TPTP Problem File: DAT213^1.p
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%------------------------------------------------------------------------------
% File : DAT213^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Splay tree analysis 186
% 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__186.p [Bla16]
% Status : Theorem
% Rating : 0.00 v7.2.0, 0.25 v7.1.0
% Syntax : Number of formulae : 368 ( 88 unt; 60 typ; 0 def)
% Number of atoms : 988 ( 256 equ; 0 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 4601 ( 113 ~; 12 |; 69 &;3855 @)
% ( 0 <=>; 552 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 9 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 222 ( 222 >; 0 *; 0 +; 0 <<)
% Number of symbols : 60 ( 57 usr; 5 con; 0-5 aty)
% Number of variables : 1181 ( 40 ^;1064 !; 23 ?;1181 :)
% ( 54 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:49:47.790
%------------------------------------------------------------------------------
%----Could-be-implicit typings (6)
thf(ty_t_Tree_Otree,type,
tree: $tType > $tType ).
thf(ty_t_Real_Oreal,type,
real: $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (54)
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_Orderings_Oord,type,
ord:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Num_Oneg__numeral,type,
neg_numeral:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Ono__top,type,
no_top:
!>[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_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[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_Oordered__ab__semigroup__add,type,
ordere779506340up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere236663937imp_le:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
condit1037483654norder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
thf(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > 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_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Splay__Tree_Ois__root,type,
splay_is_root:
!>[A: $tType] : ( A > ( tree @ A ) > $o ) ).
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__Mirabelle__pcaxyvimtd_OA,type,
splay_266122055elle_A:
!>[A: $tType] : ( A > ( tree @ A ) > real ) ).
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_Olinorder__class_Obst__eq,type,
linorder_bst_eq:
!>[A: $tType] : ( ( tree @ A ) > $o ) ).
thf(sy_c_Tree_Omirror,type,
mirror:
!>[A: $tType] : ( ( tree @ A ) > ( tree @ A ) ) ).
thf(sy_c_Tree_Osubtrees,type,
subtrees:
!>[A: $tType] : ( ( tree @ A ) > ( set @ ( tree @ A ) ) ) ).
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_Ocase__tree,type,
case_tree:
!>[B: $tType,A: $tType] : ( B > ( ( tree @ A ) > A > ( tree @ A ) > B ) > ( tree @ A ) > B ) ).
thf(sy_c_Tree_Otree_Ois__Leaf,type,
is_Leaf:
!>[A: $tType] : ( ( tree @ A ) > $o ) ).
thf(sy_c_Tree_Otree_Oleft,type,
left:
!>[A: $tType] : ( ( tree @ A ) > ( tree @ A ) ) ).
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_Orel__tree,type,
rel_tree:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( tree @ A ) > ( tree @ B ) > $o ) ).
thf(sy_c_Tree_Otree_Oright,type,
right:
!>[A: $tType] : ( ( tree @ A ) > ( tree @ A ) ) ).
thf(sy_c_Tree_Otree_Oset__tree,type,
set_tree:
!>[A: $tType] : ( ( tree @ A ) > ( set @ A ) ) ).
thf(sy_c_Tree_Otree_Oval,type,
val:
!>[A: $tType] : ( ( tree @ A ) > A ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_a,type,
a2: a ).
thf(sy_v_t,type,
t: tree @ a ).
thf(sy_v_thesis____,type,
thesis: $o ).
%----Relevant facts (254)
thf(fact_0_assms_I2_J,axiom,
member @ a @ a2 @ ( set_tree @ a @ t ) ).
% assms(2)
thf(fact_1_assms_I1_J,axiom,
linorder_bst @ a @ t ).
% assms(1)
thf(fact_2_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_3_set__treeE,axiom,
! [A: $tType,A2: A,T: tree @ A] :
( ( member @ A @ A2 @ ( set_tree @ A @ T ) )
=> ? [L: tree @ A,R: tree @ A] : ( member @ ( tree @ A ) @ ( node @ A @ L @ A2 @ R ) @ ( subtrees @ A @ T ) ) ) ).
% set_treeE
thf(fact_4_in__set__tree__if,axiom,
! [A: $tType,L2: tree @ A,A2: A,R2: tree @ A,T: tree @ A] :
( ( member @ ( tree @ A ) @ ( node @ A @ L2 @ A2 @ R2 ) @ ( subtrees @ A @ T ) )
=> ( member @ A @ A2 @ ( set_tree @ A @ T ) ) ) ).
% in_set_tree_if
thf(fact_5_Node__notin__subtrees__if,axiom,
! [A: $tType,A2: A,T: tree @ A,L2: tree @ A,R2: tree @ A] :
( ~ ( member @ A @ A2 @ ( set_tree @ A @ T ) )
=> ~ ( member @ ( tree @ A ) @ ( node @ A @ L2 @ A2 @ R2 ) @ ( subtrees @ A @ T ) ) ) ).
% Node_notin_subtrees_if
thf(fact_6_tree_Oset__intros_I3_J,axiom,
! [A: $tType,Xa: A,A3: tree @ A,A1: tree @ A,A22: A] :
( ( member @ A @ Xa @ ( set_tree @ A @ A3 ) )
=> ( member @ A @ Xa @ ( set_tree @ A @ ( node @ A @ A1 @ A22 @ A3 ) ) ) ) ).
% tree.set_intros(3)
thf(fact_7_tree_Oset__intros_I2_J,axiom,
! [A: $tType,A22: A,A1: tree @ A,A3: tree @ A] : ( member @ A @ A22 @ ( set_tree @ A @ ( node @ A @ A1 @ A22 @ A3 ) ) ) ).
% tree.set_intros(2)
thf(fact_8_tree_Oset__intros_I1_J,axiom,
! [A: $tType,X: A,A1: tree @ A,A22: A,A3: tree @ A] :
( ( member @ A @ X @ ( set_tree @ A @ A1 ) )
=> ( member @ A @ X @ ( set_tree @ A @ ( node @ A @ A1 @ A22 @ A3 ) ) ) ) ).
% tree.set_intros(1)
thf(fact_9_tree_Oset__cases,axiom,
! [A: $tType,E: A,A2: tree @ A] :
( ( member @ A @ E @ ( set_tree @ A @ A2 ) )
=> ( ! [Z1: tree @ A] :
( ? [Z2: A,Z3: tree @ A] :
( A2
= ( node @ A @ Z1 @ Z2 @ Z3 ) )
=> ~ ( member @ A @ E @ ( set_tree @ A @ Z1 ) ) )
=> ( ! [Z1: tree @ A,Z3: tree @ A] :
( A2
!= ( node @ A @ Z1 @ E @ Z3 ) )
=> ~ ! [Z1: tree @ A,Z2: A,Z3: tree @ A] :
( ( A2
= ( node @ A @ Z1 @ Z2 @ Z3 ) )
=> ~ ( member @ A @ E @ ( set_tree @ A @ Z3 ) ) ) ) ) ) ).
% tree.set_cases
thf(fact_10_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_11_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_12_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_13_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_14_tree_Opred__cong,axiom,
! [A: $tType,X: tree @ A,Ya: tree @ A,P: A > $o,Pa: A > $o] :
( ( X = Ya )
=> ( ! [Z: A] :
( ( member @ A @ Z @ ( set_tree @ A @ Ya ) )
=> ( ( P @ Z )
= ( Pa @ Z ) ) )
=> ( ( pred_tree @ A @ P @ X )
= ( pred_tree @ A @ Pa @ Ya ) ) ) ) ).
% tree.pred_cong
thf(fact_15_tree_Opred__mono__strong,axiom,
! [A: $tType,P: A > $o,X: tree @ A,Pa: A > $o] :
( ( pred_tree @ A @ P @ X )
=> ( ! [Z: A] :
( ( member @ A @ Z @ ( set_tree @ A @ X ) )
=> ( ( P @ Z )
=> ( Pa @ Z ) ) )
=> ( pred_tree @ A @ Pa @ X ) ) ) ).
% tree.pred_mono_strong
thf(fact_16_one__reorient,axiom,
! [A: $tType] :
( ( one @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( ( one_one @ A )
= X )
= ( X
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_17_splay__to__root,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: tree @ A,A2: A,T2: tree @ A] :
( ( linorder_bst @ A @ T )
=> ( ( ( splay_splay @ A @ A2 @ T )
= T2 )
=> ( ( member @ A @ A2 @ ( set_tree @ A @ T ) )
= ( ? [L3: tree @ A,R3: tree @ A] :
( T2
= ( node @ A @ L3 @ A2 @ R3 ) ) ) ) ) ) ) ).
% splay_to_root
thf(fact_18_tree_Opred__set,axiom,
! [A: $tType] :
( ( pred_tree @ A )
= ( ^ [P2: A > $o,X2: tree @ A] :
! [Y: A] :
( ( member @ A @ Y @ ( set_tree @ A @ X2 ) )
=> ( P2 @ Y ) ) ) ) ).
% tree.pred_set
thf(fact_19_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_20_tree_Osel_I2_J,axiom,
! [A: $tType,X21: tree @ A,X22: A,X23: tree @ A] :
( ( left @ A @ ( node @ A @ X21 @ X22 @ X23 ) )
= X21 ) ).
% tree.sel(2)
thf(fact_21_tree_Osel_I5_J,axiom,
! [A: $tType,X21: tree @ A,X22: A,X23: tree @ A] :
( ( right @ A @ ( node @ A @ X21 @ X22 @ X23 ) )
= X23 ) ).
