TPTP Problem File: ITP070^1.p
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%------------------------------------------------------------------------------
% File : ITP070^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer HeapImperative problem prob_824__5349520_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_824__5349520_1 [Des21]
% Status : Theorem
% Rating : 0.40 v8.2.0, 0.31 v8.1.0, 0.27 v7.5.0
% Syntax : Number of formulae : 269 ( 98 unt; 54 typ; 0 def)
% Number of atoms : 653 ( 280 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 2679 ( 90 ~; 7 |; 54 &;2166 @)
% ( 0 <=>; 362 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 8 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 343 ( 343 >; 0 *; 0 +; 0 <<)
% Number of symbols : 50 ( 47 usr; 11 con; 0-6 aty)
% Number of variables : 852 ( 52 ^; 787 !; 13 ?; 852 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:31:22.775
%------------------------------------------------------------------------------
% Could-be-implicit typings (7)
thf(ty_n_t__Product____Type__Oprod_It__Heap__OTree_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
produc768687417list_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Heap__OTree_Itf__a_J_J,type,
produc143150363Tree_a: $tType ).
thf(ty_n_t__Multiset__Omultiset_Itf__a_J,type,
multiset_a: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_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 (47)
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_HeapImperative__Mirabelle__ksbqzsoydx_Oheapify_001tf__a,type,
heapIm970322378pify_a: tree_a > tree_a ).
thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_Ohs__is__empty_001tf__a,type,
heapIm229596386mpty_a: tree_a > $o ).
thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_Ohs__of__list_001tf__a,type,
heapIm1057938560list_a: list_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_Oof__list__tree_001tf__a,type,
heapIm1637418125tree_a: list_a > tree_a ).
thf(sy_c_HeapImperative__Mirabelle__ksbqzsoydx_OremoveLeaf_001tf__a,type,
heapIm837449470Leaf_a: tree_a > produc143150363Tree_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_OHeap_001t__Heap__OTree_Itf__a_J_001tf__a,type,
heap_Tree_a_a: tree_a > ( tree_a > $o ) > ( list_a > tree_a ) > ( tree_a > multiset_a ) > ( tree_a > tree_a ) > ( tree_a > produc143150363Tree_a ) > $o ).
thf(sy_c_Heap_OHeap__axioms_001t__Heap__OTree_Itf__a_J_001tf__a,type,
heap_axioms_Tree_a_a: ( tree_a > $o ) > ( list_a > tree_a ) > ( tree_a > multiset_a ) > ( tree_a > tree_a ) > ( tree_a > produc143150363Tree_a ) > $o ).
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_List_Olist_OCons_001tf__a,type,
cons_a: a > list_a > list_a ).
thf(sy_c_Multiset_Oadd__mset_001tf__a,type,
add_mset_a: a > multiset_a > multiset_a ).
thf(sy_c_Multiset_Ois__empty_001tf__a,type,
is_empty_a: multiset_a > $o ).
thf(sy_c_Multiset_Oset__mset_001tf__a,type,
set_mset_a: multiset_a > set_a ).
thf(sy_c_Multiset_Osubseteq__mset_001tf__a,type,
subseteq_mset_a: multiset_a > multiset_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_062_I_Eo_Mtf__a_J_J,type,
ord_less_eq_o_o_a: ( $o > $o > a ) > ( $o > $o > a ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mtf__a_J,type,
ord_less_eq_o_a: ( $o > a ) > ( $o > a ) > $o ).
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_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_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_001_062_I_Eo_Mtf__a_J,type,
order_Greatest_o_a: ( ( $o > a ) > $o ) > $o > a ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001tf__a,type,
order_Greatest_a: ( a > $o ) > a ).
thf(sy_c_Product__Type_OPair_001t__Heap__OTree_Itf__a_J_001t__List__Olist_Itf__a_J,type,
produc1352981801list_a: tree_a > list_a > produc768687417list_a ).
thf(sy_c_Product__Type_OPair_001tf__a_001t__Heap__OTree_Itf__a_J,type,
produc686083979Tree_a: a > tree_a > produc143150363Tree_a ).
thf(sy_c_RemoveMax_OCollection_Oset_001t__Heap__OTree_Itf__a_J_001tf__a,type,
set_Tree_a_a: ( tree_a > multiset_a ) > tree_a > set_a ).
thf(sy_c_RemoveMax_ORemoveMax_001t__Heap__OTree_Itf__a_J_001tf__a,type,
removeMax_Tree_a_a: tree_a > ( tree_a > $o ) > ( list_a > tree_a ) > ( tree_a > multiset_a ) > ( tree_a > produc143150363Tree_a ) > ( tree_a > $o ) > $o ).
thf(sy_c_RemoveMax_ORemoveMax_Ossort_H_001t__Heap__OTree_Itf__a_J_001tf__a,type,
ssort_Tree_a_a: ( tree_a > $o ) > ( tree_a > produc143150363Tree_a ) > tree_a > list_a > list_a ).
thf(sy_c_RemoveMax_ORemoveMax_Ossort_H__dom_001t__Heap__OTree_Itf__a_J_001tf__a,type,
ssort_dom_Tree_a_a: ( tree_a > $o ) > ( tree_a > produc143150363Tree_a ) > produc768687417list_a > $o ).
thf(sy_c_RemoveMax_ORemoveMax__axioms_001t__Heap__OTree_Itf__a_J_001tf__a,type,
remove301631099ee_a_a: ( tree_a > $o ) > ( list_a > tree_a ) > ( tree_a > multiset_a ) > ( tree_a > produc143150363Tree_a ) > ( tree_a > $o ) > $o ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_l1____,type,
l1: tree_a ).
thf(sy_v_l2____,type,
l2: tree_a ).
thf(sy_v_r1____,type,
r1: tree_a ).
thf(sy_v_r2____,type,
r2: tree_a ).
thf(sy_v_v1____,type,
v1: a ).
thf(sy_v_v2____,type,
v2: a ).
thf(sy_v_v____,type,
v: a ).
% Relevant facts (214)
thf(fact_0__092_060open_062v2_A_092_060le_062_Av1_092_060close_062,axiom,
ord_less_eq_a @ v2 @ v1 ).
% \<open>v2 \<le> v1\<close>
thf(fact_1_True,axiom,
ord_less_eq_a @ v1 @ v ).
% True
thf(fact_2__C5__2_Ohyps_C_I2_J,axiom,
( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ v2 @ l2 @ r2 ) ) @ ( val_a @ ( t_a @ v1 @ l1 @ r1 ) ) )
=> ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ v2 @ l2 @ r2 ) ) @ v )
=> ( ( multiset_a2 @ ( heapIm1091024090Down_a @ ( t_a @ v @ ( heapIm1140443833left_a @ ( t_a @ v2 @ l2 @ r2 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ v2 @ l2 @ r2 ) ) ) ) )
= ( multiset_a2 @ ( t_a @ v @ ( heapIm1140443833left_a @ ( t_a @ v2 @ l2 @ r2 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ v2 @ l2 @ r2 ) ) ) ) ) ) ) ).
% "5_2.hyps"(2)
thf(fact_3__C5__2_Ohyps_C_I1_J,axiom,
( ( ord_less_eq_a @ ( val_a @ ( t_a @ v2 @ l2 @ r2 ) ) @ ( val_a @ ( t_a @ v1 @ l1 @ r1 ) ) )
=> ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ v1 @ l1 @ r1 ) ) @ v )
=> ( ( multiset_a2 @ ( heapIm1091024090Down_a @ ( t_a @ v @ ( heapIm1140443833left_a @ ( t_a @ v1 @ l1 @ r1 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ v1 @ l1 @ r1 ) ) ) ) )
= ( multiset_a2 @ ( t_a @ v @ ( heapIm1140443833left_a @ ( t_a @ v1 @ l1 @ r1 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ v1 @ l1 @ r1 ) ) ) ) ) ) ) ).
% "5_2.hyps"(1)
thf(fact_4_siftDown__Node,axiom,
! [T: tree_a,V: a,L: tree_a,R: tree_a] :
( ( T
= ( t_a @ V @ L @ R ) )
=> ? [L2: tree_a,V2: a,R2: tree_a] :
( ( ( heapIm1091024090Down_a @ T )
= ( t_a @ V2 @ L2 @ R2 ) )
& ( ord_less_eq_a @ V @ V2 ) ) ) ).
% siftDown_Node
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_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_7_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_8_siftDown__in__tree__set,axiom,
( in_tree_a
= ( ^ [V3: a,T2: tree_a] : ( in_tree_a @ V3 @ ( heapIm1091024090Down_a @ T2 ) ) ) ) ).
% siftDown_in_tree_set
thf(fact_9_left_Osimps,axiom,
! [V: a,L: tree_a,R: tree_a] :
( ( heapIm1140443833left_a @ ( t_a @ V @ L @ R ) )
= L ) ).
% left.simps
thf(fact_10_right_Osimps,axiom,
! [V: a,L: tree_a,R: tree_a] :
( ( heapIm1257206334ight_a @ ( t_a @ V @ L @ R ) )
= R ) ).
% right.simps
thf(fact_11_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_12_siftDown_Osimps_I1_J,axiom,
( ( heapIm1091024090Down_a @ e_a )
= e_a ) ).
% siftDown.simps(1)
thf(fact_13_in__tree_Osimps_I1_J,axiom,
! [V: a] :
~ ( in_tree_a @ V @ e_a ) ).
