zero @ A ) )
=> ( ( A2
!= ( zero_zero @ A ) )
& ( B
!= ( zero_zero @ A ) ) ) ) ) ).
% mult_not_zero
thf(fact_179_zero__neq__one,axiom,
! [A: $tType] :
( ( zero_neq_one @ A @ ( type @ A ) )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ) ).
% zero_neq_one
thf(fact_180_combine__common__factor,axiom,
! [A: $tType] :
( ( semiring @ A @ ( type @ A ) )
=> ! [A2: A,E2: A,B: A,C: A] :
( ( plus_plus @ A @ ( times_times @ A @ A2 @ E2 ) @ ( plus_plus @ A @ ( times_times @ A @ B @ E2 ) @ C ) )
= ( plus_plus @ A @ ( times_times @ A @ ( plus_plus @ A @ A2 @ B ) @ E2 ) @ C ) ) ) ).
% combine_common_factor
thf(fact_181_distrib__right,axiom,
! [A: $tType] :
( ( semiring @ A @ ( type @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C ) @ ( times_times @ A @ B @ C ) ) ) ) ).
% distrib_right
thf(fact_182_distrib__left,axiom,
! [A: $tType] :
( ( semiring @ A @ ( type @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( times_times @ A @ A2 @ ( plus_plus @ A @ B @ C ) )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ B ) @ ( times_times @ A @ A2 @ C ) ) ) ) ).
% distrib_left
thf(fact_183_comm__semiring__class_Odistrib,axiom,
! [A: $tType] :
( ( comm_semiring @ A @ ( type @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C ) @ ( times_times @ A @ B @ C ) ) ) ) ).
% comm_semiring_class.distrib
thf(fact_184_ring__class_Oring__distribs_I1_J,axiom,
! [A: $tType] :
( ( ring @ A @ ( type @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( times_times @ A @ A2 @ ( plus_plus @ A @ B @ C ) )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ B ) @ ( times_times @ A @ A2 @ C ) ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_185_ring__class_Oring__distribs_I2_J,axiom,
! [A: $tType] :
( ( ring @ A @ ( type @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C ) @ ( times_times @ A @ B @ C ) ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_186_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: $tType] :
( ( ordere1490568538miring @ A @ ( type @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C )
=> ( ord_less_eq @ A @ ( times_times @ A @ C @ A2 ) @ ( times_times @ A @ C @ B ) ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_187_zero__le__mult__iff,axiom,
! [A: $tType] :
( ( linord581940658strict @ A @ ( type @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_le_mult_iff
thf(fact_188_mult__nonneg__nonpos2,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A @ ( type @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ B @ A2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_189_mult__nonpos__nonneg,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A @ ( type @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonpos_nonneg
thf(fact_190_mult__nonneg__nonpos,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A @ ( type @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonneg_nonpos
thf(fact_191_mult__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A @ ( type @ 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 ) @ ( times_times @ A @ A2 @ B ) ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_192_split__mult__neg__le,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A @ ( type @ A ) )
=> ! [A2: A,B: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ B @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B ) ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B ) @ ( zero_zero @ A ) ) ) ) ).
% split_mult_neg_le
thf(fact_193_mult__le__0__iff,axiom,
! [A: $tType] :
( ( linord581940658strict @ A @ ( type @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B ) @ ( zero_zero @ A ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ B @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B ) ) ) ) ) ).
% mult_le_0_iff
thf(fact_194_mult__right__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A @ ( type @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C ) @ ( times_times @ A @ B @ C ) ) ) ) ) ).
% mult_right_mono
thf(fact_195_mult__right__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A @ ( type @ A ) )
=> ! [B: A,A2: A,C: A] :
( ( ord_less_eq @ A @ B @ A2 )
=> ( ( ord_less_eq @ A @ C @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C ) @ ( times_times @ A @ B @ C ) ) ) ) ) ).