% tree.sel(5)
thf(fact_22_bst__eq__if__bst,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: tree @ A] :
( ( linorder_bst @ A @ T )
=> ( linorder_bst_eq @ A @ T ) ) ) ).
% bst_eq_if_bst
thf(fact_23_mirror_Osimps_I2_J,axiom,
! [A: $tType,L2: tree @ A,X: A,R2: tree @ A] :
( ( mirror @ A @ ( node @ A @ L2 @ X @ R2 ) )
= ( node @ A @ ( mirror @ A @ R2 ) @ X @ ( mirror @ A @ L2 ) ) ) ).
% mirror.simps(2)
thf(fact_24_tree_Osimps_I6_J,axiom,
! [A: $tType,C: $tType,F1: C,F2: ( tree @ A ) > A > ( tree @ A ) > C > C > C] :
( ( rec_tree @ C @ A @ F1 @ F2 @ ( leaf @ A ) )
= F1 ) ).
% tree.simps(6)
thf(fact_25_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_26_mirror__mirror,axiom,
! [A: $tType,T: tree @ A] :
( ( mirror @ A @ ( mirror @ A @ T ) )
= T ) ).
% mirror_mirror
thf(fact_27_splay__Leaf__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: A,T: tree @ A] :
( ( ( splay_splay @ A @ A2 @ T )
= ( leaf @ A ) )
= ( T
= ( leaf @ A ) ) ) ) ).
% splay_Leaf_iff
thf(fact_28_mirror__Leaf,axiom,
! [A: $tType,T: tree @ A] :
( ( ( mirror @ A @ T )
= ( leaf @ A ) )
= ( T
= ( leaf @ A ) ) ) ).
% mirror_Leaf
thf(fact_29_tree_Osel_I4_J,axiom,
! [A: $tType] :
( ( right @ A @ ( leaf @ A ) )
= ( leaf @ A ) ) ).
% tree.sel(4)
thf(fact_30_tree_Osel_I1_J,axiom,
! [A: $tType] :
( ( left @ A @ ( leaf @ A ) )
= ( leaf @ A ) ) ).
% tree.sel(1)
thf(fact_31_mirror_Osimps_I1_J,axiom,
! [A: $tType] :
( ( mirror @ A @ ( leaf @ A ) )
= ( leaf @ A ) ) ).
% mirror.simps(1)
thf(fact_32_bst__eq_Osimps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ( linorder_bst_eq @ A @ ( leaf @ A ) ) ) ).
% bst_eq.simps(1)
thf(fact_33_tree_Oexhaust__sel,axiom,
! [A: $tType,Tree: tree @ A] :
( ( Tree
!= ( leaf @ A ) )
=> ( Tree
= ( node @ A @ ( left @ A @ Tree ) @ ( val @ A @ Tree ) @ ( right @ A @ Tree ) ) ) ) ).
% tree.exhaust_sel
thf(fact_34_splay__not__Leaf,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: tree @ A,A2: A] :
( ( T
!= ( leaf @ A ) )
=> ? [L: tree @ A,X3: A,R: tree @ A] :
( ( splay_splay @ A @ A2 @ T )
= ( node @ A @ L @ X3 @ R ) ) ) ) ).
% splay_not_Leaf
thf(fact_35_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_36_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_37_mirror_Oelims,axiom,
! [A: $tType,X: tree @ A,Y2: tree @ A] :
( ( ( mirror @ A @ X )
= Y2 )
=> ( ( ( X
= ( leaf @ A ) )
=> ( Y2
!= ( leaf @ A ) ) )
=> ~ ! [L: tree @ A,X3: A,R: tree @ A] :
( ( X
= ( node @ A @ L @ X3 @ R ) )
=> ( Y2
!= ( node @ A @ ( mirror @ A @ R ) @ X3 @ ( mirror @ A @ L ) ) ) ) ) ) ).
% mirror.elims
thf(fact_38_tree_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F2: ( tree @ A ) > A > ( tree @ A ) > B,Tree: tree @ A] :
( ( P @ ( case_tree @ B @ A @ F1 @ F2 @ Tree ) )
= ( ~ ( ( ( Tree
= ( leaf @ A ) )
& ~ ( P @ F1 ) )
| ( ( Tree
= ( node @ A @ ( left @ A @ Tree ) @ ( val @ A @ Tree ) @ ( right @ A @ Tree ) ) )
& ~ ( P @ ( F2 @ ( left @ A @ Tree ) @ ( val @ A @ Tree ) @ ( right @ A @ Tree ) ) ) ) ) ) ) ).
% tree.split_sel_asm
thf(fact_39_tree_Osplit__sel,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F2: ( tree @ A ) > A > ( tree @ A ) > B,Tree: tree @ A] :
( ( P @ ( case_tree @ B @ A @ F1 @ F2 @ Tree ) )
= ( ( ( Tree
= ( leaf @ A ) )
=> ( P @ F1 ) )
& ( ( Tree
= ( node @ A @ ( left @ A @ Tree ) @ ( val @ A @ Tree ) @ ( right @ A @ Tree ) ) )
=> ( P @ ( F2 @ ( left @ A @ Tree ) @ ( val @ A @ Tree ) @ ( right @ A @ Tree ) ) ) ) ) ) ).
% tree.split_sel
thf(fact_40_set__splay,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: A,T: tree @ A] :
( ( set_tree @ A @ ( splay_splay @ A @ A2 @ T ) )
= ( set_tree @ A @ T ) ) ) ).
% set_splay
thf(fact_41_bst__splay,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: tree @ A,A2: A] :
( ( linorder_bst @ A @ T )
=> ( linorder_bst @ A @ ( splay_splay @ A @ A2 @ T ) ) ) ) ).
% bst_splay
thf(fact_42_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_43_tree_Oinduct,axiom,
! [A: $tType,P: ( tree @ A ) > $o,Tree: tree @ A] :
( ( P @ ( leaf @ A ) )
=> ( ! [X1: tree @ A,X24: A,X32: tree @ A] :
( ( P @ X1 )
=> ( ( P @ X32 )
=> ( P @ ( node @ A @ X1 @ X24 @ X32 ) ) ) )
=> ( P @ Tree ) ) ) ).
% tree.induct
thf(fact_44_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_46_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_47_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_48_neq__Leaf__iff,axiom,
! [A: $tType,T: tree @ A] :
( ( T
!= ( leaf @ A ) )
= ( ? [L3: tree @ A,A5: A,R3: tree @ A] :
( T
= ( node @ A @ L3 @ A5 @ R3 ) ) ) ) ).
% neq_Leaf_iff
thf(fact_49_tree_Oexhaust,axiom,
! [A: $tType,Y2: tree @ A] :
( ( Y2
!= ( leaf @ A ) )
=> ~ ! [X212: tree @ A,X222: A,X232: tree @ A] :
( Y2
!= ( node @ A @ X212 @ X222 @ X232 ) ) ) ).
% tree.exhaust
thf(fact_50_mirror_Oinduct,axiom,
! [A: $tType,P: ( tree @ A ) > $o,A0: tree @ A] :
( ( P @ ( leaf @ A ) )
=> ( ! [L: tree @ A,X3: A,R: tree @ A] :
( ( P @ R )
=> ( ( P @ L )
=> ( P @ ( node @ A @ L @ X3 @ R ) ) ) )
=> ( P @ A0 ) ) ) ).
% mirror.induct
thf(fact_51_bst__eq_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: tree @ A] :
( ( X
!= ( leaf @ A ) )
=> ~ ! [L: tree @ A,A6: A,R: tree @ A] :
( X
!= ( node @ A @ L @ A6 @ R ) ) ) ) ).
% bst_eq.cases
thf(fact_52_bst__eq_Oinduct,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: ( tree @ A ) > $o,A0: tree @ A] :
( ( P @ ( leaf @ A ) )
=> ( ! [L: tree @ A,A6: A,R: tree @ A] :
( ( P @ L )
=> ( ( P @ R )
=> ( P @ ( node @ A @ L @ A6 @ R ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% bst_eq.induct
thf(fact_53_bst_Osimps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ( linorder_bst @ A @ ( leaf @ A ) ) ) ).
% bst.simps(1)
thf(fact_54_tree_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,F1: B,F2: ( tree @ A ) > A > ( tree @ A ) > B] :
( ( case_tree @ B @ A @ F1 @ F2 @ ( leaf @ A ) )
= F1 ) ).
% tree.simps(4)
thf(fact_55_tree_Opred__inject_I1_J,axiom,
! [A: $tType,P: A > $o] : ( pred_tree @ A @ P @ ( leaf @ A ) ) ).
% tree.pred_inject(1)
thf(fact_56_is__root__splay,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: tree @ A,A2: A] :
( ( linorder_bst @ A @ T )
=> ( ( splay_is_root @ A @ A2 @ ( splay_splay @ A @ A2 @ T ) )
= ( member @ A @ A2 @ ( set_tree @ A @ T ) ) ) ) ) ).
% is_root_splay
thf(fact_57_tree_Ocollapse_I2_J,axiom,
! [A: $tType,Tree: tree @ A] :
( ~ ( is_Leaf @ A @ Tree )
=> ( ( node @ A @ ( left @ A @ Tree ) @ ( val @ A @ Tree ) @ ( right @ A @ Tree ) )
= Tree ) ) ).
