TPTP Problem File: ITP067^2.p
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%------------------------------------------------------------------------------
% File : ITP067^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer HeapImperative problem prob_172__5340680_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : HeapImperative/prob_172__5340680_1 [Des21]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.50 v7.5.0
% Syntax : Number of formulae : 350 ( 91 unt; 67 typ; 0 def)
% Number of atoms : 881 ( 271 equ; 0 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 4701 ( 93 ~; 11 |; 55 &;4102 @)
% ( 0 <=>; 440 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 8 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 249 ( 249 >; 0 *; 0 +; 0 <<)
% Number of symbols : 69 ( 66 usr; 11 con; 0-8 aty)
% Number of variables : 1149 ( 53 ^;1028 !; 9 ?;1149 :)
% ( 59 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:15:21.931
%------------------------------------------------------------------------------
% Could-be-implicit typings (6)
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Multiset_Omultiset,type,
multiset: $tType > $tType ).
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Heap_OTree,type,
tree: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (61)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1352612707id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
ordere216010020id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere779506340up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
linord1659791738miring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere236663937imp_le:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__comm__monoid__diff,type,
ordere623563068d_diff:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
ordere516151231imp_le:
!>[A: $tType] : $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
bNF_Ca1785829860lChain:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_HeapImperative__Mirabelle__oitemzccmr_Oleft,type,
heapIm1271749598e_left:
!>[A: $tType] : ( ( tree @ A ) > ( tree @ A ) ) ).
thf(sy_c_HeapImperative__Mirabelle__oitemzccmr_Oright,type,
heapIm1434396069_right:
!>[A: $tType] : ( ( tree @ A ) > ( tree @ A ) ) ).
thf(sy_c_HeapImperative__Mirabelle__oitemzccmr_OsiftDown,type,
heapIm748920189ftDown:
!>[A: $tType] : ( ( tree @ A ) > ( tree @ A ) ) ).
thf(sy_c_Heap_OHeap,type,
heap:
!>[B: $tType,A: $tType] : ( B > ( B > $o ) > ( ( list @ A ) > B ) > ( B > ( multiset @ A ) ) > ( B > ( tree @ A ) ) > ( B > ( product_prod @ A @ B ) ) > $o ) ).
thf(sy_c_Heap_OHeap__axioms,type,
heap_axioms:
!>[B: $tType,A: $tType] : ( ( B > $o ) > ( ( list @ A ) > B ) > ( B > ( multiset @ A ) ) > ( B > ( tree @ A ) ) > ( B > ( product_prod @ A @ B ) ) > $o ) ).
thf(sy_c_Heap_OTree_OE,type,
e:
!>[A: $tType] : ( tree @ A ) ).
thf(sy_c_Heap_OTree_OT,type,
t:
!>[A: $tType] : ( A > ( tree @ A ) > ( tree @ A ) > ( tree @ A ) ) ).
thf(sy_c_Heap_OTree_Ocase__Tree,type,
case_Tree:
!>[B: $tType,A: $tType] : ( B > ( A > ( tree @ A ) > ( tree @ A ) > B ) > ( tree @ A ) > B ) ).
thf(sy_c_Heap_OTree_Opred__Tree,type,
pred_Tree:
!>[A: $tType] : ( ( A > $o ) > ( tree @ A ) > $o ) ).
thf(sy_c_Heap_OTree_Orec__Tree,type,
rec_Tree:
!>[C: $tType,A: $tType] : ( C > ( A > ( tree @ A ) > ( tree @ A ) > C > C > C ) > ( tree @ A ) > C ) ).
thf(sy_c_Heap_OTree_Oset__Tree,type,
set_Tree:
!>[A: $tType] : ( ( tree @ A ) > ( set @ A ) ) ).
thf(sy_c_Heap_Oin__tree,type,
in_tree:
!>[A: $tType] : ( A > ( tree @ A ) > $o ) ).
thf(sy_c_Heap_Ois__heap,type,
is_heap:
!>[A: $tType] : ( ( tree @ A ) > $o ) ).
thf(sy_c_Heap_Omultiset,type,
multiset2:
!>[A: $tType] : ( ( tree @ A ) > ( multiset @ A ) ) ).
thf(sy_c_Heap_Oval,type,
val:
!>[A: $tType] : ( ( tree @ A ) > A ) ).
thf(sy_c_Lattices__Big_Olinorder__class_OMax,type,
lattic929149872er_Max:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Multiset_Oadd__mset,type,
add_mset:
!>[A: $tType] : ( A > ( multiset @ A ) > ( multiset @ A ) ) ).
thf(sy_c_Multiset_Oset__mset,type,
set_mset:
!>[A: $tType] : ( ( multiset @ A ) > ( set @ A ) ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oorder__class_OGreatest,type,
order_Greatest:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Relation_OPowp,type,
powp:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_RemoveMax_OCollection,type,
collection:
!>[B: $tType,A: $tType] : ( B > ( B > $o ) > ( ( list @ A ) > B ) > ( B > ( multiset @ A ) ) > $o ) ).
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_Set_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Ois__empty,type,
is_empty:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Ois__singleton,type,
is_singleton:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_l1____,type,
l1: tree @ a ).
thf(sy_v_l2____,type,
l2: tree @ a ).
thf(sy_v_r1____,type,
r1: tree @ a ).
thf(sy_v_r2____,type,
r2: tree @ a ).
thf(sy_v_t,type,
t2: tree @ a ).
thf(sy_v_v,type,
v: a ).
thf(sy_v_v1____,type,
v1: a ).
thf(sy_v_v2____,type,
v2: a ).
thf(sy_v_v_H____,type,
v3: a ).
% Relevant facts (256)
thf(fact_0_False,axiom,
~ ( ( v = v3 )
| ( v = v1 )
| ( v = v2 ) ) ).
% False
thf(fact_1__092_060open_062in__tree_Av_A_IsiftDown_At_J_092_060close_062,axiom,
in_tree @ a @ v @ ( heapIm748920189ftDown @ a @ t2 ) ).
% \<open>in_tree v (siftDown t)\<close>
thf(fact_2_True,axiom,
ord_less_eq @ a @ v2 @ v1 ).
% True
thf(fact_3__C5__1_Oprems_C,axiom,
in_tree @ a @ v @ ( heapIm748920189ftDown @ a @ ( t @ a @ v3 @ ( t @ a @ v1 @ l1 @ r1 ) @ ( t @ a @ v2 @ l2 @ r2 ) ) ) ).
% "5_1.prems"
thf(fact_4_Tree_Oinject,axiom,
! [A: $tType,X21: A,X22: tree @ A,X23: tree @ A,Y21: A,Y22: tree @ A,Y23: tree @ A] :
( ( ( t @ A @ X21 @ X22 @ X23 )
= ( t @ A @ Y21 @ Y22 @ Y23 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 )
& ( X23 = Y23 ) ) ) ).
% Tree.inject
thf(fact_5_in__tree_Osimps_I2_J,axiom,
! [A: $tType,V: A,V2: A,L: tree @ A,R: tree @ A] :
( ( in_tree @ A @ V @ ( t @ A @ V2 @ L @ R ) )
= ( ( V = V2 )
| ( in_tree @ A @ V @ L )
| ( in_tree @ A @ V @ R ) ) ) ).
% in_tree.simps(2)
thf(fact_6__C5__1_Ohyps_C_I2_J,axiom,
( ~ ( ord_less_eq @ a @ ( val @ a @ ( t @ a @ v2 @ l2 @ r2 ) ) @ ( val @ a @ ( t @ a @ v1 @ l1 @ r1 ) ) )
=> ( ~ ( ord_less_eq @ a @ ( val @ a @ ( t @ a @ v2 @ l2 @ r2 ) ) @ v3 )
=> ( ( in_tree @ a @ v @ ( heapIm748920189ftDown @ a @ ( t @ a @ v3 @ ( heapIm1271749598e_left @ a @ ( t @ a @ v2 @ l2 @ r2 ) ) @ ( heapIm1434396069_right @ a @ ( t @ a @ v2 @ l2 @ r2 ) ) ) ) )
=> ( in_tree @ a @ v @ ( t @ a @ v3 @ ( heapIm1271749598e_left @ a @ ( t @ a @ v2 @ l2 @ r2 ) ) @ ( heapIm1434396069_right @ a @ ( t @ a @ v2 @ l2 @ r2 ) ) ) ) ) ) ) ).
