TPTP Problem File: DAT214^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : DAT214^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Splay tree analysis 231
% 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__231.p [Bla16]
% Status : Theorem
% Rating : 0.00 v7.1.0
% Syntax : Number of formulae : 416 ( 146 unt; 80 typ; 0 def)
% Number of atoms : 808 ( 320 equ; 0 cnn)
% Maximal formula atoms : 19 ( 2 avg)
% Number of connectives : 3902 ( 88 ~; 4 |; 24 &;3425 @)
% ( 0 <=>; 361 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 6 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 136 ( 136 >; 0 *; 0 +; 0 <<)
% Number of symbols : 79 ( 76 usr; 10 con; 0-5 aty)
% Number of variables : 852 ( 15 ^; 759 !; 17 ?; 852 :)
% ( 61 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:50:17.933
%------------------------------------------------------------------------------
%----Could-be-implicit typings (9)
thf(ty_t_Splay__Tree__Analysis__Base_Oop_092_060_094sub_062s_092_060_094sub_062t,type,
splay_2011811431op_s_t: $tType > $tType ).
thf(ty_t_Code__Numeral_Onatural,type,
code_natural: $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Sum__Type_Osum,type,
sum_sum: $tType > $tType > $tType ).
thf(ty_t_Tree_Otree,type,
tree: $tType > $tType ).
thf(ty_t_Real_Oreal,type,
real: $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (71)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Nat_Osize,type,
size:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Num_Oneg__numeral,type,
neg_numeral:
!>[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_Transcendental_Oln,type,
ln:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Ozero__less__one,type,
zero_less_one:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Ocomm__semiring__1,type,
comm_semiring_1:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[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_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1352612707id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
ordere216010020id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere779506340up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_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_Ostrict__ordered__comm__monoid__add,type,
strict797366125id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere236663937imp_le:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
ordere223160158up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
strict2144017051up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
ordere516151231imp_le:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
semiri456707255roduct:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Code__Numeral_Onatural_Ocase__natural,type,
code_case_natural:
!>[T: $tType] : ( T > ( code_natural > T ) > code_natural > T ) ).
thf(sy_c_Code__Numeral_Onatural_Orec__natural,type,
code_rec_natural:
!>[T: $tType] : ( T > ( code_natural > T > T ) > code_natural > T ) ).
thf(sy_c_Code__Numeral_Onatural_Osize__natural,type,
code_size_natural: code_natural > nat ).
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_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_HOL_Obool_Osize__bool,type,
size_bool: $o > nat ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
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_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Product__Type_Oold_Obool_Orec__bool,type,
product_rec_bool:
!>[T: $tType] : ( T > T > $o > T ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat,type,
set_fo292404081st_nat:
!>[A: $tType] : ( ( nat > A > A ) > nat > nat > A > A ) ).
thf(sy_c_Splay__Tree_Osplay,type,
splay_splay:
!>[A: $tType] : ( A > ( tree @ A ) > ( tree @ A ) ) ).
thf(sy_c_Splay__Tree_Osplay__max,type,
splay_splay_max:
!>[A: $tType] : ( ( tree @ A ) > ( tree @ A ) ) ).
thf(sy_c_Splay__Tree__Analysis__Base_Ot__splay,type,
splay_914434265_splay:
!>[A: $tType] : ( A > ( tree @ A ) > nat ) ).
thf(sy_c_Splay__Tree__Analysis__Base_Ot__splay__max,type,
splay_878424299ay_max:
!>[A: $tType] : ( ( tree @ A ) > nat ) ).
thf(sy_c_Splay__Tree__Analysis__Base_Ot__splay__max__rel,type,
splay_1816415694ax_rel:
!>[A: $tType] : ( ( tree @ A ) > ( tree @ A ) > $o ) ).
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_Olinorder__class_Obst,type,
linorder_bst:
!>[A: $tType] : ( ( tree @ A ) > $o ) ).
thf(sy_c_Tree_Opath__len,type,
path_len:
!>[A: $tType] : ( ( tree @ A ) > nat ) ).
thf(sy_c_Tree_Opath__len__rel,type,
path_len_rel:
!>[A: $tType] : ( ( tree @ A ) > ( tree @ A ) > $o ) ).
thf(sy_c_Tree_Osize1,type,
size1:
!>[A: $tType] : ( ( tree @ A ) > nat ) ).
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_Tree_Otree_Osize__tree,type,
size_tree:
!>[A: $tType] : ( ( A > nat ) > ( tree @ A ) > nat ) ).
thf(sy_c_Wellfounded_Oaccp,type,
accp:
!>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
thf(sy_v_b____,type,
b: a ).
thf(sy_v_c____,type,
c: a ).
thf(sy_v_l_H____,type,
l: tree @ a ).
thf(sy_v_l____,type,
l2: tree @ a ).
thf(sy_v_r_H____,type,
r: tree @ a ).
thf(sy_v_rl____,type,
rl: tree @ a ).
thf(sy_v_rr____,type,
rr: tree @ a ).
thf(sy_v_u____,type,
u: a ).
%----Relevant facts (253)
thf(fact_0__092_060open_062size_Arr_A_061_Asize_A_Isplay__max_Arr_J_092_060close_062,axiom,
( ( size_size @ ( tree @ a ) @ rr )
= ( size_size @ ( tree @ a ) @ ( splay_splay_max @ a @ rr ) ) ) ).
% \<open>size rr = size (splay_max rr)\<close>
thf(fact_1_sp,axiom,
( ( splay_splay_max @ a @ rr )
= ( node @ a @ l @ u @ r ) ) ).
% sp
thf(fact_2__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062l_H_Au_Ar_H_O_Asplay__max_Arr_A_061_A_092_060langle_062l_H_M_Au_M_Ar_H_092_060rangle_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [L: tree @ a,U: a,R: tree @ a] :
( ( splay_splay_max @ a @ rr )
!= ( node @ a @ L @ U @ R ) ) ).
% \<open>\<And>thesis. (\<And>l' u r'. splay_max rr = \<langle>l', u, r'\<rangle> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_3__092_060open_062rr_A_092_060noteq_062_A_092_060langle_062_092_060rangle_062_092_060close_062,axiom,
( rr
!= ( leaf @ a ) ) ).
% \<open>rr \<noteq> \<langle>\<rangle>\<close>
thf(fact_4_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( ( plus_plus @ A @ A2 @ B )
= ( plus_plus @ A @ A2 @ C ) )
= ( B = C ) ) ) ).
% add_left_cancel
thf(fact_5_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A,C: A] :
( ( ( plus_plus @ A @ B @ A2 )
= ( plus_plus @ A @ C @ A2 ) )
= ( B = C ) ) ) ).
% add_right_cancel
thf(fact_6_size__if__splay__max,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: tree @ A,L2: tree @ A,U2: A,R2: tree @ A] :
( ( ( splay_splay_max @ A @ T2 )
= ( node @ A @ L2 @ U2 @ R2 ) )
=> ( ( size_size @ ( tree @ A ) @ T2 )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( size_size @ ( tree @ A ) @ L2 ) @ ( size_size @ ( tree @ A ) @ R2 ) ) @ ( one_one @ nat ) ) ) ) ) ).
% size_if_splay_max
thf(fact_7_one__natural_Orsp,axiom,
( ( one_one @ nat )
= ( one_one @ nat ) ) ).
% one_natural.rsp
thf(fact_8_size__ne__size__imp__ne,axiom,
! [A: $tType] :
( ( size @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ( size_size @ A @ X )
!= ( size_size @ A @ Y ) )
=> ( X != Y ) ) ) ).
% size_ne_size_imp_ne
thf(fact_9_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_10_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_11_one__reorient,axiom,
! [A: $tType] :
( ( one @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( ( one_one @ A )
= X )
= ( X
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_12_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type2 @ 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_13_is__num__normalize_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A @ ( type2 @ 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_14_linordered__field__class_Osign__simps_I28_J,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ 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 ) ) ) ) ).
