TPTP Problem File: ITP069^2.p
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%------------------------------------------------------------------------------
% File : ITP069^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer HeapImperative problem prob_637__5346010_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_637__5346010_1 [Des21]
% Status : Theorem
% Rating : 0.00 v7.5.0
% Syntax : Number of formulae : 342 ( 85 unt; 66 typ; 0 def)
% Number of atoms : 859 ( 326 equ; 0 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 5739 ( 91 ~; 10 |; 67 &;5112 @)
% ( 0 <=>; 459 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 10 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 328 ( 328 >; 0 *; 0 +; 0 <<)
% Number of symbols : 66 ( 65 usr; 3 con; 0-8 aty)
% Number of variables : 1407 ( 52 ^;1264 !; 18 ?;1407 :)
% ( 73 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:16:28.444
%------------------------------------------------------------------------------
% 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 (60)
thf(sy_cl_HOL_Otype,type,
type:
!>[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_Groups_Oordered__ab__group__add,type,
ordered_ab_group_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_Divides_Ounique__euclidean__semiring__numeral,type,
unique1598680935umeral:
!>[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_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux,type,
unique455577585es_aux:
!>[A: $tType] : ( ( product_prod @ A @ A ) > $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_HOL_ONO__MATCH,type,
nO_MATCH:
!>[A: $tType,B: $tType] : ( A > B > $o ) ).
thf(sy_c_HeapImperative__Mirabelle__oitemzccmr_Oheapify,type,
heapIm818251801eapify:
!>[A: $tType] : ( ( tree @ A ) > ( tree @ 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_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_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_Ocomm__monoid__add_Osum__mset,type,
comm_monoid_sum_mset:
!>[A: $tType] : ( ( A > A > A ) > A > ( multiset @ A ) > A ) ).
thf(sy_c_Multiset_Ofold__mset,type,
fold_mset:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( multiset @ A ) > B ) ).
thf(sy_c_Multiset_Ois__empty,type,
is_empty:
!>[A: $tType] : ( ( multiset @ A ) > $o ) ).
thf(sy_c_Multiset_Omult,type,
mult:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) ) ) ).
thf(sy_c_Multiset_Omult1,type,
mult1:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) ) ) ).
thf(sy_c_Multiset_Omultp,type,
multp:
!>[A: $tType] : ( ( A > A > $o ) > ( multiset @ A ) > ( multiset @ A ) > $o ) ).
thf(sy_c_Multiset_Oset__mset,type,
set_mset:
!>[A: $tType] : ( ( multiset @ A ) > ( set @ 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_Product__Type_Ointernal__case__prod,type,
produc2004651681e_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
product_rec_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
thf(sy_c_Relation_Oirrefl,type,
irrefl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Otrans,type,
trans:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ 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_RemoveMax_OCollection_Oset,type,
set2:
!>[B: $tType,A: $tType] : ( ( B > ( multiset @ A ) ) > B > ( set @ A ) ) ).
thf(sy_c_RemoveMax_ORemoveMax,type,
removeMax:
!>[B: $tType,A: $tType] : ( B > ( B > $o ) > ( ( list @ A ) > B ) > ( B > ( multiset @ A ) ) > ( B > ( product_prod @ A @ B ) ) > ( B > $o ) > $o ) ).
thf(sy_c_RemoveMax_ORemoveMax__axioms,type,
removeMax_axioms:
!>[B: $tType,A: $tType] : ( ( B > $o ) > ( ( list @ A ) > B ) > ( B > ( multiset @ A ) ) > ( B > ( product_prod @ A @ B ) ) > ( B > $o ) > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_l____,type,
l: tree @ a ).
thf(sy_v_r____,type,
r: tree @ a ).
thf(sy_v_v____,type,
v: a ).
% Relevant facts (256)
thf(fact_0_T_Ohyps_I2_J,axiom,
is_heap @ a @ ( heapIm818251801eapify @ a @ r ) ).
% T.hyps(2)
thf(fact_1_T_Ohyps_I1_J,axiom,
is_heap @ a @ ( heapIm818251801eapify @ a @ l ) ).
% T.hyps(1)
thf(fact_2__092_060open_062_092_060lbrakk_062is__heap_A_Iheapify_Al_J_059_Ais__heap_A_Iheapify_Ar_J_059_AT_Av_A_Iheapify_Al_J_A_Iheapify_Ar_J_A_061_AT_Av_A_Iheapify_Al_J_A_Iheapify_Ar_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Ais__heap_A_IsiftDown_A_IT_Av_A_Iheapify_Al_J_A_Iheapify_Ar_J_J_J_092_060close_062,axiom,
( ( is_heap @ a @ ( heapIm818251801eapify @ a @ l ) )
=> ( ( is_heap @ a @ ( heapIm818251801eapify @ a @ r ) )
=> ( ( ( t @ a @ v @ ( heapIm818251801eapify @ a @ l ) @ ( heapIm818251801eapify @ a @ r ) )
= ( t @ a @ v @ ( heapIm818251801eapify @ a @ l ) @ ( heapIm818251801eapify @ a @ r ) ) )
=> ( is_heap @ a @ ( heapIm748920189ftDown @ a @ ( t @ a @ v @ ( heapIm818251801eapify @ a @ l ) @ ( heapIm818251801eapify @ a @ r ) ) ) ) ) ) ) ).
% \<open>\<lbrakk>is_heap (heapify l); is_heap (heapify r); T v (heapify l) (heapify r) = T v (heapify l) (heapify r)\<rbrakk> \<Longrightarrow> is_heap (siftDown (T v (heapify l) (heapify r)))\<close>
thf(fact_3_heapify_Osimps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [V: A,L: tree @ A,R: tree @ A] :
( ( heapIm818251801eapify @ A @ ( t @ A @ V @ L @ R ) )
= ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( heapIm818251801eapify @ A @ L ) @ ( heapIm818251801eapify @ A @ R ) ) ) ) ) ).
% heapify.simps(2)
thf(fact_4_siftDown__heap__is__heap,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: tree @ A,R: tree @ A,T2: tree @ A,V: A] :
( ( is_heap @ A @ L )
=> ( ( is_heap @ A @ R )
=> ( ( T2
= ( t @ A @ V @ L @ R ) )
=> ( is_heap @ A @ ( heapIm748920189ftDown @ A @ T2 ) ) ) ) ) ) ).
% siftDown_heap_is_heap
thf(fact_5_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_6_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_7_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_8_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_9_heapify_Osimps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( heapIm818251801eapify @ A @ ( e @ A ) )
= ( e @ A ) ) ) ).
% heapify.simps(1)
thf(fact_10_siftDown_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: tree @ A] :
( ( X
!= ( e @ A ) )
=> ( ! [V2: A] :
( X
!= ( t @ A @ V2 @ ( e @ A ) @ ( e @ A ) ) )
=> ( ! [V2: A,Va: A,Vb: tree @ A,Vc: tree @ A] :
( X
!= ( t @ A @ V2 @ ( t @ A @ Va @ Vb @ Vc ) @ ( e @ A ) ) )
=> ( ! [V2: A,Va: A,Vb: tree @ A,Vc: tree @ A] :
( X
!= ( t @ A @ V2 @ ( e @ A ) @ ( t @ A @ Va @ Vb @ Vc ) ) )
=> ~ ! [V2: A,Va: A,Vb: tree @ A,Vc: tree @ A,Vd: A,Ve: tree @ A,Vf: tree @ A] :
( X
!= ( t @ A @ V2 @ ( 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_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_13_in__tree_Osimps_I2_J,axiom,
! [A: $tType,V: A,V3: A,L: tree @ A,R: tree @ A] :
( ( in_tree @ A @ V @ ( t @ A @ V3 @ L @ R ) )
= ( ( V = V3 )
| ( in_tree @ A @ V @ L )
| ( in_tree @ A @ V @ R ) ) ) ).
