TPTP Problem File: DAT212^1.p
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%------------------------------------------------------------------------------
% File : DAT212^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Splay tree analysis 136
% 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__136.p [Bla16]
% Status : Theorem
% Rating : 1.00 v7.2.0, 0.75 v7.1.0
% Syntax : Number of formulae : 343 ( 47 unt; 57 typ; 0 def)
% Number of atoms : 1084 ( 230 equ; 0 cnn)
% Maximal formula atoms : 16 ( 3 avg)
% Number of connectives : 4824 ( 123 ~; 18 |; 91 &;3949 @)
% ( 0 <=>; 643 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 10 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 150 ( 150 >; 0 *; 0 +; 0 <<)
% Number of symbols : 56 ( 55 usr; 8 con; 0-4 aty)
% Number of variables : 1183 ( 34 ^;1058 !; 46 ?;1183 :)
% ( 45 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:49:24.843
%------------------------------------------------------------------------------
%----Could-be-implicit typings (5)
thf(ty_t_Tree_Otree,type,
tree: $tType > $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 (52)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[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_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_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[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_Divides_Osemiring__numeral__div,type,
semiring_numeral_div:
!>[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_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_Conditionally__Complete__Lattices_Olinear__continuum,type,
condit1656338222tinuum:
!>[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_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_OBall,type,
ball:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
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__Base_Ot__splay__max,type,
splay_878424299ay_max:
!>[A: $tType] : ( ( tree @ A ) > nat ) ).
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_Olinorder__class_Obst__eq__rel,type,
linorder_bst_eq_rel:
!>[A: $tType] : ( ( tree @ A ) > ( tree @ A ) > $o ) ).
thf(sy_c_Tree_Olinorder__class_Obst__rel,type,
linorder_bst_rel:
!>[A: $tType] : ( ( tree @ A ) > ( tree @ A ) > $o ) ).
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_Omap__tree,type,
map_tree:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( tree @ A ) > ( tree @ Aa ) ) ).
thf(sy_c_Tree_Otree_Opred__tree,type,
pred_tree:
!>[A: $tType] : ( ( A > $o ) > ( tree @ A ) > $o ) ).
thf(sy_c_Tree_Otree_Oset__tree,type,
set_tree:
!>[A: $tType] : ( ( tree @ A ) > ( set @ A ) ) ).
thf(sy_c_Wellfounded_Oaccp,type,
accp:
!>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_a____,type,
a2: a ).
thf(sy_v_b____,type,
b: a ).
thf(sy_v_la____,type,
la: tree @ a ).
thf(sy_v_lb____,type,
lb: tree @ a ).
thf(sy_v_lx,type,
lx: tree @ a ).
thf(sy_v_rb____,type,
rb: tree @ a ).
thf(sy_v_rx,type,
rx: tree @ a ).
thf(sy_v_xa____,type,
xa: a ).
%----Relevant facts (256)
thf(fact_0__C11_Ohyps_C_I2_J,axiom,
ord_less @ a @ xa @ b ).
% "11.hyps"(2)
thf(fact_1__C11_Ohyps_C_I1_J,axiom,
ord_less @ a @ a2 @ xa ).
% "11.hyps"(1)
thf(fact_2__C0_C,axiom,
( ~ ( member @ a @ xa @ ( set_tree @ a @ rb ) )
& ~ ( member @ a @ xa @ ( set_tree @ a @ la ) ) ) ).
% "0"
thf(fact_3__C11_Oprems_C_I1_J,axiom,
linorder_bst @ a @ ( node @ a @ la @ a2 @ ( node @ a @ lb @ b @ rb ) ) ).
% "11.prems"(1)
thf(fact_4__C11_Oprems_C_I2_J,axiom,
member @ ( tree @ a ) @ ( node @ a @ lx @ xa @ rx ) @ ( subtrees @ a @ ( node @ a @ la @ a2 @ ( node @ a @ lb @ b @ rb ) ) ) ).
% "11.prems"(2)
thf(fact_5__C11_Ohyps_C_I3_J,axiom,
( lb
!= ( leaf @ a ) ) ).
% "11.hyps"(3)
thf(fact_6_tree_Oset__intros_I3_J,axiom,
! [A: $tType,Xa: A,A3: tree @ A,A1: tree @ A,A2: A] :
( ( member @ A @ Xa @ ( set_tree @ A @ A3 ) )
=> ( member @ A @ Xa @ ( set_tree @ A @ ( node @ A @ A1 @ A2 @ A3 ) ) ) ) ).
% tree.set_intros(3)
thf(fact_7_tree_Oset__intros_I2_J,axiom,
! [A: $tType,A2: A,A1: tree @ A,A3: tree @ A] : ( member @ A @ A2 @ ( set_tree @ A @ ( node @ A @ A1 @ A2 @ A3 ) ) ) ).
% tree.set_intros(2)
thf(fact_8_tree_Oset__intros_I1_J,axiom,
! [A: $tType,X: A,A1: tree @ A,A2: A,A3: tree @ A] :
( ( member @ A @ X @ ( set_tree @ A @ A1 ) )
=> ( member @ A @ X @ ( set_tree @ A @ ( node @ A @ A1 @ A2 @ A3 ) ) ) ) ).
% tree.set_intros(1)
thf(fact_9_tree_Oset__cases,axiom,
! [A: $tType,E: A,A4: tree @ A] :
( ( member @ A @ E @ ( set_tree @ A @ A4 ) )
=> ( ! [Z1: tree @ A] :
( ? [Z2: A,Z3: tree @ A] :
( A4
= ( node @ A @ Z1 @ Z2 @ Z3 ) )
=> ~ ( member @ A @ E @ ( set_tree @ A @ Z1 ) ) )
=> ( ! [Z1: tree @ A,Z3: tree @ A] :
( A4
!= ( node @ A @ Z1 @ E @ Z3 ) )
=> ~ ! [Z1: tree @ A,Z2: A,Z3: tree @ A] :
( ( A4
= ( node @ A @ Z1 @ Z2 @ Z3 ) )
=> ~ ( member @ A @ E @ ( set_tree @ A @ Z3 ) ) ) ) ) ) ).
% tree.set_cases
thf(fact_10_set__treeE,axiom,
! [A: $tType,A4: A,T: tree @ A] :
( ( member @ A @ A4 @ ( set_tree @ A @ T ) )
=> ? [L: tree @ A,R: tree @ A] : ( member @ ( tree @ A ) @ ( node @ A @ L @ A4 @ R ) @ ( subtrees @ A @ T ) ) ) ).
% set_treeE
thf(fact_11_in__set__tree__if,axiom,
! [A: $tType,L2: tree @ A,A4: A,R2: tree @ A,T: tree @ A] :
( ( member @ ( tree @ A ) @ ( node @ A @ L2 @ A4 @ R2 ) @ ( subtrees @ A @ T ) )
=> ( member @ A @ A4 @ ( set_tree @ A @ T ) ) ) ).
% in_set_tree_if
thf(fact_12_Node__notin__subtrees__if,axiom,
! [A: $tType,A4: A,T: tree @ A,L2: tree @ A,R2: tree @ A] :
( ~ ( member @ A @ A4 @ ( set_tree @ A @ T ) )
=> ~ ( member @ ( tree @ A ) @ ( node @ A @ L2 @ A4 @ R2 ) @ ( subtrees @ A @ T ) ) ) ).