% linordered_field_class.sign_simps(28)
thf(fact_15_splay__max__Leaf__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: tree @ A] :
( ( ( splay_splay_max @ A @ T2 )
= ( leaf @ A ) )
= ( T2
= ( leaf @ A ) ) ) ) ).
% splay_max_Leaf_iff
thf(fact_16_splay__max_Osimps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [La: tree @ A,A2: A] :
( ( splay_splay_max @ A @ ( node @ A @ La @ A2 @ ( leaf @ A ) ) )
= ( node @ A @ La @ A2 @ ( leaf @ A ) ) ) ) ).
% splay_max.simps(2)
thf(fact_17_splay__max_Osimps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ( ( splay_splay_max @ A @ ( leaf @ A ) )
= ( leaf @ A ) ) ) ).
% splay_max.simps(1)
thf(fact_18_splay__max__Leaf,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: tree @ A,L2: tree @ A,A2: A,R2: tree @ A] :
( ( ( splay_splay_max @ A @ T2 )
= ( node @ A @ L2 @ A2 @ R2 ) )
=> ( R2
= ( leaf @ A ) ) ) ) ).
% splay_max_Leaf
thf(fact_19_size__splay__max,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: tree @ A] :
( ( size_size @ ( tree @ A ) @ ( splay_splay_max @ A @ T2 ) )
= ( size_size @ ( tree @ A ) @ T2 ) ) ) ).
% size_splay_max
thf(fact_20_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ 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_21_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ 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_22_add_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type2 @ 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_23_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type2 @ A ) )
=> ( ( plus_plus @ A )
= ( ^ [A3: A,B2: A] : ( plus_plus @ A @ B2 @ A3 ) ) ) ) ).
% add.commute
thf(fact_24_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A,C: A] :
( ( ( plus_plus @ A @ B @ A2 )
= ( plus_plus @ A @ C @ A2 ) )
= ( B = C ) ) ) ).
% add.right_cancel
thf(fact_25_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( ( plus_plus @ A @ A2 @ B )
= ( plus_plus @ A @ A2 @ C ) )
= ( B = C ) ) ) ).
% add.left_cancel
thf(fact_26_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A @ ( type2 @ 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_27_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ 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_28_linordered__field__class_Osign__simps_I26_J,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ 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 ) ) ) ) ).
% linordered_field_class.sign_simps(26)
thf(fact_29_linordered__field__class_Osign__simps_I27_J,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ( ( plus_plus @ A )
= ( ^ [A3: A,B2: A] : ( plus_plus @ A @ B2 @ A3 ) ) ) ) ).
% linordered_field_class.sign_simps(27)
thf(fact_30__C3_Oprems_C_I2_J,axiom,
( ( node @ a @ l2 @ b @ ( node @ a @ rl @ c @ rr ) )
!= ( leaf @ a ) ) ).
% "3.prems"(2)
thf(fact_31_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_32_tree_Odistinct_I1_J,axiom,
! [A: $tType,X21: tree @ A,X22: A,X23: tree @ A] :
( ( leaf @ A )
!= ( node @ A @ X21 @ X22 @ X23 ) ) ).
% tree.distinct(1)
thf(fact_33_tree_Oinduct,axiom,
! [A: $tType,P: ( tree @ A ) > $o,Tree: tree @ A] :
( ( P @ ( leaf @ A ) )
=> ( ! [X1: tree @ A,X2: A,X3: tree @ A] :
( ( P @ X1 )
=> ( ( P @ X3 )
=> ( P @ ( node @ A @ X1 @ X2 @ X3 ) ) ) )
=> ( P @ Tree ) ) ) ).
% tree.induct
thf(fact_34_neq__Leaf__iff,axiom,
! [A: $tType,T2: tree @ A] :
( ( T2
!= ( leaf @ A ) )
= ( ? [L3: tree @ A,A3: A,R3: tree @ A] :
( T2
= ( node @ A @ L3 @ A3 @ R3 ) ) ) ) ).
% neq_Leaf_iff
thf(fact_35_tree_Oexhaust,axiom,
! [A: $tType,Y: tree @ A] :
( ( Y
!= ( leaf @ A ) )
=> ~ ! [X212: tree @ A,X222: A,X232: tree @ A] :
( Y
!= ( node @ A @ X212 @ X222 @ X232 ) ) ) ).
% tree.exhaust
thf(fact_36_mirror_Oinduct,axiom,
! [A: $tType,P: ( tree @ A ) > $o,A0: tree @ A] :
( ( P @ ( leaf @ A ) )
=> ( ! [L4: tree @ A,X4: A,R4: tree @ A] :
( ( P @ R4 )
=> ( ( P @ L4 )
=> ( P @ ( node @ A @ L4 @ X4 @ R4 ) ) ) )
=> ( P @ A0 ) ) ) ).
% mirror.induct
thf(fact_37_bst__eq_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: tree @ A] :
( ( X
!= ( leaf @ A ) )
=> ~ ! [L4: tree @ A,A4: A,R4: tree @ A] :
( X
!= ( node @ A @ L4 @ A4 @ R4 ) ) ) ) ).
% bst_eq.cases
thf(fact_38_bst__eq_Oinduct,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: ( tree @ A ) > $o,A0: tree @ A] :
( ( P @ ( leaf @ A ) )
=> ( ! [L4: tree @ A,A4: A,R4: tree @ A] :
( ( P @ L4 )
=> ( ( P @ R4 )
=> ( P @ ( node @ A @ L4 @ A4 @ R4 ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% bst_eq.induct
thf(fact_39__092_060Phi_062_Ocases,axiom,
! [A: $tType,X: tree @ A] :
( ( X
!= ( leaf @ A ) )
=> ~ ! [L4: tree @ A,A4: A,R4: tree @ A] :
( X
!= ( node @ A @ L4 @ A4 @ R4 ) ) ) ).
% \<Phi>.cases
thf(fact_40__C3_Oprems_C_I1_J,axiom,
linorder_bst @ a @ ( node @ a @ l2 @ b @ ( node @ a @ rl @ c @ rr ) ) ).
% "3.prems"(1)
thf(fact_41_t__splay__max_Oinduct,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: ( tree @ A ) > $o,A0: tree @ A] :
( ( P @ ( leaf @ A ) )
=> ( ! [L4: tree @ A,B3: A] : ( P @ ( node @ A @ L4 @ B3 @ ( leaf @ A ) ) )
=> ( ! [L4: tree @ A,B3: A,Rl: tree @ A,C2: A,Rr: tree @ A] :
( ( ( Rr
!= ( leaf @ A ) )
=> ( P @ Rr ) )
=> ( P @ ( node @ A @ L4 @ B3 @ ( node @ A @ Rl @ C2 @ Rr ) ) ) )
=> ( P @ A0 ) ) ) ) ) ).
% t_splay_max.induct
thf(fact_42_t__splay__max_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: tree @ A] :
( ( X
!= ( leaf @ A ) )
=> ( ! [L4: tree @ A,B3: A] :
( X
!= ( node @ A @ L4 @ B3 @ ( leaf @ A ) ) )
=> ~ ! [L4: tree @ A,B3: A,Rl: tree @ A,C2: A,Rr: tree @ A] :
( X
!= ( node @ A @ L4 @ B3 @ ( node @ A @ Rl @ C2 @ Rr ) ) ) ) ) ) ).
% t_splay_max.cases
thf(fact_43__092_060Phi_062_Oinduct,axiom,
! [A: $tType,P: ( tree @ A ) > $o,A0: tree @ A] :
( ( P @ ( leaf @ A ) )
=> ( ! [L4: tree @ A,A4: A,R4: tree @ A] :
( ( P @ L4 )
=> ( ( P @ R4 )
=> ( P @ ( node @ A @ L4 @ A4 @ R4 ) ) ) )
=> ( P @ A0 ) ) ) ).
% \<Phi>.induct
thf(fact_44_ext,axiom,
! [B4: $tType,A: $tType,F: A > B4,G: A > B4] :
( ! [X4: A] :
( ( F @ X4 )
= ( G @ X4 ) )
=> ( F = G ) ) ).