% "5_1.hyps"(2)
thf(fact_7__C5__1_Ohyps_C_I1_J,axiom,
( ( ord_less_eq @ a @ ( val @ a @ ( t @ a @ v2 @ l2 @ r2 ) ) @ ( val @ a @ ( t @ a @ v1 @ l1 @ r1 ) ) )
=> ( ~ ( ord_less_eq @ a @ ( val @ a @ ( t @ a @ v1 @ l1 @ r1 ) ) @ v3 )
=> ( ( in_tree @ a @ v @ ( heapIm748920189ftDown @ a @ ( t @ a @ v3 @ ( heapIm1271749598e_left @ a @ ( t @ a @ v1 @ l1 @ r1 ) ) @ ( heapIm1434396069_right @ a @ ( t @ a @ v1 @ l1 @ r1 ) ) ) ) )
=> ( in_tree @ a @ v @ ( t @ a @ v3 @ ( heapIm1271749598e_left @ a @ ( t @ a @ v1 @ l1 @ r1 ) ) @ ( heapIm1434396069_right @ a @ ( t @ a @ v1 @ l1 @ r1 ) ) ) ) ) ) ) ).
% "5_1.hyps"(1)
thf(fact_8_left_Osimps,axiom,
! [A: $tType,V: A,L: tree @ A,R: tree @ A] :
( ( heapIm1271749598e_left @ A @ ( t @ A @ V @ L @ R ) )
= L ) ).
% left.simps
thf(fact_9_right_Osimps,axiom,
! [A: $tType,V: A,L: tree @ A,R: tree @ A] :
( ( heapIm1434396069_right @ A @ ( t @ A @ V @ L @ R ) )
= R ) ).
% right.simps
thf(fact_10_siftDown_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: tree @ A] :
( ( X
!= ( e @ A ) )
=> ( ! [V3: A] :
( X
!= ( t @ A @ V3 @ ( e @ A ) @ ( e @ A ) ) )
=> ( ! [V3: A,Va: A,Vb: tree @ A,Vc: tree @ A] :
( X
!= ( t @ A @ V3 @ ( t @ A @ Va @ Vb @ Vc ) @ ( e @ A ) ) )
=> ( ! [V3: A,Va: A,Vb: tree @ A,Vc: tree @ A] :
( X
!= ( t @ A @ V3 @ ( e @ A ) @ ( t @ A @ Va @ Vb @ Vc ) ) )
=> ~ ! [V3: A,Va: A,Vb: tree @ A,Vc: tree @ A,Vd: A,Ve: tree @ A,Vf: tree @ A] :
( X
!= ( t @ A @ V3 @ ( t @ A @ Va @ Vb @ Vc ) @ ( t @ A @ Vd @ Ve @ Vf ) ) ) ) ) ) ) ) ).
% siftDown.cases
thf(fact_11_Tree_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F1: B,F2: A > ( tree @ A ) > ( tree @ A ) > B,X21: A,X22: tree @ A,X23: tree @ A] :
( ( case_Tree @ B @ A @ F1 @ F2 @ ( t @ A @ X21 @ X22 @ X23 ) )
= ( F2 @ X21 @ X22 @ X23 ) ) ).
% Tree.simps(5)
thf(fact_12_Tree_Osimps_I7_J,axiom,
! [C: $tType,A: $tType,F1: C,F2: A > ( tree @ A ) > ( tree @ A ) > C > C > C,X21: A,X22: tree @ A,X23: tree @ A] :
( ( rec_Tree @ C @ A @ F1 @ F2 @ ( t @ A @ X21 @ X22 @ X23 ) )
= ( F2 @ X21 @ X22 @ X23 @ ( rec_Tree @ C @ A @ F1 @ F2 @ X22 ) @ ( rec_Tree @ C @ A @ F1 @ F2 @ X23 ) ) ) ).
% Tree.simps(7)
thf(fact_13_Tree_Opred__inject_I2_J,axiom,
! [A: $tType,P: A > $o,A2: A,Aa: tree @ A,Ab: tree @ A] :
( ( pred_Tree @ A @ P @ ( t @ A @ A2 @ Aa @ Ab ) )
= ( ( P @ A2 )
& ( pred_Tree @ A @ P @ Aa )
& ( pred_Tree @ A @ P @ Ab ) ) ) ).
% Tree.pred_inject(2)
thf(fact_14_in__tree_Osimps_I1_J,axiom,
! [A: $tType,V: A] :
~ ( in_tree @ A @ V @ ( e @ A ) ) ).
% in_tree.simps(1)
thf(fact_15_Tree_Oset__intros_I3_J,axiom,
! [A: $tType,Ya: A,X23: tree @ A,X21: A,X22: tree @ A] :
( ( member @ A @ Ya @ ( set_Tree @ A @ X23 ) )
=> ( member @ A @ Ya @ ( set_Tree @ A @ ( t @ A @ X21 @ X22 @ X23 ) ) ) ) ).
% Tree.set_intros(3)
thf(fact_16_Tree_Osimps_I6_J,axiom,
! [A: $tType,C: $tType,F1: C,F2: A > ( tree @ A ) > ( tree @ A ) > C > C > C] :
( ( rec_Tree @ C @ A @ F1 @ F2 @ ( e @ A ) )
= F1 ) ).
% Tree.simps(6)
thf(fact_17_Tree_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,F1: B,F2: A > ( tree @ A ) > ( tree @ A ) > B] :
( ( case_Tree @ B @ A @ F1 @ F2 @ ( e @ A ) )
= F1 ) ).
% Tree.simps(4)
thf(fact_18_Tree_Opred__inject_I1_J,axiom,
! [A: $tType,P: A > $o] : ( pred_Tree @ A @ P @ ( e @ A ) ) ).
% Tree.pred_inject(1)
thf(fact_19_siftDown_Osimps_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Va2: A,Vb2: tree @ A,Vc2: tree @ A,Vd2: A,Ve2: tree @ A,Vf2: tree @ A,V: A] :
( ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) @ ( val @ A @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) )
=> ( ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) @ V )
=> ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) )
= ( t @ A @ V @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) @ V )
=> ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) )
= ( t @ A @ ( val @ A @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) @ ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( heapIm1271749598e_left @ A @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) @ ( heapIm1434396069_right @ A @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) ) ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) @ ( val @ A @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) )
=> ( ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) @ V )
=> ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) )
= ( t @ A @ V @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) @ V )
=> ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) )
= ( t @ A @ ( val @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) @ ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( heapIm1271749598e_left @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) @ ( heapIm1434396069_right @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ) ).
% siftDown.simps(6)
thf(fact_20_siftDown_Osimps_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Vd2: A,Ve2: tree @ A,Vf2: tree @ A,Va2: A,Vb2: tree @ A,Vc2: tree @ A,V: A] :
( ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) @ ( val @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) )
=> ( ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) @ V )
=> ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) )
= ( t @ A @ V @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) @ V )
=> ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) )
= ( t @ A @ ( val @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) @ ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( heapIm1271749598e_left @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) @ ( heapIm1434396069_right @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) ) ) @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) @ ( val @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) )
=> ( ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) @ V )
=> ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) )
= ( t @ A @ V @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) @ V )
=> ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) )
= ( t @ A @ ( val @ A @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( heapIm1271749598e_left @ A @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) @ ( heapIm1434396069_right @ A @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) ) ) ) ) ) ) ) ) ) ).
