TPTP Problem File: DAT209^1.p
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%------------------------------------------------------------------------------
% File : DAT209^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Splay tree analysis 17
% 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__17.p [Bla16]
% Status : Theorem
% Rating : 1.00 v7.5.0, 0.33 v7.2.0, 0.50 v7.1.0
% Syntax : Number of formulae : 434 ( 111 unt; 87 typ; 0 def)
% Number of atoms : 762 ( 429 equ; 0 cnn)
% Maximal formula atoms : 5 ( 2 avg)
% Number of connectives : 3395 ( 29 ~; 20 |; 20 &;3066 @)
% ( 0 <=>; 260 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 5 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 126 ( 126 >; 0 *; 0 +; 0 <<)
% Number of symbols : 87 ( 84 usr; 5 con; 0-5 aty)
% Number of variables : 784 ( 24 ^; 678 !; 0 ?; 784 :)
% ( 82 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:48:22.648
%------------------------------------------------------------------------------
%----Could-be-implicit typings (5)
thf(ty_t_Tree_Otree,type,
tree: $tType > $tType ).
thf(ty_t_Real_Oreal,type,
real: $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 (82)
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Oidom,type,
idom:
!>[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_Rings_Oring__1,type,
ring_1:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Osgn__if,type,
sgn_if:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ouminus,type,
uminus:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Osemiring,type,
semiring:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Nat_Oring__char__0,type,
ring_char_0:
!>[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_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Ocomm__ring__1,type,
comm_ring_1:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Transcendental_Oln,type,
ln:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Omonoid__mult,type,
monoid_mult:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oab__group__add,type,
ab_group_add:
!>[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_Groups_Osemigroup__mult,type,
semigroup_mult:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Ocomm__semiring__1,type,
comm_semiring_1:
!>[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_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Fields_Olinordered__field,type,
linordered_field:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocomm__monoid__mult,type,
comm_monoid_mult:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oab__semigroup__mult,type,
ab_semigroup_mult:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Oboolean__algebra,type,
boolean_algebra:
!>[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__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_Real__Vector__Spaces_Oreal__vector,type,
real_V1076094709vector:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Archimedean__Field_Ofloor__ceiling,type,
archim1727834104eiling:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__group__add__abs,type,
ordere142940540dd_abs:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere779506340up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Real__Vector__Spaces_Oreal__algebra,type,
real_V148923926lgebra:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
semiri1193490041visors:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Real__Vector__Spaces_Oreal__algebra__1,type,
real_V84468443ebra_1:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__vector,type,
real_V55928688vector:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors__cancel,type,
semiri1923998003cancel:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__algebra__1,type,
real_V1229719638ebra_1:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__div__algebra,type,
real_V68988228lgebra:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
semiri456707255roduct:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Archimedean__Field_Oceiling,type,
archimedean_ceiling:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor,type,
archim2085082626_floor:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_Archimedean__Field_Oround,type,
archimedean_round:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_Groups_Oabs__class_Oabs,type,
abs_abs:
!>[A: $tType] : ( A > A ) ).
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_Osgn__class_Osgn,type,
sgn_sgn:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Otimes__class_Otimes,type,
times_times:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Odbl,type,
neg_numeral_dbl:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec,type,
neg_numeral_dbl_dec:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc,type,
neg_numeral_dbl_inc:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Real__Vector__Spaces_Olinear__axioms,type,
real_V2128096322axioms:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm,type,
real_V1127708846m_norm:
!>[A: $tType] : ( A > real ) ).
thf(sy_c_Real__Vector__Spaces_Oof__real,type,
real_Vector_of_real:
!>[A: $tType] : ( real > A ) ).
thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR,type,
real_V1908273582scaleR:
!>[A: $tType] : ( real > A > A ) ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
zero_neq_one_of_bool:
!>[A: $tType] : ( $o > A ) ).
thf(sy_c_Splay__Tree__Analysis__Mirabelle__pcaxyvimtd_OA,type,
splay_266122055elle_A:
!>[A: $tType] : ( A > ( tree @ A ) > real ) ).
thf(sy_c_Transcendental_Oln__class_Oln,type,
ln_ln:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Opowr,type,
powr:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Tree_Otree_ONode,type,
node:
!>[A: $tType] : ( ( tree @ A ) > A > ( tree @ A ) > ( tree @ A ) ) ).
thf(sy_c_Tree_Otree_Ocase__tree,type,
case_tree:
!>[B: $tType,A: $tType] : ( B > ( ( tree @ A ) > A > ( tree @ A ) > B ) > ( tree @ A ) > B ) ).
thf(sy_c_Tree_Otree_Opred__tree,type,
pred_tree:
!>[A: $tType] : ( ( A > $o ) > ( tree @ A ) > $o ) ).
thf(sy_c_Tree_Otree_Orec__tree,type,
rec_tree:
!>[C: $tType,A: $tType] : ( C > ( ( tree @ A ) > A > ( tree @ A ) > C > C > C ) > ( tree @ A ) > C ) ).
thf(sy_c_Tree_Otree_Oval,type,
val:
!>[A: $tType] : ( ( tree @ A ) > A ) ).
thf(sy_v_a,type,
a2: a ).
thf(sy_v_l,type,
l: tree @ a ).
thf(sy_v_r,type,
r: tree @ a ).
%----Relevant facts (254)
thf(fact_0_tree_Oinject,axiom,
! [A: $tType,X21: tree @ A,X22: A,X23: tree @ A,Y21: tree @ A,Y22: A,Y23: tree @ A] :
( ( ( node @ A @ X21 @ X22 @ X23 )
= ( node @ A @ Y21 @ Y22 @ Y23 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 )
& ( X23 = Y23 ) ) ) ).
% tree.inject
thf(fact_1_one__reorient,axiom,
! [A: $tType] :
( ( one @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( ( one_one @ A )
= X )
= ( X
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_2_powr__one__eq__one,axiom,
! [A: $tType] :
( ( ln @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( powr @ A @ ( one_one @ A ) @ A2 )
= ( one_one @ A ) ) ) ).
% powr_one_eq_one
thf(fact_3_norm__one,axiom,
! [A: $tType] :
( ( real_V1229719638ebra_1 @ A @ ( type2 @ A ) )
=> ( ( real_V1127708846m_norm @ A @ ( one_one @ A ) )
= ( one_one @ real ) ) ) ).
% norm_one
thf(fact_4_of__real__1,axiom,
! [A: $tType] :
( ( real_V84468443ebra_1 @ A @ ( type2 @ A ) )
=> ( ( real_Vector_of_real @ A @ ( one_one @ real ) )
= ( one_one @ A ) ) ) ).
% of_real_1
thf(fact_5_of__bool__eq_I2_J,axiom,
! [A: $tType] :
( ( zero_neq_one @ A @ ( type2 @ A ) )
=> ( ( zero_neq_one_of_bool @ A @ $true )
= ( one_one @ A ) ) ) ).
% of_bool_eq(2)
thf(fact_6_dbl__dec__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A @ ( type2 @ A ) )
=> ( ( neg_numeral_dbl_dec @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% dbl_dec_simps(3)
thf(fact_7_tree_Osimps_I7_J,axiom,
! [C: $tType,A: $tType,F1: C,F2: ( tree @ A ) > A > ( tree @ A ) > C > C > C,X21: tree @ A,X22: A,X23: tree @ A] :
( ( rec_tree @ C @ A @ F1 @ F2 @ ( node @ A @ X21 @ X22 @ X23 ) )
= ( F2 @ X21 @ X22 @ X23 @ ( rec_tree @ C @ A @ F1 @ F2 @ X21 ) @ ( rec_tree @ C @ A @ F1 @ F2 @ X23 ) ) ) ).
% tree.simps(7)
thf(fact_8_tree_Opred__inject_I2_J,axiom,
! [A: $tType,P: A > $o,A2: tree @ A,Aa: A,Ab: tree @ A] :
( ( pred_tree @ A @ P @ ( node @ A @ A2 @ Aa @ Ab ) )
= ( ( pred_tree @ A @ P @ A2 )
& ( P @ Aa )
& ( pred_tree @ A @ P @ Ab ) ) ) ).
% tree.pred_inject(2)
thf(fact_9_round__1,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A @ ( type2 @ A ) )
=> ( ( archimedean_round @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% round_1
thf(fact_10_tree_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F1: B,F2: ( tree @ A ) > A > ( tree @ A ) > B,X21: tree @ A,X22: A,X23: tree @ A] :
( ( case_tree @ B @ A @ F1 @ F2 @ ( node @ A @ X21 @ X22 @ X23 ) )
= ( F2 @ X21 @ X22 @ X23 ) ) ).
