TPTP Problem File: ITP071^2.p
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%------------------------------------------------------------------------------
% File : ITP071^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer HeapImperative problem prob_933__5350368_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_933__5350368_1 [Des21]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.25 v7.5.0
% Syntax : Number of formulae : 335 ( 131 unt; 57 typ; 0 def)
% Number of atoms : 673 ( 284 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 5517 ( 81 ~; 5 |; 52 &;5016 @)
% ( 0 <=>; 363 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 9 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 527 ( 527 >; 0 *; 0 +; 0 <<)
% Number of symbols : 59 ( 56 usr; 4 con; 0-9 aty)
% Number of variables : 1594 ( 140 ^;1360 !; 7 ?;1594 :)
% ( 87 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:16:59.117
%------------------------------------------------------------------------------
% 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 (51)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
bNF_Ca1785829860lChain:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
thf(sy_c_BNF__Def_Ocsquare,type,
bNF_csquare:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( set @ A ) > ( B > C ) > ( D > C ) > ( A > B ) > ( A > D ) > $o ) ).
thf(sy_c_BNF__Def_OfstOp,type,
bNF_fstOp:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > ( product_prod @ A @ C ) > ( product_prod @ A @ B ) ) ).
thf(sy_c_BNF__Def_Opick__middlep,type,
bNF_pick_middlep:
!>[B: $tType,A: $tType,C: $tType] : ( ( B > A > $o ) > ( A > C > $o ) > B > C > A ) ).
thf(sy_c_BNF__Def_OsndOp,type,
bNF_sndOp:
!>[C: $tType,A: $tType,B: $tType] : ( ( C > A > $o ) > ( A > B > $o ) > ( product_prod @ C @ B ) > ( product_prod @ A @ B ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
bNF_Greatest_image2:
!>[C: $tType,A: $tType,B: $tType] : ( ( set @ C ) > ( C > A ) > ( C > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Fun_Ocomp,type,
comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
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__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_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_Oset__mset,type,
set_mset:
!>[A: $tType] : ( ( multiset @ A ) > ( set @ A ) ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oorder__class_OGreatest,type,
order_Greatest:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_Oapfst,type,
product_apfst:
!>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( product_prod @ A @ B ) > ( product_prod @ C @ B ) ) ).
thf(sy_c_Product__Type_Oapsnd,type,
product_apsnd:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( product_prod @ A @ B ) > ( product_prod @ A @ C ) ) ).
thf(sy_c_Product__Type_Ocurry,type,
product_curry:
!>[A: $tType,B: $tType,C: $tType] : ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
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_Ocase__prod,type,
product_case_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
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_OPowp,type,
powp:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Relation_Orelcompp,type,
relcompp:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > A > C > $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_t,type,
t2: tree @ a ).
thf(sy_v_v____,type,
v: a ).
% Relevant facts (256)
thf(fact_0_assms_I2_J,axiom,
( t2
!= ( e @ a ) ) ).
% assms(2)
thf(fact_1_assms_I1_J,axiom,
is_heap @ a @ t2 ).
% assms(1)
thf(fact_2__C1_Oprems_C_I2_J,axiom,
( ( t @ a @ v @ ( e @ a ) @ ( e @ a ) )
!= ( e @ a ) ) ).
% "1.prems"(2)
thf(fact_3__C1_Oprems_C_I1_J,axiom,
is_heap @ a @ ( t @ a @ v @ ( e @ a ) @ ( e @ a ) ) ).
% "1.prems"(1)
thf(fact_4_siftDown_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: tree @ A] :
( ( X
!= ( e @ A ) )
=> ( ! [V: A] :
( X
!= ( t @ A @ V @ ( e @ A ) @ ( e @ A ) ) )
=> ( ! [V: A,Va: A,Vb: tree @ A,Vc: tree @ A] :
( X
!= ( t @ A @ V @ ( t @ A @ Va @ Vb @ Vc ) @ ( e @ A ) ) )
=> ( ! [V: A,Va: A,Vb: tree @ A,Vc: tree @ A] :
( X
!= ( t @ A @ V @ ( e @ A ) @ ( t @ A @ Va @ Vb @ Vc ) ) )
=> ~ ! [V: A,Va: A,Vb: tree @ A,Vc: tree @ A,Vd: A,Ve: tree @ A,Vf: tree @ A] :
( X
!= ( t @ A @ V @ ( t @ A @ Va @ Vb @ Vc ) @ ( t @ A @ Vd @ Ve @ Vf ) ) ) ) ) ) ) ) ).
% siftDown.cases
thf(fact_5_removeLeaf_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: tree @ A] :
( ! [V: A] :
( X
!= ( t @ A @ V @ ( e @ A ) @ ( e @ A ) ) )
=> ( ! [V: A,Va: A,Vb: tree @ A,Vc: tree @ A] :
( X
!= ( t @ A @ V @ ( t @ A @ Va @ Vb @ Vc ) @ ( e @ A ) ) )
=> ( ! [V: A,Va: A,Vb: tree @ A,Vc: tree @ A] :
( X
!= ( t @ A @ V @ ( e @ A ) @ ( t @ A @ Va @ Vb @ Vc ) ) )
=> ( ! [V: A,Va: A,Vb: tree @ A,Vc: tree @ A,Vd: A,Ve: tree @ A,Vf: tree @ A] :
( X
!= ( t @ A @ V @ ( t @ A @ Va @ Vb @ Vc ) @ ( t @ A @ Vd @ Ve @ Vf ) ) )
=> ( ! [V: A,Vd: A,Ve: tree @ A,Vf: tree @ A,Va: A,Vb: tree @ A,Vc: tree @ A] :
( X
!= ( t @ A @ V @ ( t @ A @ Vd @ Ve @ Vf ) @ ( t @ A @ Va @ Vb @ Vc ) ) )
=> ( X
= ( e @ A ) ) ) ) ) ) ) ) ).
% removeLeaf.cases
thf(fact_6_removeLeaf_Oinduct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: ( tree @ A ) > $o,A0: tree @ A] :
( ! [V: A] : ( P @ ( t @ A @ V @ ( e @ A ) @ ( e @ A ) ) )
=> ( ! [V: A,Va: A,Vb: tree @ A,Vc: tree @ A] :
( ( P @ ( t @ A @ Va @ Vb @ Vc ) )
=> ( ( P @ ( t @ A @ Va @ Vb @ Vc ) )
=> ( P @ ( t @ A @ V @ ( t @ A @ Va @ Vb @ Vc ) @ ( e @ A ) ) ) ) )
=> ( ! [V: A,Va: A,Vb: tree @ A,Vc: tree @ A] :
( ( P @ ( t @ A @ Va @ Vb @ Vc ) )
=> ( ( P @ ( t @ A @ Va @ Vb @ Vc ) )
=> ( P @ ( t @ A @ V @ ( e @ A ) @ ( t @ A @ Va @ Vb @ Vc ) ) ) ) )
=> ( ! [V: A,Va: A,Vb: tree @ A,Vc: tree @ A,Vd: A,Ve: tree @ A,Vf: tree @ A] :
( ( P @ ( t @ A @ Va @ Vb @ Vc ) )
=> ( ( P @ ( t @ A @ Va @ Vb @ Vc ) )
=> ( P @ ( t @ A @ V @ ( t @ A @ Va @ Vb @ Vc ) @ ( t @ A @ Vd @ Ve @ Vf ) ) ) ) )
=> ( ! [V: A,Vd: A,Ve: tree @ A,Vf: tree @ A,Va: A,Vb: tree @ A,Vc: tree @ A] :
( ( P @ ( t @ A @ Vd @ Ve @ Vf ) )
=> ( ( P @ ( t @ A @ Vd @ Ve @ Vf ) )
=> ( P @ ( t @ A @ V @ ( t @ A @ Vd @ Ve @ Vf ) @ ( t @ A @ Va @ Vb @ Vc ) ) ) ) )
=> ( ( P @ ( e @ A ) )
=> ( P @ A0 ) ) ) ) ) ) ) ) ).
% removeLeaf.induct
thf(fact_7_is__heap_Osimps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [V2: A] : ( is_heap @ A @ ( t @ A @ V2 @ ( e @ A ) @ ( e @ A ) ) ) ) ).
% is_heap.simps(2)
thf(fact_8_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_9_hs__is__empty__def,axiom,
! [A: $tType] :
( ( heapIm721255937_empty @ A )
= ( ^ [T2: tree @ A] :
( T2
= ( e @ A ) ) ) ) ).
% hs_is_empty_def
thf(fact_10_is__heap_Osimps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( is_heap @ A @ ( e @ A ) ) ) ).
% is_heap.simps(1)
thf(fact_11_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_12_Tree_Oinduct,axiom,
! [A: $tType,P: ( tree @ A ) > $o,Tree: tree @ A] :
( ( P @ ( e @ A ) )
=> ( ! [X1: A,X2: tree @ A,X3: tree @ A] :
( ( P @ X2 )
=> ( ( P @ X3 )
=> ( P @ ( t @ A @ X1 @ X2 @ X3 ) ) ) )
=> ( P @ Tree ) ) ) ).
% Tree.induct
thf(fact_13_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_14_is__heap_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: tree @ A] :
( ( X
!= ( e @ A ) )
=> ( ! [V: A] :
( X
!= ( t @ A @ V @ ( e @ A ) @ ( e @ A ) ) )
=> ( ! [V: A,Va: A,Vb: tree @ A,Vc: tree @ A] :
( X
!= ( t @ A @ V @ ( e @ A ) @ ( t @ A @ Va @ Vb @ Vc ) ) )
=> ( ! [V: A,Va: A,Vb: tree @ A,Vc: tree @ A] :
( X
!= ( t @ A @ V @ ( t @ A @ Va @ Vb @ Vc ) @ ( e @ A ) ) )
=> ~ ! [V: A,Va: A,Vb: tree @ A,Vc: tree @ A,Vd: A,Ve: tree @ A,Vf: tree @ A] :
( X
!= ( t @ A @ V @ ( t @ A @ Va @ Vb @ Vc ) @ ( t @ A @ Vd @ Ve @ Vf ) ) ) ) ) ) ) ) ).