% siftDown.simps(5)
thf(fact_21_siftDown_Osimps_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Va2: A,Vb2: tree @ A,Vc2: tree @ A,V: A] :
( ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) @ V )
=> ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( e @ A ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) )
= ( t @ A @ V @ ( e @ A ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) @ V )
=> ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( e @ A ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) )
= ( t @ A @ ( val @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) @ ( e @ A ) @ ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( heapIm1271749598e_left @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) @ ( heapIm1434396069_right @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ).
% siftDown.simps(4)
thf(fact_22_siftDown_Osimps_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Va2: A,Vb2: tree @ A,Vc2: tree @ A,V: A] :
( ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) @ V )
=> ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( e @ A ) ) )
= ( t @ A @ V @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( e @ A ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) @ V )
=> ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( e @ A ) ) )
= ( t @ A @ ( val @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) @ ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( heapIm1271749598e_left @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) @ ( heapIm1434396069_right @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) ) ) @ ( e @ A ) ) ) ) ) ) ).
% siftDown.simps(3)
thf(fact_23_siftDown_Osimps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( heapIm748920189ftDown @ A @ ( e @ A ) )
= ( e @ A ) ) ) ).
% siftDown.simps(1)
thf(fact_24_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_25_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_26_siftDown__in__tree,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: tree @ A] :
( ( T
!= ( e @ A ) )
=> ( in_tree @ A @ ( val @ A @ ( heapIm748920189ftDown @ A @ T ) ) @ T ) ) ) ).
% siftDown_in_tree
thf(fact_27_siftDown__Node,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: tree @ A,V: A,L: tree @ A,R: tree @ A] :
( ( T
= ( t @ A @ V @ L @ R ) )
=> ? [L2: tree @ A,V4: A,R2: tree @ A] :
( ( ( heapIm748920189ftDown @ A @ T )
= ( t @ A @ V4 @ L2 @ R2 ) )
& ( ord_less_eq @ A @ V @ V4 ) ) ) ) ).
% siftDown_Node
thf(fact_28_is__heap_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: tree @ A] :
( ( X
!= ( e @ A ) )
=> ( ! [V3: A] :
( X
!= ( t @ A @ V3 @ ( e @ A ) @ ( e @ A ) ) )
=> ( ! [V3: A,Va: A,Vb: tree @ A,Vc: tree @ A] :
( X
!= ( t @ A @ V3 @ ( e @ A ) @ ( t @ A @ Va @ Vb @ Vc ) ) )
=> ( ! [V3: A,Va: A,Vb: tree @ A,Vc: tree @ A] :
( X
!= ( t @ A @ V3 @ ( t @ A @ Va @ Vb @ Vc ) @ ( e @ A ) ) )
=> ~ ! [V3: A,Va: A,Vb: tree @ A,Vc: tree @ A,Vd: A,Ve: tree @ A,Vf: tree @ A] :
( X
!= ( t @ A @ V3 @ ( t @ A @ Va @ Vb @ Vc ) @ ( t @ A @ Vd @ Ve @ Vf ) ) ) ) ) ) ) ) ).
% is_heap.cases
thf(fact_29_Tree_Oexhaust,axiom,
! [A: $tType,Y: tree @ A] :
( ( Y
!= ( e @ A ) )
=> ~ ! [X212: A,X222: tree @ A,X232: tree @ A] :
( Y
!= ( t @ A @ X212 @ X222 @ X232 ) ) ) ).
% Tree.exhaust
thf(fact_30_Tree_Oinduct,axiom,
! [A: $tType,P: ( tree @ A ) > $o,Tree: tree @ A] :
( ( P @ ( e @ A ) )
=> ( ! [X1: A,X2: tree @ A,X3: tree @ A] :
( ( P @ X2 )
=> ( ( P @ X3 )
=> ( P @ ( t @ A @ X1 @ X2 @ X3 ) ) ) )
=> ( P @ Tree ) ) ) ).
% Tree.induct
thf(fact_31_Tree_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: tree @ A,X23: tree @ A] :
( ( e @ A )
!= ( t @ A @ X21 @ X22 @ X23 ) ) ).
% Tree.distinct(1)
thf(fact_32_siftDown_Osimps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [V: A] :
( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( e @ A ) @ ( e @ A ) ) )
= ( t @ A @ V @ ( e @ A ) @ ( e @ A ) ) ) ) ).
% siftDown.simps(2)
thf(fact_33_val_Osimps,axiom,
! [A: $tType,V: A,Uu: tree @ A,Uv: tree @ A] :
( ( val @ A @ ( t @ A @ V @ Uu @ Uv ) )
= V ) ).
% val.simps
thf(fact_34_Tree_Oset__cases,axiom,
! [A: $tType,E: A,A2: tree @ A] :
( ( member @ A @ E @ ( set_Tree @ A @ A2 ) )
=> ( ! [Z2: tree @ A,Z3: tree @ A] :
( A2
!= ( t @ A @ E @ Z2 @ Z3 ) )
=> ( ! [Z1: A,Z2: tree @ A] :
( ? [Z3: tree @ A] :
( A2
= ( t @ A @ Z1 @ Z2 @ Z3 ) )
=> ~ ( member @ A @ E @ ( set_Tree @ A @ Z2 ) ) )
=> ~ ! [Z1: A,Z2: tree @ A,Z3: tree @ A] :
( ( A2
= ( t @ A @ Z1 @ Z2 @ Z3 ) )
=> ~ ( member @ A @ E @ ( set_Tree @ A @ Z3 ) ) ) ) ) ) ).
% Tree.set_cases
thf(fact_35_Tree_Oset__intros_I1_J,axiom,
! [A: $tType,X21: A,X22: tree @ A,X23: tree @ A] : ( member @ A @ X21 @ ( set_Tree @ A @ ( t @ A @ X21 @ X22 @ X23 ) ) ) ).
% Tree.set_intros(1)
thf(fact_36_Tree_Oset__intros_I2_J,axiom,
! [A: $tType,Y: A,X22: tree @ A,X21: A,X23: tree @ A] :
( ( member @ A @ Y @ ( set_Tree @ A @ X22 ) )
=> ( member @ A @ Y @ ( set_Tree @ A @ ( t @ A @ X21 @ X22 @ X23 ) ) ) ) ).
% Tree.set_intros(2)
thf(fact_37_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_38_is__heap_Osimps_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [V: A,Va2: A,Vb2: tree @ A,Vc2: tree @ A] :
( ( is_heap @ A @ ( t @ A @ V @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( e @ A ) ) )
= ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) @ V )
& ( is_heap @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ).
% is_heap.simps(4)
thf(fact_39_is__heap_Osimps_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [V: A,Va2: A,Vb2: tree @ A,Vc2: tree @ A] :
( ( is_heap @ A @ ( t @ A @ V @ ( e @ A ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) )
= ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) @ V )
& ( is_heap @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ).
% is_heap.simps(3)
thf(fact_40_is__heap__max,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [V: A,T: tree @ A] :
( ( in_tree @ A @ V @ T )
=> ( ( is_heap @ A @ T )
=> ( ord_less_eq @ A @ V @ ( val @ A @ T ) ) ) ) ) ).
% is_heap_max
thf(fact_41_is__heap_Osimps_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [V: A,Vd2: A,Ve2: tree @ A,Vf2: tree @ A,Va2: A,Vb2: tree @ A,Vc2: tree @ A] :
( ( is_heap @ A @ ( t @ A @ V @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) )
= ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) @ V )
& ( is_heap @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) )
& ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) @ V )
& ( is_heap @ A @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) ) ) ) ).
% is_heap.simps(6)
thf(fact_42_is__heap_Osimps_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [V: A,Va2: A,Vb2: tree @ A,Vc2: tree @ A,Vd2: A,Ve2: tree @ A,Vf2: tree @ A] :
( ( is_heap @ A @ ( t @ A @ V @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) )
= ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) @ V )
& ( is_heap @ A @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) )
& ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) @ V )
& ( is_heap @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ).