% tree.simps(5)
thf(fact_11_of__real__eq__iff,axiom,
! [A: $tType] :
( ( real_V84468443ebra_1 @ A @ ( type2 @ A ) )
=> ! [X: real,Y: real] :
( ( ( real_Vector_of_real @ A @ X )
= ( real_Vector_of_real @ A @ Y ) )
= ( X = Y ) ) ) ).
% of_real_eq_iff
thf(fact_12_of__bool__eq__iff,axiom,
! [A: $tType] :
( ( zero_neq_one @ A @ ( type2 @ A ) )
=> ! [P2: $o,Q: $o] :
( ( ( zero_neq_one_of_bool @ A @ P2 )
= ( zero_neq_one_of_bool @ A @ Q ) )
= ( P2 = Q ) ) ) ).
% of_bool_eq_iff
thf(fact_13_one__integer_Orsp,axiom,
( ( one_one @ int )
= ( one_one @ int ) ) ).
% one_integer.rsp
thf(fact_14_powr__zero__eq__one,axiom,
! [A: $tType] :
( ( ln @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( ( X
= ( zero_zero @ A ) )
=> ( ( powr @ A @ X @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) )
& ( ( X
!= ( zero_zero @ A ) )
=> ( ( powr @ A @ X @ ( zero_zero @ A ) )
= ( one_one @ A ) ) ) ) ) ).
% powr_zero_eq_one
thf(fact_15_of__bool__def,axiom,
! [A: $tType] :
( ( zero_neq_one @ A @ ( type2 @ A ) )
=> ( ( zero_neq_one_of_bool @ A )
= ( ^ [P3: $o] : ( if @ A @ P3 @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ) ).
% of_bool_def
thf(fact_16_split__of__bool,axiom,
! [A: $tType] :
( ( zero_neq_one @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P2 ) )
= ( ( P2
=> ( P @ ( one_one @ A ) ) )
& ( ~ P2
=> ( P @ ( zero_zero @ A ) ) ) ) ) ) ).
% split_of_bool
thf(fact_17_split__of__bool__asm,axiom,
! [A: $tType] :
( ( zero_neq_one @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P2 ) )
= ( ~ ( ( P2
& ~ ( P @ ( one_one @ A ) ) )
| ( ~ P2
& ~ ( P @ ( zero_zero @ A ) ) ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_18_norm__of__real,axiom,
! [A: $tType] :
( ( real_V1229719638ebra_1 @ A @ ( type2 @ A ) )
=> ! [R: real] :
( ( real_V1127708846m_norm @ A @ ( real_Vector_of_real @ A @ R ) )
= ( abs_abs @ real @ R ) ) ) ).
% norm_of_real
thf(fact_19_ceiling__one,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A @ ( type2 @ A ) )
=> ( ( archimedean_ceiling @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% ceiling_one
thf(fact_20_floor__one,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A @ ( type2 @ A ) )
=> ( ( archim2085082626_floor @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% floor_one
thf(fact_21_of__bool__eq_I1_J,axiom,
! [A: $tType] :
( ( zero_neq_one @ A @ ( type2 @ A ) )
=> ( ( zero_neq_one_of_bool @ A @ $false )
= ( zero_zero @ A ) ) ) ).
% of_bool_eq(1)
thf(fact_22_of__real__def,axiom,
! [A: $tType] :
( ( real_V84468443ebra_1 @ A @ ( type2 @ A ) )
=> ( ( real_Vector_of_real @ A )
= ( ^ [R2: real] : ( real_V1908273582scaleR @ A @ R2 @ ( one_one @ A ) ) ) ) ) ).
% of_real_def
thf(fact_23_tree_Osel_I3_J,axiom,
! [A: $tType,X21: tree @ A,X22: A,X23: tree @ A] :
( ( val @ A @ ( node @ A @ X21 @ X22 @ X23 ) )
= X22 ) ).
% tree.sel(3)
thf(fact_24_abs__idempotent,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( abs_abs @ A @ ( abs_abs @ A @ A2 ) )
= ( abs_abs @ A @ A2 ) ) ) ).
% abs_idempotent
thf(fact_25_real__vector_Oscale__cancel__left,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A @ ( type2 @ A ) )
=> ! [A2: real,X: A,Y: A] :
( ( ( real_V1908273582scaleR @ A @ A2 @ X )
= ( real_V1908273582scaleR @ A @ A2 @ Y ) )
= ( ( X = Y )
| ( A2
= ( zero_zero @ real ) ) ) ) ) ).
% real_vector.scale_cancel_left
thf(fact_26_abs__0__eq,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( abs_abs @ A @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% abs_0_eq
thf(fact_27_abs__eq__0,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ( abs_abs @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% abs_eq_0
thf(fact_28_abs__zero,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A @ ( type2 @ A ) )
=> ( ( abs_abs @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% abs_zero
thf(fact_29_norm__eq__zero,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( ( real_V1127708846m_norm @ A @ X )
= ( zero_zero @ real ) )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% norm_eq_zero
thf(fact_30_norm__zero,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A @ ( type2 @ A ) )
=> ( ( real_V1127708846m_norm @ A @ ( zero_zero @ A ) )
= ( zero_zero @ real ) ) ) ).
% norm_zero
thf(fact_31_abs__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_one
thf(fact_32_real__vector_Oscale__eq__0__iff,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A @ ( type2 @ A ) )
=> ! [A2: real,X: A] :
( ( ( real_V1908273582scaleR @ A @ A2 @ X )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ real ) )
| ( X
= ( zero_zero @ A ) ) ) ) ) ).
% real_vector.scale_eq_0_iff
thf(fact_33_real__vector_Oscale__zero__left,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( real_V1908273582scaleR @ A @ ( zero_zero @ real ) @ X )
= ( zero_zero @ A ) ) ) ).
% real_vector.scale_zero_left
thf(fact_34_real__vector_Oscale__zero__right,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A @ ( type2 @ A ) )
=> ! [A2: real] :
( ( real_V1908273582scaleR @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% real_vector.scale_zero_right
thf(fact_35_real__vector_Oscale__cancel__right,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A @ ( type2 @ A ) )
=> ! [A2: real,X: A,B2: real] :
( ( ( real_V1908273582scaleR @ A @ A2 @ X )
= ( real_V1908273582scaleR @ A @ B2 @ X ) )
= ( ( A2 = B2 )
| ( X
= ( zero_zero @ A ) ) ) ) ) ).
% real_vector.scale_cancel_right
thf(fact_36_of__real__eq__0__iff,axiom,
! [A: $tType] :
( ( real_V84468443ebra_1 @ A @ ( type2 @ A ) )
=> ! [X: real] :
( ( ( real_Vector_of_real @ A @ X )
= ( zero_zero @ A ) )
= ( X
= ( zero_zero @ real ) ) ) ) ).
% of_real_eq_0_iff
thf(fact_37_of__real__0,axiom,
! [A: $tType] :
( ( real_V84468443ebra_1 @ A @ ( type2 @ A ) )
=> ( ( real_Vector_of_real @ A @ ( zero_zero @ real ) )
= ( zero_zero @ A ) ) ) ).
% of_real_0
thf(fact_38_floor__zero,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A @ ( type2 @ A ) )
=> ( ( archim2085082626_floor @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% floor_zero
thf(fact_39_powr__eq__0__iff,axiom,
! [A: $tType] :
( ( ln @ A @ ( type2 @ A ) )
=> ! [W: A,Z: A] :
( ( ( powr @ A @ W @ Z )
= ( zero_zero @ A ) )
= ( W
= ( zero_zero @ A ) ) ) ) ).
% powr_eq_0_iff
thf(fact_40_powr__0,axiom,
! [A: $tType] :
( ( ln @ A @ ( type2 @ A ) )
=> ! [Z: A] :
( ( powr @ A @ ( zero_zero @ A ) @ Z )
= ( zero_zero @ A ) ) ) ).
% powr_0
thf(fact_41_scaleR__one,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( real_V1908273582scaleR @ A @ ( one_one @ real ) @ X )
= X ) ) ).
% scaleR_one
thf(fact_42_ceiling__zero,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A @ ( type2 @ A ) )
=> ( ( archimedean_ceiling @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% ceiling_zero
thf(fact_43_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X2: A] :
( ( F @ X2 )
= ( G @ X2 ) )
=> ( F = G ) ) ).