% Node_notin_subtrees_if
thf(fact_13_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_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_tree_Omap__cong,axiom,
! [B: $tType,A: $tType,X: tree @ A,Ya: tree @ A,F: A > B,G: A > B] :
( ( X = Ya )
=> ( ! [Z: A] :
( ( member @ A @ Z @ ( set_tree @ A @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_tree @ A @ B @ F @ X )
= ( map_tree @ A @ B @ G @ Ya ) ) ) ) ).
% tree.map_cong
thf(fact_17_tree_Omap__cong0,axiom,
! [B: $tType,A: $tType,X: tree @ A,F: A > B,G: A > B] :
( ! [Z: A] :
( ( member @ A @ Z @ ( set_tree @ A @ X ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_tree @ A @ B @ F @ X )
= ( map_tree @ A @ B @ G @ X ) ) ) ).
% tree.map_cong0
thf(fact_18_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_19_tree_Opred__inject_I2_J,axiom,
! [A: $tType,P: A > $o,A4: tree @ A,Aa2: A,Ab: tree @ A] :
( ( pred_tree @ A @ P @ ( node @ A @ A4 @ Aa2 @ Ab ) )
= ( ( pred_tree @ A @ P @ A4 )
& ( P @ Aa2 )
& ( pred_tree @ A @ P @ Ab ) ) ) ).
% tree.pred_inject(2)
thf(fact_20_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_21_tree_Osimps_I9_J,axiom,
! [B: $tType,A: $tType,F: A > B,X21: tree @ A,X22: A,X23: tree @ A] :
( ( map_tree @ A @ B @ F @ ( node @ A @ X21 @ X22 @ X23 ) )
= ( node @ B @ ( map_tree @ A @ B @ F @ X21 ) @ ( F @ X22 ) @ ( map_tree @ A @ B @ F @ X23 ) ) ) ).
% tree.simps(9)
thf(fact_22_tree_Osimps_I8_J,axiom,
! [A: $tType,B: $tType,F: A > B] :
( ( map_tree @ A @ B @ F @ ( leaf @ A ) )
= ( leaf @ B ) ) ).
% tree.simps(8)
thf(fact_23_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_24_tree_Opred__inject_I1_J,axiom,
! [A: $tType,P: A > $o] : ( pred_tree @ A @ P @ ( leaf @ A ) ) ).
% tree.pred_inject(1)
thf(fact_25_splay__max_Osimps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [La: tree @ A,A4: A] :
( ( splay_splay_max @ A @ ( node @ A @ La @ A4 @ ( leaf @ A ) ) )
= ( node @ A @ La @ A4 @ ( leaf @ A ) ) ) ) ).
% splay_max.simps(2)
thf(fact_26_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_27_bst_Osimps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ( linorder_bst @ A @ ( leaf @ A ) ) ) ).
% bst.simps(1)
thf(fact_28_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_29_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_30_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_31_mirror_Oinduct,axiom,
! [A: $tType,P: ( tree @ A ) > $o,A0: tree @ A] :
( ( P @ ( leaf @ A ) )
=> ( ! [L: tree @ A,X4: A,R: tree @ A] :
( ( P @ R )
=> ( ( P @ L )
=> ( P @ ( node @ A @ L @ X4 @ R ) ) ) )
=> ( P @ A0 ) ) ) ).
% mirror.induct
thf(fact_32_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_33_splay__max__Leaf,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: tree @ A,L2: tree @ A,A4: A,R2: tree @ A] :
( ( ( splay_splay_max @ A @ T )
= ( node @ A @ L2 @ A4 @ R2 ) )
=> ( R2
= ( leaf @ A ) ) ) ) ).
% splay_max_Leaf
thf(fact_34_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_35_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_36_tree_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,X: tree @ A,Xa: tree @ A,F: A > B,Fa: A > B] :
( ! [Z: A,Za: A] :
( ( member @ A @ Z @ ( set_tree @ A @ X ) )
=> ( ( member @ A @ Za @ ( set_tree @ A @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_tree @ A @ B @ F @ X )
= ( map_tree @ A @ B @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% tree.inj_map_strong
thf(fact_37__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_38__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_39_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,C: A,Rr: tree @ A] :
( X
!= ( node @ A @ L @ B2 @ ( node @ A @ Rl @ C @ Rr ) ) ) ) ) ) ).
% t_splay_max.cases
thf(fact_40_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,C: A,Rr: tree @ A] :
( ( ( Rr
!= ( leaf @ A ) )
=> ( P @ Rr ) )
=> ( P @ ( node @ A @ L @ B2 @ ( node @ A @ Rl @ C @ Rr ) ) ) )
=> ( P @ A0 ) ) ) ) ) ).
% t_splay_max.induct
thf(fact_41_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 )
& ! [X5: A] :
( ( member @ A @ X5 @ ( set_tree @ A @ L ) )
=> ( ord_less @ A @ X5 @ A6 ) )
& ! [X5: A] :
( ( member @ A @ X5 @ ( set_tree @ A @ R ) )
=> ( ord_less @ A @ A6 @ X5 ) ) ) ) ) ) ) ).
% bst.elims(2)
thf(fact_42_bst_Oelims_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: tree @ A,Y: $o] :
( ( ( linorder_bst @ A @ X )
= Y )
=> ( ( ( X
= ( leaf @ A ) )
=> ~ Y )
=> ~ ! [L: tree @ A,A6: A,R: tree @ A] :
( ( X
= ( node @ A @ L @ A6 @ R ) )
=> ( Y
= ( ~ ( ( linorder_bst @ A @ L )
& ( linorder_bst @ A @ R )
& ! [X6: A] :
( ( member @ A @ X6 @ ( set_tree @ A @ L ) )
=> ( ord_less @ A @ X6 @ A6 ) )
& ! [X6: A] :
( ( member @ A @ X6 @ ( set_tree @ A @ R ) )
=> ( ord_less @ A @ A6 @ X6 ) ) ) ) ) ) ) ) ) ).
% bst.elims(1)
thf(fact_43_splay__bstL,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: tree @ A,A4: A,L2: tree @ A,E: A,R2: tree @ A,X: A] :
( ( linorder_bst @ A @ T )
=> ( ( ( splay_splay @ A @ A4 @ T )
= ( node @ A @ L2 @ E @ R2 ) )
=> ( ( member @ A @ X @ ( set_tree @ A @ L2 ) )
=> ( ord_less @ A @ X @ A4 ) ) ) ) ) ).
% splay_bstL
thf(fact_44_splay__bstR,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: tree @ A,A4: A,L2: tree @ A,E: A,R2: tree @ A,X: A] :
( ( linorder_bst @ A @ T )
=> ( ( ( splay_splay @ A @ A4 @ T )
= ( node @ A @ L2 @ E @ R2 ) )
=> ( ( member @ A @ X @ ( set_tree @ A @ R2 ) )
=> ( ord_less @ A @ A4 @ X ) ) ) ) ) ).
% splay_bstR
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A4: A,P: A > $o] :
( ( member @ A @ A4 @ ( collect @ A @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A7: set @ A] :
( ( collect @ A
@ ^ [X6: A] : ( member @ A @ X6 @ A7 ) )
= A7 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X4: A] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X4: A] :
( ( F @ X4 )
= ( G @ X4 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_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 )
& ! [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(3)
thf(fact_50_bst_Osimps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [L2: tree @ A,A4: A,R2: tree @ A] :
( ( linorder_bst @ A @ ( node @ A @ L2 @ A4 @ R2 ) )
= ( ( linorder_bst @ A @ L2 )
& ( linorder_bst @ A @ R2 )
& ! [X6: A] :
( ( member @ A @ X6 @ ( set_tree @ A @ L2 ) )
=> ( ord_less @ A @ X6 @ A4 ) )
& ! [X6: A] :
( ( member @ A @ X6 @ ( set_tree @ A @ R2 ) )
=> ( ord_less @ A @ A4 @ X6 ) ) ) ) ) ).