% is_heap.cases
thf(fact_15_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_16_removeLeaf_Osimps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [V2: A] :
( ( heapIm970386777veLeaf @ A @ ( t @ A @ V2 @ ( e @ A ) @ ( e @ A ) ) )
= ( product_Pair @ A @ ( tree @ A ) @ V2 @ ( e @ A ) ) ) ) ).
% removeLeaf.simps(1)
thf(fact_17_siftDown__heap__is__heap,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: tree @ A,R: tree @ A,T3: tree @ A,V2: A] :
( ( is_heap @ A @ L )
=> ( ( is_heap @ A @ R )
=> ( ( T3
= ( t @ A @ V2 @ L @ R ) )
=> ( is_heap @ A @ ( heapIm748920189ftDown @ A @ T3 ) ) ) ) ) ) ).
% siftDown_heap_is_heap
thf(fact_18_siftDown_Osimps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [V2: A] :
( ( heapIm748920189ftDown @ A @ ( t @ A @ V2 @ ( e @ A ) @ ( e @ A ) ) )
= ( t @ A @ V2 @ ( e @ A ) @ ( e @ A ) ) ) ) ).
% siftDown.simps(2)
thf(fact_19_val_Osimps,axiom,
! [A: $tType,V2: A,Uu: tree @ A,Uv: tree @ A] :
( ( val @ A @ ( t @ A @ V2 @ Uu @ Uv ) )
= V2 ) ).
% val.simps
thf(fact_20_siftDown_Osimps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( heapIm748920189ftDown @ A @ ( e @ A ) )
= ( e @ A ) ) ) ).
% siftDown.simps(1)
thf(fact_21_prod_Oinject,axiom,
! [A: $tType,B: $tType,X12: A,X24: B,Y1: A,Y2: B] :
( ( ( product_Pair @ A @ B @ X12 @ X24 )
= ( product_Pair @ A @ B @ Y1 @ Y2 ) )
= ( ( X12 = Y1 )
& ( X24 = Y2 ) ) ) ).
% prod.inject
thf(fact_22_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_23_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_24_eq__snd__iff,axiom,
! [A: $tType,B: $tType,B2: A,P2: product_prod @ B @ A] :
( ( B2
= ( product_snd @ B @ A @ P2 ) )
= ( ? [A4: B] :
( P2
= ( product_Pair @ B @ A @ A4 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_25_snd__conv,axiom,
! [Aa: $tType,A: $tType,X12: Aa,X24: A] :
( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X12 @ X24 ) )
= X24 ) ).
% snd_conv
thf(fact_26_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_27_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_28_removeLeaf_Osimps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [V2: A,Va2: A,Vb2: tree @ A,Vc2: tree @ A] :
( ( heapIm970386777veLeaf @ A @ ( t @ A @ V2 @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( e @ A ) ) )
= ( product_Pair @ A @ ( tree @ A ) @ ( product_fst @ A @ ( tree @ A ) @ ( heapIm970386777veLeaf @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) ) @ ( t @ A @ V2 @ ( product_snd @ A @ ( tree @ A ) @ ( heapIm970386777veLeaf @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) ) @ ( e @ A ) ) ) ) ) ).
% removeLeaf.simps(2)
thf(fact_29_removeLeaf_Osimps_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [V2: A,Va2: A,Vb2: tree @ A,Vc2: tree @ A] :
( ( heapIm970386777veLeaf @ A @ ( t @ A @ V2 @ ( e @ A ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) )
= ( product_Pair @ A @ ( tree @ A ) @ ( product_fst @ A @ ( tree @ A ) @ ( heapIm970386777veLeaf @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) ) @ ( t @ A @ V2 @ ( e @ A ) @ ( product_snd @ A @ ( tree @ A ) @ ( heapIm970386777veLeaf @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ).
% removeLeaf.simps(3)
thf(fact_30_removeLeaf_Osimps_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [V2: A,Va2: A,Vb2: tree @ A,Vc2: tree @ A,Vd2: A,Ve2: tree @ A,Vf2: tree @ A] :
( ( heapIm970386777veLeaf @ A @ ( t @ A @ V2 @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) )
= ( product_Pair @ A @ ( tree @ A ) @ ( product_fst @ A @ ( tree @ A ) @ ( heapIm970386777veLeaf @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) ) @ ( t @ A @ V2 @ ( product_snd @ A @ ( tree @ A ) @ ( heapIm970386777veLeaf @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) ) @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) ) ) ) ).
% removeLeaf.simps(4)
thf(fact_31_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_32_eq__fst__iff,axiom,
! [A: $tType,B: $tType,A2: A,P2: product_prod @ A @ B] :
( ( A2
= ( product_fst @ A @ B @ P2 ) )
= ( ? [B4: B] :
( P2
= ( product_Pair @ A @ B @ A2 @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_33_fst__conv,axiom,
! [B: $tType,A: $tType,X12: A,X24: B] :
( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X12 @ X24 ) )
= X12 ) ).
% fst_conv
thf(fact_34_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_35_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_36_prod__eqI,axiom,
! [B: $tType,A: $tType,P2: product_prod @ A @ B,Q: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ P2 )
= ( product_fst @ A @ B @ Q ) )
=> ( ( ( product_snd @ A @ B @ P2 )
= ( product_snd @ A @ B @ Q ) )
=> ( P2 = Q ) ) ) ).
% prod_eqI
thf(fact_37_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_38_prod__eq__iff,axiom,
! [B: $tType,A: $tType] :
( ( ^ [Y3: product_prod @ A @ B,Z2: product_prod @ A @ B] : Y3 = Z2 )
= ( ^ [S: product_prod @ A @ B,T2: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ S )
= ( product_fst @ A @ B @ T2 ) )
& ( ( product_snd @ A @ B @ S )
= ( product_snd @ A @ B @ T2 ) ) ) ) ) ).
% prod_eq_iff
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_in__tree_Osimps_I2_J,axiom,
! [A: $tType,V2: A,V3: A,L: tree @ A,R: tree @ A] :
( ( in_tree @ A @ V2 @ ( t @ A @ V3 @ L @ R ) )
= ( ( V2 = V3 )
| ( in_tree @ A @ V2 @ L )
| ( in_tree @ A @ V2 @ R ) ) ) ).
% in_tree.simps(2)
thf(fact_42_in__tree_Osimps_I1_J,axiom,
! [A: $tType,V2: A] :
~ ( in_tree @ A @ V2 @ ( e @ A ) ) ).
% in_tree.simps(1)
thf(fact_43_siftDown__in__tree__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( in_tree @ A )
= ( ^ [V4: A,T2: tree @ A] : ( in_tree @ A @ V4 @ ( heapIm748920189ftDown @ A @ T2 ) ) ) ) ) ).
% siftDown_in_tree_set
thf(fact_44_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A5: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B5 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A6: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A6 ) )
= A6 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q2: A > $o] :
( ! [X5: A] :
( ( P @ X5 )
= ( Q2 @ X5 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q2 ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X5: A] :
( ( F @ X5 )
= ( G @ X5 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A5: A,B5: B] :
( Y
!= ( product_Pair @ A @ B @ A5 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_50_prod__induct7,axiom,
! [G2: $tType,F2: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) )] :
( ! [A5: A,B5: B,C2: C,D2: D,E2: E,F3: F2,G3: G2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F2 @ G2 ) @ E2 @ ( product_Pair @ F2 @ G2 @ F3 @ G3 ) ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct7
thf(fact_51_prod__induct6,axiom,
! [F2: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) )] :
( ! [A5: A,B5: B,C2: C,D2: D,E2: E,F3: F2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F2 ) @ D2 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct6
thf(fact_52_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
( ! [A5: A,B5: B,C2: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct5
thf(fact_53_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A5: A,B5: B,C2: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
=> ( P @ X ) ) ).
% prod_induct4
thf(fact_54_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A5: A,B5: B,C2: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C2 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_55_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,G2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) )] :
~ ! [A5: A,B5: B,C2: C,D2: D,E2: E,F3: F2,G3: G2] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F2 @ G2 ) @ E2 @ ( product_Pair @ F2 @ G2 @ F3 @ G3 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_56_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) )] :
~ ! [A5: A,B5: B,C2: C,D2: D,E2: E,F3: F2] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F2 ) @ D2 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_57_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 ) ) )] :
~ ! [A5: A,B5: B,C2: C,D2: D,E2: E] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_58_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A5: A,B5: B,C2: C,D2: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_59_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A5: A,B5: B,C2: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C2 ) ) ) ).
% prod_cases3
thf(fact_60_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_61_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
( ! [A5: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B5 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_62_surj__pair,axiom,
! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
? [X5: A,Y4: B] :
( P2
= ( product_Pair @ A @ B @ X5 @ Y4 ) ) ).
% surj_pair
thf(fact_63_removeLeaf_Osimps_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [V2: A,Vd2: A,Ve2: tree @ A,Vf2: tree @ A,Va2: A,Vb2: tree @ A,Vc2: tree @ A] :
( ( heapIm970386777veLeaf @ A @ ( t @ A @ V2 @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) )
= ( product_Pair @ A @ ( tree @ A ) @ ( product_fst @ A @ ( tree @ A ) @ ( heapIm970386777veLeaf @ A @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) ) @ ( t @ A @ V2 @ ( product_snd @ A @ ( tree @ A ) @ ( heapIm970386777veLeaf @ A @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ).
% removeLeaf.simps(5)
thf(fact_64_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_65_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [B: $tType,A: $tType,P: A > B > $o,X: A,Y: B,A2: product_prod @ A @ B] :
( ( P @ X @ Y )
=> ( ( A2
= ( product_Pair @ A @ B @ X @ Y ) )
=> ( P @ ( product_fst @ A @ B @ A2 ) @ ( product_snd @ A @ B @ A2 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_66_conjI__realizer,axiom,
! [A: $tType,B: $tType,P: A > $o,P2: A,Q2: B > $o,Q: B] :
( ( P @ P2 )
=> ( ( Q2 @ Q )
=> ( ( P @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q ) ) )
& ( Q2 @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_67_exI__realizer,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Y: A,X: B] :
( ( P @ Y @ X )
=> ( P @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_68_exE__realizer_H,axiom,
! [A: $tType,B: $tType,P: A > B > $o,P2: product_prod @ B @ A] :
( ( P @ ( product_snd @ B @ A @ P2 ) @ ( product_fst @ B @ A @ P2 ) )
=> ~ ! [X5: B,Y4: A] :
~ ( P @ Y4 @ X5 ) ) ).