% ext
thf(fact_44_abs__norm__cancel,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( abs_abs @ real @ ( real_V1127708846m_norm @ A @ A2 ) )
= ( real_V1127708846m_norm @ A @ A2 ) ) ) ).
% abs_norm_cancel
thf(fact_45_round__0,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A @ ( type2 @ A ) )
=> ( ( archimedean_round @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% round_0
thf(fact_46_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_47_real__norm__def,axiom,
( ( real_V1127708846m_norm @ real )
= ( abs_abs @ real ) ) ).
% real_norm_def
thf(fact_48_real__vector_Oscale__left__imp__eq,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A @ ( type2 @ A ) )
=> ! [A2: real,X: A,Y: A] :
( ( A2
!= ( zero_zero @ real ) )
=> ( ( ( real_V1908273582scaleR @ A @ A2 @ X )
= ( real_V1908273582scaleR @ A @ A2 @ Y ) )
=> ( X = Y ) ) ) ) ).
% real_vector.scale_left_imp_eq
thf(fact_49_real__vector_Oscale__left__commute,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A @ ( type2 @ A ) )
=> ! [A2: real,B2: real,X: A] :
( ( real_V1908273582scaleR @ A @ A2 @ ( real_V1908273582scaleR @ A @ B2 @ X ) )
= ( real_V1908273582scaleR @ A @ B2 @ ( real_V1908273582scaleR @ A @ A2 @ X ) ) ) ) ).
% real_vector.scale_left_commute
thf(fact_50_real__vector_Oscale__right__imp__eq,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A @ ( type2 @ A ) )
=> ! [X: A,A2: real,B2: real] :
( ( X
!= ( zero_zero @ A ) )
=> ( ( ( real_V1908273582scaleR @ A @ A2 @ X )
= ( real_V1908273582scaleR @ A @ B2 @ X ) )
=> ( A2 = B2 ) ) ) ) ).
% real_vector.scale_right_imp_eq
thf(fact_51_zero__neq__one,axiom,
! [A: $tType] :
( ( zero_neq_one @ A @ ( type2 @ A ) )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ) ).
% zero_neq_one
thf(fact_52_ln__one,axiom,
! [A: $tType] :
( ( ln @ A @ ( type2 @ A ) )
=> ( ( ln_ln @ A @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% ln_one
thf(fact_53_dbl__inc__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A @ ( type2 @ A ) )
=> ( ( neg_numeral_dbl_inc @ A @ ( zero_zero @ A ) )
= ( one_one @ A ) ) ) ).
% dbl_inc_simps(2)
thf(fact_54_dbl__dec__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A @ ( type2 @ A ) )
=> ( ( neg_numeral_dbl_dec @ A @ ( zero_zero @ A ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% dbl_dec_simps(2)
thf(fact_55_norm__scaleR,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A @ ( type2 @ A ) )
=> ! [A2: real,X: A] :
( ( real_V1127708846m_norm @ A @ ( real_V1908273582scaleR @ A @ A2 @ X ) )
= ( times_times @ real @ ( abs_abs @ real @ A2 ) @ ( real_V1127708846m_norm @ A @ X ) ) ) ) ).
% norm_scaleR
thf(fact_56_norm__sgn,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( ( X
= ( zero_zero @ A ) )
=> ( ( real_V1127708846m_norm @ A @ ( sgn_sgn @ A @ X ) )
= ( zero_zero @ real ) ) )
& ( ( X
!= ( zero_zero @ A ) )
=> ( ( real_V1127708846m_norm @ A @ ( sgn_sgn @ A @ X ) )
= ( one_one @ real ) ) ) ) ) ).
% norm_sgn
thf(fact_57_abs__sgn__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ( A2
= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A2 ) )
= ( zero_zero @ A ) ) )
& ( ( A2
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A2 ) )
= ( one_one @ A ) ) ) ) ) ).
% abs_sgn_eq
thf(fact_58_dbl__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A @ ( type2 @ A ) )
=> ( ( neg_numeral_dbl @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% dbl_simps(2)
thf(fact_59_linear__axioms__def,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V1076094709vector @ A @ ( type2 @ A ) )
& ( real_V1076094709vector @ B @ ( type2 @ B ) ) )
=> ( ( real_V2128096322axioms @ A @ B )
= ( ^ [F3: A > B] :
! [R2: real,X3: A] :
( ( F3 @ ( real_V1908273582scaleR @ A @ R2 @ X3 ) )
= ( real_V1908273582scaleR @ B @ R2 @ ( F3 @ X3 ) ) ) ) ) ) ).
% linear_axioms_def
thf(fact_60_linear__axioms_Ointro,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V1076094709vector @ A @ ( type2 @ A ) )
& ( real_V1076094709vector @ B @ ( type2 @ B ) ) )
=> ! [F: A > B] :
( ! [R3: real,X2: A] :
( ( F @ ( real_V1908273582scaleR @ A @ R3 @ X2 ) )
= ( real_V1908273582scaleR @ B @ R3 @ ( F @ X2 ) ) )
=> ( real_V2128096322axioms @ A @ B @ F ) ) ) ).
% linear_axioms.intro
thf(fact_61_norm__of__real__add1,axiom,
! [A: $tType] :
( ( real_V68988228lgebra @ A @ ( type2 @ A ) )
=> ! [X: real] :
( ( real_V1127708846m_norm @ A @ ( plus_plus @ A @ ( real_Vector_of_real @ A @ X ) @ ( one_one @ A ) ) )
= ( abs_abs @ real @ ( plus_plus @ real @ X @ ( one_one @ real ) ) ) ) ) ).
% norm_of_real_add1
thf(fact_62_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C2 @ A2 ) )
= ( B2 = C2 ) ) ) ).
% add_right_cancel
thf(fact_63_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C2 ) )
= ( B2 = C2 ) ) ) ).
% add_left_cancel
thf(fact_64_neg__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ( uminus_uminus @ A @ A2 )
= ( uminus_uminus @ A @ B2 ) )
= ( A2 = B2 ) ) ) ).
% neg_equal_iff_equal
thf(fact_65_add_Oinverse__inverse,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A2 ) )
= A2 ) ) ).
% add.inverse_inverse
thf(fact_66_sgn__sgn,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( sgn_sgn @ A @ ( sgn_sgn @ A @ A2 ) )
= ( sgn_sgn @ A @ A2 ) ) ) ).
% sgn_sgn
thf(fact_67_mult__cancel__right,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A @ ( type2 @ A ) )
=> ! [A2: A,C2: A,B2: A] :
( ( ( times_times @ A @ A2 @ C2 )
= ( times_times @ A @ B2 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A2 = B2 ) ) ) ) ).
% mult_cancel_right
thf(fact_68_mult__cancel__left,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A @ ( type2 @ A ) )
=> ! [C2: A,A2: A,B2: A] :
( ( ( times_times @ A @ C2 @ A2 )
= ( times_times @ A @ C2 @ B2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A2 = B2 ) ) ) ) ).
% mult_cancel_left
thf(fact_69_mult__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri1193490041visors @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ( times_times @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% mult_eq_0_iff
thf(fact_70_mult__zero__right,axiom,
! [A: $tType] :
( ( mult_zero @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% mult_zero_right
thf(fact_71_mult__zero__left,axiom,
! [A: $tType] :
( ( mult_zero @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( times_times @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% mult_zero_left
thf(fact_72_add__cancel__right__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( A2
= ( plus_plus @ A @ A2 @ B2 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_right
thf(fact_73_add__cancel__right__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( A2
= ( plus_plus @ A @ B2 @ A2 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_left
thf(fact_74_add__cancel__left__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= A2 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_right
thf(fact_75_add__cancel__left__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= A2 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_left
thf(fact_76_double__zero__sym,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ A2 @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% double_zero_sym
thf(fact_77_double__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ( plus_plus @ A @ A2 @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% double_zero
thf(fact_78_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.right_neutral
thf(fact_79_add_Oleft__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add.left_neutral
thf(fact_80_mult_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_mult @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% mult.right_neutral
thf(fact_81_mult_Oleft__neutral,axiom,
! [A: $tType] :
( ( monoid_mult @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( times_times @ A @ ( one_one @ A ) @ A2 )
= A2 ) ) ).