% bst.simps(2)
thf(fact_51_splay__Leaf__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A4: A,T: tree @ A] :
( ( ( splay_splay @ A @ A4 @ T )
= ( leaf @ A ) )
= ( T
= ( leaf @ A ) ) ) ) ).
% splay_Leaf_iff
thf(fact_52_splay_Osimps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A4: A,L2: tree @ A,R2: tree @ A] :
( ( splay_splay @ A @ A4 @ ( node @ A @ L2 @ A4 @ R2 ) )
= ( node @ A @ L2 @ A4 @ R2 ) ) ) ).
% splay.simps(2)
thf(fact_53_splay_Osimps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A4: A] :
( ( splay_splay @ A @ A4 @ ( leaf @ A ) )
= ( leaf @ A ) ) ) ).
% splay.simps(1)
thf(fact_54_set__splay,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A4: A,T: tree @ A] :
( ( set_tree @ A @ ( splay_splay @ A @ A4 @ T ) )
= ( set_tree @ A @ T ) ) ) ).
% set_splay
thf(fact_55_bst__splay,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: tree @ A,A4: A] :
( ( linorder_bst @ A @ T )
=> ( linorder_bst @ A @ ( splay_splay @ A @ A4 @ T ) ) ) ) ).
% bst_splay
thf(fact_56_splay_Osimps_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A,La: tree @ A,Ra: tree @ A,Rb: tree @ A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( splay_splay @ A @ A4 @ ( node @ A @ ( node @ A @ La @ A4 @ Ra ) @ B3 @ Rb ) )
= ( node @ A @ La @ A4 @ ( node @ A @ Ra @ B3 @ Rb ) ) ) ) ) ).
% splay.simps(3)
thf(fact_57_splay_Osimps_I9_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [B3: A,A4: A,Lb: tree @ A,La: tree @ A,Ra: tree @ A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ( splay_splay @ A @ A4 @ ( node @ A @ Lb @ B3 @ ( node @ A @ La @ A4 @ Ra ) ) )
= ( node @ A @ ( node @ A @ Lb @ B3 @ La ) @ A4 @ Ra ) ) ) ) ).
% splay.simps(9)
thf(fact_58_splay__not__Leaf,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: tree @ A,A4: A] :
( ( T
!= ( leaf @ A ) )
=> ? [L: tree @ A,X4: A,R: tree @ A] :
( ( splay_splay @ A @ A4 @ T )
= ( node @ A @ L @ X4 @ R ) ) ) ) ).
% splay_not_Leaf
thf(fact_59_tree_Opred__set,axiom,
! [A: $tType] :
( ( pred_tree @ A )
= ( ^ [P2: A > $o,X6: tree @ A] :
! [Y2: A] :
( ( member @ A @ Y2 @ ( set_tree @ A @ X6 ) )
=> ( P2 @ Y2 ) ) ) ) ).
% tree.pred_set
thf(fact_60_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_61_splay_Osimps_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A,R2: tree @ A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( splay_splay @ A @ A4 @ ( node @ A @ ( leaf @ A ) @ B3 @ R2 ) )
= ( node @ A @ ( leaf @ A ) @ B3 @ R2 ) ) ) ) ).
% splay.simps(4)
thf(fact_62_splay_Osimps_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,A4: A,B3: A,Ra: tree @ A,Rb: tree @ A] :
( ( ord_less @ A @ X @ A4 )
=> ( ( ord_less @ A @ X @ B3 )
=> ( ( splay_splay @ A @ X @ ( node @ A @ ( node @ A @ ( leaf @ A ) @ A4 @ Ra ) @ B3 @ Rb ) )
= ( node @ A @ ( leaf @ A ) @ A4 @ ( node @ A @ Ra @ B3 @ Rb ) ) ) ) ) ) ).
% splay.simps(5)
thf(fact_63_splay_Osimps_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,B3: A,A4: A,La: tree @ A,Rb: tree @ A] :
( ( ord_less @ A @ X @ B3 )
=> ( ( ord_less @ A @ A4 @ X )
=> ( ( splay_splay @ A @ X @ ( node @ A @ ( node @ A @ La @ A4 @ ( leaf @ A ) ) @ B3 @ Rb ) )
= ( node @ A @ La @ A4 @ ( node @ A @ ( leaf @ A ) @ B3 @ Rb ) ) ) ) ) ) ).
% splay.simps(7)
thf(fact_64_splay_Osimps_I10_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A4: A,X: A,L2: tree @ A] :
( ( ord_less @ A @ A4 @ X )
=> ( ( splay_splay @ A @ X @ ( node @ A @ L2 @ A4 @ ( leaf @ A ) ) )
= ( node @ A @ L2 @ A4 @ ( leaf @ A ) ) ) ) ) ).
% splay.simps(10)
thf(fact_65_splay_Osimps_I12_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A4: A,X: A,B3: A,La: tree @ A,Rb: tree @ A] :
( ( ord_less @ A @ A4 @ X )
=> ( ( ord_less @ A @ X @ B3 )
=> ( ( splay_splay @ A @ X @ ( node @ A @ La @ A4 @ ( node @ A @ ( leaf @ A ) @ B3 @ Rb ) ) )
= ( node @ A @ ( node @ A @ La @ A4 @ ( leaf @ A ) ) @ B3 @ Rb ) ) ) ) ) ).
% splay.simps(12)
thf(fact_66_splay_Osimps_I13_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A4: A,X: A,B3: A,La: tree @ A,Lb: tree @ A] :
( ( ord_less @ A @ A4 @ X )
=> ( ( ord_less @ A @ B3 @ X )
=> ( ( splay_splay @ A @ X @ ( node @ A @ La @ A4 @ ( node @ A @ Lb @ B3 @ ( leaf @ A ) ) ) )
= ( node @ A @ ( node @ A @ La @ A4 @ Lb ) @ B3 @ ( leaf @ A ) ) ) ) ) ) ).
% splay.simps(13)
thf(fact_67_splay__to__root,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: tree @ A,A4: A,T2: tree @ A] :
( ( linorder_bst @ A @ T )
=> ( ( ( splay_splay @ A @ A4 @ T )
= T2 )
=> ( ( member @ A @ A4 @ ( set_tree @ A @ T ) )
= ( ? [L3: tree @ A,R3: tree @ A] :
( T2
= ( node @ A @ L3 @ A4 @ R3 ) ) ) ) ) ) ) ).
% splay_to_root
thf(fact_68_is__root__splay,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: tree @ A,A4: A] :
( ( linorder_bst @ A @ T )
=> ( ( splay_is_root @ A @ A4 @ ( splay_splay @ A @ A4 @ T ) )
= ( member @ A @ A4 @ ( set_tree @ A @ T ) ) ) ) ) ).
% is_root_splay
thf(fact_69_ex__in__set__tree,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: tree @ A,A4: A] :
( ( T
!= ( leaf @ A ) )
=> ( ( linorder_bst @ A @ T )
=> ? [X4: A] :
( ( member @ A @ X4 @ ( set_tree @ A @ T ) )
& ( ( splay_splay @ A @ X4 @ T )
= ( splay_splay @ A @ A4 @ T ) )
& ( ( splay_914434265_splay @ A @ X4 @ T )
= ( splay_914434265_splay @ A @ A4 @ T ) ) ) ) ) ) ).
