TPTP Problem File: ITP069^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP069^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer HeapImperative problem prob_637__5346010_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : HeapImperative/prob_637__5346010_1 [Des21]
% Status : Theorem
% Rating : 0.30 v8.2.0, 0.23 v8.1.0, 0.27 v7.5.0
% Syntax : Number of formulae : 192 ( 72 unt; 49 typ; 0 def)
% Number of atoms : 402 ( 200 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 1738 ( 55 ~; 10 |; 49 &;1455 @)
% ( 0 <=>; 169 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 7 avg)
% Number of types : 11 ( 10 usr)
% Number of type conns : 79 ( 79 >; 0 *; 0 +; 0 <<)
% Number of symbols : 40 ( 39 usr; 5 con; 0-3 aty)
% Number of variables : 469 ( 20 ^; 434 !; 15 ?; 469 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:31:05.115
%------------------------------------------------------------------------------
% Could-be-implicit typings (10)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J,type,
set_Pr158363655iset_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J,type,
produc1127127335iset_a: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
set_Product_prod_a_a: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Multiset__Omultiset_Itf__a_J_J,type,
multiset_multiset_a: $tType ).
thf(ty_n_t__Set__Oset_It__Multiset__Omultiset_Itf__a_J_J,type,
set_multiset_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
product_prod_a_a: $tType ).
thf(ty_n_t__Multiset__Omultiset_Itf__a_J,type,
multiset_a: $tType ).
thf(ty_n_t__Heap__OTree_Itf__a_J,type,
tree_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (39)
thf(sy_c_Groups_Ominus__class_Ominus_001t__Multiset__Omultiset_Itf__a_J,type,
minus_1649712171iset_a: multiset_a > multiset_a > multiset_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_Itf__a_J,type,
plus_plus_multiset_a: multiset_a > multiset_a > multiset_a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_Itf__a_J,type,
zero_zero_multiset_a: multiset_a ).
thf(sy_c_HOL_ONO__MATCH_001t__Multiset__Omultiset_Itf__a_J_001t__Multiset__Omultiset_Itf__a_J,type,
nO_MAT1617603563iset_a: multiset_a > multiset_a > $o ).
thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_Oheapify_001tf__a,type,
heapIm970322378pify_a: tree_a > tree_a ).
thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_Oleft_001tf__a,type,
heapIm1140443833left_a: tree_a > tree_a ).
thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_Oright_001tf__a,type,
heapIm1257206334ight_a: tree_a > tree_a ).
thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_OsiftDown_001tf__a,type,
heapIm1091024090Down_a: tree_a > tree_a ).
thf(sy_c_Heap_OTree_OE_001tf__a,type,
e_a: tree_a ).
thf(sy_c_Heap_OTree_OT_001tf__a,type,
t_a: a > tree_a > tree_a > tree_a ).
thf(sy_c_Heap_Oin__tree_001tf__a,type,
in_tree_a: a > tree_a > $o ).
thf(sy_c_Heap_Ois__heap_001tf__a,type,
is_heap_a: tree_a > $o ).
thf(sy_c_Heap_Omultiset_001tf__a,type,
multiset_a2: tree_a > multiset_a ).
thf(sy_c_Heap_Oval_001tf__a,type,
val_a: tree_a > a ).
thf(sy_c_Lattices__Big_Olinorder__class_OMax_001tf__a,type,
lattic146396397_Max_a: set_a > a ).
thf(sy_c_Multiset_Oadd__mset_001tf__a,type,
add_mset_a: a > multiset_a > multiset_a ).
thf(sy_c_Multiset_Ocomm__monoid__add_Osum__mset_001t__Multiset__Omultiset_Itf__a_J,type,
comm_m315775925iset_a: ( multiset_a > multiset_a > multiset_a ) > multiset_a > multiset_multiset_a > multiset_a ).
thf(sy_c_Multiset_Ofold__mset_001t__Multiset__Omultiset_Itf__a_J_001t__Multiset__Omultiset_Itf__a_J,type,
fold_m382157835iset_a: ( multiset_a > multiset_a > multiset_a ) > multiset_a > multiset_multiset_a > multiset_a ).
thf(sy_c_Multiset_Ofold__mset_001tf__a_001t__Multiset__Omultiset_Itf__a_J,type,
fold_m364285649iset_a: ( a > multiset_a > multiset_a ) > multiset_a > multiset_a > multiset_a ).
thf(sy_c_Multiset_Ois__empty_001tf__a,type,
is_empty_a: multiset_a > $o ).
thf(sy_c_Multiset_Omult1_001tf__a,type,
mult1_a: set_Product_prod_a_a > set_Pr158363655iset_a ).
thf(sy_c_Multiset_Omult_001tf__a,type,
mult_a: set_Product_prod_a_a > set_Pr158363655iset_a ).
thf(sy_c_Multiset_Oset__mset_001t__Multiset__Omultiset_Itf__a_J,type,
set_mset_multiset_a: multiset_multiset_a > set_multiset_a ).
thf(sy_c_Multiset_Oset__mset_001tf__a,type,
set_mset_a: multiset_a > set_a ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Multiset__Omultiset_Itf__a_J,type,
ord_le1199012836iset_a: multiset_a > multiset_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__a,type,
ord_less_eq_a: a > a > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001tf__a,type,
order_Greatest_a: ( a > $o ) > a ).
thf(sy_c_Product__Type_OPair_001t__Multiset__Omultiset_Itf__a_J_001t__Multiset__Omultiset_Itf__a_J,type,
produc2037245207iset_a: multiset_a > multiset_a > produc1127127335iset_a ).
thf(sy_c_Product__Type_OPair_001tf__a_001tf__a,type,
product_Pair_a_a: a > a > product_prod_a_a ).
thf(sy_c_Relation_Oirrefl_001tf__a,type,
irrefl_a: set_Product_prod_a_a > $o ).
thf(sy_c_Relation_Otrans_001tf__a,type,
trans_a: set_Product_prod_a_a > $o ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_member_001t__Multiset__Omultiset_Itf__a_J,type,
member_multiset_a: multiset_a > set_multiset_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J,type,
member340150864iset_a: produc1127127335iset_a > set_Pr158363655iset_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
member449909584od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_l____,type,
l: tree_a ).
thf(sy_v_r____,type,
r: tree_a ).
thf(sy_v_v____,type,
v: a ).
% Relevant facts (142)
thf(fact_0_T_Ohyps_I2_J,axiom,
is_heap_a @ ( heapIm970322378pify_a @ r ) ).