% exE_realizer'
thf(fact_69_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_70_heapify_Osimps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [V2: A,L: tree @ A,R: tree @ A] :
( ( heapIm818251801eapify @ A @ ( t @ A @ V2 @ L @ R ) )
= ( heapIm748920189ftDown @ A @ ( t @ A @ V2 @ ( heapIm818251801eapify @ A @ L ) @ ( heapIm818251801eapify @ A @ R ) ) ) ) ) ).
% heapify.simps(2)
thf(fact_71_is__heap_Osimps_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [V2: A,Va2: A,Vb2: tree @ A,Vc2: tree @ A] :
( ( is_heap @ A @ ( t @ A @ V2 @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( e @ A ) ) )
= ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) @ V2 )
& ( is_heap @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ).
% is_heap.simps(4)
thf(fact_72_is__heap_Osimps_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [V2: A,Va2: A,Vb2: tree @ A,Vc2: tree @ A] :
( ( is_heap @ A @ ( t @ A @ V2 @ ( e @ A ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) )
= ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) @ V2 )
& ( is_heap @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ).
% is_heap.simps(3)
thf(fact_73_swap__swap,axiom,
! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P2 ) )
= P2 ) ).
% swap_swap
thf(fact_74_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_75_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_76_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_77_siftDown__Node,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T3: tree @ A,V2: A,L: tree @ A,R: tree @ A] :
( ( T3
= ( t @ A @ V2 @ L @ R ) )
=> ? [L2: tree @ A,V5: A,R2: tree @ A] :
( ( ( heapIm748920189ftDown @ A @ T3 )
= ( t @ A @ V5 @ L2 @ R2 ) )
& ( ord_less_eq @ A @ V2 @ V5 ) ) ) ) ).
% siftDown_Node
thf(fact_78_heapify_Osimps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( heapIm818251801eapify @ A @ ( e @ A ) )
= ( e @ A ) ) ) ).
% heapify.simps(1)
thf(fact_79_heapify__heap__is__heap,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T3: tree @ A] : ( is_heap @ A @ ( heapIm818251801eapify @ A @ T3 ) ) ) ).
% heapify_heap_is_heap
thf(fact_80_is__heap_Osimps_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [V2: A,Va2: A,Vb2: tree @ A,Vc2: tree @ A,Vd2: A,Ve2: tree @ A,Vf2: tree @ A] :
( ( is_heap @ A @ ( t @ A @ V2 @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) )
= ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) @ V2 )
& ( is_heap @ A @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) )
& ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) @ V2 )
& ( is_heap @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ).
% is_heap.simps(5)
thf(fact_81_is__heap_Osimps_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [V2: A,Vd2: A,Ve2: tree @ A,Vf2: tree @ A,Va2: A,Vb2: tree @ A,Vc2: tree @ A] :
( ( is_heap @ A @ ( t @ A @ V2 @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) )
= ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) @ V2 )
& ( is_heap @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) )
& ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) @ V2 )
& ( is_heap @ A @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) ) ) ) ).
% is_heap.simps(6)
thf(fact_82_is__heap__max,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [V2: A,T3: tree @ A] :
( ( in_tree @ A @ V2 @ T3 )
=> ( ( is_heap @ A @ T3 )
=> ( ord_less_eq @ A @ V2 @ ( val @ A @ T3 ) ) ) ) ) ).
% is_heap_max
thf(fact_83_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_84_siftDown_Osimps_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Va2: A,Vb2: tree @ A,Vc2: tree @ A,V2: A] :
( ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) @ V2 )
=> ( ( heapIm748920189ftDown @ A @ ( t @ A @ V2 @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( e @ A ) ) )
= ( t @ A @ V2 @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( e @ A ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) @ V2 )
=> ( ( heapIm748920189ftDown @ A @ ( t @ A @ V2 @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( e @ A ) ) )
= ( t @ A @ ( val @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) @ ( heapIm748920189ftDown @ A @ ( t @ A @ V2 @ ( heapIm1271749598e_left @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) @ ( heapIm1434396069_right @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) ) ) @ ( e @ A ) ) ) ) ) ) ).
% siftDown.simps(3)
thf(fact_85_siftDown_Osimps_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Va2: A,Vb2: tree @ A,Vc2: tree @ A,V2: A] :
( ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) @ V2 )
=> ( ( heapIm748920189ftDown @ A @ ( t @ A @ V2 @ ( e @ A ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) )
= ( t @ A @ V2 @ ( e @ A ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) @ V2 )
=> ( ( heapIm748920189ftDown @ A @ ( t @ A @ V2 @ ( e @ A ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) )
= ( t @ A @ ( val @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) @ ( e @ A ) @ ( heapIm748920189ftDown @ A @ ( t @ A @ V2 @ ( heapIm1271749598e_left @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) @ ( heapIm1434396069_right @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ).
% siftDown.simps(4)
thf(fact_86_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_87_siftDown_Osimps_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Va2: A,Vb2: tree @ A,Vc2: tree @ A,Vd2: A,Ve2: tree @ A,Vf2: tree @ A,V2: A] :
( ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) @ ( val @ A @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) )
=> ( ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) @ V2 )
=> ( ( heapIm748920189ftDown @ A @ ( t @ A @ V2 @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) )
= ( t @ A @ V2 @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) @ V2 )
=> ( ( heapIm748920189ftDown @ A @ ( t @ A @ V2 @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) )
= ( t @ A @ ( val @ A @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) @ ( heapIm748920189ftDown @ A @ ( t @ A @ V2 @ ( heapIm1271749598e_left @ A @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) @ ( heapIm1434396069_right @ A @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) ) ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) @ ( val @ A @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) )
=> ( ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) @ V2 )
=> ( ( heapIm748920189ftDown @ A @ ( t @ A @ V2 @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) )
= ( t @ A @ V2 @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) @ V2 )
=> ( ( heapIm748920189ftDown @ A @ ( t @ A @ V2 @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) )
= ( t @ A @ ( val @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) @ ( heapIm748920189ftDown @ A @ ( t @ A @ V2 @ ( heapIm1271749598e_left @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) @ ( heapIm1434396069_right @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ) ).
% siftDown.simps(6)
thf(fact_88_siftDown_Osimps_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Vd2: A,Ve2: tree @ A,Vf2: tree @ A,Va2: A,Vb2: tree @ A,Vc2: tree @ A,V2: A] :
( ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) @ ( val @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) )
=> ( ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) @ V2 )
=> ( ( heapIm748920189ftDown @ A @ ( t @ A @ V2 @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) )
= ( t @ A @ V2 @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) @ V2 )
=> ( ( heapIm748920189ftDown @ A @ ( t @ A @ V2 @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) )
= ( t @ A @ ( val @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) @ ( heapIm748920189ftDown @ A @ ( t @ A @ V2 @ ( heapIm1271749598e_left @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) @ ( heapIm1434396069_right @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) ) ) @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) @ ( val @ A @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) ) )
=> ( ( ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) @ V2 )
=> ( ( heapIm748920189ftDown @ A @ ( t @ A @ V2 @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) )
= ( t @ A @ V2 @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( val @ A @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) @ V2 )
=> ( ( heapIm748920189ftDown @ A @ ( t @ A @ V2 @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) )
= ( t @ A @ ( val @ A @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) @ ( t @ A @ Va2 @ Vb2 @ Vc2 ) @ ( heapIm748920189ftDown @ A @ ( t @ A @ V2 @ ( heapIm1271749598e_left @ A @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) @ ( heapIm1434396069_right @ A @ ( t @ A @ Vd2 @ Ve2 @ Vf2 ) ) ) ) ) ) ) ) ) ) ) ).
% siftDown.simps(5)
thf(fact_89_left_Osimps,axiom,
! [A: $tType,V2: A,L: tree @ A,R: tree @ A] :
( ( heapIm1271749598e_left @ A @ ( t @ A @ V2 @ L @ R ) )
= L ) ).
% left.simps
thf(fact_90_right_Osimps,axiom,
! [A: $tType,V2: A,L: tree @ A,R: tree @ A] :
( ( heapIm1434396069_right @ A @ ( t @ A @ V2 @ L @ R ) )
= R ) ).
% right.simps
thf(fact_91_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% dual_order.antisym
thf(fact_92_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y3: A,Z2: A] : Y3 = Z2 )
= ( ^ [A4: A,B4: A] :
( ( ord_less_eq @ A @ B4 @ A4 )
& ( ord_less_eq @ A @ A4 @ B4 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_93_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C3 @ B2 )
=> ( ord_less_eq @ A @ C3 @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_94_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A5: A,B5: A] :
( ( ord_less_eq @ A @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: A,B5: A] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_95_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_96_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% order_trans
thf(fact_97_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ) ).
% order_class.order.antisym
thf(fact_98_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( B2 = C3 )
=> ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_99_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_100_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y3: A,Z2: A] : Y3 = Z2 )
= ( ^ [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
& ( ord_less_eq @ A @ B4 @ A4 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_101_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv
thf(fact_102_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z ) )
=> ( ( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less_eq @ A @ X @ Z ) )
=> ( ( ( ord_less_eq @ A @ X @ Z )
=> ~ ( ord_less_eq @ A @ Z @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z )
=> ~ ( ord_less_eq @ A @ Z @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z @ X )
=> ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_103_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% order.trans
thf(fact_104_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% le_cases
thf(fact_105_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( X = Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% eq_refl
thf(fact_106_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linear
thf(fact_107_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( X = Y ) ) ) ) ).
% antisym
thf(fact_108_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y3: A,Z2: A] : Y3 = Z2 )
= ( ^ [X4: A,Y5: A] :
( ( ord_less_eq @ A @ X4 @ Y5 )
& ( ord_less_eq @ A @ Y5 @ X4 ) ) ) ) ) ).