% ext
thf(fact_45_t__splay__max_Osimps_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Rr2: tree @ A,L2: tree @ A,B: A,Rl2: tree @ A,C: A] :
( ( ( Rr2
= ( leaf @ A ) )
=> ( ( splay_878424299ay_max @ A @ ( node @ A @ L2 @ B @ ( node @ A @ Rl2 @ C @ Rr2 ) ) )
= ( one_one @ nat ) ) )
& ( ( Rr2
!= ( leaf @ A ) )
=> ( ( splay_878424299ay_max @ A @ ( node @ A @ L2 @ B @ ( node @ A @ Rl2 @ C @ Rr2 ) ) )
= ( plus_plus @ nat @ ( splay_878424299ay_max @ A @ Rr2 ) @ ( one_one @ nat ) ) ) ) ) ) ).
% t_splay_max.simps(3)
thf(fact_46_t__splay__max_Oelims,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: tree @ A,Y: nat] :
( ( ( splay_878424299ay_max @ A @ X )
= Y )
=> ( ( ( X
= ( leaf @ A ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ( ( ? [L4: tree @ A,B3: A] :
( X
= ( node @ A @ L4 @ B3 @ ( leaf @ A ) ) )
=> ( Y
!= ( one_one @ nat ) ) )
=> ~ ! [L4: tree @ A,B3: A,Rl: tree @ A,C2: A,Rr: tree @ A] :
( ( X
= ( node @ A @ L4 @ B3 @ ( node @ A @ Rl @ C2 @ Rr ) ) )
=> ~ ( ( ( Rr
= ( leaf @ A ) )
=> ( Y
= ( one_one @ nat ) ) )
& ( ( Rr
!= ( leaf @ A ) )
=> ( Y
= ( plus_plus @ nat @ ( splay_878424299ay_max @ A @ Rr ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ).
% t_splay_max.elims
thf(fact_47_path__len_Osimps_I2_J,axiom,
! [A: $tType,L2: tree @ A,Uu: A,R2: tree @ A] :
( ( path_len @ A @ ( node @ A @ L2 @ Uu @ R2 ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( path_len @ A @ L2 ) @ ( size_size @ ( tree @ A ) @ L2 ) ) @ ( path_len @ A @ R2 ) ) @ ( size_size @ ( tree @ A ) @ R2 ) ) ) ).
% path_len.simps(2)
thf(fact_48_t__splay__max_Osimps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [L2: tree @ A,B: A] :
( ( splay_878424299ay_max @ A @ ( node @ A @ L2 @ B @ ( leaf @ A ) ) )
= ( one_one @ nat ) ) ) ).
% t_splay_max.simps(2)
thf(fact_49_size__if__splay,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: A,T2: tree @ A,L2: tree @ A,U2: A,R2: tree @ A] :
( ( ( splay_splay @ A @ A2 @ T2 )
= ( node @ A @ L2 @ U2 @ R2 ) )
=> ( ( size_size @ ( tree @ A ) @ T2 )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( size_size @ ( tree @ A ) @ L2 ) @ ( size_size @ ( tree @ A ) @ R2 ) ) @ ( one_one @ nat ) ) ) ) ) ).
% size_if_splay
thf(fact_50_size1__def,axiom,
! [A: $tType] :
( ( size1 @ A )
= ( ^ [T3: tree @ A] : ( plus_plus @ nat @ ( size_size @ ( tree @ A ) @ T3 ) @ ( one_one @ nat ) ) ) ) ).
% size1_def
thf(fact_51_dbl__inc__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A @ ( type2 @ A ) )
=> ( ( neg_numeral_dbl_inc @ A )
= ( ^ [X5: A] : ( plus_plus @ A @ ( plus_plus @ A @ X5 @ X5 ) @ ( one_one @ A ) ) ) ) ) ).
% dbl_inc_def
thf(fact_52_t__splay__max_Osimps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ( ( splay_878424299ay_max @ A @ ( leaf @ A ) )
= ( one_one @ nat ) ) ) ).
% t_splay_max.simps(1)
thf(fact_53_splay__Leaf__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: A,T2: tree @ A] :
( ( ( splay_splay @ A @ A2 @ T2 )
= ( leaf @ A ) )
= ( T2
= ( leaf @ A ) ) ) ) ).
% splay_Leaf_iff
thf(fact_54_size__splay,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: A,T2: tree @ A] :
( ( size_size @ ( tree @ A ) @ ( splay_splay @ A @ A2 @ T2 ) )
= ( size_size @ ( tree @ A ) @ T2 ) ) ) ).
% size_splay
thf(fact_55_size1__simps_I2_J,axiom,
! [B4: $tType,L2: tree @ B4,X: B4,R2: tree @ B4] :
( ( size1 @ B4 @ ( node @ B4 @ L2 @ X @ R2 ) )
= ( plus_plus @ nat @ ( size1 @ B4 @ L2 ) @ ( size1 @ B4 @ R2 ) ) ) ).
% size1_simps(2)
thf(fact_56_size1__simps_I1_J,axiom,
! [A: $tType] :
( ( size1 @ A @ ( leaf @ A ) )
= ( one_one @ nat ) ) ).
% size1_simps(1)
thf(fact_57_bst__splay,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: tree @ A,A2: A] :
( ( linorder_bst @ A @ T2 )
=> ( linorder_bst @ A @ ( splay_splay @ A @ A2 @ T2 ) ) ) ) ).
% bst_splay
thf(fact_58_splay__max__eq__splay__ex,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: tree @ A] :
( ( linorder_bst @ A @ T2 )
=> ? [A4: A] :
( ( splay_splay_max @ A @ T2 )
= ( splay_splay @ A @ A4 @ T2 ) ) ) ) ).
% splay_max_eq_splay_ex
thf(fact_59_splay_Osimps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: A,L2: tree @ A,R2: tree @ A] :
( ( splay_splay @ A @ A2 @ ( node @ A @ L2 @ A2 @ R2 ) )
= ( node @ A @ L2 @ A2 @ R2 ) ) ) ).
% splay.simps(2)
thf(fact_60_splay_Osimps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( splay_splay @ A @ A2 @ ( leaf @ A ) )
= ( leaf @ A ) ) ) ).
% splay.simps(1)
thf(fact_61_bst_Osimps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ( linorder_bst @ A @ ( leaf @ A ) ) ) ).
% bst.simps(1)
thf(fact_62_bst__splay__max,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: tree @ A] :
( ( linorder_bst @ A @ T2 )
=> ( linorder_bst @ A @ ( splay_splay_max @ A @ T2 ) ) ) ) ).
% bst_splay_max
thf(fact_63_splay__not__Leaf,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: tree @ A,A2: A] :
( ( T2
!= ( leaf @ A ) )
=> ? [L4: tree @ A,X4: A,R4: tree @ A] :
( ( splay_splay @ A @ A2 @ T2 )
= ( node @ A @ L4 @ X4 @ R4 ) ) ) ) ).
% splay_not_Leaf
thf(fact_64_path__len_Oelims,axiom,
! [A: $tType,X: tree @ A,Y: nat] :
( ( ( path_len @ A @ X )
= Y )
=> ( ( ( X
= ( leaf @ A ) )
=> ( Y
!= ( zero_zero @ nat ) ) )
=> ~ ! [L4: tree @ A,Uu2: A,R4: tree @ A] :
( ( X
= ( node @ A @ L4 @ Uu2 @ R4 ) )
=> ( Y
!= ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( path_len @ A @ L4 ) @ ( size_size @ ( tree @ A ) @ L4 ) ) @ ( path_len @ A @ R4 ) ) @ ( size_size @ ( tree @ A ) @ R4 ) ) ) ) ) ) ).
% path_len.elims
thf(fact_65_A__simps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: A,L2: tree @ A,R2: tree @ A] :
( ( splay_266122055elle_A @ A @ A2 @ ( node @ A @ L2 @ A2 @ R2 ) )
= ( one_one @ real ) ) ) ).