% in_tree.simps(1)
thf(fact_14_is__heap_Osimps_I6_J,axiom,
! [V: a,Vd: a,Ve: tree_a,Vf: tree_a,Va: a,Vb: tree_a,Vc: tree_a] :
( ( is_heap_a @ ( t_a @ V @ ( t_a @ Vd @ Ve @ Vf ) @ ( t_a @ Va @ Vb @ Vc ) ) )
= ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
& ( is_heap_a @ ( t_a @ Va @ Vb @ Vc ) )
& ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ V )
& ( is_heap_a @ ( t_a @ Vd @ Ve @ Vf ) ) ) ) ).
% is_heap.simps(6)
thf(fact_15_is__heap_Osimps_I5_J,axiom,
! [V: a,Va: a,Vb: tree_a,Vc: tree_a,Vd: a,Ve: tree_a,Vf: tree_a] :
( ( is_heap_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) )
= ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ V )
& ( is_heap_a @ ( t_a @ Vd @ Ve @ Vf ) )
& ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
& ( is_heap_a @ ( t_a @ Va @ Vb @ Vc ) ) ) ) ).
% is_heap.simps(5)
thf(fact_16_is__heap_Osimps_I4_J,axiom,
! [V: a,Va: a,Vb: tree_a,Vc: tree_a] :
( ( is_heap_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ e_a ) )
= ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
& ( is_heap_a @ ( t_a @ Va @ Vb @ Vc ) ) ) ) ).
% is_heap.simps(4)
thf(fact_17_is__heap_Osimps_I3_J,axiom,
! [V: a,Va: a,Vb: tree_a,Vc: tree_a] :
( ( is_heap_a @ ( t_a @ V @ e_a @ ( t_a @ Va @ Vb @ Vc ) ) )
= ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
& ( is_heap_a @ ( t_a @ Va @ Vb @ Vc ) ) ) ) ).
% is_heap.simps(3)
thf(fact_18_is__heap_Osimps_I2_J,axiom,
! [V: a] : ( is_heap_a @ ( t_a @ V @ e_a @ e_a ) ) ).
% is_heap.simps(2)
thf(fact_19_is__heap_Osimps_I1_J,axiom,
is_heap_a @ e_a ).
% is_heap.simps(1)
thf(fact_20_heapify_Osimps_I1_J,axiom,
( ( heapIm970322378pify_a @ e_a )
= e_a ) ).
% heapify.simps(1)
thf(fact_21_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_22_heapify__heap__is__heap,axiom,
! [T: tree_a] : ( is_heap_a @ ( heapIm970322378pify_a @ T ) ) ).
% heapify_heap_is_heap
thf(fact_23_siftDown__in__tree,axiom,
! [T: tree_a] :
( ( T != e_a )
=> ( in_tree_a @ ( val_a @ ( heapIm1091024090Down_a @ T ) ) @ T ) ) ).
% siftDown_in_tree
thf(fact_24_is__heap_Ocases,axiom,
! [X: tree_a] :
( ( X != e_a )
=> ( ! [V4: a] :
( X
!= ( t_a @ V4 @ e_a @ e_a ) )
=> ( ! [V4: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
( X
!= ( t_a @ V4 @ e_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
=> ( ! [V4: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
( X
!= ( t_a @ V4 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ e_a ) )
=> ~ ! [V4: a,Va2: a,Vb2: tree_a,Vc2: tree_a,Vd2: a,Ve2: tree_a,Vf2: tree_a] :
( X
!= ( t_a @ V4 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) ) ) ) ) ) ).
% is_heap.cases
thf(fact_25_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_26_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_27_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_28_siftDown_Osimps_I3_J,axiom,
! [Va: a,Vb: tree_a,Vc: tree_a,V: a] :
( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
=> ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ e_a ) )
= ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ e_a ) ) )
& ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
=> ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ e_a ) )
= ( t_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Va @ Vb @ Vc ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Va @ Vb @ Vc ) ) ) ) @ e_a ) ) ) ) ).
% siftDown.simps(3)
thf(fact_29_siftDown_Osimps_I4_J,axiom,
! [Va: a,Vb: tree_a,Vc: tree_a,V: a] :
( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
=> ( ( heapIm1091024090Down_a @ ( t_a @ V @ e_a @ ( t_a @ Va @ Vb @ Vc ) ) )
= ( t_a @ V @ e_a @ ( t_a @ Va @ Vb @ Vc ) ) ) )
& ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
=> ( ( heapIm1091024090Down_a @ ( t_a @ V @ e_a @ ( t_a @ Va @ Vb @ Vc ) ) )
= ( t_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ e_a @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Va @ Vb @ Vc ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Va @ Vb @ Vc ) ) ) ) ) ) ) ) ).
% siftDown.simps(4)
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_siftDown_Osimps_I5_J,axiom,
! [Vd: a,Ve: tree_a,Vf: tree_a,Va: a,Vb: tree_a,Vc: tree_a,V: a] :
( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) )
=> ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
=> ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) )
= ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) ) )
& ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
=> ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) )
= ( t_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Va @ Vb @ Vc ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Va @ Vb @ Vc ) ) ) ) @ ( t_a @ Vd @ Ve @ Vf ) ) ) ) ) )
& ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) )
=> ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ V )
=> ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) )
= ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) ) )
& ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ V )
=> ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Va @ Vb @ Vc ) @ ( t_a @ Vd @ Ve @ Vf ) ) )
= ( t_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ ( t_a @ Va @ Vb @ Vc ) @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Vd @ Ve @ Vf ) ) ) ) ) ) ) ) ) ) ).
% siftDown.simps(5)
thf(fact_32_siftDown_Osimps_I6_J,axiom,
! [Va: a,Vb: tree_a,Vc: tree_a,Vd: a,Ve: tree_a,Vf: tree_a,V: a] :
( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) )
=> ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ V )
=> ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Vd @ Ve @ Vf ) @ ( t_a @ Va @ Vb @ Vc ) ) )
= ( t_a @ V @ ( t_a @ Vd @ Ve @ Vf ) @ ( t_a @ Va @ Vb @ Vc ) ) ) )
& ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ V )
=> ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Vd @ Ve @ Vf ) @ ( t_a @ Va @ Vb @ Vc ) ) )
= ( t_a @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Vd @ Ve @ Vf ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Vd @ Ve @ Vf ) ) ) ) @ ( t_a @ Va @ Vb @ Vc ) ) ) ) ) )
& ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ ( val_a @ ( t_a @ Vd @ Ve @ Vf ) ) )
=> ( ( ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
=> ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Vd @ Ve @ Vf ) @ ( t_a @ Va @ Vb @ Vc ) ) )
= ( t_a @ V @ ( t_a @ Vd @ Ve @ Vf ) @ ( t_a @ Va @ Vb @ Vc ) ) ) )
& ( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ V )
=> ( ( heapIm1091024090Down_a @ ( t_a @ V @ ( t_a @ Vd @ Ve @ Vf ) @ ( t_a @ Va @ Vb @ Vc ) ) )
= ( t_a @ ( val_a @ ( t_a @ Va @ Vb @ Vc ) ) @ ( t_a @ Vd @ Ve @ Vf ) @ ( heapIm1091024090Down_a @ ( t_a @ V @ ( heapIm1140443833left_a @ ( t_a @ Va @ Vb @ Vc ) ) @ ( heapIm1257206334ight_a @ ( t_a @ Va @ Vb @ Vc ) ) ) ) ) ) ) ) ) ) ).
% siftDown.simps(6)
thf(fact_33_in__tree_Osimps_I2_J,axiom,
! [V: a,V5: a,L: tree_a,R: tree_a] :
( ( in_tree_a @ V @ ( t_a @ V5 @ L @ R ) )
= ( ( V = V5 )
| ( in_tree_a @ V @ L )
| ( in_tree_a @ V @ R ) ) ) ).
% in_tree.simps(2)
thf(fact_34_removeLeaf_Oinduct,axiom,
! [P: tree_a > $o,A0: tree_a] :
( ! [V4: a] : ( P @ ( t_a @ V4 @ e_a @ e_a ) )
=> ( ! [V4: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
( ( P @ ( t_a @ Va2 @ Vb2 @ Vc2 ) )
=> ( ( P @ ( t_a @ Va2 @ Vb2 @ Vc2 ) )
=> ( P @ ( t_a @ V4 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ e_a ) ) ) )
=> ( ! [V4: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
( ( P @ ( t_a @ Va2 @ Vb2 @ Vc2 ) )
=> ( ( P @ ( t_a @ Va2 @ Vb2 @ Vc2 ) )
=> ( P @ ( t_a @ V4 @ e_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) ) )
=> ( ! [V4: a,Va2: a,Vb2: tree_a,Vc2: tree_a,Vd2: a,Ve2: tree_a,Vf2: tree_a] :
( ( P @ ( t_a @ Va2 @ Vb2 @ Vc2 ) )
=> ( ( P @ ( t_a @ Va2 @ Vb2 @ Vc2 ) )
=> ( P @ ( t_a @ V4 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) ) ) )
=> ( ! [V4: a,Vd2: a,Ve2: tree_a,Vf2: tree_a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
( ( P @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) )
=> ( ( P @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) )
=> ( P @ ( t_a @ V4 @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) ) ) )
=> ( ( P @ e_a )
=> ( P @ A0 ) ) ) ) ) ) ) ).