% mult.left_neutral
thf(fact_82_neg__equal__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ( uminus_uminus @ A @ A2 )
= A2 )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% neg_equal_zero
thf(fact_83_equal__neg__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( A2
= ( uminus_uminus @ A @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% equal_neg_zero
thf(fact_84_neg__equal__0__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ( uminus_uminus @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% neg_equal_0_iff_equal
thf(fact_85_neg__0__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( uminus_uminus @ A @ A2 ) )
= ( ( zero_zero @ A )
= A2 ) ) ) ).
% neg_0_equal_iff_equal
thf(fact_86_add_Oinverse__neutral,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% add.inverse_neutral
thf(fact_87_mult__minus__right,axiom,
! [A: $tType] :
( ( ring @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( times_times @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( uminus_uminus @ A @ ( times_times @ A @ A2 @ B2 ) ) ) ) ).
% mult_minus_right
thf(fact_88_minus__mult__minus,axiom,
! [A: $tType] :
( ( ring @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) )
= ( times_times @ A @ A2 @ B2 ) ) ) ).
% minus_mult_minus
thf(fact_89_mult__minus__left,axiom,
! [A: $tType] :
( ( ring @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( uminus_uminus @ A @ ( times_times @ A @ A2 @ B2 ) ) ) ) ).
% mult_minus_left
thf(fact_90_minus__add__distrib,axiom,
! [A: $tType] :
( ( ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% minus_add_distrib
thf(fact_91_minus__add__cancel,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( plus_plus @ A @ A2 @ B2 ) )
= B2 ) ) ).
% minus_add_cancel
thf(fact_92_add__minus__cancel,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) )
= B2 ) ) ).
% add_minus_cancel
thf(fact_93_abs__mult__self,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ A2 ) )
= ( times_times @ A @ A2 @ A2 ) ) ) ).
% abs_mult_self
thf(fact_94_abs__add__abs,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( abs_abs @ A @ ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) )
= ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_add_abs
thf(fact_95_abs__minus__cancel,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( abs_abs @ A @ ( uminus_uminus @ A @ A2 ) )
= ( abs_abs @ A @ A2 ) ) ) ).
% abs_minus_cancel
thf(fact_96_mult__scaleR__right,axiom,
! [A: $tType] :
( ( real_V148923926lgebra @ A @ ( type2 @ A ) )
=> ! [X: A,A2: real,Y: A] :
( ( times_times @ A @ X @ ( real_V1908273582scaleR @ A @ A2 @ Y ) )
= ( real_V1908273582scaleR @ A @ A2 @ ( times_times @ A @ X @ Y ) ) ) ) ).
% mult_scaleR_right
thf(fact_97_mult__scaleR__left,axiom,
! [A: $tType] :
( ( real_V148923926lgebra @ A @ ( type2 @ A ) )
=> ! [A2: real,X: A,Y: A] :
( ( times_times @ A @ ( real_V1908273582scaleR @ A @ A2 @ X ) @ Y )
= ( real_V1908273582scaleR @ A @ A2 @ ( times_times @ A @ X @ Y ) ) ) ) ).
% mult_scaleR_left
thf(fact_98_norm__minus__cancel,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( real_V1127708846m_norm @ A @ ( uminus_uminus @ A @ X ) )
= ( real_V1127708846m_norm @ A @ X ) ) ) ).
% norm_minus_cancel
thf(fact_99_real__vector_Oscale__minus__left,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A @ ( type2 @ A ) )
=> ! [A2: real,X: A] :
( ( real_V1908273582scaleR @ A @ ( uminus_uminus @ real @ A2 ) @ X )
= ( uminus_uminus @ A @ ( real_V1908273582scaleR @ A @ A2 @ X ) ) ) ) ).
% real_vector.scale_minus_left
thf(fact_100_scaleR__left_Ominus,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A @ ( type2 @ A ) )
=> ! [X: real,Xa: A] :
( ( real_V1908273582scaleR @ A @ ( uminus_uminus @ real @ X ) @ Xa )
= ( uminus_uminus @ A @ ( real_V1908273582scaleR @ A @ X @ Xa ) ) ) ) ).
% scaleR_left.minus
thf(fact_101_real__vector_Oscale__minus__right,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A @ ( type2 @ A ) )
=> ! [A2: real,X: A] :
( ( real_V1908273582scaleR @ A @ A2 @ ( uminus_uminus @ A @ X ) )
= ( uminus_uminus @ A @ ( real_V1908273582scaleR @ A @ A2 @ X ) ) ) ) ).
% real_vector.scale_minus_right
thf(fact_102_sgn__zero,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A @ ( type2 @ A ) )
=> ( ( sgn_sgn @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% sgn_zero
thf(fact_103_sgn0,axiom,
! [A: $tType] :
( ( sgn_if @ A @ ( type2 @ A ) )
=> ( ( sgn_sgn @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% sgn0
thf(fact_104_of__real__minus,axiom,
! [A: $tType] :
( ( real_V84468443ebra_1 @ A @ ( type2 @ A ) )
=> ! [X: real] :
( ( real_Vector_of_real @ A @ ( uminus_uminus @ real @ X ) )
= ( uminus_uminus @ A @ ( real_Vector_of_real @ A @ X ) ) ) ) ).
% of_real_minus
thf(fact_105_scaleR__scaleR,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A @ ( type2 @ A ) )
=> ! [A2: real,B2: real,X: A] :
( ( real_V1908273582scaleR @ A @ A2 @ ( real_V1908273582scaleR @ A @ B2 @ X ) )
= ( real_V1908273582scaleR @ A @ ( times_times @ real @ A2 @ B2 ) @ X ) ) ) ).
% scaleR_scaleR
thf(fact_106_sgn__one,axiom,
! [A: $tType] :
( ( real_V1229719638ebra_1 @ A @ ( type2 @ A ) )
=> ( ( sgn_sgn @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% sgn_one
thf(fact_107_mult__cancel__left1,axiom,
! [A: $tType] :
( ( ring_11004092258visors @ A @ ( type2 @ A ) )
=> ! [C2: A,B2: A] :
( ( C2
= ( times_times @ A @ C2 @ B2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( B2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left1
thf(fact_108_mult__cancel__left2,axiom,
! [A: $tType] :
( ( ring_11004092258visors @ A @ ( type2 @ A ) )
=> ! [C2: A,A2: A] :
( ( ( times_times @ A @ C2 @ A2 )
= C2 )
= ( ( C2
= ( zero_zero @ A ) )
| ( A2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left2
thf(fact_109_mult__cancel__right1,axiom,
! [A: $tType] :
( ( ring_11004092258visors @ A @ ( type2 @ A ) )
=> ! [C2: A,B2: A] :
( ( C2
= ( times_times @ A @ B2 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( B2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right1
thf(fact_110_mult__cancel__right2,axiom,
! [A: $tType] :
( ( ring_11004092258visors @ A @ ( type2 @ A ) )
=> ! [A2: A,C2: A] :
( ( ( times_times @ A @ A2 @ C2 )
= C2 )
= ( ( C2
= ( zero_zero @ A ) )
| ( A2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right2
thf(fact_111_add_Oleft__inverse,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% add.left_inverse
thf(fact_112_add_Oright__inverse,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
= ( zero_zero @ A ) ) ) ).
% add.right_inverse
thf(fact_113_mult__minus1,axiom,
! [A: $tType] :
( ( ring_1 @ A @ ( type2 @ A ) )
=> ! [Z: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ Z )
= ( uminus_uminus @ A @ Z ) ) ) ).
% mult_minus1
thf(fact_114_mult__minus1__right,axiom,
! [A: $tType] :
( ( ring_1 @ A @ ( type2 @ A ) )
=> ! [Z: A] :
( ( times_times @ A @ Z @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ Z ) ) ) ).
% mult_minus1_right
thf(fact_115_abs__neg__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ( ( abs_abs @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( one_one @ A ) ) ) ).
% abs_neg_one
thf(fact_116_scaleR__eq__iff,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A @ ( type2 @ A ) )
=> ! [B2: A,U: real,A2: A] :
( ( ( plus_plus @ A @ B2 @ ( real_V1908273582scaleR @ A @ U @ A2 ) )
= ( plus_plus @ A @ A2 @ ( real_V1908273582scaleR @ A @ U @ B2 ) ) )
= ( ( A2 = B2 )
| ( U
= ( one_one @ real ) ) ) ) ) ).
% scaleR_eq_iff
thf(fact_117_scaleR__minus1__left,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( real_V1908273582scaleR @ A @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
= ( uminus_uminus @ A @ X ) ) ) ).