% A_simps(1)
thf(fact_66_t__splay__max_Opelims,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: tree @ A,Y: nat] :
( ( ( splay_878424299ay_max @ A @ X )
= Y )
=> ( ( accp @ ( tree @ A ) @ ( splay_1816415694ax_rel @ A ) @ X )
=> ( ( ( X
= ( leaf @ A ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( tree @ A ) @ ( splay_1816415694ax_rel @ A ) @ ( leaf @ A ) ) ) )
=> ( ! [L4: tree @ A,B3: A] :
( ( X
= ( node @ A @ L4 @ B3 @ ( leaf @ A ) ) )
=> ( ( Y
= ( one_one @ nat ) )
=> ~ ( accp @ ( tree @ A ) @ ( splay_1816415694ax_rel @ A ) @ ( node @ A @ L4 @ B3 @ ( leaf @ A ) ) ) ) )
=> ~ ! [L4: tree @ A,B3: A,Rl: tree @ A,C2: A,Rr: tree @ A] :
( ( X
= ( node @ A @ L4 @ B3 @ ( node @ A @ Rl @ C2 @ Rr ) ) )
=> ( ( ( ( Rr
= ( leaf @ A ) )
=> ( Y
= ( one_one @ nat ) ) )
& ( ( Rr
!= ( leaf @ A ) )
=> ( Y
= ( plus_plus @ nat @ ( splay_878424299ay_max @ A @ Rr ) @ ( one_one @ nat ) ) ) ) )
=> ~ ( accp @ ( tree @ A ) @ ( splay_1816415694ax_rel @ A ) @ ( node @ A @ L4 @ B3 @ ( node @ A @ Rl @ C2 @ Rr ) ) ) ) ) ) ) ) ) ) ).
% t_splay_max.pelims
thf(fact_67_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_68_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_69_t__splay__simps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ 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_70_splay_Osimps_I13_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: A,X: A,B: A,La: tree @ A,Lb: tree @ A] :
( ( ord_less @ A @ A2 @ X )
=> ( ( ord_less @ A @ B @ X )
=> ( ( splay_splay @ A @ X @ ( node @ A @ La @ A2 @ ( node @ A @ Lb @ B @ ( leaf @ A ) ) ) )
= ( node @ A @ ( node @ A @ La @ A2 @ Lb ) @ B @ ( leaf @ A ) ) ) ) ) ) ).
% splay.simps(13)
thf(fact_71_natural_Osize_I3_J,axiom,
( ( size_size @ code_natural @ ( zero_zero @ code_natural ) )
= ( zero_zero @ nat ) ) ).
% natural.size(3)
thf(fact_72_neq0__conv,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% neq0_conv
thf(fact_73_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
= ( ord_less @ nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_74_not__gr__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% not_gr_zero
thf(fact_75_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_76_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_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_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_79_add__cancel__left__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A] :
( ( ( plus_plus @ A @ B @ A2 )
= A2 )
= ( B
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_left
thf(fact_80_add__cancel__left__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A] :
( ( ( plus_plus @ A @ A2 @ B )
= A2 )
= ( B
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_right
thf(fact_81_add__cancel__right__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A] :
( ( A2
= ( plus_plus @ A @ B @ A2 ) )
= ( B
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_left
thf(fact_82_add__cancel__right__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A] :
( ( A2
= ( plus_plus @ A @ A2 @ B ) )
= ( B
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_right
thf(fact_83_add__less__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [C: A,A2: A,B: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
= ( ord_less @ A @ A2 @ B ) ) ) ).
% add_less_cancel_left
thf(fact_84_add__less__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,B: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
= ( ord_less @ A @ A2 @ B ) ) ) ).
% add_less_cancel_right
thf(fact_85_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ M @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
| ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% add_gr_0
thf(fact_86_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_87_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% Nat.add_0_right
thf(fact_88_less__one,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( one_one @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_one
thf(fact_89_add__less__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ B @ A2 ) @ B )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel1
thf(fact_90_add__less__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ B ) @ B )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel2
thf(fact_91_less__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ B ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% less_add_same_cancel1
thf(fact_92_less__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ A2 @ ( plus_plus @ A @ B @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B ) ) ) ).
% less_add_same_cancel2
thf(fact_93_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_94_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_95_t__splay__simps_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,Ll: tree @ A,Lr: tree @ A,R2: tree @ A] :
( ( ord_less @ A @ A2 @ B )
=> ( ( splay_914434265_splay @ A @ A2 @ ( node @ A @ ( node @ A @ Ll @ A2 @ Lr ) @ B @ R2 ) )
= ( one_one @ nat ) ) ) ) ).
% t_splay_simps(3)
thf(fact_96_t__splay__simps_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A,L2: tree @ A,Rl2: tree @ A,Rr2: tree @ A] :
( ( ord_less @ A @ B @ A2 )
=> ( ( splay_914434265_splay @ A @ A2 @ ( node @ A @ L2 @ B @ ( node @ A @ Rl2 @ A2 @ Rr2 ) ) )
= ( one_one @ nat ) ) ) ) ).
% t_splay_simps(7)
thf(fact_97_t__splay__simps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,R2: tree @ A] :
( ( ord_less @ A @ A2 @ B )
=> ( ( splay_914434265_splay @ A @ A2 @ ( node @ A @ ( leaf @ A ) @ B @ R2 ) )
= ( one_one @ nat ) ) ) ) ).
% t_splay_simps(2)
thf(fact_98_t__splay__simps_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A,L2: tree @ A] :
( ( ord_less @ A @ B @ A2 )
=> ( ( splay_914434265_splay @ A @ A2 @ ( node @ A @ L2 @ B @ ( leaf @ A ) ) )
= ( one_one @ nat ) ) ) ) ).
% t_splay_simps(6)
thf(fact_99_t__splay__simps_I9_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A,C: A,Rr2: tree @ A,L2: tree @ A,Rl2: tree @ A] :
( ( ord_less @ A @ B @ A2 )
=> ( ( ord_less @ A @ C @ A2 )
=> ( ( ( Rr2
= ( leaf @ A ) )
=> ( ( splay_914434265_splay @ A @ A2 @ ( node @ A @ L2 @ B @ ( node @ A @ Rl2 @ C @ Rr2 ) ) )
= ( one_one @ nat ) ) )
& ( ( Rr2
!= ( leaf @ A ) )
=> ( ( splay_914434265_splay @ A @ A2 @ ( node @ A @ L2 @ B @ ( node @ A @ Rl2 @ C @ Rr2 ) ) )
= ( plus_plus @ nat @ ( splay_914434265_splay @ A @ A2 @ Rr2 ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% t_splay_simps(9)
thf(fact_100_t__splay__simps_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A,C: A,Rl2: tree @ A,L2: tree @ A,Rr2: tree @ A] :
( ( ord_less @ A @ B @ A2 )
=> ( ( ord_less @ A @ A2 @ C )
=> ( ( ( Rl2
= ( leaf @ A ) )
=> ( ( splay_914434265_splay @ A @ A2 @ ( node @ A @ L2 @ B @ ( node @ A @ Rl2 @ C @ Rr2 ) ) )
= ( one_one @ nat ) ) )
& ( ( Rl2
!= ( leaf @ A ) )
=> ( ( splay_914434265_splay @ A @ A2 @ ( node @ A @ L2 @ B @ ( node @ A @ Rl2 @ C @ Rr2 ) ) )
= ( plus_plus @ nat @ ( splay_914434265_splay @ A @ A2 @ Rl2 ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% t_splay_simps(8)
thf(fact_101_t__splay__simps_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A,Lr: tree @ A,Ll: tree @ A,R2: tree @ A] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ord_less @ A @ C @ A2 )
=> ( ( ( Lr
= ( leaf @ A ) )
=> ( ( splay_914434265_splay @ A @ A2 @ ( node @ A @ ( node @ A @ Ll @ C @ Lr ) @ B @ R2 ) )
= ( one_one @ nat ) ) )
& ( ( Lr
!= ( leaf @ A ) )
=> ( ( splay_914434265_splay @ A @ A2 @ ( node @ A @ ( node @ A @ Ll @ C @ Lr ) @ B @ R2 ) )
= ( plus_plus @ nat @ ( splay_914434265_splay @ A @ A2 @ Lr ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% t_splay_simps(5)
thf(fact_102_t__splay__simps_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A,Ll: tree @ A,Lr: tree @ A,R2: tree @ A] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ord_less @ A @ A2 @ C )
=> ( ( ( Ll
= ( leaf @ A ) )
=> ( ( splay_914434265_splay @ A @ A2 @ ( node @ A @ ( node @ A @ Ll @ C @ Lr ) @ B @ R2 ) )
= ( one_one @ nat ) ) )
& ( ( Ll
!= ( leaf @ A ) )
=> ( ( splay_914434265_splay @ A @ A2 @ ( node @ A @ ( node @ A @ Ll @ C @ Lr ) @ B @ R2 ) )
= ( plus_plus @ nat @ ( splay_914434265_splay @ A @ A2 @ Ll ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% t_splay_simps(4)
thf(fact_103_size1__ge0,axiom,
! [A: $tType,T2: tree @ A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size1 @ A @ T2 ) ) ).