% tree.collapse(2)
thf(fact_58__092_060Phi_062_Ocases,axiom,
! [A: $tType,X: tree @ A] :
( ( X
!= ( leaf @ A ) )
=> ~ ! [L: tree @ A,A6: A,R: tree @ A] :
( X
!= ( node @ A @ L @ A6 @ R ) ) ) ).
% \<Phi>.cases
thf(fact_59__092_060Phi_062_Oinduct,axiom,
! [A: $tType,P: ( tree @ A ) > $o,A0: tree @ A] :
( ( P @ ( leaf @ A ) )
=> ( ! [L: tree @ A,A6: A,R: tree @ A] :
( ( P @ L )
=> ( ( P @ R )
=> ( P @ ( node @ A @ L @ A6 @ R ) ) ) )
=> ( P @ A0 ) ) ) ).
% \<Phi>.induct
thf(fact_60_t__splay__max_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: tree @ A] :
( ( X
!= ( leaf @ A ) )
=> ( ! [L: tree @ A,B2: A] :
( X
!= ( node @ A @ L @ B2 @ ( leaf @ A ) ) )
=> ~ ! [L: tree @ A,B2: A,Rl: tree @ A,C2: A,Rr: tree @ A] :
( X
!= ( node @ A @ L @ B2 @ ( node @ A @ Rl @ C2 @ Rr ) ) ) ) ) ) ).
% t_splay_max.cases
thf(fact_61_t__splay__max_Oinduct,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: ( tree @ A ) > $o,A0: tree @ A] :
( ( P @ ( leaf @ A ) )
=> ( ! [L: tree @ A,B2: A] : ( P @ ( node @ A @ L @ B2 @ ( leaf @ A ) ) )
=> ( ! [L: tree @ A,B2: A,Rl: tree @ A,C2: A,Rr: tree @ A] :
( ( ( Rr
!= ( leaf @ A ) )
=> ( P @ Rr ) )
=> ( P @ ( node @ A @ L @ B2 @ ( node @ A @ Rl @ C2 @ Rr ) ) ) )
=> ( P @ A0 ) ) ) ) ) ).
% t_splay_max.induct
thf(fact_62_tree_Ocase__eq__if,axiom,
! [A: $tType,B: $tType] :
( ( case_tree @ B @ A )
= ( ^ [F12: B,F22: ( tree @ A ) > A > ( tree @ A ) > B,Tree2: tree @ A] : ( if @ B @ ( is_Leaf @ A @ Tree2 ) @ F12 @ ( F22 @ ( left @ A @ Tree2 ) @ ( val @ A @ Tree2 ) @ ( right @ A @ Tree2 ) ) ) ) ) ).
% tree.case_eq_if
thf(fact_63_bst__eq_Oelims_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: tree @ A] :
( ( linorder_bst_eq @ A @ X )
=> ( ( X
!= ( leaf @ A ) )
=> ~ ! [L: tree @ A,A6: A,R: tree @ A] :
( ( X
= ( node @ A @ L @ A6 @ R ) )
=> ~ ( ( linorder_bst_eq @ A @ L )
& ( linorder_bst_eq @ A @ R )
& ! [X4: A] :
( ( member @ A @ X4 @ ( set_tree @ A @ L ) )
=> ( ord_less_eq @ A @ X4 @ A6 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ ( set_tree @ A @ R ) )
=> ( ord_less_eq @ A @ A6 @ X4 ) ) ) ) ) ) ) ).
% bst_eq.elims(2)
thf(fact_64_bst__eq_Oelims_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: tree @ A,Y2: $o] :
( ( ( linorder_bst_eq @ A @ X )
= Y2 )
=> ( ( ( X
= ( leaf @ A ) )
=> ~ Y2 )
=> ~ ! [L: tree @ A,A6: A,R: tree @ A] :
( ( X
= ( node @ A @ L @ A6 @ R ) )
=> ( Y2
= ( ~ ( ( linorder_bst_eq @ A @ L )
& ( linorder_bst_eq @ A @ R )
& ! [X2: A] :
( ( member @ A @ X2 @ ( set_tree @ A @ L ) )
=> ( ord_less_eq @ A @ X2 @ A6 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ ( set_tree @ A @ R ) )
=> ( ord_less_eq @ A @ A6 @ X2 ) ) ) ) ) ) ) ) ) ).
% bst_eq.elims(1)
thf(fact_65_powr__mono,axiom,
! [A2: real,B3: real,X: real] :
( ( ord_less_eq @ real @ A2 @ B3 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ord_less_eq @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ X @ B3 ) ) ) ) ).
% powr_mono
thf(fact_66_tree_OdiscI_I2_J,axiom,
! [A: $tType,Tree: tree @ A,X21: tree @ A,X22: A,X23: tree @ A] :
( ( Tree
= ( node @ A @ X21 @ X22 @ X23 ) )
=> ~ ( is_Leaf @ A @ Tree ) ) ).
% tree.discI(2)
thf(fact_67_tree_Odisc_I2_J,axiom,
! [A: $tType,X21: tree @ A,X22: A,X23: tree @ A] :
~ ( is_Leaf @ A @ ( node @ A @ X21 @ X22 @ X23 ) ) ).
% tree.disc(2)
thf(fact_68_is__Leaf__def,axiom,
! [A: $tType] :
( ( is_Leaf @ A )
= ( ^ [Tree2: tree @ A] :
( Tree2
= ( leaf @ A ) ) ) ) ).
% is_Leaf_def
thf(fact_69_tree_Ocollapse_I1_J,axiom,
! [A: $tType,Tree: tree @ A] :
( ( is_Leaf @ A @ Tree )
=> ( Tree
= ( leaf @ A ) ) ) ).
% tree.collapse(1)
thf(fact_70_tree_OdiscI_I1_J,axiom,
! [A: $tType,Tree: tree @ A] :
( ( Tree
= ( leaf @ A ) )
=> ( is_Leaf @ A @ Tree ) ) ).
% tree.discI(1)
thf(fact_71_tree_Odisc_I1_J,axiom,
! [A: $tType] : ( is_Leaf @ A @ ( leaf @ A ) ) ).
% tree.disc(1)
thf(fact_72_tree_Oset__sel_I3_J,axiom,
! [A: $tType,A2: tree @ A,Xa: A] :
( ~ ( is_Leaf @ A @ A2 )
=> ( ( member @ A @ Xa @ ( set_tree @ A @ ( right @ A @ A2 ) ) )
=> ( member @ A @ Xa @ ( set_tree @ A @ A2 ) ) ) ) ).
% tree.set_sel(3)
thf(fact_73_tree_Oset__sel_I1_J,axiom,
! [A: $tType,A2: tree @ A,X: A] :
( ~ ( is_Leaf @ A @ A2 )
=> ( ( member @ A @ X @ ( set_tree @ A @ ( left @ A @ A2 ) ) )
=> ( member @ A @ X @ ( set_tree @ A @ A2 ) ) ) ) ).
% tree.set_sel(1)
thf(fact_74_tree_Oset__sel_I2_J,axiom,
! [A: $tType,A2: tree @ A] :
( ~ ( is_Leaf @ A @ A2 )
=> ( member @ A @ ( val @ A @ A2 ) @ ( set_tree @ A @ A2 ) ) ) ).
% tree.set_sel(2)
thf(fact_75_bst__eq_Oelims_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: tree @ A] :
( ~ ( linorder_bst_eq @ A @ X )
=> ~ ! [L: tree @ A,A6: A,R: tree @ A] :
( ( X
= ( node @ A @ L @ A6 @ R ) )
=> ( ( linorder_bst_eq @ A @ L )
& ( linorder_bst_eq @ A @ R )
& ! [X3: A] :
( ( member @ A @ X3 @ ( set_tree @ A @ L ) )
=> ( ord_less_eq @ A @ X3 @ A6 ) )
& ! [X3: A] :
( ( member @ A @ X3 @ ( set_tree @ A @ R ) )
=> ( ord_less_eq @ A @ A6 @ X3 ) ) ) ) ) ) ).
% bst_eq.elims(3)
thf(fact_76_bst__eq_Osimps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [L2: tree @ A,A2: A,R2: tree @ A] :
( ( linorder_bst_eq @ A @ ( node @ A @ L2 @ A2 @ R2 ) )
= ( ( linorder_bst_eq @ A @ L2 )
& ( linorder_bst_eq @ A @ R2 )
& ! [X2: A] :
( ( member @ A @ X2 @ ( set_tree @ A @ L2 ) )
=> ( ord_less_eq @ A @ X2 @ A2 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ ( set_tree @ A @ R2 ) )
=> ( ord_less_eq @ A @ A2 @ X2 ) ) ) ) ) ).
% bst_eq.simps(2)
thf(fact_77_tree_Oexpand,axiom,
! [A: $tType,Tree: tree @ A,Tree3: tree @ A] :
( ( ( is_Leaf @ A @ Tree )
= ( is_Leaf @ A @ Tree3 ) )
=> ( ( ~ ( is_Leaf @ A @ Tree )
=> ( ~ ( is_Leaf @ A @ Tree3 )
=> ( ( ( left @ A @ Tree )
= ( left @ A @ Tree3 ) )
& ( ( val @ A @ Tree )
= ( val @ A @ Tree3 ) )
& ( ( right @ A @ Tree )
= ( right @ A @ Tree3 ) ) ) ) )
=> ( Tree = Tree3 ) ) ) ).