% T.hyps(2)
thf(fact_1_T_Ohyps_I1_J,axiom,
is_heap_a @ ( heapIm970322378pify_a @ l ) ).
% T.hyps(1)
thf(fact_2__092_060open_062_092_060lbrakk_062is__heap_A_Iheapify_Al_J_059_Ais__heap_A_Iheapify_Ar_J_059_AT_Av_A_Iheapify_Al_J_A_Iheapify_Ar_J_A_061_AT_Av_A_Iheapify_Al_J_A_Iheapify_Ar_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Ais__heap_A_IsiftDown_A_IT_Av_A_Iheapify_Al_J_A_Iheapify_Ar_J_J_J_092_060close_062,axiom,
( ( is_heap_a @ ( heapIm970322378pify_a @ l ) )
=> ( ( is_heap_a @ ( heapIm970322378pify_a @ r ) )
=> ( ( ( t_a @ v @ ( heapIm970322378pify_a @ l ) @ ( heapIm970322378pify_a @ r ) )
= ( t_a @ v @ ( heapIm970322378pify_a @ l ) @ ( heapIm970322378pify_a @ r ) ) )
=> ( is_heap_a @ ( heapIm1091024090Down_a @ ( t_a @ v @ ( heapIm970322378pify_a @ l ) @ ( heapIm970322378pify_a @ r ) ) ) ) ) ) ) ).
% \<open>\<lbrakk>is_heap (heapify l); is_heap (heapify r); T v (heapify l) (heapify r) = T v (heapify l) (heapify r)\<rbrakk> \<Longrightarrow> is_heap (siftDown (T v (heapify l) (heapify r)))\<close>
thf(fact_3_heapify_Osimps_I2_J,axiom,
! [V: a,L: tree_a,R: tree_a] :
( ( heapIm970322378pify_a @ ( t_a @ V @ L @ R ) )
= ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm970322378pify_a @ L ) @ ( heapIm970322378pify_a @ R ) ) ) ) ).
% heapify.simps(2)
thf(fact_4_siftDown__heap__is__heap,axiom,
! [L: tree_a,R: tree_a,T: tree_a,V: a] :
( ( is_heap_a @ L )
=> ( ( is_heap_a @ R )
=> ( ( T
= ( t_a @ V @ L @ R ) )
=> ( is_heap_a @ ( heapIm1091024090Down_a @ T ) ) ) ) ) ).
% siftDown_heap_is_heap
thf(fact_5_Tree_Oinject,axiom,
! [X21: a,X22: tree_a,X23: tree_a,Y21: a,Y22: tree_a,Y23: tree_a] :
( ( ( t_a @ X21 @ X22 @ X23 )
= ( t_a @ Y21 @ Y22 @ Y23 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 )
& ( X23 = Y23 ) ) ) ).
% Tree.inject
thf(fact_6_left_Osimps,axiom,
! [V: a,L: tree_a,R: tree_a] :
( ( heapIm1140443833left_a @ ( t_a @ V @ L @ R ) )
= L ) ).
% left.simps
thf(fact_7_right_Osimps,axiom,
! [V: a,L: tree_a,R: tree_a] :
( ( heapIm1257206334ight_a @ ( t_a @ V @ L @ R ) )
= R ) ).
% right.simps
thf(fact_8_is__heap_Osimps_I2_J,axiom,
! [V: a] : ( is_heap_a @ ( t_a @ V @ e_a @ e_a ) ) ).
% is_heap.simps(2)
thf(fact_9_heapify_Osimps_I1_J,axiom,
( ( heapIm970322378pify_a @ e_a )
= e_a ) ).
% heapify.simps(1)
thf(fact_10_siftDown_Ocases,axiom,
! [X: tree_a] :
( ( X != e_a )
=> ( ! [V2: a] :
( X
!= ( t_a @ V2 @ e_a @ e_a ) )
=> ( ! [V2: a,Va: a,Vb: tree_a,Vc: tree_a] :
( X
!= ( t_a @ V2 @ ( t_a @ Va @ Vb @ Vc ) @ e_a ) )
=> ( ! [V2: a,Va: a,Vb: tree_a,Vc: tree_a] :
( X
!= ( t_a @ V2 @ e_a @ ( t_a @ Va @ Vb @ Vc ) ) )
=> ~ ! [V2: a,Va: a,Vb: tree_a,Vc: tree_a,Vd: a,Ve: tree_a,Vf: tree_a] :
( X
!= ( t_a @ V2 @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) ) ) ) ) ) ).
% siftDown.cases
thf(fact_11_in__tree_Osimps_I2_J,axiom,
! [V: a,V3: a,L: tree_a,R: tree_a] :
( ( in_tree_a @ V @ ( t_a @ V3 @ L @ R ) )
= ( ( V = V3 )
| ( in_tree_a @ V @ L )
| ( in_tree_a @ V @ R ) ) ) ).
% in_tree.simps(2)
thf(fact_12_siftDown_Osimps_I2_J,axiom,
! [V: a] :
( ( heapIm1091024090Down_a @ ( t_a @ V @ e_a @ e_a ) )
= ( t_a @ V @ e_a @ e_a ) ) ).
% siftDown.simps(2)
thf(fact_13_in__tree_Osimps_I1_J,axiom,
! [V: a] :
~ ( in_tree_a @ V @ e_a ) ).
% in_tree.simps(1)
thf(fact_14_siftDown_Osimps_I1_J,axiom,
( ( heapIm1091024090Down_a @ e_a )
= e_a ) ).
% siftDown.simps(1)
thf(fact_15_siftDown__in__tree__set,axiom,
( in_tree_a
= ( ^ [V4: a,T2: tree_a] : ( in_tree_a @ V4 @ ( heapIm1091024090Down_a @ T2 ) ) ) ) ).
% siftDown_in_tree_set
thf(fact_16_is__heap_Ocases,axiom,
! [X: tree_a] :
( ( X != e_a )
=> ( ! [V2: a] :
( X
!= ( t_a @ V2 @ e_a @ e_a ) )
=> ( ! [V2: a,Va: a,Vb: tree_a,Vc: tree_a] :
( X
!= ( t_a @ V2 @ e_a @ ( t_a @ Va @ Vb @ Vc ) ) )
=> ( ! [V2: a,Va: a,Vb: tree_a,Vc: tree_a] :
( X
!= ( t_a @ V2 @ ( t_a @ Va @ Vb @ Vc ) @ e_a ) )
=> ~ ! [V2: a,Va: a,Vb: tree_a,Vc: tree_a,Vd: a,Ve: tree_a,Vf: tree_a] :
( X
!= ( t_a @ V2 @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) ) ) ) ) ) ).