% in_tree.simps(2)
thf(fact_14_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_15_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_16_in__tree_Osimps_I1_J,axiom,
! [A: $tType,V: A] :
~ ( in_tree @ A @ V @ ( e @ A ) ) ).
% in_tree.simps(1)
thf(fact_17_siftDown_Osimps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( heapIm748920189ftDown @ A @ ( e @ A ) )
= ( e @ A ) ) ) ).
% siftDown.simps(1)
thf(fact_18_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 ),T2: B] :
( ( heap @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max )
=> ( ( ( As_tree @ T2 )
= ( e @ A ) )
= ( Is_empty @ T2 ) ) ) ) ).
% Heap.as_tree_empty
thf(fact_19_siftDown__in__tree__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( in_tree @ A )
= ( ^ [V4: A,T3: tree @ A] : ( in_tree @ A @ V4 @ ( heapIm748920189ftDown @ A @ T3 ) ) ) ) ) ).
% siftDown_in_tree_set
thf(fact_20_is__heap_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: tree @ A] :
( ( X
!= ( e @ A ) )
=> ( ! [V2: A] :
( X
!= ( t @ A @ V2 @ ( e @ A ) @ ( e @ A ) ) )
=> ( ! [V2: A,Va: A,Vb: tree @ A,Vc: tree @ A] :
( X
!= ( t @ A @ V2 @ ( e @ A ) @ ( t @ A @ Va @ Vb @ Vc ) ) )
=> ( ! [V2: A,Va: A,Vb: tree @ A,Vc: tree @ A] :
( X
!= ( t @ A @ V2 @ ( t @ A @ Va @ Vb @ Vc ) @ ( e @ A ) ) )
=> ~ ! [V2: A,Va: A,Vb: tree @ A,Vc: tree @ A,Vd: A,Ve: tree @ A,Vf: tree @ A] :
( X
!= ( t @ A @ V2 @ ( t @ A @ Va @ Vb @ Vc ) @ ( t @ A @ Vd @ Ve @ Vf ) ) ) ) ) ) ) ) ).
% is_heap.cases
thf(fact_21_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_22_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_23_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_24_is__heap_Osimps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( is_heap @ A @ ( e @ A ) ) ) ).
% is_heap.simps(1)
thf(fact_25_siftDown__in__tree,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: tree @ A] :
( ( T2
!= ( e @ A ) )
=> ( in_tree @ A @ ( val @ A @ ( heapIm748920189ftDown @ A @ T2 ) ) @ T2 ) ) ) ).
% siftDown_in_tree
thf(fact_26_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_27_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_28_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_29_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_30_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_31_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_32_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,L2: 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 @ L2 )
= ( Remove_max @ L ) )
=> ( is_heap @ A @ ( As_tree @ L2 ) ) ) ) ) ) ) ).
% Heap.remove_max_is_heap
thf(fact_33_siftDown__Node,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: tree @ A,V: A,L: tree @ A,R: tree @ A] :
( ( T2
= ( t @ A @ V @ L @ R ) )
=> ? [L3: tree @ A,V5: A,R2: tree @ A] :
( ( ( heapIm748920189ftDown @ A @ T2 )
= ( t @ A @ V5 @ L3 @ R2 ) )
& ( ord_less_eq @ A @ V @ V5 ) ) ) ) ).
% siftDown_Node
thf(fact_34_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_35_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 ),T2: B,M: A,T4: B] :
( ( heap @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max )
=> ( ~ ( Is_empty @ T2 )
=> ( ( ( product_Pair @ A @ B @ M @ T4 )
= ( Remove_max @ T2 ) )
=> ( M
= ( val @ A @ ( As_tree @ T2 ) ) ) ) ) ) ) ).
% Heap.remove_max_val
thf(fact_36_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_37_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_38_is__heap__max,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [V: A,T2: tree @ A] :
( ( in_tree @ A @ V @ T2 )
=> ( ( is_heap @ A @ T2 )
=> ( ord_less_eq @ A @ V @ ( val @ A @ T2 ) ) ) ) ) ).
% is_heap_max
thf(fact_39_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_40_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_41_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_42_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_43_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_44_prod_Oinject,axiom,
! [A: $tType,B: $tType,X12: A,X24: B,Y1: A,Y2: B] :
( ( ( product_Pair @ A @ B @ X12 @ X24 )
= ( product_Pair @ A @ B @ Y1 @ Y2 ) )
= ( ( X12 = Y1 )
& ( X24 = Y2 ) ) ) ).
% prod.inject
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,A4: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A4 ) )
= A4 ) ).
% 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_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_50_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 ) ) )
=> ( ! [T5: B] :
( ( ( As_tree @ T5 )
= ( e @ A ) )
= ( Is_empty @ T5 ) )
=> ( ! [L4: B,M2: A,L3: B] :
( ~ ( Is_empty @ L4 )
=> ( ( ( product_Pair @ A @ B @ M2 @ L3 )
= ( Remove_max @ L4 ) )
=> ( ( add_mset @ A @ M2 @ ( Multiset @ L3 ) )
= ( Multiset @ L4 ) ) ) )
=> ( ! [L4: B,M2: A,L3: B] :
( ~ ( Is_empty @ L4 )
=> ( ( is_heap @ A @ ( As_tree @ L4 ) )
=> ( ( ( product_Pair @ A @ B @ M2 @ L3 )
= ( Remove_max @ L4 ) )
=> ( is_heap @ A @ ( As_tree @ L3 ) ) ) ) )
=> ( ! [T5: B,M2: A,T6: B] :
( ~ ( Is_empty @ T5 )
=> ( ( ( product_Pair @ A @ B @ M2 @ T6 )
= ( Remove_max @ T5 ) )
=> ( M2
= ( val @ A @ ( As_tree @ T5 ) ) ) ) )
=> ( heap_axioms @ B @ A @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max ) ) ) ) ) ) ) ) ).
% Heap_axioms.intro
thf(fact_51_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 ) ) )
& ! [T3: B] :
( ( ( As_tree2 @ T3 )
= ( e @ A ) )
= ( Is_empty2 @ T3 ) )
& ! [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 ) ) ) ) )
& ! [T3: B,M3: A,T7: B] :
( ~ ( Is_empty2 @ T3 )
=> ( ( ( product_Pair @ A @ B @ M3 @ T7 )
= ( Remove_max2 @ T3 ) )
=> ( M3
= ( val @ A @ ( As_tree2 @ T3 ) ) ) ) ) ) ) ) ) ).
% Heap_axioms_def
thf(fact_52_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_53_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_54_heap__top__geq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,T2: tree @ A] :
( ( member @ A @ A2 @ ( set_mset @ A @ ( multiset2 @ A @ T2 ) ) )
=> ( ( is_heap @ A @ T2 )
=> ( ord_less_eq @ A @ A2 @ ( val @ A @ T2 ) ) ) ) ) ).
% heap_top_geq
thf(fact_55_relChain__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( bNF_Ca1785829860lChain @ A @ B )
= ( ^ [R3: set @ ( product_prod @ A @ A ),As: A > B] :
! [I3: A,J: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I3 @ J ) @ R3 )
=> ( ord_less_eq @ B @ ( As @ I3 ) @ ( As @ J ) ) ) ) ) ) ).
% relChain_def
thf(fact_56_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_57_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,L2: B] :
( ( heap @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max )
=> ( ~ ( Is_empty @ L )
=> ( ( ( product_Pair @ A @ B @ M @ L2 )
= ( Remove_max @ L ) )
=> ( ( add_mset @ A @ M @ ( Multiset @ L2 ) )
= ( Multiset @ L ) ) ) ) ) ) ).