% tree.expand
thf(fact_78_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_79_le__numeral__extra_I4_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% le_numeral_extra(4)
thf(fact_80_splay__max__eq__splay,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: tree @ A,A2: A] :
( ( linorder_bst @ A @ T )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set_tree @ A @ T ) )
=> ( ord_less_eq @ A @ X3 @ A2 ) )
=> ( ( splay_splay_max @ A @ T )
= ( splay_splay @ A @ A2 @ T ) ) ) ) ) ).
% splay_max_eq_splay
thf(fact_81_ex__in__set__tree,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: tree @ A,A2: A] :
( ( T
!= ( leaf @ A ) )
=> ( ( linorder_bst @ A @ T )
=> ? [X3: A] :
( ( member @ A @ X3 @ ( set_tree @ A @ T ) )
& ( ( splay_splay @ A @ X3 @ T )
= ( splay_splay @ A @ A2 @ T ) )
& ( ( splay_914434265_splay @ A @ X3 @ T )
= ( splay_914434265_splay @ A @ A2 @ T ) ) ) ) ) ) ).
% ex_in_set_tree
thf(fact_82_bst_Oelims_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: tree @ A] :
( ( linorder_bst @ A @ X )
=> ( ( X
!= ( leaf @ A ) )
=> ~ ! [L: tree @ A,A6: A,R: tree @ A] :
( ( X
= ( node @ A @ L @ A6 @ R ) )
=> ~ ( ( linorder_bst @ A @ L )
& ( linorder_bst @ A @ R )
& ! [X4: A] :
( ( member @ A @ X4 @ ( set_tree @ A @ L ) )
=> ( ord_less @ A @ X4 @ A6 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ ( set_tree @ A @ R ) )
=> ( ord_less @ A @ A6 @ X4 ) ) ) ) ) ) ) ).
% bst.elims(2)
thf(fact_83_bst_Oelims_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: tree @ A,Y2: $o] :
( ( ( linorder_bst @ A @ X )
= Y2 )
=> ( ( ( X
= ( leaf @ A ) )
=> ~ Y2 )
=> ~ ! [L: tree @ A,A6: A,R: tree @ A] :
( ( X
= ( node @ A @ L @ A6 @ R ) )
=> ( Y2
= ( ~ ( ( linorder_bst @ A @ L )
& ( linorder_bst @ A @ R )
& ! [X2: A] :
( ( member @ A @ X2 @ ( set_tree @ A @ L ) )
=> ( ord_less @ A @ X2 @ A6 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ ( set_tree @ A @ R ) )
=> ( ord_less @ A @ A6 @ X2 ) ) ) ) ) ) ) ) ) ).
% bst.elims(1)
thf(fact_84_tree_Orel__sel,axiom,
! [B: $tType,A: $tType] :
( ( rel_tree @ A @ B )
= ( ^ [R4: A > B > $o,A5: tree @ A,B4: tree @ B] :
( ( ( is_Leaf @ A @ A5 )
= ( is_Leaf @ B @ B4 ) )
& ( ~ ( is_Leaf @ A @ A5 )
=> ( ~ ( is_Leaf @ B @ B4 )
=> ( ( rel_tree @ A @ B @ R4 @ ( left @ A @ A5 ) @ ( left @ B @ B4 ) )
& ( R4 @ ( val @ A @ A5 ) @ ( val @ B @ B4 ) )
& ( rel_tree @ A @ B @ R4 @ ( right @ A @ A5 ) @ ( right @ B @ B4 ) ) ) ) ) ) ) ) ).
% tree.rel_sel
thf(fact_85_powr__less__cancel__iff,axiom,
! [X: real,A2: real,B3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ X @ B3 ) )
= ( ord_less @ real @ A2 @ B3 ) ) ) ).
% powr_less_cancel_iff
thf(fact_86_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_87_splay__max__Leaf__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: tree @ A] :
( ( ( splay_splay_max @ A @ T )
= ( leaf @ A ) )
= ( T
= ( leaf @ A ) ) ) ) ).
% splay_max_Leaf_iff
thf(fact_88_powr__le__cancel__iff,axiom,
! [X: real,A2: real,B3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ X @ B3 ) )
= ( ord_less_eq @ real @ A2 @ B3 ) ) ) ).
% powr_le_cancel_iff
thf(fact_89_t__splay__simps_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,L2: tree @ A,Rl2: tree @ A,Rr2: tree @ A] :
( ( ord_less @ A @ B3 @ A2 )
=> ( ( splay_914434265_splay @ A @ A2 @ ( node @ A @ L2 @ B3 @ ( node @ A @ Rl2 @ A2 @ Rr2 ) ) )
= ( one_one @ nat ) ) ) ) ).
% t_splay_simps(7)
thf(fact_90_t__splay__simps_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,Ll: tree @ A,Lr: tree @ A,R2: tree @ A] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( splay_914434265_splay @ A @ A2 @ ( node @ A @ ( node @ A @ Ll @ A2 @ Lr ) @ B3 @ R2 ) )
= ( one_one @ nat ) ) ) ) ).
% t_splay_simps(3)
thf(fact_91_t__splay__simps_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,L2: tree @ A] :
( ( ord_less @ A @ B3 @ A2 )
=> ( ( splay_914434265_splay @ A @ A2 @ ( node @ A @ L2 @ B3 @ ( leaf @ A ) ) )
= ( one_one @ nat ) ) ) ) ).
% t_splay_simps(6)
thf(fact_92_t__splay__simps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,R2: tree @ A] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( splay_914434265_splay @ A @ A2 @ ( node @ A @ ( leaf @ A ) @ B3 @ R2 ) )
= ( one_one @ nat ) ) ) ) ).
% t_splay_simps(2)
thf(fact_93_leD,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y2: A,X: A] :
( ( ord_less_eq @ A @ Y2 @ X )
=> ~ ( ord_less @ A @ X @ Y2 ) ) ) ).
% leD
thf(fact_94_leI,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ~ ( ord_less @ A @ X @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% leI
thf(fact_95_le__less,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
| ( X2 = Y ) ) ) ) ) ).
% le_less
thf(fact_96_less__le,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
& ( X2 != Y ) ) ) ) ) ).
% less_le
thf(fact_97_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B3: B,C3: B] :
( ( ord_less_eq @ A @ A2 @ ( F @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C3 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less @ B @ X3 @ Y3 )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C3 ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_98_order__le__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C @ ( type2 @ C ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B3: A,F: A > C,C3: C] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ord_less @ C @ ( F @ B3 ) @ C3 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ C @ ( F @ A2 ) @ C3 ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_99_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B3: B,C3: B] :
( ( ord_less @ A @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C3 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less_eq @ B @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C3 ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_100_order__less__le__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C @ ( type2 @ C ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B3: A,F: A > C,C3: C] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( ord_less_eq @ C @ ( F @ B3 ) @ C3 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less @ C @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ C @ ( F @ A2 ) @ C3 ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_101_not__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ~ ( ord_less_eq @ A @ X @ Y2 ) )
= ( ord_less @ A @ Y2 @ X ) ) ) ).
% not_le
thf(fact_102_not__less,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ~ ( ord_less @ A @ X @ Y2 ) )
= ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% not_less
thf(fact_103_le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_less @ A @ A2 @ B3 ) ) ) ) ).
% le_neq_trans
thf(fact_104_less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ord_less_eq @ A @ X @ Y2 ) ) ) ).
% less_imp_le
thf(fact_105_antisym__conv1,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ~ ( ord_less @ A @ X @ Y2 )
=> ( ( ord_less_eq @ A @ X @ Y2 )
= ( X = Y2 ) ) ) ) ).
% antisym_conv1
thf(fact_106_antisym__conv2,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ~ ( ord_less @ A @ X @ Y2 ) )
= ( X = Y2 ) ) ) ) ).
% antisym_conv2
thf(fact_107_le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A,Z4: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ord_less @ A @ Y2 @ Z4 )
=> ( ord_less @ A @ X @ Z4 ) ) ) ) ).
% le_less_trans
thf(fact_108_less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A,Z4: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ Z4 )
=> ( ord_less @ A @ X @ Z4 ) ) ) ) ).
% less_le_trans
thf(fact_109_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Z4: A,Y2: A] :
( ! [X3: A] :
( ( ord_less @ A @ Z4 @ X3 )
=> ( ord_less_eq @ A @ Y2 @ X3 ) )
=> ( ord_less_eq @ A @ Y2 @ Z4 ) ) ) ).
% dense_ge
thf(fact_110_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Y2: A,Z4: A] :
( ! [X3: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less_eq @ A @ X3 @ Z4 ) )
=> ( ord_less_eq @ A @ Y2 @ Z4 ) ) ) ).
% dense_le
thf(fact_111_le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
| ( ord_less @ A @ Y2 @ X ) ) ) ).
% le_less_linear
thf(fact_112_le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ord_less @ A @ X @ Y2 )
| ( X = Y2 ) ) ) ) ).
% le_imp_less_or_eq
thf(fact_113_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
& ~ ( ord_less_eq @ A @ Y @ X2 ) ) ) ) ) ).
% less_le_not_le
thf(fact_114_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y2: A,X: A] :
( ~ ( ord_less_eq @ A @ Y2 @ X )
=> ( ord_less @ A @ X @ Y2 ) ) ) ).
% not_le_imp_less
thf(fact_115_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ord_less @ A @ B3 @ C3 )
=> ( ord_less @ A @ A2 @ C3 ) ) ) ) ).
% order.strict_trans1
thf(fact_116_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C3: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C3 )
=> ( ord_less @ A @ A2 @ C3 ) ) ) ) ).