% is_heap.cases
thf(fact_17_Tree_Oexhaust,axiom,
! [Y: tree_a] :
( ( Y != e_a )
=> ~ ! [X212: a,X222: tree_a,X232: tree_a] :
( Y
!= ( t_a @ X212 @ X222 @ X232 ) ) ) ).
% Tree.exhaust
thf(fact_18_Tree_Oinduct,axiom,
! [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_19_Tree_Odistinct_I1_J,axiom,
! [X21: a,X22: tree_a,X23: tree_a] :
( e_a
!= ( t_a @ X21 @ X22 @ X23 ) ) ).
% Tree.distinct(1)
thf(fact_20_is__heap_Osimps_I1_J,axiom,
is_heap_a @ e_a ).
% is_heap.simps(1)
thf(fact_21_siftDown__in__tree,axiom,
! [T: tree_a] :
( ( T != e_a )
=> ( in_tree_a @ ( val_a @ ( heapIm1091024090Down_a @ T ) ) @ T ) ) ).
% siftDown_in_tree
thf(fact_22_siftDown_Osimps_I4_J,axiom,
! [Va2: a,Vb2: tree_a,Vc2: tree_a,V: a] :
( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
=> ( ( heapIm1091024090Down_a @ ( t_a @ V @ e_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
= ( t_a @ V @ e_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) )
& ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
=> ( ( heapIm1091024090Down_a @ ( t_a @ V @ e_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
= ( t_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ e_a @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ).
% siftDown.simps(4)
thf(fact_23_siftDown_Osimps_I3_J,axiom,
! [Va2: a,Vb2: tree_a,Vc2: tree_a,V: a] :
( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
=> ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ e_a ) )
= ( t_a @ V @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ e_a ) ) )
& ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
=> ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ e_a ) )
= ( t_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) ) @ e_a ) ) ) ) ).
% siftDown.simps(3)
thf(fact_24_siftDown_Osimps_I6_J,axiom,
! [Va2: a,Vb2: tree_a,Vc2: tree_a,Vd2: a,Ve2: tree_a,Vf2: tree_a,V: a] :
( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) )
=> ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ V )
=> ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
= ( t_a @ V @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) )
& ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ V )
=> ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
= ( t_a @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) ) ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) )
& ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) )
=> ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
=> ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
= ( t_a @ V @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) )
& ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
=> ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
= ( t_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).
% siftDown.simps(6)
thf(fact_25_siftDown_Osimps_I5_J,axiom,
! [Vd2: a,Ve2: tree_a,Vf2: tree_a,Va2: a,Vb2: tree_a,Vc2: tree_a,V: a] :
( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
=> ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
=> ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) )
= ( t_a @ V @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) ) )
& ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
=> ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) )
= ( t_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) ) ) ) )
& ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
=> ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ V )
=> ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) )
= ( t_a @ V @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) ) )
& ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ V )
=> ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) )
= ( t_a @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) ) ) ) ) ) ) ) ) ).
% siftDown.simps(5)
thf(fact_26_siftDown__Node,axiom,
! [T: tree_a,V: a,L: tree_a,R: tree_a] :
( ( T
= ( t_a @ V @ L @ R ) )
=> ? [L2: tree_a,V5: a,R2: tree_a] :
( ( ( heapIm1091024090Down_a @ T )
= ( t_a @ V5 @ L2 @ R2 ) )
& ( ord_less_eq_a @ V @ V5 ) ) ) ).
% siftDown_Node
thf(fact_27_is__heap_Osimps_I6_J,axiom,
! [V: a,Vd2: a,Ve2: tree_a,Vf2: tree_a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
( ( is_heap_a @ ( t_a @ V @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
= ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
& ( is_heap_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) )
& ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ V )
& ( is_heap_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) ) ) ).
% is_heap.simps(6)
thf(fact_28_is__heap_Osimps_I5_J,axiom,
! [V: a,Va2: a,Vb2: tree_a,Vc2: tree_a,Vd2: a,Ve2: tree_a,Vf2: tree_a] :
( ( is_heap_a @ ( t_a @ V @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) )
= ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) @ V )
& ( is_heap_a @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) )
& ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
& ( is_heap_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) ) ).
% is_heap.simps(5)
thf(fact_29_is__heap__max,axiom,
! [V: a,T: tree_a] :
( ( in_tree_a @ V @ T )
=> ( ( is_heap_a @ T )
=> ( ord_less_eq_a @ V @ ( val_a @ T ) ) ) ) ).
% is_heap_max
thf(fact_30_val_Osimps,axiom,
! [V: a,Uu: tree_a,Uv: tree_a] :
( ( val_a @ ( t_a @ V @ Uu @ Uv ) )
= V ) ).
% val.simps
thf(fact_31_is__heap_Osimps_I4_J,axiom,
! [V: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
( ( is_heap_a @ ( t_a @ V @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ e_a ) )
= ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
& ( is_heap_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) ) ).
% is_heap.simps(4)
thf(fact_32_is__heap_Osimps_I3_J,axiom,
! [V: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
( ( is_heap_a @ ( t_a @ V @ e_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
= ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) @ V )
& ( is_heap_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) ) ).
% is_heap.simps(3)
thf(fact_33_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_34_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X4: a] : ( member_a @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_35_order__refl,axiom,
! [X: a] : ( ord_less_eq_a @ X @ X ) ).
% order_refl
thf(fact_36_heap__top__geq,axiom,
! [A: a,T: tree_a] :
( ( member_a @ A @ ( set_mset_a @ ( multiset_a2 @ T ) ) )
=> ( ( is_heap_a @ T )
=> ( ord_less_eq_a @ A @ ( val_a @ T ) ) ) ) ).
% heap_top_geq
thf(fact_37_order__subst1,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X5: a,Y2: a] :
( ( ord_less_eq_a @ X5 @ Y2 )
=> ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_38_order__subst2,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X5: a,Y2: a] :
( ( ord_less_eq_a @ X5 @ Y2 )
=> ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_39_ord__eq__le__subst,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X5: a,Y2: a] :
( ( ord_less_eq_a @ X5 @ Y2 )
=> ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_40_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X5: a,Y2: a] :
( ( ord_less_eq_a @ X5 @ Y2 )
=> ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_41_eq__iff,axiom,
( ( ^ [Y3: a,Z: a] : Y3 = Z )
= ( ^ [X4: a,Y4: a] :
( ( ord_less_eq_a @ X4 @ Y4 )
& ( ord_less_eq_a @ Y4 @ X4 ) ) ) ) ).
