TPTP Problem File: ITP066^2.p
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%------------------------------------------------------------------------------
% File : ITP066^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer HeapImperative problem prob_1195__5351920_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_1195__5351920_1 [Des21]
% Status : Theorem
% Rating : 0.33 v8.2.0, 0.67 v8.1.0, 0.25 v7.5.0
% Syntax : Number of formulae : 356 ( 98 unt; 81 typ; 0 def)
% Number of atoms : 779 ( 389 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 5485 ( 84 ~; 10 |; 61 &;4971 @)
% ( 0 <=>; 359 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 8 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 223 ( 223 >; 0 *; 0 +; 0 <<)
% Number of symbols : 83 ( 80 usr; 14 con; 0-8 aty)
% Number of variables : 1341 ( 56 ^;1188 !; 19 ?;1341 :)
% ( 78 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:17:35.238
%------------------------------------------------------------------------------
% 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 (75)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Oneg__numeral,type,
neg_numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1352612707id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
ordere216010020id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere779506340up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
linord1659791738miring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere236663937imp_le:
!>[A: $tType] : $o ).
thf(sy_cl_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_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
semiri456707255roduct:
!>[A: $tType] : $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_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_Ohs__is__empty,type,
heapIm721255937_empty:
!>[A: $tType] : ( ( tree @ A ) > $o ) ).
thf(sy_c_HeapImperative__Mirabelle__oitemzccmr_Ohs__of__list,type,
heapIm874063447f_list:
!>[A: $tType] : ( ( list @ A ) > ( tree @ A ) ) ).
thf(sy_c_HeapImperative__Mirabelle__oitemzccmr_Ohs__remove__max,type,
heapIm1542349758ve_max:
!>[A: $tType] : ( ( tree @ A ) > ( product_prod @ 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_Oof__list__tree,type,
heapIm1912108042t_tree:
!>[A: $tType] : ( ( list @ A ) > ( tree @ A ) ) ).
thf(sy_c_HeapImperative__Mirabelle__oitemzccmr_OremoveLeaf,type,
heapIm970386777veLeaf:
!>[A: $tType] : ( ( tree @ A ) > ( product_prod @ 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_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_If,type,
if:
!>[A: $tType] : ( $o > A > 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_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_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_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_Product__Type_Oprod_Ofst,type,
product_fst:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
thf(sy_c_Product__Type_Oprod_Osnd,type,
product_snd:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
thf(sy_c_Product__Type_Oprod_Oswap,type,
product_swap:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
thf(sy_c_Product__Type_Oscomp,type,
product_scomp:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( A > ( product_prod @ B @ C ) ) > ( B > C > D ) > A > D ) ).
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_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_l1____,type,
l1: tree @ a ).
thf(sy_v_l2____,type,
l2: tree @ a ).
thf(sy_v_r1____,type,
r1: tree @ a ).
thf(sy_v_r2____,type,
r2: tree @ a ).
thf(sy_v_t,type,
t2: tree @ a ).
thf(sy_v_t_H,type,
t3: tree @ a ).
thf(sy_v_t_Ha____,type,
t_a: tree @ a ).
thf(sy_v_v1____,type,
v1: a ).
thf(sy_v_v2____,type,
v2: a ).
thf(sy_v_v_H,type,
v: a ).
thf(sy_v_v_Ha____,type,
v_a: a ).
thf(sy_v_v____,type,
v3: a ).
% Relevant facts (255)
thf(fact_0__C4__2_Oprems_C_I1_J,axiom,
( ( product_Pair @ a @ ( tree @ a ) @ v_a @ t_a )
= ( heapIm970386777veLeaf @ a @ ( t @ a @ v3 @ ( t @ a @ v1 @ l1 @ r1 ) @ ( t @ a @ v2 @ l2 @ r2 ) ) ) ) ).
% "4_2.prems"(1)
thf(fact_1_assms_I1_J,axiom,
( ( product_Pair @ a @ ( tree @ a ) @ v @ t3 )
= ( heapIm970386777veLeaf @ a @ t2 ) ) ).
% assms(1)
thf(fact_2__C4__2_Oprems_C_I2_J,axiom,
( ( t @ a @ v3 @ ( t @ a @ v1 @ l1 @ r1 ) @ ( t @ a @ v2 @ l2 @ r2 ) )
!= ( e @ a ) ) ).
% "4_2.prems"(2)
thf(fact_3__092_060open_062t_H_A_061_AT_Av_A_Isnd_A_IremoveLeaf_A_IT_Av1_Al1_Ar1_J_J_J_A_IT_Av2_Al2_Ar2_J_092_060close_062,axiom,
( t_a
= ( t @ a @ v3 @ ( product_snd @ a @ ( tree @ a ) @ ( heapIm970386777veLeaf @ a @ ( t @ a @ v1 @ l1 @ r1 ) ) ) @ ( t @ a @ v2 @ l2 @ r2 ) ) ) ).
% \<open>t' = T v (snd (removeLeaf (T v1 l1 r1))) (T v2 l2 r2)\<close>
thf(fact_4_Tree_Oinject,axiom,
! [A: $tType,X21: A,X22: tree @ A,X23: tree @ A,Y21: A,Y22: tree @ A,Y23: tree @ A] :
( ( ( t @ A @ X21 @ X22 @ X23 )
= ( t @ A @ Y21 @ Y22 @ Y23 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 )
& ( X23 = Y23 ) ) ) ).
% Tree.inject
thf(fact_5_removeLeaf_Osimps_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [V: A,Vd: A,Ve: tree @ A,Vf: tree @ A,Va: A,Vb: tree @ A,Vc: tree @ A] :
( ( heapIm970386777veLeaf @ A @ ( t @ A @ V @ ( t @ A @ Vd @ Ve @ Vf ) @ ( t @ A @ Va @ Vb @ Vc ) ) )
= ( product_Pair @ A @ ( tree @ A ) @ ( product_fst @ A @ ( tree @ A ) @ ( heapIm970386777veLeaf @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) ) @ ( t @ A @ V @ ( product_snd @ A @ ( tree @ A ) @ ( heapIm970386777veLeaf @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) ) @ ( t @ A @ Va @ Vb @ Vc ) ) ) ) ) ).
% removeLeaf.simps(5)
thf(fact_6_removeLeaf_Osimps_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [V: A,Va: A,Vb: tree @ A,Vc: tree @ A,Vd: A,Ve: tree @ A,Vf: tree @ A] :
( ( heapIm970386777veLeaf @ A @ ( t @ A @ V @ ( t @ A @ Va @ Vb @ Vc ) @ ( t @ A @ Vd @ Ve @ Vf ) ) )
= ( product_Pair @ A @ ( tree @ A ) @ ( product_fst @ A @ ( tree @ A ) @ ( heapIm970386777veLeaf @ A @ ( t @ A @ Va @ Vb @ Vc ) ) ) @ ( t @ A @ V @ ( product_snd @ A @ ( tree @ A ) @ ( heapIm970386777veLeaf @ A @ ( t @ A @ Va @ Vb @ Vc ) ) ) @ ( t @ A @ Vd @ Ve @ Vf ) ) ) ) ) ).
% removeLeaf.simps(4)
thf(fact_7_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_8_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_9__C4__2_Ohyps_C_I1_J,axiom,
! [V2: a,T2: tree @ a] :
( ( ( product_Pair @ a @ ( tree @ a ) @ V2 @ T2 )
= ( heapIm970386777veLeaf @ a @ ( t @ a @ v1 @ l1 @ r1 ) ) )
=> ( ( ( t @ a @ v1 @ l1 @ r1 )
!= ( e @ a ) )
=> ( ( plus_plus @ ( multiset @ a ) @ ( add_mset @ a @ V2 @ ( zero_zero @ ( multiset @ a ) ) ) @ ( multiset2 @ a @ T2 ) )
= ( multiset2 @ a @ ( t @ a @ v1 @ l1 @ r1 ) ) ) ) ) ).
% "4_2.hyps"(1)
thf(fact_10_removeLeaf_Osimps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [V: A] :
( ( heapIm970386777veLeaf @ A @ ( t @ A @ V @ ( e @ A ) @ ( e @ A ) ) )
= ( product_Pair @ A @ ( tree @ A ) @ V @ ( e @ A ) ) ) ) ).
% removeLeaf.simps(1)
thf(fact_11_fstI,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,Y: A,Z: B] :
( ( X
= ( product_Pair @ A @ B @ Y @ Z ) )
=> ( ( product_fst @ A @ B @ X )
= Y ) ) ).
% fstI
thf(fact_12_fst__eqD,axiom,
! [B: $tType,A: $tType,X: A,Y: B,A2: A] :
( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_13_fst__conv,axiom,
! [B: $tType,A: $tType,X1: A,X2: B] :
( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_14_assms_I2_J,axiom,
( t2
!= ( e @ a ) ) ).
% assms(2)
thf(fact_15_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
( ( ( product_Pair @ A @ B @ X1 @ X2 )
= ( product_Pair @ A @ B @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_16_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_17_prod_Ocollapse,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_18_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_19_multiset_Osimps_I1_J,axiom,
! [A: $tType] :
( ( multiset2 @ A @ ( e @ A ) )
= ( zero_zero @ ( multiset @ A ) ) ) ).
% multiset.simps(1)
thf(fact_20_sndI,axiom,
! [A: $tType,B: $tType,X: product_prod @ A @ B,Y: A,Z: B] :
( ( X
= ( product_Pair @ A @ B @ Y @ Z ) )
=> ( ( product_snd @ A @ B @ X )
= Z ) ) ).
% sndI
thf(fact_21_snd__eqD,axiom,
! [B: $tType,A: $tType,X: B,Y: A,A2: A] :
( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
= A2 )
=> ( Y = A2 ) ) ).