% order.strict_trans2
thf(fact_117_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
( ( ord_less @ A @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_118_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_119_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,C3: A] :
( ( ord_less_eq @ A @ B3 @ A2 )
=> ( ( ord_less @ A @ C3 @ B3 )
=> ( ord_less @ A @ C3 @ A2 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_120_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,C3: A] :
( ( ord_less @ A @ B3 @ A2 )
=> ( ( ord_less_eq @ A @ C3 @ B3 )
=> ( ord_less @ A @ C3 @ A2 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_121_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Z4: A,X: A,Y2: A] :
( ( ord_less @ A @ Z4 @ X )
=> ( ! [W: A] :
( ( ord_less @ A @ Z4 @ W )
=> ( ( ord_less @ A @ W @ X )
=> ( ord_less_eq @ A @ Y2 @ W ) ) )
=> ( ord_less_eq @ A @ Y2 @ Z4 ) ) ) ) ).
% dense_ge_bounded
thf(fact_122_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A,Z4: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ! [W: A] :
( ( ord_less @ A @ X @ W )
=> ( ( ord_less @ A @ W @ Y2 )
=> ( ord_less_eq @ A @ W @ Z4 ) ) )
=> ( ord_less_eq @ A @ Y2 @ Z4 ) ) ) ) ).
% dense_le_bounded
thf(fact_123_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ord_less_eq @ A @ A2 @ B3 ) ) ) ).
% order.strict_implies_order
thf(fact_124_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A5: A] :
( ( ord_less @ A @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_125_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [B4: A,A5: A] :
( ( ord_less_eq @ A @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_126_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A] :
( ( ord_less @ A @ B3 @ A2 )
=> ( ord_less_eq @ A @ B3 @ A2 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_127_order_Onot__eq__order__implies__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( A2 != B3 )
=> ( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ord_less @ A @ A2 @ B3 ) ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_128_tree_Orel__mono,axiom,
! [B: $tType,A: $tType,R5: A > B > $o,Ra: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ R5 @ Ra )
=> ( ord_less_eq @ ( ( tree @ A ) > ( tree @ B ) > $o ) @ ( rel_tree @ A @ B @ R5 ) @ ( rel_tree @ A @ B @ Ra ) ) ) ).
% tree.rel_mono
thf(fact_129_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_130_tree_Orel__eq,axiom,
! [A: $tType] :
( ( rel_tree @ A @ A
@ ^ [Y4: A,Z5: A] : Y4 = Z5 )
= ( ^ [Y4: tree @ A,Z5: tree @ A] : Y4 = Z5 ) ) ).
% tree.rel_eq
thf(fact_131_tree_Orel__refl,axiom,
! [B: $tType,Ra: B > B > $o,X: tree @ B] :
( ! [X3: B] : ( Ra @ X3 @ X3 )
=> ( rel_tree @ B @ B @ Ra @ X @ X ) ) ).
% tree.rel_refl
thf(fact_132_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A] :
( ( ord_less @ A @ B3 @ A2 )
=> ( A2 != B3 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_133_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ( A2 != B3 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_134_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ~ ( ord_less @ A @ X @ Y2 ) )
= ( ( ord_less @ A @ Y2 @ X )
| ( X = Y2 ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_135_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,C3: A] :
( ( ord_less @ A @ B3 @ A2 )
=> ( ( ord_less @ A @ C3 @ B3 )
=> ( ord_less @ A @ C3 @ A2 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_136_less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ~ ( ord_less @ A @ Y2 @ X ) ) ) ).
% less_imp_not_less
thf(fact_137_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C3: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( ord_less @ A @ B3 @ C3 )
=> ( ord_less @ A @ A2 @ C3 ) ) ) ) ).
% order.strict_trans
thf(fact_138_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% dual_order.irrefl
thf(fact_139_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ~ ( ord_less @ A @ X @ Y2 )
=> ( ( X != Y2 )
=> ( ord_less @ A @ Y2 @ X ) ) ) ) ).
% linorder_cases
thf(fact_140_less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A,P: $o] :
( ( ord_less @ A @ X @ Y2 )
=> ( ( ord_less @ A @ Y2 @ X )
=> P ) ) ) ).
% less_imp_triv
thf(fact_141_less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( Y2 != X ) ) ) ).
% less_imp_not_eq2
thf(fact_142_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y2: A,X: A] :
( ~ ( ord_less @ A @ Y2 @ X )
=> ( ( ~ ( ord_less @ A @ X @ Y2 ) )
= ( X = Y2 ) ) ) ) ).
% antisym_conv3
thf(fact_143_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,A2: A] :
( ! [X3: A] :
( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ) ).
% less_induct
thf(fact_144_less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ~ ( ord_less @ A @ Y2 @ X ) ) ) ).
% less_not_sym
thf(fact_145_less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( X != Y2 ) ) ) ).
% less_imp_not_eq
thf(fact_146_dual__order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A] :
( ( ord_less @ A @ B3 @ A2 )
=> ~ ( ord_less @ A @ A2 @ B3 ) ) ) ).
% dual_order.asym
thf(fact_147_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C3: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( B3 = C3 )
=> ( ord_less @ A @ A2 @ C3 ) ) ) ) ).
% ord_less_eq_trans
thf(fact_148_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C3: A] :
( ( A2 = B3 )
=> ( ( ord_less @ A @ B3 @ C3 )
=> ( ord_less @ A @ A2 @ C3 ) ) ) ) ).
% ord_eq_less_trans
thf(fact_149_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] :
~ ( ord_less @ A @ X @ X ) ) ).
% less_irrefl
thf(fact_150_less__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
| ( X = Y2 )
| ( ord_less @ A @ Y2 @ X ) ) ) ).
% less_linear
thf(fact_151_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A,Z4: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ( ord_less @ A @ Y2 @ Z4 )
=> ( ord_less @ A @ X @ Z4 ) ) ) ) ).
% less_trans
thf(fact_152_less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ~ ( ord_less @ A @ B3 @ A2 ) ) ) ).
% less_asym'
thf(fact_153_less__asym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ~ ( ord_less @ A @ Y2 @ X ) ) ) ).
% less_asym
thf(fact_154_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( X != Y2 ) ) ) ).
% less_imp_neq
thf(fact_155_dense,axiom,
! [A: $tType] :
( ( dense_order @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ? [Z: A] :
( ( ord_less @ A @ X @ Z )
& ( ord_less @ A @ Z @ Y2 ) ) ) ) ).
% dense
thf(fact_156_order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ~ ( ord_less @ A @ B3 @ A2 ) ) ) ).
% order.asym
thf(fact_157_neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( X != Y2 )
= ( ( ord_less @ A @ X @ Y2 )
| ( ord_less @ A @ Y2 @ X ) ) ) ) ).
% neq_iff
thf(fact_158_neqE,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( X != Y2 )
=> ( ~ ( ord_less @ A @ X @ Y2 )
=> ( ord_less @ A @ Y2 @ X ) ) ) ) ).
% neqE
thf(fact_159_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [X1: A] : ( ord_less @ A @ X @ X1 ) ) ).
% gt_ex
thf(fact_160_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [Y3: A] : ( ord_less @ A @ Y3 @ X ) ) ).
% lt_ex
thf(fact_161_order__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C @ ( type2 @ C ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B3: A,F: A > C,C3: C] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( ord_less @ C @ ( F @ B3 ) @ C3 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less @ C @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ C @ ( F @ A2 ) @ C3 ) ) ) ) ) ).
% order_less_subst2
thf(fact_162_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B3: B,C3: B] :
( ( ord_less @ A @ A2 @ ( F @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C3 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less @ B @ X3 @ Y3 )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C3 ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_163_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B3: A,F: A > B,C3: B] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C3 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less @ B @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ B @ ( F @ A2 ) @ C3 ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_164_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B3: B,C3: B] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C3 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less @ B @ X3 @ Y3 )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C3 ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_165_powr__powr__swap,axiom,
! [X: real,A2: real,B3: real] :
( ( powr @ real @ ( powr @ real @ X @ A2 ) @ B3 )
= ( powr @ real @ ( powr @ real @ X @ B3 ) @ A2 ) ) ).
% powr_powr_swap
thf(fact_166_tree_Orel__inject_I2_J,axiom,
! [A: $tType,B: $tType,R5: A > B > $o,X21: tree @ A,X22: A,X23: tree @ A,Y21: tree @ B,Y22: B,Y23: tree @ B] :
( ( rel_tree @ A @ B @ R5 @ ( node @ A @ X21 @ X22 @ X23 ) @ ( node @ B @ Y21 @ Y22 @ Y23 ) )
= ( ( rel_tree @ A @ B @ R5 @ X21 @ Y21 )
& ( R5 @ X22 @ Y22 )
& ( rel_tree @ A @ B @ R5 @ X23 @ Y23 ) ) ) ).
% tree.rel_inject(2)
thf(fact_167_tree_Orel__intros_I2_J,axiom,
! [A: $tType,B: $tType,R5: A > B > $o,X21: tree @ A,Y21: tree @ B,X22: A,Y22: B,X23: tree @ A,Y23: tree @ B] :
( ( rel_tree @ A @ B @ R5 @ X21 @ Y21 )
=> ( ( R5 @ X22 @ Y22 )
=> ( ( rel_tree @ A @ B @ R5 @ X23 @ Y23 )
=> ( rel_tree @ A @ B @ R5 @ ( node @ A @ X21 @ X22 @ X23 ) @ ( node @ B @ Y21 @ Y22 @ Y23 ) ) ) ) ) ).