% mult_right_mono_neg
thf(fact_196_mult__left__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A @ ( type @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C )
=> ( ord_less_eq @ A @ ( times_times @ A @ C @ A2 ) @ ( times_times @ A @ C @ B ) ) ) ) ) ).
% mult_left_mono
thf(fact_197_mult__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordered_ring @ A @ ( type @ 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 @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B ) ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_198_mult__left__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A @ ( type @ A ) )
=> ! [B: A,A2: A,C: A] :
( ( ord_less_eq @ A @ B @ A2 )
=> ( ( ord_less_eq @ A @ C @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ C @ A2 ) @ ( times_times @ A @ C @ B ) ) ) ) ) ).
% mult_left_mono_neg
thf(fact_199_split__mult__pos__le,axiom,
! [A: $tType] :
( ( ordered_ring @ A @ ( type @ 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 @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B @ ( zero_zero @ A ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B ) ) ) ) ).
% split_mult_pos_le
thf(fact_200_zero__le__square,axiom,
! [A: $tType] :
( ( linordered_ring @ A @ ( type @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ A2 ) ) ) ).
% zero_le_square
thf(fact_201_mult__mono_H,axiom,
! [A: $tType] :
( ( ordered_semiring @ A @ ( type @ 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 @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C ) @ ( times_times @ A @ B @ D ) ) ) ) ) ) ) ).
% mult_mono'
thf(fact_202_mult__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A @ ( type @ 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 @ ( zero_zero @ A ) @ B )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C ) @ ( times_times @ A @ B @ D ) ) ) ) ) ) ) ).
% mult_mono
thf(fact_203_not__one__le__zero,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A @ ( type @ A ) )
=> ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_le_zero
thf(fact_204_zero__le__one,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A @ ( type @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_le_one
thf(fact_205_sum__squares__ge__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A @ ( type @ A ) )
=> ! [X5: A,Y: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X5 @ X5 ) @ ( times_times @ A @ Y @ Y ) ) ) ) ).
% sum_squares_ge_zero
thf(fact_206_mult__left__le,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A @ ( type @ A ) )
=> ! [C: A,A2: A] :
( ( ord_less_eq @ A @ C @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C ) @ A2 ) ) ) ) ).
% mult_left_le
thf(fact_207_mult__le__one,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A @ ( type @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B )
=> ( ( ord_less_eq @ A @ B @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B ) @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_one
thf(fact_208_mult__right__le__one__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type @ A ) )
=> ! [X5: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X5 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ Y @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ X5 @ Y ) @ X5 ) ) ) ) ) ).
% mult_right_le_one_le
thf(fact_209_mult__left__le__one__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type @ A ) )
=> ! [X5: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X5 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ Y @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ Y @ X5 ) @ X5 ) ) ) ) ) ).
% mult_left_le_one_le
thf(fact_210_convex__bound__le,axiom,
! [A: $tType] :
( ( linord1278240602ring_1 @ A @ ( type @ A ) )
=> ! [X5: A,A2: A,Y: A,U: A,V: A] :
( ( ord_less_eq @ A @ X5 @ A2 )
=> ( ( ord_less_eq @ A @ Y @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V )
=> ( ( ( plus_plus @ A @ U @ V )
= ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X5 ) @ ( times_times @ A @ V @ Y ) ) @ A2 ) ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_211_sum__squares__eq__zero__iff,axiom,
! [A: $tType] :
( ( linord581940658strict @ A @ ( type @ A ) )
=> ! [X5: A,Y: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ X5 @ X5 ) @ ( times_times @ A @ Y @ Y ) )
= ( zero_zero @ A ) )
= ( ( X5
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_212_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type @ 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_213_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type @ 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_214_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( ( plus_plus @ A @ A2 @ B )
= ( plus_plus @ A @ A2 @ C ) )
= ( B = C ) ) ) ).