% Heap.remove_max_multiset'
thf(fact_58_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_59_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_60_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_61_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_62_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,Y3: B] :
( ( ord_less_eq @ B @ X5 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_63_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,Y3: A] :
( ( ord_less_eq @ A @ X5 @ Y3 )
=> ( ord_less_eq @ C @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_64_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,Y3: B] :
( ( ord_less_eq @ B @ X5 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_65_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,Y3: A] :
( ( ord_less_eq @ A @ X5 @ Y3 )
=> ( ord_less_eq @ B @ ( F @ X5 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_66_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z: A] : Y4 = Z )
= ( ^ [X4: A,Y5: A] :
( ( ord_less_eq @ A @ X4 @ Y5 )
& ( ord_less_eq @ A @ Y5 @ X4 ) ) ) ) ) ).
% eq_iff
thf(fact_67_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_68_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_69_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( X = Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% eq_refl
thf(fact_70_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_71_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_72_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less_eq @ A @ X @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X )
=> ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_73_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_74_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z: A] : Y4 = Z )
= ( ^ [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
& ( ord_less_eq @ A @ B4 @ A5 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_75_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_76_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_77_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_78_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z2 )
=> ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% order_trans
thf(fact_79_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_80_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A6: A,B5: A] :
( ( ord_less_eq @ A @ A6 @ B5 )
=> ( P @ A6 @ B5 ) )
=> ( ! [A6: A,B5: A] :
( ( P @ B5 @ A6 )
=> ( P @ A6 @ B5 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_81_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_82_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z: A] : Y4 = Z )
= ( ^ [A5: A,B4: A] :
( ( ord_less_eq @ A @ B4 @ A5 )
& ( ord_less_eq @ A @ A5 @ B4 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_83_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_84_surj__pair,axiom,
! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
? [X5: A,Y3: B] :
( P2
= ( product_Pair @ A @ B @ X5 @ Y3 ) ) ).
% surj_pair
thf(fact_85_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
( ! [A6: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A6 @ B5 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_86_Pair__inject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_87_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A6: A,B5: B,C3: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) ) ).
% prod_cases3
thf(fact_88_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A6: A,B5: B,C3: C,D2: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_89_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
~ ! [A6: A,B5: B,C3: C,D2: D,E2: E] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_90_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F4: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) )] :
~ ! [A6: A,B5: B,C3: C,D2: D,E2: E,F5: F4] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F4 ) @ D2 @ ( product_Pair @ E @ F4 @ E2 @ F5 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_91_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F4: $tType,G3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) )] :
~ ! [A6: A,B5: B,C3: C,D2: D,E2: E,F5: F4,G4: G3] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F4 @ G3 ) @ E2 @ ( product_Pair @ F4 @ G3 @ F5 @ G4 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_92_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A6: A,B5: B,C3: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_93_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A6: A,B5: B,C3: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) )
=> ( P @ X ) ) ).
% prod_induct4
thf(fact_94_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
( ! [A6: A,B5: B,C3: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct5
thf(fact_95_prod__induct6,axiom,
! [F4: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) )] :
( ! [A6: A,B5: B,C3: C,D2: D,E2: E,F5: F4] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F4 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F4 ) @ D2 @ ( product_Pair @ E @ F4 @ E2 @ F5 ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct6
thf(fact_96_prod__induct7,axiom,
! [G3: $tType,F4: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) )] :
( ! [A6: A,B5: B,C3: C,D2: D,E2: E,F5: F4,G4: G3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F4 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F4 @ G3 ) @ E2 @ ( product_Pair @ F4 @ G3 @ F5 @ G4 ) ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct7
thf(fact_97_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A6: A,B5: B] :
( Y
!= ( product_Pair @ A @ B @ A6 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_98_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A6: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A6 @ B5 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_99_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
= ( F1 @ A2 @ B2 ) ) ).
% old.prod.rec
thf(fact_100_add__mset__add__mset__same__iff,axiom,
! [A: $tType,A2: A,A4: multiset @ A,B6: multiset @ A] :
( ( ( add_mset @ A @ A2 @ A4 )
= ( add_mset @ A @ A2 @ B6 ) )
= ( A4 = B6 ) ) ).
% add_mset_add_mset_same_iff
thf(fact_101_multi__self__add__other__not__self,axiom,
! [A: $tType,M4: multiset @ A,X: A] :
( M4
!= ( add_mset @ A @ X @ M4 ) ) ).
% multi_self_add_other_not_self
thf(fact_102_mset__add,axiom,
! [A: $tType,A2: A,A4: multiset @ A] :
( ( member @ A @ A2 @ ( set_mset @ A @ A4 ) )
=> ~ ! [B7: multiset @ A] :
( A4
!= ( add_mset @ A @ A2 @ B7 ) ) ) ).
% mset_add
thf(fact_103_multi__member__split,axiom,
! [A: $tType,X: A,M4: multiset @ A] :
( ( member @ A @ X @ ( set_mset @ A @ M4 ) )
=> ? [A7: multiset @ A] :
( M4
= ( add_mset @ A @ X @ A7 ) ) ) ).
% multi_member_split
thf(fact_104_insert__noteq__member,axiom,
! [A: $tType,B2: A,B6: multiset @ A,C2: A,C4: multiset @ A] :
( ( ( add_mset @ A @ B2 @ B6 )
= ( add_mset @ A @ C2 @ C4 ) )
=> ( ( B2 != C2 )
=> ( member @ A @ C2 @ ( set_mset @ A @ B6 ) ) ) ) ).
% insert_noteq_member
thf(fact_105_union__single__eq__member,axiom,
! [A: $tType,X: A,M4: multiset @ A,N: multiset @ A] :
( ( ( add_mset @ A @ X @ M4 )
= N )
=> ( member @ A @ X @ ( set_mset @ A @ N ) ) ) ).
% union_single_eq_member
thf(fact_106_heap__top__max,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: tree @ A] :
( ( T2
!= ( e @ A ) )
=> ( ( is_heap @ A @ T2 )
=> ( ( val @ A @ T2 )
= ( lattic929149872er_Max @ A @ ( set_mset @ A @ ( multiset2 @ A @ T2 ) ) ) ) ) ) ) ).
% heap_top_max
thf(fact_107_add__mset__commute,axiom,
! [A: $tType,X: A,Y: A,M4: multiset @ A] :
( ( add_mset @ A @ X @ ( add_mset @ A @ Y @ M4 ) )
= ( add_mset @ A @ Y @ ( add_mset @ A @ X @ M4 ) ) ) ).
% add_mset_commute
thf(fact_108_add__eq__conv__ex,axiom,
! [A: $tType,A2: A,M4: multiset @ A,B2: A,N: multiset @ A] :
( ( ( add_mset @ A @ A2 @ M4 )
= ( add_mset @ A @ B2 @ N ) )
= ( ( ( M4 = N )
& ( A2 = B2 ) )
| ? [K: multiset @ A] :
( ( M4
= ( add_mset @ A @ B2 @ K ) )
& ( N
= ( add_mset @ A @ A2 @ K ) ) ) ) ) ).
% add_eq_conv_ex
thf(fact_109_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A2: B,B2: C] :
( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
= ( C2 @ A2 @ B2 ) ) ).
% internal_case_prod_conv
thf(fact_110_Collection_Ois__empty__empty,axiom,
! [A: $tType,B: $tType,Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A )] :
( ( collection @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset )
=> ( Is_empty @ Empty ) ) ).
% Collection.is_empty_empty
thf(fact_111_Collection_Ois__empty__inj,axiom,
! [A: $tType,B: $tType,Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),E3: B] :
( ( collection @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset )
=> ( ( Is_empty @ E3 )
=> ( E3 = Empty ) ) ) ).
% Collection.is_empty_inj
thf(fact_112_Greatest__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A] :
( ( P @ X )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ( order_Greatest @ A @ P )
= X ) ) ) ) ).