% eq_iff
thf(fact_109_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B2: A,F: A > B,C3: B] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C3 )
=> ( ! [X5: A,Y4: A] :
( ( ord_less_eq @ A @ X5 @ Y4 )
=> ( ord_less_eq @ B @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ B @ ( F @ A2 ) @ C3 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_110_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,F: B > A,B2: B,C3: B] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C3 )
=> ( ! [X5: B,Y4: B] :
( ( ord_less_eq @ B @ X5 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C3 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_111_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F: A > C,C3: C] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C @ ( F @ B2 ) @ C3 )
=> ( ! [X5: A,Y4: A] :
( ( ord_less_eq @ A @ X5 @ Y4 )
=> ( ord_less_eq @ C @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ C @ ( F @ A2 ) @ C3 ) ) ) ) ) ).
% order_subst2
thf(fact_112_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F: B > A,B2: B,C3: B] :
( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C3 )
=> ( ! [X5: B,Y4: B] :
( ( ord_less_eq @ B @ X5 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C3 ) ) ) ) ) ) ).
% order_subst1
thf(fact_113_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F4: A > B,G4: A > B] :
! [X4: A] : ( ord_less_eq @ B @ ( F4 @ X4 ) @ ( G4 @ X4 ) ) ) ) ) ).
% le_fun_def
thf(fact_114_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B] :
( ! [X5: A] : ( ord_less_eq @ B @ ( F @ X5 ) @ ( G @ X5 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_115_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funE
thf(fact_116_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funD
thf(fact_117_hs__remove__max__def,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( heapIm1542349758ve_max @ A )
= ( ^ [T2: tree @ A] :
( if @ ( product_prod @ A @ ( tree @ A ) )
@ ( ( product_snd @ A @ ( tree @ A ) @ ( heapIm970386777veLeaf @ A @ T2 ) )
= ( e @ A ) )
@ ( product_Pair @ A @ ( tree @ A ) @ ( val @ A @ T2 ) @ ( e @ A ) )
@ ( product_Pair @ A @ ( tree @ A ) @ ( val @ A @ T2 ) @ ( heapIm748920189ftDown @ A @ ( t @ A @ ( product_fst @ A @ ( tree @ A ) @ ( heapIm970386777veLeaf @ A @ T2 ) ) @ ( heapIm1271749598e_left @ A @ ( product_snd @ A @ ( tree @ A ) @ ( heapIm970386777veLeaf @ A @ T2 ) ) ) @ ( heapIm1434396069_right @ A @ ( product_snd @ A @ ( tree @ A ) @ ( heapIm970386777veLeaf @ A @ T2 ) ) ) ) ) ) ) ) ) ) ).
% hs_remove_max_def
thf(fact_118_relChain__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( bNF_Ca1785829860lChain @ A @ B )
= ( ^ [R3: set @ ( product_prod @ A @ A ),As: A > B] :
! [I: A,J: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R3 )
=> ( ord_less_eq @ B @ ( As @ I ) @ ( As @ J ) ) ) ) ) ) ).
% relChain_def
thf(fact_119_Heap_Oremove__max__is__heap,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),As_tree: B > ( tree @ A ),Remove_max: B > ( product_prod @ A @ B ),L: B,M: A,L3: B] :
( ( heap @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max )
=> ( ~ ( Is_empty @ L )
=> ( ( is_heap @ A @ ( As_tree @ L ) )
=> ( ( ( product_Pair @ A @ B @ M @ L3 )
= ( Remove_max @ L ) )
=> ( is_heap @ A @ ( As_tree @ L3 ) ) ) ) ) ) ) ).
% Heap.remove_max_is_heap
thf(fact_120_Heap_Oremove__max__val,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),As_tree: B > ( tree @ A ),Remove_max: B > ( product_prod @ A @ B ),T3: B,M: A,T4: B] :
( ( heap @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max )
=> ( ~ ( Is_empty @ T3 )
=> ( ( ( product_Pair @ A @ B @ M @ T4 )
= ( Remove_max @ T3 ) )
=> ( M
= ( val @ A @ ( As_tree @ T3 ) ) ) ) ) ) ) ).
% Heap.remove_max_val
thf(fact_121_sndOp__def,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( bNF_sndOp @ C @ A @ B )
= ( ^ [P4: C > A > $o,Q3: A > B > $o,Ac: product_prod @ C @ B] : ( product_Pair @ A @ B @ ( bNF_pick_middlep @ C @ A @ B @ P4 @ Q3 @ ( product_fst @ C @ B @ Ac ) @ ( product_snd @ C @ B @ Ac ) ) @ ( product_snd @ C @ B @ Ac ) ) ) ) ).
% sndOp_def
thf(fact_122_fstOp__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( bNF_fstOp @ A @ B @ C )
= ( ^ [P4: A > B > $o,Q3: B > C > $o,Ac: product_prod @ A @ C] : ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Ac ) @ ( bNF_pick_middlep @ A @ B @ C @ P4 @ Q3 @ ( product_fst @ A @ C @ Ac ) @ ( product_snd @ A @ C @ Ac ) ) ) ) ) ).
% fstOp_def
thf(fact_123_Heap_Oas__tree__empty,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),As_tree: B > ( tree @ A ),Remove_max: B > ( product_prod @ A @ B ),T3: B] :
( ( heap @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max )
=> ( ( ( As_tree @ T3 )
= ( e @ A ) )
= ( Is_empty @ T3 ) ) ) ) ).
% Heap.as_tree_empty
thf(fact_124_Heap_Ois__heap__of__list,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),As_tree: B > ( tree @ A ),Remove_max: B > ( product_prod @ A @ B ),I2: list @ A] :
( ( heap @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max )
=> ( is_heap @ A @ ( As_tree @ ( Of_list @ I2 ) ) ) ) ) ).
% Heap.is_heap_of_list
thf(fact_125_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_126_pred__subset__eq2,axiom,
! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( A > B > $o )
@ ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R4 )
@ ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ S2 ) )
= ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R4 @ S2 ) ) ).
% pred_subset_eq2
thf(fact_127_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R4 ) )
= ( ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ S2 ) ) )
= ( R4 = S2 ) ) ).
% pred_equals_eq2
thf(fact_128_csquare__fstOp__sndOp,axiom,
! [A: $tType,B: $tType,C: $tType,F: ( A > B > $o ) > ( product_prod @ A @ B ) > $o,P: A > C > $o,Q2: C > B > $o] : ( bNF_csquare @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ C @ ( product_prod @ C @ B ) @ ( collect @ ( product_prod @ A @ B ) @ ( F @ ( relcompp @ A @ C @ B @ P @ Q2 ) ) ) @ ( product_snd @ A @ C ) @ ( product_fst @ C @ B ) @ ( bNF_fstOp @ A @ C @ B @ P @ Q2 ) @ ( bNF_sndOp @ A @ C @ B @ P @ Q2 ) ) ).
% csquare_fstOp_sndOp
thf(fact_129_predicate2I,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Q2: A > B > $o] :
( ! [X5: A,Y4: B] :
( ( P @ X5 @ Y4 )
=> ( Q2 @ X5 @ Y4 ) )
=> ( ord_less_eq @ ( A > B > $o ) @ P @ Q2 ) ) ).
% predicate2I
thf(fact_130_predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q2: A > B > $o,X: A,Y: B] :
( ( ord_less_eq @ ( A > B > $o ) @ P @ Q2 )
=> ( ( P @ X @ Y )
=> ( Q2 @ X @ Y ) ) ) ).
% predicate2D
thf(fact_131_rev__predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,X: A,Y: B,Q2: A > B > $o] :
( ( P @ X @ Y )
=> ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q2 )
=> ( Q2 @ X @ Y ) ) ) ).
% rev_predicate2D
thf(fact_132_scomp__scomp,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F2: $tType,E: $tType,F: A > ( product_prod @ E @ F2 ),G: E > F2 > ( product_prod @ C @ D ),H: C > D > B] :
( ( product_scomp @ A @ C @ D @ B @ ( product_scomp @ A @ E @ F2 @ ( product_prod @ C @ D ) @ F @ G ) @ H )
= ( product_scomp @ A @ E @ F2 @ B @ F
@ ^ [X4: E] : ( product_scomp @ F2 @ C @ D @ B @ ( G @ X4 ) @ H ) ) ) ).
% scomp_scomp
thf(fact_133_subset__Collect__iff,axiom,
! [A: $tType,B6: set @ A,A6: set @ A,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A6 )
& ( P @ X4 ) ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ B6 )
=> ( P @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_134_subset__CollectI,axiom,
! [A: $tType,B6: set @ A,A6: set @ A,Q2: A > $o,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ A6 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ B6 )
=> ( ( Q2 @ X5 )
=> ( P @ X5 ) ) )
=> ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ B6 )
& ( Q2 @ X4 ) ) )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A6 )
& ( P @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_135_pred__subset__eq,axiom,
! [A: $tType,R4: set @ A,S2: set @ A] :
( ( ord_less_eq @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ R4 )
@ ^ [X4: A] : ( member @ A @ X4 @ S2 ) )
= ( ord_less_eq @ ( set @ A ) @ R4 @ S2 ) ) ).
% pred_subset_eq
thf(fact_136_relcompp__mono,axiom,
! [A: $tType,C: $tType,B: $tType,R5: A > B > $o,R: A > B > $o,S3: B > C > $o,S4: B > C > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ R5 @ R )
=> ( ( ord_less_eq @ ( B > C > $o ) @ S3 @ S4 )
=> ( ord_less_eq @ ( A > C > $o ) @ ( relcompp @ A @ B @ C @ R5 @ S3 ) @ ( relcompp @ A @ B @ C @ R @ S4 ) ) ) ) ).
% relcompp_mono
thf(fact_137_leq__OOI,axiom,
! [A: $tType,R4: A > A > $o] :
( ( R4
= ( ^ [Y3: A,Z2: A] : Y3 = Z2 ) )
=> ( ord_less_eq @ ( A > A > $o ) @ R4 @ ( relcompp @ A @ A @ A @ R4 @ R4 ) ) ) ).
% leq_OOI
thf(fact_138_ge__eq__refl,axiom,
! [A: $tType,R4: A > A > $o,X: A] :
( ( ord_less_eq @ ( A > A > $o )
@ ^ [Y3: A,Z2: A] : Y3 = Z2
@ R4 )
=> ( R4 @ X @ X ) ) ).