% eq_iff
thf(fact_42_antisym,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
=> ( ( ord_less_eq_a @ Y @ X )
=> ( X = Y ) ) ) ).
% antisym
thf(fact_43_linear,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
| ( ord_less_eq_a @ Y @ X ) ) ).
% linear
thf(fact_44_eq__refl,axiom,
! [X: a,Y: a] :
( ( X = Y )
=> ( ord_less_eq_a @ X @ Y ) ) ).
% eq_refl
thf(fact_45_le__cases,axiom,
! [X: a,Y: a] :
( ~ ( ord_less_eq_a @ X @ Y )
=> ( ord_less_eq_a @ Y @ X ) ) ).
% le_cases
thf(fact_46_order_Otrans,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_eq_a @ A @ C ) ) ) ).
% order.trans
thf(fact_47_le__cases3,axiom,
! [X: a,Y: a,Z2: a] :
( ( ( ord_less_eq_a @ X @ Y )
=> ~ ( ord_less_eq_a @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_a @ Y @ X )
=> ~ ( ord_less_eq_a @ X @ Z2 ) )
=> ( ( ( ord_less_eq_a @ X @ Z2 )
=> ~ ( ord_less_eq_a @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_a @ Z2 @ Y )
=> ~ ( ord_less_eq_a @ Y @ X ) )
=> ( ( ( ord_less_eq_a @ Y @ Z2 )
=> ~ ( ord_less_eq_a @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_a @ Z2 @ X )
=> ~ ( ord_less_eq_a @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_48_antisym__conv,axiom,
! [Y: a,X: a] :
( ( ord_less_eq_a @ Y @ X )
=> ( ( ord_less_eq_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv
thf(fact_49_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y3: a,Z: a] : Y3 = Z )
= ( ^ [A3: a,B2: a] :
( ( ord_less_eq_a @ A3 @ B2 )
& ( ord_less_eq_a @ B2 @ A3 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_50_ord__eq__le__trans,axiom,
! [A: a,B: a,C: a] :
( ( A = B )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ord_less_eq_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_51_ord__le__eq__trans,axiom,
! [A: a,B: a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_52_order__class_Oorder_Oantisym,axiom,
! [A: a,B: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_a @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_53_order__trans,axiom,
! [X: a,Y: a,Z2: a] :
( ( ord_less_eq_a @ X @ Y )
=> ( ( ord_less_eq_a @ Y @ Z2 )
=> ( ord_less_eq_a @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_54_dual__order_Orefl,axiom,
! [A: a] : ( ord_less_eq_a @ A @ A ) ).
% dual_order.refl
thf(fact_55_linorder__wlog,axiom,
! [P: a > a > $o,A: a,B: a] :
( ! [A4: a,B3: a] :
( ( ord_less_eq_a @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: a,B3: a] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_56_dual__order_Otrans,axiom,
! [B: a,A: a,C: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_eq_a @ C @ B )
=> ( ord_less_eq_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_57_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: a,Z: a] : Y3 = Z )
= ( ^ [A3: a,B2: a] :
( ( ord_less_eq_a @ B2 @ A3 )
& ( ord_less_eq_a @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_58_dual__order_Oantisym,axiom,
! [B: a,A: a] :
( ( ord_less_eq_a @ B @ A )
=> ( ( ord_less_eq_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_59_add__mset__add__mset__same__iff,axiom,
! [A: a,A2: multiset_a,B4: multiset_a] :
( ( ( add_mset_a @ A @ A2 )
= ( add_mset_a @ A @ B4 ) )
= ( A2 = B4 ) ) ).
% add_mset_add_mset_same_iff
thf(fact_60_multi__self__add__other__not__self,axiom,
! [M: multiset_a,X: a] :
( M
!= ( add_mset_a @ X @ M ) ) ).
% multi_self_add_other_not_self
thf(fact_61_mset__add,axiom,
! [A: a,A2: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ A2 ) )
=> ~ ! [B5: multiset_a] :
( A2
!= ( add_mset_a @ A @ B5 ) ) ) ).
% mset_add
thf(fact_62_multi__member__split,axiom,
! [X: a,M: multiset_a] :
( ( member_a @ X @ ( set_mset_a @ M ) )
=> ? [A5: multiset_a] :
( M
= ( add_mset_a @ X @ A5 ) ) ) ).
% multi_member_split
thf(fact_63_insert__noteq__member,axiom,
! [B: a,B4: multiset_a,C: a,C2: multiset_a] :
( ( ( add_mset_a @ B @ B4 )
= ( add_mset_a @ C @ C2 ) )
=> ( ( B != C )
=> ( member_a @ C @ ( set_mset_a @ B4 ) ) ) ) ).
% insert_noteq_member
thf(fact_64_union__single__eq__member,axiom,
! [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_65_heap__top__max,axiom,
! [T: tree_a] :
( ( T != e_a )
=> ( ( is_heap_a @ T )
=> ( ( val_a @ T )
= ( lattic146396397_Max_a @ ( set_mset_a @ ( multiset_a2 @ T ) ) ) ) ) ) ).
% heap_top_max
thf(fact_66_add__mset__commute,axiom,
! [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_67_add__eq__conv__ex,axiom,
! [A: a,M: multiset_a,B: a,N: multiset_a] :
( ( ( add_mset_a @ A @ M )
= ( add_mset_a @ B @ N ) )
= ( ( ( M = N )
& ( A = B ) )
| ? [K: multiset_a] :
( ( M
= ( add_mset_a @ B @ K ) )
& ( N
= ( add_mset_a @ A @ K ) ) ) ) ) ).