% is_heap.simps(5)
thf(fact_43_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ 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_44_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ 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_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X5: A] :
( ( P @ X5 )
= ( Q @ X5 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X5: A] :
( ( F @ X5 )
= ( G @ X5 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B] :
( ! [X5: A] : ( ord_less_eq @ B @ ( F @ X5 ) @ ( G @ X5 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_50_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F3: A > B,G2: A > B] :
! [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% le_fun_def
thf(fact_51_Tree_Opred__mono,axiom,
! [A: $tType,P: A > $o,Pa: A > $o] :
( ( ord_less_eq @ ( A > $o ) @ P @ Pa )
=> ( ord_less_eq @ ( ( tree @ A ) > $o ) @ ( pred_Tree @ A @ P ) @ ( pred_Tree @ A @ Pa ) ) ) ).
% Tree.pred_mono
thf(fact_52_is__heap_Osimps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( is_heap @ A @ ( e @ A ) ) ) ).
% is_heap.simps(1)
thf(fact_53_is__heap_Osimps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [V: A] : ( is_heap @ A @ ( t @ A @ V @ ( e @ A ) @ ( e @ A ) ) ) ) ).
% is_heap.simps(2)
thf(fact_54_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% dual_order.antisym
thf(fact_55_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y2: A,Z4: A] : Y2 = Z4 )
= ( ^ [A4: A,B3: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
& ( ord_less_eq @ A @ A4 @ B3 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_56_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_57_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
=> ( P @ A5 @ B4 ) )
=> ( ! [A5: A,B4: A] :
( ( P @ B4 @ A5 )
=> ( P @ A5 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_58_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_59_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z5: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z5 )
=> ( ord_less_eq @ A @ X @ Z5 ) ) ) ) ).
% order_trans
thf(fact_60_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ) ).
% order_class.order.antisym
thf(fact_61_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_62_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_63_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y2: A,Z4: A] : Y2 = Z4 )
= ( ^ [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
& ( ord_less_eq @ A @ B3 @ A4 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_64_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv
thf(fact_65_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z5: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z5 ) )
=> ( ( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less_eq @ A @ X @ Z5 ) )
=> ( ( ( ord_less_eq @ A @ X @ Z5 )
=> ~ ( ord_less_eq @ A @ Z5 @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z5 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z5 )
=> ~ ( ord_less_eq @ A @ Z5 @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z5 @ X )
=> ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_66_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% order.trans
thf(fact_67_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% le_cases
thf(fact_68_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( X = Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% eq_refl
thf(fact_69_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linear
thf(fact_70_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( X = Y ) ) ) ) ).
% antisym
thf(fact_71_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y2: A,Z4: A] : Y2 = Z4 )
= ( ^ [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
& ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ) ) ).
% eq_iff
thf(fact_72_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B2: A,F: A > B,C2: B] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X5: A,Y4: A] :
( ( ord_less_eq @ A @ X5 @ Y4 )
=> ( ord_less_eq @ B @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_73_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,F: B > A,B2: B,C2: B] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X5: B,Y4: B] :
( ( ord_less_eq @ B @ X5 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_74_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F: A > C,C2: C] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
=> ( ! [X5: A,Y4: A] :
( ( ord_less_eq @ A @ X5 @ Y4 )
=> ( ord_less_eq @ C @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_75_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F: B > A,B2: B,C2: B] :
( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X5: B,Y4: B] :
( ( ord_less_eq @ B @ X5 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_76_Greatest__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A] :
( ( P @ X )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X ) )
=> ( ( order_Greatest @ A @ P )
= X ) ) ) ) ).
% Greatest_equality
thf(fact_77_GreatestI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A,Q: A > $o] :
( ( P @ X )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X ) )
=> ( ! [X5: A] :
( ( P @ X5 )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ A @ Y5 @ X5 ) )
=> ( Q @ X5 ) ) )
=> ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% GreatestI2_order
thf(fact_78_le__rel__bool__arg__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_less_eq @ ( $o > A ) )
= ( ^ [X6: $o > A,Y6: $o > A] :
( ( ord_less_eq @ A @ ( X6 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq @ A @ ( X6 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_79_verit__la__disequality,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( A2 = B2 )
| ~ ( ord_less_eq @ A @ A2 @ B2 )
| ~ ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% verit_la_disequality
thf(fact_80_Tree_Osimps_I14_J,axiom,
! [A: $tType] :
( ( set_Tree @ A @ ( e @ A ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Tree.simps(14)
thf(fact_81_Heap_Ois__heap__of__list,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),As_tree: B > ( tree @ A ),Remove_max: B > ( product_prod @ A @ B ),I: list @ A] :
( ( heap @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max )
=> ( is_heap @ A @ ( As_tree @ ( Of_list @ I ) ) ) ) ) ).
% Heap.is_heap_of_list
thf(fact_82_Tree_Opred__set,axiom,
! [A: $tType] :
( ( pred_Tree @ A )
= ( ^ [P2: A > $o,X4: tree @ A] :
! [Y3: A] :
( ( member @ A @ Y3 @ ( set_Tree @ A @ X4 ) )
=> ( P2 @ Y3 ) ) ) ) ).
% Tree.pred_set
thf(fact_83_Heap_Oas__tree__empty,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),As_tree: B > ( tree @ A ),Remove_max: B > ( product_prod @ A @ B ),T: B] :
( ( heap @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max )
=> ( ( ( As_tree @ T )
= ( e @ A ) )
= ( Is_empty @ T ) ) ) ) ).
% Heap.as_tree_empty
thf(fact_84_bot__apply,axiom,
! [C: $tType,D: $tType] :
( ( bot @ C )
=> ( ( bot_bot @ ( D > C ) )
= ( ^ [X4: D] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_85_predicate1I,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X5: A] :
( ( P @ X5 )
=> ( Q @ X5 ) )
=> ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% predicate1I
thf(fact_86_predicate1D,axiom,
! [A: $tType,P: A > $o,Q: A > $o,X: A] :
( ( ord_less_eq @ ( A > $o ) @ P @ Q )
=> ( ( P @ X )
=> ( Q @ X ) ) ) ).
% predicate1D
thf(fact_87_rev__predicate1D,axiom,
! [A: $tType,P: A > $o,X: A,Q: A > $o] :
( ( P @ X )
=> ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
=> ( Q @ X ) ) ) ).
% rev_predicate1D
thf(fact_88_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X4: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_89_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ).
% bot.extremum
thf(fact_90_bot_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
= ( A2
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_unique
thf(fact_91_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
=> ( A2
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_uniqueI
thf(fact_92_ball__empty,axiom,
! [A: $tType,P: A > $o,X7: A] :
( ( member @ A @ X7 @ ( bot_bot @ ( set @ A ) ) )
=> ( P @ X7 ) ) ).
% ball_empty
thf(fact_93_subset__empty,axiom,
! [A: $tType,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_94_empty__subsetI,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 ) ).
% empty_subsetI
thf(fact_95_empty__iff,axiom,
! [A: $tType,C2: A] :
~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_96_all__not__in__conv,axiom,
! [A: $tType,A3: set @ A] :
( ( ! [X4: A] :
~ ( member @ A @ X4 @ A3 ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_97_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X4: A] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_98_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X4: A] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_99_subsetI,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( member @ A @ X5 @ B5 ) )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ).
% subsetI
thf(fact_100_subset__antisym,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A3 )
=> ( A3 = B5 ) ) ) ).
% subset_antisym
thf(fact_101_in__mono,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( member @ A @ X @ A3 )
=> ( member @ A @ X @ B5 ) ) ) ).
% in_mono
thf(fact_102_subsetD,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B5 ) ) ) ).
% subsetD
thf(fact_103_equalityE,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( A3 = B5 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A3 ) ) ) ).
% equalityE
thf(fact_104_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( member @ A @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_105_equalityD1,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( A3 = B5 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ).
% equalityD1
thf(fact_106_equalityD2,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( A3 = B5 )
=> ( ord_less_eq @ ( set @ A ) @ B5 @ A3 ) ) ).