% removeLeaf.induct
thf(fact_35_removeLeaf_Ocases,axiom,
! [X: tree_a] :
( ! [V4: a] :
( X
!= ( t_a @ V4 @ e_a @ e_a ) )
=> ( ! [V4: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
( X
!= ( t_a @ V4 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ e_a ) )
=> ( ! [V4: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
( X
!= ( t_a @ V4 @ e_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
=> ( ! [V4: a,Va2: a,Vb2: tree_a,Vc2: tree_a,Vd2: a,Ve2: tree_a,Vf2: tree_a] :
( X
!= ( t_a @ V4 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) )
=> ( ! [V4: a,Vd2: a,Ve2: tree_a,Vf2: tree_a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
( X
!= ( t_a @ V4 @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
=> ( X = e_a ) ) ) ) ) ) ).
% removeLeaf.cases
thf(fact_36_siftDown_Ocases,axiom,
! [X: tree_a] :
( ( X != e_a )
=> ( ! [V4: a] :
( X
!= ( t_a @ V4 @ e_a @ e_a ) )
=> ( ! [V4: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
( X
!= ( t_a @ V4 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ e_a ) )
=> ( ! [V4: a,Va2: a,Vb2: tree_a,Vc2: tree_a] :
( X
!= ( t_a @ V4 @ e_a @ ( t_a @ Va2 @ Vb2 @ Vc2 ) ) )
=> ~ ! [V4: a,Va2: a,Vb2: tree_a,Vc2: tree_a,Vd2: a,Ve2: tree_a,Vf2: tree_a] :
( X
!= ( t_a @ V4 @ ( t_a @ Va2 @ Vb2 @ Vc2 ) @ ( t_a @ Vd2 @ Ve2 @ Vf2 ) ) ) ) ) ) ) ).
% siftDown.cases
thf(fact_37_hs__is__empty__def,axiom,
( heapIm229596386mpty_a
= ( ^ [T2: tree_a] : T2 = e_a ) ) ).
% hs_is_empty_def
thf(fact_38_order__refl,axiom,
! [X: $o > a] : ( ord_less_eq_o_a @ X @ X ) ).
% order_refl
thf(fact_39_order__refl,axiom,
! [X: a] : ( ord_less_eq_a @ X @ X ) ).
% order_refl
thf(fact_40_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_41_hs__of__list__def,axiom,
( heapIm1057938560list_a
= ( ^ [L3: list_a] : ( heapIm970322378pify_a @ ( heapIm1637418125tree_a @ L3 ) ) ) ) ).
% hs_of_list_def
thf(fact_42_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_43_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X4: a] : ( member_a @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_44_le__funD,axiom,
! [F: $o > a,G: $o > a,X: $o] :
( ( ord_less_eq_o_a @ F @ G )
=> ( ord_less_eq_a @ ( F @ X ) @ ( G @ X ) ) ) ).
% le_funD
thf(fact_45_le__funE,axiom,
! [F: $o > a,G: $o > a,X: $o] :
( ( ord_less_eq_o_a @ F @ G )
=> ( ord_less_eq_a @ ( F @ X ) @ ( G @ X ) ) ) ).
% le_funE
thf(fact_46_le__funI,axiom,
! [F: $o > a,G: $o > a] :
( ! [X5: $o] : ( ord_less_eq_a @ ( F @ X5 ) @ ( G @ X5 ) )
=> ( ord_less_eq_o_a @ F @ G ) ) ).
% le_funI
thf(fact_47_le__fun__def,axiom,
( ord_less_eq_o_a
= ( ^ [F2: $o > a,G2: $o > a] :
! [X4: $o] : ( ord_less_eq_a @ ( F2 @ X4 ) @ ( G2 @ X4 ) ) ) ) ).
% le_fun_def
thf(fact_48_order__subst1,axiom,
! [A: a,F: ( $o > a ) > a,B: $o > a,C: $o > a] :
( ( ord_less_eq_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_o_a @ B @ C )
=> ( ! [X5: $o > a,Y2: $o > a] :
( ( ord_less_eq_o_a @ X5 @ Y2 )
=> ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_49_order__subst1,axiom,
! [A: $o > a,F: a > $o > a,B: a,C: a] :
( ( ord_less_eq_o_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X5: a,Y2: a] :
( ( ord_less_eq_a @ X5 @ Y2 )
=> ( ord_less_eq_o_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_o_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_50_order__subst1,axiom,
! [A: $o > a,F: ( $o > a ) > $o > a,B: $o > a,C: $o > a] :
( ( ord_less_eq_o_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_o_a @ B @ C )
=> ( ! [X5: $o > a,Y2: $o > a] :
( ( ord_less_eq_o_a @ X5 @ Y2 )
=> ( ord_less_eq_o_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_o_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_51_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_52_order__subst2,axiom,
! [A: a,B: a,F: a > $o > a,C: $o > a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ord_less_eq_o_a @ ( F @ B ) @ C )
=> ( ! [X5: a,Y2: a] :
( ( ord_less_eq_a @ X5 @ Y2 )
=> ( ord_less_eq_o_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_o_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_53_order__subst2,axiom,
! [A: $o > a,B: $o > a,F: ( $o > a ) > a,C: a] :
( ( ord_less_eq_o_a @ A @ B )
=> ( ( ord_less_eq_a @ ( F @ B ) @ C )
=> ( ! [X5: $o > a,Y2: $o > a] :
( ( ord_less_eq_o_a @ X5 @ Y2 )
=> ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_54_order__subst2,axiom,
! [A: $o > a,B: $o > a,F: ( $o > a ) > $o > a,C: $o > a] :
( ( ord_less_eq_o_a @ A @ B )
=> ( ( ord_less_eq_o_a @ ( F @ B ) @ C )
=> ( ! [X5: $o > a,Y2: $o > a] :
( ( ord_less_eq_o_a @ X5 @ Y2 )
=> ( ord_less_eq_o_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_o_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_55_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_56_dual__order_Oantisym,axiom,
! [B: $o > a,A: $o > a] :
( ( ord_less_eq_o_a @ B @ A )
=> ( ( ord_less_eq_o_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_57_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_58_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: $o > a,Z: $o > a] : Y3 = Z )
= ( ^ [A3: $o > a,B2: $o > a] :
( ( ord_less_eq_o_a @ B2 @ A3 )
& ( ord_less_eq_o_a @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_59_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_60_dual__order_Otrans,axiom,
! [B: $o > a,A: $o > a,C: $o > a] :
( ( ord_less_eq_o_a @ B @ A )
=> ( ( ord_less_eq_o_a @ C @ B )
=> ( ord_less_eq_o_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_61_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_62_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_63_dual__order_Orefl,axiom,
! [A: $o > a] : ( ord_less_eq_o_a @ A @ A ) ).
% dual_order.refl
thf(fact_64_dual__order_Orefl,axiom,
! [A: a] : ( ord_less_eq_a @ A @ A ) ).
% dual_order.refl
thf(fact_65_order__trans,axiom,
! [X: $o > a,Y: $o > a,Z2: $o > a] :
( ( ord_less_eq_o_a @ X @ Y )
=> ( ( ord_less_eq_o_a @ Y @ Z2 )
=> ( ord_less_eq_o_a @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_66_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_67_order__class_Oorder_Oantisym,axiom,
! [A: $o > a,B: $o > a] :
( ( ord_less_eq_o_a @ A @ B )
=> ( ( ord_less_eq_o_a @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_68_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_69_ord__le__eq__trans,axiom,
! [A: $o > a,B: $o > a,C: $o > a] :
( ( ord_less_eq_o_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_o_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_70_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_71_ord__eq__le__trans,axiom,
! [A: $o > a,B: $o > a,C: $o > a] :
( ( A = B )
=> ( ( ord_less_eq_o_a @ B @ C )
=> ( ord_less_eq_o_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_72_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_73_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y3: $o > a,Z: $o > a] : Y3 = Z )
= ( ^ [A3: $o > a,B2: $o > a] :
( ( ord_less_eq_o_a @ A3 @ B2 )
& ( ord_less_eq_o_a @ B2 @ A3 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_74_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_75_antisym__conv,axiom,
! [Y: $o > a,X: $o > a] :
( ( ord_less_eq_o_a @ Y @ X )
=> ( ( ord_less_eq_o_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv
thf(fact_76_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_77_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_78_order_Otrans,axiom,
! [A: $o > a,B: $o > a,C: $o > a] :
( ( ord_less_eq_o_a @ A @ B )
=> ( ( ord_less_eq_o_a @ B @ C )
=> ( ord_less_eq_o_a @ A @ C ) ) ) ).
% order.trans
thf(fact_79_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_80_le__cases,axiom,
! [X: a,Y: a] :
( ~ ( ord_less_eq_a @ X @ Y )
=> ( ord_less_eq_a @ Y @ X ) ) ).
% le_cases
thf(fact_81_eq__refl,axiom,
! [X: $o > a,Y: $o > a] :
( ( X = Y )
=> ( ord_less_eq_o_a @ X @ Y ) ) ).
% eq_refl
thf(fact_82_eq__refl,axiom,
! [X: a,Y: a] :
( ( X = Y )
=> ( ord_less_eq_a @ X @ Y ) ) ).
% eq_refl
thf(fact_83_linear,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
| ( ord_less_eq_a @ Y @ X ) ) ).