% snd_eqD
thf(fact_22_snd__conv,axiom,
! [Aa: $tType,A: $tType,X1: Aa,X2: A] :
( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X2 ) )
= X2 ) ).
% snd_conv
thf(fact_23_surj__pair,axiom,
! [A: $tType,B: $tType,P: product_prod @ A @ B] :
? [X3: A,Y3: B] :
( P
= ( product_Pair @ A @ B @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_24_prod__cases,axiom,
! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,P: product_prod @ A @ B] :
( ! [A4: A,B4: B] : ( P2 @ ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_25_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_26_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A4: A,B4: B,C2: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) ) ).
% prod_cases3
thf(fact_27_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A4: A,B4: B,C2: C,D2: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_28_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 ) ) )] :
~ ! [A4: A,B4: B,C2: C,D2: D,E2: E] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_29_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
~ ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_30_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,G: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
~ ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F,G2: G] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_31_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A4: A,B4: B,C2: C] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) )
=> ( P2 @ X ) ) ).
% prod_induct3
thf(fact_32_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A4: A,B4: B,C2: C,D2: D] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
=> ( P2 @ X ) ) ).
% prod_induct4
thf(fact_33_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( 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 ) ) )] :
( ! [A4: A,B4: B,C2: C,D2: D,E2: E] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P2 @ X ) ) ).
% prod_induct5
thf(fact_34_prod__induct6,axiom,
! [F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
( ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) )
=> ( P2 @ X ) ) ).
% prod_induct6
thf(fact_35_prod__induct7,axiom,
! [G: $tType,F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
( ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F,G2: G] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) )
=> ( P2 @ X ) ) ).
% prod_induct7
thf(fact_36_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A4: A,B4: B] :
( Y
!= ( product_Pair @ A @ B @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_37_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A4: A,B4: B] : ( P2 @ ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ( P2 @ Prod ) ) ).
% old.prod.inducts
thf(fact_38_HS_Omultiset__empty,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( multiset2 @ A @ ( e @ A ) )
= ( zero_zero @ ( multiset @ A ) ) ) ) ).
% HS.multiset_empty
thf(fact_39_surjective__pairing,axiom,
! [B: $tType,A: $tType,T3: product_prod @ A @ B] :
( T3
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T3 ) @ ( product_snd @ A @ B @ T3 ) ) ) ).
% surjective_pairing
thf(fact_40_prod_Oexhaust__sel,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_41_prod__eq__iff,axiom,
! [B: $tType,A: $tType] :
( ( ^ [Y4: product_prod @ A @ B,Z2: product_prod @ A @ B] : Y4 = Z2 )
= ( ^ [S: product_prod @ A @ B,T4: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ S )
= ( product_fst @ A @ B @ T4 ) )
& ( ( product_snd @ A @ B @ S )
= ( product_snd @ A @ B @ T4 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_42_prod_Oexpand,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
( ( ( ( product_fst @ A @ B @ Prod )
= ( product_fst @ A @ B @ Prod2 ) )
& ( ( product_snd @ A @ B @ Prod )
= ( product_snd @ A @ B @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_43_prod__eqI,axiom,
! [B: $tType,A: $tType,P: product_prod @ A @ B,Q: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ P )
= ( product_fst @ A @ B @ Q ) )
=> ( ( ( product_snd @ A @ B @ P )
= ( product_snd @ A @ B @ Q ) )
=> ( P = Q ) ) ) ).
% prod_eqI
thf(fact_44_is__heap_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: tree @ A] :
( ( X
!= ( e @ A ) )
=> ( ! [V3: A] :
( X
!= ( t @ A @ V3 @ ( e @ A ) @ ( e @ A ) ) )
=> ( ! [V3: A,Va2: A,Vb2: tree @ A,Vc2: tree @ A] :
( X
!= ( t @ A @ V3 @ ( e @ A ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) )
=> ( ! [V3: A,Va2: A,Vb2: tree @ A,Vc2: tree @ A] :
( X
!= ( t @ A @ V3 @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( e @ A ) ) )
=> ~ ! [V3: A,Va2: A,Vb2: tree @ A,Vc2: tree @ A,Vd2: A,Ve2: tree @ A,Vf2: tree @ A] :
( X
!= ( t @ A @ V3 @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) ) ) ) ) ) ) ).
% is_heap.cases
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P2: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P2: A > $o,Q2: A > $o] :
( ! [X3: A] :
( ( P2 @ X3 )
= ( Q2 @ X3 ) )
=> ( ( collect @ A @ P2 )
= ( collect @ A @ Q2 ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F3: A > B,G3: A > B] :
( ! [X3: A] :
( ( F3 @ X3 )
= ( G3 @ X3 ) )
=> ( F3 = G3 ) ) ).
% ext
thf(fact_49_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_50_Tree_Oinduct,axiom,
! [A: $tType,P2: ( tree @ A ) > $o,Tree: tree @ A] :
( ( P2 @ ( e @ A ) )
=> ( ! [X12: A,X24: tree @ A,X32: tree @ A] :
( ( P2 @ X24 )
=> ( ( P2 @ X32 )
=> ( P2 @ ( t @ A @ X12 @ X24 @ X32 ) ) ) )
=> ( P2 @ Tree ) ) ) ).
% Tree.induct
thf(fact_51_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_52_removeLeaf_Oinduct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P2: ( tree @ A ) > $o,A0: tree @ A] :
( ! [V3: A] : ( P2 @ ( t @ A @ V3 @ ( e @ A ) @ ( e @ A ) ) )
=> ( ! [V3: A,Va2: A,Vb2: tree @ A,Vc2: tree @ A] :
( ( P2 @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) )
=> ( ( P2 @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) )
=> ( P2 @ ( t @ A @ V3 @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( e @ A ) ) ) ) )
=> ( ! [V3: A,Va2: A,Vb2: tree @ A,Vc2: tree @ A] :
( ( P2 @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) )
=> ( ( P2 @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) )
=> ( P2 @ ( t @ A @ V3 @ ( e @ A ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) ) ) )
=> ( ! [V3: A,Va2: A,Vb2: tree @ A,Vc2: tree @ A,Vd2: A,Ve2: tree @ A,Vf2: tree @ A] :
( ( P2 @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) )
=> ( ( P2 @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) )
=> ( P2 @ ( t @ A @ V3 @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) ) ) )
=> ( ! [V3: A,Vd2: A,Ve2: tree @ A,Vf2: tree @ A,Va2: A,Vb2: tree @ A,Vc2: tree @ A] :
( ( P2 @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) )
=> ( ( P2 @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) )
=> ( P2 @ ( t @ A @ V3 @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) ) ) )
=> ( ( P2 @ ( e @ A ) )
=> ( P2 @ A0 ) ) ) ) ) ) ) ) ).
% removeLeaf.induct
thf(fact_53_removeLeaf_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: tree @ A] :
( ! [V3: A] :
( X
!= ( t @ A @ V3 @ ( e @ A ) @ ( e @ A ) ) )
=> ( ! [V3: A,Va2: A,Vb2: tree @ A,Vc2: tree @ A] :
( X
!= ( t @ A @ V3 @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( e @ A ) ) )
=> ( ! [V3: A,Va2: A,Vb2: tree @ A,Vc2: tree @ A] :
( X
!= ( t @ A @ V3 @ ( e @ A ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) )
=> ( ! [V3: A,Va2: A,Vb2: tree @ A,Vc2: tree @ A,Vd2: A,Ve2: tree @ A,Vf2: tree @ A] :
( X
!= ( t @ A @ V3 @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) )
=> ( ! [V3: A,Vd2: A,Ve2: tree @ A,Vf2: tree @ A,Va2: A,Vb2: tree @ A,Vc2: tree @ A] :
( X
!= ( t @ A @ V3 @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) )
=> ( X
= ( e @ A ) ) ) ) ) ) ) ) ).
% removeLeaf.cases
thf(fact_54_siftDown_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: tree @ A] :
( ( X
!= ( e @ A ) )
=> ( ! [V3: A] :
( X
!= ( t @ A @ V3 @ ( e @ A ) @ ( e @ A ) ) )
=> ( ! [V3: A,Va2: A,Vb2: tree @ A,Vc2: tree @ A] :
( X
!= ( t @ A @ V3 @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( e @ A ) ) )
=> ( ! [V3: A,Va2: A,Vb2: tree @ A,Vc2: tree @ A] :
( X
!= ( t @ A @ V3 @ ( e @ A ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) )
=> ~ ! [V3: A,Va2: A,Vb2: tree @ A,Vc2: tree @ A,Vd2: A,Ve2: tree @ A,Vf2: tree @ A] :
( X
!= ( t @ A @ V3 @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) ) ) ) ) ) ) ).
% siftDown.cases
thf(fact_55_removeLeaf_Osimps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [V: A,Va: A,Vb: tree @ A,Vc: tree @ A] :
( ( heapIm970386777veLeaf @ A @ ( t @ A @ V @ ( t @ A @ Va @ Vb @ Vc ) @ ( e @ A ) ) )
= ( product_Pair @ A @ ( tree @ A ) @ ( product_fst @ A @ ( tree @ A ) @ ( heapIm970386777veLeaf @ A @ ( t @ A @ Va @ Vb @ Vc ) ) ) @ ( t @ A @ V @ ( product_snd @ A @ ( tree @ A ) @ ( heapIm970386777veLeaf @ A @ ( t @ A @ Va @ Vb @ Vc ) ) ) @ ( e @ A ) ) ) ) ) ).