% ge_eq_refl
thf(fact_139_refl__ge__eq,axiom,
! [A: $tType,R4: A > A > $o] :
( ! [X5: A] : ( R4 @ X5 @ X5 )
=> ( ord_less_eq @ ( A > A > $o )
@ ^ [Y3: A,Z2: A] : Y3 = Z2
@ R4 ) ) ).
% refl_ge_eq
thf(fact_140_pick__middlep,axiom,
! [B: $tType,A: $tType,C: $tType,P: A > B > $o,Q2: B > C > $o,A2: A,C3: C] :
( ( relcompp @ A @ B @ C @ P @ Q2 @ A2 @ C3 )
=> ( ( P @ A2 @ ( bNF_pick_middlep @ A @ B @ C @ P @ Q2 @ A2 @ C3 ) )
& ( Q2 @ ( bNF_pick_middlep @ A @ B @ C @ P @ Q2 @ A2 @ C3 ) @ C3 ) ) ) ).
% pick_middlep
thf(fact_141_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_142_Pair__scomp,axiom,
! [A: $tType,B: $tType,C: $tType,X: C,F: C > A > B] :
( ( product_scomp @ A @ C @ A @ B @ ( product_Pair @ C @ A @ X ) @ F )
= ( F @ X ) ) ).
% Pair_scomp
thf(fact_143_csquare__def,axiom,
! [B: $tType,C: $tType,D: $tType,A: $tType] :
( ( bNF_csquare @ A @ B @ C @ D )
= ( ^ [A7: set @ A,F12: B > C,F22: D > C,P1: A > B,P22: A > D] :
! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( ( F12 @ ( P1 @ X4 ) )
= ( F22 @ ( P22 @ X4 ) ) ) ) ) ) ).
% csquare_def
thf(fact_144_subrelI,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S4: set @ ( product_prod @ A @ B )] :
( ! [X5: A,Y4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y4 ) @ R )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y4 ) @ S4 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S4 ) ) ).
% subrelI
thf(fact_145_subset__antisym,axiom,
! [A: $tType,A6: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ A6 )
=> ( A6 = B6 ) ) ) ).
% subset_antisym
thf(fact_146_subsetI,axiom,
! [A: $tType,A6: set @ A,B6: set @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ A6 )
=> ( member @ A @ X5 @ B6 ) )
=> ( ord_less_eq @ ( set @ A ) @ A6 @ B6 ) ) ).
% subsetI
thf(fact_147_eq__subset,axiom,
! [A: $tType,P: A > A > $o] :
( ord_less_eq @ ( A > A > $o )
@ ^ [Y3: A,Z2: A] : Y3 = Z2
@ ^ [A4: A,B4: A] :
( ( P @ A4 @ B4 )
| ( A4 = B4 ) ) ) ).
% eq_subset
thf(fact_148_Collect__restrict,axiom,
! [A: $tType,X6: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ X6 )
& ( P @ X4 ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_149_predicate1I,axiom,
! [A: $tType,P: A > $o,Q2: A > $o] :
( ! [X5: A] :
( ( P @ X5 )
=> ( Q2 @ X5 ) )
=> ( ord_less_eq @ ( A > $o ) @ P @ Q2 ) ) ).
% predicate1I
thf(fact_150_predicate1D,axiom,
! [A: $tType,P: A > $o,Q2: A > $o,X: A] :
( ( ord_less_eq @ ( A > $o ) @ P @ Q2 )
=> ( ( P @ X )
=> ( Q2 @ X ) ) ) ).
% predicate1D
thf(fact_151_rev__predicate1D,axiom,
! [A: $tType,P: A > $o,X: A,Q2: A > $o] :
( ( P @ X )
=> ( ( ord_less_eq @ ( A > $o ) @ P @ Q2 )
=> ( Q2 @ X ) ) ) ).
% rev_predicate1D
thf(fact_152_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R: A,S4: B,R4: set @ ( product_prod @ A @ B ),S3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S4 ) @ R4 )
=> ( ( S3 = S4 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S3 ) @ R4 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_153_in__mono,axiom,
! [A: $tType,A6: set @ A,B6: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ( member @ A @ X @ A6 )
=> ( member @ A @ X @ B6 ) ) ) ).
% in_mono
thf(fact_154_subsetD,axiom,
! [A: $tType,A6: set @ A,B6: set @ A,C3: A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ( member @ A @ C3 @ A6 )
=> ( member @ A @ C3 @ B6 ) ) ) ).
% subsetD
thf(fact_155_equalityE,axiom,
! [A: $tType,A6: set @ A,B6: set @ A] :
( ( A6 = B6 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ).
% equalityE
thf(fact_156_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( member @ A @ X4 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_157_equalityD1,axiom,
! [A: $tType,A6: set @ A,B6: set @ A] :
( ( A6 = B6 )
=> ( ord_less_eq @ ( set @ A ) @ A6 @ B6 ) ) ).
% equalityD1
thf(fact_158_equalityD2,axiom,
! [A: $tType,A6: set @ A,B6: set @ A] :
( ( A6 = B6 )
=> ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ).
% equalityD2
thf(fact_159_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
! [T2: A] :
( ( member @ A @ T2 @ A7 )
=> ( member @ A @ T2 @ B7 ) ) ) ) ).
% subset_iff
thf(fact_160_subset__refl,axiom,
! [A: $tType,A6: set @ A] : ( ord_less_eq @ ( set @ A ) @ A6 @ A6 ) ).
% subset_refl
thf(fact_161_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q2: A > $o] :
( ! [X5: A] :
( ( P @ X5 )
=> ( Q2 @ X5 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q2 ) ) ) ).
% Collect_mono
thf(fact_162_subset__trans,axiom,
! [A: $tType,A6: set @ A,B6: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ C4 )
=> ( ord_less_eq @ ( set @ A ) @ A6 @ C4 ) ) ) ).
% subset_trans
thf(fact_163_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y3: set @ A,Z2: set @ A] : Y3 = Z2 )
= ( ^ [A7: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
& ( ord_less_eq @ ( set @ A ) @ B7 @ A7 ) ) ) ) ).
% set_eq_subset
thf(fact_164_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q2: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q2 ) )
= ( ! [X4: A] :
( ( P @ X4 )
=> ( Q2 @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_165_predicate2D__conj,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q2: A > B > $o,R4: $o,X: A,Y: B] :
( ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q2 )
& R4 )
=> ( R4
& ( ( P @ X @ Y )
=> ( Q2 @ X @ Y ) ) ) ) ).
% predicate2D_conj
thf(fact_166_Collect__subset,axiom,
! [A: $tType,A6: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A6 )
& ( P @ X4 ) ) )
@ A6 ) ).
% Collect_subset
thf(fact_167_less__eq__set__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
( ord_less_eq @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A7 )
@ ^ [X4: A] : ( member @ A @ X4 @ B7 ) ) ) ) ).
% less_eq_set_def
thf(fact_168_prop__restrict,axiom,
! [A: $tType,X: A,Z3: set @ A,X6: set @ A,P: A > $o] :
( ( member @ A @ X @ Z3 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z3
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ X6 )
& ( P @ X4 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_169_conj__subset__def,axiom,
! [A: $tType,A6: set @ A,P: A > $o,Q2: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A6
@ ( collect @ A
@ ^ [X4: A] :
( ( P @ X4 )
& ( Q2 @ X4 ) ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A6 @ ( collect @ A @ P ) )
& ( ord_less_eq @ ( set @ A ) @ A6 @ ( collect @ A @ Q2 ) ) ) ) ).
% conj_subset_def
thf(fact_170_Heap_Oaxioms_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),As_tree: B > ( tree @ A ),Remove_max: B > ( product_prod @ A @ B )] :
( ( heap @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max )
=> ( heap_axioms @ B @ A @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max ) ) ) ).
% Heap.axioms(2)
thf(fact_171_Powp__mono,axiom,
! [A: $tType,A6: A > $o,B6: A > $o] :
( ( ord_less_eq @ ( A > $o ) @ A6 @ B6 )
=> ( ord_less_eq @ ( ( set @ A ) > $o ) @ ( powp @ A @ A6 ) @ ( powp @ A @ B6 ) ) ) ).
% Powp_mono
thf(fact_172_Heap_Ointro,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),As_tree: B > ( tree @ A ),Remove_max: B > ( product_prod @ A @ B )] :
( ( collection @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset )
=> ( ( heap_axioms @ B @ A @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max )
=> ( heap @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max ) ) ) ) ).
% Heap.intro
thf(fact_173_Heap__def,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( heap @ B @ A )
= ( ^ [Empty2: B,Is_empty2: B > $o,Of_list2: ( list @ A ) > B,Multiset2: B > ( multiset @ A ),As_tree2: B > ( tree @ A ),Remove_max2: B > ( product_prod @ A @ B )] :
( ( collection @ B @ A @ Empty2 @ Is_empty2 @ Of_list2 @ Multiset2 )
& ( heap_axioms @ B @ A @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 ) ) ) ) ) ).
% Heap_def
thf(fact_174_image2__eqI,axiom,
! [A: $tType,C: $tType,B: $tType,B2: A,F: B > A,X: B,C3: C,G: B > C,A6: set @ B] :
( ( B2
= ( F @ X ) )
=> ( ( C3
= ( G @ X ) )
=> ( ( member @ B @ X @ A6 )
=> ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B2 @ C3 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A6 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_175_Heap_Oaxioms_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),As_tree: B > ( tree @ A ),Remove_max: B > ( product_prod @ A @ B )] :
( ( heap @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max )
=> ( collection @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset ) ) ) ).
% Heap.axioms(1)
thf(fact_176_GreatestI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A,Q2: A > $o] :
( ( P @ X )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X ) )
=> ( ! [X5: A] :
( ( P @ X5 )
=> ( ! [Y6: A] :
( ( P @ Y6 )
=> ( ord_less_eq @ A @ Y6 @ X5 ) )
=> ( Q2 @ X5 ) ) )
=> ( Q2 @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% GreatestI2_order
thf(fact_177_Greatest__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A] :
( ( P @ X )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X ) )
=> ( ( order_Greatest @ A @ P )
= X ) ) ) ) ).