% Greatest_equality
thf(fact_113_GreatestI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A,Q: A > $o] :
( ( P @ X )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ! [X5: A] :
( ( P @ X5 )
=> ( ! [Y6: A] :
( ( P @ Y6 )
=> ( ord_less_eq @ A @ Y6 @ X5 ) )
=> ( Q @ X5 ) ) )
=> ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% GreatestI2_order
thf(fact_114_Collection_Oset__def,axiom,
! [A: $tType,B: $tType,Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),L: B] :
( ( collection @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset )
=> ( ( set2 @ B @ A @ Multiset @ L )
= ( set_mset @ A @ ( Multiset @ L ) ) ) ) ).
% Collection.set_def
thf(fact_115_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_116_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_117_subrelI,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ! [X5: A,Y3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y3 ) @ R )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y3 ) @ S ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% subrelI
thf(fact_118_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R: A,S: B,R4: set @ ( product_prod @ A @ B ),S2: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S ) @ R4 )
=> ( ( S2 = S )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S2 ) @ R4 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_119_add__mset__eq__singleton__iff,axiom,
! [A: $tType,X: A,M4: multiset @ A,Y: A] :
( ( ( add_mset @ A @ X @ M4 )
= ( add_mset @ A @ Y @ ( zero_zero @ ( multiset @ A ) ) ) )
= ( ( M4
= ( zero_zero @ ( multiset @ A ) ) )
& ( X = Y ) ) ) ).
% add_mset_eq_singleton_iff
thf(fact_120_single__eq__add__mset,axiom,
! [A: $tType,A2: A,B2: A,M4: multiset @ A] :
( ( ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) )
= ( add_mset @ A @ B2 @ M4 ) )
= ( ( B2 = A2 )
& ( M4
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% single_eq_add_mset
thf(fact_121_add__mset__eq__single,axiom,
! [A: $tType,B2: A,M4: multiset @ A,A2: A] :
( ( ( add_mset @ A @ B2 @ M4 )
= ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) )
= ( ( B2 = A2 )
& ( M4
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% add_mset_eq_single
thf(fact_122_single__eq__single,axiom,
! [A: $tType,A2: A,B2: A] :
( ( ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) )
= ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) )
= ( A2 = B2 ) ) ).
% single_eq_single
thf(fact_123_Collection_Oset_Ocong,axiom,
! [A: $tType,B: $tType] :
( ( set2 @ B @ A )
= ( set2 @ B @ A ) ) ).
% Collection.set.cong
thf(fact_124_multiset__cases,axiom,
! [A: $tType,M4: multiset @ A] :
( ( M4
!= ( zero_zero @ ( multiset @ A ) ) )
=> ~ ! [X5: A,N2: multiset @ A] :
( M4
!= ( add_mset @ A @ X5 @ N2 ) ) ) ).
% multiset_cases
thf(fact_125_multiset__induct,axiom,
! [A: $tType,P: ( multiset @ A ) > $o,M4: multiset @ A] :
( ( P @ ( zero_zero @ ( multiset @ A ) ) )
=> ( ! [X5: A,M5: multiset @ A] :
( ( P @ M5 )
=> ( P @ ( add_mset @ A @ X5 @ M5 ) ) )
=> ( P @ M4 ) ) ) ).
% multiset_induct
thf(fact_126_multiset__induct2,axiom,
! [A: $tType,B: $tType,P: ( multiset @ A ) > ( multiset @ B ) > $o,M4: multiset @ A,N: multiset @ B] :
( ( P @ ( zero_zero @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ B ) ) )
=> ( ! [A6: A,M5: multiset @ A,N2: multiset @ B] :
( ( P @ M5 @ N2 )
=> ( P @ ( add_mset @ A @ A6 @ M5 ) @ N2 ) )
=> ( ! [A6: B,M5: multiset @ A,N2: multiset @ B] :
( ( P @ M5 @ N2 )
=> ( P @ M5 @ ( add_mset @ B @ A6 @ N2 ) ) )
=> ( P @ M4 @ N ) ) ) ) ).
% multiset_induct2
thf(fact_127_empty__not__add__mset,axiom,
! [A: $tType,A2: A,A4: multiset @ A] :
( ( zero_zero @ ( multiset @ A ) )
!= ( add_mset @ A @ A2 @ A4 ) ) ).
% empty_not_add_mset
thf(fact_128_multi__nonempty__split,axiom,
! [A: $tType,M4: multiset @ A] :
( ( M4
!= ( zero_zero @ ( multiset @ A ) ) )
=> ? [A7: multiset @ A,A6: A] :
( M4
= ( add_mset @ A @ A6 @ A7 ) ) ) ).
% multi_nonempty_split
thf(fact_129_multiset__nonemptyE,axiom,
! [A: $tType,A4: multiset @ A] :
( ( A4
!= ( zero_zero @ ( multiset @ A ) ) )
=> ~ ! [X5: A] :
~ ( member @ A @ X5 @ ( set_mset @ A @ A4 ) ) ) ).
% multiset_nonemptyE
thf(fact_130_Collection_Omultiset__empty,axiom,
! [B: $tType,A: $tType,Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A )] :
( ( collection @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset )
=> ( ( Multiset @ Empty )
= ( zero_zero @ ( multiset @ A ) ) ) ) ).
% Collection.multiset_empty
thf(fact_131_Collection_Ois__empty__as__list,axiom,
! [B: $tType,A: $tType,Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),E3: B] :
( ( collection @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset )
=> ( ( Is_empty @ E3 )
=> ( ( Multiset @ E3 )
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% Collection.is_empty_as_list
thf(fact_132_multi__member__last,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( set_mset @ A @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% multi_member_last
thf(fact_133_multiset_Osimps_I1_J,axiom,
! [A: $tType] :
( ( multiset2 @ A @ ( e @ A ) )
= ( zero_zero @ ( multiset @ A ) ) ) ).
% multiset.simps(1)
thf(fact_134_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N3: A] :
( ( ord_less_eq @ A @ N3 @ ( zero_zero @ A ) )
= ( N3
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_135_RemoveMax__axioms_Ointro,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Inv: B > $o,Of_list: ( list @ A ) > B,Is_empty: B > $o,Remove_max: B > ( product_prod @ A @ B ),Multiset: B > ( multiset @ A )] :
( ! [X5: list @ A] : ( Inv @ ( Of_list @ X5 ) )
=> ( ! [L4: B,M2: A,L3: B] :
( ~ ( Is_empty @ L4 )
=> ( ( Inv @ L4 )
=> ( ( ( product_Pair @ A @ B @ M2 @ L3 )
= ( Remove_max @ L4 ) )
=> ( M2
= ( lattic929149872er_Max @ A @ ( set2 @ B @ A @ Multiset @ L4 ) ) ) ) ) )
=> ( ! [L4: B,M2: A,L3: B] :
( ~ ( Is_empty @ L4 )
=> ( ( Inv @ L4 )
=> ( ( ( product_Pair @ A @ B @ M2 @ L3 )
= ( Remove_max @ L4 ) )
=> ( ( add_mset @ A @ M2 @ ( Multiset @ L3 ) )
= ( Multiset @ L4 ) ) ) ) )
=> ( ! [L4: B,M2: A,L3: B] :
( ~ ( Is_empty @ L4 )
=> ( ( Inv @ L4 )
=> ( ( ( product_Pair @ A @ B @ M2 @ L3 )
= ( Remove_max @ L4 ) )
=> ( Inv @ L3 ) ) ) )
=> ( removeMax_axioms @ B @ A @ Is_empty @ Of_list @ Multiset @ Remove_max @ Inv ) ) ) ) ) ) ).