% removeLeaf.simps(2)
thf(fact_56_removeLeaf_Osimps_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [V: A,Va: A,Vb: tree @ A,Vc: tree @ A] :
( ( heapIm970386777veLeaf @ A @ ( t @ A @ V @ ( e @ A ) @ ( t @ A @ Va @ Vb @ Vc ) ) )
= ( product_Pair @ A @ ( tree @ A ) @ ( product_fst @ A @ ( tree @ A ) @ ( heapIm970386777veLeaf @ A @ ( t @ A @ Va @ Vb @ Vc ) ) ) @ ( t @ A @ V @ ( e @ A ) @ ( product_snd @ A @ ( tree @ A ) @ ( heapIm970386777veLeaf @ A @ ( t @ A @ Va @ Vb @ Vc ) ) ) ) ) ) ) ).
% removeLeaf.simps(3)
thf(fact_57_union__mset__add__mset__left,axiom,
! [A: $tType,A2: A,A5: multiset @ A,B5: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A2 @ A5 ) @ B5 )
= ( add_mset @ A @ A2 @ ( plus_plus @ ( multiset @ A ) @ A5 @ B5 ) ) ) ).
% union_mset_add_mset_left
thf(fact_58_union__mset__add__mset__right,axiom,
! [A: $tType,A5: multiset @ A,A2: A,B5: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ A5 @ ( add_mset @ A @ A2 @ B5 ) )
= ( add_mset @ A @ A2 @ ( plus_plus @ ( multiset @ A ) @ A5 @ B5 ) ) ) ).
% union_mset_add_mset_right
thf(fact_59_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_60_add__mset__eq__single,axiom,
! [A: $tType,B2: A,M: multiset @ A,A2: A] :
( ( ( add_mset @ A @ B2 @ M )
= ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) )
= ( ( B2 = A2 )
& ( M
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% add_mset_eq_single
thf(fact_61_single__eq__add__mset,axiom,
! [A: $tType,A2: A,B2: A,M: multiset @ A] :
( ( ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) )
= ( add_mset @ A @ B2 @ M ) )
= ( ( B2 = A2 )
& ( M
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% single_eq_add_mset
thf(fact_62_add__mset__eq__singleton__iff,axiom,
! [A: $tType,X: A,M: multiset @ A,Y: A] :
( ( ( add_mset @ A @ X @ M )
= ( add_mset @ A @ Y @ ( zero_zero @ ( multiset @ A ) ) ) )
= ( ( M
= ( zero_zero @ ( multiset @ A ) ) )
& ( X = Y ) ) ) ).
% add_mset_eq_singleton_iff
thf(fact_63_empty__eq__union,axiom,
! [A: $tType,M: multiset @ A,N: multiset @ A] :
( ( ( zero_zero @ ( multiset @ A ) )
= ( plus_plus @ ( multiset @ A ) @ M @ N ) )
= ( ( M
= ( zero_zero @ ( multiset @ A ) ) )
& ( N
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% empty_eq_union
thf(fact_64_union__eq__empty,axiom,
! [A: $tType,M: multiset @ A,N: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ M @ N )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ( M
= ( zero_zero @ ( multiset @ A ) ) )
& ( N
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% union_eq_empty
thf(fact_65_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_66_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_67_add_Oleft__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add.left_neutral
thf(fact_68_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C3 @ A2 ) )
= ( B2 = C3 ) ) ) ).
% add_right_cancel
thf(fact_69_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C3 ) )
= ( B2 = C3 ) ) ) ).
% add_left_cancel
thf(fact_70_multi__self__add__other__not__self,axiom,
! [A: $tType,M: multiset @ A,X: A] :
( M
!= ( add_mset @ A @ X @ M ) ) ).
% multi_self_add_other_not_self
thf(fact_71_add__mset__add__mset__same__iff,axiom,
! [A: $tType,A2: A,A5: multiset @ A,B5: multiset @ A] :
( ( ( add_mset @ A @ A2 @ A5 )
= ( add_mset @ A @ A2 @ B5 ) )
= ( A5 = B5 ) ) ).
% add_mset_add_mset_same_iff
thf(fact_72_zero__eq__add__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [X: A,Y: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ X @ Y ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_73_add__eq__0__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [X: A,Y: A] :
( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_74_add__cancel__right__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( plus_plus @ A @ A2 @ B2 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_right
thf(fact_75_add__cancel__right__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( plus_plus @ A @ B2 @ A2 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_left
thf(fact_76_add__cancel__left__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A2: A,B2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= A2 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_right
thf(fact_77_add__cancel__left__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [B2: A,A2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= A2 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_left
thf(fact_78_double__zero__sym,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ A2 @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% double_zero_sym
thf(fact_79_double__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ( plus_plus @ A @ A2 @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% double_zero
thf(fact_80_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.right_neutral
thf(fact_81_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_82_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C3 @ A2 ) )
=> ( B2 = C3 ) ) ) ).
% add_right_imp_eq
thf(fact_83_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C3 ) )
=> ( B2 = C3 ) ) ) ).
% add_left_imp_eq
thf(fact_84_add_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C3 ) )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C3 ) ) ) ) ).
% add.left_commute
thf(fact_85_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( ( plus_plus @ A )
= ( ^ [A6: A,B6: A] : ( plus_plus @ A @ B6 @ A6 ) ) ) ) ).
% add.commute
thf(fact_86_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C3 @ A2 ) )
= ( B2 = C3 ) ) ) ).
% add.right_cancel
thf(fact_87_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C3 ) )
= ( B2 = C3 ) ) ) ).
% add.left_cancel
thf(fact_88_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C3 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C3 ) ) ) ) ).
% add.assoc
thf(fact_89_group__cancel_Oadd2,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [B5: A,K: A,B2: A,A2: A] :
( ( B5
= ( plus_plus @ A @ K @ B2 ) )
=> ( ( plus_plus @ A @ A2 @ B5 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.add2
thf(fact_90_group__cancel_Oadd1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: A,K: A,A2: A,B2: A] :
( ( A5
= ( plus_plus @ A @ K @ A2 ) )
=> ( ( plus_plus @ A @ A5 @ B2 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.add1
thf(fact_91_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus @ A @ I @ K )
= ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_92_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C3 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C3 ) ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_93_multi__union__self__other__eq,axiom,
! [A: $tType,A5: multiset @ A,X5: multiset @ A,Y5: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ A5 @ X5 )
= ( plus_plus @ ( multiset @ A ) @ A5 @ Y5 ) )
=> ( X5 = Y5 ) ) ).
% multi_union_self_other_eq
thf(fact_94_union__right__cancel,axiom,
! [A: $tType,M: multiset @ A,K2: multiset @ A,N: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ M @ K2 )
= ( plus_plus @ ( multiset @ A ) @ N @ K2 ) )
= ( M = N ) ) ).
% union_right_cancel
thf(fact_95_union__left__cancel,axiom,
! [A: $tType,K2: multiset @ A,M: multiset @ A,N: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ K2 @ M )
= ( plus_plus @ ( multiset @ A ) @ K2 @ N ) )
= ( M = N ) ) ).
% union_left_cancel
thf(fact_96_union__commute,axiom,
! [A: $tType] :
( ( plus_plus @ ( multiset @ A ) )
= ( ^ [M2: multiset @ A,N2: multiset @ A] : ( plus_plus @ ( multiset @ A ) @ N2 @ M2 ) ) ) ).
% union_commute
thf(fact_97_union__lcomm,axiom,
! [A: $tType,M: multiset @ A,N: multiset @ A,K2: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ M @ ( plus_plus @ ( multiset @ A ) @ N @ K2 ) )
= ( plus_plus @ ( multiset @ A ) @ N @ ( plus_plus @ ( multiset @ A ) @ M @ K2 ) ) ) ).
% union_lcomm
thf(fact_98_union__assoc,axiom,
! [A: $tType,M: multiset @ A,N: multiset @ A,K2: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ M @ N ) @ K2 )
= ( plus_plus @ ( multiset @ A ) @ M @ ( plus_plus @ ( multiset @ A ) @ N @ K2 ) ) ) ).
% union_assoc
thf(fact_99_add__mset__commute,axiom,
! [A: $tType,X: A,Y: A,M: multiset @ A] :
( ( add_mset @ A @ X @ ( add_mset @ A @ Y @ M ) )
= ( add_mset @ A @ Y @ ( add_mset @ A @ X @ M ) ) ) ).
% add_mset_commute
thf(fact_100_add__eq__conv__ex,axiom,
! [A: $tType,A2: A,M: multiset @ A,B2: A,N: multiset @ A] :
( ( ( add_mset @ A @ A2 @ M )
= ( add_mset @ A @ B2 @ N ) )
= ( ( ( M = N )
& ( A2 = B2 ) )
| ? [K3: multiset @ A] :
( ( M
= ( add_mset @ A @ B2 @ K3 ) )
& ( N
= ( add_mset @ A @ A2 @ K3 ) ) ) ) ) ).
% add_eq_conv_ex
thf(fact_101_add_Ogroup__left__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add.group_left_neutral
thf(fact_102_add_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.comm_neutral
thf(fact_103_comm__monoid__add__class_Oadd__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% comm_monoid_add_class.add_0
thf(fact_104_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_105_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_106_multi__nonempty__split,axiom,
! [A: $tType,M: multiset @ A] :
( ( M
!= ( zero_zero @ ( multiset @ A ) ) )
=> ? [A7: multiset @ A,A4: A] :
( M
= ( add_mset @ A @ A4 @ A7 ) ) ) ).
% multi_nonempty_split
thf(fact_107_empty__not__add__mset,axiom,
! [A: $tType,A2: A,A5: multiset @ A] :
( ( zero_zero @ ( multiset @ A ) )
!= ( add_mset @ A @ A2 @ A5 ) ) ).