% Greatest_equality
thf(fact_178_Heap__axioms__def,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( heap_axioms @ B @ A )
= ( ^ [Is_empty2: B > $o,Of_list2: ( list @ A ) > B,Multiset2: B > ( multiset @ A ),As_tree2: B > ( tree @ A ),Remove_max2: B > ( product_prod @ A @ B )] :
( ! [L4: B] :
( ( Multiset2 @ L4 )
= ( multiset2 @ A @ ( As_tree2 @ L4 ) ) )
& ! [I: list @ A] : ( is_heap @ A @ ( As_tree2 @ ( Of_list2 @ I ) ) )
& ! [T2: B] :
( ( ( As_tree2 @ T2 )
= ( e @ A ) )
= ( Is_empty2 @ T2 ) )
& ! [L4: B,M2: A,L5: B] :
( ~ ( Is_empty2 @ L4 )
=> ( ( ( product_Pair @ A @ B @ M2 @ L5 )
= ( Remove_max2 @ L4 ) )
=> ( ( add_mset @ A @ M2 @ ( Multiset2 @ L5 ) )
= ( Multiset2 @ L4 ) ) ) )
& ! [L4: B,M2: A,L5: B] :
( ~ ( Is_empty2 @ L4 )
=> ( ( is_heap @ A @ ( As_tree2 @ L4 ) )
=> ( ( ( product_Pair @ A @ B @ M2 @ L5 )
= ( Remove_max2 @ L4 ) )
=> ( is_heap @ A @ ( As_tree2 @ L5 ) ) ) ) )
& ! [T2: B,M2: A,T5: B] :
( ~ ( Is_empty2 @ T2 )
=> ( ( ( product_Pair @ A @ B @ M2 @ T5 )
= ( Remove_max2 @ T2 ) )
=> ( M2
= ( val @ A @ ( As_tree2 @ T2 ) ) ) ) ) ) ) ) ) ).
% Heap_axioms_def
thf(fact_179_siftDown__multiset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T3: tree @ A] :
( ( multiset2 @ A @ ( heapIm748920189ftDown @ A @ T3 ) )
= ( multiset2 @ A @ T3 ) ) ) ).
% siftDown_multiset
thf(fact_180_Heap_Oremove__max__multiset_H,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),As_tree: B > ( tree @ A ),Remove_max: B > ( product_prod @ A @ B ),L: B,M: A,L3: B] :
( ( heap @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max )
=> ( ~ ( Is_empty @ L )
=> ( ( ( product_Pair @ A @ B @ M @ L3 )
= ( Remove_max @ L ) )
=> ( ( add_mset @ A @ M @ ( Multiset @ L3 ) )
= ( Multiset @ L ) ) ) ) ) ) ).
% Heap.remove_max_multiset'
thf(fact_181_Heap_Omultiset,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Empty: B,Is_empty: B > $o,Of_list: ( list @ A ) > B,Multiset: B > ( multiset @ A ),As_tree: B > ( tree @ A ),Remove_max: B > ( product_prod @ A @ B ),L: B] :
( ( heap @ B @ A @ Empty @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max )
=> ( ( Multiset @ L )
= ( multiset2 @ A @ ( As_tree @ L ) ) ) ) ) ).
% Heap.multiset
thf(fact_182_multiset__heapify,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T3: tree @ A] :
( ( multiset2 @ A @ ( heapIm818251801eapify @ A @ T3 ) )
= ( multiset2 @ A @ T3 ) ) ) ).
% multiset_heapify
thf(fact_183_Heap__axioms_Ointro,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Multiset: B > ( multiset @ A ),As_tree: B > ( tree @ A ),Of_list: ( list @ A ) > B,Is_empty: B > $o,Remove_max: B > ( product_prod @ A @ B )] :
( ! [L6: B] :
( ( Multiset @ L6 )
= ( multiset2 @ A @ ( As_tree @ L6 ) ) )
=> ( ! [I3: list @ A] : ( is_heap @ A @ ( As_tree @ ( Of_list @ I3 ) ) )
=> ( ! [T6: B] :
( ( ( As_tree @ T6 )
= ( e @ A ) )
= ( Is_empty @ T6 ) )
=> ( ! [L6: B,M3: A,L2: B] :
( ~ ( Is_empty @ L6 )
=> ( ( ( product_Pair @ A @ B @ M3 @ L2 )
= ( Remove_max @ L6 ) )
=> ( ( add_mset @ A @ M3 @ ( Multiset @ L2 ) )
= ( Multiset @ L6 ) ) ) )
=> ( ! [L6: B,M3: A,L2: B] :
( ~ ( Is_empty @ L6 )
=> ( ( is_heap @ A @ ( As_tree @ L6 ) )
=> ( ( ( product_Pair @ A @ B @ M3 @ L2 )
= ( Remove_max @ L6 ) )
=> ( is_heap @ A @ ( As_tree @ L2 ) ) ) ) )
=> ( ! [T6: B,M3: A,T7: B] :
( ~ ( Is_empty @ T6 )
=> ( ( ( product_Pair @ A @ B @ M3 @ T7 )
= ( Remove_max @ T6 ) )
=> ( M3
= ( val @ A @ ( As_tree @ T6 ) ) ) ) )
=> ( heap_axioms @ B @ A @ Is_empty @ Of_list @ Multiset @ As_tree @ Remove_max ) ) ) ) ) ) ) ) ).
% Heap_axioms.intro
thf(fact_184_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_185_prod_Osplit__sel__asm,axiom,
! [C: $tType,B: $tType,A: $tType,P: C > $o,F: A > B > C,Prod: product_prod @ A @ B] :
( ( P @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
= ( ~ ( ( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
& ~ ( P @ ( F @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ) ).
% prod.split_sel_asm
thf(fact_186_prod_Osplit__sel,axiom,
! [C: $tType,B: $tType,A: $tType,P: C > $o,F: A > B > C,Prod: product_prod @ A @ B] :
( ( P @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
= ( ( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
=> ( P @ ( F @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ).
% prod.split_sel
thf(fact_187_mem__case__prodI,axiom,
! [A: $tType,B: $tType,C: $tType,Z: A,C3: B > C > ( set @ A ),A2: B,B2: C] :
( ( member @ A @ Z @ ( C3 @ A2 @ B2 ) )
=> ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ ( product_Pair @ B @ C @ A2 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_188_mem__case__prodI2,axiom,
! [C: $tType,B: $tType,A: $tType,P2: product_prod @ A @ B,Z: C,C3: A > B > ( set @ C )] :
( ! [A5: A,B5: B] :
( ( P2
= ( product_Pair @ A @ B @ A5 @ B5 ) )
=> ( member @ C @ Z @ ( C3 @ A5 @ B5 ) ) )
=> ( member @ C @ Z @ ( product_case_prod @ A @ B @ ( set @ C ) @ C3 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_189_case__prodI2_H,axiom,
! [A: $tType,B: $tType,C: $tType,P2: product_prod @ A @ B,C3: A > B > C > $o,X: C] :
( ! [A5: A,B5: B] :
( ( ( product_Pair @ A @ B @ A5 @ B5 )
= P2 )
=> ( C3 @ A5 @ B5 @ X ) )
=> ( product_case_prod @ A @ B @ ( C > $o ) @ C3 @ P2 @ X ) ) ).
% case_prodI2'
thf(fact_190_case__prodI,axiom,
! [A: $tType,B: $tType,F: A > B > $o,A2: A,B2: B] :
( ( F @ A2 @ B2 )
=> ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% case_prodI
thf(fact_191_case__prodI2,axiom,
! [B: $tType,A: $tType,P2: product_prod @ A @ B,C3: A > B > $o] :
( ! [A5: A,B5: B] :
( ( P2
= ( product_Pair @ A @ B @ A5 @ B5 ) )
=> ( C3 @ A5 @ B5 ) )
=> ( product_case_prod @ A @ B @ $o @ C3 @ P2 ) ) ).
% case_prodI2
thf(fact_192_scomp__apply,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType] :
( ( product_scomp @ B @ C @ D @ A )
= ( ^ [F4: B > ( product_prod @ C @ D ),G4: C > D > A,X4: B] : ( product_case_prod @ C @ D @ A @ G4 @ ( F4 @ X4 ) ) ) ) ).
% scomp_apply
thf(fact_193_case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,F: B > C > A,A2: B,B2: C] :
( ( product_case_prod @ B @ C @ A @ F @ ( product_Pair @ B @ C @ A2 @ B2 ) )
= ( F @ A2 @ B2 ) ) ).
% case_prod_conv
thf(fact_194_case__swap,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > B > A,P2: product_prod @ C @ B] :
( ( product_case_prod @ B @ C @ A
@ ^ [Y5: B,X4: C] : ( F @ X4 @ Y5 )
@ ( product_swap @ C @ B @ P2 ) )
= ( product_case_prod @ C @ B @ A @ F @ P2 ) ) ).
% case_swap
thf(fact_195_Product__Type_OCollect__case__prodD,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,A6: A > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ X @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A6 ) ) )
=> ( A6 @ ( product_fst @ A @ B @ X ) @ ( product_snd @ A @ B @ X ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_196_prod_Ocase__distrib,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,H: C > D,F: A > B > C,Prod: product_prod @ A @ B] :
( ( H @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
= ( product_case_prod @ A @ B @ D
@ ^ [X13: A,X25: B] : ( H @ ( F @ X13 @ X25 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_197_case__prodE,axiom,
! [A: $tType,B: $tType,C3: A > B > $o,P2: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ $o @ C3 @ P2 )
=> ~ ! [X5: A,Y4: B] :
( ( P2
= ( product_Pair @ A @ B @ X5 @ Y4 ) )
=> ~ ( C3 @ X5 @ Y4 ) ) ) ).
% case_prodE
thf(fact_198_case__prodD,axiom,
! [A: $tType,B: $tType,F: A > B > $o,A2: A,B2: B] :
( ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A2 @ B2 ) )
=> ( F @ A2 @ B2 ) ) ).
% case_prodD
thf(fact_199_old_Oprod_Ocase,axiom,
! [A: $tType,C: $tType,B: $tType,F: A > B > C,X12: A,X24: B] :
( ( product_case_prod @ A @ B @ C @ F @ ( product_Pair @ A @ B @ X12 @ X24 ) )
= ( F @ X12 @ X24 ) ) ).