% add_eq_conv_ex
thf(fact_68_Greatest__equality,axiom,
! [P: a > $o,X: a] :
( ( P @ X )
=> ( ! [Y2: a] :
( ( P @ Y2 )
=> ( ord_less_eq_a @ Y2 @ X ) )
=> ( ( order_Greatest_a @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_69_GreatestI2__order,axiom,
! [P: a > $o,X: a,Q: a > $o] :
( ( P @ X )
=> ( ! [Y2: a] :
( ( P @ Y2 )
=> ( ord_less_eq_a @ Y2 @ X ) )
=> ( ! [X5: a] :
( ( P @ X5 )
=> ( ! [Y5: a] :
( ( P @ Y5 )
=> ( ord_less_eq_a @ Y5 @ X5 ) )
=> ( Q @ X5 ) ) )
=> ( Q @ ( order_Greatest_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_70_multiset__induct__min,axiom,
! [P: multiset_a > $o,M: multiset_a] :
( ( P @ zero_zero_multiset_a )
=> ( ! [X5: a,M2: multiset_a] :
( ( P @ M2 )
=> ( ! [Xa: a] :
( ( member_a @ Xa @ ( set_mset_a @ M2 ) )
=> ( ord_less_eq_a @ X5 @ Xa ) )
=> ( P @ ( add_mset_a @ X5 @ M2 ) ) ) )
=> ( P @ M ) ) ) ).
% multiset_induct_min
thf(fact_71_multiset__induct__max,axiom,
! [P: multiset_a > $o,M: multiset_a] :
( ( P @ zero_zero_multiset_a )
=> ( ! [X5: a,M2: multiset_a] :
( ( P @ M2 )
=> ( ! [Xa: a] :
( ( member_a @ Xa @ ( set_mset_a @ M2 ) )
=> ( ord_less_eq_a @ Xa @ X5 ) )
=> ( P @ ( add_mset_a @ X5 @ M2 ) ) ) )
=> ( P @ M ) ) ) ).
% multiset_induct_max
thf(fact_72_add__mset__eq__singleton__iff,axiom,
! [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_73_single__eq__add__mset,axiom,
! [A: a,B: a,M: multiset_a] :
( ( ( add_mset_a @ A @ zero_zero_multiset_a )
= ( add_mset_a @ B @ M ) )
= ( ( B = A )
& ( M = zero_zero_multiset_a ) ) ) ).
% single_eq_add_mset
thf(fact_74_add__mset__eq__single,axiom,
! [B: a,M: multiset_a,A: a] :
( ( ( add_mset_a @ B @ M )
= ( add_mset_a @ A @ zero_zero_multiset_a ) )
= ( ( B = A )
& ( M = zero_zero_multiset_a ) ) ) ).
% add_mset_eq_single
thf(fact_75_single__eq__single,axiom,
! [A: a,B: a] :
( ( ( add_mset_a @ A @ zero_zero_multiset_a )
= ( add_mset_a @ B @ zero_zero_multiset_a ) )
= ( A = B ) ) ).
% single_eq_single
thf(fact_76_multiset__cases,axiom,
! [M: multiset_a] :
( ( M != zero_zero_multiset_a )
=> ~ ! [X5: a,N2: multiset_a] :
( M
!= ( add_mset_a @ X5 @ N2 ) ) ) ).
% multiset_cases
thf(fact_77_multiset__induct,axiom,
! [P: multiset_a > $o,M: multiset_a] :
( ( P @ zero_zero_multiset_a )
=> ( ! [X5: a,M2: multiset_a] :
( ( P @ M2 )
=> ( P @ ( add_mset_a @ X5 @ M2 ) ) )
=> ( P @ M ) ) ) ).
% multiset_induct
thf(fact_78_multiset__induct2,axiom,
! [P: multiset_a > multiset_a > $o,M: multiset_a,N: multiset_a] :
( ( P @ zero_zero_multiset_a @ zero_zero_multiset_a )
=> ( ! [A4: a,M2: multiset_a,N2: multiset_a] :
( ( P @ M2 @ N2 )
=> ( P @ ( add_mset_a @ A4 @ M2 ) @ N2 ) )
=> ( ! [A4: a,M2: multiset_a,N2: multiset_a] :
( ( P @ M2 @ N2 )
=> ( P @ M2 @ ( add_mset_a @ A4 @ N2 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% multiset_induct2
thf(fact_79_empty__not__add__mset,axiom,
! [A: a,A2: multiset_a] :
( zero_zero_multiset_a
!= ( add_mset_a @ A @ A2 ) ) ).
% empty_not_add_mset
thf(fact_80_multi__nonempty__split,axiom,
! [M: multiset_a] :
( ( M != zero_zero_multiset_a )
=> ? [A5: multiset_a,A4: a] :
( M
= ( add_mset_a @ A4 @ A5 ) ) ) ).
% multi_nonempty_split
thf(fact_81_multiset__nonemptyE,axiom,
! [A2: multiset_a] :
( ( A2 != zero_zero_multiset_a )
=> ~ ! [X5: a] :
~ ( member_a @ X5 @ ( set_mset_a @ A2 ) ) ) ).
% multiset_nonemptyE
thf(fact_82_multi__member__last,axiom,
! [X: a] : ( member_a @ X @ ( set_mset_a @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) ) ).
% multi_member_last
thf(fact_83_multiset_Osimps_I1_J,axiom,
( ( multiset_a2 @ e_a )
= zero_zero_multiset_a ) ).
% multiset.simps(1)
thf(fact_84_Multiset_Ois__empty__def,axiom,
( is_empty_a
= ( ^ [A6: multiset_a] : A6 = zero_zero_multiset_a ) ) ).
% Multiset.is_empty_def
thf(fact_85_multiset_Osimps_I2_J,axiom,
! [V: a,L: tree_a,R: tree_a] :
( ( multiset_a2 @ ( t_a @ V @ L @ R ) )
= ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ ( multiset_a2 @ L ) @ ( add_mset_a @ V @ zero_zero_multiset_a ) ) @ ( multiset_a2 @ R ) ) ) ).
% multiset.simps(2)
thf(fact_86_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
! [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_87_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
! [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_88_union__eq__empty,axiom,
! [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_89_empty__eq__union,axiom,
! [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_90_union__mset__add__mset__right,axiom,
! [A2: multiset_a,A: a,B4: multiset_a] :
( ( plus_plus_multiset_a @ A2 @ ( add_mset_a @ A @ B4 ) )
= ( add_mset_a @ A @ ( plus_plus_multiset_a @ A2 @ B4 ) ) ) ).
% union_mset_add_mset_right
thf(fact_91_union__mset__add__mset__left,axiom,
! [A: a,A2: multiset_a,B4: multiset_a] :
( ( plus_plus_multiset_a @ ( add_mset_a @ A @ A2 ) @ B4 )
= ( add_mset_a @ A @ ( plus_plus_multiset_a @ A2 @ B4 ) ) ) ).