% linear
thf(fact_84_antisym,axiom,
! [X: $o > a,Y: $o > a] :
( ( ord_less_eq_o_a @ X @ Y )
=> ( ( ord_less_eq_o_a @ Y @ X )
=> ( X = Y ) ) ) ).
% antisym
thf(fact_85_antisym,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
=> ( ( ord_less_eq_a @ Y @ X )
=> ( X = Y ) ) ) ).
% antisym
thf(fact_86_eq__iff,axiom,
( ( ^ [Y3: $o > a,Z: $o > a] : Y3 = Z )
= ( ^ [X4: $o > a,Y4: $o > a] :
( ( ord_less_eq_o_a @ X4 @ Y4 )
& ( ord_less_eq_o_a @ Y4 @ X4 ) ) ) ) ).
% eq_iff
thf(fact_87_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_88_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > $o > a,C: $o > a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X5: a,Y2: a] :
( ( ord_less_eq_a @ X5 @ Y2 )
=> ( ord_less_eq_o_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_o_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_89_ord__le__eq__subst,axiom,
! [A: $o > a,B: $o > a,F: ( $o > a ) > a,C: a] :
( ( ord_less_eq_o_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X5: $o > a,Y2: $o > a] :
( ( ord_less_eq_o_a @ X5 @ Y2 )
=> ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_90_ord__le__eq__subst,axiom,
! [A: $o > a,B: $o > a,F: ( $o > a ) > $o > a,C: $o > a] :
( ( ord_less_eq_o_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X5: $o > a,Y2: $o > a] :
( ( ord_less_eq_o_a @ X5 @ Y2 )
=> ( ord_less_eq_o_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_o_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_91_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_92_ord__eq__le__subst,axiom,
! [A: $o > a,F: a > $o > 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_o_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_o_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_93_ord__eq__le__subst,axiom,
! [A: a,F: ( $o > a ) > a,B: $o > a,C: $o > a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_o_a @ B @ C )
=> ( ! [X5: $o > a,Y2: $o > a] :
( ( ord_less_eq_o_a @ X5 @ Y2 )
=> ( ord_less_eq_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_94_ord__eq__le__subst,axiom,
! [A: $o > a,F: ( $o > a ) > $o > a,B: $o > a,C: $o > a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_o_a @ B @ C )
=> ( ! [X5: $o > a,Y2: $o > a] :
( ( ord_less_eq_o_a @ X5 @ Y2 )
=> ( ord_less_eq_o_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_o_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_95_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_96_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_97_of__list__tree_Osimps_I2_J,axiom,
! [V: a,Tail: list_a] :
( ( heapIm1637418125tree_a @ ( cons_a @ V @ Tail ) )
= ( t_a @ V @ ( heapIm1637418125tree_a @ Tail ) @ e_a ) ) ).
% of_list_tree.simps(2)
thf(fact_98_Greatest__equality,axiom,
! [P: ( $o > a ) > $o,X: $o > a] :
( ( P @ X )
=> ( ! [Y2: $o > a] :
( ( P @ Y2 )
=> ( ord_less_eq_o_a @ Y2 @ X ) )
=> ( ( order_Greatest_o_a @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_99_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_100_GreatestI2__order,axiom,
! [P: ( $o > a ) > $o,X: $o > a,Q: ( $o > a ) > $o] :
( ( P @ X )
=> ( ! [Y2: $o > a] :
( ( P @ Y2 )
=> ( ord_less_eq_o_a @ Y2 @ X ) )
=> ( ! [X5: $o > a] :
( ( P @ X5 )
=> ( ! [Y5: $o > a] :
( ( P @ Y5 )
=> ( ord_less_eq_o_a @ Y5 @ X5 ) )
=> ( Q @ X5 ) ) )
=> ( Q @ ( order_Greatest_o_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_101_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_102_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_o_a
= ( ^ [X6: $o > $o > a,Y6: $o > $o > a] :
( ( ord_less_eq_o_a @ ( X6 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_o_a @ ( X6 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_103_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_a
= ( ^ [X6: $o > a,Y6: $o > a] :
( ( ord_less_eq_a @ ( X6 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_a @ ( X6 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_104_verit__la__disequality,axiom,
! [A: a,B: a] :
( ( A = B )
| ~ ( ord_less_eq_a @ A @ B )
| ~ ( ord_less_eq_a @ B @ A ) ) ).
% verit_la_disequality
thf(fact_105_list_Oinject,axiom,
! [X21: a,X22: list_a,Y21: a,Y22: list_a] :
( ( ( cons_a @ X21 @ X22 )
= ( cons_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_106_not__Cons__self2,axiom,
! [X: a,Xs: list_a] :
( ( cons_a @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_107_Heap__axioms__def,axiom,
( heap_axioms_Tree_a_a
= ( ^ [Is_empty: tree_a > $o,Of_list: list_a > tree_a,Multiset: tree_a > multiset_a,As_tree: tree_a > tree_a,Remove_max: tree_a > produc143150363Tree_a] :
( ! [L3: tree_a] :
( ( Multiset @ L3 )
= ( multiset_a2 @ ( As_tree @ L3 ) ) )
& ! [I: list_a] : ( is_heap_a @ ( As_tree @ ( Of_list @ I ) ) )
& ! [T2: tree_a] :
( ( ( As_tree @ T2 )
= e_a )
= ( Is_empty @ T2 ) )
& ! [L3: tree_a,M: a,L4: tree_a] :
( ~ ( Is_empty @ L3 )
=> ( ( ( produc686083979Tree_a @ M @ L4 )
= ( Remove_max @ L3 ) )
=> ( ( add_mset_a @ M @ ( Multiset @ L4 ) )
= ( Multiset @ L3 ) ) ) )
& ! [L3: tree_a,M: a,L4: tree_a] :
( ~ ( Is_empty @ L3 )
=> ( ( is_heap_a @ ( As_tree @ L3 ) )
=> ( ( ( produc686083979Tree_a @ M @ L4 )
= ( Remove_max @ L3 ) )
=> ( is_heap_a @ ( As_tree @ L4 ) ) ) ) )
& ! [T2: tree_a,M: a,T3: tree_a] :
( ~ ( Is_empty @ T2 )
=> ( ( ( produc686083979Tree_a @ M @ T3 )
= ( Remove_max @ T2 ) )
=> ( M
= ( val_a @ ( As_tree @ T2 ) ) ) ) ) ) ) ) ).
% Heap_axioms_def
thf(fact_108_Heap__axioms_Ointro,axiom,
! [Multiset2: tree_a > multiset_a,As_tree2: tree_a > tree_a,Of_list2: list_a > tree_a,Is_empty2: tree_a > $o,Remove_max2: tree_a > produc143150363Tree_a] :
( ! [L5: tree_a] :
( ( Multiset2 @ L5 )
= ( multiset_a2 @ ( As_tree2 @ L5 ) ) )
=> ( ! [I2: list_a] : ( is_heap_a @ ( As_tree2 @ ( Of_list2 @ I2 ) ) )
=> ( ! [T4: tree_a] :
( ( ( As_tree2 @ T4 )
= e_a )
= ( Is_empty2 @ T4 ) )
=> ( ! [L5: tree_a,M2: a,L2: tree_a] :
( ~ ( Is_empty2 @ L5 )
=> ( ( ( produc686083979Tree_a @ M2 @ L2 )
= ( Remove_max2 @ L5 ) )
=> ( ( add_mset_a @ M2 @ ( Multiset2 @ L2 ) )
= ( Multiset2 @ L5 ) ) ) )
=> ( ! [L5: tree_a,M2: a,L2: tree_a] :
( ~ ( Is_empty2 @ L5 )
=> ( ( is_heap_a @ ( As_tree2 @ L5 ) )
=> ( ( ( produc686083979Tree_a @ M2 @ L2 )
= ( Remove_max2 @ L5 ) )
=> ( is_heap_a @ ( As_tree2 @ L2 ) ) ) ) )
=> ( ! [T4: tree_a,M2: a,T5: tree_a] :
( ~ ( Is_empty2 @ T4 )
=> ( ( ( produc686083979Tree_a @ M2 @ T5 )
= ( Remove_max2 @ T4 ) )
=> ( M2
= ( val_a @ ( As_tree2 @ T4 ) ) ) ) )
=> ( heap_axioms_Tree_a_a @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 ) ) ) ) ) ) ) ).
% Heap_axioms.intro
thf(fact_109_Heap_Oremove__max__multiset_H,axiom,
! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,As_tree2: tree_a > tree_a,Remove_max2: tree_a > produc143150363Tree_a,L: tree_a,M3: a,L6: tree_a] :
( ( heap_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
=> ( ~ ( Is_empty2 @ L )
=> ( ( ( produc686083979Tree_a @ M3 @ L6 )
= ( Remove_max2 @ L ) )
=> ( ( add_mset_a @ M3 @ ( Multiset2 @ L6 ) )
= ( Multiset2 @ L ) ) ) ) ) ).
% Heap.remove_max_multiset'
thf(fact_110_Heap_Oremove__max__val,axiom,
! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,As_tree2: tree_a > tree_a,Remove_max2: tree_a > produc143150363Tree_a,T: tree_a,M3: a,T6: tree_a] :
( ( heap_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
=> ( ~ ( Is_empty2 @ T )
=> ( ( ( produc686083979Tree_a @ M3 @ T6 )
= ( Remove_max2 @ T ) )
=> ( M3
= ( val_a @ ( As_tree2 @ T ) ) ) ) ) ) ).