% equalityD2
thf(fact_107_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
! [T2: A] :
( ( member @ A @ T2 @ A6 )
=> ( member @ A @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_108_subset__refl,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% subset_refl
thf(fact_109_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X5: A] :
( ( P @ X5 )
=> ( Q @ X5 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_110_subset__trans,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% subset_trans
thf(fact_111_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y2: set @ A,Z4: set @ A] : Y2 = Z4 )
= ( ^ [A6: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
& ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_112_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_113_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_114_ex__in__conv,axiom,
! [A: $tType,A3: set @ A] :
( ( ? [X4: A] : ( member @ A @ X4 @ A3 ) )
= ( A3
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_115_equals0I,axiom,
! [A: $tType,A3: set @ A] :
( ! [Y4: A] :
~ ( member @ A @ Y4 @ A3 )
=> ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_116_equals0D,axiom,
! [A: $tType,A3: set @ A,A2: A] :
( ( A3
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A2 @ A3 ) ) ).
% equals0D
thf(fact_117_emptyE,axiom,
! [A: $tType,A2: A] :
~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_118_Ball__def,axiom,
! [A: $tType] :
( ( ball @ A )
= ( ^ [A6: set @ A,P2: A > $o] :
! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( P2 @ X4 ) ) ) ) ).
% Ball_def
thf(fact_119_Ball__Collect,axiom,
! [A: $tType] :
( ( ball @ A )
= ( ^ [A6: set @ A,P2: A > $o] : ( ord_less_eq @ ( set @ A ) @ A6 @ ( collect @ A @ P2 ) ) ) ) ).
% Ball_Collect
thf(fact_120_subset__emptyI,axiom,
! [A: $tType,A3: set @ A] :
( ! [X5: A] :
~ ( member @ A @ X5 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_emptyI
thf(fact_121_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A6: set @ A] :
( A6
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_122_Heap_Oaxioms_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),As_tree: B > ( tree @ A ),Remove_max: B > ( product_prod @ A @ B )] :
( ( heap @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max )
=> ( heap_axioms @ B @ A @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max ) ) ) ).
% Heap.axioms(2)
thf(fact_123_ball__reg,axiom,
! [A: $tType,R3: set @ A,P: A > $o,Q: A > $o] :
( ! [X5: A] :
( ( member @ A @ X5 @ R3 )
=> ( ( P @ X5 )
=> ( Q @ X5 ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ R3 )
=> ( P @ X5 ) )
=> ! [X7: A] :
( ( member @ A @ X7 @ R3 )
=> ( Q @ X7 ) ) ) ) ).
% ball_reg
thf(fact_124_Collect__empty__eq__bot,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( P
= ( bot_bot @ ( A > $o ) ) ) ) ).
% Collect_empty_eq_bot
thf(fact_125_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_126_Heap_Ointro,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),As_tree: B > ( tree @ A ),Remove_max: B > ( product_prod @ A @ B )] :
( ( collection @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset )
=> ( ( heap_axioms @ B @ A @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max )
=> ( heap @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max ) ) ) ) ).
% Heap.intro
thf(fact_127_Heap__def,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( heap @ B @ A )
= ( ^ [Empty2: B,Is_empty2: B > $o,Of_list2: ( list @ A ) > B,Multiset2: B > ( multiset @ A ),As_tree2: B > ( tree @ A ),Remove_max2: B > ( product_prod @ A @ B )] :
( ( collection @ B @ A @ Empty2 @ Is_empty2 @ Of_list2 @ Multiset2 )
& ( heap_axioms @ B @ A @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 ) ) ) ) ) ).
% Heap_def
thf(fact_128_Heap_Oaxioms_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),As_tree: B > ( tree @ A ),Remove_max: B > ( product_prod @ A @ B )] :
( ( heap @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max )
=> ( collection @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset ) ) ) ).
% Heap.axioms(1)
thf(fact_129_Powp__mono,axiom,
! [A: $tType,A3: A > $o,B5: A > $o] :
( ( ord_less_eq @ ( A > $o ) @ A3 @ B5 )
=> ( ord_less_eq @ ( ( set @ A ) > $o ) @ ( powp @ A @ A3 ) @ ( powp @ A @ B5 ) ) ) ).
% Powp_mono
thf(fact_130_Heap_Omultiset,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),As_tree: B > ( tree @ A ),Remove_max: B > ( product_prod @ A @ B ),L: B] :
( ( heap @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max )
=> ( ( Multiset @ L )
= ( multiset2 @ A @ ( As_tree @ L ) ) ) ) ) ).
% Heap.multiset
thf(fact_131_Heap_Oremove__max__is__heap,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),As_tree: B > ( tree @ A ),Remove_max: B > ( product_prod @ A @ B ),L: B,M: A,L3: B] :
( ( heap @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max )
=> ( ~ ( Is_empty @ L )
=> ( ( is_heap @ A @ ( As_tree @ L ) )
=> ( ( ( product_Pair @ A @ B @ M @ L3 )
= ( Remove_max @ L ) )
=> ( is_heap @ A @ ( As_tree @ L3 ) ) ) ) ) ) ) ).
% Heap.remove_max_is_heap
thf(fact_132_Heap_Oremove__max__val,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),As_tree: B > ( tree @ A ),Remove_max: B > ( product_prod @ A @ B ),T: B,M: A,T3: B] :
( ( heap @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max )
=> ( ~ ( Is_empty @ T )
=> ( ( ( product_Pair @ A @ B @ M @ T3 )
= ( Remove_max @ T ) )
=> ( M
= ( val @ A @ ( As_tree @ T ) ) ) ) ) ) ) ).
% Heap.remove_max_val
thf(fact_133_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R: A,S: B,R3: set @ ( product_prod @ A @ B ),S2: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S ) @ R3 )
=> ( ( S2 = S )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S2 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_134_Heap__axioms_Ointro,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Multiset: B > ( multiset @ A ),As_tree: B > ( tree @ A ),Of_list: ( list @ A ) > B,Is_empty: B > $o,Remove_max: B > ( product_prod @ A @ B )] :
( ! [L4: B] :
( ( Multiset @ L4 )
= ( multiset2 @ A @ ( As_tree @ L4 ) ) )
=> ( ! [I2: list @ A] : ( is_heap @ A @ ( As_tree @ ( Of_list @ I2 ) ) )
=> ( ! [T4: B] :
( ( ( As_tree @ T4 )
= ( e @ A ) )
= ( Is_empty @ T4 ) )
=> ( ! [L4: B,M2: A,L2: B] :
( ~ ( Is_empty @ L4 )
=> ( ( ( product_Pair @ A @ B @ M2 @ L2 )
= ( Remove_max @ L4 ) )
=> ( ( add_mset @ A @ M2 @ ( Multiset @ L2 ) )
= ( Multiset @ L4 ) ) ) )
=> ( ! [L4: B,M2: A,L2: B] :
( ~ ( Is_empty @ L4 )
=> ( ( is_heap @ A @ ( As_tree @ L4 ) )
=> ( ( ( product_Pair @ A @ B @ M2 @ L2 )
= ( Remove_max @ L4 ) )
=> ( is_heap @ A @ ( As_tree @ L2 ) ) ) ) )
=> ( ! [T4: B,M2: A,T5: B] :
( ~ ( Is_empty @ T4 )
=> ( ( ( product_Pair @ A @ B @ M2 @ T5 )
= ( Remove_max @ T4 ) )
=> ( M2
= ( val @ A @ ( As_tree @ T4 ) ) ) ) )
=> ( heap_axioms @ B @ A @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max ) ) ) ) ) ) ) ) ).