% union_mset_add_mset_left
thf(fact_92_add__right__mono,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a] :
( ( ord_le1199012836iset_a @ A @ B )
=> ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ A @ C ) @ ( plus_plus_multiset_a @ B @ C ) ) ) ).
% add_right_mono
thf(fact_93_add__left__mono,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a] :
( ( ord_le1199012836iset_a @ A @ B )
=> ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ C @ A ) @ ( plus_plus_multiset_a @ C @ B ) ) ) ).
% add_left_mono
thf(fact_94_add__mono,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a,D: multiset_a] :
( ( ord_le1199012836iset_a @ A @ B )
=> ( ( ord_le1199012836iset_a @ C @ D )
=> ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ A @ C ) @ ( plus_plus_multiset_a @ B @ D ) ) ) ) ).
% add_mono
thf(fact_95_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: multiset_a,J: multiset_a,K2: multiset_a,L: multiset_a] :
( ( ( ord_le1199012836iset_a @ I @ J )
& ( ord_le1199012836iset_a @ K2 @ L ) )
=> ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ I @ K2 ) @ ( plus_plus_multiset_a @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_96_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: multiset_a,J: multiset_a,K2: multiset_a,L: multiset_a] :
( ( ( I = J )
& ( ord_le1199012836iset_a @ K2 @ L ) )
=> ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ I @ K2 ) @ ( plus_plus_multiset_a @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_97_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: multiset_a,J: multiset_a,K2: multiset_a,L: multiset_a] :
( ( ( ord_le1199012836iset_a @ I @ J )
& ( K2 = L ) )
=> ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ I @ K2 ) @ ( plus_plus_multiset_a @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_98_empty__neutral_I2_J,axiom,
! [X: multiset_a] :
( ( plus_plus_multiset_a @ X @ zero_zero_multiset_a )
= X ) ).
% empty_neutral(2)
thf(fact_99_empty__neutral_I1_J,axiom,
! [X: multiset_a] :
( ( plus_plus_multiset_a @ zero_zero_multiset_a @ X )
= X ) ).
% empty_neutral(1)
thf(fact_100_union__iff,axiom,
! [A: a,A2: multiset_a,B4: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ ( plus_plus_multiset_a @ A2 @ B4 ) ) )
= ( ( member_a @ A @ ( set_mset_a @ A2 ) )
| ( member_a @ A @ ( set_mset_a @ B4 ) ) ) ) ).
% union_iff
thf(fact_101_union__assoc,axiom,
! [M: multiset_a,N: multiset_a,K3: multiset_a] :
( ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ M @ N ) @ K3 )
= ( plus_plus_multiset_a @ M @ ( plus_plus_multiset_a @ N @ K3 ) ) ) ).
% union_assoc
thf(fact_102_union__lcomm,axiom,
! [M: multiset_a,N: multiset_a,K3: multiset_a] :
( ( plus_plus_multiset_a @ M @ ( plus_plus_multiset_a @ N @ K3 ) )
= ( plus_plus_multiset_a @ N @ ( plus_plus_multiset_a @ M @ K3 ) ) ) ).
% union_lcomm
thf(fact_103_union__commute,axiom,
( plus_plus_multiset_a
= ( ^ [M3: multiset_a,N3: multiset_a] : ( plus_plus_multiset_a @ N3 @ M3 ) ) ) ).
% union_commute
thf(fact_104_union__left__cancel,axiom,
! [K3: multiset_a,M: multiset_a,N: multiset_a] :
( ( ( plus_plus_multiset_a @ K3 @ M )
= ( plus_plus_multiset_a @ K3 @ N ) )
= ( M = N ) ) ).
% union_left_cancel
thf(fact_105_union__right__cancel,axiom,
! [M: multiset_a,K3: multiset_a,N: multiset_a] :
( ( ( plus_plus_multiset_a @ M @ K3 )
= ( plus_plus_multiset_a @ N @ K3 ) )
= ( M = N ) ) ).
% union_right_cancel
thf(fact_106_multi__union__self__other__eq,axiom,
! [A2: multiset_a,X6: multiset_a,Y6: multiset_a] :
( ( ( plus_plus_multiset_a @ A2 @ X6 )
= ( plus_plus_multiset_a @ A2 @ Y6 ) )
=> ( X6 = Y6 ) ) ).
% multi_union_self_other_eq
thf(fact_107_add__mset__add__single,axiom,
( add_mset_a
= ( ^ [A3: a,A6: multiset_a] : ( plus_plus_multiset_a @ A6 @ ( add_mset_a @ A3 @ zero_zero_multiset_a ) ) ) ) ).
% add_mset_add_single
thf(fact_108_union__is__single,axiom,
! [M: multiset_a,N: multiset_a,A: a] :
( ( ( plus_plus_multiset_a @ M @ N )
= ( add_mset_a @ A @ zero_zero_multiset_a ) )
= ( ( ( M
= ( add_mset_a @ A @ zero_zero_multiset_a ) )
& ( N = zero_zero_multiset_a ) )
| ( ( M = zero_zero_multiset_a )
& ( N
= ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) ) ) ).
% union_is_single
thf(fact_109_single__is__union,axiom,
! [A: a,M: multiset_a,N: multiset_a] :
( ( ( add_mset_a @ A @ zero_zero_multiset_a )
= ( plus_plus_multiset_a @ M @ N ) )
= ( ( ( ( add_mset_a @ A @ zero_zero_multiset_a )
= M )
& ( N = zero_zero_multiset_a ) )
| ( ( M = zero_zero_multiset_a )
& ( ( add_mset_a @ A @ zero_zero_multiset_a )
= N ) ) ) ) ).
% single_is_union
thf(fact_110_multi__member__skip,axiom,
! [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_111_multi__member__this,axiom,
! [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_112_mult1E,axiom,
! [N: multiset_a,M: multiset_a,R: set_Product_prod_a_a] :
( ( member340150864iset_a @ ( produc2037245207iset_a @ N @ M ) @ ( mult1_a @ R ) )
=> ~ ! [A4: a,M0: multiset_a] :
( ( M
= ( add_mset_a @ A4 @ M0 ) )
=> ! [K4: multiset_a] :
( ( N
= ( plus_plus_multiset_a @ M0 @ K4 ) )
=> ~ ! [B6: a] :
( ( member_a @ B6 @ ( set_mset_a @ K4 ) )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ B6 @ A4 ) @ R ) ) ) ) ) ).