% ex_in_set_tree
thf(fact_70_splay__max__eq__splay,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: tree @ A,A4: A] :
( ( linorder_bst @ A @ T )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ ( set_tree @ A @ T ) )
=> ( ord_less_eq @ A @ X4 @ A4 ) )
=> ( ( splay_splay_max @ A @ T )
= ( splay_splay @ A @ A4 @ T ) ) ) ) ) ).
% splay_max_eq_splay
thf(fact_71_bst_Opelims_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: tree @ A,Y: $o] :
( ( ( linorder_bst @ A @ X )
= Y )
=> ( ( accp @ ( tree @ A ) @ ( linorder_bst_rel @ A ) @ X )
=> ( ( ( X
= ( leaf @ A ) )
=> ( Y
=> ~ ( accp @ ( tree @ A ) @ ( linorder_bst_rel @ A ) @ ( leaf @ A ) ) ) )
=> ~ ! [L: tree @ A,A6: A,R: tree @ A] :
( ( X
= ( node @ A @ L @ A6 @ R ) )
=> ( ( Y
= ( ( linorder_bst @ A @ L )
& ( linorder_bst @ A @ R )
& ! [X6: A] :
( ( member @ A @ X6 @ ( set_tree @ A @ L ) )
=> ( ord_less @ A @ X6 @ A6 ) )
& ! [X6: A] :
( ( member @ A @ X6 @ ( set_tree @ A @ R ) )
=> ( ord_less @ A @ A6 @ X6 ) ) ) )
=> ~ ( accp @ ( tree @ A ) @ ( linorder_bst_rel @ A ) @ ( node @ A @ L @ A6 @ R ) ) ) ) ) ) ) ) ).
% bst.pelims(1)
thf(fact_72_bst_Opelims_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: tree @ A] :
( ( linorder_bst @ A @ X )
=> ( ( accp @ ( tree @ A ) @ ( linorder_bst_rel @ A ) @ X )
=> ( ( ( X
= ( leaf @ A ) )
=> ~ ( accp @ ( tree @ A ) @ ( linorder_bst_rel @ A ) @ ( leaf @ A ) ) )
=> ~ ! [L: tree @ A,A6: A,R: tree @ A] :
( ( X
= ( node @ A @ L @ A6 @ R ) )
=> ( ( accp @ ( tree @ A ) @ ( linorder_bst_rel @ A ) @ ( node @ A @ L @ A6 @ R ) )
=> ~ ( ( linorder_bst @ A @ L )
& ( linorder_bst @ A @ R )
& ! [X5: A] :
( ( member @ A @ X5 @ ( set_tree @ A @ L ) )
=> ( ord_less @ A @ X5 @ A6 ) )
& ! [X5: A] :
( ( member @ A @ X5 @ ( set_tree @ A @ R ) )
=> ( ord_less @ A @ A6 @ X5 ) ) ) ) ) ) ) ) ) ).
% bst.pelims(2)
thf(fact_73_ball__reg,axiom,
! [A: $tType,R4: set @ A,P: A > $o,Q: A > $o] :
( ! [X4: A] :
( ( member @ A @ X4 @ R4 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ R4 )
=> ( P @ X4 ) )
=> ! [X5: A] :
( ( member @ A @ X5 @ R4 )
=> ( Q @ X5 ) ) ) ) ).
% ball_reg
thf(fact_74_Ball__def,axiom,
! [A: $tType] :
( ( ball @ A )
= ( ^ [A8: set @ A,P2: A > $o] :
! [X6: A] :
( ( member @ A @ X6 @ A8 )
=> ( P2 @ X6 ) ) ) ) ).
% Ball_def
thf(fact_75_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
=> ! [A4: A] :
? [B2: A] :
( ( ord_less @ A @ A4 @ B2 )
| ( ord_less @ A @ B2 @ A4 ) ) ) ).
% ex_gt_or_lt
thf(fact_76_complete__interval,axiom,
! [A: $tType] :
( ( condit1037483654norder @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A,P: A > $o] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( P @ A4 )
=> ( ~ ( P @ B3 )
=> ? [C: A] :
( ( ord_less_eq @ A @ A4 @ C )
& ( ord_less_eq @ A @ C @ B3 )
& ! [X5: A] :
( ( ( ord_less_eq @ A @ A4 @ X5 )
& ( ord_less @ A @ X5 @ C ) )
=> ( P @ X5 ) )
& ! [D: A] :
( ! [X4: A] :
( ( ( ord_less_eq @ A @ A4 @ X4 )
& ( ord_less @ A @ X4 @ D ) )
=> ( P @ X4 ) )
=> ( ord_less_eq @ A @ D @ C ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_77_bst_Opelims_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: tree @ A] :
( ~ ( linorder_bst @ A @ X )
=> ( ( accp @ ( tree @ A ) @ ( linorder_bst_rel @ A ) @ X )
=> ~ ! [L: tree @ A,A6: A,R: tree @ A] :
( ( X
= ( node @ A @ L @ A6 @ R ) )
=> ( ( accp @ ( tree @ A ) @ ( linorder_bst_rel @ A ) @ ( 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.pelims(3)
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_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 )
& ! [X5: A] :
( ( member @ A @ X5 @ ( set_tree @ A @ L ) )
=> ( ord_less_eq @ A @ X5 @ A6 ) )
& ! [X5: A] :
( ( member @ A @ X5 @ ( set_tree @ A @ R ) )
=> ( ord_less_eq @ A @ A6 @ X5 ) ) ) ) ) ) ) ).
% bst_eq.elims(2)
thf(fact_80_bst__eq_Oelims_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: tree @ A,Y: $o] :
( ( ( linorder_bst_eq @ A @ X )
= Y )
=> ( ( ( X
= ( leaf @ A ) )
=> ~ Y )
=> ~ ! [L: tree @ A,A6: A,R: tree @ A] :
( ( X
= ( node @ A @ L @ A6 @ R ) )
=> ( Y
= ( ~ ( ( linorder_bst_eq @ A @ L )
& ( linorder_bst_eq @ A @ R )
& ! [X6: A] :
( ( member @ A @ X6 @ ( set_tree @ A @ L ) )
=> ( ord_less_eq @ A @ X6 @ A6 ) )
& ! [X6: A] :
( ( member @ A @ X6 @ ( set_tree @ A @ R ) )
=> ( ord_less_eq @ A @ A6 @ X6 ) ) ) ) ) ) ) ) ) ).
% bst_eq.elims(1)
thf(fact_81_order_Onot__eq__order__implies__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A] :
( ( A4 != B3 )
=> ( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ord_less @ A @ A4 @ B3 ) ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_82_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A4: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ord_less_eq @ A @ B3 @ A4 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_83_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_84_bst__eq_Osimps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ( linorder_bst_eq @ A @ ( leaf @ A ) ) ) ).
% bst_eq.simps(1)
thf(fact_85_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_86_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A4: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( ord_less_eq @ A @ A4 @ B3 )
=> ( A4 = B3 ) ) ) ) ).
% dual_order.antisym
thf(fact_87_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A4: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( ord_less_eq @ A @ C2 @ B3 )
=> ( ord_less_eq @ A @ C2 @ A4 ) ) ) ) ).
% dual_order.trans
thf(fact_88_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,A4: 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 @ A4 @ B3 ) ) ) ) ).
% linorder_wlog
thf(fact_89_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A] : ( ord_less_eq @ A @ A4 @ A4 ) ) ).
% dual_order.refl
thf(fact_90_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z4: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z4 )
=> ( ord_less_eq @ A @ X @ Z4 ) ) ) ) ).
% order_trans
thf(fact_91_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ A4 )
=> ( A4 = B3 ) ) ) ) ).