% add_left_cancel
thf(fact_215_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type @ A ) )
=> ! [B: A,A2: A,C: A] :
( ( ( plus_plus @ A @ B @ A2 )
= ( plus_plus @ A @ C @ A2 ) )
= ( B = C ) ) ) ).
% add_right_cancel
thf(fact_216_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type @ A ) )
=> ! [N: A] :
( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_217_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type @ 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_218_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type @ 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_219_add_Oleft__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A @ ( type @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add.left_neutral
thf(fact_220_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A @ ( type @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.right_neutral
thf(fact_221_double__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type @ A ) )
=> ! [A2: A] :
( ( ( plus_plus @ A @ A2 @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% double_zero
thf(fact_222_double__zero__sym,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type @ A ) )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ A2 @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% double_zero_sym
thf(fact_223_add__cancel__left__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type @ A ) )
=> ! [B: A,A2: A] :
( ( ( plus_plus @ A @ B @ A2 )
= A2 )
= ( B
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_left
thf(fact_224_add__cancel__left__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type @ A ) )
=> ! [A2: A,B: A] :
( ( ( plus_plus @ A @ A2 @ B )
= A2 )
= ( B
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_right
thf(fact_225_add__cancel__right__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type @ A ) )
=> ! [A2: A,B: A] :
( ( A2
= ( plus_plus @ A @ B @ A2 ) )
= ( B
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_left
thf(fact_226_add__cancel__right__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type @ A ) )
=> ! [A2: A,B: A] :
( ( A2
= ( plus_plus @ A @ A2 @ B ) )
= ( B
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_right
thf(fact_227_mult_Oleft__neutral,axiom,
! [A: $tType] :
( ( monoid_mult @ A @ ( type @ A ) )
=> ! [A2: A] :
( ( times_times @ A @ ( one_one @ A ) @ A2 )
= A2 ) ) ).
% mult.left_neutral
thf(fact_228_mult_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_mult @ A @ ( type @ A ) )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% mult.right_neutral
thf(fact_229_add__le__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type @ 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_230_add__le__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type @ 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_231_le__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type @ 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_232_le__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type @ 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_233_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A @ ( type @ A ) )
=> ! [X5: A] :
( ( ( zero_zero @ A )
= X5 )
= ( X5
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_234_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A @ ( type @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( times_times @ A @ ( times_times @ A @ A2 @ B ) @ C )
= ( times_times @ A @ A2 @ ( times_times @ A @ B @ C ) ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_235_mult_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_mult @ A @ ( type @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( times_times @ A @ ( times_times @ A @ A2 @ B ) @ C )
= ( times_times @ A @ A2 @ ( times_times @ A @ B @ C ) ) ) ) ).
% mult.assoc
thf(fact_236_mult_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A @ ( type @ A ) )
=> ( ( times_times @ A )
= ( ^ [A5: A,B3: A] : ( times_times @ A @ B3 @ A5 ) ) ) ) ).
% mult.commute
thf(fact_237_mult_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A @ ( type @ A ) )
=> ! [B: A,A2: A,C: A] :
( ( times_times @ A @ B @ ( times_times @ A @ A2 @ C ) )
= ( times_times @ A @ A2 @ ( times_times @ A @ B @ C ) ) ) ) ).
% mult.left_commute
thf(fact_238_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type @ 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_239_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type @ A ) )
=> ! [I: A,J: A,K: A,L2: A] :
( ( ( I = J )
& ( K = L2 ) )
=> ( ( plus_plus @ A @ I @ K )
= ( plus_plus @ A @ J @ L2 ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_240_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A @ ( type @ 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_241_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( ( plus_plus @ A @ A2 @ B )
= ( plus_plus @ A @ A2 @ C ) )
= ( B = C ) ) ) ).
% add.left_cancel
thf(fact_242_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type @ A ) )
=> ! [B: A,A2: A,C: A] :
( ( ( plus_plus @ A @ B @ A2 )
= ( plus_plus @ A @ C @ A2 ) )
= ( B = C ) ) ) ).