% empty_not_add_mset
thf(fact_108_multiset__induct2,axiom,
! [A: $tType,B: $tType,P2: ( multiset @ A ) > ( multiset @ B ) > $o,M: multiset @ A,N: multiset @ B] :
( ( P2 @ ( zero_zero @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ B ) ) )
=> ( ! [A4: A,M3: multiset @ A,N3: multiset @ B] :
( ( P2 @ M3 @ N3 )
=> ( P2 @ ( add_mset @ A @ A4 @ M3 ) @ N3 ) )
=> ( ! [A4: B,M3: multiset @ A,N3: multiset @ B] :
( ( P2 @ M3 @ N3 )
=> ( P2 @ M3 @ ( add_mset @ B @ A4 @ N3 ) ) )
=> ( P2 @ M @ N ) ) ) ) ).
% multiset_induct2
thf(fact_109_multiset__induct,axiom,
! [A: $tType,P2: ( multiset @ A ) > $o,M: multiset @ A] :
( ( P2 @ ( zero_zero @ ( multiset @ A ) ) )
=> ( ! [X3: A,M3: multiset @ A] :
( ( P2 @ M3 )
=> ( P2 @ ( add_mset @ A @ X3 @ M3 ) ) )
=> ( P2 @ M ) ) ) ).
% multiset_induct
thf(fact_110_multiset__cases,axiom,
! [A: $tType,M: multiset @ A] :
( ( M
!= ( zero_zero @ ( multiset @ A ) ) )
=> ~ ! [X3: A,N3: multiset @ A] :
( M
!= ( add_mset @ A @ X3 @ N3 ) ) ) ).
% multiset_cases
thf(fact_111_add__mset__add__single,axiom,
! [A: $tType] :
( ( add_mset @ A )
= ( ^ [A6: A,A8: multiset @ A] : ( plus_plus @ ( multiset @ A ) @ A8 @ ( add_mset @ A @ A6 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% add_mset_add_single
thf(fact_112_union__is__single,axiom,
! [A: $tType,M: multiset @ A,N: multiset @ A,A2: A] :
( ( ( plus_plus @ ( multiset @ A ) @ M @ N )
= ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) )
= ( ( ( M
= ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) )
& ( N
= ( zero_zero @ ( multiset @ A ) ) ) )
| ( ( M
= ( zero_zero @ ( multiset @ A ) ) )
& ( N
= ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% union_is_single
thf(fact_113_single__is__union,axiom,
! [A: $tType,A2: A,M: multiset @ A,N: multiset @ A] :
( ( ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) )
= ( plus_plus @ ( multiset @ A ) @ M @ N ) )
= ( ( ( ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) )
= M )
& ( N
= ( zero_zero @ ( multiset @ A ) ) ) )
| ( ( M
= ( zero_zero @ ( multiset @ A ) ) )
& ( ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) )
= N ) ) ) ) ).
% single_is_union
thf(fact_114_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_115_exI__realizer,axiom,
! [B: $tType,A: $tType,P2: A > B > $o,Y: A,X: B] :
( ( P2 @ Y @ X )
=> ( P2 @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_116_conjI__realizer,axiom,
! [A: $tType,B: $tType,P2: A > $o,P: A,Q2: B > $o,Q: B] :
( ( P2 @ P )
=> ( ( Q2 @ Q )
=> ( ( P2 @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P @ Q ) ) )
& ( Q2 @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_117_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [B: $tType,A: $tType,P2: A > B > $o,X: A,Y: B,A2: product_prod @ A @ B] :
( ( P2 @ X @ Y )
=> ( ( A2
= ( product_Pair @ A @ B @ X @ Y ) )
=> ( P2 @ ( product_fst @ A @ B @ A2 ) @ ( product_snd @ A @ B @ A2 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_118_Multiset_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A8: multiset @ A] :
( A8
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% Multiset.is_empty_def
thf(fact_119_exE__realizer_H,axiom,
! [A: $tType,B: $tType,P2: A > B > $o,P: product_prod @ B @ A] :
( ( P2 @ ( product_snd @ B @ A @ P ) @ ( product_fst @ B @ A @ P ) )
=> ~ ! [X3: B,Y3: A] :
~ ( P2 @ Y3 @ X3 ) ) ).
% exE_realizer'
thf(fact_120_eq__snd__iff,axiom,
! [A: $tType,B: $tType,B2: A,P: product_prod @ B @ A] :
( ( B2
= ( product_snd @ B @ A @ P ) )
= ( ? [A6: B] :
( P
= ( product_Pair @ B @ A @ A6 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_121_eq__fst__iff,axiom,
! [A: $tType,B: $tType,A2: A,P: product_prod @ A @ B] :
( ( A2
= ( product_fst @ A @ B @ P ) )
= ( ? [B6: B] :
( P
= ( product_Pair @ A @ B @ A2 @ B6 ) ) ) ) ).
% eq_fst_iff
thf(fact_122_verit__sum__simplify,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% verit_sum_simplify
thf(fact_123_add__0__iff,axiom,
! [A: $tType] :
( ( semiri456707255roduct @ A )
=> ! [B2: A,A2: A] :
( ( B2
= ( plus_plus @ A @ B2 @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% add_0_iff
thf(fact_124_HS_Ois__empty__as__list,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [E3: tree @ A] :
( ( heapIm721255937_empty @ A @ E3 )
=> ( ( multiset2 @ A @ E3 )
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% HS.is_empty_as_list
thf(fact_125_add__mset__replicate__mset__safe,axiom,
! [A: $tType,B: $tType,M: multiset @ B,A2: B] :
( ( nO_MATCH @ ( multiset @ A ) @ ( multiset @ B ) @ ( zero_zero @ ( multiset @ A ) ) @ M )
=> ( ( add_mset @ B @ A2 @ M )
= ( plus_plus @ ( multiset @ B ) @ ( add_mset @ B @ A2 @ ( zero_zero @ ( multiset @ B ) ) ) @ M ) ) ) ).
% add_mset_replicate_mset_safe
thf(fact_126_HS_Ois__empty__empty,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( heapIm721255937_empty @ A @ ( e @ A ) ) ) ).
% HS.is_empty_empty
thf(fact_127_hs__is__empty__def,axiom,
! [A: $tType] :
( ( heapIm721255937_empty @ A )
= ( ^ [T4: tree @ A] :
( T4
= ( e @ A ) ) ) ) ).
% hs_is_empty_def
thf(fact_128_HS_Ois__empty__inj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [E3: tree @ A] :
( ( heapIm721255937_empty @ A @ E3 )
=> ( E3
= ( e @ A ) ) ) ) ).
% HS.is_empty_inj
thf(fact_129_prod_Oswap__def,axiom,
! [B: $tType,A: $tType] :
( ( product_swap @ A @ B )
= ( ^ [P3: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P3 ) @ ( product_fst @ A @ B @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_130_removeLeaf__heap__is__heap,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T3: tree @ A] :
( ( is_heap @ A @ T3 )
=> ( ( T3
!= ( e @ A ) )
=> ( is_heap @ A @ ( product_snd @ A @ ( tree @ A ) @ ( heapIm970386777veLeaf @ A @ T3 ) ) ) ) ) ) ).
% removeLeaf_heap_is_heap
thf(fact_131_NO__MATCH__cong,axiom,
! [B: $tType,A: $tType] :
( ( nO_MATCH @ A @ B )
= ( nO_MATCH @ A @ B ) ) ).
% NO_MATCH_cong
thf(fact_132_NO__MATCH__def,axiom,
! [B: $tType,A: $tType] :
( ( nO_MATCH @ A @ B )
= ( ^ [Pat: A,Val: B] : $true ) ) ).
% NO_MATCH_def
thf(fact_133_swap__swap,axiom,
! [B: $tType,A: $tType,P: product_prod @ A @ B] :
( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P ) )
= P ) ).
% swap_swap
thf(fact_134_swap__simp,axiom,
! [A: $tType,B: $tType,X: B,Y: A] :
( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
= ( product_Pair @ A @ B @ Y @ X ) ) ).
% swap_simp
thf(fact_135_snd__swap,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B] :
( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X ) )
= ( product_fst @ A @ B @ X ) ) ).
% snd_swap
thf(fact_136_fst__swap,axiom,
! [A: $tType,B: $tType,X: product_prod @ B @ A] :
( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X ) )
= ( product_snd @ B @ A @ X ) ) ).
% fst_swap
thf(fact_137_is__heap_Osimps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( is_heap @ A @ ( e @ A ) ) ) ).
% is_heap.simps(1)
thf(fact_138_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_139_removeLeaf__val__val,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T3: tree @ A] :
( ( ( product_snd @ A @ ( tree @ A ) @ ( heapIm970386777veLeaf @ A @ T3 ) )
!= ( e @ A ) )
=> ( ( T3
!= ( e @ A ) )
=> ( ( val @ A @ T3 )
= ( val @ A @ ( product_snd @ A @ ( tree @ A ) @ ( heapIm970386777veLeaf @ A @ T3 ) ) ) ) ) ) ) ).
% removeLeaf_val_val
thf(fact_140_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A2: B,B2: C] :
( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
= ( C3 @ A2 @ B2 ) ) ).