% old.prod.case
thf(fact_200_mem__case__prodE,axiom,
! [B: $tType,A: $tType,C: $tType,Z: A,C3: B > C > ( set @ A ),P2: product_prod @ B @ C] :
( ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ P2 ) )
=> ~ ! [X5: B,Y4: C] :
( ( P2
= ( product_Pair @ B @ C @ X5 @ Y4 ) )
=> ~ ( member @ A @ Z @ ( C3 @ X5 @ Y4 ) ) ) ) ).
% mem_case_prodE
thf(fact_201_snd__def,axiom,
! [B: $tType,A: $tType] :
( ( product_snd @ A @ B )
= ( product_case_prod @ A @ B @ B
@ ^ [X13: A,X25: B] : X25 ) ) ).
% snd_def
thf(fact_202_fst__def,axiom,
! [B: $tType,A: $tType] :
( ( product_fst @ A @ B )
= ( product_case_prod @ A @ B @ A
@ ^ [X13: A,X25: B] : X13 ) ) ).
% fst_def
thf(fact_203_case__prodE2,axiom,
! [B: $tType,A: $tType,C: $tType,Q2: A > $o,P: B > C > A,Z: product_prod @ B @ C] :
( ( Q2 @ ( product_case_prod @ B @ C @ A @ P @ Z ) )
=> ~ ! [X5: B,Y4: C] :
( ( Z
= ( product_Pair @ B @ C @ X5 @ Y4 ) )
=> ~ ( Q2 @ ( P @ X5 @ Y4 ) ) ) ) ).
% case_prodE2
thf(fact_204_case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C
@ ^ [X4: A,Y5: B] : ( F @ ( product_Pair @ A @ B @ X4 @ Y5 ) ) )
= F ) ).
% case_prod_eta
thf(fact_205_cond__case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > B > C,G: ( product_prod @ A @ B ) > C] :
( ! [X5: A,Y4: B] :
( ( F @ X5 @ Y4 )
= ( G @ ( product_Pair @ A @ B @ X5 @ Y4 ) ) )
=> ( ( product_case_prod @ A @ B @ C @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_206_case__prodD_H,axiom,
! [B: $tType,A: $tType,C: $tType,R4: A > B > C > $o,A2: A,B2: B,C3: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ R4 @ ( product_Pair @ A @ B @ A2 @ B2 ) @ C3 )
=> ( R4 @ A2 @ B2 @ C3 ) ) ).
% case_prodD'
thf(fact_207_case__prodE_H,axiom,
! [B: $tType,A: $tType,C: $tType,C3: A > B > C > $o,P2: product_prod @ A @ B,Z: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ C3 @ P2 @ Z )
=> ~ ! [X5: A,Y4: B] :
( ( P2
= ( product_Pair @ A @ B @ X5 @ Y4 ) )
=> ~ ( C3 @ X5 @ Y4 @ Z ) ) ) ).
% case_prodE'
thf(fact_208_scomp__def,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] : ( product_case_prod @ B @ C @ D @ G4 @ ( F4 @ X4 ) ) ) ) ).
% scomp_def
thf(fact_209_case__prod__Pair__iden,axiom,
! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P2 )
= P2 ) ).
% case_prod_Pair_iden
thf(fact_210_case__prod__app,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ ( D > A ) )
= ( ^ [F4: B > C > D > A,X4: product_prod @ B @ C,Y5: D] :
( product_case_prod @ B @ C @ A
@ ^ [L4: B,R3: C] : ( F4 @ L4 @ R3 @ Y5 )
@ X4 ) ) ) ).
% case_prod_app
thf(fact_211_Collect__case__prod__mono,axiom,
! [B: $tType,A: $tType,A6: A > B > $o,B6: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ A6 @ B6 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A6 ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ B6 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_212_prod_Ocase__eq__if,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [F4: A > B > C,Prod3: product_prod @ A @ B] : ( F4 @ ( product_fst @ A @ B @ Prod3 ) @ ( product_snd @ A @ B @ Prod3 ) ) ) ) ).
% prod.case_eq_if
thf(fact_213_case__prod__beta,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ A )
= ( ^ [F4: B > C > A,P3: product_prod @ B @ C] : ( F4 @ ( product_fst @ B @ C @ P3 ) @ ( product_snd @ B @ C @ P3 ) ) ) ) ).
% case_prod_beta
thf(fact_214_internal__case__prod__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( produc2004651681e_prod @ A @ B @ C )
= ( product_case_prod @ A @ B @ C ) ) ).
% internal_case_prod_def
thf(fact_215_fstOp__in,axiom,
! [B: $tType,C: $tType,A: $tType,Ac2: product_prod @ A @ B,P: A > C > $o,Q2: C > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ Ac2 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( relcompp @ A @ C @ B @ P @ Q2 ) ) ) )
=> ( member @ ( product_prod @ A @ C ) @ ( bNF_fstOp @ A @ C @ B @ P @ Q2 @ Ac2 ) @ ( collect @ ( product_prod @ A @ C ) @ ( product_case_prod @ A @ C @ $o @ P ) ) ) ) ).
% fstOp_in
thf(fact_216_sndOp__in,axiom,
! [A: $tType,B: $tType,C: $tType,Ac2: product_prod @ A @ B,P: A > C > $o,Q2: C > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ Ac2 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( relcompp @ A @ C @ B @ P @ Q2 ) ) ) )
=> ( member @ ( product_prod @ C @ B ) @ ( bNF_sndOp @ A @ C @ B @ P @ Q2 @ Ac2 ) @ ( collect @ ( product_prod @ C @ B ) @ ( product_case_prod @ C @ B @ $o @ Q2 ) ) ) ) ).
% sndOp_in
thf(fact_217_split__comp__eq,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,F: A > B > C,G: D > A] :
( ( ^ [U: product_prod @ D @ B] : ( F @ ( G @ ( product_fst @ D @ B @ U ) ) @ ( product_snd @ D @ B @ U ) ) )
= ( product_case_prod @ D @ B @ C
@ ^ [X4: D] : ( F @ ( G @ X4 ) ) ) ) ).
% split_comp_eq
thf(fact_218_case__prod__beta_H,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [F4: A > B > C,X4: product_prod @ A @ B] : ( F4 @ ( product_fst @ A @ B @ X4 ) @ ( product_snd @ A @ B @ X4 ) ) ) ) ).
% case_prod_beta'
thf(fact_219_case__prod__unfold,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [C5: A > B > C,P3: product_prod @ A @ B] : ( C5 @ ( product_fst @ A @ B @ P3 ) @ ( product_snd @ A @ B @ P3 ) ) ) ) ).
% case_prod_unfold
thf(fact_220_exE__realizer,axiom,
! [C: $tType,A: $tType,B: $tType,P: A > B > $o,P2: product_prod @ B @ A,Q2: C > $o,F: B > A > C] :
( ( P @ ( product_snd @ B @ A @ P2 ) @ ( product_fst @ B @ A @ P2 ) )
=> ( ! [X5: B,Y4: A] :
( ( P @ Y4 @ X5 )
=> ( Q2 @ ( F @ X5 @ Y4 ) ) )
=> ( Q2 @ ( product_case_prod @ B @ A @ C @ F @ P2 ) ) ) ) ).
% exE_realizer
thf(fact_221_split__cong,axiom,
! [C: $tType,B: $tType,A: $tType,Q: product_prod @ A @ B,F: A > B > C,G: A > B > C,P2: product_prod @ A @ B] :
( ! [X5: A,Y4: B] :
( ( ( product_Pair @ A @ B @ X5 @ Y4 )
= Q )
=> ( ( F @ X5 @ Y4 )
= ( G @ X5 @ Y4 ) ) )
=> ( ( P2 = Q )
=> ( ( product_case_prod @ A @ B @ C @ F @ P2 )
= ( product_case_prod @ A @ B @ C @ G @ Q ) ) ) ) ).
% split_cong
thf(fact_222_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_223_curry__case__prod,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > B > C] :
( ( product_curry @ A @ B @ C @ ( product_case_prod @ A @ B @ C @ F ) )
= F ) ).
% curry_case_prod
thf(fact_224_curryI,axiom,
! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
( ( F @ ( product_Pair @ A @ B @ A2 @ B2 ) )
=> ( product_curry @ A @ B @ $o @ F @ A2 @ B2 ) ) ).
% curryI
thf(fact_225_split__part,axiom,
! [B: $tType,A: $tType,P: $o,Q2: A > B > $o] :
( ( product_case_prod @ A @ B @ $o
@ ^ [A4: A,B4: B] :
( P
& ( Q2 @ A4 @ B4 ) ) )
= ( ^ [Ab: product_prod @ A @ B] :
( P
& ( product_case_prod @ A @ B @ $o @ Q2 @ Ab ) ) ) ) ).
% split_part
thf(fact_226_curry__conv,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( product_curry @ B @ C @ A )
= ( ^ [F4: ( product_prod @ B @ C ) > A,A4: B,B4: C] : ( F4 @ ( product_Pair @ B @ C @ A4 @ B4 ) ) ) ) ).
% curry_conv
thf(fact_227_case__prod__curry,axiom,
! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C @ ( product_curry @ A @ B @ C @ F ) )
= F ) ).
% case_prod_curry
thf(fact_228_prod_Odisc__eq__case,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( product_case_prod @ A @ B @ $o
@ ^ [Uu2: A,Uv2: B] : $true
@ Prod ) ).
% prod.disc_eq_case
thf(fact_229_curry__K,axiom,
! [B: $tType,C: $tType,A: $tType,C3: C] :
( ( product_curry @ A @ B @ C
@ ^ [X4: product_prod @ A @ B] : C3 )
= ( ^ [X4: A,Y5: B] : C3 ) ) ).
% curry_K
thf(fact_230_curryD,axiom,
! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
( ( product_curry @ A @ B @ $o @ F @ A2 @ B2 )
=> ( F @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% curryD
thf(fact_231_curryE,axiom,
! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A2: A,B2: B] :
( ( product_curry @ A @ B @ $o @ F @ A2 @ B2 )
=> ( F @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% curryE
thf(fact_232_curry__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_curry @ A @ B @ C )
= ( ^ [C5: ( product_prod @ A @ B ) > C,X4: A,Y5: B] : ( C5 @ ( product_Pair @ A @ B @ X4 @ Y5 ) ) ) ) ).