% scaleR_minus1_left
thf(fact_118_of__real__mult,axiom,
! [A: $tType] :
( ( real_V84468443ebra_1 @ A @ ( type2 @ A ) )
=> ! [X: real,Y: real] :
( ( real_Vector_of_real @ A @ ( times_times @ real @ X @ Y ) )
= ( times_times @ A @ ( real_Vector_of_real @ A @ X ) @ ( real_Vector_of_real @ A @ Y ) ) ) ) ).
% of_real_mult
thf(fact_119_of__real__add,axiom,
! [A: $tType] :
( ( real_V84468443ebra_1 @ A @ ( type2 @ A ) )
=> ! [X: real,Y: real] :
( ( real_Vector_of_real @ A @ ( plus_plus @ real @ X @ Y ) )
= ( plus_plus @ A @ ( real_Vector_of_real @ A @ X ) @ ( real_Vector_of_real @ A @ Y ) ) ) ) ).
% of_real_add
thf(fact_120_dbl__inc__simps_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A @ ( type2 @ A ) )
=> ( ( neg_numeral_dbl_inc @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% dbl_inc_simps(4)
thf(fact_121_add__neg__numeral__special_I7_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A @ ( type2 @ A ) )
=> ( ( plus_plus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% add_neg_numeral_special(7)
thf(fact_122_add__neg__numeral__special_I8_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A @ ( type2 @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% add_neg_numeral_special(8)
thf(fact_123_floor__add__one,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( archim2085082626_floor @ A @ ( plus_plus @ A @ X @ ( one_one @ A ) ) )
= ( plus_plus @ int @ ( archim2085082626_floor @ A @ X ) @ ( one_one @ int ) ) ) ) ).
% floor_add_one
thf(fact_124_ceiling__add__one,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ ( one_one @ A ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) ) ) ) ).
% ceiling_add_one
thf(fact_125_ln__powr,axiom,
! [X: real,Y: real] :
( ( X
!= ( zero_zero @ real ) )
=> ( ( ln_ln @ real @ ( powr @ real @ X @ Y ) )
= ( times_times @ real @ Y @ ( ln_ln @ real @ X ) ) ) ) ).
% ln_powr
thf(fact_126_real__scaleR__def,axiom,
( ( real_V1908273582scaleR @ real )
= ( times_times @ real ) ) ).
% real_scaleR_def
thf(fact_127_scaleR__left_Oadd,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A @ ( type2 @ A ) )
=> ! [X: real,Y: real,Xa: A] :
( ( real_V1908273582scaleR @ A @ ( plus_plus @ real @ X @ Y ) @ Xa )
= ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ X @ Xa ) @ ( real_V1908273582scaleR @ A @ Y @ Xa ) ) ) ) ).
% scaleR_left.add
thf(fact_128_add__eq__0__iff,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( B2
= ( uminus_uminus @ A @ A2 ) ) ) ) ).
% add_eq_0_iff
thf(fact_129_ab__group__add__class_Oab__left__minus,axiom,
! [A: $tType] :
( ( ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% ab_group_add_class.ab_left_minus
thf(fact_130_add_Oinverse__unique,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ A2 )
= B2 ) ) ) ).
% add.inverse_unique
thf(fact_131_eq__neg__iff__add__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( A2
= ( uminus_uminus @ A @ B2 ) )
= ( ( plus_plus @ A @ A2 @ B2 )
= ( zero_zero @ A ) ) ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_132_neg__eq__iff__add__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ( uminus_uminus @ A @ A2 )
= B2 )
= ( ( plus_plus @ A @ A2 @ B2 )
= ( zero_zero @ A ) ) ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_133_scaleR__add__left,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A @ ( type2 @ A ) )
=> ! [A2: real,B2: real,X: A] :
( ( real_V1908273582scaleR @ A @ ( plus_plus @ real @ A2 @ B2 ) @ X )
= ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ A2 @ X ) @ ( real_V1908273582scaleR @ A @ B2 @ X ) ) ) ) ).
% scaleR_add_left
thf(fact_134_zero__integer_Orsp,axiom,
( ( zero_zero @ int )
= ( zero_zero @ int ) ) ).
% zero_integer.rsp
thf(fact_135_square__eq__1__iff,axiom,
! [A: $tType] :
( ( ring_11004092258visors @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( ( times_times @ A @ X @ X )
= ( one_one @ A ) )
= ( ( X
= ( one_one @ A ) )
| ( X
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% square_eq_1_iff
thf(fact_136_norm__mult,axiom,
! [A: $tType] :
( ( real_V68988228lgebra @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( real_V1127708846m_norm @ A @ ( times_times @ A @ X @ Y ) )
= ( times_times @ real @ ( real_V1127708846m_norm @ A @ X ) @ ( real_V1127708846m_norm @ A @ Y ) ) ) ) ).
% norm_mult
thf(fact_137_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C2 @ A2 ) )
=> ( B2 = C2 ) ) ) ).
% add_right_imp_eq
thf(fact_138_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C2 ) )
=> ( B2 = C2 ) ) ) ).
% add_left_imp_eq
thf(fact_139_mult_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C2: A] :
( ( times_times @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) )
= ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% mult.left_commute
thf(fact_140_add_Oinverse__distrib__swap,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% add.inverse_distrib_swap
thf(fact_141_add_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C2: A] :
( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add.left_commute
thf(fact_142_mult_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A @ ( type2 @ A ) )
=> ( ( times_times @ A )
= ( ^ [A3: A,B3: A] : ( times_times @ A @ B3 @ A3 ) ) ) ) ).
% mult.commute
thf(fact_143_combine__common__factor,axiom,
! [A: $tType] :
( ( semiring @ A @ ( type2 @ A ) )
=> ! [A2: A,E: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( times_times @ A @ A2 @ E ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ E ) @ C2 ) ) ) ).
% combine_common_factor
thf(fact_144_minus__equation__iff,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ( uminus_uminus @ A @ A2 )
= B2 )
= ( ( uminus_uminus @ A @ B2 )
= A2 ) ) ) ).
% minus_equation_iff
thf(fact_145_equation__minus__iff,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( A2
= ( uminus_uminus @ A @ B2 ) )
= ( B2
= ( uminus_uminus @ A @ A2 ) ) ) ) ).
% equation_minus_iff
thf(fact_146_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type2 @ A ) )
=> ( ( plus_plus @ A )
= ( ^ [A3: A,B3: A] : ( plus_plus @ A @ B3 @ A3 ) ) ) ) ).
% add.commute
thf(fact_147_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C2 @ A2 ) )
= ( B2 = C2 ) ) ) ).
% add.right_cancel
thf(fact_148_mult_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_mult @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
= ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% mult.assoc
thf(fact_149_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C2 ) )
= ( B2 = C2 ) ) ) ).
% add.left_cancel
thf(fact_150_sgn__times,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( sgn_sgn @ A @ ( times_times @ A @ A2 @ B2 ) )
= ( times_times @ A @ ( sgn_sgn @ A @ A2 ) @ ( sgn_sgn @ A @ B2 ) ) ) ) ).
% sgn_times
thf(fact_151_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add.assoc
thf(fact_152_minus__mult__commute,axiom,
! [A: $tType] :
( ( ring @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( times_times @ A @ A2 @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% minus_mult_commute
thf(fact_153_distrib__right,axiom,
! [A: $tType] :
( ( semiring @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% distrib_right
thf(fact_154_distrib__left,axiom,
! [A: $tType] :
( ( semiring @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ).
% distrib_left
thf(fact_155_comm__semiring__class_Odistrib,axiom,
! [A: $tType] :
( ( comm_semiring @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% comm_semiring_class.distrib
thf(fact_156_square__eq__iff,axiom,
! [A: $tType] :
( ( idom @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ( times_times @ A @ A2 @ A2 )
= ( times_times @ A @ B2 @ B2 ) )
= ( ( A2 = B2 )
| ( A2
= ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% square_eq_iff
thf(fact_157_dbl__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A @ ( type2 @ A ) )
=> ( ( neg_numeral_dbl @ A )
= ( ^ [X3: A] : ( plus_plus @ A @ X3 @ X3 ) ) ) ) ).
% dbl_def
thf(fact_158_sgn__minus,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( sgn_sgn @ A @ ( uminus_uminus @ A @ X ) )
= ( uminus_uminus @ A @ ( sgn_sgn @ A @ X ) ) ) ) ).