% size1_ge0
thf(fact_104_less__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% less_numeral_extra(3)
thf(fact_105_gr0I,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% gr0I
thf(fact_106_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% not_gr0
thf(fact_107_not__less0,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% not_less0
thf(fact_108_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_zeroE
thf(fact_109_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( N
!= ( zero_zero @ nat ) ) ) ).
% gr_implies_not0
thf(fact_110_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_111_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_112_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_nat_zero_code
thf(fact_113_zero__natural_Orsp,axiom,
( ( zero_zero @ nat )
= ( zero_zero @ nat ) ) ).
% zero_natural.rsp
thf(fact_114_infinite__descent0__measure,axiom,
! [A: $tType,V: A > nat,P: A > $o,X: A] :
( ! [X4: A] :
( ( ( V @ X4 )
= ( zero_zero @ nat ) )
=> ( P @ X4 ) )
=> ( ! [X4: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X4 ) )
=> ( ~ ( P @ X4 )
=> ? [Y2: A] :
( ( ord_less @ nat @ ( V @ Y2 ) @ ( V @ X4 ) )
& ~ ( P @ Y2 ) ) ) )
=> ( P @ X ) ) ) ).
% infinite_descent0_measure
thf(fact_115_linordered__field__no__lb,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [X6: A] :
? [Y3: A] : ( ord_less @ A @ Y3 @ X6 ) ) ).
% linordered_field_no_lb
thf(fact_116_linordered__field__no__ub,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [X6: A] :
? [X1: A] : ( ord_less @ A @ X6 @ X1 ) ) ).
% linordered_field_no_ub
thf(fact_117_gr__zeroI,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
( ( N
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% gr_zeroI
thf(fact_118_not__less__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_119_gr__implies__not__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [M: A,N: A] :
( ( ord_less @ A @ M @ N )
=> ( N
!= ( zero_zero @ A ) ) ) ) ).
% gr_implies_not_zero
thf(fact_120_zero__less__iff__neq__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ N )
= ( N
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_iff_neq_zero
thf(fact_121_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
& ( ( plus_plus @ nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_122_less__numeral__extra_I2_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% less_numeral_extra(2)
thf(fact_123_less__numeral__extra_I1_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% less_numeral_extra(1)
thf(fact_124_add__neg__neg,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ B ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_neg
thf(fact_125_add__pos__pos,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B ) ) ) ) ) ).
% add_pos_pos
thf(fact_126_pos__add__strict,axiom,
! [A: $tType] :
( ( strict797366125id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ B @ C )
=> ( ord_less @ A @ B @ ( plus_plus @ A @ A2 @ C ) ) ) ) ) ).
% pos_add_strict
thf(fact_127_op_092_060_094sub_062s_092_060_094sub_062t_Osize__neq,axiom,
! [A: $tType,X: splay_2011811431op_s_t @ A] :
( ( size_size @ ( splay_2011811431op_s_t @ A ) @ X )
!= ( zero_zero @ nat ) ) ).
% op\<^sub>s\<^sub>t.size_neq
thf(fact_128_add__mono__thms__linordered__field_I5_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K: A,L2: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less @ A @ K @ L2 ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L2 ) ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_129_add__mono__thms__linordered__field_I2_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K: A,L2: A] :
( ( ( I = J )
& ( ord_less @ A @ K @ L2 ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L2 ) ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_130_add__mono__thms__linordered__field_I1_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K: A,L2: A] :
( ( ( ord_less @ A @ I @ J )
& ( K = L2 ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L2 ) ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_131_add__strict__mono,axiom,
! [A: $tType] :
( ( strict2144017051up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A,D: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ord_less @ A @ C @ D )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ D ) ) ) ) ) ).
% add_strict_mono
thf(fact_132_add__strict__left__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) ) ) ) ).
% add_strict_left_mono
thf(fact_133_add__strict__right__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% add_strict_right_mono
thf(fact_134_add__less__imp__less__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [C: A,A2: A,B: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
=> ( ord_less @ A @ A2 @ B ) ) ) ).
% add_less_imp_less_left
thf(fact_135_add__less__imp__less__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,B: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
=> ( ord_less @ A @ A2 @ B ) ) ) ).
% add_less_imp_less_right
thf(fact_136_less__numeral__extra_I4_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% less_numeral_extra(4)
thf(fact_137_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ K )
=> ( ord_less @ nat @ I @ K ) ) ).
% add_lessD1
thf(fact_138_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L2: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ K @ L2 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L2 ) ) ) ) ).
% add_less_mono
thf(fact_139_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_140_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_141_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_142_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ord_less @ nat @ I @ ( plus_plus @ nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_143_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ord_less @ nat @ I @ ( plus_plus @ nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_144_less__add__eq__less,axiom,
! [K: nat,L2: nat,M: nat,N: nat] :
( ( ord_less @ nat @ K @ L2 )
=> ( ( ( plus_plus @ nat @ M @ L2 )
= ( plus_plus @ nat @ K @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_145_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_146_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_147_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_148_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_149_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% plus_nat.add_0
thf(fact_150_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_151_splay_Osimps_I9_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A,Lb: tree @ A,La: tree @ A,Ra: tree @ A] :
( ( ord_less @ A @ B @ A2 )
=> ( ( splay_splay @ A @ A2 @ ( node @ A @ Lb @ B @ ( node @ A @ La @ A2 @ Ra ) ) )
= ( node @ A @ ( node @ A @ Lb @ B @ La ) @ A2 @ Ra ) ) ) ) ).
% splay.simps(9)
thf(fact_152_splay_Osimps_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,La: tree @ A,Ra: tree @ A,Rb: tree @ A] :
( ( ord_less @ A @ A2 @ B )
=> ( ( splay_splay @ A @ A2 @ ( node @ A @ ( node @ A @ La @ A2 @ Ra ) @ B @ Rb ) )
= ( node @ A @ La @ A2 @ ( node @ A @ Ra @ B @ Rb ) ) ) ) ) ).
% splay.simps(3)
thf(fact_153_size__0__iff__Leaf,axiom,
! [A: $tType,T2: tree @ A] :
( ( ( size_size @ ( tree @ A ) @ T2 )
= ( zero_zero @ nat ) )
= ( T2
= ( leaf @ A ) ) ) ).
% size_0_iff_Leaf
thf(fact_154_tree_Osize_I3_J,axiom,
! [A: $tType] :
( ( size_size @ ( tree @ A ) @ ( leaf @ A ) )
= ( zero_zero @ nat ) ) ).
% tree.size(3)
thf(fact_155_t__splay_Osimps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( splay_914434265_splay @ A @ A2 @ ( leaf @ A ) )
= ( one_one @ nat ) ) ) ).
% t_splay.simps(1)
thf(fact_156_path__len_Osimps_I1_J,axiom,
! [A: $tType] :
( ( path_len @ A @ ( leaf @ A ) )
= ( zero_zero @ nat ) ) ).