% mult1E
thf(fact_113_mult1I,axiom,
! [M: multiset_a,A: a,M02: multiset_a,N: multiset_a,K3: multiset_a,R: set_Product_prod_a_a] :
( ( M
= ( add_mset_a @ A @ M02 ) )
=> ( ( N
= ( plus_plus_multiset_a @ M02 @ K3 ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ ( set_mset_a @ K3 ) )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ B3 @ A ) @ R ) )
=> ( member340150864iset_a @ ( produc2037245207iset_a @ N @ M ) @ ( mult1_a @ R ) ) ) ) ) ).
% mult1I
thf(fact_114_less__add,axiom,
! [N: multiset_a,A: a,M02: multiset_a,R: set_Product_prod_a_a] :
( ( member340150864iset_a @ ( produc2037245207iset_a @ N @ ( add_mset_a @ A @ M02 ) ) @ ( mult1_a @ R ) )
=> ( ? [M2: multiset_a] :
( ( member340150864iset_a @ ( produc2037245207iset_a @ M2 @ M02 ) @ ( mult1_a @ R ) )
& ( N
= ( add_mset_a @ A @ M2 ) ) )
| ? [K4: multiset_a] :
( ! [B6: a] :
( ( member_a @ B6 @ ( set_mset_a @ K4 ) )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ B6 @ A ) @ R ) )
& ( N
= ( plus_plus_multiset_a @ M02 @ K4 ) ) ) ) ) ).
% less_add
thf(fact_115_one__step__implies__mult,axiom,
! [J2: multiset_a,K3: multiset_a,R: set_Product_prod_a_a,I2: multiset_a] :
( ( J2 != zero_zero_multiset_a )
=> ( ! [X5: a] :
( ( member_a @ X5 @ ( set_mset_a @ K3 ) )
=> ? [Xa: a] :
( ( member_a @ Xa @ ( set_mset_a @ J2 ) )
& ( member449909584od_a_a @ ( product_Pair_a_a @ X5 @ Xa ) @ R ) ) )
=> ( member340150864iset_a @ ( produc2037245207iset_a @ ( plus_plus_multiset_a @ I2 @ K3 ) @ ( plus_plus_multiset_a @ I2 @ J2 ) ) @ ( mult_a @ R ) ) ) ) ).
% one_step_implies_mult
thf(fact_116_not__less__empty,axiom,
! [M: multiset_a,R: set_Product_prod_a_a] :
~ ( member340150864iset_a @ ( produc2037245207iset_a @ M @ zero_zero_multiset_a ) @ ( mult1_a @ R ) ) ).
% not_less_empty
thf(fact_117_mult1__union,axiom,
! [B4: multiset_a,D2: multiset_a,R: set_Product_prod_a_a,C2: multiset_a] :
( ( member340150864iset_a @ ( produc2037245207iset_a @ B4 @ D2 ) @ ( mult1_a @ R ) )
=> ( member340150864iset_a @ ( produc2037245207iset_a @ ( plus_plus_multiset_a @ C2 @ B4 ) @ ( plus_plus_multiset_a @ C2 @ D2 ) ) @ ( mult1_a @ R ) ) ) ).
% mult1_union
thf(fact_118_mult__implies__one__step,axiom,
! [R: set_Product_prod_a_a,M: multiset_a,N: multiset_a] :
( ( trans_a @ R )
=> ( ( member340150864iset_a @ ( produc2037245207iset_a @ M @ N ) @ ( mult_a @ R ) )
=> ? [I3: multiset_a,J3: multiset_a] :
( ( N
= ( plus_plus_multiset_a @ I3 @ J3 ) )
& ? [K4: multiset_a] :
( ( M
= ( plus_plus_multiset_a @ I3 @ K4 ) )
& ( J3 != zero_zero_multiset_a )
& ! [X7: a] :
( ( member_a @ X7 @ ( set_mset_a @ K4 ) )
=> ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_mset_a @ J3 ) )
& ( member449909584od_a_a @ ( product_Pair_a_a @ X7 @ Xa2 ) @ R ) ) ) ) ) ) ) ).
% mult_implies_one_step
thf(fact_119_subset__mset_Osum__mset__0__iff,axiom,
! [M: multiset_multiset_a] :
( ( ( comm_m315775925iset_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_120_mult__cancel__add__mset,axiom,
! [S: set_Product_prod_a_a,Uu: a,X6: multiset_a,Y6: multiset_a] :
( ( trans_a @ S )
=> ( ( irrefl_a @ S )
=> ( ( member340150864iset_a @ ( produc2037245207iset_a @ ( add_mset_a @ Uu @ X6 ) @ ( add_mset_a @ Uu @ Y6 ) ) @ ( mult_a @ S ) )
= ( member340150864iset_a @ ( produc2037245207iset_a @ X6 @ Y6 ) @ ( mult_a @ S ) ) ) ) ) ).
% mult_cancel_add_mset
thf(fact_121_mult__cancel,axiom,
! [S: set_Product_prod_a_a,X6: multiset_a,Z3: multiset_a,Y6: multiset_a] :
( ( trans_a @ S )
=> ( ( irrefl_a @ S )
=> ( ( member340150864iset_a @ ( produc2037245207iset_a @ ( plus_plus_multiset_a @ X6 @ Z3 ) @ ( plus_plus_multiset_a @ Y6 @ Z3 ) ) @ ( mult_a @ S ) )
= ( member340150864iset_a @ ( produc2037245207iset_a @ X6 @ Y6 ) @ ( mult_a @ S ) ) ) ) ) ).
% mult_cancel
thf(fact_122_in__mset__fold__plus__iff,axiom,
! [X: a,M: multiset_a,NN: multiset_multiset_a] :
( ( member_a @ X @ ( set_mset_a @ ( fold_m382157835iset_a @ plus_plus_multiset_a @ M @ NN ) ) )
= ( ( member_a @ X @ ( set_mset_a @ M ) )
| ? [N3: multiset_a] :
( ( member_multiset_a @ N3 @ ( set_mset_multiset_a @ NN ) )
& ( member_a @ X @ ( set_mset_a @ N3 ) ) ) ) ) ).
% in_mset_fold_plus_iff
thf(fact_123_union__fold__mset__add__mset,axiom,
( plus_plus_multiset_a
= ( fold_m364285649iset_a @ add_mset_a ) ) ).
% union_fold_mset_add_mset
thf(fact_124_add__mset__replicate__mset__safe,axiom,
! [M: multiset_a,A: a] :
( ( nO_MAT1617603563iset_a @ zero_zero_multiset_a @ M )
=> ( ( add_mset_a @ A @ M )
= ( plus_plus_multiset_a @ ( add_mset_a @ A @ zero_zero_multiset_a ) @ M ) ) ) ).