% internal_case_prod_conv
thf(fact_141_Heap__axioms__def,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( heap_axioms @ B @ A )
= ( ^ [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 )] :
( ! [L2: B] :
( ( Multiset @ L2 )
= ( multiset2 @ A @ ( As_tree @ L2 ) ) )
& ! [I2: list @ A] : ( is_heap @ A @ ( As_tree @ ( Of_list @ I2 ) ) )
& ! [T4: B] :
( ( ( As_tree @ T4 )
= ( e @ A ) )
= ( Is_empty @ T4 ) )
& ! [L2: B,M4: A,L3: B] :
( ~ ( Is_empty @ L2 )
=> ( ( ( product_Pair @ A @ B @ M4 @ L3 )
= ( Remove_max @ L2 ) )
=> ( ( add_mset @ A @ M4 @ ( Multiset @ L3 ) )
= ( Multiset @ L2 ) ) ) )
& ! [L2: B,M4: A,L3: B] :
( ~ ( Is_empty @ L2 )
=> ( ( is_heap @ A @ ( As_tree @ L2 ) )
=> ( ( ( product_Pair @ A @ B @ M4 @ L3 )
= ( Remove_max @ L2 ) )
=> ( is_heap @ A @ ( As_tree @ L3 ) ) ) ) )
& ! [T4: B,M4: A,T5: B] :
( ~ ( Is_empty @ T4 )
=> ( ( ( product_Pair @ A @ B @ M4 @ T5 )
= ( Remove_max @ T4 ) )
=> ( M4
= ( val @ A @ ( As_tree @ T4 ) ) ) ) ) ) ) ) ) ).
% Heap_axioms_def
thf(fact_142_Heap__axioms_Ointro,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Multiset2: B > ( multiset @ A ),As_tree2: B > ( tree @ A ),Of_list2: ( list @ A ) > B,Is_empty2: B > $o,Remove_max2: B > ( product_prod @ A @ B )] :
( ! [L4: B] :
( ( Multiset2 @ L4 )
= ( multiset2 @ A @ ( As_tree2 @ L4 ) ) )
=> ( ! [I3: list @ A] : ( is_heap @ A @ ( As_tree2 @ ( Of_list2 @ I3 ) ) )
=> ( ! [T6: B] :
( ( ( As_tree2 @ T6 )
= ( e @ A ) )
= ( Is_empty2 @ T6 ) )
=> ( ! [L4: B,M5: A,L5: B] :
( ~ ( Is_empty2 @ L4 )
=> ( ( ( product_Pair @ A @ B @ M5 @ L5 )
= ( Remove_max2 @ L4 ) )
=> ( ( add_mset @ A @ M5 @ ( Multiset2 @ L5 ) )
= ( Multiset2 @ L4 ) ) ) )
=> ( ! [L4: B,M5: A,L5: B] :
( ~ ( Is_empty2 @ L4 )
=> ( ( is_heap @ A @ ( As_tree2 @ L4 ) )
=> ( ( ( product_Pair @ A @ B @ M5 @ L5 )
= ( Remove_max2 @ L4 ) )
=> ( is_heap @ A @ ( As_tree2 @ L5 ) ) ) ) )
=> ( ! [T6: B,M5: A,T7: B] :
( ~ ( Is_empty2 @ T6 )
=> ( ( ( product_Pair @ A @ B @ M5 @ T7 )
= ( Remove_max2 @ T6 ) )
=> ( M5
= ( val @ A @ ( As_tree2 @ T6 ) ) ) ) )
=> ( heap_axioms @ B @ A @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 ) ) ) ) ) ) ) ) ).
% Heap_axioms.intro
thf(fact_143_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_144_is__heap_Osimps_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [V: A,Va: A,Vb: tree @ A,Vc: tree @ A] :
( ( is_heap @ A @ ( t @ A @ V @ ( t @ A @ Va @ Vb @ Vc ) @ ( e @ A ) ) )
= ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ V )
& ( is_heap @ A @ ( t @ A @ Va @ Vb @ Vc ) ) ) ) ) ).
% is_heap.simps(4)
thf(fact_145_is__heap_Osimps_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [V: A,Va: A,Vb: tree @ A,Vc: tree @ A] :
( ( is_heap @ A @ ( t @ A @ V @ ( e @ A ) @ ( t @ A @ Va @ Vb @ Vc ) ) )
= ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ V )
& ( is_heap @ A @ ( t @ A @ Va @ Vb @ Vc ) ) ) ) ) ).
% is_heap.simps(3)
thf(fact_146_divides__aux__eq,axiom,
! [A: $tType] :
( ( unique1598680935umeral @ A )
=> ! [Q: A,R: A] :
( ( unique455577585es_aux @ A @ ( product_Pair @ A @ A @ Q @ R ) )
= ( R
= ( zero_zero @ A ) ) ) ) ).
% divides_aux_eq
thf(fact_147_Heap_Oremove__max__is__heap,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Empty: 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 ),L: B,M6: A,L6: B] :
( ( heap @ B @ A @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
=> ( ~ ( Is_empty2 @ L )
=> ( ( is_heap @ A @ ( As_tree2 @ L ) )
=> ( ( ( product_Pair @ A @ B @ M6 @ L6 )
= ( Remove_max2 @ L ) )
=> ( is_heap @ A @ ( As_tree2 @ L6 ) ) ) ) ) ) ) ).
% Heap.remove_max_is_heap
thf(fact_148_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N4: A] :
( ( ord_less_eq @ A @ N4 @ ( zero_zero @ A ) )
= ( N4
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_149_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B2 @ C3 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_cancel_right
thf(fact_150_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A2 ) @ ( plus_plus @ A @ C3 @ B2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_cancel_left
thf(fact_151_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_152_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_153_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_154_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_155_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_156_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_157_verit__la__disequality,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( A2 = B2 )
| ~ ( ord_less_eq @ A @ A2 @ B2 )
| ~ ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% verit_la_disequality
thf(fact_158_add__le__imp__le__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B2 @ C3 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_imp_le_right
thf(fact_159_add__le__imp__le__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A2 ) @ ( plus_plus @ A @ C3 @ B2 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_imp_le_left
thf(fact_160_le__iff__add,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A6: A,B6: A] :
? [C4: A] :
( B6
= ( plus_plus @ A @ A6 @ C4 ) ) ) ) ) ).
% le_iff_add
thf(fact_161_add__right__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B2 @ C3 ) ) ) ) ).
% add_right_mono
thf(fact_162_less__eqE,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ~ ! [C2: A] :
( B2
!= ( plus_plus @ A @ A2 @ C2 ) ) ) ) ).
% less_eqE
thf(fact_163_add__left__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A2 ) @ ( plus_plus @ A @ C3 @ B2 ) ) ) ) ).
% add_left_mono
thf(fact_164_add__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [A2: A,B2: A,C3: A,D3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ D3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% add_mono
thf(fact_165_add__mono__thms__linordered__semiring_I1_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_166_add__mono__thms__linordered__semiring_I2_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_167_add__mono__thms__linordered__semiring_I3_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( K = L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_168_zero__le,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% zero_le
thf(fact_169_add__decreasing,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ C3 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C3 ) @ B2 ) ) ) ) ).
% add_decreasing
thf(fact_170_add__increasing,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A2 @ C3 ) ) ) ) ) ).
% add_increasing
thf(fact_171_add__decreasing2,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C3 ) @ B2 ) ) ) ) ).
% add_decreasing2
thf(fact_172_add__increasing2,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A )
=> ! [C3: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A2 @ C3 ) ) ) ) ) ).
% add_increasing2
thf(fact_173_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_174_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_175_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_176_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_177_Heap_Oas__tree__empty,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Empty: 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 ),T3: B] :
( ( heap @ B @ A @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
=> ( ( ( As_tree2 @ T3 )
= ( e @ A ) )
= ( Is_empty2 @ T3 ) ) ) ) ).
% Heap.as_tree_empty
thf(fact_178_Heap_Ois__heap__of__list,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [Empty: 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 ),I: list @ A] :
( ( heap @ B @ A @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
=> ( is_heap @ A @ ( As_tree2 @ ( Of_list2 @ I ) ) ) ) ) ).
% Heap.is_heap_of_list
thf(fact_179_Heap_Omultiset,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Empty: 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 ),L: B] :
( ( heap @ B @ A @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
=> ( ( Multiset2 @ L )
= ( multiset2 @ A @ ( As_tree2 @ L ) ) ) ) ) ).
% Heap.multiset
thf(fact_180_divides__aux__def,axiom,
! [A: $tType] :
( ( unique1598680935umeral @ A )
=> ( ( unique455577585es_aux @ A )
= ( ^ [Qr: product_prod @ A @ A] :
( ( product_snd @ A @ A @ Qr )
= ( zero_zero @ A ) ) ) ) ) ).
% divides_aux_def
thf(fact_181_Heap_Oaxioms_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Empty: 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 )] :
( ( heap @ B @ A @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
=> ( heap_axioms @ B @ A @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 ) ) ) ).
% Heap.axioms(2)
thf(fact_182_is__heap_Osimps_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [V: A,Vd: A,Ve: tree @ A,Vf: tree @ A,Va: A,Vb: tree @ A,Vc: tree @ A] :
( ( is_heap @ A @ ( t @ A @ V @ ( t @ A @ Vd @ Ve @ Vf ) @ ( t @ A @ Va @ Vb @ Vc ) ) )
= ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ V )
& ( is_heap @ A @ ( t @ A @ Va @ Vb @ Vc ) )
& ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) @ V )
& ( is_heap @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) ) ) ) ).
% is_heap.simps(6)
thf(fact_183_is__heap_Osimps_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [V: A,Va: A,Vb: tree @ A,Vc: tree @ A,Vd: A,Ve: tree @ A,Vf: tree @ A] :
( ( is_heap @ A @ ( t @ A @ V @ ( t @ A @ Va @ Vb @ Vc ) @ ( t @ A @ Vd @ Ve @ Vf ) ) )
= ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) @ V )
& ( is_heap @ A @ ( t @ A @ Vd @ Ve @ Vf ) )
& ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ V )
& ( is_heap @ A @ ( t @ A @ Va @ Vb @ Vc ) ) ) ) ) ).