% order_class.order.antisym
thf(fact_92_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_93_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A,C2: A] :
( ( A4 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_94_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv
thf(fact_95_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z4: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z4 ) )
=> ( ( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less_eq @ A @ X @ Z4 ) )
=> ( ( ( ord_less_eq @ A @ X @ Z4 )
=> ~ ( ord_less_eq @ A @ Z4 @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z4 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z4 )
=> ~ ( ord_less_eq @ A @ Z4 @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z4 @ X )
=> ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_96_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% order.trans
thf(fact_97_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% le_cases
thf(fact_98_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( X = Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% eq_refl
thf(fact_99_linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linear
thf(fact_100_antisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( X = Y ) ) ) ) ).
% antisym
thf(fact_101_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y3: A,Z5: A] : Y3 = Z5 )
= ( ^ [X6: A,Y2: A] :
( ( ord_less_eq @ A @ X6 @ Y2 )
& ( ord_less_eq @ A @ Y2 @ X6 ) ) ) ) ) ).
% eq_iff
thf(fact_102_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B3: A,F: A > B,C2: B] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ord_less_eq @ B @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ B @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_103_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B > A,B3: B,C2: B] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X4: B,Y4: B] :
( ( ord_less_eq @ B @ X4 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_104_order__subst2,axiom,
! [A: $tType,C3: $tType] :
( ( ( order @ C3 @ ( type2 @ C3 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B3: A,F: A > C3,C2: C3] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ C3 @ ( F @ B3 ) @ C2 )
=> ( ! [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ord_less_eq @ C3 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ C3 @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_105_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B > A,B3: B,C2: B] :
( ( ord_less_eq @ A @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X4: B,Y4: B] :
( ( ord_less_eq @ B @ X4 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_106_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
! [X6: A] : ( ord_less_eq @ B @ ( F2 @ X6 ) @ ( G2 @ X6 ) ) ) ) ) ).
% le_fun_def
thf(fact_107_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B] :
( ! [X4: A] : ( ord_less_eq @ B @ ( F @ X4 ) @ ( G @ X4 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_108_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_109_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_110_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A4: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( A4 != B3 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_111_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( A4 != B3 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_112_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less @ A @ X @ Y ) )
= ( ( ord_less @ A @ Y @ X )
| ( X = Y ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_113_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A4: A,C2: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ( ord_less @ A @ C2 @ B3 )
=> ( ord_less @ A @ C2 @ A4 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_114_less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% less_imp_not_less
thf(fact_115_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ B3 @ C2 )
=> ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% order.strict_trans
thf(fact_116_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A] :
~ ( ord_less @ A @ A4 @ A4 ) ) ).
% dual_order.irrefl
thf(fact_117_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( X != Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_cases
thf(fact_118_less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,P: $o] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ X )
=> P ) ) ) ).
% less_imp_triv
thf(fact_119_less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( Y != X ) ) ) ).
% less_imp_not_eq2
thf(fact_120_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ~ ( ord_less @ A @ Y @ X )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv3
thf(fact_121_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,A4: A] :
( ! [X4: A] :
( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X4 )
=> ( P @ Y5 ) )
=> ( P @ X4 ) )
=> ( P @ A4 ) ) ) ).
% less_induct
thf(fact_122_less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% less_not_sym
thf(fact_123_less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% less_imp_not_eq
thf(fact_124_dual__order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A4: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ~ ( ord_less @ A @ A4 @ B3 ) ) ) ).
% dual_order.asym
thf(fact_125_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% ord_less_eq_trans
thf(fact_126_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A,C2: A] :
( ( A4 = B3 )
=> ( ( ord_less @ A @ B3 @ C2 )
=> ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% ord_eq_less_trans
thf(fact_127_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] :
~ ( ord_less @ A @ X @ X ) ) ).
% less_irrefl
thf(fact_128_less__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
| ( X = Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% less_linear
thf(fact_129_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z4: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z4 )
=> ( ord_less @ A @ X @ Z4 ) ) ) ) ).
% less_trans
thf(fact_130_less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ~ ( ord_less @ A @ B3 @ A4 ) ) ) ).
% less_asym'
thf(fact_131_less__asym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% less_asym
thf(fact_132_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% less_imp_neq
thf(fact_133_dense,axiom,
! [A: $tType] :
( ( dense_order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ? [Z: A] :
( ( ord_less @ A @ X @ Z )
& ( ord_less @ A @ Z @ Y ) ) ) ) ).
% dense
thf(fact_134_order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ~ ( ord_less @ A @ B3 @ A4 ) ) ) ).
% order.asym
thf(fact_135_neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( X != Y )
= ( ( ord_less @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ) ).
% neq_iff
thf(fact_136_neqE,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% neqE
thf(fact_137_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [X1: A] : ( ord_less @ A @ X @ X1 ) ) ).
% gt_ex
thf(fact_138_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [Y4: A] : ( ord_less @ A @ Y4 @ X ) ) ).
% lt_ex
thf(fact_139_order__less__subst2,axiom,
! [A: $tType,C3: $tType] :
( ( ( order @ C3 @ ( type2 @ C3 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B3: A,F: A > C3,C2: C3] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ C3 @ ( F @ B3 ) @ C2 )
=> ( ! [X4: A,Y4: A] :
( ( ord_less @ A @ X4 @ Y4 )
=> ( ord_less @ C3 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ C3 @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% order_less_subst2
thf(fact_140_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B > A,B3: B,C2: B] :
( ( ord_less @ A @ A4 @ ( F @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C2 )
=> ( ! [X4: B,Y4: B] :
( ( ord_less @ B @ X4 @ Y4 )
=> ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_141_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B3: A,F: A > B,C2: B] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X4: A,Y4: A] :
( ( ord_less @ A @ X4 @ Y4 )
=> ( ord_less @ B @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ B @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_142_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B > A,B3: B,C2: B] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C2 )
=> ( ! [X4: B,Y4: B] :
( ( ord_less @ B @ X4 @ Y4 )
=> ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_143_bst__eq_Osimps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [L2: tree @ A,A4: A,R2: tree @ A] :
( ( linorder_bst_eq @ A @ ( node @ A @ L2 @ A4 @ R2 ) )
= ( ( linorder_bst_eq @ A @ L2 )
& ( linorder_bst_eq @ A @ R2 )
& ! [X6: A] :
( ( member @ A @ X6 @ ( set_tree @ A @ L2 ) )
=> ( ord_less_eq @ A @ X6 @ A4 ) )
& ! [X6: A] :
( ( member @ A @ X6 @ ( set_tree @ A @ R2 ) )
=> ( ord_less_eq @ A @ A4 @ X6 ) ) ) ) ) ).
% bst_eq.simps(2)
thf(fact_144_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 )
& ! [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(3)
thf(fact_145_leD,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less @ A @ X @ Y ) ) ) ).
% leD
thf(fact_146_leI,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% leI
thf(fact_147_le__less,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [X6: A,Y2: A] :
( ( ord_less @ A @ X6 @ Y2 )
| ( X6 = Y2 ) ) ) ) ) ).
% le_less
thf(fact_148_less__le,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [X6: A,Y2: A] :
( ( ord_less_eq @ A @ X6 @ Y2 )
& ( X6 != Y2 ) ) ) ) ) ).