% path_len.simps(1)
thf(fact_157_splay_Osimps_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,R2: tree @ A] :
( ( ord_less @ A @ A2 @ B )
=> ( ( splay_splay @ A @ A2 @ ( node @ A @ ( leaf @ A ) @ B @ R2 ) )
= ( node @ A @ ( leaf @ A ) @ B @ R2 ) ) ) ) ).
% splay.simps(4)
thf(fact_158_splay_Osimps_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,A2: A,B: A,Ra: tree @ A,Rb: tree @ A] :
( ( ord_less @ A @ X @ A2 )
=> ( ( ord_less @ A @ X @ B )
=> ( ( splay_splay @ A @ X @ ( node @ A @ ( node @ A @ ( leaf @ A ) @ A2 @ Ra ) @ B @ Rb ) )
= ( node @ A @ ( leaf @ A ) @ A2 @ ( node @ A @ Ra @ B @ Rb ) ) ) ) ) ) ).
% splay.simps(5)
thf(fact_159_splay_Osimps_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,B: A,A2: A,La: tree @ A,Rb: tree @ A] :
( ( ord_less @ A @ X @ B )
=> ( ( ord_less @ A @ A2 @ X )
=> ( ( splay_splay @ A @ X @ ( node @ A @ ( node @ A @ La @ A2 @ ( leaf @ A ) ) @ B @ Rb ) )
= ( node @ A @ La @ A2 @ ( node @ A @ ( leaf @ A ) @ B @ Rb ) ) ) ) ) ) ).
% splay.simps(7)
thf(fact_160_splay_Osimps_I10_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: A,X: A,L2: tree @ A] :
( ( ord_less @ A @ A2 @ X )
=> ( ( splay_splay @ A @ X @ ( node @ A @ L2 @ A2 @ ( leaf @ A ) ) )
= ( node @ A @ L2 @ A2 @ ( leaf @ A ) ) ) ) ) ).
% splay.simps(10)
thf(fact_161_splay_Osimps_I12_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: A,X: A,B: A,La: tree @ A,Rb: tree @ A] :
( ( ord_less @ A @ A2 @ X )
=> ( ( ord_less @ A @ X @ B )
=> ( ( splay_splay @ A @ X @ ( node @ A @ La @ A2 @ ( node @ A @ ( leaf @ A ) @ B @ Rb ) ) )
= ( node @ A @ ( node @ A @ La @ A2 @ ( leaf @ A ) ) @ B @ Rb ) ) ) ) ) ).
% splay.simps(12)
thf(fact_162_path__len_Opelims,axiom,
! [A: $tType,X: tree @ A,Y: nat] :
( ( ( path_len @ A @ X )
= Y )
=> ( ( accp @ ( tree @ A ) @ ( path_len_rel @ A ) @ X )
=> ( ( ( X
= ( leaf @ A ) )
=> ( ( Y
= ( zero_zero @ nat ) )
=> ~ ( accp @ ( tree @ A ) @ ( path_len_rel @ A ) @ ( leaf @ A ) ) ) )
=> ~ ! [L4: tree @ A,Uu2: A,R4: tree @ A] :
( ( X
= ( node @ A @ L4 @ Uu2 @ R4 ) )
=> ( ( Y
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( path_len @ A @ L4 ) @ ( size_size @ ( tree @ A ) @ L4 ) ) @ ( path_len @ A @ R4 ) ) @ ( size_size @ ( tree @ A ) @ R4 ) ) )
=> ~ ( accp @ ( tree @ A ) @ ( path_len_rel @ A ) @ ( node @ A @ L4 @ Uu2 @ R4 ) ) ) ) ) ) ) ).
% path_len.pelims
thf(fact_163_zero__less__two,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A @ ( type2 @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ) ).
% zero_less_two
thf(fact_164_less__add__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) ) ) ).
% less_add_one
thf(fact_165_zero__less__one,axiom,
! [A: $tType] :
( ( zero_less_one @ A @ ( type2 @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_less_one
thf(fact_166_not__one__less__zero,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A @ ( type2 @ A ) )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_less_zero
thf(fact_167_bool_Osize_I4_J,axiom,
( ( size_size @ $o @ $false )
= ( zero_zero @ nat ) ) ).
% bool.size(4)
thf(fact_168_bool_Osize_I3_J,axiom,
( ( size_size @ $o @ $true )
= ( zero_zero @ nat ) ) ).
% bool.size(3)
thf(fact_169_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less @ nat @ M @ N )
| ( ord_less @ nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_170_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_not_refl
thf(fact_171_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_172_less__not__refl3,axiom,
! [S: nat,T2: nat] :
( ( ord_less @ nat @ S @ T2 )
=> ( S != T2 ) ) ).
% less_not_refl3
thf(fact_173_measure__induct,axiom,
! [A: $tType,F: A > nat,P: A > $o,A2: A] :
( ! [X4: A] :
( ! [Y2: A] :
( ( ord_less @ nat @ ( F @ Y2 ) @ ( F @ X4 ) )
=> ( P @ Y2 ) )
=> ( P @ X4 ) )
=> ( P @ A2 ) ) ).
% measure_induct
thf(fact_174_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_175_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_176_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_177_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less @ nat @ X @ Y )
=> ( ord_less @ nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_178_measure__induct__rule,axiom,
! [A: $tType,F: A > nat,P: A > $o,A2: A] :
( ! [X4: A] :
( ! [Y2: A] :
( ( ord_less @ nat @ ( F @ Y2 ) @ ( F @ X4 ) )
=> ( P @ Y2 ) )
=> ( P @ X4 ) )
=> ( P @ A2 ) ) ).
% measure_induct_rule
thf(fact_179_infinite__descent__measure,axiom,
! [A: $tType,P: A > $o,V: A > nat,X: A] :
( ! [X4: A] :
( ~ ( P @ X4 )
=> ? [Y2: A] :
( ( ord_less @ nat @ ( V @ Y2 ) @ ( V @ X4 ) )
& ~ ( P @ Y2 ) ) )
=> ( P @ X ) ) ).
% infinite_descent_measure
thf(fact_180_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_181_prod_Osize__neq,axiom,
! [A: $tType,B4: $tType,X: product_prod @ A @ B4] :
( ( size_size @ ( product_prod @ A @ B4 ) @ X )
!= ( zero_zero @ nat ) ) ).
% prod.size_neq
thf(fact_182_sum_Osize__neq,axiom,
! [A: $tType,B4: $tType,X: sum_sum @ A @ B4] :
( ( size_size @ ( sum_sum @ A @ B4 ) @ X )
!= ( zero_zero @ nat ) ) ).
% sum.size_neq
thf(fact_183_size__bool,axiom,
( ( size_size @ $o )
= ( ^ [B2: $o] : ( zero_zero @ nat ) ) ) ).
% size_bool
thf(fact_184_zero__neq__one,axiom,
! [A: $tType] :
( ( zero_neq_one @ A @ ( type2 @ A ) )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ) ).
% zero_neq_one
thf(fact_185_natural_Osize_I1_J,axiom,
( ( code_size_natural @ ( zero_zero @ code_natural ) )
= ( zero_zero @ nat ) ) ).
% natural.size(1)
thf(fact_186_fold__atLeastAtMost__nat_Oinduct,axiom,
! [A: $tType,P: ( nat > A > A ) > nat > nat > A > $o,A0: nat > A > A,A1: nat,A22: nat,A32: A] :
( ! [F2: nat > A > A,A4: nat,B3: nat,Acc: A] :
( ( ~ ( ord_less @ nat @ B3 @ A4 )
=> ( P @ F2 @ ( plus_plus @ nat @ A4 @ ( one_one @ nat ) ) @ B3 @ ( F2 @ A4 @ Acc ) ) )
=> ( P @ F2 @ A4 @ B3 @ Acc ) )
=> ( P @ A0 @ A1 @ A22 @ A32 ) ) ).