% sgn_minus
thf(fact_159_sgn__mult,axiom,
! [A: $tType] :
( ( real_V68988228lgebra @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( sgn_sgn @ A @ ( times_times @ A @ X @ Y ) )
= ( times_times @ A @ ( sgn_sgn @ A @ X ) @ ( sgn_sgn @ A @ Y ) ) ) ) ).
% sgn_mult
thf(fact_160_powr__powr,axiom,
! [X: real,A2: real,B2: real] :
( ( powr @ real @ ( powr @ real @ X @ A2 ) @ B2 )
= ( powr @ real @ X @ ( times_times @ real @ A2 @ B2 ) ) ) ).
% powr_powr
thf(fact_161_powr__add,axiom,
! [X: real,A2: real,B2: real] :
( ( powr @ real @ X @ ( plus_plus @ real @ A2 @ B2 ) )
= ( times_times @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ X @ B2 ) ) ) ).
% powr_add
thf(fact_162_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ 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_163_is__num__normalize_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% is_num_normalize(1)
thf(fact_164_is__num__normalize_I8_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% is_num_normalize(8)
thf(fact_165_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
= ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_166_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_167_ring__class_Oring__distribs_I1_J,axiom,
! [A: $tType] :
( ( ring @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_168_ring__class_Oring__distribs_I2_J,axiom,
! [A: $tType] :
( ( ring @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_169_mult__sgn__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( times_times @ A @ ( sgn_sgn @ A @ X ) @ ( abs_abs @ A @ X ) )
= X ) ) ).
% mult_sgn_abs
thf(fact_170_abs__sgn,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ( ( abs_abs @ A )
= ( ^ [K2: A] : ( times_times @ A @ K2 @ ( sgn_sgn @ A @ K2 ) ) ) ) ) ).
% abs_sgn
thf(fact_171_sgn__scaleR,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A @ ( type2 @ A ) )
=> ! [R: real,X: A] :
( ( sgn_sgn @ A @ ( real_V1908273582scaleR @ A @ R @ X ) )
= ( real_V1908273582scaleR @ A @ ( sgn_sgn @ real @ R ) @ ( sgn_sgn @ A @ X ) ) ) ) ).
% sgn_scaleR
thf(fact_172_sgn__of__real,axiom,
! [A: $tType] :
( ( real_V1229719638ebra_1 @ A @ ( type2 @ A ) )
=> ! [R: real] :
( ( sgn_sgn @ A @ ( real_Vector_of_real @ A @ R ) )
= ( real_Vector_of_real @ A @ ( sgn_sgn @ real @ R ) ) ) ) ).
% sgn_of_real
thf(fact_173_dbl__inc__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A @ ( type2 @ A ) )
=> ( ( neg_numeral_dbl_inc @ A )
= ( ^ [X3: A] : ( plus_plus @ A @ ( plus_plus @ A @ X3 @ X3 ) @ ( one_one @ A ) ) ) ) ) ).
% dbl_inc_def
thf(fact_174_ceiling__minus,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( archimedean_ceiling @ A @ ( uminus_uminus @ A @ X ) )
= ( uminus_uminus @ int @ ( archim2085082626_floor @ A @ X ) ) ) ) ).
% ceiling_minus
thf(fact_175_floor__minus,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( archim2085082626_floor @ A @ ( uminus_uminus @ A @ X ) )
= ( uminus_uminus @ int @ ( archimedean_ceiling @ A @ X ) ) ) ) ).
% floor_minus
thf(fact_176_ceiling__def,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A @ ( type2 @ A ) )
=> ( ( archimedean_ceiling @ A )
= ( ^ [X3: A] : ( uminus_uminus @ int @ ( archim2085082626_floor @ A @ ( uminus_uminus @ A @ X3 ) ) ) ) ) ) ).
% ceiling_def
thf(fact_177_add__eq__0__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_178_add_Ogroup__left__neutral,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add.group_left_neutral
thf(fact_179_add_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.comm_neutral
thf(fact_180_comm__monoid__add__class_Oadd__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% comm_monoid_add_class.add_0
thf(fact_181_one__neq__neg__one,axiom,
! [A: $tType] :
( ( ring_char_0 @ A @ ( type2 @ A ) )
=> ( ( one_one @ A )
!= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% one_neq_neg_one
thf(fact_182_mult__right__cancel,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A @ ( type2 @ A ) )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A2 @ C2 )
= ( times_times @ A @ B2 @ C2 ) )
= ( A2 = B2 ) ) ) ) ).
% mult_right_cancel
thf(fact_183_mult__left__cancel,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A @ ( type2 @ A ) )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ C2 @ A2 )
= ( times_times @ A @ C2 @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% mult_left_cancel
thf(fact_184_no__zero__divisors,axiom,
! [A: $tType] :
( ( semiri1193490041visors @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ B2 )
!= ( zero_zero @ A ) ) ) ) ) ).
% no_zero_divisors
thf(fact_185_divisors__zero,axiom,
! [A: $tType] :
( ( semiri1193490041visors @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ( times_times @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
=> ( ( A2
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divisors_zero
thf(fact_186_mult__not__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ( times_times @ A @ A2 @ B2 )
!= ( zero_zero @ A ) )
=> ( ( A2
!= ( zero_zero @ A ) )
& ( B2
!= ( zero_zero @ A ) ) ) ) ) ).
% mult_not_zero
thf(fact_187_mult_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% mult.comm_neutral
thf(fact_188_comm__monoid__mult__class_Omult__1,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( times_times @ A @ ( one_one @ A ) @ A2 )
= A2 ) ) ).
% comm_monoid_mult_class.mult_1
thf(fact_189_abs__eq__iff,axiom,
! [A: $tType] :
( ( linordered_ring @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ( abs_abs @ A @ X )
= ( abs_abs @ A @ Y ) )
= ( ( X = Y )
| ( X
= ( uminus_uminus @ A @ Y ) ) ) ) ) ).
% abs_eq_iff
thf(fact_190_abs__mult,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( abs_abs @ A @ ( times_times @ A @ A2 @ B2 ) )
= ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_mult
thf(fact_191_scaleR__add__right,axiom,
! [A: $tType] :
( ( real_V1076094709vector @ A @ ( type2 @ A ) )
=> ! [A2: real,X: A,Y: A] :
( ( real_V1908273582scaleR @ A @ A2 @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ A @ ( real_V1908273582scaleR @ A @ A2 @ X ) @ ( real_V1908273582scaleR @ A @ A2 @ Y ) ) ) ) ).
% scaleR_add_right
thf(fact_192_sgn__0__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ( sgn_sgn @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% sgn_0_0
thf(fact_193_sgn__zero__iff,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( ( sgn_sgn @ A @ X )
= ( zero_zero @ A ) )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% sgn_zero_iff
thf(fact_194_zero__neq__neg__one,axiom,
! [A: $tType] :
( ( ring_char_0 @ A @ ( type2 @ A ) )
=> ( ( zero_zero @ A )
!= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% zero_neq_neg_one
thf(fact_195_scaleR__conv__of__real,axiom,
! [A: $tType] :
( ( real_V84468443ebra_1 @ A @ ( type2 @ A ) )
=> ( ( real_V1908273582scaleR @ A )
= ( ^ [R2: real] : ( times_times @ A @ ( real_Vector_of_real @ A @ R2 ) ) ) ) ) ).
% scaleR_conv_of_real
thf(fact_196_sum__squares__eq__zero__iff,axiom,
! [A: $tType] :
( ( linord581940658strict @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_197_real__add__minus__iff,axiom,
! [X: real,A2: real] :
( ( ( plus_plus @ real @ X @ ( uminus_uminus @ real @ A2 ) )
= ( zero_zero @ real ) )
= ( X = A2 ) ) ).
% real_add_minus_iff
thf(fact_198_ring__normalization__rules_I1_J,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A @ ( type2 @ A ) )
=> ( ( uminus_uminus @ A )
= ( times_times @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% ring_normalization_rules(1)
thf(fact_199_semiring__normalization__rules_I2_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A,M: A] :
( ( plus_plus @ A @ ( times_times @ A @ A2 @ M ) @ M )
= ( times_times @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ M ) ) ) ).
% semiring_normalization_rules(2)
thf(fact_200_semiring__normalization__rules_I3_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [M: A,A2: A] :
( ( plus_plus @ A @ M @ ( times_times @ A @ A2 @ M ) )
= ( times_times @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ M ) ) ) ).
% semiring_normalization_rules(3)
thf(fact_201_semiring__normalization__rules_I4_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [M: A] :
( ( plus_plus @ A @ M @ M )
= ( times_times @ A @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) @ M ) ) ) ).