% is_heap.simps(5)
thf(fact_184_Heap_Oremove__max__multiset_H,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Empty: 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 ),L: B,M6: A,L6: B] :
( ( heap @ B @ A @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
=> ( ~ ( Is_empty2 @ L )
=> ( ( ( product_Pair @ A @ B @ M6 @ L6 )
= ( Remove_max2 @ L ) )
=> ( ( add_mset @ A @ M6 @ ( Multiset2 @ L6 ) )
= ( Multiset2 @ L ) ) ) ) ) ) ).
% Heap.remove_max_multiset'
thf(fact_185_Heap_Oremove__max__val,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Empty: 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 ),T3: B,M6: A,T2: B] :
( ( heap @ B @ A @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
=> ( ~ ( Is_empty2 @ T3 )
=> ( ( ( product_Pair @ A @ B @ M6 @ T2 )
= ( Remove_max2 @ T3 ) )
=> ( M6
= ( val @ A @ ( As_tree2 @ T3 ) ) ) ) ) ) ) ).
% Heap.remove_max_val
thf(fact_186_siftDown_Osimps_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Va: A,Vb: tree @ A,Vc: tree @ A,V: A] :
( ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ V )
=> ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( e @ A ) @ ( t @ A @ Va @ Vb @ Vc ) ) )
= ( t @ A @ V @ ( e @ A ) @ ( t @ A @ Va @ Vb @ Vc ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ V )
=> ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( e @ A ) @ ( t @ A @ Va @ Vb @ Vc ) ) )
= ( t @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ ( e @ A ) @ ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( heapIm1271749598e_left @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ ( heapIm1434396069_right @ A @ ( t @ A @ Va @ Vb @ Vc ) ) ) ) ) ) ) ) ) ).
% siftDown.simps(4)
thf(fact_187_siftDown_Osimps_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Va: A,Vb: tree @ A,Vc: tree @ A,V: A] :
( ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ V )
=> ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( t @ A @ Va @ Vb @ Vc ) @ ( e @ A ) ) )
= ( t @ A @ V @ ( t @ A @ Va @ Vb @ Vc ) @ ( e @ A ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ V )
=> ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( t @ A @ Va @ Vb @ Vc ) @ ( e @ A ) ) )
= ( t @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( heapIm1271749598e_left @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ ( heapIm1434396069_right @ A @ ( t @ A @ Va @ Vb @ Vc ) ) ) ) @ ( e @ A ) ) ) ) ) ) ).
% siftDown.simps(3)
thf(fact_188_siftDown_Osimps_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Vd: A,Ve: tree @ A,Vf: tree @ A,Va: A,Vb: tree @ A,Vc: tree @ A,V: A] :
( ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) )
=> ( ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ V )
=> ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( t @ A @ Va @ Vb @ Vc ) @ ( t @ A @ Vd @ Ve @ Vf ) ) )
= ( t @ A @ V @ ( t @ A @ Va @ Vb @ Vc ) @ ( t @ A @ Vd @ Ve @ Vf ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ V )
=> ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( t @ A @ Va @ Vb @ Vc ) @ ( t @ A @ Vd @ Ve @ Vf ) ) )
= ( t @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( heapIm1271749598e_left @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ ( heapIm1434396069_right @ A @ ( t @ A @ Va @ Vb @ Vc ) ) ) ) @ ( t @ A @ Vd @ Ve @ Vf ) ) ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) )
=> ( ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) @ V )
=> ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( t @ A @ Va @ Vb @ Vc ) @ ( t @ A @ Vd @ Ve @ Vf ) ) )
= ( t @ A @ V @ ( t @ A @ Va @ Vb @ Vc ) @ ( t @ A @ Vd @ Ve @ Vf ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) @ V )
=> ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( t @ A @ Va @ Vb @ Vc ) @ ( t @ A @ Vd @ Ve @ Vf ) ) )
= ( t @ A @ ( val @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) @ ( t @ A @ Va @ Vb @ Vc ) @ ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( heapIm1271749598e_left @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) @ ( heapIm1434396069_right @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) ) ) ) ) ) ) ) ) ) ).
% siftDown.simps(5)
thf(fact_189_siftDown_Osimps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( heapIm748920189ftDown @ A @ ( e @ A ) )
= ( e @ A ) ) ) ).
% siftDown.simps(1)
thf(fact_190_siftDown__multiset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T3: tree @ A] :
( ( multiset2 @ A @ ( heapIm748920189ftDown @ A @ T3 ) )
= ( multiset2 @ A @ T3 ) ) ) ).
% siftDown_multiset
thf(fact_191_siftDown__Node,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T3: tree @ A,V: A,L: tree @ A,R: tree @ A] :
( ( T3
= ( t @ A @ V @ L @ R ) )
=> ? [L5: tree @ A,V4: A,R2: tree @ A] :
( ( ( heapIm748920189ftDown @ A @ T3 )
= ( t @ A @ V4 @ L5 @ R2 ) )
& ( ord_less_eq @ A @ V @ V4 ) ) ) ) ).
% siftDown_Node
thf(fact_192_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_193_siftDown__heap__is__heap,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: tree @ A,R: tree @ A,T3: tree @ A,V: A] :
( ( is_heap @ A @ L )
=> ( ( is_heap @ A @ R )
=> ( ( T3
= ( t @ A @ V @ L @ R ) )
=> ( is_heap @ A @ ( heapIm748920189ftDown @ A @ T3 ) ) ) ) ) ) ).
% siftDown_heap_is_heap
thf(fact_194_siftDown_Osimps_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Va: A,Vb: tree @ A,Vc: tree @ A,Vd: A,Ve: tree @ A,Vf: tree @ A,V: A] :
( ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ ( val @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) )
=> ( ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) @ V )
=> ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( t @ A @ Vd @ Ve @ Vf ) @ ( t @ A @ Va @ Vb @ Vc ) ) )
= ( t @ A @ V @ ( t @ A @ Vd @ Ve @ Vf ) @ ( t @ A @ Va @ Vb @ Vc ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) @ V )
=> ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( t @ A @ Vd @ Ve @ Vf ) @ ( t @ A @ Va @ Vb @ Vc ) ) )
= ( t @ A @ ( val @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) @ ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( heapIm1271749598e_left @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) @ ( heapIm1434396069_right @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) ) ) @ ( t @ A @ Va @ Vb @ Vc ) ) ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ ( val @ A @ ( t @ A @ Vd @ Ve @ Vf ) ) )
=> ( ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ V )
=> ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( t @ A @ Vd @ Ve @ Vf ) @ ( t @ A @ Va @ Vb @ Vc ) ) )
= ( t @ A @ V @ ( t @ A @ Vd @ Ve @ Vf ) @ ( t @ A @ Va @ Vb @ Vc ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ V )
=> ( ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( t @ A @ Vd @ Ve @ Vf ) @ ( t @ A @ Va @ Vb @ Vc ) ) )
= ( t @ A @ ( val @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ ( t @ A @ Vd @ Ve @ Vf ) @ ( heapIm748920189ftDown @ A @ ( t @ A @ V @ ( heapIm1271749598e_left @ A @ ( t @ A @ Va @ Vb @ Vc ) ) @ ( heapIm1434396069_right @ A @ ( t @ A @ Va @ Vb @ Vc ) ) ) ) ) ) ) ) ) ) ) ).
% siftDown.simps(6)
thf(fact_195_siftDown__in__tree,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T3: tree @ A] :
( ( T3
!= ( e @ A ) )
=> ( in_tree @ A @ ( val @ A @ ( heapIm748920189ftDown @ A @ T3 ) ) @ T3 ) ) ) ).
% siftDown_in_tree
thf(fact_196_is__heap__max,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [V: A,T3: tree @ A] :
( ( in_tree @ A @ V @ T3 )
=> ( ( is_heap @ A @ T3 )
=> ( ord_less_eq @ A @ V @ ( val @ A @ T3 ) ) ) ) ) ).
% is_heap_max
thf(fact_197_in__tree_Osimps_I2_J,axiom,
! [A: $tType,V: A,V2: A,L: tree @ A,R: tree @ A] :
( ( in_tree @ A @ V @ ( t @ A @ V2 @ L @ R ) )
= ( ( V = V2 )
| ( in_tree @ A @ V @ L )
| ( in_tree @ A @ V @ R ) ) ) ).
% in_tree.simps(2)
thf(fact_198_in__tree_Osimps_I1_J,axiom,
! [A: $tType,V: A] :
~ ( in_tree @ A @ V @ ( e @ A ) ) ).
% in_tree.simps(1)
thf(fact_199_siftDown__in__tree__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( in_tree @ A )
= ( ^ [V5: A,T4: tree @ A] : ( in_tree @ A @ V5 @ ( heapIm748920189ftDown @ A @ T4 ) ) ) ) ) ).
% siftDown_in_tree_set
thf(fact_200_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_201_hs__remove__max__def,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( heapIm1542349758ve_max @ A )
= ( ^ [T4: tree @ A] :
( if @ ( product_prod @ A @ ( tree @ A ) )
@ ( ( product_snd @ A @ ( tree @ A ) @ ( heapIm970386777veLeaf @ A @ T4 ) )
= ( e @ A ) )
@ ( product_Pair @ A @ ( tree @ A ) @ ( val @ A @ T4 ) @ ( e @ A ) )
@ ( product_Pair @ A @ ( tree @ A ) @ ( val @ A @ T4 ) @ ( heapIm748920189ftDown @ A @ ( t @ A @ ( product_fst @ A @ ( tree @ A ) @ ( heapIm970386777veLeaf @ A @ T4 ) ) @ ( heapIm1271749598e_left @ A @ ( product_snd @ A @ ( tree @ A ) @ ( heapIm970386777veLeaf @ A @ T4 ) ) ) @ ( heapIm1434396069_right @ A @ ( product_snd @ A @ ( tree @ A ) @ ( heapIm970386777veLeaf @ A @ T4 ) ) ) ) ) ) ) ) ) ) ).