% add.right_cancel
thf(fact_243_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type @ A ) )
=> ( ( plus_plus @ A )
= ( ^ [A5: A,B3: A] : ( plus_plus @ A @ B3 @ A5 ) ) ) ) ).
% add.commute
thf(fact_244_add_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type @ 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_245_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type @ 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_246_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type @ 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_247_one__reorient,axiom,
! [A: $tType] :
( ( one @ A @ ( type @ A ) )
=> ! [X5: A] :
( ( ( one_one @ A )
= X5 )
= ( X5
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_248_zero__le,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type @ A ) )
=> ! [X5: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X5 ) ) ).
% zero_le
thf(fact_249_add__mono__thms__linordered__semiring_I3_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type @ A ) )
=> ! [I: A,J: A,K: A,L2: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( K = L2 ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L2 ) ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_250_add__mono__thms__linordered__semiring_I2_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type @ A ) )
=> ! [I: A,J: A,K: A,L2: A] :
( ( ( I = J )
& ( ord_less_eq @ A @ K @ L2 ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L2 ) ) ) ) ).
% add_mono_thms_linordered_semiring(2)
%----Type constructors (110)
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
ordere516151231imp_le @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors__cancel,axiom,
semiri1923998003cancel @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
linord1659791738miring @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors,axiom,
semiri1193490041visors @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
linord219039673up_add @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Rings_Oring__1__no__zero__divisors,axiom,
ring_11004092258visors @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
cancel1352612707id_add @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Rings_Olinordered__ring__strict,axiom,
linord581940658strict @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Rings_Oordered__comm__semiring,axiom,
ordere1490568538miring @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring__1,axiom,
linord1278240602ring_1 @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Rings_Oordered__semiring__0,axiom,
ordered_semiring_0 @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
linordered_semidom @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Rings_Oordered__semiring,axiom,
ordered_semiring @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Rings_Olinordered__ring,axiom,
linordered_ring @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Groups_Osemigroup__mult,axiom,
semigroup_mult @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Num_Osemiring__numeral,axiom,
semiring_numeral @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
semigroup_add @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring,axiom,
comm_semiring @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
semiring_char_0 @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Rings_Ozero__neq__one,axiom,
zero_neq_one @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Rings_Oordered__ring,axiom,
ordered_ring @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
linorder @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Groups_Omonoid__mult,axiom,
monoid_mult @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Groups_Omonoid__add,axiom,
monoid_add @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1 @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
group_add @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Rings_Omult__zero,axiom,
mult_zero @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Num_Oneg__numeral,axiom,
neg_numeral @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Rings_Osemiring,axiom,
semiring @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Num_Onumeral,axiom,
numeral @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Groups_Ozero,axiom,
zero @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Rings_Oring,axiom,
ring @ int @ ( type @ int ) ).
thf(tcon_Int_Oint___Groups_Oone,axiom,
one @ int @ ( type @ int ) ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_1,axiom,
ordere516151231imp_le @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_2,axiom,
semiri1923998003cancel @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_3,axiom,
ordere236663937imp_le @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_4,axiom,
linord1659791738miring @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_5,axiom,
semiri1193490041visors @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_6,axiom,
ordere779506340up_add @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_7,axiom,
cancel1352612707id_add @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Rings_Oordered__comm__semiring_8,axiom,
ordere1490568538miring @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_9,axiom,
cancel_semigroup_add @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring__0_10,axiom,
ordered_semiring_0 @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom_11,axiom,
linordered_semidom @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult_12,axiom,
ab_semigroup_mult @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__add_13,axiom,
ab_semigroup_add @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring_14,axiom,
ordered_semiring @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Groups_Osemigroup__mult_15,axiom,
semigroup_mult @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Num_Osemiring__numeral_16,axiom,
semiring_numeral @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add_17,axiom,
semigroup_add @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring_18,axiom,
comm_semiring @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Nat_Osemiring__char__0_19,axiom,
semiring_char_0 @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Rings_Ozero__neq__one_20,axiom,
zero_neq_one @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Olinorder_21,axiom,
linorder @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Groups_Omonoid__mult_22,axiom,
monoid_mult @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Groups_Omonoid__add_23,axiom,
monoid_add @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Rings_Osemiring__1_24,axiom,
semiring_1 @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Rings_Omult__zero_25,axiom,
mult_zero @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Rings_Osemiring_26,axiom,
semiring @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Num_Onumeral_27,axiom,
numeral @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ozero_28,axiom,
zero @ nat @ ( type @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oone_29,axiom,
one @ nat @ ( type @ nat ) ).