% RemoveMax_axioms.intro
thf(fact_136_RemoveMax__axioms__def,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( removeMax_axioms @ B @ A )
= ( ^ [Is_empty2: B > $o,Of_list2: ( list @ A ) > B,Multiset2: B > ( multiset @ A ),Remove_max2: B > ( product_prod @ A @ B ),Inv2: B > $o] :
( ! [X4: list @ A] : ( Inv2 @ ( Of_list2 @ X4 ) )
& ! [L5: B,M3: A,L6: B] :
( ~ ( Is_empty2 @ L5 )
=> ( ( Inv2 @ L5 )
=> ( ( ( product_Pair @ A @ B @ M3 @ L6 )
= ( Remove_max2 @ L5 ) )
=> ( M3
= ( lattic929149872er_Max @ A @ ( set2 @ B @ A @ Multiset2 @ L5 ) ) ) ) ) )
& ! [L5: B,M3: A,L6: B] :
( ~ ( Is_empty2 @ L5 )
=> ( ( Inv2 @ 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 )
=> ( ( Inv2 @ L5 )
=> ( ( ( product_Pair @ A @ B @ M3 @ L6 )
= ( Remove_max2 @ L5 ) )
=> ( Inv2 @ L6 ) ) ) ) ) ) ) ) ).
% RemoveMax_axioms_def
thf(fact_137_Multiset_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A8: multiset @ A] :
( A8
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% Multiset.is_empty_def
thf(fact_138_zero__le,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% zero_le
thf(fact_139_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_140_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_141_RemoveMax_Oremove__max__max,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),Remove_max: B > ( product_prod @ A @ B ),Inv: B > $o,L: B,M: A,L2: B] :
( ( removeMax @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ Remove_max @ Inv )
=> ( ~ ( Is_empty @ L )
=> ( ( Inv @ L )
=> ( ( ( product_Pair @ A @ B @ M @ L2 )
= ( Remove_max @ L ) )
=> ( M
= ( lattic929149872er_Max @ A @ ( set2 @ B @ A @ Multiset @ L ) ) ) ) ) ) ) ) ).
% RemoveMax.remove_max_max
thf(fact_142_divides__aux__eq,axiom,
! [A: $tType] :
( ( unique1598680935umeral @ A )
=> ! [Q2: A,R: A] :
( ( unique455577585es_aux @ A @ ( product_Pair @ A @ A @ Q2 @ R ) )
= ( R
= ( zero_zero @ A ) ) ) ) ).
% divides_aux_eq
thf(fact_143_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_144_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_145_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
! [A: $tType,X: multiset @ A,Y: multiset @ A] :
( ( ( zero_zero @ ( multiset @ A ) )
= ( plus_plus @ ( multiset @ A ) @ X @ Y ) )
= ( ( X
= ( zero_zero @ ( multiset @ A ) ) )
& ( Y
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% subset_mset.zero_eq_add_iff_both_eq_0
thf(fact_146_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
! [A: $tType,X: multiset @ A,Y: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ X @ Y )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ( X
= ( zero_zero @ ( multiset @ A ) ) )
& ( Y
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% subset_mset.add_eq_0_iff_both_eq_0
thf(fact_147_union__eq__empty,axiom,
! [A: $tType,M4: multiset @ A,N: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ M4 @ N )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ( M4
= ( zero_zero @ ( multiset @ A ) ) )
& ( N
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% union_eq_empty
thf(fact_148_empty__eq__union,axiom,
! [A: $tType,M4: multiset @ A,N: multiset @ A] :
( ( ( zero_zero @ ( multiset @ A ) )
= ( plus_plus @ ( multiset @ A ) @ M4 @ N ) )
= ( ( M4
= ( zero_zero @ ( multiset @ A ) ) )
& ( N
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% empty_eq_union
thf(fact_149_union__mset__add__mset__right,axiom,
! [A: $tType,A4: multiset @ A,A2: A,B6: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ A4 @ ( add_mset @ A @ A2 @ B6 ) )
= ( add_mset @ A @ A2 @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) ) ) ).
% union_mset_add_mset_right
thf(fact_150_union__mset__add__mset__left,axiom,
! [A: $tType,A2: A,A4: multiset @ A,B6: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A2 @ A4 ) @ B6 )
= ( add_mset @ A @ A2 @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) ) ) ).
% union_mset_add_mset_left
thf(fact_151_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_152_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_153_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_154_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_155_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_156_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_157_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_158_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_159_le__iff__add,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
? [C5: A] :
( B4
= ( plus_plus @ A @ A5 @ C5 ) ) ) ) ) ).
% le_iff_add
thf(fact_160_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_161_less__eqE,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ~ ! [C3: A] :
( B2
!= ( plus_plus @ A @ A2 @ C3 ) ) ) ) ).
% less_eqE
thf(fact_162_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_163_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_164_add__mono__thms__linordered__semiring_I1_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J2: A,K2: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J2 )
& ( ord_less_eq @ A @ K2 @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K2 ) @ ( plus_plus @ A @ J2 @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_165_add__mono__thms__linordered__semiring_I2_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J2: A,K2: A,L: A] :
( ( ( I = J2 )
& ( ord_less_eq @ A @ K2 @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K2 ) @ ( plus_plus @ A @ J2 @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_166_add__mono__thms__linordered__semiring_I3_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J2: A,K2: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J2 )
& ( K2 = L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K2 ) @ ( plus_plus @ A @ J2 @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_167_empty__neutral_I2_J,axiom,
! [A: $tType,X: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ X @ ( zero_zero @ ( multiset @ A ) ) )
= X ) ).
% empty_neutral(2)
thf(fact_168_empty__neutral_I1_J,axiom,
! [A: $tType,X: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ ( zero_zero @ ( multiset @ A ) ) @ X )
= X ) ).
% empty_neutral(1)
thf(fact_169_RemoveMax_Oremove__max__inv,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),Remove_max: B > ( product_prod @ A @ B ),Inv: B > $o,L: B,M: A,L2: B] :
( ( removeMax @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ Remove_max @ Inv )
=> ( ~ ( Is_empty @ L )
=> ( ( Inv @ L )
=> ( ( ( product_Pair @ A @ B @ M @ L2 )
= ( Remove_max @ L ) )
=> ( Inv @ L2 ) ) ) ) ) ) ).
% RemoveMax.remove_max_inv
thf(fact_170_RemoveMax_Oof__list__inv,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),Remove_max: B > ( product_prod @ A @ B ),Inv: B > $o,X: list @ A] :
( ( removeMax @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ Remove_max @ Inv )
=> ( Inv @ ( Of_list @ X ) ) ) ) ).
% RemoveMax.of_list_inv
thf(fact_171_union__iff,axiom,
! [A: $tType,A2: A,A4: multiset @ A,B6: multiset @ A] :
( ( member @ A @ A2 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) ) )
= ( ( member @ A @ A2 @ ( set_mset @ A @ A4 ) )
| ( member @ A @ A2 @ ( set_mset @ A @ B6 ) ) ) ) ).
% union_iff
thf(fact_172_union__assoc,axiom,
! [A: $tType,M4: multiset @ A,N: multiset @ A,K3: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ M4 @ N ) @ K3 )
= ( plus_plus @ ( multiset @ A ) @ M4 @ ( plus_plus @ ( multiset @ A ) @ N @ K3 ) ) ) ).
% union_assoc
thf(fact_173_union__lcomm,axiom,
! [A: $tType,M4: multiset @ A,N: multiset @ A,K3: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ M4 @ ( plus_plus @ ( multiset @ A ) @ N @ K3 ) )
= ( plus_plus @ ( multiset @ A ) @ N @ ( plus_plus @ ( multiset @ A ) @ M4 @ K3 ) ) ) ).
% union_lcomm
thf(fact_174_union__commute,axiom,
! [A: $tType] :
( ( plus_plus @ ( multiset @ A ) )
= ( ^ [M6: multiset @ A,N4: multiset @ A] : ( plus_plus @ ( multiset @ A ) @ N4 @ M6 ) ) ) ).