% less_le
thf(fact_149_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B > A,B3: B,C2: B] :
( ( ord_less_eq @ A @ A4 @ ( F @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C2 )
=> ( ! [X4: B,Y4: B] :
( ( ord_less @ B @ X4 @ Y4 )
=> ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_150_order__le__less__subst2,axiom,
! [A: $tType,C3: $tType] :
( ( ( order @ C3 @ ( type2 @ C3 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B3: A,F: A > C3,C2: C3] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less @ C3 @ ( F @ B3 ) @ C2 )
=> ( ! [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ord_less_eq @ C3 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ C3 @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_151_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B > A,B3: B,C2: B] :
( ( ord_less @ A @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X4: B,Y4: B] :
( ( ord_less_eq @ B @ X4 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_152_order__less__le__subst2,axiom,
! [A: $tType,C3: $tType] :
( ( ( order @ C3 @ ( type2 @ C3 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B3: A,F: A > C3,C2: C3] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ C3 @ ( F @ B3 ) @ C2 )
=> ( ! [X4: A,Y4: A] :
( ( ord_less @ A @ X4 @ Y4 )
=> ( ord_less @ C3 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ C3 @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_153_not__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less_eq @ A @ X @ Y ) )
= ( ord_less @ A @ Y @ X ) ) ) ).
% not_le
thf(fact_154_not__less,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less @ A @ X @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% not_less
thf(fact_155_le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( A4 != B3 )
=> ( ord_less @ A @ A4 @ B3 ) ) ) ) ).
% le_neq_trans
thf(fact_156_less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% less_imp_le
thf(fact_157_antisym__conv1,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv1
thf(fact_158_antisym__conv2,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv2
thf(fact_159_le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z4: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z4 )
=> ( ord_less @ A @ X @ Z4 ) ) ) ) ).
% le_less_trans
thf(fact_160_less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z4: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z4 )
=> ( ord_less @ A @ X @ Z4 ) ) ) ) ).
% less_le_trans
thf(fact_161_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Z4: A,Y: A] :
( ! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ord_less_eq @ A @ Y @ X4 ) )
=> ( ord_less_eq @ A @ Y @ Z4 ) ) ) ).
% dense_ge
thf(fact_162_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Y: A,Z4: A] :
( ! [X4: A] :
( ( ord_less @ A @ X4 @ Y )
=> ( ord_less_eq @ A @ X4 @ Z4 ) )
=> ( ord_less_eq @ A @ Y @ Z4 ) ) ) ).
% dense_le
thf(fact_163_le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% le_less_linear
thf(fact_164_le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ X @ Y )
| ( X = Y ) ) ) ) ).
% le_imp_less_or_eq
thf(fact_165_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [X6: A,Y2: A] :
( ( ord_less_eq @ A @ X6 @ Y2 )
& ~ ( ord_less_eq @ A @ Y2 @ X6 ) ) ) ) ) ).
% less_le_not_le
thf(fact_166_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ~ ( ord_less_eq @ A @ Y @ X )
=> ( ord_less @ A @ X @ Y ) ) ) ).
% not_le_imp_less
thf(fact_167_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ B3 @ C2 )
=> ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% order.strict_trans1
thf(fact_168_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% order.strict_trans2
thf(fact_169_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_170_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_171_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A4: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( ord_less @ A @ C2 @ B3 )
=> ( ord_less @ A @ C2 @ A4 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_172_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A4: A,C2: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ( ord_less_eq @ A @ C2 @ B3 )
=> ( ord_less @ A @ C2 @ A4 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_173_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Z4: A,X: A,Y: A] :
( ( ord_less @ A @ Z4 @ X )
=> ( ! [W: A] :
( ( ord_less @ A @ Z4 @ W )
=> ( ( ord_less @ A @ W @ X )
=> ( ord_less_eq @ A @ Y @ W ) ) )
=> ( ord_less_eq @ A @ Y @ Z4 ) ) ) ) ).
% dense_ge_bounded
thf(fact_174_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z4: A] :
( ( ord_less @ A @ X @ Y )
=> ( ! [W: A] :
( ( ord_less @ A @ X @ W )
=> ( ( ord_less @ A @ W @ Y )
=> ( ord_less_eq @ A @ W @ Z4 ) ) )
=> ( ord_less_eq @ A @ Y @ Z4 ) ) ) ) ).
% dense_le_bounded
thf(fact_175_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ord_less_eq @ A @ A4 @ B3 ) ) ) ).
% order.strict_implies_order
thf(fact_176_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_177_bst__eq_Opelims_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: tree @ A,Y: $o] :
( ( ( linorder_bst_eq @ A @ X )
= Y )
=> ( ( accp @ ( tree @ A ) @ ( linorder_bst_eq_rel @ A ) @ X )
=> ( ( ( X
= ( leaf @ A ) )
=> ( Y
=> ~ ( accp @ ( tree @ A ) @ ( linorder_bst_eq_rel @ A ) @ ( leaf @ A ) ) ) )
=> ~ ! [L: tree @ A,A6: A,R: tree @ A] :
( ( X
= ( node @ A @ L @ A6 @ R ) )
=> ( ( Y
= ( ( linorder_bst_eq @ A @ L )
& ( linorder_bst_eq @ A @ R )
& ! [X6: A] :
( ( member @ A @ X6 @ ( set_tree @ A @ L ) )
=> ( ord_less_eq @ A @ X6 @ A6 ) )
& ! [X6: A] :
( ( member @ A @ X6 @ ( set_tree @ A @ R ) )
=> ( ord_less_eq @ A @ A6 @ X6 ) ) ) )
=> ~ ( accp @ ( tree @ A ) @ ( linorder_bst_eq_rel @ A ) @ ( node @ A @ L @ A6 @ R ) ) ) ) ) ) ) ) ).
% bst_eq.pelims(1)
thf(fact_178_bst__eq_Opelims_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: tree @ A] :
( ( linorder_bst_eq @ A @ X )
=> ( ( accp @ ( tree @ A ) @ ( linorder_bst_eq_rel @ A ) @ X )
=> ( ( ( X
= ( leaf @ A ) )
=> ~ ( accp @ ( tree @ A ) @ ( linorder_bst_eq_rel @ A ) @ ( leaf @ A ) ) )
=> ~ ! [L: tree @ A,A6: A,R: tree @ A] :
( ( X
= ( node @ A @ L @ A6 @ R ) )
=> ( ( accp @ ( tree @ A ) @ ( linorder_bst_eq_rel @ A ) @ ( node @ A @ L @ A6 @ R ) )
=> ~ ( ( linorder_bst_eq @ A @ L )
& ( linorder_bst_eq @ A @ R )
& ! [X5: A] :
( ( member @ A @ X5 @ ( set_tree @ A @ L ) )
=> ( ord_less_eq @ A @ X5 @ A6 ) )
& ! [X5: A] :
( ( member @ A @ X5 @ ( set_tree @ A @ R ) )
=> ( ord_less_eq @ A @ A6 @ X5 ) ) ) ) ) ) ) ) ) ).
% bst_eq.pelims(2)
thf(fact_179_bst__eq_Opelims_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: tree @ A] :
( ~ ( linorder_bst_eq @ A @ X )
=> ( ( accp @ ( tree @ A ) @ ( linorder_bst_eq_rel @ A ) @ X )
=> ~ ! [L: tree @ A,A6: A,R: tree @ A] :
( ( X
= ( node @ A @ L @ A6 @ R ) )
=> ( ( accp @ ( tree @ A ) @ ( linorder_bst_eq_rel @ A ) @ ( 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.pelims(3)
thf(fact_180_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z )
=> ~ ( ord_less_eq @ A @ T @ X5 ) ) ) ).
% minf(8)
thf(fact_181_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z )
=> ( ord_less_eq @ A @ X5 @ T ) ) ) ).
% minf(6)
thf(fact_182_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ Z @ X5 )
=> ( ord_less_eq @ A @ T @ X5 ) ) ) ).