% tree.rel_intros(2)
thf(fact_168_tree_Octr__transfer_I1_J,axiom,
! [A: $tType,B: $tType,R5: A > B > $o] : ( rel_tree @ A @ B @ R5 @ ( leaf @ A ) @ ( leaf @ B ) ) ).
% tree.ctr_transfer(1)
thf(fact_169_tree_Orel__refl__strong,axiom,
! [A: $tType,X: tree @ A,Ra: A > A > $o] :
( ! [Z: A] :
( ( member @ A @ Z @ ( set_tree @ A @ X ) )
=> ( Ra @ Z @ Z ) )
=> ( rel_tree @ A @ A @ Ra @ X @ X ) ) ).
% tree.rel_refl_strong
thf(fact_170_tree_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,R5: A > B > $o,X: tree @ A,Y2: tree @ B,Ra: A > B > $o] :
( ( rel_tree @ A @ B @ R5 @ X @ Y2 )
=> ( ! [Z: A,Yb: B] :
( ( member @ A @ Z @ ( set_tree @ A @ X ) )
=> ( ( member @ B @ Yb @ ( set_tree @ B @ Y2 ) )
=> ( ( R5 @ Z @ Yb )
=> ( Ra @ Z @ Yb ) ) ) )
=> ( rel_tree @ A @ B @ Ra @ X @ Y2 ) ) ) ).
% tree.rel_mono_strong
thf(fact_171_tree_Orel__cong,axiom,
! [A: $tType,B: $tType,X: tree @ A,Ya: tree @ A,Y2: tree @ B,Xa: tree @ B,R5: A > B > $o,Ra: A > B > $o] :
( ( X = Ya )
=> ( ( Y2 = Xa )
=> ( ! [Z: A,Yb: B] :
( ( member @ A @ Z @ ( set_tree @ A @ Ya ) )
=> ( ( member @ B @ Yb @ ( set_tree @ B @ Xa ) )
=> ( ( R5 @ Z @ Yb )
= ( Ra @ Z @ Yb ) ) ) )
=> ( ( rel_tree @ A @ B @ R5 @ X @ Y2 )
= ( rel_tree @ A @ B @ Ra @ Ya @ Xa ) ) ) ) ) ).
% tree.rel_cong
thf(fact_172_powr__less__mono,axiom,
! [A2: real,B3: real,X: real] :
( ( ord_less @ real @ A2 @ B3 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ord_less @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ X @ B3 ) ) ) ) ).
% powr_less_mono
thf(fact_173_powr__less__cancel,axiom,
! [X: real,A2: real,B3: real] :
( ( ord_less @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ X @ B3 ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ord_less @ real @ A2 @ B3 ) ) ) ).
% powr_less_cancel
thf(fact_174_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_175_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_176_set__splay__max,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: tree @ A] :
( ( set_tree @ A @ ( splay_splay_max @ A @ T ) )
= ( set_tree @ A @ T ) ) ) ).
% set_splay_max
thf(fact_177_bst__splay__max,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: tree @ A] :
( ( linorder_bst @ A @ T )
=> ( linorder_bst @ A @ ( splay_splay_max @ A @ T ) ) ) ) ).
% bst_splay_max
thf(fact_178_tree_Orel__induct,axiom,
! [A: $tType,B: $tType,R5: A > B > $o,X: tree @ A,Y2: tree @ B,Q: ( tree @ A ) > ( tree @ B ) > $o] :
( ( rel_tree @ A @ B @ R5 @ X @ Y2 )
=> ( ( Q @ ( leaf @ A ) @ ( leaf @ B ) )
=> ( ! [A21: tree @ A,A222: A,A23: tree @ A,B21: tree @ B,B22: B,B23: tree @ B] :
( ( Q @ A21 @ B21 )
=> ( ( R5 @ A222 @ B22 )
=> ( ( Q @ A23 @ B23 )
=> ( Q @ ( node @ A @ A21 @ A222 @ A23 ) @ ( node @ B @ B21 @ B22 @ B23 ) ) ) ) )
=> ( Q @ X @ Y2 ) ) ) ) ).
% tree.rel_induct
thf(fact_179_tree_Orel__cases,axiom,
! [A: $tType,B: $tType,R5: A > B > $o,A2: tree @ A,B3: tree @ B] :
( ( rel_tree @ A @ B @ R5 @ A2 @ B3 )
=> ( ( ( A2
= ( leaf @ A ) )
=> ( B3
!= ( leaf @ B ) ) )
=> ~ ! [X1: tree @ A,X24: A,X32: tree @ A] :
( ( A2
= ( node @ A @ X1 @ X24 @ X32 ) )
=> ! [Y1: tree @ B,Y24: B,Y32: tree @ B] :
( ( B3
= ( node @ B @ Y1 @ Y24 @ Y32 ) )
=> ( ( rel_tree @ A @ B @ R5 @ X1 @ Y1 )
=> ( ( R5 @ X24 @ Y24 )
=> ~ ( rel_tree @ A @ B @ R5 @ X32 @ Y32 ) ) ) ) ) ) ) ).
% tree.rel_cases
thf(fact_180_tree_Orel__distinct_I1_J,axiom,
! [A: $tType,B: $tType,R5: A > B > $o,Y21: tree @ B,Y22: B,Y23: tree @ B] :
~ ( rel_tree @ A @ B @ R5 @ ( leaf @ A ) @ ( node @ B @ Y21 @ Y22 @ Y23 ) ) ).
% tree.rel_distinct(1)
thf(fact_181_tree_Orel__distinct_I2_J,axiom,
! [A: $tType,B: $tType,R5: A > B > $o,Y21: tree @ A,Y22: A,Y23: tree @ A] :
~ ( rel_tree @ A @ B @ R5 @ ( node @ A @ Y21 @ Y22 @ Y23 ) @ ( leaf @ B ) ) ).
% tree.rel_distinct(2)
thf(fact_182_splay_Osimps_I9_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,Lb: tree @ A,La: tree @ A,Ra2: tree @ A] :
( ( ord_less @ A @ B3 @ A2 )
=> ( ( splay_splay @ A @ A2 @ ( node @ A @ Lb @ B3 @ ( node @ A @ La @ A2 @ Ra2 ) ) )
= ( node @ A @ ( node @ A @ Lb @ B3 @ La ) @ A2 @ Ra2 ) ) ) ) ).
% splay.simps(9)
thf(fact_183_splay_Osimps_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,La: tree @ A,Ra2: tree @ A,Rb: tree @ A] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( splay_splay @ A @ A2 @ ( node @ A @ ( node @ A @ La @ A2 @ Ra2 ) @ B3 @ Rb ) )
= ( node @ A @ La @ A2 @ ( node @ A @ Ra2 @ B3 @ Rb ) ) ) ) ) ).
% splay.simps(3)
thf(fact_184_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A] :
( ( ord_less_eq @ A @ B3 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ) ).
% dual_order.antisym
thf(fact_185_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,C3: A] :
( ( ord_less_eq @ A @ B3 @ A2 )
=> ( ( ord_less_eq @ A @ C3 @ B3 )
=> ( ord_less_eq @ A @ C3 @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_186_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,A2: A,B3: A] :
( ! [A6: A,B2: A] :
( ( ord_less_eq @ A @ A6 @ B2 )
=> ( P @ A6 @ B2 ) )
=> ( ! [A6: A,B2: A] :
( ( P @ B2 @ A6 )
=> ( P @ A6 @ B2 ) )
=> ( P @ A2 @ B3 ) ) ) ) ).
% linorder_wlog
thf(fact_187_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_188_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A,Z4: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ Z4 )
=> ( ord_less_eq @ A @ X @ Z4 ) ) ) ) ).
% order_trans
thf(fact_189_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ) ).
% order_class.order.antisym
thf(fact_190_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( B3 = C3 )
=> ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_191_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C3: A] :
( ( A2 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C3 )
=> ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_192_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Y2: A,X: A] :
( ( ord_less_eq @ A @ Y2 @ X )
=> ( ( ord_less_eq @ A @ X @ Y2 )
= ( X = Y2 ) ) ) ) ).
% antisym_conv
thf(fact_193_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A,Z4: A] :
( ( ( ord_less_eq @ A @ X @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ Z4 ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ X )
=> ~ ( ord_less_eq @ A @ X @ Z4 ) )
=> ( ( ( ord_less_eq @ A @ X @ Z4 )
=> ~ ( ord_less_eq @ A @ Z4 @ Y2 ) )
=> ( ( ( ord_less_eq @ A @ Z4 @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ X ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ Z4 )
=> ~ ( ord_less_eq @ A @ Z4 @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z4 @ X )
=> ~ ( ord_less_eq @ A @ X @ Y2 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_194_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C3 )
=> ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% order.trans
thf(fact_195_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ~ ( ord_less_eq @ A @ X @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% le_cases
thf(fact_196_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( X = Y2 )
=> ( ord_less_eq @ A @ X @ Y2 ) ) ) ).
% eq_refl
thf(fact_197_linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
| ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% linear
thf(fact_198_antisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ X )
=> ( X = Y2 ) ) ) ) ).
% antisym
thf(fact_199_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y4: A,Z5: A] : Y4 = Z5 )
= ( ^ [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
& ( ord_less_eq @ A @ Y @ X2 ) ) ) ) ) ).