% fold_atLeastAtMost_nat.induct
thf(fact_187_natural_Osimps_I4_J,axiom,
! [T: $tType,F1: T,F22: code_natural > T] :
( ( code_case_natural @ T @ F1 @ F22 @ ( zero_zero @ code_natural ) )
= F1 ) ).
% natural.simps(4)
thf(fact_188_bool_Osize_I1_J,axiom,
( ( size_bool @ $true )
= ( zero_zero @ nat ) ) ).
% bool.size(1)
thf(fact_189_bool_Osize_I2_J,axiom,
( ( size_bool @ $false )
= ( zero_zero @ nat ) ) ).
% bool.size(2)
thf(fact_190_size__bool__def,axiom,
( size_bool
= ( product_rec_bool @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) ) ).
% size_bool_def
thf(fact_191_ind__euclid,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( P @ A4 @ B3 )
= ( P @ B3 @ A4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ ( zero_zero @ nat ) )
=> ( ! [A4: nat,B3: nat] :
( ( P @ A4 @ B3 )
=> ( P @ A4 @ ( plus_plus @ nat @ A4 @ B3 ) ) )
=> ( P @ A2 @ B ) ) ) ) ).
% ind_euclid
thf(fact_192_size__bool__overloaded__def,axiom,
( ( size_size @ $o )
= ( product_rec_bool @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) ) ).
% size_bool_overloaded_def
thf(fact_193_add__0__iff,axiom,
! [A: $tType] :
( ( semiri456707255roduct @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A] :
( ( B
= ( plus_plus @ A @ B @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% add_0_iff
thf(fact_194_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_195_semiring__normalization__rules_I20_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A,D: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B ) @ ( plus_plus @ A @ C @ D ) )
= ( plus_plus @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ D ) ) ) ) ).
% semiring_normalization_rules(20)
thf(fact_196_semiring__normalization__rules_I21_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ 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 ) ) ) ) ).
% semiring_normalization_rules(21)
thf(fact_197_semiring__normalization__rules_I22_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,D: A] :
( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ C @ D ) )
= ( plus_plus @ A @ C @ ( plus_plus @ A @ A2 @ D ) ) ) ) ).
% semiring_normalization_rules(22)
thf(fact_198_semiring__normalization__rules_I23_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
= ( plus_plus @ A @ ( plus_plus @ A @ A2 @ C ) @ B ) ) ) ).
% semiring_normalization_rules(23)
thf(fact_199_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_200_semiring__normalization__rules_I25_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,D: A] :
( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ C @ D ) )
= ( plus_plus @ A @ ( plus_plus @ A @ A2 @ C ) @ D ) ) ) ).
% semiring_normalization_rules(25)
thf(fact_201_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_202_natural_Osimps_I6_J,axiom,
! [T: $tType,F1: T,F22: code_natural > T > T] :
( ( code_rec_natural @ T @ F1 @ F22 @ ( zero_zero @ code_natural ) )
= F1 ) ).
% natural.simps(6)
thf(fact_203_fold__atLeastAtMost__nat_Osimps,axiom,
! [A: $tType] :
( ( set_fo292404081st_nat @ A )
= ( ^ [F3: nat > A > A,A3: nat,B2: nat,Acc2: A] : ( if @ A @ ( ord_less @ nat @ B2 @ A3 ) @ Acc2 @ ( set_fo292404081st_nat @ A @ F3 @ ( plus_plus @ nat @ A3 @ ( one_one @ nat ) ) @ B2 @ ( F3 @ A3 @ Acc2 ) ) ) ) ) ).
% fold_atLeastAtMost_nat.simps
thf(fact_204_fold__atLeastAtMost__nat_Oelims,axiom,
! [A: $tType,X: nat > A > A,Xa: nat,Xb: nat,Xc: A,Y: A] :
( ( ( set_fo292404081st_nat @ A @ X @ Xa @ Xb @ Xc )
= Y )
=> ( ( ( ord_less @ nat @ Xb @ Xa )
=> ( Y = Xc ) )
& ( ~ ( ord_less @ nat @ Xb @ Xa )
=> ( Y
= ( set_fo292404081st_nat @ A @ X @ ( plus_plus @ nat @ Xa @ ( one_one @ nat ) ) @ Xb @ ( X @ Xa @ Xc ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.elims
thf(fact_205_ln__one,axiom,
! [A: $tType] :
( ( ln @ A @ ( type2 @ A ) )
=> ( ( ln_ln @ A @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% ln_one
thf(fact_206_tree_Osize__gen_I1_J,axiom,
! [A: $tType,X: A > nat] :
( ( size_tree @ A @ X @ ( leaf @ A ) )
= ( zero_zero @ nat ) ) ).
% tree.size_gen(1)
thf(fact_207_ln__less__zero__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( ln_ln @ real @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( one_one @ real ) ) ) ) ).
% ln_less_zero_iff
thf(fact_208_ln__gt__zero__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X ) )
= ( ord_less @ real @ ( one_one @ real ) @ X ) ) ) ).
% ln_gt_zero_iff
thf(fact_209_ln__eq__zero__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( ln_ln @ real @ X )
= ( zero_zero @ real ) )
= ( X
= ( one_one @ real ) ) ) ) ).
% ln_eq_zero_iff
thf(fact_210_ln__gt__zero__imp__gt__one,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less @ real @ ( one_one @ real ) @ X ) ) ) ).
% ln_gt_zero_imp_gt_one
thf(fact_211_ln__less__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( ord_less @ real @ ( ln_ln @ real @ X ) @ ( zero_zero @ real ) ) ) ) ).
% ln_less_zero
thf(fact_212_ln__gt__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X ) ) ) ).
% ln_gt_zero
thf(fact_213_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_214_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_215_powr__gt__zero,axiom,
! [X: real,A2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( powr @ real @ X @ A2 ) )
= ( X
!= ( zero_zero @ real ) ) ) ).
% powr_gt_zero
thf(fact_216_powr__less__cancel__iff,axiom,
! [X: real,A2: real,B: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ X @ B ) )
= ( ord_less @ real @ A2 @ B ) ) ) ).
% powr_less_cancel_iff
thf(fact_217_powr__0,axiom,
! [A: $tType] :
( ( ln @ A @ ( type2 @ A ) )
=> ! [Z: A] :
( ( powr @ A @ ( zero_zero @ A ) @ Z )
= ( zero_zero @ A ) ) ) ).
% powr_0
thf(fact_218_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_219_ln__inj__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ( ln_ln @ real @ X )
= ( ln_ln @ real @ Y ) )
= ( X = Y ) ) ) ) ).
% ln_inj_iff
thf(fact_220_ln__less__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ ( ln_ln @ real @ X ) @ ( ln_ln @ real @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ) ) ).
% ln_less_cancel_iff
thf(fact_221_ln__less__self,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less @ real @ ( ln_ln @ real @ X ) @ X ) ) ).
% ln_less_self
thf(fact_222_powr__less__mono2__neg,axiom,
! [A2: real,X: real,Y: real] :
( ( ord_less @ real @ A2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( powr @ real @ Y @ A2 ) @ ( powr @ real @ X @ A2 ) ) ) ) ) ).
% powr_less_mono2_neg
thf(fact_223_powr__less__cancel,axiom,
! [X: real,A2: real,B: real] :
( ( ord_less @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ X @ B ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ord_less @ real @ A2 @ B ) ) ) ).
% powr_less_cancel
thf(fact_224_powr__less__mono,axiom,
! [A2: real,B: real,X: real] :
( ( ord_less @ real @ A2 @ B )
=> ( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ord_less @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ X @ B ) ) ) ) ).
% powr_less_mono
thf(fact_225_powr__inj,axiom,
! [A2: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ( powr @ real @ A2 @ X )
= ( powr @ real @ A2 @ Y ) )
= ( X = Y ) ) ) ) ).
% powr_inj
thf(fact_226_real__lbound__gt__zero,axiom,
! [D1: real,D2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D1 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ? [E: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E )
& ( ord_less @ real @ E @ D1 )
& ( ord_less @ real @ E @ D2 ) ) ) ) ).