% pinf(8)
thf(fact_183_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
& ~ ( ord_less_eq @ ( A > B ) @ G2 @ F2 ) ) ) ) ) ).
% less_fun_def
thf(fact_184_minf_I11_J,axiom,
! [C3: $tType,D2: $tType] :
( ( ord @ C3 @ ( type2 @ C3 ) )
=> ! [F3: D2] :
? [Z: C3] :
! [X5: C3] :
( ( ord_less @ C3 @ X5 @ Z )
=> ( F3 = F3 ) ) ) ).
% minf(11)
thf(fact_185_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z )
=> ~ ( ord_less @ A @ T @ X5 ) ) ) ).
% minf(7)
thf(fact_186_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z )
=> ( ord_less @ A @ X5 @ T ) ) ) ).
% minf(5)
thf(fact_187_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z )
=> ( X5 != T ) ) ) ).
% minf(4)
thf(fact_188_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z )
=> ( X5 != T ) ) ) ).
% minf(3)
thf(fact_189_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P3: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z6: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z6 )
=> ( ( P @ X4 )
= ( P3 @ X4 ) ) )
=> ( ? [Z6: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z6 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P3 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_190_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P3: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z6: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z6 )
=> ( ( P @ X4 )
= ( P3 @ X4 ) ) )
=> ( ? [Z6: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z6 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P3 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_191_pinf_I11_J,axiom,
! [C3: $tType,D2: $tType] :
( ( ord @ C3 @ ( type2 @ C3 ) )
=> ! [F3: D2] :
? [Z: C3] :
! [X5: C3] :
( ( ord_less @ C3 @ Z @ X5 )
=> ( F3 = F3 ) ) ) ).
% pinf(11)
thf(fact_192_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ Z @ X5 )
=> ( ord_less @ A @ T @ X5 ) ) ) ).
% pinf(7)
thf(fact_193_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ Z @ X5 )
=> ~ ( ord_less @ A @ X5 @ T ) ) ) ).
% pinf(5)
thf(fact_194_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ Z @ X5 )
=> ( X5 != T ) ) ) ).
% pinf(4)
thf(fact_195_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ Z @ X5 )
=> ( X5 != T ) ) ) ).
% pinf(3)
thf(fact_196_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P3: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z6: A] :
! [X4: A] :
( ( ord_less @ A @ Z6 @ X4 )
=> ( ( P @ X4 )
= ( P3 @ X4 ) ) )
=> ( ? [Z6: A] :
! [X4: A] :
( ( ord_less @ A @ Z6 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ Z @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P3 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_197_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P3: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z6: A] :
! [X4: A] :
( ( ord_less @ A @ Z6 @ X4 )
=> ( ( P @ X4 )
= ( P3 @ X4 ) ) )
=> ( ? [Z6: A] :
! [X4: A] :
( ( ord_less @ A @ Z6 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ Z @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P3 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_198_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X5: A] :
( ( ord_less @ A @ Z @ X5 )
=> ~ ( ord_less_eq @ A @ X5 @ T ) ) ) ).
% pinf(6)
thf(fact_199_t__splay__simps_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [B3: A,A4: A,L2: tree @ A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ( splay_914434265_splay @ A @ A4 @ ( node @ A @ L2 @ B3 @ ( leaf @ A ) ) )
= ( one_one @ nat ) ) ) ) ).
% t_splay_simps(6)
thf(fact_200_t__splay__simps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A,R2: tree @ A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( splay_914434265_splay @ A @ A4 @ ( node @ A @ ( leaf @ A ) @ B3 @ R2 ) )
= ( one_one @ nat ) ) ) ) ).
% t_splay_simps(2)
thf(fact_201_t__splay__simps_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [B3: A,A4: A,L2: tree @ A,Rl2: tree @ A,Rr2: tree @ A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ( splay_914434265_splay @ A @ A4 @ ( node @ A @ L2 @ B3 @ ( node @ A @ Rl2 @ A4 @ Rr2 ) ) )
= ( one_one @ nat ) ) ) ) ).
% t_splay_simps(7)
thf(fact_202_t__splay__simps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A4: A,L2: tree @ A,R2: tree @ A] :
( ( splay_914434265_splay @ A @ A4 @ ( node @ A @ L2 @ A4 @ R2 ) )
= ( one_one @ nat ) ) ) ).
% t_splay_simps(1)
thf(fact_203_t__splay__simps_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A,Ll: tree @ A,Lr: tree @ A,R2: tree @ A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( splay_914434265_splay @ A @ A4 @ ( node @ A @ ( node @ A @ Ll @ A4 @ Lr ) @ B3 @ R2 ) )
= ( one_one @ nat ) ) ) ) ).
% t_splay_simps(3)
thf(fact_204_t__splay_Osimps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A4: A] :
( ( splay_914434265_splay @ A @ A4 @ ( leaf @ A ) )
= ( one_one @ nat ) ) ) ).
% t_splay.simps(1)
thf(fact_205_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_206_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_207_t__splay__simps_I9_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [B3: A,A4: A,C2: A,Rr2: tree @ A,L2: tree @ A,Rl2: tree @ A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ( ord_less @ A @ C2 @ A4 )
=> ( ( ( Rr2
= ( leaf @ A ) )
=> ( ( splay_914434265_splay @ A @ A4 @ ( node @ A @ L2 @ B3 @ ( node @ A @ Rl2 @ C2 @ Rr2 ) ) )
= ( one_one @ nat ) ) )
& ( ( Rr2
!= ( leaf @ A ) )
=> ( ( splay_914434265_splay @ A @ A4 @ ( node @ A @ L2 @ B3 @ ( node @ A @ Rl2 @ C2 @ Rr2 ) ) )
= ( plus_plus @ nat @ ( splay_914434265_splay @ A @ A4 @ Rr2 ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% t_splay_simps(9)
thf(fact_208_t__splay__simps_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [B3: A,A4: A,C2: A,Rl2: tree @ A,L2: tree @ A,Rr2: tree @ A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ( ord_less @ A @ A4 @ C2 )
=> ( ( ( Rl2
= ( leaf @ A ) )
=> ( ( splay_914434265_splay @ A @ A4 @ ( node @ A @ L2 @ B3 @ ( node @ A @ Rl2 @ C2 @ Rr2 ) ) )
= ( one_one @ nat ) ) )
& ( ( Rl2
!= ( leaf @ A ) )
=> ( ( splay_914434265_splay @ A @ A4 @ ( node @ A @ L2 @ B3 @ ( node @ A @ Rl2 @ C2 @ Rr2 ) ) )
= ( plus_plus @ nat @ ( splay_914434265_splay @ A @ A4 @ Rl2 ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% t_splay_simps(8)
thf(fact_209_t__splay__simps_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A,C2: A,Ll: tree @ A,Lr: tree @ A,R2: tree @ A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ A4 @ C2 )
=> ( ( ( Ll
= ( leaf @ A ) )
=> ( ( splay_914434265_splay @ A @ A4 @ ( node @ A @ ( node @ A @ Ll @ C2 @ Lr ) @ B3 @ R2 ) )
= ( one_one @ nat ) ) )
& ( ( Ll
!= ( leaf @ A ) )
=> ( ( splay_914434265_splay @ A @ A4 @ ( node @ A @ ( node @ A @ Ll @ C2 @ Lr ) @ B3 @ R2 ) )
= ( plus_plus @ nat @ ( splay_914434265_splay @ A @ A4 @ Ll ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% t_splay_simps(4)
thf(fact_210_t__splay__simps_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A,C2: A,Lr: tree @ A,Ll: tree @ A,R2: tree @ A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ C2 @ A4 )
=> ( ( ( Lr
= ( leaf @ A ) )
=> ( ( splay_914434265_splay @ A @ A4 @ ( node @ A @ ( node @ A @ Ll @ C2 @ Lr ) @ B3 @ R2 ) )
= ( one_one @ nat ) ) )
& ( ( Lr
!= ( leaf @ A ) )
=> ( ( splay_914434265_splay @ A @ A4 @ ( node @ A @ ( node @ A @ Ll @ C2 @ Lr ) @ B3 @ R2 ) )
= ( plus_plus @ nat @ ( splay_914434265_splay @ A @ A4 @ Lr ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% t_splay_simps(5)
thf(fact_211_add__less__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [A4: A,C2: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A4 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) )
= ( ord_less @ A @ A4 @ B3 ) ) ) ).