% semiring_normalization_rules(4)
thf(fact_202_add__scale__eq__noteq,axiom,
! [A: $tType] :
( ( semiri456707255roduct @ A @ ( type2 @ A ) )
=> ! [R: A,A2: A,B2: A,C2: A,D: A] :
( ( R
!= ( zero_zero @ A ) )
=> ( ( ( A2 = B2 )
& ( C2 != D ) )
=> ( ( plus_plus @ A @ A2 @ ( times_times @ A @ R @ C2 ) )
!= ( plus_plus @ A @ B2 @ ( times_times @ A @ R @ D ) ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_203_powr__powr__swap,axiom,
! [X: real,A2: real,B2: real] :
( ( powr @ real @ ( powr @ real @ X @ A2 ) @ B2 )
= ( powr @ real @ ( powr @ real @ X @ B2 ) @ A2 ) ) ).
% powr_powr_swap
thf(fact_204_semiring__normalization__rules_I7_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ( ( times_times @ A )
= ( ^ [A3: A,B3: A] : ( times_times @ A @ B3 @ A3 ) ) ) ) ).
% semiring_normalization_rules(7)
thf(fact_205_semiring__normalization__rules_I13_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [Lx: A,Ly: A,Rx: A,Ry: A] :
( ( times_times @ A @ ( times_times @ A @ Lx @ Ly ) @ ( times_times @ A @ Rx @ Ry ) )
= ( times_times @ A @ ( times_times @ A @ Lx @ Rx ) @ ( times_times @ A @ Ly @ Ry ) ) ) ) ).
% semiring_normalization_rules(13)
thf(fact_206_semiring__normalization__rules_I14_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [Lx: A,Ly: A,Rx: A,Ry: A] :
( ( times_times @ A @ ( times_times @ A @ Lx @ Ly ) @ ( times_times @ A @ Rx @ Ry ) )
= ( times_times @ A @ Lx @ ( times_times @ A @ Ly @ ( times_times @ A @ Rx @ Ry ) ) ) ) ) ).
% semiring_normalization_rules(14)
thf(fact_207_semiring__normalization__rules_I15_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [Lx: A,Ly: A,Rx: A,Ry: A] :
( ( times_times @ A @ ( times_times @ A @ Lx @ Ly ) @ ( times_times @ A @ Rx @ Ry ) )
= ( times_times @ A @ Rx @ ( times_times @ A @ ( times_times @ A @ Lx @ Ly ) @ Ry ) ) ) ) ).
% semiring_normalization_rules(15)
thf(fact_208_semiring__normalization__rules_I16_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [Lx: A,Ly: A,Rx: A] :
( ( times_times @ A @ ( times_times @ A @ Lx @ Ly ) @ Rx )
= ( times_times @ A @ ( times_times @ A @ Lx @ Rx ) @ Ly ) ) ) ).
% semiring_normalization_rules(16)
thf(fact_209_semiring__normalization__rules_I17_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [Lx: A,Ly: A,Rx: A] :
( ( times_times @ A @ ( times_times @ A @ Lx @ Ly ) @ Rx )
= ( times_times @ A @ Lx @ ( times_times @ A @ Ly @ Rx ) ) ) ) ).
% semiring_normalization_rules(17)
thf(fact_210_semiring__normalization__rules_I18_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [Lx: A,Rx: A,Ry: A] :
( ( times_times @ A @ Lx @ ( times_times @ A @ Rx @ Ry ) )
= ( times_times @ A @ ( times_times @ A @ Lx @ Rx ) @ Ry ) ) ) ).
% semiring_normalization_rules(18)
thf(fact_211_semiring__normalization__rules_I19_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [Lx: A,Rx: A,Ry: A] :
( ( times_times @ A @ Lx @ ( times_times @ A @ Rx @ Ry ) )
= ( times_times @ A @ Rx @ ( times_times @ A @ Lx @ Ry ) ) ) ) ).
% semiring_normalization_rules(19)
thf(fact_212_semiring__normalization__rules_I20_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ C2 @ D ) )
= ( plus_plus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D ) ) ) ) ).
% semiring_normalization_rules(20)
thf(fact_213_semiring__normalization__rules_I21_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% semiring_normalization_rules(21)
thf(fact_214_semiring__normalization__rules_I22_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A,C2: A,D: A] :
( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ C2 @ D ) )
= ( plus_plus @ A @ C2 @ ( plus_plus @ A @ A2 @ D ) ) ) ) ).
% semiring_normalization_rules(22)
thf(fact_215_semiring__normalization__rules_I23_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% semiring_normalization_rules(23)
thf(fact_216_semiring__normalization__rules_I24_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ( ( plus_plus @ A )
= ( ^ [A3: A,C3: A] : ( plus_plus @ A @ C3 @ A3 ) ) ) ) ).
% semiring_normalization_rules(24)
thf(fact_217_semiring__normalization__rules_I25_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A,C2: A,D: A] :
( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ C2 @ D ) )
= ( plus_plus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ D ) ) ) ).
% semiring_normalization_rules(25)
thf(fact_218_semiring__normalization__rules_I9_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( times_times @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% semiring_normalization_rules(9)
thf(fact_219_semiring__normalization__rules_I10_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% semiring_normalization_rules(10)
thf(fact_220_add__0__iff,axiom,
! [A: $tType] :
( ( semiri456707255roduct @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( B2
= ( plus_plus @ A @ B2 @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% add_0_iff
thf(fact_221_semiring__normalization__rules_I5_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% semiring_normalization_rules(5)
thf(fact_222_semiring__normalization__rules_I6_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% semiring_normalization_rules(6)
thf(fact_223_crossproduct__noteq,axiom,
! [A: $tType] :
( ( semiri456707255roduct @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( ( A2 != B2 )
& ( C2 != D ) )
= ( ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D ) )
!= ( plus_plus @ A @ ( times_times @ A @ A2 @ D ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% crossproduct_noteq
thf(fact_224_crossproduct__eq,axiom,
! [A: $tType] :
( ( semiri456707255roduct @ A @ ( type2 @ A ) )
=> ! [W: A,Y: A,X: A,Z: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ W @ Y ) @ ( times_times @ A @ X @ Z ) )
= ( plus_plus @ A @ ( times_times @ A @ W @ Z ) @ ( times_times @ A @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z ) ) ) ) ).
% crossproduct_eq
thf(fact_225_semiring__normalization__rules_I1_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A,M: A,B2: A] :
( ( plus_plus @ A @ ( times_times @ A @ A2 @ M ) @ ( times_times @ A @ B2 @ M ) )
= ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ M ) ) ) ).
% semiring_normalization_rules(1)
thf(fact_226_semiring__normalization__rules_I8_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% semiring_normalization_rules(8)
thf(fact_227_semiring__normalization__rules_I34_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( times_times @ A @ X @ ( plus_plus @ A @ Y @ Z ) )
= ( plus_plus @ A @ ( times_times @ A @ X @ Y ) @ ( times_times @ A @ X @ Z ) ) ) ) ).
% semiring_normalization_rules(34)
thf(fact_228_semiring__normalization__rules_I12_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% semiring_normalization_rules(12)
thf(fact_229_semiring__normalization__rules_I11_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( times_times @ A @ ( one_one @ A ) @ A2 )
= A2 ) ) ).
% semiring_normalization_rules(11)
thf(fact_230_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ Z ) @ Z )
!= ( zero_zero @ int ) ) ).
% odd_nonzero
thf(fact_231_compl__eq__compl__iff,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ( uminus_uminus @ A @ X )
= ( uminus_uminus @ A @ Y ) )
= ( X = Y ) ) ) ).
% compl_eq_compl_iff
thf(fact_232_uminus__apply,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B @ ( type2 @ B ) )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A4: A > B,X3: A] : ( uminus_uminus @ B @ ( A4 @ X3 ) ) ) ) ) ).
% uminus_apply
thf(fact_233_double__compl,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X ) )
= X ) ) ).
% double_compl
thf(fact_234_fun__Compl__def,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B @ ( type2 @ B ) )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A4: A > B,X3: A] : ( uminus_uminus @ B @ ( A4 @ X3 ) ) ) ) ) ).
% fun_Compl_def
thf(fact_235_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times @ int @ ( zero_zero @ int ) @ L )
= ( zero_zero @ int ) ) ).
% times_int_code(2)
thf(fact_236_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times @ int @ K @ ( zero_zero @ int ) )
= ( zero_zero @ int ) ) ).