% union_commute
thf(fact_175_union__left__cancel,axiom,
! [A: $tType,K3: multiset @ A,M4: multiset @ A,N: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ K3 @ M4 )
= ( plus_plus @ ( multiset @ A ) @ K3 @ N ) )
= ( M4 = N ) ) ).
% union_left_cancel
thf(fact_176_union__right__cancel,axiom,
! [A: $tType,M4: multiset @ A,K3: multiset @ A,N: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ M4 @ K3 )
= ( plus_plus @ ( multiset @ A ) @ N @ K3 ) )
= ( M4 = N ) ) ).
% union_right_cancel
thf(fact_177_multi__union__self__other__eq,axiom,
! [A: $tType,A4: multiset @ A,X6: multiset @ A,Y7: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ A4 @ X6 )
= ( plus_plus @ ( multiset @ A ) @ A4 @ Y7 ) )
=> ( X6 = Y7 ) ) ).
% multi_union_self_other_eq
thf(fact_178_RemoveMax_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 ),Remove_max: B > ( product_prod @ A @ B ),Inv: B > $o] :
( ( removeMax @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ Remove_max @ Inv )
=> ( collection @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset ) ) ) ).
% RemoveMax.axioms(1)
thf(fact_179_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_180_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_181_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_182_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_183_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_184_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_185_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_186_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_187_RemoveMax_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 ),Remove_max: B > ( product_prod @ A @ B ),Inv: B > $o] :
( ( removeMax @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ Remove_max @ Inv )
=> ( removeMax_axioms @ B @ A @ Is_empty @ Of_list @ Multiset @ Remove_max @ Inv ) ) ) ).
% RemoveMax.axioms(2)
thf(fact_188_add__mset__add__single,axiom,
! [A: $tType] :
( ( add_mset @ A )
= ( ^ [A5: A,A8: multiset @ A] : ( plus_plus @ ( multiset @ A ) @ A8 @ ( add_mset @ A @ A5 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% add_mset_add_single
thf(fact_189_union__is__single,axiom,
! [A: $tType,M4: multiset @ A,N: multiset @ A,A2: A] :
( ( ( plus_plus @ ( multiset @ A ) @ M4 @ N )
= ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) )
= ( ( ( M4
= ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) )
& ( N
= ( zero_zero @ ( multiset @ A ) ) ) )
| ( ( M4
= ( zero_zero @ ( multiset @ A ) ) )
& ( N
= ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% union_is_single
thf(fact_190_single__is__union,axiom,
! [A: $tType,A2: A,M4: multiset @ A,N: multiset @ A] :
( ( ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) )
= ( plus_plus @ ( multiset @ A ) @ M4 @ N ) )
= ( ( ( ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) )
= M4 )
& ( N
= ( zero_zero @ ( multiset @ A ) ) ) )
| ( ( M4
= ( zero_zero @ ( multiset @ A ) ) )
& ( ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) )
= N ) ) ) ) ).
% single_is_union
thf(fact_191_RemoveMax_Oremove__max__multiset,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),Remove_max: B > ( product_prod @ A @ B ),Inv: B > $o,L: B,M: A,L2: B] :
( ( removeMax @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ Remove_max @ Inv )
=> ( ~ ( Is_empty @ L )
=> ( ( Inv @ L )
=> ( ( ( product_Pair @ A @ B @ M @ L2 )
= ( Remove_max @ L ) )
=> ( ( add_mset @ A @ M @ ( Multiset @ L2 ) )
= ( Multiset @ L ) ) ) ) ) ) ) ).
% RemoveMax.remove_max_multiset
thf(fact_192_multi__member__skip,axiom,
! [A: $tType,X: A,XS: multiset @ A,Y: A] :
( ( member @ A @ X @ ( set_mset @ A @ XS ) )
=> ( member @ A @ X @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ Y @ ( zero_zero @ ( multiset @ A ) ) ) @ XS ) ) ) ) ).
% multi_member_skip
thf(fact_193_multi__member__this,axiom,
! [A: $tType,X: A,XS: multiset @ A] : ( member @ A @ X @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) @ XS ) ) ) ).
% multi_member_this
thf(fact_194_RemoveMax_Ointro,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),Remove_max: B > ( product_prod @ A @ B ),Inv: B > $o] :
( ( collection @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset )
=> ( ( removeMax_axioms @ B @ A @ Is_empty @ Of_list @ Multiset @ Remove_max @ Inv )
=> ( removeMax @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ Remove_max @ Inv ) ) ) ) ).
% RemoveMax.intro
thf(fact_195_RemoveMax__def,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( removeMax @ B @ A )
= ( ^ [Empty2: B,Is_empty2: B > $o,Of_list2: ( list @ A ) > B,Multiset2: B > ( multiset @ A ),Remove_max2: B > ( product_prod @ A @ B ),Inv2: B > $o] :
( ( collection @ B @ A @ Empty2 @ Is_empty2 @ Of_list2 @ Multiset2 )
& ( removeMax_axioms @ B @ A @ Is_empty2 @ Of_list2 @ Multiset2 @ Remove_max2 @ Inv2 ) ) ) ) ) ).
% RemoveMax_def
thf(fact_196_mult1E,axiom,
! [A: $tType,N: multiset @ A,M4: multiset @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N @ M4 ) @ ( mult1 @ A @ R ) )
=> ~ ! [A6: A,M0: multiset @ A] :
( ( M4
= ( add_mset @ A @ A6 @ M0 ) )
=> ! [K4: multiset @ A] :
( ( N
= ( plus_plus @ ( multiset @ A ) @ M0 @ K4 ) )
=> ~ ! [B8: A] :
( ( member @ A @ B8 @ ( set_mset @ A @ K4 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B8 @ A6 ) @ R ) ) ) ) ) ).
% mult1E
thf(fact_197_mult1I,axiom,
! [A: $tType,M4: multiset @ A,A2: A,M02: multiset @ A,N: multiset @ A,K3: multiset @ A,R: set @ ( product_prod @ A @ A )] :
( ( M4
= ( add_mset @ A @ A2 @ M02 ) )
=> ( ( N
= ( plus_plus @ ( multiset @ A ) @ M02 @ K3 ) )
=> ( ! [B5: A] :
( ( member @ A @ B5 @ ( set_mset @ A @ K3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A2 ) @ R ) )
=> ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N @ M4 ) @ ( mult1 @ A @ R ) ) ) ) ) ).
% mult1I
thf(fact_198_less__add,axiom,
! [A: $tType,N: multiset @ A,A2: A,M02: multiset @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N @ ( add_mset @ A @ A2 @ M02 ) ) @ ( mult1 @ A @ R ) )
=> ( ? [M5: multiset @ A] :
( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ M5 @ M02 ) @ ( mult1 @ A @ R ) )
& ( N
= ( add_mset @ A @ A2 @ M5 ) ) )
| ? [K4: multiset @ A] :
( ! [B8: A] :
( ( member @ A @ B8 @ ( set_mset @ A @ K4 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B8 @ A2 ) @ R ) )
& ( N
= ( plus_plus @ ( multiset @ A ) @ M02 @ K4 ) ) ) ) ) ).
% less_add
thf(fact_199_one__step__implies__mult,axiom,
! [A: $tType,J3: multiset @ A,K3: multiset @ A,R: set @ ( product_prod @ A @ A ),I4: multiset @ A] :
( ( J3
!= ( zero_zero @ ( multiset @ A ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( set_mset @ A @ K3 ) )
=> ? [Xa: A] :
( ( member @ A @ Xa @ ( set_mset @ A @ J3 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Xa ) @ R ) ) )
=> ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ I4 @ K3 ) @ ( plus_plus @ ( multiset @ A ) @ I4 @ J3 ) ) @ ( mult @ A @ R ) ) ) ) ).