% curry_def
thf(fact_233_fst__snd__flip,axiom,
! [B: $tType,A: $tType] :
( ( product_fst @ A @ B )
= ( comp @ ( product_prod @ B @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_snd @ B @ A )
@ ( product_case_prod @ A @ B @ ( product_prod @ B @ A )
@ ^ [X4: A,Y5: B] : ( product_Pair @ B @ A @ Y5 @ X4 ) ) ) ) ).
% fst_snd_flip
thf(fact_234_snd__fst__flip,axiom,
! [A: $tType,B: $tType] :
( ( product_snd @ B @ A )
= ( comp @ ( product_prod @ A @ B ) @ A @ ( product_prod @ B @ A ) @ ( product_fst @ A @ B )
@ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
@ ^ [X4: B,Y5: A] : ( product_Pair @ A @ B @ Y5 @ X4 ) ) ) ) ).
% snd_fst_flip
thf(fact_235_fst__fstOp,axiom,
! [A: $tType,B: $tType,C: $tType,P: A > C > $o,Q2: C > B > $o] :
( ( product_fst @ A @ B )
= ( comp @ ( product_prod @ A @ C ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ C ) @ ( bNF_fstOp @ A @ C @ B @ P @ Q2 ) ) ) ).
% fst_fstOp
thf(fact_236_snd__sndOp,axiom,
! [B: $tType,A: $tType,C: $tType,P: B > C > $o,Q2: C > A > $o] :
( ( product_snd @ B @ A )
= ( comp @ ( product_prod @ C @ A ) @ A @ ( product_prod @ B @ A ) @ ( product_snd @ C @ A ) @ ( bNF_sndOp @ B @ C @ A @ P @ Q2 ) ) ) ).
% snd_sndOp
thf(fact_237_comp__apply__eq,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,F: B > A,G: C > B,X: C,H: D > A,K: C > D] :
( ( ( F @ ( G @ X ) )
= ( H @ ( K @ X ) ) )
=> ( ( comp @ B @ A @ C @ F @ G @ X )
= ( comp @ D @ A @ C @ H @ K @ X ) ) ) ).
% comp_apply_eq
thf(fact_238_snd__comp__apsnd,axiom,
! [C: $tType,B: $tType,A: $tType,F: B > C] :
( ( comp @ ( product_prod @ A @ C ) @ C @ ( product_prod @ A @ B ) @ ( product_snd @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F ) )
= ( comp @ B @ C @ ( product_prod @ A @ B ) @ F @ ( product_snd @ A @ B ) ) ) ).
% snd_comp_apsnd
thf(fact_239_fst__comp__apfst,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > C] :
( ( comp @ ( product_prod @ C @ B ) @ C @ ( product_prod @ A @ B ) @ ( product_fst @ C @ B ) @ ( product_apfst @ A @ C @ B @ F ) )
= ( comp @ A @ C @ ( product_prod @ A @ B ) @ F @ ( product_fst @ A @ B ) ) ) ).
% fst_comp_apfst
thf(fact_240_apfst__conv,axiom,
! [C: $tType,A: $tType,B: $tType,F: C > A,X: C,Y: B] :
( ( product_apfst @ C @ A @ B @ F @ ( product_Pair @ C @ B @ X @ Y ) )
= ( product_Pair @ A @ B @ ( F @ X ) @ Y ) ) ).
% apfst_conv
thf(fact_241_apsnd__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > B,X: A,Y: C] :
( ( product_apsnd @ C @ B @ A @ F @ ( product_Pair @ A @ C @ X @ Y ) )
= ( product_Pair @ A @ B @ X @ ( F @ Y ) ) ) ).
% apsnd_conv
thf(fact_242_fst__apfst,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > A,X: product_prod @ C @ B] :
( ( product_fst @ A @ B @ ( product_apfst @ C @ A @ B @ F @ X ) )
= ( F @ ( product_fst @ C @ B @ X ) ) ) ).
% fst_apfst
thf(fact_243_apfst__eq__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > A,X: product_prod @ C @ B,G: C > A] :
( ( ( product_apfst @ C @ A @ B @ F @ X )
= ( product_apfst @ C @ A @ B @ G @ X ) )
= ( ( F @ ( product_fst @ C @ B @ X ) )
= ( G @ ( product_fst @ C @ B @ X ) ) ) ) ).
% apfst_eq_conv
thf(fact_244_snd__apfst,axiom,
! [B: $tType,A: $tType,C: $tType,F: C > B,X: product_prod @ C @ A] :
( ( product_snd @ B @ A @ ( product_apfst @ C @ B @ A @ F @ X ) )
= ( product_snd @ C @ A @ X ) ) ).
% snd_apfst
thf(fact_245_fst__apsnd,axiom,
! [B: $tType,C: $tType,A: $tType,F: C > B,X: product_prod @ A @ C] :
( ( product_fst @ A @ B @ ( product_apsnd @ C @ B @ A @ F @ X ) )
= ( product_fst @ A @ C @ X ) ) ).
% fst_apsnd
thf(fact_246_apsnd__eq__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F: C > B,X: product_prod @ A @ C,G: C > B] :
( ( ( product_apsnd @ C @ B @ A @ F @ X )
= ( product_apsnd @ C @ B @ A @ G @ X ) )
= ( ( F @ ( product_snd @ A @ C @ X ) )
= ( G @ ( product_snd @ A @ C @ X ) ) ) ) ).
% apsnd_eq_conv
thf(fact_247_snd__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F: C > A,X: product_prod @ B @ C] :
( ( product_snd @ B @ A @ ( product_apsnd @ C @ A @ B @ F @ X ) )
= ( F @ ( product_snd @ B @ C @ X ) ) ) ).
% snd_apsnd
thf(fact_248_snd__comp__apfst,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > C] :
( ( comp @ ( product_prod @ C @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_snd @ C @ B ) @ ( product_apfst @ A @ C @ B @ F ) )
= ( product_snd @ A @ B ) ) ).
% snd_comp_apfst
thf(fact_249_fst__comp__apsnd,axiom,
! [C: $tType,B: $tType,A: $tType,F: B > C] :
( ( comp @ ( product_prod @ A @ C ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F ) )
= ( product_fst @ A @ B ) ) ).
% fst_comp_apsnd
thf(fact_250_apfst__apsnd,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,F: C > A,G: D > B,X: product_prod @ C @ D] :
( ( product_apfst @ C @ A @ B @ F @ ( product_apsnd @ D @ B @ C @ G @ X ) )
= ( product_Pair @ A @ B @ ( F @ ( product_fst @ C @ D @ X ) ) @ ( G @ ( product_snd @ C @ D @ X ) ) ) ) ).
% apfst_apsnd
thf(fact_251_apsnd__apfst,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F: C > B,G: D > A,X: product_prod @ D @ C] :
( ( product_apsnd @ C @ B @ A @ F @ ( product_apfst @ D @ A @ C @ G @ X ) )
= ( product_Pair @ A @ B @ ( G @ ( product_fst @ D @ C @ X ) ) @ ( F @ ( product_snd @ D @ C @ X ) ) ) ) ).
% apsnd_apfst
thf(fact_252_apsnd__compose,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,F: C > B,G: D > C,X: product_prod @ A @ D] :
( ( product_apsnd @ C @ B @ A @ F @ ( product_apsnd @ D @ C @ A @ G @ X ) )
= ( product_apsnd @ D @ B @ A @ ( comp @ C @ B @ D @ F @ G ) @ X ) ) ).
% apsnd_compose
thf(fact_253_apfst__compose,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G: D > C,X: product_prod @ D @ B] :
( ( product_apfst @ C @ A @ B @ F @ ( product_apfst @ D @ C @ B @ G @ X ) )
= ( product_apfst @ D @ A @ B @ ( comp @ C @ A @ D @ F @ G ) @ X ) ) ).
% apfst_compose
thf(fact_254_apsnd__apfst__commute,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F: C > B,G: D > A,P2: product_prod @ D @ C] :
( ( product_apsnd @ C @ B @ A @ F @ ( product_apfst @ D @ A @ C @ G @ P2 ) )
= ( product_apfst @ D @ A @ B @ G @ ( product_apsnd @ C @ B @ D @ F @ P2 ) ) ) ).
% apsnd_apfst_commute
thf(fact_255_case__prod__comp,axiom,
! [D: $tType,A: $tType,C: $tType,B: $tType,F: D > C > A,G: B > D,X: product_prod @ B @ C] :
( ( product_case_prod @ B @ C @ A @ ( comp @ D @ ( C > A ) @ B @ F @ G ) @ X )
= ( F @ ( G @ ( product_fst @ B @ C @ X ) ) @ ( product_snd @ B @ C @ X ) ) ) ).
% case_prod_comp
% Subclasses (4)
thf(subcl_Orderings_Olinorder___HOL_Otype,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( type @ A ) ) ).
thf(subcl_Orderings_Olinorder___Orderings_Oord,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ord @ A ) ) ).
thf(subcl_Orderings_Olinorder___Orderings_Oorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( order @ A ) ) ).
thf(subcl_Orderings_Olinorder___Orderings_Opreorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( preorder @ A ) ) ).
% Type constructors (13)
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A8: $tType,A9: $tType] :
( ( preorder @ A9 )
=> ( preorder @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A8: $tType,A9: $tType] :
( ( order @ A9 )
=> ( order @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A8: $tType,A9: $tType] :
( ( ord @ A9 )
=> ( ord @ ( A8 > A9 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_1,axiom,
! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_2,axiom,
! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_3,axiom,
! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_4,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_5,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_6,axiom,
ord @ $o ).
thf(tcon_Multiset_Omultiset___Orderings_Opreorder_7,axiom,
! [A8: $tType] :
( ( preorder @ A8 )
=> ( preorder @ ( multiset @ A8 ) ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Oorder_8,axiom,
! [A8: $tType] :
( ( preorder @ A8 )
=> ( order @ ( multiset @ A8 ) ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Oord_9,axiom,
! [A8: $tType] :
( ( preorder @ A8 )
=> ( ord @ ( multiset @ A8 ) ) ) ).
% Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,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,
is_heap @ a @ ( product_snd @ a @ ( tree @ a ) @ ( heapIm970386777veLeaf @ a @ ( t @ a @ v @ ( e @ a ) @ ( e @ a ) ) ) ) ).
%------------------------------------------------------------------------------