% Heap.remove_max_val
thf(fact_111_Heap_Oremove__max__is__heap,axiom,
! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,As_tree2: tree_a > tree_a,Remove_max2: tree_a > produc143150363Tree_a,L: tree_a,M3: a,L6: tree_a] :
( ( heap_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ As_tree2 @ Remove_max2 )
=> ( ~ ( Is_empty2 @ L )
=> ( ( is_heap_a @ ( As_tree2 @ L ) )
=> ( ( ( produc686083979Tree_a @ M3 @ L6 )
= ( Remove_max2 @ L ) )
=> ( is_heap_a @ ( As_tree2 @ L6 ) ) ) ) ) ) ).
% Heap.remove_max_is_heap
thf(fact_112_multi__self__add__other__not__self,axiom,
! [M4: multiset_a,X: a] :
( M4
!= ( add_mset_a @ X @ M4 ) ) ).
% multi_self_add_other_not_self
thf(fact_113_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_114_old_Oprod_Oinject,axiom,
! [A: a,B: tree_a,A5: a,B5: tree_a] :
( ( ( produc686083979Tree_a @ A @ B )
= ( produc686083979Tree_a @ A5 @ B5 ) )
= ( ( A = A5 )
& ( B = B5 ) ) ) ).
% old.prod.inject
thf(fact_115_prod_Oinject,axiom,
! [X12: a,X24: tree_a,Y1: a,Y24: tree_a] :
( ( ( produc686083979Tree_a @ X12 @ X24 )
= ( produc686083979Tree_a @ Y1 @ Y24 ) )
= ( ( X12 = Y1 )
& ( X24 = Y24 ) ) ) ).
% prod.inject
thf(fact_116_mset__add,axiom,
! [A: a,A2: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ A2 ) )
=> ~ ! [B6: multiset_a] :
( A2
!= ( add_mset_a @ A @ B6 ) ) ) ).
% mset_add
thf(fact_117_multi__member__split,axiom,
! [X: a,M4: multiset_a] :
( ( member_a @ X @ ( set_mset_a @ M4 ) )
=> ? [A6: multiset_a] :
( M4
= ( add_mset_a @ X @ A6 ) ) ) ).
% multi_member_split
thf(fact_118_surj__pair,axiom,
! [P2: produc143150363Tree_a] :
? [X5: a,Y2: tree_a] :
( P2
= ( produc686083979Tree_a @ X5 @ Y2 ) ) ).
% surj_pair
thf(fact_119_prod__cases,axiom,
! [P: produc143150363Tree_a > $o,P2: produc143150363Tree_a] :
( ! [A4: a,B3: tree_a] : ( P @ ( produc686083979Tree_a @ A4 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_120_Pair__inject,axiom,
! [A: a,B: tree_a,A5: a,B5: tree_a] :
( ( ( produc686083979Tree_a @ A @ B )
= ( produc686083979Tree_a @ A5 @ B5 ) )
=> ~ ( ( A = A5 )
=> ( B != B5 ) ) ) ).
% Pair_inject
thf(fact_121_old_Oprod_Oexhaust,axiom,
! [Y: produc143150363Tree_a] :
~ ! [A4: a,B3: tree_a] :
( Y
!= ( produc686083979Tree_a @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_122_old_Oprod_Oinducts,axiom,
! [P: produc143150363Tree_a > $o,Prod: produc143150363Tree_a] :
( ! [A4: a,B3: tree_a] : ( P @ ( produc686083979Tree_a @ A4 @ B3 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_123_add__eq__conv__ex,axiom,
! [A: a,M4: multiset_a,B: a,N: multiset_a] :
( ( ( add_mset_a @ A @ M4 )
= ( add_mset_a @ B @ N ) )
= ( ( ( M4 = N )
& ( A = B ) )
| ? [K: multiset_a] :
( ( M4
= ( add_mset_a @ B @ K ) )
& ( N
= ( add_mset_a @ A @ K ) ) ) ) ) ).
% add_eq_conv_ex
thf(fact_124_add__mset__commute,axiom,
! [X: a,Y: a,M4: multiset_a] :
( ( add_mset_a @ X @ ( add_mset_a @ Y @ M4 ) )
= ( add_mset_a @ Y @ ( add_mset_a @ X @ M4 ) ) ) ).
% add_mset_commute
thf(fact_125_union__single__eq__member,axiom,
! [X: a,M4: multiset_a,N: multiset_a] :
( ( ( add_mset_a @ X @ M4 )
= N )
=> ( member_a @ X @ ( set_mset_a @ N ) ) ) ).
% union_single_eq_member
thf(fact_126_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_127_removeLeaf_Osimps_I1_J,axiom,
! [V: a] :
( ( heapIm837449470Leaf_a @ ( t_a @ V @ e_a @ e_a ) )
= ( produc686083979Tree_a @ V @ e_a ) ) ).
% removeLeaf.simps(1)
thf(fact_128_RemoveMax__axioms__def,axiom,
( remove301631099ee_a_a
= ( ^ [Is_empty: tree_a > $o,Of_list: list_a > tree_a,Multiset: tree_a > multiset_a,Remove_max: tree_a > produc143150363Tree_a,Inv: tree_a > $o] :
( ! [X4: list_a] : ( Inv @ ( Of_list @ X4 ) )
& ! [L3: tree_a,M: a,L4: tree_a] :
( ~ ( Is_empty @ L3 )
=> ( ( Inv @ L3 )
=> ( ( ( produc686083979Tree_a @ M @ L4 )
= ( Remove_max @ L3 ) )
=> ( M
= ( lattic146396397_Max_a @ ( set_Tree_a_a @ Multiset @ L3 ) ) ) ) ) )
& ! [L3: tree_a,M: a,L4: tree_a] :
( ~ ( Is_empty @ L3 )
=> ( ( Inv @ L3 )
=> ( ( ( produc686083979Tree_a @ M @ L4 )
= ( Remove_max @ L3 ) )
=> ( ( add_mset_a @ M @ ( Multiset @ L4 ) )
= ( Multiset @ L3 ) ) ) ) )
& ! [L3: tree_a,M: a,L4: tree_a] :
( ~ ( Is_empty @ L3 )
=> ( ( Inv @ L3 )
=> ( ( ( produc686083979Tree_a @ M @ L4 )
= ( Remove_max @ L3 ) )
=> ( Inv @ L4 ) ) ) ) ) ) ) ).
% RemoveMax_axioms_def
thf(fact_129_RemoveMax__axioms_Ointro,axiom,
! [Inv2: tree_a > $o,Of_list2: list_a > tree_a,Is_empty2: tree_a > $o,Remove_max2: tree_a > produc143150363Tree_a,Multiset2: tree_a > multiset_a] :
( ! [X5: list_a] : ( Inv2 @ ( Of_list2 @ X5 ) )
=> ( ! [L5: tree_a,M2: a,L2: tree_a] :
( ~ ( Is_empty2 @ L5 )
=> ( ( Inv2 @ L5 )
=> ( ( ( produc686083979Tree_a @ M2 @ L2 )
= ( Remove_max2 @ L5 ) )
=> ( M2
= ( lattic146396397_Max_a @ ( set_Tree_a_a @ Multiset2 @ L5 ) ) ) ) ) )
=> ( ! [L5: tree_a,M2: a,L2: tree_a] :
( ~ ( Is_empty2 @ L5 )
=> ( ( Inv2 @ L5 )
=> ( ( ( produc686083979Tree_a @ M2 @ L2 )
= ( Remove_max2 @ L5 ) )
=> ( ( add_mset_a @ M2 @ ( Multiset2 @ L2 ) )
= ( Multiset2 @ L5 ) ) ) ) )
=> ( ! [L5: tree_a,M2: a,L2: tree_a] :
( ~ ( Is_empty2 @ L5 )
=> ( ( Inv2 @ L5 )
=> ( ( ( produc686083979Tree_a @ M2 @ L2 )
= ( Remove_max2 @ L5 ) )
=> ( Inv2 @ L2 ) ) ) )
=> ( remove301631099ee_a_a @ Is_empty2 @ Of_list2 @ Multiset2 @ Remove_max2 @ Inv2 ) ) ) ) ) ).
% RemoveMax_axioms.intro
thf(fact_130_RemoveMax_Oremove__max__max,axiom,
! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,Remove_max2: tree_a > produc143150363Tree_a,Inv2: tree_a > $o,L: tree_a,M3: a,L6: tree_a] :
( ( removeMax_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ Remove_max2 @ Inv2 )
=> ( ~ ( Is_empty2 @ L )
=> ( ( Inv2 @ L )
=> ( ( ( produc686083979Tree_a @ M3 @ L6 )
= ( Remove_max2 @ L ) )
=> ( M3
= ( lattic146396397_Max_a @ ( set_Tree_a_a @ Multiset2 @ L ) ) ) ) ) ) ) ).
% RemoveMax.remove_max_max
thf(fact_131_multiset__induct__max,axiom,
! [P: multiset_a > $o,M4: multiset_a] :
( ( P @ zero_zero_multiset_a )
=> ( ! [X5: a,M5: multiset_a] :
( ( P @ M5 )
=> ( ! [Xa: a] :
( ( member_a @ Xa @ ( set_mset_a @ M5 ) )
=> ( ord_less_eq_a @ Xa @ X5 ) )
=> ( P @ ( add_mset_a @ X5 @ M5 ) ) ) )
=> ( P @ M4 ) ) ) ).