% add_mset_replicate_mset_safe
thf(fact_125_insert__DiffM2,axiom,
! [X: a,M: multiset_a] :
( ( member_a @ X @ ( set_mset_a @ M ) )
=> ( ( plus_plus_multiset_a @ ( minus_1649712171iset_a @ M @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) @ ( add_mset_a @ X @ zero_zero_multiset_a ) )
= M ) ) ).
% insert_DiffM2
thf(fact_126_diff__diff__add__mset,axiom,
! [M: multiset_a,N: multiset_a,P: multiset_a] :
( ( minus_1649712171iset_a @ ( minus_1649712171iset_a @ M @ N ) @ P )
= ( minus_1649712171iset_a @ M @ ( plus_plus_multiset_a @ N @ P ) ) ) ).
% diff_diff_add_mset
thf(fact_127_add__mset__remove__trivial,axiom,
! [X: a,M: multiset_a] :
( ( minus_1649712171iset_a @ ( add_mset_a @ X @ M ) @ ( add_mset_a @ X @ zero_zero_multiset_a ) )
= M ) ).
% add_mset_remove_trivial
thf(fact_128_diff__add__mset__swap,axiom,
! [B: a,A2: multiset_a,M: multiset_a] :
( ~ ( member_a @ B @ ( set_mset_a @ A2 ) )
=> ( ( minus_1649712171iset_a @ ( add_mset_a @ B @ M ) @ A2 )
= ( add_mset_a @ B @ ( minus_1649712171iset_a @ M @ A2 ) ) ) ) ).
% diff_add_mset_swap
thf(fact_129_diff__union__swap2,axiom,
! [Y: a,M: multiset_a,X: a] :
( ( member_a @ Y @ ( set_mset_a @ M ) )
=> ( ( minus_1649712171iset_a @ ( add_mset_a @ X @ M ) @ ( add_mset_a @ Y @ zero_zero_multiset_a ) )
= ( add_mset_a @ X @ ( minus_1649712171iset_a @ M @ ( add_mset_a @ Y @ zero_zero_multiset_a ) ) ) ) ) ).
% diff_union_swap2
thf(fact_130_insert__DiffM,axiom,
! [X: a,M: multiset_a] :
( ( member_a @ X @ ( set_mset_a @ M ) )
=> ( ( add_mset_a @ X @ ( minus_1649712171iset_a @ M @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) )
= M ) ) ).
% insert_DiffM
thf(fact_131_in__diffD,axiom,
! [A: a,M: multiset_a,N: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ ( minus_1649712171iset_a @ M @ N ) ) )
=> ( member_a @ A @ ( set_mset_a @ M ) ) ) ).
% in_diffD
thf(fact_132_add__mset__diff__bothsides,axiom,
! [A: a,M: multiset_a,A2: multiset_a] :
( ( minus_1649712171iset_a @ ( add_mset_a @ A @ M ) @ ( add_mset_a @ A @ A2 ) )
= ( minus_1649712171iset_a @ M @ A2 ) ) ).
% add_mset_diff_bothsides
thf(fact_133_Multiset_Odiff__cancel,axiom,
! [A2: multiset_a] :
( ( minus_1649712171iset_a @ A2 @ A2 )
= zero_zero_multiset_a ) ).
% Multiset.diff_cancel
thf(fact_134_diff__empty,axiom,
! [M: multiset_a] :
( ( ( minus_1649712171iset_a @ M @ zero_zero_multiset_a )
= M )
& ( ( minus_1649712171iset_a @ zero_zero_multiset_a @ M )
= zero_zero_multiset_a ) ) ).
% diff_empty
thf(fact_135_union__single__eq__diff,axiom,
! [X: a,M: multiset_a,N: multiset_a] :
( ( ( add_mset_a @ X @ M )
= N )
=> ( M
= ( minus_1649712171iset_a @ N @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) ) ) ).
% union_single_eq_diff
thf(fact_136_add__eq__conv__diff,axiom,
! [A: a,M: multiset_a,B: a,N: multiset_a] :
( ( ( add_mset_a @ A @ M )
= ( add_mset_a @ B @ N ) )
= ( ( ( M = N )
& ( A = B ) )
| ( ( M
= ( add_mset_a @ B @ ( minus_1649712171iset_a @ N @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) )
& ( N
= ( add_mset_a @ A @ ( minus_1649712171iset_a @ M @ ( add_mset_a @ B @ zero_zero_multiset_a ) ) ) ) ) ) ) ).
% add_eq_conv_diff
thf(fact_137_diff__union__swap,axiom,
! [A: a,B: a,M: multiset_a] :
( ( A != B )
=> ( ( add_mset_a @ B @ ( minus_1649712171iset_a @ M @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) )
= ( minus_1649712171iset_a @ ( add_mset_a @ B @ M ) @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) ) ).
% diff_union_swap
thf(fact_138_diff__union__cancelR,axiom,
! [M: multiset_a,N: multiset_a] :
( ( minus_1649712171iset_a @ ( plus_plus_multiset_a @ M @ N ) @ N )
= M ) ).
% diff_union_cancelR
thf(fact_139_diff__union__cancelL,axiom,
! [N: multiset_a,M: multiset_a] :
( ( minus_1649712171iset_a @ ( plus_plus_multiset_a @ N @ M ) @ N )
= M ) ).
% diff_union_cancelL
thf(fact_140_Multiset_Odiff__add,axiom,
! [M: multiset_a,N: multiset_a,Q: multiset_a] :
( ( minus_1649712171iset_a @ M @ ( plus_plus_multiset_a @ N @ Q ) )
= ( minus_1649712171iset_a @ ( minus_1649712171iset_a @ M @ N ) @ Q ) ) ).
% Multiset.diff_add
thf(fact_141_diff__single__trivial,axiom,
! [X: a,M: multiset_a] :
( ~ ( member_a @ X @ ( set_mset_a @ M ) )
=> ( ( minus_1649712171iset_a @ M @ ( add_mset_a @ X @ zero_zero_multiset_a ) )
= M ) ) ).
% diff_single_trivial
% Conjectures (1)
thf(conj_0,conjecture,
is_heap_a @ ( heapIm970322378pify_a @ ( t_a @ v @ l @ r ) ) ).
%------------------------------------------------------------------------------