thf(tcon_Num_Onum___Orderings_Olinorder_30,axiom,
linorder @ num @ ( type @ num ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder_31,axiom,
linorder @ $o @ ( type @ $o ) ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__monoid__add__imp__le_32,axiom,
ordere516151231imp_le @ real @ ( type @ real ) ).
thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors__cancel_33,axiom,
semiri1923998003cancel @ real @ ( type @ real ) ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_34,axiom,
ordere236663937imp_le @ real @ ( type @ real ) ).
thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_35,axiom,
linord1659791738miring @ real @ ( type @ real ) ).
thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_36,axiom,
semiri1193490041visors @ real @ ( type @ real ) ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add_37,axiom,
ordere779506340up_add @ real @ ( type @ real ) ).
thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_38,axiom,
linord219039673up_add @ real @ ( type @ real ) ).
thf(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors_39,axiom,
ring_11004092258visors @ real @ ( type @ real ) ).
thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_40,axiom,
cancel1352612707id_add @ real @ ( type @ real ) ).
thf(tcon_Real_Oreal___Rings_Olinordered__ring__strict_41,axiom,
linord581940658strict @ real @ ( type @ real ) ).
thf(tcon_Real_Oreal___Rings_Oordered__comm__semiring_42,axiom,
ordere1490568538miring @ real @ ( type @ real ) ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring__1_43,axiom,
linord1278240602ring_1 @ real @ ( type @ real ) ).
thf(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_44,axiom,
cancel_semigroup_add @ real @ ( type @ real ) ).
thf(tcon_Real_Oreal___Rings_Oordered__semiring__0_45,axiom,
ordered_semiring_0 @ real @ ( type @ real ) ).
thf(tcon_Real_Oreal___Rings_Olinordered__semidom_46,axiom,
linordered_semidom @ real @ ( type @ real ) ).
thf(tcon_Real_Oreal___Groups_Oab__semigroup__mult_47,axiom,
ab_semigroup_mult @ real @ ( type @ real ) ).
thf(tcon_Real_Oreal___Groups_Oab__semigroup__add_48,axiom,
ab_semigroup_add @ real @ ( type @ real ) ).
thf(tcon_Real_Oreal___Rings_Oordered__semiring_49,axiom,
ordered_semiring @ real @ ( type @ real ) ).
thf(tcon_Real_Oreal___Rings_Olinordered__ring_50,axiom,
linordered_ring @ real @ ( type @ real ) ).
thf(tcon_Real_Oreal___Rings_Olinordered__idom_51,axiom,
linordered_idom @ real @ ( type @ real ) ).
thf(tcon_Real_Oreal___Groups_Osemigroup__mult_52,axiom,
semigroup_mult @ real @ ( type @ real ) ).
thf(tcon_Real_Oreal___Num_Osemiring__numeral_53,axiom,
semiring_numeral @ real @ ( type @ real ) ).
thf(tcon_Real_Oreal___Groups_Osemigroup__add_54,axiom,
semigroup_add @ real @ ( type @ real ) ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring_55,axiom,
comm_semiring @ real @ ( type @ real ) ).
thf(tcon_Real_Oreal___Nat_Osemiring__char__0_56,axiom,
semiring_char_0 @ real @ ( type @ real ) ).