% eq_iff
thf(fact_200_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B3: A,F: A > B,C3: B] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C3 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ B @ ( F @ A2 ) @ C3 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_201_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B3: B,C3: B] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C3 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less_eq @ B @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C3 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_202_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C @ ( type2 @ C ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B3: A,F: A > C,C3: C] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ord_less_eq @ C @ ( F @ B3 ) @ C3 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ C @ ( F @ A2 ) @ C3 ) ) ) ) ) ).
% order_subst2
thf(fact_203_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B3: B,C3: B] :
( ( ord_less_eq @ A @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C3 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less_eq @ B @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C3 ) ) ) ) ) ) ).
% order_subst1
thf(fact_204_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F3: A > B,G2: A > B] :
! [X2: A] : ( ord_less_eq @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% le_fun_def
thf(fact_205_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B] :
( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_206_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funE
thf(fact_207_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funD
thf(fact_208_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_209_splay__max__Leaf,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: tree @ A,L2: tree @ A,A2: A,R2: tree @ A] :
( ( ( splay_splay_max @ A @ T )
= ( node @ A @ L2 @ A2 @ R2 ) )
=> ( R2
= ( leaf @ A ) ) ) ) ).
% splay_max_Leaf
thf(fact_210_splay_Osimps_I13_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: A,X: A,B3: A,La: tree @ A,Lb: tree @ A] :
( ( ord_less @ A @ A2 @ X )
=> ( ( ord_less @ A @ B3 @ X )
=> ( ( splay_splay @ A @ X @ ( node @ A @ La @ A2 @ ( node @ A @ Lb @ B3 @ ( leaf @ A ) ) ) )
= ( node @ A @ ( node @ A @ La @ A2 @ Lb ) @ B3 @ ( leaf @ A ) ) ) ) ) ) ).
% splay.simps(13)
thf(fact_211_splay_Osimps_I12_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: A,X: A,B3: A,La: tree @ A,Rb: tree @ A] :
( ( ord_less @ A @ A2 @ X )
=> ( ( ord_less @ A @ X @ B3 )
=> ( ( splay_splay @ A @ X @ ( node @ A @ La @ A2 @ ( node @ A @ ( leaf @ A ) @ B3 @ Rb ) ) )
= ( node @ A @ ( node @ A @ La @ A2 @ ( leaf @ A ) ) @ B3 @ Rb ) ) ) ) ) ).
% splay.simps(12)
thf(fact_212_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_213_splay_Osimps_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,B3: A,A2: A,La: tree @ A,Rb: tree @ A] :
( ( ord_less @ A @ X @ B3 )
=> ( ( ord_less @ A @ A2 @ X )
=> ( ( splay_splay @ A @ X @ ( node @ A @ ( node @ A @ La @ A2 @ ( leaf @ A ) ) @ B3 @ Rb ) )
= ( node @ A @ La @ A2 @ ( node @ A @ ( leaf @ A ) @ B3 @ Rb ) ) ) ) ) ) ).
% splay.simps(7)
thf(fact_214_splay_Osimps_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,A2: A,B3: A,Ra2: tree @ A,Rb: tree @ A] :
( ( ord_less @ A @ X @ A2 )
=> ( ( ord_less @ A @ X @ B3 )
=> ( ( splay_splay @ A @ X @ ( node @ A @ ( node @ A @ ( leaf @ A ) @ A2 @ Ra2 ) @ B3 @ Rb ) )
= ( node @ A @ ( leaf @ A ) @ A2 @ ( node @ A @ Ra2 @ B3 @ Rb ) ) ) ) ) ) ).
% splay.simps(5)
thf(fact_215_splay_Osimps_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,R2: tree @ A] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( splay_splay @ A @ A2 @ ( node @ A @ ( leaf @ A ) @ B3 @ R2 ) )
= ( node @ A @ ( leaf @ A ) @ B3 @ R2 ) ) ) ) ).
% splay.simps(4)
thf(fact_216_splay__max__eq__splay__ex,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: tree @ A] :
( ( linorder_bst @ A @ T )
=> ? [A6: A] :
( ( splay_splay_max @ A @ T )
= ( splay_splay @ A @ A6 @ T ) ) ) ) ).
% splay_max_eq_splay_ex
thf(fact_217_bst_Osimps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [L2: tree @ A,A2: A,R2: tree @ A] :
( ( linorder_bst @ A @ ( node @ A @ L2 @ A2 @ R2 ) )
= ( ( linorder_bst @ A @ L2 )
& ( linorder_bst @ A @ R2 )
& ! [X2: A] :
( ( member @ A @ X2 @ ( set_tree @ A @ L2 ) )
=> ( ord_less @ A @ X2 @ A2 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ ( set_tree @ A @ R2 ) )
=> ( ord_less @ A @ A2 @ X2 ) ) ) ) ) ).
% bst.simps(2)
thf(fact_218_bst_Oelims_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: tree @ A] :
( ~ ( linorder_bst @ A @ X )
=> ~ ! [L: tree @ A,A6: A,R: tree @ A] :
( ( X
= ( node @ A @ L @ A6 @ R ) )
=> ( ( linorder_bst @ A @ L )
& ( linorder_bst @ A @ R )
& ! [X3: A] :
( ( member @ A @ X3 @ ( set_tree @ A @ L ) )
=> ( ord_less @ A @ X3 @ A6 ) )
& ! [X3: A] :
( ( member @ A @ X3 @ ( set_tree @ A @ R ) )
=> ( ord_less @ A @ A6 @ X3 ) ) ) ) ) ) ).
% bst.elims(3)
thf(fact_219_splay__bstR,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: tree @ A,A2: A,L2: tree @ A,E: A,R2: tree @ A,X: A] :
( ( linorder_bst @ A @ T )
=> ( ( ( splay_splay @ A @ A2 @ T )
= ( node @ A @ L2 @ E @ R2 ) )
=> ( ( member @ A @ X @ ( set_tree @ A @ R2 ) )
=> ( ord_less @ A @ A2 @ X ) ) ) ) ) ).
% splay_bstR
thf(fact_220_splay__bstL,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: tree @ A,A2: A,L2: tree @ A,E: A,R2: tree @ A,X: A] :
( ( linorder_bst @ A @ T )
=> ( ( ( splay_splay @ A @ A2 @ T )
= ( node @ A @ L2 @ E @ R2 ) )
=> ( ( member @ A @ X @ ( set_tree @ A @ L2 ) )
=> ( ord_less @ A @ X @ A2 ) ) ) ) ) ).
% splay_bstL
thf(fact_221_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z )
=> ~ ( ord_less_eq @ A @ T @ X4 ) ) ) ).
% minf(8)
thf(fact_222_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z )
=> ( ord_less_eq @ A @ X4 @ T ) ) ) ).
% minf(6)
thf(fact_223_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X4: A] :
( ( ord_less @ A @ Z @ X4 )
=> ( ord_less_eq @ A @ T @ X4 ) ) ) ).
% pinf(8)
thf(fact_224_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X4: A] :
( ( ord_less @ A @ Z @ X4 )
=> ~ ( ord_less_eq @ A @ X4 @ T ) ) ) ).
% pinf(6)
thf(fact_225_complete__interval,axiom,
! [A: $tType] :
( ( condit1037483654norder @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,P: A > $o] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B3 )
=> ? [C2: A] :
( ( ord_less_eq @ A @ A2 @ C2 )
& ( ord_less_eq @ A @ C2 @ B3 )
& ! [X4: A] :
( ( ( ord_less_eq @ A @ A2 @ X4 )
& ( ord_less @ A @ X4 @ C2 ) )
=> ( P @ X4 ) )
& ! [D: A] :
( ! [X3: A] :
( ( ( ord_less_eq @ A @ A2 @ X3 )
& ( ord_less @ A @ X3 @ D ) )
=> ( P @ X3 ) )
=> ( ord_less_eq @ A @ D @ C2 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_226_t__splay__simps_I9_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,C3: A,Rr2: tree @ A,L2: tree @ A,Rl2: tree @ A] :
( ( ord_less @ A @ B3 @ A2 )
=> ( ( ord_less @ A @ C3 @ A2 )
=> ( ( ( Rr2
= ( leaf @ A ) )
=> ( ( splay_914434265_splay @ A @ A2 @ ( node @ A @ L2 @ B3 @ ( node @ A @ Rl2 @ C3 @ Rr2 ) ) )
= ( one_one @ nat ) ) )
& ( ( Rr2
!= ( leaf @ A ) )
=> ( ( splay_914434265_splay @ A @ A2 @ ( node @ A @ L2 @ B3 @ ( node @ A @ Rl2 @ C3 @ Rr2 ) ) )
= ( plus_plus @ nat @ ( splay_914434265_splay @ A @ A2 @ Rr2 ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% t_splay_simps(9)
thf(fact_227_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,C3: A] :
( ( ( plus_plus @ A @ B3 @ A2 )
= ( plus_plus @ A @ C3 @ A2 ) )
= ( B3 = C3 ) ) ) ).
% add_right_cancel
thf(fact_228_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C3: A] :
( ( ( plus_plus @ A @ A2 @ B3 )
= ( plus_plus @ A @ A2 @ C3 ) )
= ( B3 = C3 ) ) ) ).
% add_left_cancel
thf(fact_229_predicate2I,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
( ! [X3: A,Y3: B] :
( ( P @ X3 @ Y3 )
=> ( Q @ X3 @ Y3 ) )
=> ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).
% predicate2I
thf(fact_230_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,C3: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B3 @ C3 ) )
= ( ord_less_eq @ A @ A2 @ B3 ) ) ) ).