% hs_remove_max_def
thf(fact_202_is__num__normalize_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C3 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C3 ) ) ) ) ).
% is_num_normalize(1)
thf(fact_203_scomp__unfold,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType] :
( ( product_scomp @ A @ B @ C @ D )
= ( ^ [F4: A > ( product_prod @ B @ C ),G4: B > C > D,X4: A] : ( G4 @ ( product_fst @ B @ C @ ( F4 @ X4 ) ) @ ( product_snd @ B @ C @ ( F4 @ X4 ) ) ) ) ) ).
% scomp_unfold
thf(fact_204_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_205_scomp__scomp,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F: $tType,E: $tType,F3: A > ( product_prod @ E @ F ),G3: E > F > ( product_prod @ C @ D ),H: C > D > B] :
( ( product_scomp @ A @ C @ D @ B @ ( product_scomp @ A @ E @ F @ ( product_prod @ C @ D ) @ F3 @ G3 ) @ H )
= ( product_scomp @ A @ E @ F @ B @ F3
@ ^ [X4: E] : ( product_scomp @ F @ C @ D @ B @ ( G3 @ X4 ) @ H ) ) ) ).
% scomp_scomp
thf(fact_206_Pair__scomp,axiom,
! [A: $tType,B: $tType,C: $tType,X: C,F3: C > A > B] :
( ( product_scomp @ A @ C @ A @ B @ ( product_Pair @ C @ A @ X ) @ F3 )
= ( F3 @ X ) ) ).
% Pair_scomp
thf(fact_207_scomp__Pair,axiom,
! [C: $tType,B: $tType,A: $tType,X: A > ( product_prod @ B @ C )] :
( ( product_scomp @ A @ B @ C @ ( product_prod @ B @ C ) @ X @ ( product_Pair @ B @ C ) )
= X ) ).
% scomp_Pair
thf(fact_208_heapify_Osimps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( heapIm818251801eapify @ A @ ( e @ A ) )
= ( e @ A ) ) ) ).
% heapify.simps(1)
thf(fact_209_heapify__heap__is__heap,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T3: tree @ A] : ( is_heap @ A @ ( heapIm818251801eapify @ A @ T3 ) ) ) ).
% heapify_heap_is_heap
thf(fact_210_multiset__heapify,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T3: tree @ A] :
( ( multiset2 @ A @ ( heapIm818251801eapify @ A @ T3 ) )
= ( multiset2 @ A @ T3 ) ) ) ).
% multiset_heapify
thf(fact_211_Heap__def,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( heap @ B @ A )
= ( ^ [Empty2: 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 @ Empty2 @ Is_empty @ Of_list @ Multiset )
& ( heap_axioms @ B @ A @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max ) ) ) ) ) ).
% Heap_def
thf(fact_212_Heap_Ointro,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Empty: 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 @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 )
=> ( ( heap_axioms @ B @ A @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
=> ( heap @ B @ A @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 ) ) ) ) ).
% Heap.intro
thf(fact_213_Heap_Oaxioms_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Empty: 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 )] :
( ( heap @ B @ A @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
=> ( collection @ B @ A @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 ) ) ) ).
% Heap.axioms(1)
thf(fact_214_HS_OCollection__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( collection @ ( tree @ A ) @ A @ ( e @ A ) @ ( heapIm721255937_empty @ A ) @ ( heapIm874063447f_list @ A ) @ ( multiset2 @ A ) ) ) ).
% HS.Collection_axioms
thf(fact_215_Collection_Omultiset__empty,axiom,
! [B: $tType,A: $tType,Empty: B,Is_empty2: B > $o,Of_list2: ( list @ A ) > B,Multiset2: B > ( multiset @ A )] :
( ( collection @ B @ A @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 )
=> ( ( Multiset2 @ Empty )
= ( zero_zero @ ( multiset @ A ) ) ) ) ).
% Collection.multiset_empty
thf(fact_216_Collection_Ois__empty__as__list,axiom,
! [B: $tType,A: $tType,Empty: B,Is_empty2: B > $o,Of_list2: ( list @ A ) > B,Multiset2: B > ( multiset @ A ),E3: B] :
( ( collection @ B @ A @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 )
=> ( ( Is_empty2 @ E3 )
=> ( ( Multiset2 @ E3 )
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% Collection.is_empty_as_list
thf(fact_217_hs__of__list__def,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( heapIm874063447f_list @ A )
= ( ^ [L2: list @ A] : ( heapIm818251801eapify @ A @ ( heapIm1912108042t_tree @ A @ L2 ) ) ) ) ) ).
% hs_of_list_def
thf(fact_218_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X4: A,Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y6 ) @ R3 ) )
= ( ^ [X4: A,Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y6 ) @ S2 ) ) )
= ( R3 = S2 ) ) ).
% pred_equals_eq2
thf(fact_219_subrelI,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ! [X3: A,Y3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S3 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S3 ) ) ).
% subrelI
thf(fact_220_pred__subset__eq2,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( A > B > $o )
@ ^ [X4: A,Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y6 ) @ R3 )
@ ^ [X4: A,Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y6 ) @ S2 ) )
= ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S2 ) ) ).
% pred_subset_eq2
thf(fact_221_fun__cong__unused__0,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( zero @ B )
=> ! [F3: ( A > B ) > C,G3: C] :
( ( F3
= ( ^ [X4: A > B] : G3 ) )
=> ( ( F3
@ ^ [X4: A] : ( zero_zero @ B ) )
= G3 ) ) ) ).
% fun_cong_unused_0
thf(fact_222_heap__top__geq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,T3: tree @ A] :
( ( member @ A @ A2 @ ( set_mset @ A @ ( multiset2 @ A @ T3 ) ) )
=> ( ( is_heap @ A @ T3 )
=> ( ord_less_eq @ A @ A2 @ ( val @ A @ T3 ) ) ) ) ) ).
% heap_top_geq
thf(fact_223_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_224_union__single__eq__member,axiom,
! [A: $tType,X: A,M: multiset @ A,N: multiset @ A] :
( ( ( add_mset @ A @ X @ M )
= N )
=> ( member @ A @ X @ ( set_mset @ A @ N ) ) ) ).
% union_single_eq_member
thf(fact_225_insert__noteq__member,axiom,
! [A: $tType,B2: A,B5: multiset @ A,C3: A,C5: multiset @ A] :
( ( ( add_mset @ A @ B2 @ B5 )
= ( add_mset @ A @ C3 @ C5 ) )
=> ( ( B2 != C3 )
=> ( member @ A @ C3 @ ( set_mset @ A @ B5 ) ) ) ) ).
% insert_noteq_member
thf(fact_226_multi__member__split,axiom,
! [A: $tType,X: A,M: multiset @ A] :
( ( member @ A @ X @ ( set_mset @ A @ M ) )
=> ? [A7: multiset @ A] :
( M
= ( add_mset @ A @ X @ A7 ) ) ) ).
% multi_member_split
thf(fact_227_mset__add,axiom,
! [A: $tType,A2: A,A5: multiset @ A] :
( ( member @ A @ A2 @ ( set_mset @ A @ A5 ) )
=> ~ ! [B7: multiset @ A] :
( A5
!= ( add_mset @ A @ A2 @ B7 ) ) ) ).
% mset_add
thf(fact_228_union__iff,axiom,
! [A: $tType,A2: A,A5: multiset @ A,B5: multiset @ A] :
( ( member @ A @ A2 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A5 @ B5 ) ) )
= ( ( member @ A @ A2 @ ( set_mset @ A @ A5 ) )
| ( member @ A @ A2 @ ( set_mset @ A @ B5 ) ) ) ) ).
% union_iff
thf(fact_229_multiset__nonemptyE,axiom,
! [A: $tType,A5: multiset @ A] :
( ( A5
!= ( zero_zero @ ( multiset @ A ) ) )
=> ~ ! [X3: A] :
~ ( member @ A @ X3 @ ( set_mset @ A @ A5 ) ) ) ).
% multiset_nonemptyE
thf(fact_230_ge__eq__refl,axiom,
! [A: $tType,R3: A > A > $o,X: A] :
( ( ord_less_eq @ ( A > A > $o )
@ ^ [Y4: A,Z2: A] : Y4 = Z2
@ R3 )
=> ( R3 @ X @ X ) ) ).
% ge_eq_refl
thf(fact_231_refl__ge__eq,axiom,
! [A: $tType,R3: A > A > $o] :
( ! [X3: A] : ( R3 @ X3 @ X3 )
=> ( ord_less_eq @ ( A > A > $o )
@ ^ [Y4: A,Z2: A] : Y4 = Z2
@ R3 ) ) ).
% refl_ge_eq
thf(fact_232_subset__CollectI,axiom,
! [A: $tType,B5: set @ A,A5: set @ A,Q2: A > $o,P2: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B5 )
=> ( ( Q2 @ X3 )
=> ( P2 @ X3 ) ) )
=> ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ B5 )
& ( Q2 @ X4 ) ) )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A5 )
& ( P2 @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_233_subset__Collect__iff,axiom,
! [A: $tType,B5: set @ A,A5: set @ A,P2: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A5 )
& ( P2 @ X4 ) ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ B5 )
=> ( P2 @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_234_multiset__induct__min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P2: ( multiset @ A ) > $o,M: multiset @ A] :
( ( P2 @ ( zero_zero @ ( multiset @ A ) ) )
=> ( ! [X3: A,M3: multiset @ A] :
( ( P2 @ M3 )
=> ( ! [Xa: A] :
( ( member @ A @ Xa @ ( set_mset @ A @ M3 ) )
=> ( ord_less_eq @ A @ X3 @ Xa ) )
=> ( P2 @ ( add_mset @ A @ X3 @ M3 ) ) ) )
=> ( P2 @ M ) ) ) ) ).