% Heap_axioms.intro
thf(fact_135_Heap__axioms__def,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( heap_axioms @ B @ A )
= ( ^ [Is_empty2: B > $o,Of_list2: ( list @ A ) > B,Multiset2: B > ( multiset @ A ),As_tree2: B > ( tree @ A ),Remove_max2: B > ( product_prod @ A @ B )] :
( ! [L5: B] :
( ( Multiset2 @ L5 )
= ( multiset2 @ A @ ( As_tree2 @ L5 ) ) )
& ! [I3: list @ A] : ( is_heap @ A @ ( As_tree2 @ ( Of_list2 @ I3 ) ) )
& ! [T2: B] :
( ( ( As_tree2 @ T2 )
= ( e @ A ) )
= ( Is_empty2 @ T2 ) )
& ! [L5: B,M3: A,L6: B] :
( ~ ( Is_empty2 @ L5 )
=> ( ( ( product_Pair @ A @ B @ M3 @ L6 )
= ( Remove_max2 @ L5 ) )
=> ( ( add_mset @ A @ M3 @ ( Multiset2 @ L6 ) )
= ( Multiset2 @ L5 ) ) ) )
& ! [L5: B,M3: A,L6: B] :
( ~ ( Is_empty2 @ L5 )
=> ( ( is_heap @ A @ ( As_tree2 @ L5 ) )
=> ( ( ( product_Pair @ A @ B @ M3 @ L6 )
= ( Remove_max2 @ L5 ) )
=> ( is_heap @ A @ ( As_tree2 @ L6 ) ) ) ) )
& ! [T2: B,M3: A,T6: B] :
( ~ ( Is_empty2 @ T2 )
=> ( ( ( product_Pair @ A @ B @ M3 @ T6 )
= ( Remove_max2 @ T2 ) )
=> ( M3
= ( val @ A @ ( As_tree2 @ T2 ) ) ) ) ) ) ) ) ) ).
% Heap_axioms_def
thf(fact_136_heap__top__geq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,T: tree @ A] :
( ( member @ A @ A2 @ ( set_mset @ A @ ( multiset2 @ A @ T ) ) )
=> ( ( is_heap @ A @ T )
=> ( ord_less_eq @ A @ A2 @ ( val @ A @ T ) ) ) ) ) ).
% heap_top_geq
thf(fact_137_relChain__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( bNF_Ca1785829860lChain @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ A ),As: A > B] :
! [I3: A,J: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I3 @ J ) @ R4 )
=> ( ord_less_eq @ B @ ( As @ I3 ) @ ( As @ J ) ) ) ) ) ) ).
% relChain_def
thf(fact_138_Heap_Oremove__max__multiset_H,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),As_tree: B > ( tree @ A ),Remove_max: B > ( product_prod @ A @ B ),L: B,M: A,L3: B] :
( ( heap @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max )
=> ( ~ ( Is_empty @ L )
=> ( ( ( product_Pair @ A @ B @ M @ L3 )
= ( Remove_max @ L ) )
=> ( ( add_mset @ A @ M @ ( Multiset @ L3 ) )
= ( Multiset @ L ) ) ) ) ) ) ).
% Heap.remove_max_multiset'
thf(fact_139_heap__top__max,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: tree @ A] :
( ( T
!= ( e @ A ) )
=> ( ( is_heap @ A @ T )
=> ( ( val @ A @ T )
= ( lattic929149872er_Max @ A @ ( set_mset @ A @ ( multiset2 @ A @ T ) ) ) ) ) ) ) ).
% heap_top_max
thf(fact_140_multiset_Osimps_I1_J,axiom,
! [A: $tType] :
( ( multiset2 @ A @ ( e @ A ) )
= ( zero_zero @ ( multiset @ A ) ) ) ).
% multiset.simps(1)
thf(fact_141_is__singletonI_H,axiom,
! [A: $tType,A3: set @ A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A,Y4: A] :
( ( member @ A @ X5 @ A3 )
=> ( ( member @ A @ Y4 @ A3 )
=> ( X5 = Y4 ) ) )
=> ( is_singleton @ A @ A3 ) ) ) ).
% is_singletonI'
thf(fact_142_Diff__eq__empty__iff,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A3 @ B5 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ).
% Diff_eq_empty_iff
thf(fact_143_Diff__idemp,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) @ B5 )
= ( minus_minus @ ( set @ A ) @ A3 @ B5 ) ) ).
% Diff_idemp
thf(fact_144_Diff__iff,axiom,
! [A: $tType,C2: A,A3: set @ A,B5: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) )
= ( ( member @ A @ C2 @ A3 )
& ~ ( member @ A @ C2 @ B5 ) ) ) ).
% Diff_iff
thf(fact_145_DiffI,axiom,
! [A: $tType,C2: A,A3: set @ A,B5: set @ A] :
( ( member @ A @ C2 @ A3 )
=> ( ~ ( member @ A @ C2 @ B5 )
=> ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) ) ) ) ).
% DiffI
thf(fact_146_Diff__empty,axiom,
! [A: $tType,A3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= A3 ) ).
% Diff_empty
thf(fact_147_empty__Diff,axiom,
! [A: $tType,A3: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_Diff
thf(fact_148_Diff__cancel,axiom,
! [A: $tType,A3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ A3 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_cancel
thf(fact_149_DiffD2,axiom,
! [A: $tType,C2: A,A3: set @ A,B5: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) )
=> ~ ( member @ A @ C2 @ B5 ) ) ).
% DiffD2
thf(fact_150_DiffD1,axiom,
! [A: $tType,C2: A,A3: set @ A,B5: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) )
=> ( member @ A @ C2 @ A3 ) ) ).
% DiffD1
thf(fact_151_DiffE,axiom,
! [A: $tType,C2: A,A3: set @ A,B5: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) )
=> ~ ( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B5 ) ) ) ).
% DiffE
thf(fact_152_double__diff,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C3 )
=> ( ( minus_minus @ ( set @ A ) @ B5 @ ( minus_minus @ ( set @ A ) @ C3 @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_153_Diff__subset,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) @ A3 ) ).
% Diff_subset
thf(fact_154_Diff__mono,axiom,
! [A: $tType,A3: set @ A,C3: set @ A,D2: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ D2 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) @ ( minus_minus @ ( set @ A ) @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_155_set__mset__eq__empty__iff,axiom,
! [A: $tType,M4: multiset @ A] :
( ( ( set_mset @ A @ M4 )
= ( bot_bot @ ( set @ A ) ) )
= ( M4
= ( zero_zero @ ( multiset @ A ) ) ) ) ).
% set_mset_eq_empty_iff
thf(fact_156_set__mset__empty,axiom,
! [A: $tType] :
( ( set_mset @ A @ ( zero_zero @ ( multiset @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% set_mset_empty
thf(fact_157_diff__ge__0__iff__ge,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% diff_ge_0_iff_ge
thf(fact_158_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_159_zero__le,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% zero_le
thf(fact_160_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A,D3: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C2 @ D3 ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
= ( ord_less_eq @ A @ C2 @ D3 ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_161_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% diff_right_mono
thf(fact_162_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C2 @ A2 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% diff_left_mono
thf(fact_163_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,D3: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ D3 @ C2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ) ).
% diff_mono
thf(fact_164_le__iff__diff__le__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B3: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A4 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% le_iff_diff_le_0
thf(fact_165_multiset__induct__max,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: ( multiset @ A ) > $o,M4: multiset @ A] :
( ( P @ ( zero_zero @ ( multiset @ A ) ) )
=> ( ! [X5: A,M5: multiset @ A] :
( ( P @ M5 )
=> ( ! [Xa: A] :
( ( member @ A @ Xa @ ( set_mset @ A @ M5 ) )
=> ( ord_less_eq @ A @ Xa @ X5 ) )
=> ( P @ ( add_mset @ A @ X5 @ M5 ) ) ) )
=> ( P @ M4 ) ) ) ) ).
% multiset_induct_max
thf(fact_166_multiset__induct__min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: ( multiset @ A ) > $o,M4: multiset @ A] :
( ( P @ ( zero_zero @ ( multiset @ A ) ) )
=> ( ! [X5: A,M5: multiset @ A] :
( ( P @ M5 )
=> ( ! [Xa: A] :
( ( member @ A @ Xa @ ( set_mset @ A @ M5 ) )
=> ( ord_less_eq @ A @ X5 @ Xa ) )
=> ( P @ ( add_mset @ A @ X5 @ M5 ) ) ) )
=> ( P @ M4 ) ) ) ) ).