% multiset_induct_max
thf(fact_132_add__mset__eq__singleton__iff,axiom,
! [X: a,M4: multiset_a,Y: a] :
( ( ( add_mset_a @ X @ M4 )
= ( add_mset_a @ Y @ zero_zero_multiset_a ) )
= ( ( M4 = zero_zero_multiset_a )
& ( X = Y ) ) ) ).
% add_mset_eq_singleton_iff
thf(fact_133_single__eq__add__mset,axiom,
! [A: a,B: a,M4: multiset_a] :
( ( ( add_mset_a @ A @ zero_zero_multiset_a )
= ( add_mset_a @ B @ M4 ) )
= ( ( B = A )
& ( M4 = zero_zero_multiset_a ) ) ) ).
% single_eq_add_mset
thf(fact_134_add__mset__eq__single,axiom,
! [B: a,M4: multiset_a,A: a] :
( ( ( add_mset_a @ B @ M4 )
= ( add_mset_a @ A @ zero_zero_multiset_a ) )
= ( ( B = A )
& ( M4 = zero_zero_multiset_a ) ) ) ).
% add_mset_eq_single
thf(fact_135_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_136_multiset__cases,axiom,
! [M4: multiset_a] :
( ( M4 != zero_zero_multiset_a )
=> ~ ! [X5: a,N2: multiset_a] :
( M4
!= ( add_mset_a @ X5 @ N2 ) ) ) ).
% multiset_cases
thf(fact_137_multiset__induct,axiom,
! [P: multiset_a > $o,M4: multiset_a] :
( ( P @ zero_zero_multiset_a )
=> ( ! [X5: a,M5: multiset_a] :
( ( P @ M5 )
=> ( P @ ( add_mset_a @ X5 @ M5 ) ) )
=> ( P @ M4 ) ) ) ).
% multiset_induct
thf(fact_138_multiset__induct2,axiom,
! [P: multiset_a > multiset_a > $o,M4: multiset_a,N: multiset_a] :
( ( P @ zero_zero_multiset_a @ zero_zero_multiset_a )
=> ( ! [A4: a,M5: multiset_a,N2: multiset_a] :
( ( P @ M5 @ N2 )
=> ( P @ ( add_mset_a @ A4 @ M5 ) @ N2 ) )
=> ( ! [A4: a,M5: multiset_a,N2: multiset_a] :
( ( P @ M5 @ N2 )
=> ( P @ M5 @ ( add_mset_a @ A4 @ N2 ) ) )
=> ( P @ M4 @ N ) ) ) ) ).
% multiset_induct2
thf(fact_139_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_140_multi__nonempty__split,axiom,
! [M4: multiset_a] :
( ( M4 != zero_zero_multiset_a )
=> ? [A6: multiset_a,A4: a] :
( M4
= ( add_mset_a @ A4 @ A6 ) ) ) ).
% multi_nonempty_split
thf(fact_141_multiset__nonemptyE,axiom,
! [A2: multiset_a] :
( ( A2 != zero_zero_multiset_a )
=> ~ ! [X5: a] :
~ ( member_a @ X5 @ ( set_mset_a @ A2 ) ) ) ).
% multiset_nonemptyE
thf(fact_142_RemoveMax_Oremove__max__inv,axiom,
! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,Remove_max2: tree_a > produc143150363Tree_a,Inv2: tree_a > $o,L: tree_a,M3: a,L6: tree_a] :
( ( removeMax_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ Remove_max2 @ Inv2 )
=> ( ~ ( Is_empty2 @ L )
=> ( ( Inv2 @ L )
=> ( ( ( produc686083979Tree_a @ M3 @ L6 )
= ( Remove_max2 @ L ) )
=> ( Inv2 @ L6 ) ) ) ) ) ).
% RemoveMax.remove_max_inv
thf(fact_143_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_144_RemoveMax_Oremove__max__multiset,axiom,
! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,Remove_max2: tree_a > produc143150363Tree_a,Inv2: tree_a > $o,L: tree_a,M3: a,L6: tree_a] :
( ( removeMax_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ Remove_max2 @ Inv2 )
=> ( ~ ( Is_empty2 @ L )
=> ( ( Inv2 @ L )
=> ( ( ( produc686083979Tree_a @ M3 @ L6 )
= ( Remove_max2 @ L ) )
=> ( ( add_mset_a @ M3 @ ( Multiset2 @ L6 ) )
= ( Multiset2 @ L ) ) ) ) ) ) ).
% RemoveMax.remove_max_multiset
thf(fact_145_multiset_Osimps_I1_J,axiom,
( ( multiset_a2 @ e_a )
= zero_zero_multiset_a ) ).
% multiset.simps(1)
thf(fact_146_multiset__induct__min,axiom,
! [P: multiset_a > $o,M4: multiset_a] :
( ( P @ zero_zero_multiset_a )
=> ( ! [X5: a,M5: multiset_a] :
( ( P @ M5 )
=> ( ! [Xa: a] :
( ( member_a @ Xa @ ( set_mset_a @ M5 ) )
=> ( ord_less_eq_a @ X5 @ Xa ) )
=> ( P @ ( add_mset_a @ X5 @ M5 ) ) ) )
=> ( P @ M4 ) ) ) ).
% multiset_induct_min
thf(fact_147_Multiset_Ois__empty__def,axiom,
( is_empty_a
= ( ^ [A7: multiset_a] : A7 = zero_zero_multiset_a ) ) ).
% Multiset.is_empty_def
thf(fact_148_RemoveMax_Ossort_HInduct,axiom,
! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,Remove_max2: tree_a > produc143150363Tree_a,Inv2: tree_a > $o,L: tree_a,P: tree_a > list_a > $o,Sl: list_a] :
( ( removeMax_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ Remove_max2 @ Inv2 )
=> ( ( Inv2 @ L )
=> ( ( P @ L @ Sl )
=> ( ! [L5: tree_a,Sl2: list_a,M2: a,L2: tree_a] :
( ~ ( Is_empty2 @ L5 )
=> ( ( Inv2 @ L5 )
=> ( ( ( produc686083979Tree_a @ M2 @ L2 )
= ( Remove_max2 @ L5 ) )
=> ( ( P @ L5 @ Sl2 )
=> ( P @ L2 @ ( cons_a @ M2 @ Sl2 ) ) ) ) ) )
=> ( P @ Empty @ ( ssort_Tree_a_a @ Is_empty2 @ Remove_max2 @ L @ Sl ) ) ) ) ) ) ).
% RemoveMax.ssort'Induct
thf(fact_149_RemoveMax_Ossort_H__dom_Ocases,axiom,
! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,Remove_max2: tree_a > produc143150363Tree_a,Inv2: tree_a > $o,A: produc768687417list_a] :
( ( removeMax_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ Remove_max2 @ Inv2 )
=> ( ( ssort_dom_Tree_a_a @ Is_empty2 @ Remove_max2 @ A )
=> ~ ! [L5: tree_a,Sl2: list_a] :
( ( A
= ( produc1352981801list_a @ L5 @ Sl2 ) )
=> ~ ( ~ ( Is_empty2 @ L5 )
=> ! [M6: a,L7: tree_a] :
( ( ( produc686083979Tree_a @ M6 @ L7 )
= ( Remove_max2 @ L5 ) )
=> ( ssort_dom_Tree_a_a @ Is_empty2 @ Remove_max2 @ ( produc1352981801list_a @ L7 @ ( cons_a @ M6 @ Sl2 ) ) ) ) ) ) ) ) ).
% RemoveMax.ssort'_dom.cases
thf(fact_150_RemoveMax_Ossort_H__dom_Osimps,axiom,
! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,Remove_max2: tree_a > produc143150363Tree_a,Inv2: tree_a > $o,A: produc768687417list_a] :
( ( removeMax_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ Remove_max2 @ Inv2 )
=> ( ( ssort_dom_Tree_a_a @ Is_empty2 @ Remove_max2 @ A )
= ( ? [L3: tree_a,Sl3: list_a] :
( ( A
= ( produc1352981801list_a @ L3 @ Sl3 ) )
& ! [X4: a,Y4: tree_a] :
( ~ ( Is_empty2 @ L3 )
=> ( ( ( produc686083979Tree_a @ X4 @ Y4 )
= ( Remove_max2 @ L3 ) )
=> ( ssort_dom_Tree_a_a @ Is_empty2 @ Remove_max2 @ ( produc1352981801list_a @ Y4 @ ( cons_a @ X4 @ Sl3 ) ) ) ) ) ) ) ) ) ).