thf(tcon_Real_Oreal___Rings_Ozero__neq__one_57,axiom,
zero_neq_one @ real @ ( type @ real ) ).
thf(tcon_Real_Oreal___Rings_Oordered__ring_58,axiom,
ordered_ring @ real @ ( type @ real ) ).
thf(tcon_Real_Oreal___Orderings_Olinorder_59,axiom,
linorder @ real @ ( type @ real ) ).
thf(tcon_Real_Oreal___Groups_Omonoid__mult_60,axiom,
monoid_mult @ real @ ( type @ real ) ).
thf(tcon_Real_Oreal___Groups_Omonoid__add_61,axiom,
monoid_add @ real @ ( type @ real ) ).
thf(tcon_Real_Oreal___Rings_Osemiring__1_62,axiom,
semiring_1 @ real @ ( type @ real ) ).
thf(tcon_Real_Oreal___Groups_Ogroup__add_63,axiom,
group_add @ real @ ( type @ real ) ).
thf(tcon_Real_Oreal___Rings_Omult__zero_64,axiom,
mult_zero @ real @ ( type @ real ) ).
thf(tcon_Real_Oreal___Num_Oneg__numeral_65,axiom,
neg_numeral @ real @ ( type @ real ) ).
thf(tcon_Real_Oreal___Rings_Osemiring_66,axiom,
semiring @ real @ ( type @ real ) ).
thf(tcon_Real_Oreal___Num_Onumeral_67,axiom,
numeral @ real @ ( type @ real ) ).
thf(tcon_Real_Oreal___Groups_Ozero_68,axiom,
zero @ real @ ( type @ real ) ).
thf(tcon_Real_Oreal___Rings_Oring_69,axiom,
ring @ real @ ( type @ real ) ).
thf(tcon_Real_Oreal___Groups_Oone_70,axiom,
one @ real @ ( type @ real ) ).
%----Free types (1)
thf(tfree_0,hypothesis,
linorder @ a @ ( type @ a ) ).
%----Conjectures (3)
thf(conj_0,hypothesis,
$true ).
thf(conj_1,hypothesis,
ord_less_eq @ real @ ( plus_plus @ real @ ( splay_1414789891is_Phi @ a @ r ) @ ( plus_plus @ real @ ( semiring_1_of_nat @ real @ ( splay_914434265_splay @ a @ a2 @ ( node @ a @ ls @ x @ rs ) ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( plus_plus @ real @ ( semiring_1_of_nat @ real @ ( size_size @ ( tree @ a ) @ r ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) @ ( plus_plus @ real @ ( one_one @ real ) @ ( plus_plus @ real @ ( splay_1414789891is_Phi @ a @ ls ) @ ( plus_plus @ real @ ( splay_1414789891is_Phi @ a @ rs ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( plus_plus @ real @ ( semiring_1_of_nat @ real @ ( size_size @ ( tree @ a ) @ ls ) ) @ ( plus_plus @ real @ ( semiring_1_of_nat @ real @ ( size_size @ ( tree @ a ) @ rs ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
thf(conj_2,conjecture,
ord_less_eq @ real @ ( plus_plus @ real @ ( splay_1414789891is_Phi @ a @ r ) @ ( semiring_1_of_nat @ real @ ( splay_914434265_splay @ a @ a2 @ ( node @ a @ ls @ x @ rs ) ) ) ) @ ( plus_plus @ real @ ( splay_1414789891is_Phi @ a @ ls ) @ ( plus_plus @ real @ ( splay_1414789891is_Phi @ a @ rs ) @ ( plus_plus @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit1 @ ( bit1 @ one2 ) ) ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( plus_plus @ real @ ( semiring_1_of_nat @ real @ ( size_size @ ( tree @ a ) @ ls ) ) @ ( plus_plus @ real @ ( semiring_1_of_nat @ real @ ( size_size @ ( tree @ a ) @ rs ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------