% multiset_induct_min
thf(fact_167_le__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% le_numeral_extra(3)
thf(fact_168_at__most__one__mset__mset__diff,axiom,
! [A: $tType,A2: A,M4: multiset @ A] :
( ~ ( member @ A @ A2 @ ( set_mset @ A @ ( minus_minus @ ( multiset @ A ) @ M4 @ ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( ( set_mset @ A @ ( minus_minus @ ( multiset @ A ) @ M4 @ ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
= ( minus_minus @ ( set @ A ) @ ( set_mset @ A @ M4 ) @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% at_most_one_mset_mset_diff
thf(fact_169_multiset_Osimps_I2_J,axiom,
! [A: $tType,V: A,L: tree @ A,R: tree @ A] :
( ( multiset2 @ A @ ( t @ A @ V @ L @ R ) )
= ( plus_plus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ ( multiset2 @ A @ L ) @ ( add_mset @ A @ V @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( multiset2 @ A @ R ) ) ) ).
% multiset.simps(2)
thf(fact_170_insert__absorb2,axiom,
! [A: $tType,X: A,A3: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ X @ A3 ) )
= ( insert @ A @ X @ A3 ) ) ).
% insert_absorb2
thf(fact_171_insert__iff,axiom,
! [A: $tType,A2: A,B2: A,A3: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B2 @ A3 ) )
= ( ( A2 = B2 )
| ( member @ A @ A2 @ A3 ) ) ) ).
% insert_iff
thf(fact_172_insertCI,axiom,
! [A: $tType,A2: A,B5: set @ A,B2: A] :
( ( ~ ( member @ A @ A2 @ B5 )
=> ( A2 = B2 ) )
=> ( member @ A @ A2 @ ( insert @ A @ B2 @ B5 ) ) ) ).
% insertCI
thf(fact_173_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_cancel_right
thf(fact_174_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_cancel_left
thf(fact_175_singletonI,axiom,
! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_176_insert__subset,axiom,
! [A: $tType,X: A,A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A3 ) @ B5 )
= ( ( member @ A @ X @ B5 )
& ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ) ).
% insert_subset
thf(fact_177_insert__Diff1,axiom,
! [A: $tType,X: A,B5: set @ A,A3: set @ A] :
( ( member @ A @ X @ B5 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A3 ) @ B5 )
= ( minus_minus @ ( set @ A ) @ A3 @ B5 ) ) ) ).
% insert_Diff1
thf(fact_178_Diff__insert0,axiom,
! [A: $tType,X: A,A3: set @ A,B5: set @ A] :
( ~ ( member @ A @ X @ A3 )
=> ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ B5 ) )
= ( minus_minus @ ( set @ A ) @ A3 @ B5 ) ) ) ).
% Diff_insert0
thf(fact_179_add__le__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel1
thf(fact_180_add__le__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel2
thf(fact_181_le__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% le_add_same_cancel1
thf(fact_182_le__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% le_add_same_cancel2
thf(fact_183_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_184_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_185_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A2: A,A3: set @ A,B2: A] :
( ( ( insert @ A @ A2 @ A3 )
= ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A2 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_186_singleton__insert__inj__eq,axiom,
! [A: $tType,B2: A,A2: A,A3: set @ A] :
( ( ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ A2 @ A3 ) )
= ( ( A2 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_187_insert__Diff__single,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( insert @ A @ A2 @ A3 ) ) ).
% insert_Diff_single
thf(fact_188_is__singletonI,axiom,
! [A: $tType,X: A] : ( is_singleton @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% is_singletonI
thf(fact_189_insert__Diff__if,axiom,
! [A: $tType,X: A,B5: set @ A,A3: set @ A] :
( ( ( member @ A @ X @ B5 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A3 ) @ B5 )
= ( minus_minus @ ( set @ A ) @ A3 @ B5 ) ) )
& ( ~ ( member @ A @ X @ B5 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A3 ) @ B5 )
= ( insert @ A @ X @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_190_verit__sum__simplify,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% verit_sum_simplify
thf(fact_191_add__le__imp__le__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_imp_le_right
thf(fact_192_add__le__imp__le__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_imp_le_left
thf(fact_193_le__iff__add,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B3: A] :
? [C4: A] :
( B3
= ( plus_plus @ A @ A4 @ C4 ) ) ) ) ) ).
% le_iff_add
thf(fact_194_add__right__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add_right_mono
thf(fact_195_less__eqE,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ~ ! [C5: A] :
( B2
!= ( plus_plus @ A @ A2 @ C5 ) ) ) ) ).
% less_eqE
thf(fact_196_add__left__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% add_left_mono
thf(fact_197_add__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [A2: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% add_mono
thf(fact_198_add__mono__thms__linordered__semiring_I1_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J2: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J2 )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J2 @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_199_add__mono__thms__linordered__semiring_I2_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J2: A,K: A,L: A] :
( ( ( I = J2 )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J2 @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_200_add__mono__thms__linordered__semiring_I3_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J2: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J2 )
& ( K = L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J2 @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_201_mk__disjoint__insert,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( member @ A @ A2 @ A3 )
=> ? [B7: set @ A] :
( ( A3
= ( insert @ A @ A2 @ B7 ) )
& ~ ( member @ A @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_202_insert__commute,axiom,
! [A: $tType,X: A,Y: A,A3: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ Y @ A3 ) )
= ( insert @ A @ Y @ ( insert @ A @ X @ A3 ) ) ) ).
% insert_commute
thf(fact_203_insert__eq__iff,axiom,
! [A: $tType,A2: A,A3: set @ A,B2: A,B5: set @ A] :
( ~ ( member @ A @ A2 @ A3 )
=> ( ~ ( member @ A @ B2 @ B5 )
=> ( ( ( insert @ A @ A2 @ A3 )
= ( insert @ A @ B2 @ B5 ) )
= ( ( ( A2 = B2 )
=> ( A3 = B5 ) )
& ( ( A2 != B2 )
=> ? [C6: set @ A] :
( ( A3
= ( insert @ A @ B2 @ C6 ) )
& ~ ( member @ A @ B2 @ C6 )
& ( B5
= ( insert @ A @ A2 @ C6 ) )
& ~ ( member @ A @ A2 @ C6 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_204_insert__absorb,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( member @ A @ A2 @ A3 )
=> ( ( insert @ A @ A2 @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_205_insert__ident,axiom,
! [A: $tType,X: A,A3: set @ A,B5: set @ A] :
( ~ ( member @ A @ X @ A3 )
=> ( ~ ( member @ A @ X @ B5 )
=> ( ( ( insert @ A @ X @ A3 )
= ( insert @ A @ X @ B5 ) )
= ( A3 = B5 ) ) ) ) ).
% insert_ident
thf(fact_206_Set_Oset__insert,axiom,
! [A: $tType,X: A,A3: set @ A] :
( ( member @ A @ X @ A3 )
=> ~ ! [B7: set @ A] :
( ( A3
= ( insert @ A @ X @ B7 ) )
=> ( member @ A @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_207_insertI2,axiom,
! [A: $tType,A2: A,B5: set @ A,B2: A] :
( ( member @ A @ A2 @ B5 )
=> ( member @ A @ A2 @ ( insert @ A @ B2 @ B5 ) ) ) ).
% insertI2
thf(fact_208_insertI1,axiom,
! [A: $tType,A2: A,B5: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B5 ) ) ).
% insertI1
thf(fact_209_insertE,axiom,
! [A: $tType,A2: A,B2: A,A3: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B2 @ A3 ) )
=> ( ( A2 != B2 )
=> ( member @ A @ A2 @ A3 ) ) ) ).