% RemoveMax.ssort'_dom.simps
thf(fact_151_RemoveMax_Ossort_H__dom_Oinducts,axiom,
! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,Remove_max2: tree_a > produc143150363Tree_a,Inv2: tree_a > $o,X: produc768687417list_a,P: produc768687417list_a > $o] :
( ( removeMax_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ Remove_max2 @ Inv2 )
=> ( ( ssort_dom_Tree_a_a @ Is_empty2 @ Remove_max2 @ X )
=> ( ! [L5: tree_a,Sl2: list_a] :
( ! [M6: a,L7: tree_a] :
( ~ ( Is_empty2 @ L5 )
=> ( ( ( produc686083979Tree_a @ M6 @ L7 )
= ( Remove_max2 @ L5 ) )
=> ( ssort_dom_Tree_a_a @ Is_empty2 @ Remove_max2 @ ( produc1352981801list_a @ L7 @ ( cons_a @ M6 @ Sl2 ) ) ) ) )
=> ( ! [M6: a,L7: tree_a] :
( ~ ( Is_empty2 @ L5 )
=> ( ( ( produc686083979Tree_a @ M6 @ L7 )
= ( Remove_max2 @ L5 ) )
=> ( P @ ( produc1352981801list_a @ L7 @ ( cons_a @ M6 @ Sl2 ) ) ) ) )
=> ( P @ ( produc1352981801list_a @ L5 @ Sl2 ) ) ) )
=> ( P @ X ) ) ) ) ).
% RemoveMax.ssort'_dom.inducts
thf(fact_152_RemoveMax_Ossort_H__dom_Ointros,axiom,
! [Empty: tree_a,Is_empty2: tree_a > $o,Of_list2: list_a > tree_a,Multiset2: tree_a > multiset_a,Remove_max2: tree_a > produc143150363Tree_a,Inv2: tree_a > $o,L: tree_a,Sl: list_a] :
( ( removeMax_Tree_a_a @ Empty @ Is_empty2 @ Of_list2 @ Multiset2 @ Remove_max2 @ Inv2 )
=> ( ! [M2: a,L2: tree_a] :
( ~ ( Is_empty2 @ L )
=> ( ( ( produc686083979Tree_a @ M2 @ L2 )
= ( Remove_max2 @ L ) )
=> ( ssort_dom_Tree_a_a @ Is_empty2 @ Remove_max2 @ ( produc1352981801list_a @ L2 @ ( cons_a @ M2 @ Sl ) ) ) ) )
=> ( ssort_dom_Tree_a_a @ Is_empty2 @ Remove_max2 @ ( produc1352981801list_a @ L @ Sl ) ) ) ) ).
% RemoveMax.ssort'_dom.intros
thf(fact_153_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_154_single__subset__iff,axiom,
! [A: a,M4: multiset_a] :
( ( subseteq_mset_a @ ( add_mset_a @ A @ zero_zero_multiset_a ) @ M4 )
= ( member_a @ A @ ( set_mset_a @ M4 ) ) ) ).
% single_subset_iff
thf(fact_155_subset__mset_Obot_Oextremum__unique,axiom,
! [A: multiset_a] :
( ( subseteq_mset_a @ A @ zero_zero_multiset_a )
= ( A = zero_zero_multiset_a ) ) ).
% subset_mset.bot.extremum_unique
thf(fact_156_subset__mset_Ole__zero__eq,axiom,
! [N3: multiset_a] :
( ( subseteq_mset_a @ N3 @ zero_zero_multiset_a )
= ( N3 = zero_zero_multiset_a ) ) ).
% subset_mset.le_zero_eq
thf(fact_157_subset__mset_Oadd__le__cancel__left,axiom,
! [C: multiset_a,A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ C @ A ) @ ( plus_plus_multiset_a @ C @ B ) )
= ( subseteq_mset_a @ A @ B ) ) ).
% subset_mset.add_le_cancel_left
thf(fact_158_subset__mset_Oadd__le__cancel__right,axiom,
! [A: multiset_a,C: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ C ) @ ( plus_plus_multiset_a @ B @ C ) )
= ( subseteq_mset_a @ A @ B ) ) ).
% subset_mset.add_le_cancel_right
thf(fact_159_mset__subset__eq__mono__add__left__cancel,axiom,
! [C2: multiset_a,A2: multiset_a,B4: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ C2 @ A2 ) @ ( plus_plus_multiset_a @ C2 @ B4 ) )
= ( subseteq_mset_a @ A2 @ B4 ) ) ).
% mset_subset_eq_mono_add_left_cancel
thf(fact_160_mset__subset__eq__mono__add__right__cancel,axiom,
! [A2: multiset_a,C2: multiset_a,B4: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ A2 @ C2 ) @ ( plus_plus_multiset_a @ B4 @ C2 ) )
= ( subseteq_mset_a @ A2 @ B4 ) ) ).
% mset_subset_eq_mono_add_right_cancel
thf(fact_161_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_162_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_163_union__eq__empty,axiom,
! [M4: multiset_a,N: multiset_a] :
( ( ( plus_plus_multiset_a @ M4 @ N )
= zero_zero_multiset_a )
= ( ( M4 = zero_zero_multiset_a )
& ( N = zero_zero_multiset_a ) ) ) ).
% union_eq_empty
thf(fact_164_empty__eq__union,axiom,
! [M4: multiset_a,N: multiset_a] :
( ( zero_zero_multiset_a
= ( plus_plus_multiset_a @ M4 @ N ) )
= ( ( M4 = zero_zero_multiset_a )
& ( N = zero_zero_multiset_a ) ) ) ).
% empty_eq_union
thf(fact_165_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_166_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_167_subset__mset_Oadd__le__same__cancel1,axiom,
! [B: multiset_a,A: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ B @ A ) @ B )
= ( subseteq_mset_a @ A @ zero_zero_multiset_a ) ) ).
% subset_mset.add_le_same_cancel1
thf(fact_168_subset__mset_Oadd__le__same__cancel2,axiom,
! [A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ B ) @ B )
= ( subseteq_mset_a @ A @ zero_zero_multiset_a ) ) ).
% subset_mset.add_le_same_cancel2
thf(fact_169_subset__mset_Ole__add__same__cancel1,axiom,
! [A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ A @ ( plus_plus_multiset_a @ A @ B ) )
= ( subseteq_mset_a @ zero_zero_multiset_a @ B ) ) ).
% subset_mset.le_add_same_cancel1
thf(fact_170_subset__mset_Ole__add__same__cancel2,axiom,
! [A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ A @ ( plus_plus_multiset_a @ B @ A ) )
= ( subseteq_mset_a @ zero_zero_multiset_a @ B ) ) ).
% subset_mset.le_add_same_cancel2
thf(fact_171_add__mset__subseteq__single__iff,axiom,
! [A: a,M4: multiset_a,B: a] :
( ( subseteq_mset_a @ ( add_mset_a @ A @ M4 ) @ ( add_mset_a @ B @ zero_zero_multiset_a ) )
= ( ( M4 = zero_zero_multiset_a )
& ( A = B ) ) ) ).
% add_mset_subseteq_single_iff
thf(fact_172_union__assoc,axiom,
! [M4: multiset_a,N: multiset_a,K2: multiset_a] :
( ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ M4 @ N ) @ K2 )
= ( plus_plus_multiset_a @ M4 @ ( plus_plus_multiset_a @ N @ K2 ) ) ) ).
% union_assoc
thf(fact_173_union__lcomm,axiom,
! [M4: multiset_a,N: multiset_a,K2: multiset_a] :
( ( plus_plus_multiset_a @ M4 @ ( plus_plus_multiset_a @ N @ K2 ) )
= ( plus_plus_multiset_a @ N @ ( plus_plus_multiset_a @ M4 @ K2 ) ) ) ).
% union_lcomm
thf(fact_174_union__commute,axiom,
( plus_plus_multiset_a
= ( ^ [M7: multiset_a,N4: multiset_a] : ( plus_plus_multiset_a @ N4 @ M7 ) ) ) ).
% union_commute
thf(fact_175_union__left__cancel,axiom,
! [K2: multiset_a,M4: multiset_a,N: multiset_a] :
( ( ( plus_plus_multiset_a @ K2 @ M4 )
= ( plus_plus_multiset_a @ K2 @ N ) )
= ( M4 = N ) ) ).
% union_left_cancel
thf(fact_176_union__right__cancel,axiom,
! [M4: multiset_a,K2: multiset_a,N: multiset_a] :
( ( ( plus_plus_multiset_a @ M4 @ K2 )
= ( plus_plus_multiset_a @ N @ K2 ) )
= ( M4 = N ) ) ).
% union_right_cancel
thf(fact_177_subset__mset_Oadd__mono,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a,D: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( ( subseteq_mset_a @ C @ D )
=> ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ C ) @ ( plus_plus_multiset_a @ B @ D ) ) ) ) ).
% subset_mset.add_mono
thf(fact_178_subset__mset_Oless__eqE,axiom,
! [A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ~ ! [C3: multiset_a] :
( B
!= ( plus_plus_multiset_a @ A @ C3 ) ) ) ).
% subset_mset.less_eqE
thf(fact_179_subset__mset_Ole__iff__add,axiom,
( subseteq_mset_a
= ( ^ [A3: multiset_a,B2: multiset_a] :
? [C4: multiset_a] :
( B2
= ( plus_plus_multiset_a @ A3 @ C4 ) ) ) ) ).
% subset_mset.le_iff_add
thf(fact_180_mset__subset__eq__add__left,axiom,
! [A2: multiset_a,B4: multiset_a] : ( subseteq_mset_a @ A2 @ ( plus_plus_multiset_a @ A2 @ B4 ) ) ).
% mset_subset_eq_add_left
thf(fact_181_mset__subset__eq__mono__add,axiom,
! [A2: multiset_a,B4: multiset_a,C2: multiset_a,D2: multiset_a] :
( ( subseteq_mset_a @ A2 @ B4 )
=> ( ( subseteq_mset_a @ C2 @ D2 )
=> ( subseteq_mset_a @ ( plus_plus_multiset_a @ A2 @ C2 ) @ ( plus_plus_multiset_a @ B4 @ D2 ) ) ) ) ).