% one_step_implies_mult
thf(fact_200_mono__mult,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ R5 )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) ) @ ( mult @ A @ R ) @ ( mult @ A @ R5 ) ) ) ).
% mono_mult
thf(fact_201_not__less__empty,axiom,
! [A: $tType,M4: multiset @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ M4 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( mult1 @ A @ R ) ) ).
% not_less_empty
thf(fact_202_mult1__union,axiom,
! [A: $tType,B6: multiset @ A,D4: multiset @ A,R: set @ ( product_prod @ A @ A ),C4: multiset @ A] :
( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ B6 @ D4 ) @ ( mult1 @ A @ R ) )
=> ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ C4 @ B6 ) @ ( plus_plus @ ( multiset @ A ) @ C4 @ D4 ) ) @ ( mult1 @ A @ R ) ) ) ).
% mult1_union
thf(fact_203_mono__mult1,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ R5 )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) ) @ ( mult1 @ A @ R ) @ ( mult1 @ A @ R5 ) ) ) ).
% mono_mult1
thf(fact_204_mult__implies__one__step,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),M4: multiset @ A,N: multiset @ A] :
( ( trans @ A @ R )
=> ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ M4 @ N ) @ ( mult @ A @ R ) )
=> ? [I5: multiset @ A,J4: multiset @ A] :
( ( N
= ( plus_plus @ ( multiset @ A ) @ I5 @ J4 ) )
& ? [K4: multiset @ A] :
( ( M4
= ( plus_plus @ ( multiset @ A ) @ I5 @ K4 ) )
& ( J4
!= ( zero_zero @ ( multiset @ A ) ) )
& ! [X7: A] :
( ( member @ A @ X7 @ ( set_mset @ A @ K4 ) )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ ( set_mset @ A @ J4 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X7 @ Xa2 ) @ R ) ) ) ) ) ) ) ).
% mult_implies_one_step
thf(fact_205_subset__mset_Osum__mset__0__iff,axiom,
! [A: $tType,M4: multiset @ ( multiset @ A )] :
( ( ( comm_monoid_sum_mset @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ M4 )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ! [X4: multiset @ A] :
( ( member @ ( multiset @ A ) @ X4 @ ( set_mset @ ( multiset @ A ) @ M4 ) )
=> ( X4
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% subset_mset.sum_mset_0_iff
thf(fact_206_trans__def,axiom,
! [A: $tType] :
( ( trans @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A )] :
! [X4: A,Y5: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ Z3 ) @ R3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Z3 ) @ R3 ) ) ) ) ) ).
% trans_def
thf(fact_207_transI,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ! [X5: A,Y3: A,Z4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y3 ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z4 ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Z4 ) @ R ) ) )
=> ( trans @ A @ R ) ) ).
% transI
thf(fact_208_transE,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X: A,Y: A,Z2: A] :
( ( trans @ A @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ R ) ) ) ) ).
% transE
thf(fact_209_transD,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X: A,Y: A,Z2: A] :
( ( trans @ A @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ R ) ) ) ) ).
% transD
thf(fact_210_comm__monoid__add_Osum__mset_Ocong,axiom,
! [A: $tType] :
( ( comm_monoid_sum_mset @ A )
= ( comm_monoid_sum_mset @ A ) ) ).
% comm_monoid_add.sum_mset.cong
thf(fact_211_mult__cancel__add__mset,axiom,
! [A: $tType,S: set @ ( product_prod @ A @ A ),Uu: A,X6: multiset @ A,Y7: multiset @ A] :
( ( trans @ A @ S )
=> ( ( irrefl @ A @ S )
=> ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( add_mset @ A @ Uu @ X6 ) @ ( add_mset @ A @ Uu @ Y7 ) ) @ ( mult @ A @ S ) )
= ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ X6 @ Y7 ) @ ( mult @ A @ S ) ) ) ) ) ).
% mult_cancel_add_mset
thf(fact_212_mult__cancel,axiom,
! [A: $tType,S: set @ ( product_prod @ A @ A ),X6: multiset @ A,Z5: multiset @ A,Y7: multiset @ A] :
( ( trans @ A @ S )
=> ( ( irrefl @ A @ S )
=> ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ X6 @ Z5 ) @ ( plus_plus @ ( multiset @ A ) @ Y7 @ Z5 ) ) @ ( mult @ A @ S ) )
= ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ X6 @ Y7 ) @ ( mult @ A @ S ) ) ) ) ) ).
% mult_cancel
thf(fact_213_irrefl__def,axiom,
! [A: $tType] :
( ( irrefl @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A )] :
! [A5: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ A5 ) @ R3 ) ) ) ).
% irrefl_def
thf(fact_214_irreflI,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A )] :
( ! [A6: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ A6 ) @ R4 )
=> ( irrefl @ A @ R4 ) ) ).
% irreflI
thf(fact_215_multp__iff,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A ),P: A > A > $o,N: multiset @ A,M4: multiset @ A] :
( ( irrefl @ A @ R4 )
=> ( ( trans @ A @ R4 )
=> ( ! [X5: A,Y3: A] :
( ( P @ X5 @ Y3 )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y3 ) @ R4 ) )
=> ( ( multp @ A @ P @ N @ M4 )
= ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N @ M4 ) @ ( mult @ A @ R4 ) ) ) ) ) ) ).
% multp_iff
thf(fact_216_in__mset__fold__plus__iff,axiom,
! [A: $tType,X: A,M4: multiset @ A,NN: multiset @ ( multiset @ A )] :
( ( member @ A @ X @ ( set_mset @ A @ ( fold_mset @ ( multiset @ A ) @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ M4 @ NN ) ) )
= ( ( member @ A @ X @ ( set_mset @ A @ M4 ) )
| ? [N4: multiset @ A] :
( ( member @ ( multiset @ A ) @ N4 @ ( set_mset @ ( multiset @ A ) @ NN ) )
& ( member @ A @ X @ ( set_mset @ A @ N4 ) ) ) ) ) ).
% in_mset_fold_plus_iff
thf(fact_217_fold__mset__empty,axiom,
! [B: $tType,A: $tType,F: B > A > A,S: A] :
( ( fold_mset @ B @ A @ F @ S @ ( zero_zero @ ( multiset @ B ) ) )
= S ) ).
% fold_mset_empty
thf(fact_218_union__fold__mset__add__mset,axiom,
! [A: $tType] :
( ( plus_plus @ ( multiset @ A ) )
= ( fold_mset @ A @ ( multiset @ A ) @ ( add_mset @ A ) ) ) ).
% union_fold_mset_add_mset
thf(fact_219_add__mset__replicate__mset__safe,axiom,
! [A: $tType,B: $tType,M4: multiset @ B,A2: B] :
( ( nO_MATCH @ ( multiset @ A ) @ ( multiset @ B ) @ ( zero_zero @ ( multiset @ A ) ) @ M4 )
=> ( ( add_mset @ B @ A2 @ M4 )
= ( plus_plus @ ( multiset @ B ) @ ( add_mset @ B @ A2 @ ( zero_zero @ ( multiset @ B ) ) ) @ M4 ) ) ) ).
% add_mset_replicate_mset_safe
thf(fact_220_insert__DiffM2,axiom,
! [A: $tType,X: A,M4: multiset @ A] :
( ( member @ A @ X @ ( set_mset @ A @ M4 ) )
=> ( ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M4 @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) )
= M4 ) ) ).
% insert_DiffM2
thf(fact_221_diff__diff__add__mset,axiom,
! [A: $tType,M4: multiset @ A,N: multiset @ A,P: multiset @ A] :
( ( minus_minus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M4 @ N ) @ P )
= ( minus_minus @ ( multiset @ A ) @ M4 @ ( plus_plus @ ( multiset @ A ) @ N @ P ) ) ) ).