% insertE
thf(fact_210_singletonD,axiom,
! [A: $tType,B2: A,A2: A] :
( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_211_singleton__iff,axiom,
! [A: $tType,B2: A,A2: A] :
( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_212_doubleton__eq__iff,axiom,
! [A: $tType,A2: A,B2: A,C2: A,D3: A] :
( ( ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ C2 @ ( insert @ A @ D3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A2 = C2 )
& ( B2 = D3 ) )
| ( ( A2 = D3 )
& ( B2 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_213_insert__not__empty,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( insert @ A @ A2 @ A3 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_214_singleton__inject,axiom,
! [A: $tType,A2: A,B2: A] :
( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_215_insert__subsetI,axiom,
! [A: $tType,X: A,A3: set @ A,X8: set @ A] :
( ( member @ A @ X @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ X8 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ X8 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_216_subset__insertI2,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ B5 ) ) ) ).
% subset_insertI2
thf(fact_217_subset__insertI,axiom,
! [A: $tType,B5: set @ A,A2: A] : ( ord_less_eq @ ( set @ A ) @ B5 @ ( insert @ A @ A2 @ B5 ) ) ).
% subset_insertI
thf(fact_218_subset__insert,axiom,
! [A: $tType,X: A,A3: set @ A,B5: set @ A] :
( ~ ( member @ A @ X @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X @ B5 ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ) ).
% subset_insert
thf(fact_219_insert__mono,axiom,
! [A: $tType,C3: set @ A,D2: set @ A,A2: A] :
( ( ord_less_eq @ ( set @ A ) @ C3 @ D2 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A2 @ C3 ) @ ( insert @ A @ A2 @ D2 ) ) ) ).
% insert_mono
thf(fact_220_add__decreasing,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).
% add_decreasing
thf(fact_221_add__increasing,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ).
% add_increasing
thf(fact_222_add__decreasing2,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).
% add_decreasing2
thf(fact_223_add__increasing2,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ).
% add_increasing2
thf(fact_224_add__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_225_add__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_nonpos
thf(fact_226_add__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_227_add__nonpos__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
=> ( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_228_diff__le__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% diff_le_eq
thf(fact_229_le__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( minus_minus @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% le_diff_eq
thf(fact_230_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: $tType] :
( ( ordere623563068d_diff @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A2 ) @ A2 )
= B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_231_le__add__diff,axiom,
! [A: $tType] :
( ( ordere623563068d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A2 ) ) ) ) ).
% le_add_diff
thf(fact_232_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: $tType] :
( ( ordere623563068d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ B2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_233_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: $tType] :
( ( ordere623563068d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B2 ) @ A2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_234_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: $tType] :
( ( ordere623563068d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B2 ) @ A2 )
= ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_235_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: $tType] :
( ( ordere623563068d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A2 ) @ C2 )
= ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_236_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: $tType] :
( ( ordere623563068d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A2 )
= ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A2 ) @ C2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_237_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: $tType] :
( ( ordere623563068d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( minus_minus @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A2 ) @ B2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_238_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: $tType] :
( ( ordere623563068d_diff @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B2 @ A2 ) )
= B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_239_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: $tType] :
( ( ordere623563068d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( minus_minus @ A @ B2 @ A2 )
= C2 )
= ( B2
= ( plus_plus @ A @ C2 @ A2 ) ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_240_subset__singleton__iff,axiom,
! [A: $tType,X8: set @ A,A2: A] :
( ( ord_less_eq @ ( set @ A ) @ X8 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( X8
= ( bot_bot @ ( set @ A ) ) )
| ( X8
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singleton_iff
thf(fact_241_subset__singletonD,axiom,
! [A: $tType,A3: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( A3
= ( bot_bot @ ( set @ A ) ) )
| ( A3
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singletonD
thf(fact_242_Diff__insert__absorb,axiom,
! [A: $tType,X: A,A3: set @ A] :
( ~ ( member @ A @ X @ A3 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A3 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_243_Diff__insert2,axiom,
! [A: $tType,A3: set @ A,A2: A,B5: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B5 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) @ B5 ) ) ).
% Diff_insert2
thf(fact_244_insert__Diff,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( member @ A @ A2 @ A3 )
=> ( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_245_Diff__insert,axiom,
! [A: $tType,A3: set @ A,A2: A,B5: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B5 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Diff_insert
thf(fact_246_subset__Diff__insert,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,X: A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B5 @ ( insert @ A @ X @ C3 ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B5 @ C3 ) )
& ~ ( member @ A @ X @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_247_is__singleton__def,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A6: set @ A] :
? [X4: A] :
( A6
= ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_def
thf(fact_248_is__singletonE,axiom,
! [A: $tType,A3: set @ A] :
( ( is_singleton @ A @ A3 )
=> ~ ! [X5: A] :
( A3
!= ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% is_singletonE
thf(fact_249_subset__insert__iff,axiom,
! [A: $tType,A3: set @ A,X: A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X @ B5 ) )
= ( ( ( member @ A @ X @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B5 ) )
& ( ~ ( member @ A @ X @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ) ) ).
% subset_insert_iff
thf(fact_250_Diff__single__insert,axiom,
! [A: $tType,A3: set @ A,X: A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X @ B5 ) ) ) ).
% Diff_single_insert
thf(fact_251_set__mset__single,axiom,
! [A: $tType,B2: A] :
( ( set_mset @ A @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) )
= ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% set_mset_single
thf(fact_252_Max__singleton,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A] :
( ( lattic929149872er_Max @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% Max_singleton
thf(fact_253_le__add__diff__inverse2,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ B2 )
= A2 ) ) ) ).
% le_add_diff_inverse2
thf(fact_254_le__add__diff__inverse,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( plus_plus @ A @ B2 @ ( minus_minus @ A @ A2 @ B2 ) )
= A2 ) ) ) ).
% le_add_diff_inverse
thf(fact_255_add__le__imp__le__diff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: A,K: A,N: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N )
=> ( ord_less_eq @ A @ I @ ( minus_minus @ A @ N @ K ) ) ) ) ).
% add_le_imp_le_diff
% Subclasses (4)
thf(subcl_Orderings_Olinorder___HOL_Otype,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( type @ A ) ) ).
thf(subcl_Orderings_Olinorder___Orderings_Oord,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ord @ A ) ) ).
thf(subcl_Orderings_Olinorder___Orderings_Oorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( order @ A ) ) ).
thf(subcl_Orderings_Olinorder___Orderings_Opreorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( preorder @ A ) ) ).
% Type constructors (21)
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A7: $tType,A8: $tType] :
( ( order_bot @ A8 )
=> ( order_bot @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A7: $tType,A8: $tType] :
( ( preorder @ A8 )
=> ( preorder @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A7: $tType,A8: $tType] :
( ( order @ A8 )
=> ( order @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A7: $tType,A8: $tType] :
( ( ord @ A8 )
=> ( ord @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A7: $tType,A8: $tType] :
( ( bot @ A8 )
=> ( bot @ ( A7 > A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_1,axiom,
! [A7: $tType] : ( order_bot @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_2,axiom,
! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_3,axiom,
! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_4,axiom,
! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_5,axiom,
! [A7: $tType] : ( bot @ ( set @ A7 ) ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_6,axiom,
order_bot @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_7,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_8,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_9,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Orderings_Obot_10,axiom,
bot @ $o ).
thf(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add,axiom,
! [A7: $tType] :
( ( preorder @ A7 )
=> ( ordere779506340up_add @ ( multiset @ A7 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocancel__comm__monoid__add,axiom,
! [A7: $tType] : ( cancel1352612707id_add @ ( multiset @ A7 ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Opreorder_11,axiom,
! [A7: $tType] :
( ( preorder @ A7 )
=> ( preorder @ ( multiset @ A7 ) ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Oorder_12,axiom,
! [A7: $tType] :
( ( preorder @ A7 )
=> ( order @ ( multiset @ A7 ) ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Oord_13,axiom,
! [A7: $tType] :
( ( preorder @ A7 )
=> ( ord @ ( multiset @ A7 ) ) ) ).
% Free types (1)
thf(tfree_0,hypothesis,
linorder @ a ).
% Conjectures (1)
thf(conj_0,conjecture,
in_tree @ a @ v @ ( t @ a @ v3 @ ( t @ a @ v1 @ l1 @ r1 ) @ ( t @ a @ v2 @ l2 @ r2 ) ) ).
%------------------------------------------------------------------------------