% real_lbound_gt_zero
thf(fact_227_ln__ge__zero__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X ) )
= ( ord_less_eq @ real @ ( one_one @ real ) @ X ) ) ) ).
% ln_ge_zero_iff
thf(fact_228_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_229_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ 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_230_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ 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_231_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ 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_232_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ 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_233_le__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ 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_234_le__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ 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_235_add__le__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ 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_236_add__le__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ 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_237_ln__le__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ ( ln_ln @ real @ X ) @ ( ln_ln @ real @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ) ) ).
% ln_le_cancel_iff
thf(fact_238_powr__one__gt__zero__iff,axiom,
! [X: real] :
( ( ( powr @ real @ X @ ( one_one @ real ) )
= X )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X ) ) ).
% powr_one_gt_zero_iff
thf(fact_239_powr__one,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ X @ ( one_one @ real ) )
= X ) ) ).
% powr_one
thf(fact_240_powr__le__cancel__iff,axiom,
! [X: real,A2: real,B: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ X @ B ) )
= ( ord_less_eq @ real @ A2 @ B ) ) ) ).
% powr_le_cancel_iff
thf(fact_241_ln__bound,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ X ) @ X ) ) ).
% ln_bound
thf(fact_242_powr__mono,axiom,
! [A2: real,B: real,X: real] :
( ( ord_less_eq @ real @ A2 @ B )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ord_less_eq @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ X @ B ) ) ) ) ).
% powr_mono
thf(fact_243_powr__mono2,axiom,
! [A2: real,X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ord_less_eq @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ Y @ A2 ) ) ) ) ) ).
% powr_mono2
thf(fact_244_powr__mono2_H,axiom,
! [A2: real,X: real,Y: real] :
( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ord_less_eq @ real @ ( powr @ real @ Y @ A2 ) @ ( powr @ real @ X @ A2 ) ) ) ) ) ).
% powr_mono2'
thf(fact_245_powr__ge__pzero,axiom,
! [X: real,Y: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( powr @ real @ X @ Y ) ) ).
% powr_ge_pzero
thf(fact_246_powr__less__mono2,axiom,
! [A2: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ Y @ A2 ) ) ) ) ) ).
% powr_less_mono2
thf(fact_247_ge__one__powr__ge__zero,axiom,
! [X: real,A2: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ord_less_eq @ real @ ( one_one @ real ) @ ( powr @ real @ X @ A2 ) ) ) ) ).
% ge_one_powr_ge_zero
thf(fact_248_less__eq__real__def,axiom,
( ( ord_less_eq @ real )
= ( ^ [X5: real,Y4: real] :
( ( ord_less @ real @ X5 @ Y4 )
| ( X5 = Y4 ) ) ) ) ).
% less_eq_real_def
thf(fact_249_zero__le,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% zero_le
thf(fact_250_le__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% le_numeral_extra(3)
thf(fact_251_add__less__le__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A,D: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ord_less_eq @ A @ C @ D )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ D ) ) ) ) ) ).
% add_less_le_mono
thf(fact_252_add__le__less__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A,D: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less @ A @ C @ D )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ D ) ) ) ) ) ).
% add_le_less_mono
%----Subclasses (1)
thf(subcl_Orderings_Olinorder___HOL_Otype,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ( type @ A @ ( type2 @ A ) ) ) ).
%----Type constructors (77)
thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
semiri456707255roduct @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
ordere516151231imp_le @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
strict2144017051up_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere223160158up_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add,axiom,
strict797366125id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring,axiom,
linord1659791738miring @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add,axiom,
ordere216010020id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
cancel1352612707id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom,axiom,
linordered_semidom @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1 @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add,axiom,
semigroup_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Ozero__less__one,axiom,
zero_less_one @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Ozero__neq__one,axiom,
zero_neq_one @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Omonoid__add,axiom,
monoid_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ozero,axiom,
zero @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oone,axiom,
one @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Nat_Osize,axiom,
size @ nat @ ( type2 @ nat ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder_1,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Nat_Osize_2,axiom,
size @ $o @ ( type2 @ $o ) ).
thf(tcon_Real_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_3,axiom,
semiri456707255roduct @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__monoid__add__imp__le_4,axiom,
ordere516151231imp_le @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Ostrict__ordered__ab__semigroup__add_5,axiom,
strict2144017051up_add @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_6,axiom,
ordere223160158up_add @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_7,axiom,
ordere236663937imp_le @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Ostrict__ordered__comm__monoid__add_8,axiom,
strict797366125id_add @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_9,axiom,
linord1659791738miring @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add_10,axiom,
ordere779506340up_add @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Oordered__comm__monoid__add_11,axiom,
ordere216010020id_add @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add,axiom,
linord219039673up_add @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_12,axiom,
cancel1352612707id_add @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_13,axiom,
cancel_semigroup_add @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Rings_Olinordered__semidom_14,axiom,
linordered_semidom @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Oab__semigroup__add_15,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_16,axiom,
comm_monoid_add @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Rings_Olinordered__idom,axiom,
linordered_idom @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1_17,axiom,
comm_semiring_1 @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Osemigroup__add_18,axiom,
semigroup_add @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Rings_Ozero__less__one_19,axiom,
zero_less_one @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Rings_Ozero__neq__one_20,axiom,
zero_neq_one @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Orderings_Olinorder_21,axiom,
linorder @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Transcendental_Oln,axiom,
ln @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Omonoid__add_22,axiom,
monoid_add @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Ogroup__add,axiom,
group_add @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Num_Oneg__numeral,axiom,
neg_numeral @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Ozero_23,axiom,
zero @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Oone_24,axiom,
one @ real @ ( type2 @ real ) ).
thf(tcon_Tree_Otree___Nat_Osize_25,axiom,
! [A5: $tType] : ( size @ ( tree @ A5 ) @ ( type2 @ ( tree @ A5 ) ) ) ).
thf(tcon_Sum__Type_Osum___Nat_Osize_26,axiom,
! [A5: $tType,A6: $tType] : ( size @ ( sum_sum @ A5 @ A6 ) @ ( type2 @ ( sum_sum @ A5 @ A6 ) ) ) ).
thf(tcon_Product__Type_Oprod___Nat_Osize_27,axiom,
! [A5: $tType,A6: $tType] : ( size @ ( product_prod @ A5 @ A6 ) @ ( type2 @ ( product_prod @ A5 @ A6 ) ) ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__monoid__add__imp__le_28,axiom,
ordere516151231imp_le @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__ab__semigroup__add_29,axiom,
strict2144017051up_add @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__ab__semigroup__add_30,axiom,
ordere223160158up_add @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add__imp__le_31,axiom,
ordere236663937imp_le @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__comm__monoid__add_32,axiom,
strict797366125id_add @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add_33,axiom,
ordere779506340up_add @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__comm__monoid__add_34,axiom,
ordere216010020id_add @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__comm__monoid__add_35,axiom,
cancel1352612707id_add @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__semigroup__add_36,axiom,
cancel_semigroup_add @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__add_37,axiom,
ab_semigroup_add @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__add_38,axiom,
comm_monoid_add @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1_39,axiom,
comm_semiring_1 @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Osemigroup__add_40,axiom,
semigroup_add @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Rings_Ozero__neq__one_41,axiom,
zero_neq_one @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Orderings_Olinorder_42,axiom,
linorder @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Omonoid__add_43,axiom,
monoid_add @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Ozero_44,axiom,
zero @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Oone_45,axiom,
one @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Nat_Osize_46,axiom,
size @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Splay__Tree__Analysis__Base_Oop_092_060_094sub_062s_092_060_094sub_062t___Nat_Osize_47,axiom,
! [A5: $tType] : ( size @ ( splay_2011811431op_s_t @ A5 ) @ ( type2 @ ( splay_2011811431op_s_t @ A5 ) ) ) ).
%----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,
( ( size_size @ ( tree @ a ) @ rr )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( size_size @ ( tree @ a ) @ l ) @ ( size_size @ ( tree @ a ) @ r ) ) @ ( one_one @ nat ) ) ) ).
%------------------------------------------------------------------------------