% add_le_cancel_right
thf(fact_231_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [C3: A,A2: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A2 ) @ ( plus_plus @ A @ C3 @ B3 ) )
= ( ord_less_eq @ A @ A2 @ B3 ) ) ) ).
% add_le_cancel_left
thf(fact_232_add__less__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,C3: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B3 @ C3 ) )
= ( ord_less @ A @ A2 @ B3 ) ) ) ).
% add_less_cancel_right
thf(fact_233_add__less__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [C3: A,A2: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C3 @ A2 ) @ ( plus_plus @ A @ C3 @ B3 ) )
= ( ord_less @ A @ A2 @ B3 ) ) ) ).
% add_less_cancel_left
thf(fact_234_t__splay__simps_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C3: A,Ll: tree @ A,Lr: tree @ A,R2: tree @ A] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( ord_less @ A @ A2 @ C3 )
=> ( ( ( Ll
= ( leaf @ A ) )
=> ( ( splay_914434265_splay @ A @ A2 @ ( node @ A @ ( node @ A @ Ll @ C3 @ Lr ) @ B3 @ R2 ) )
= ( one_one @ nat ) ) )
& ( ( Ll
!= ( leaf @ A ) )
=> ( ( splay_914434265_splay @ A @ A2 @ ( node @ A @ ( node @ A @ Ll @ C3 @ Lr ) @ B3 @ R2 ) )
= ( plus_plus @ nat @ ( splay_914434265_splay @ A @ A2 @ Ll ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% t_splay_simps(4)
thf(fact_235_t__splay__simps_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C3: A,Lr: tree @ A,Ll: tree @ A,R2: tree @ A] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( ord_less @ A @ C3 @ A2 )
=> ( ( ( Lr
= ( leaf @ A ) )
=> ( ( splay_914434265_splay @ A @ A2 @ ( node @ A @ ( node @ A @ Ll @ C3 @ Lr ) @ B3 @ R2 ) )
= ( one_one @ nat ) ) )
& ( ( Lr
!= ( leaf @ A ) )
=> ( ( splay_914434265_splay @ A @ A2 @ ( node @ A @ ( node @ A @ Ll @ C3 @ Lr ) @ B3 @ R2 ) )
= ( plus_plus @ nat @ ( splay_914434265_splay @ A @ A2 @ Lr ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% t_splay_simps(5)
thf(fact_236_t__splay__simps_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,C3: A,Rl2: tree @ A,L2: tree @ A,Rr2: tree @ A] :
( ( ord_less @ A @ B3 @ A2 )
=> ( ( ord_less @ A @ A2 @ C3 )
=> ( ( ( Rl2
= ( leaf @ A ) )
=> ( ( splay_914434265_splay @ A @ A2 @ ( node @ A @ L2 @ B3 @ ( node @ A @ Rl2 @ C3 @ Rr2 ) ) )
= ( one_one @ nat ) ) )
& ( ( Rl2
!= ( leaf @ A ) )
=> ( ( splay_914434265_splay @ A @ A2 @ ( node @ A @ L2 @ B3 @ ( node @ A @ Rl2 @ C3 @ Rr2 ) ) )
= ( plus_plus @ nat @ ( splay_914434265_splay @ A @ A2 @ Rl2 ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% t_splay_simps(8)
thf(fact_237_predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,X: A,Y2: B] :
( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( ( P @ X @ Y2 )
=> ( Q @ X @ Y2 ) ) ) ).
% predicate2D
thf(fact_238_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less @ ( A > B ) )
= ( ^ [F3: A > B,G2: A > B] :
( ( ord_less_eq @ ( A > B ) @ F3 @ G2 )
& ~ ( ord_less_eq @ ( A > B ) @ G2 @ F3 ) ) ) ) ) ).
% less_fun_def
thf(fact_239_rev__predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,X: A,Y2: B,Q: A > B > $o] :
( ( P @ X @ Y2 )
=> ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( Q @ X @ Y2 ) ) ) ).
% rev_predicate2D
thf(fact_240_is__num__normalize_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C3: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B3 ) @ C3 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B3 @ C3 ) ) ) ) ).
% is_num_normalize(1)
thf(fact_241_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,C3: A] :
( ( ( plus_plus @ A @ B3 @ A2 )
= ( plus_plus @ A @ C3 @ A2 ) )
=> ( B3 = C3 ) ) ) ).
% add_right_imp_eq
thf(fact_242_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C3: A] :
( ( ( plus_plus @ A @ A2 @ B3 )
= ( plus_plus @ A @ A2 @ C3 ) )
=> ( B3 = C3 ) ) ) ).
% add_left_imp_eq
thf(fact_243_add_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,C3: A] :
( ( plus_plus @ A @ B3 @ ( plus_plus @ A @ A2 @ C3 ) )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B3 @ C3 ) ) ) ) ).
% add.left_commute
thf(fact_244_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type2 @ A ) )
=> ( ( plus_plus @ A )
= ( ^ [A5: A,B4: A] : ( plus_plus @ A @ B4 @ A5 ) ) ) ) ).
% add.commute
thf(fact_245_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,C3: A] :
( ( ( plus_plus @ A @ B3 @ A2 )
= ( plus_plus @ A @ C3 @ A2 ) )
= ( B3 = C3 ) ) ) ).
% add.right_cancel
thf(fact_246_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C3: A] :
( ( ( plus_plus @ A @ A2 @ B3 )
= ( plus_plus @ A @ A2 @ C3 ) )
= ( B3 = C3 ) ) ) ).
% add.left_cancel
thf(fact_247_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C3: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B3 ) @ C3 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B3 @ C3 ) ) ) ) ).
% add.assoc
thf(fact_248_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_249_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C3: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B3 ) @ C3 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B3 @ C3 ) ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_250_add__le__imp__le__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,C3: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B3 @ C3 ) )
=> ( ord_less_eq @ A @ A2 @ B3 ) ) ) ).
% add_le_imp_le_right
thf(fact_251_add__le__imp__le__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [C3: A,A2: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A2 ) @ ( plus_plus @ A @ C3 @ B3 ) )
=> ( ord_less_eq @ A @ A2 @ B3 ) ) ) ).
% add_le_imp_le_left
thf(fact_252_le__iff__add,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
? [C4: A] :
( B4
= ( plus_plus @ A @ A5 @ C4 ) ) ) ) ) ).
% le_iff_add
thf(fact_253_add__right__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B3 @ C3 ) ) ) ) ).
% add_right_mono
%----Subclasses (4)
thf(subcl_Orderings_Olinorder___HOL_Otype,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ( type @ A @ ( type2 @ A ) ) ) ).
thf(subcl_Orderings_Olinorder___Orderings_Oord,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ( ord @ A @ ( type2 @ A ) ) ) ).
thf(subcl_Orderings_Olinorder___Orderings_Oorder,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ( order @ A @ ( type2 @ A ) ) ) ).
thf(subcl_Orderings_Olinorder___Orderings_Opreorder,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ( preorder @ A @ ( type2 @ A ) ) ) ).
%----Type constructors (44)
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A7: $tType,A8: $tType] :
( ( preorder @ A8 @ ( type2 @ A8 ) )
=> ( preorder @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A7: $tType,A8: $tType] :
( ( order @ A8 @ ( type2 @ A8 ) )
=> ( order @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A7: $tType,A8: $tType] :
( ( ord @ A8 @ ( type2 @ A8 ) )
=> ( ord @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit1037483654norder @ 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_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_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_Osemigroup__add,axiom,
semigroup_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
preorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
no_top @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
order @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
ord @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oone,axiom,
one @ nat @ ( type2 @ nat ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
! [A7: $tType] : ( preorder @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_5,axiom,
! [A7: $tType] : ( order @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_6,axiom,
! [A7: $tType] : ( ord @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_7,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder_8,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_9,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_10,axiom,
ord @ $o @ ( type2 @ $o ) ).
thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_11,axiom,
condit1037483654norder @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_12,axiom,
ordere236663937imp_le @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add_13,axiom,
ordere779506340up_add @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_14,axiom,
cancel_semigroup_add @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Rings_Olinordered__semidom_15,axiom,
linordered_semidom @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Orderings_Odense__linorder,axiom,
dense_linorder @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Oab__semigroup__add_16,axiom,
ab_semigroup_add @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Orderings_Odense__order,axiom,
dense_order @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Osemigroup__add_17,axiom,
semigroup_add @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Orderings_Opreorder_18,axiom,
preorder @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Orderings_Olinorder_19,axiom,
linorder @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Transcendental_Oln,axiom,
ln @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Orderings_Ono__top_20,axiom,
no_top @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Orderings_Ono__bot,axiom,
no_bot @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Ogroup__add,axiom,
group_add @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Orderings_Oorder_21,axiom,
order @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Num_Oneg__numeral,axiom,
neg_numeral @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Orderings_Oord_22,axiom,
ord @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Oone_23,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,Y2: A] :
( ( if @ A @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y2: A] :
( ( if @ A @ $true @ X @ Y2 )
= X ) ).
%----Free types (1)
thf(tfree_0,hypothesis,
linorder @ a @ ( type2 @ a ) ).
%----Conjectures (2)
thf(conj_0,hypothesis,
! [La2: tree @ a,Ra3: tree @ a] :
( ( member @ ( tree @ a ) @ ( node @ a @ La2 @ a2 @ Ra3 ) @ ( subtrees @ a @ t ) )
=> thesis ) ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------