% multiset_induct_min
thf(fact_235_multiset__induct__max,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P2: ( multiset @ A ) > $o,M: multiset @ A] :
( ( P2 @ ( zero_zero @ ( multiset @ A ) ) )
=> ( ! [X3: A,M3: multiset @ A] :
( ( P2 @ M3 )
=> ( ! [Xa: A] :
( ( member @ A @ Xa @ ( set_mset @ A @ M3 ) )
=> ( ord_less_eq @ A @ Xa @ X3 ) )
=> ( P2 @ ( add_mset @ A @ X3 @ M3 ) ) ) )
=> ( P2 @ M ) ) ) ) ).
% multiset_induct_max
thf(fact_236_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_237_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_238_less__add,axiom,
! [A: $tType,N: multiset @ A,A2: A,M0: 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 @ M0 ) ) @ ( mult1 @ A @ R ) )
=> ( ? [M3: multiset @ A] :
( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ M3 @ M0 ) @ ( mult1 @ A @ R ) )
& ( N
= ( add_mset @ A @ A2 @ M3 ) ) )
| ? [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 ) @ M0 @ K4 ) ) ) ) ) ).
% less_add
thf(fact_239_mult1I,axiom,
! [A: $tType,M: multiset @ A,A2: A,M0: multiset @ A,N: multiset @ A,K2: multiset @ A,R: set @ ( product_prod @ A @ A )] :
( ( M
= ( add_mset @ A @ A2 @ M0 ) )
=> ( ( N
= ( plus_plus @ ( multiset @ A ) @ M0 @ K2 ) )
=> ( ! [B4: A] :
( ( member @ A @ B4 @ ( set_mset @ A @ K2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A2 ) @ R ) )
=> ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N @ M ) @ ( mult1 @ A @ R ) ) ) ) ) ).
% mult1I
thf(fact_240_mono__mult1,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ R4 )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) ) @ ( mult1 @ A @ R ) @ ( mult1 @ A @ R4 ) ) ) ).
% mono_mult1
thf(fact_241_mult1__union,axiom,
! [A: $tType,B5: multiset @ A,D4: multiset @ A,R: set @ ( product_prod @ A @ A ),C5: multiset @ A] :
( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ B5 @ D4 ) @ ( mult1 @ A @ R ) )
=> ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ C5 @ B5 ) @ ( plus_plus @ ( multiset @ A ) @ C5 @ D4 ) ) @ ( mult1 @ A @ R ) ) ) ).
% mult1_union
thf(fact_242_not__less__empty,axiom,
! [A: $tType,M: multiset @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ M @ ( zero_zero @ ( multiset @ A ) ) ) @ ( mult1 @ A @ R ) ) ).
% not_less_empty
thf(fact_243_mult1E,axiom,
! [A: $tType,N: multiset @ A,M: multiset @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N @ M ) @ ( mult1 @ A @ R ) )
=> ~ ! [A4: A,M02: multiset @ A] :
( ( M
= ( add_mset @ A @ A4 @ M02 ) )
=> ! [K4: multiset @ A] :
( ( N
= ( plus_plus @ ( multiset @ A ) @ M02 @ K4 ) )
=> ~ ! [B8: A] :
( ( member @ A @ B8 @ ( set_mset @ A @ K4 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B8 @ A4 ) @ R ) ) ) ) ) ).
% mult1E
thf(fact_244_one__step__implies__mult,axiom,
! [A: $tType,J2: multiset @ A,K2: multiset @ A,R: set @ ( product_prod @ A @ A ),I4: multiset @ A] :
( ( J2
!= ( zero_zero @ ( multiset @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set_mset @ A @ K2 ) )
=> ? [Xa: A] :
( ( member @ A @ Xa @ ( set_mset @ A @ J2 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Xa ) @ R ) ) )
=> ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ I4 @ K2 ) @ ( plus_plus @ ( multiset @ A ) @ I4 @ J2 ) ) @ ( mult @ A @ R ) ) ) ) ).
% one_step_implies_mult
thf(fact_245_heap__top__max,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T3: tree @ A] :
( ( T3
!= ( e @ A ) )
=> ( ( is_heap @ A @ T3 )
=> ( ( val @ A @ T3 )
= ( lattic929149872er_Max @ A @ ( set_mset @ A @ ( multiset2 @ A @ T3 ) ) ) ) ) ) ) ).
% heap_top_max
thf(fact_246_mono__mult,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ R4 )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) ) @ ( mult @ A @ R ) @ ( mult @ A @ R4 ) ) ) ).
% mono_mult
thf(fact_247_mult__implies__one__step,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),M: multiset @ A,N: multiset @ A] :
( ( trans @ A @ R )
=> ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ M @ N ) @ ( mult @ A @ R ) )
=> ? [I5: multiset @ A,J3: multiset @ A] :
( ( N
= ( plus_plus @ ( multiset @ A ) @ I5 @ J3 ) )
& ? [K4: multiset @ A] :
( ( M
= ( plus_plus @ ( multiset @ A ) @ I5 @ K4 ) )
& ( J3
!= ( zero_zero @ ( multiset @ A ) ) )
& ! [X6: A] :
( ( member @ A @ X6 @ ( set_mset @ A @ K4 ) )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ ( set_mset @ A @ J3 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X6 @ Xa2 ) @ R ) ) ) ) ) ) ) ).
% mult_implies_one_step
thf(fact_248_subset__mset_Osum__mset__0__iff,axiom,
! [A: $tType,M: multiset @ ( multiset @ A )] :
( ( ( comm_monoid_sum_mset @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ M )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ! [X4: multiset @ A] :
( ( member @ ( multiset @ A ) @ X4 @ ( set_mset @ ( multiset @ A ) @ M ) )
=> ( X4
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% subset_mset.sum_mset_0_iff
thf(fact_249_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_250_transD,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X: A,Y: A,Z: 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 @ Z ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ R ) ) ) ) ).
% transD
thf(fact_251_transE,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X: A,Y: A,Z: 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 @ Z ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ R ) ) ) ) ).
% transE
thf(fact_252_transI,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ! [X3: A,Y3: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z3 ) @ R ) ) )
=> ( trans @ A @ R ) ) ).
% transI
thf(fact_253_trans__def,axiom,
! [A: $tType] :
( ( trans @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [X4: A,Y6: A,Z4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y6 ) @ R5 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ Z4 ) @ R5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Z4 ) @ R5 ) ) ) ) ) ).
% trans_def
thf(fact_254_mult__cancel__add__mset,axiom,
! [A: $tType,S3: set @ ( product_prod @ A @ A ),Uu: A,X5: multiset @ A,Y5: multiset @ A] :
( ( trans @ A @ S3 )
=> ( ( irrefl @ A @ S3 )
=> ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( add_mset @ A @ Uu @ X5 ) @ ( add_mset @ A @ Uu @ Y5 ) ) @ ( mult @ A @ S3 ) )
= ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ X5 @ Y5 ) @ ( mult @ A @ S3 ) ) ) ) ) ).
% mult_cancel_add_mset
% Subclasses (2)
thf(subcl_Orderings_Olinorder___HOL_Otype,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( type @ A ) ) ).
thf(subcl_Orderings_Olinorder___Orderings_Opreorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( preorder @ A ) ) ).
% Type constructors (13)
thf(tcon_Multiset_Omultiset___Orderings_Opreorder,axiom,
! [A9: $tType] :
( ( preorder @ A9 )
=> ( preorder @ ( multiset @ A9 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_1,axiom,
preorder @ $o ).
thf(tcon_Set_Oset___Orderings_Opreorder_2,axiom,
! [A9: $tType] : ( preorder @ ( set @ A9 ) ) ).
thf(tcon_fun___Orderings_Opreorder_3,axiom,
! [A9: $tType,A10: $tType] :
( ( preorder @ A10 )
=> ( preorder @ ( A9 > A10 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o ).
thf(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add,axiom,
! [A9: $tType] :
( ( preorder @ A9 )
=> ( ordere779506340up_add @ ( multiset @ A9 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocancel__comm__monoid__add,axiom,
! [A9: $tType] : ( cancel1352612707id_add @ ( multiset @ A9 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocancel__semigroup__add,axiom,
! [A9: $tType] : ( cancel_semigroup_add @ ( multiset @ A9 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Oab__semigroup__add,axiom,
! [A9: $tType] : ( ab_semigroup_add @ ( multiset @ A9 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add,axiom,
! [A9: $tType] : ( comm_monoid_add @ ( multiset @ A9 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Osemigroup__add,axiom,
! [A9: $tType] : ( semigroup_add @ ( multiset @ A9 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Omonoid__add,axiom,
! [A9: $tType] : ( monoid_add @ ( multiset @ A9 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ozero,axiom,
! [A9: $tType] : ( zero @ ( multiset @ A9 ) ) ).
% Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $true @ X @ Y )
= X ) ).
% Free types (1)
thf(tfree_0,hypothesis,
linorder @ a ).
% Conjectures (1)
thf(conj_0,conjecture,
( v_a
= ( product_fst @ a @ ( tree @ a ) @ ( heapIm970386777veLeaf @ a @ ( t @ a @ v1 @ l1 @ r1 ) ) ) ) ).
%------------------------------------------------------------------------------