% mset_subset_eq_mono_add
thf(fact_182_mset__subset__eq__add__right,axiom,
! [B4: multiset_a,A2: multiset_a] : ( subseteq_mset_a @ B4 @ ( plus_plus_multiset_a @ A2 @ B4 ) ) ).
% mset_subset_eq_add_right
thf(fact_183_multi__union__self__other__eq,axiom,
! [A2: multiset_a,X7: multiset_a,Y7: multiset_a] :
( ( ( plus_plus_multiset_a @ A2 @ X7 )
= ( plus_plus_multiset_a @ A2 @ Y7 ) )
=> ( X7 = Y7 ) ) ).
% multi_union_self_other_eq
thf(fact_184_subset__mset_Oadd__left__mono,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( subseteq_mset_a @ ( plus_plus_multiset_a @ C @ A ) @ ( plus_plus_multiset_a @ C @ B ) ) ) ).
% subset_mset.add_left_mono
thf(fact_185_mset__subset__eq__exists__conv,axiom,
( subseteq_mset_a
= ( ^ [A7: multiset_a,B7: multiset_a] :
? [C5: multiset_a] :
( B7
= ( plus_plus_multiset_a @ A7 @ C5 ) ) ) ) ).
% mset_subset_eq_exists_conv
thf(fact_186_subset__mset_Oadd__right__mono,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ C ) @ ( plus_plus_multiset_a @ B @ C ) ) ) ).
% subset_mset.add_right_mono
thf(fact_187_subset__mset_Oadd__le__imp__le__left,axiom,
! [C: multiset_a,A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ C @ A ) @ ( plus_plus_multiset_a @ C @ B ) )
=> ( subseteq_mset_a @ A @ B ) ) ).
% subset_mset.add_le_imp_le_left
thf(fact_188_subset__mset_Oadd__le__imp__le__right,axiom,
! [A: multiset_a,C: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ C ) @ ( plus_plus_multiset_a @ B @ C ) )
=> ( subseteq_mset_a @ A @ B ) ) ).
% subset_mset.add_le_imp_le_right
thf(fact_189_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_190_mset__subset__eqD,axiom,
! [A2: multiset_a,B4: multiset_a,X: a] :
( ( subseteq_mset_a @ A2 @ B4 )
=> ( ( member_a @ X @ ( set_mset_a @ A2 ) )
=> ( member_a @ X @ ( set_mset_a @ B4 ) ) ) ) ).
% mset_subset_eqD
thf(fact_191_set__mset__mono,axiom,
! [A2: multiset_a,B4: multiset_a] :
( ( subseteq_mset_a @ A2 @ B4 )
=> ( ord_less_eq_set_a @ ( set_mset_a @ A2 ) @ ( set_mset_a @ B4 ) ) ) ).
% set_mset_mono
thf(fact_192_mset__subset__eq__add__mset__cancel,axiom,
! [A: a,A2: multiset_a,B4: multiset_a] :
( ( subseteq_mset_a @ ( add_mset_a @ A @ A2 ) @ ( add_mset_a @ A @ B4 ) )
= ( subseteq_mset_a @ A2 @ B4 ) ) ).
% mset_subset_eq_add_mset_cancel
thf(fact_193_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I3: multiset_a,J: multiset_a,K3: multiset_a,L: multiset_a] :
( ( ( ord_le1199012836iset_a @ I3 @ J )
& ( K3 = L ) )
=> ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ I3 @ K3 ) @ ( plus_plus_multiset_a @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_194_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I3: multiset_a,J: multiset_a,K3: multiset_a,L: multiset_a] :
( ( ( I3 = J )
& ( ord_le1199012836iset_a @ K3 @ L ) )
=> ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ I3 @ K3 ) @ ( plus_plus_multiset_a @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_195_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I3: multiset_a,J: multiset_a,K3: multiset_a,L: multiset_a] :
( ( ( ord_le1199012836iset_a @ I3 @ J )
& ( ord_le1199012836iset_a @ K3 @ L ) )
=> ( ord_le1199012836iset_a @ ( plus_plus_multiset_a @ I3 @ K3 ) @ ( plus_plus_multiset_a @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_196_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_197_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_198_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_199_empty__neutral_I2_J,axiom,
! [X: multiset_a] :
( ( plus_plus_multiset_a @ X @ zero_zero_multiset_a )
= X ) ).
% empty_neutral(2)
thf(fact_200_empty__neutral_I1_J,axiom,
! [X: multiset_a] :
( ( plus_plus_multiset_a @ zero_zero_multiset_a @ X )
= X ) ).
% empty_neutral(1)
thf(fact_201_empty__le,axiom,
! [A2: multiset_a] : ( subseteq_mset_a @ zero_zero_multiset_a @ A2 ) ).
% empty_le
thf(fact_202_subset__mset_Ozero__le,axiom,
! [X: multiset_a] : ( subseteq_mset_a @ zero_zero_multiset_a @ X ) ).
% subset_mset.zero_le
thf(fact_203_subset__mset_Obot_Oextremum,axiom,
! [A: multiset_a] : ( subseteq_mset_a @ zero_zero_multiset_a @ A ) ).
% subset_mset.bot.extremum
thf(fact_204_subset__mset_Oadd__decreasing,axiom,
! [A: multiset_a,C: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ A @ zero_zero_multiset_a )
=> ( ( subseteq_mset_a @ C @ B )
=> ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ C ) @ B ) ) ) ).
% subset_mset.add_decreasing
thf(fact_205_subset__mset_Oadd__increasing,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ zero_zero_multiset_a @ A )
=> ( ( subseteq_mset_a @ B @ C )
=> ( subseteq_mset_a @ B @ ( plus_plus_multiset_a @ A @ C ) ) ) ) ).
% subset_mset.add_increasing
thf(fact_206_subset__mset_Oadd__decreasing2,axiom,
! [C: multiset_a,A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ C @ zero_zero_multiset_a )
=> ( ( subseteq_mset_a @ A @ B )
=> ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ C ) @ B ) ) ) ).
% subset_mset.add_decreasing2
thf(fact_207_subset__mset_Oadd__increasing2,axiom,
! [C: multiset_a,B: multiset_a,A: multiset_a] :
( ( subseteq_mset_a @ zero_zero_multiset_a @ C )
=> ( ( subseteq_mset_a @ B @ A )
=> ( subseteq_mset_a @ B @ ( plus_plus_multiset_a @ A @ C ) ) ) ) ).
% subset_mset.add_increasing2
thf(fact_208_subset__mset_Oadd__nonneg__nonneg,axiom,
! [A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ zero_zero_multiset_a @ A )
=> ( ( subseteq_mset_a @ zero_zero_multiset_a @ B )
=> ( subseteq_mset_a @ zero_zero_multiset_a @ ( plus_plus_multiset_a @ A @ B ) ) ) ) ).
% subset_mset.add_nonneg_nonneg
thf(fact_209_subset__mset_Oadd__nonpos__nonpos,axiom,
! [A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ A @ zero_zero_multiset_a )
=> ( ( subseteq_mset_a @ B @ zero_zero_multiset_a )
=> ( subseteq_mset_a @ ( plus_plus_multiset_a @ A @ B ) @ zero_zero_multiset_a ) ) ) ).
% subset_mset.add_nonpos_nonpos
thf(fact_210_subset__mset_Oadd__nonneg__eq__0__iff,axiom,
! [X: multiset_a,Y: multiset_a] :
( ( subseteq_mset_a @ zero_zero_multiset_a @ X )
=> ( ( subseteq_mset_a @ zero_zero_multiset_a @ Y )
=> ( ( ( plus_plus_multiset_a @ X @ Y )
= zero_zero_multiset_a )
= ( ( X = zero_zero_multiset_a )
& ( Y = zero_zero_multiset_a ) ) ) ) ) ).
% subset_mset.add_nonneg_eq_0_iff
thf(fact_211_subset__mset_Oadd__nonpos__eq__0__iff,axiom,
! [X: multiset_a,Y: multiset_a] :
( ( subseteq_mset_a @ X @ zero_zero_multiset_a )
=> ( ( subseteq_mset_a @ Y @ zero_zero_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_nonpos_eq_0_iff
thf(fact_212_subset__mset_Obot_Oextremum__uniqueI,axiom,
! [A: multiset_a] :
( ( subseteq_mset_a @ A @ zero_zero_multiset_a )
=> ( A = zero_zero_multiset_a ) ) ).
% subset_mset.bot.extremum_uniqueI
thf(fact_213_verit__sum__simplify,axiom,
! [A: multiset_a] :
( ( plus_plus_multiset_a @ A @ zero_zero_multiset_a )
= A ) ).
% verit_sum_simplify
% Conjectures (1)
thf(conj_0,conjecture,
( ( multiset_a2 @ ( heapIm1091024090Down_a @ ( t_a @ v @ ( t_a @ v1 @ l1 @ r1 ) @ ( t_a @ v2 @ l2 @ r2 ) ) ) )
= ( multiset_a2 @ ( t_a @ v @ ( t_a @ v1 @ l1 @ r1 ) @ ( t_a @ v2 @ l2 @ r2 ) ) ) ) ).
%------------------------------------------------------------------------------