% add_less_cancel_right
thf(fact_212_add__less__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A4 ) @ ( plus_plus @ A @ C2 @ B3 ) )
= ( ord_less @ A @ A4 @ B3 ) ) ) ).
% add_less_cancel_left
thf(fact_213_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A4 ) @ ( plus_plus @ A @ C2 @ B3 ) )
= ( ord_less_eq @ A @ A4 @ B3 ) ) ) ).
% add_le_cancel_left
thf(fact_214_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [A4: A,C2: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) )
= ( ord_less_eq @ A @ A4 @ B3 ) ) ) ).
% add_le_cancel_right
thf(fact_215_add__mono__thms__linordered__semiring_I3_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K: A,L2: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( K = L2 ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L2 ) ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_216_add__mono__thms__linordered__semiring_I2_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K: A,L2: A] :
( ( ( I = J )
& ( ord_less_eq @ A @ K @ L2 ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L2 ) ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_217_add__mono__thms__linordered__semiring_I1_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K: A,L2: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less_eq @ A @ K @ L2 ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L2 ) ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_218_add__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ C2 @ D3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ C2 ) @ ( plus_plus @ A @ B3 @ D3 ) ) ) ) ) ).
% add_mono
thf(fact_219_add__left__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A4 ) @ ( plus_plus @ A @ C2 @ B3 ) ) ) ) ).
% add_left_mono
thf(fact_220_add__right__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% add_right_mono
thf(fact_221_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_222_add__le__imp__le__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A4 ) @ ( plus_plus @ A @ C2 @ B3 ) )
=> ( ord_less_eq @ A @ A4 @ B3 ) ) ) ).
% add_le_imp_le_left
thf(fact_223_add__le__imp__le__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [A4: A,C2: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A4 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) )
=> ( ord_less_eq @ A @ A4 @ B3 ) ) ) ).
% add_le_imp_le_right
thf(fact_224_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_225_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_226_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_227_add__strict__mono,axiom,
! [A: $tType] :
( ( strict2144017051up_add @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A4 @ C2 ) @ ( plus_plus @ A @ B3 @ D3 ) ) ) ) ) ).
% add_strict_mono
thf(fact_228_add__strict__left__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ord_less @ A @ ( plus_plus @ A @ C2 @ A4 ) @ ( plus_plus @ A @ C2 @ B3 ) ) ) ) ).
% add_strict_left_mono
thf(fact_229_add__strict__right__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A4 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% add_strict_right_mono
thf(fact_230_add__less__imp__less__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A4 ) @ ( plus_plus @ A @ C2 @ B3 ) )
=> ( ord_less @ A @ A4 @ B3 ) ) ) ).
% add_less_imp_less_left
thf(fact_231_add__less__imp__less__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [A4: A,C2: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A4 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) )
=> ( ord_less @ A @ A4 @ B3 ) ) ) ).
% add_less_imp_less_right
thf(fact_232_add__mono__thms__linordered__field_I4_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K: A,L2: A] :
( ( ( ord_less_eq @ 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(4)
thf(fact_233_add__mono__thms__linordered__field_I3_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K: A,L2: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less_eq @ A @ K @ L2 ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L2 ) ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_234_add__le__less__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A4 @ C2 ) @ ( plus_plus @ A @ B3 @ D3 ) ) ) ) ) ).
% add_le_less_mono
thf(fact_235_add__less__le__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A,C2: A,D3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ C2 @ D3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A4 @ C2 ) @ ( plus_plus @ A @ B3 @ D3 ) ) ) ) ) ).
% add_less_le_mono
thf(fact_236_discrete,axiom,
! [A: $tType] :
( ( semiring_numeral_div @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [A5: A] : ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ ( one_one @ A ) ) ) ) ) ) ).
% discrete
thf(fact_237_less__add__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [A4: A] : ( ord_less @ A @ A4 @ ( plus_plus @ A @ A4 @ ( one_one @ A ) ) ) ) ).
% less_add_one
thf(fact_238_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_239_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_240_t__splay__max_Osimps_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Rr2: tree @ A,L2: tree @ A,B3: A,Rl2: tree @ A,C2: A] :
( ( ( Rr2
= ( leaf @ A ) )
=> ( ( splay_878424299ay_max @ A @ ( node @ A @ L2 @ B3 @ ( node @ A @ Rl2 @ C2 @ Rr2 ) ) )
= ( one_one @ nat ) ) )
& ( ( Rr2
!= ( leaf @ A ) )
=> ( ( splay_878424299ay_max @ A @ ( node @ A @ L2 @ B3 @ ( node @ A @ Rl2 @ C2 @ Rr2 ) ) )
= ( plus_plus @ nat @ ( splay_878424299ay_max @ A @ Rr2 ) @ ( one_one @ nat ) ) ) ) ) ) ).
% t_splay_max.simps(3)
thf(fact_241_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_242_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( M != N )
=> ( ord_less @ nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_243_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less @ nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_244_le__eq__less__or__eq,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_245_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_246_nat__less__le,axiom,
( ( ord_less @ nat )
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_247_le__refl,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% le_refl
thf(fact_248_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ J @ K )
=> ( ord_less_eq @ nat @ I @ K ) ) ) ).
% le_trans
thf(fact_249_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_250_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_251_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
| ( ord_less_eq @ nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_252_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M3: nat,N3: nat] :
( ( ord_less @ nat @ M3 @ N3 )
=> ( ord_less @ nat @ ( F @ M3 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus @ nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_253_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq @ nat @ M @ N )
=> ~ ( ord_less_eq @ nat @ K @ N ) ) ) ).
% add_leE
thf(fact_254_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ N @ M ) ) ).
% le_add1
thf(fact_255_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ M @ N ) ) ).
% le_add2
%----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 (24)
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A9: $tType,A10: $tType] :
( ( preorder @ A10 @ ( type2 @ A10 ) )
=> ( preorder @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A9: $tType,A10: $tType] :
( ( order @ A10 @ ( type2 @ A10 ) )
=> ( order @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A9: $tType,A10: $tType] :
( ( ord @ A10 @ ( type2 @ A10 ) )
=> ( ord @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit1037483654norder @ 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_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___Divides_Osemiring__numeral__div,axiom,
semiring_numeral_div @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom,axiom,
linordered_semidom @ 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_Set_Oset___Orderings_Opreorder_4,axiom,
! [A9: $tType] : ( preorder @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_5,axiom,
! [A9: $tType] : ( order @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_6,axiom,
! [A9: $tType] : ( ord @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
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 ) ).
%----Free types (1)
thf(tfree_0,hypothesis,
linorder @ a @ ( type2 @ a ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
member @ a @ xa @ ( set_tree @ a @ lb ) ).
%------------------------------------------------------------------------------