% diff_diff_add_mset
thf(fact_222_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_223_add__mset__remove__trivial,axiom,
! [A: $tType,X: A,M4: multiset @ A] :
( ( minus_minus @ ( multiset @ A ) @ ( add_mset @ A @ X @ M4 ) @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) )
= M4 ) ).
% add_mset_remove_trivial
thf(fact_224_diff__add__mset__swap,axiom,
! [A: $tType,B2: A,A4: multiset @ A,M4: multiset @ A] :
( ~ ( member @ A @ B2 @ ( set_mset @ A @ A4 ) )
=> ( ( minus_minus @ ( multiset @ A ) @ ( add_mset @ A @ B2 @ M4 ) @ A4 )
= ( add_mset @ A @ B2 @ ( minus_minus @ ( multiset @ A ) @ M4 @ A4 ) ) ) ) ).
% diff_add_mset_swap
thf(fact_225_diff__union__swap2,axiom,
! [A: $tType,Y: A,M4: multiset @ A,X: A] :
( ( member @ A @ Y @ ( set_mset @ A @ M4 ) )
=> ( ( minus_minus @ ( multiset @ A ) @ ( add_mset @ A @ X @ M4 ) @ ( add_mset @ A @ Y @ ( zero_zero @ ( multiset @ A ) ) ) )
= ( add_mset @ A @ X @ ( minus_minus @ ( multiset @ A ) @ M4 @ ( add_mset @ A @ Y @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% diff_union_swap2
thf(fact_226_insert__DiffM,axiom,
! [A: $tType,X: A,M4: multiset @ A] :
( ( member @ A @ X @ ( set_mset @ A @ M4 ) )
=> ( ( add_mset @ A @ X @ ( minus_minus @ ( multiset @ A ) @ M4 @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) ) )
= M4 ) ) ).
% insert_DiffM
thf(fact_227_le__iff__diff__le__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A5 @ B4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% le_iff_diff_le_0
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_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_241_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_242_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_243_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_244_in__diffD,axiom,
! [A: $tType,A2: A,M4: multiset @ A,N: multiset @ A] :
( ( member @ A @ A2 @ ( set_mset @ A @ ( minus_minus @ ( multiset @ A ) @ M4 @ N ) ) )
=> ( member @ A @ A2 @ ( set_mset @ A @ M4 ) ) ) ).
% in_diffD
thf(fact_245_add__mset__diff__bothsides,axiom,
! [A: $tType,A2: A,M4: multiset @ A,A4: multiset @ A] :
( ( minus_minus @ ( multiset @ A ) @ ( add_mset @ A @ A2 @ M4 ) @ ( add_mset @ A @ A2 @ A4 ) )
= ( minus_minus @ ( multiset @ A ) @ M4 @ A4 ) ) ).
% add_mset_diff_bothsides
thf(fact_246_Multiset_Odiff__right__commute,axiom,
! [A: $tType,M4: multiset @ A,N: multiset @ A,Q: multiset @ A] :
( ( minus_minus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M4 @ N ) @ Q )
= ( minus_minus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M4 @ Q ) @ N ) ) ).
% Multiset.diff_right_commute
thf(fact_247_Multiset_Odiff__cancel,axiom,
! [A: $tType,A4: multiset @ A] :
( ( minus_minus @ ( multiset @ A ) @ A4 @ A4 )
= ( zero_zero @ ( multiset @ A ) ) ) ).
% Multiset.diff_cancel
thf(fact_248_diff__empty,axiom,
! [A: $tType,M4: multiset @ A] :
( ( ( minus_minus @ ( multiset @ A ) @ M4 @ ( zero_zero @ ( multiset @ A ) ) )
= M4 )
& ( ( minus_minus @ ( multiset @ A ) @ ( zero_zero @ ( multiset @ A ) ) @ M4 )
= ( zero_zero @ ( multiset @ A ) ) ) ) ).
% diff_empty
thf(fact_249_union__single__eq__diff,axiom,
! [A: $tType,X: A,M4: multiset @ A,N: multiset @ A] :
( ( ( add_mset @ A @ X @ M4 )
= N )
=> ( M4
= ( minus_minus @ ( multiset @ A ) @ N @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% union_single_eq_diff
thf(fact_250_add__eq__conv__diff,axiom,
! [A: $tType,A2: A,M4: multiset @ A,B2: A,N: multiset @ A] :
( ( ( add_mset @ A @ A2 @ M4 )
= ( add_mset @ A @ B2 @ N ) )
= ( ( ( M4 = N )
& ( A2 = B2 ) )
| ( ( M4
= ( add_mset @ A @ B2 @ ( minus_minus @ ( multiset @ A ) @ N @ ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
& ( N
= ( add_mset @ A @ A2 @ ( minus_minus @ ( multiset @ A ) @ M4 @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% add_eq_conv_diff
thf(fact_251_diff__union__swap,axiom,
! [A: $tType,A2: A,B2: A,M4: multiset @ A] :
( ( A2 != B2 )
=> ( ( add_mset @ A @ B2 @ ( minus_minus @ ( multiset @ A ) @ M4 @ ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
= ( minus_minus @ ( multiset @ A ) @ ( add_mset @ A @ B2 @ M4 ) @ ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% diff_union_swap
thf(fact_252_diff__union__cancelR,axiom,
! [A: $tType,M4: multiset @ A,N: multiset @ A] :
( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ M4 @ N ) @ N )
= M4 ) ).
% diff_union_cancelR
thf(fact_253_diff__union__cancelL,axiom,
! [A: $tType,N: multiset @ A,M4: multiset @ A] :
( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ N @ M4 ) @ N )
= M4 ) ).
% diff_union_cancelL
thf(fact_254_Multiset_Odiff__add,axiom,
! [A: $tType,M4: multiset @ A,N: multiset @ A,Q: multiset @ A] :
( ( minus_minus @ ( multiset @ A ) @ M4 @ ( plus_plus @ ( multiset @ A ) @ N @ Q ) )
= ( minus_minus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M4 @ N ) @ Q ) ) ).
% Multiset.diff_add
thf(fact_255_diff__single__trivial,axiom,
! [A: $tType,X: A,M4: multiset @ A] :
( ~ ( member @ A @ X @ ( set_mset @ A @ M4 ) )
=> ( ( minus_minus @ ( multiset @ A ) @ M4 @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) )
= M4 ) ) ).
% diff_single_trivial
% 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 (14)
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A9: $tType,A10: $tType] :
( ( preorder @ A10 )
=> ( preorder @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A9: $tType,A10: $tType] :
( ( order @ A10 )
=> ( order @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A9: $tType,A10: $tType] :
( ( ord @ A10 )
=> ( ord @ ( A9 > A10 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_1,axiom,
! [A9: $tType] : ( preorder @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_2,axiom,
! [A9: $tType] : ( order @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_3,axiom,
! [A9: $tType] : ( ord @ ( set @ A9 ) ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_4,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_5,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_6,axiom,
ord @ $o ).
thf(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add,axiom,
! [A9: $tType] :
( ( preorder @ A9 )
=> ( ordere779506340up_add @ ( multiset @ A9 ) ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Opreorder_7,axiom,
! [A9: $tType] :
( ( preorder @ A9 )
=> ( preorder @ ( multiset @ A9 ) ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Oorder_8,axiom,
! [A9: $tType] :
( ( preorder @ A9 )
=> ( order @ ( multiset @ A9 ) ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Oord_9,axiom,
! [A9: $tType] :
( ( preorder @ A9 )
=> ( ord @ ( multiset @ A9 ) ) ) ).
% Free types (1)
thf(tfree_0,hypothesis,
linorder @ a ).
% Conjectures (1)
thf(conj_0,conjecture,
is_heap @ a @ ( heapIm818251801eapify @ a @ ( t @ a @ v @ l @ r ) ) ).
%------------------------------------------------------------------------------