% times_int_code(1)
thf(fact_237_abs__zmult__eq__1,axiom,
! [M: int,N: int] :
( ( ( abs_abs @ int @ ( times_times @ int @ M @ N ) )
= ( one_one @ int ) )
=> ( ( abs_abs @ int @ M )
= ( one_one @ int ) ) ) ).
% abs_zmult_eq_1
thf(fact_238_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_239_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_240_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus @ int @ K @ ( zero_zero @ int ) )
= K ) ).
% plus_int_code(1)
thf(fact_241_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus @ int @ ( zero_zero @ int ) @ L )
= L ) ).
% plus_int_code(2)
thf(fact_242_uminus__int__code_I1_J,axiom,
( ( uminus_uminus @ int @ ( zero_zero @ int ) )
= ( zero_zero @ int ) ) ).
% uminus_int_code(1)
thf(fact_243_zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ( times_times @ int @ M @ N )
= ( one_one @ int ) )
= ( ( ( M
= ( one_one @ int ) )
& ( N
= ( one_one @ int ) ) )
| ( ( M
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
& ( N
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_244_pos__zmult__eq__1__iff__lemma,axiom,
! [M: int,N: int] :
( ( ( times_times @ int @ M @ N )
= ( one_one @ int ) )
=> ( ( M
= ( one_one @ int ) )
| ( M
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_245_linordered__field__class_Osign__simps_I36_J,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ).
% linordered_field_class.sign_simps(36)
thf(fact_246_linordered__field__class_Osign__simps_I35_J,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% linordered_field_class.sign_simps(35)
thf(fact_247_linordered__field__class_Osign__simps_I23_J,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
= ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% linordered_field_class.sign_simps(23)
thf(fact_248_linordered__field__class_Osign__simps_I24_J,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ( ( times_times @ A )
= ( ^ [A3: A,B3: A] : ( times_times @ A @ B3 @ A3 ) ) ) ) ).
% linordered_field_class.sign_simps(24)
thf(fact_249_linordered__field__class_Osign__simps_I25_J,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C2: A] :
( ( times_times @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) )
= ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% linordered_field_class.sign_simps(25)
thf(fact_250_linordered__field__class_Osign__simps_I26_J,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% linordered_field_class.sign_simps(26)
thf(fact_251_linordered__field__class_Osign__simps_I27_J,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ( ( plus_plus @ A )
= ( ^ [A3: A,B3: A] : ( plus_plus @ A @ B3 @ A3 ) ) ) ) ).
% linordered_field_class.sign_simps(27)
thf(fact_252_linordered__field__class_Osign__simps_I28_J,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C2: A] :
( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% linordered_field_class.sign_simps(28)
thf(fact_253_powr__mult__base,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( times_times @ real @ X @ ( powr @ real @ X @ Y ) )
= ( powr @ real @ X @ ( plus_plus @ real @ ( one_one @ real ) @ Y ) ) ) ) ).
% powr_mult_base
%----Subclasses (1)
thf(subcl_Orderings_Olinorder___HOL_Otype,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ( type @ A @ ( type2 @ A ) ) ) ).
%----Type constructors (87)
thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
! [A5: $tType,A6: $tType] :
( ( boolean_algebra @ A6 @ ( type2 @ A6 ) )
=> ( boolean_algebra @ ( A5 > A6 ) @ ( type2 @ ( A5 > A6 ) ) ) ) ).
thf(tcon_fun___Groups_Ouminus,axiom,
! [A5: $tType,A6: $tType] :
( ( uminus @ A6 @ ( type2 @ A6 ) )
=> ( uminus @ ( A5 > A6 ) @ ( type2 @ ( A5 > A6 ) ) ) ) ).
thf(tcon_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
semiri456707255roduct @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors__cancel,axiom,
semiri1923998003cancel @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors,axiom,
semiri1193490041visors @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Groups_Oordered__ab__group__add__abs,axiom,
ordere142940540dd_abs @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
linord219039673up_add @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Rings_Oring__1__no__zero__divisors,axiom,
ring_11004092258visors @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
cancel1352612707id_add @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Rings_Olinordered__ring__strict,axiom,
linord581940658strict @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Rings_Olinordered__ring,axiom,
linordered_ring @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1 @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Groups_Osemigroup__mult,axiom,
semigroup_mult @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
semigroup_add @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring,axiom,
comm_semiring @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Groups_Oab__group__add,axiom,
ab_group_add @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Rings_Ozero__neq__one,axiom,
zero_neq_one @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Groups_Omonoid__mult,axiom,
monoid_mult @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Rings_Ocomm__ring__1,axiom,
comm_ring_1 @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Groups_Omonoid__add,axiom,
monoid_add @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
group_add @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Rings_Omult__zero,axiom,
mult_zero @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Num_Oneg__numeral,axiom,
neg_numeral @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Nat_Oring__char__0,axiom,
ring_char_0 @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Rings_Osemiring,axiom,
semiring @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Groups_Ouminus_1,axiom,
uminus @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Groups_Osgn__if,axiom,
sgn_if @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Rings_Oring__1,axiom,
ring_1 @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Groups_Ozero,axiom,
zero @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Rings_Oring,axiom,
ring @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Rings_Oidom,axiom,
idom @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Groups_Oone,axiom,
one @ int @ ( type2 @ int ) ).
thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_2,axiom,
boolean_algebra @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Groups_Ouminus_3,axiom,
uminus @ $o @ ( type2 @ $o ) ).
thf(tcon_Real_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_4,axiom,
semiri456707255roduct @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__div__algebra,axiom,
real_V68988228lgebra @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__algebra__1,axiom,
real_V1229719638ebra_1 @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors__cancel_5,axiom,
semiri1923998003cancel @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__vector,axiom,
real_V55928688vector @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__algebra__1,axiom,
real_V84468443ebra_1 @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_6,axiom,
semiri1193490041visors @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__algebra,axiom,
real_V148923926lgebra @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add_7,axiom,
ordere779506340up_add @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_8,axiom,
ordere142940540dd_abs @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling,axiom,
archim1727834104eiling @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__vector,axiom,
real_V1076094709vector @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_9,axiom,
linord219039673up_add @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors_10,axiom,
ring_11004092258visors @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_11,axiom,
cancel1352612707id_add @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Rings_Olinordered__ring__strict_12,axiom,
linord581940658strict @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_13,axiom,
cancel_semigroup_add @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Oab__semigroup__mult_14,axiom,
ab_semigroup_mult @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_15,axiom,
comm_monoid_mult @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Oab__semigroup__add_16,axiom,
ab_semigroup_add @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Fields_Olinordered__field,axiom,
linordered_field @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Ocomm__monoid__add_17,axiom,
comm_monoid_add @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Rings_Olinordered__ring_18,axiom,
linordered_ring @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Rings_Olinordered__idom_19,axiom,
linordered_idom @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1_20,axiom,
comm_semiring_1 @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Osemigroup__mult_21,axiom,
semigroup_mult @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Osemigroup__add_22,axiom,
semigroup_add @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring_23,axiom,
comm_semiring @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Oab__group__add_24,axiom,
ab_group_add @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Rings_Ozero__neq__one_25,axiom,
zero_neq_one @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Omonoid__mult_26,axiom,
monoid_mult @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Transcendental_Oln,axiom,
ln @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Rings_Ocomm__ring__1_27,axiom,
comm_ring_1 @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Omonoid__add_28,axiom,
monoid_add @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Ogroup__add_29,axiom,
group_add @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Rings_Omult__zero_30,axiom,
mult_zero @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Num_Oneg__numeral_31,axiom,
neg_numeral @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Nat_Oring__char__0_32,axiom,
ring_char_0 @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Rings_Osemiring_33,axiom,
semiring @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Ouminus_34,axiom,
uminus @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Osgn__if_35,axiom,
sgn_if @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Rings_Oring__1_36,axiom,
ring_1 @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Ozero_37,axiom,
zero @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Rings_Oring_38,axiom,
ring @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Rings_Oidom_39,axiom,
idom @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Oone_40,axiom,
one @ real @ ( type2 @ real ) ).
%----Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $true @ X @ Y )
= X ) ).
%----Free types (1)
thf(tfree_0,hypothesis,
linorder @ a @ ( type2 @ a ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( ( splay_266122055elle_A @ a @ a2 @ ( node @ a @ l @ a2 @ r ) )
= ( one_one @ real ) ) ).
%------------------------------------------------------------------------------