TPTP Problem File: ITP068^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP068^1 : TPTP v9.0.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer HeapImperative problem prob_383__5342344_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_383__5342344_1 [Des21]
% Status : Theorem
% Rating : 0.38 v9.0.0, 0.50 v8.2.0, 0.38 v8.1.0, 0.36 v7.5.0
% Syntax : Number of formulae : 240 ( 77 unt; 64 typ; 0 def)
% Number of atoms : 543 ( 193 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 2155 ( 59 ~; 11 |; 53 &;1759 @)
% ( 0 <=>; 273 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 8 avg)
% Number of types : 11 ( 10 usr)
% Number of type conns : 278 ( 278 >; 0 *; 0 +; 0 <<)
% Number of symbols : 57 ( 54 usr; 15 con; 0-3 aty)
% Number of variables : 630 ( 43 ^; 569 !; 18 ?; 630 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:30:28.993
%------------------------------------------------------------------------------
% 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 (54)
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_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_OTree_Orel__Tree_001tf__a_001tf__a,type,
rel_Tree_a_a: ( a > a > $o ) > tree_a > tree_a > $o ).
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_Oimage__mset_001tf__a_001t__Multiset__Omultiset_Itf__a_J,type,
image_929116801iset_a: ( a > multiset_a ) > multiset_a > multiset_multiset_a ).
thf(sy_c_Multiset_Oimage__mset_001tf__a_001tf__a,type,
image_mset_a_a: ( a > a ) > multiset_a > multiset_a ).
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_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_001_062_It__Heap__OTree_Itf__a_J_M_062_It__Heap__OTree_Itf__a_J_M_Eo_J_J,type,
ord_le1530450702ee_a_o: ( tree_a > tree_a > $o ) > ( tree_a > tree_a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_062_Itf__a_M_Eo_J_J,type,
ord_less_eq_a_a_o: ( a > a > $o ) > ( a > a > $o ) > $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__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_l1____,type,
l1: tree_a ).
thf(sy_v_la____,type,
la: tree_a ).
thf(sy_v_r,type,
r: tree_a ).
thf(sy_v_r1____,type,
r1: tree_a ).
thf(sy_v_ra____,type,
ra: tree_a ).
thf(sy_v_t,type,
t: tree_a ).
thf(sy_v_v,type,
v: a ).
thf(sy_v_v1____,type,
v1: a ).
thf(sy_v_v2____,type,
v2: a ).
thf(sy_v_va____,type,
va: a ).
% Relevant facts (175)
thf(fact_0__C4_Oprems_C_I2_J,axiom,
is_heap_a @ ra ).
% "4.prems"(2)
thf(fact_1__C4_Oprems_C_I1_J,axiom,
is_heap_a @ la ).
% "4.prems"(1)
thf(fact_2_assms_I2_J,axiom,
is_heap_a @ r ).
% assms(2)
thf(fact_3_assms_I1_J,axiom,
is_heap_a @ l ).
% assms(1)
thf(fact_4_True,axiom,
ord_less_eq_a @ v1 @ v2 ).
% True
thf(fact_5__C4_Oprems_C_I3_J,axiom,
( ( t_a @ v2 @ e_a @ ( t_a @ v1 @ l1 @ r1 ) )
= ( t_a @ va @ la @ ra ) ) ).
% "4.prems"(3)
thf(fact_6_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_7_siftDown_Osimps_I1_J,axiom,
( ( heapIm1091024090Down_a @ e_a )
= e_a ) ).
% siftDown.simps(1)
thf(fact_8_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_9_assms_I3_J,axiom,
( t
= ( t_a @ v @ l @ r ) ) ).
% assms(3)
thf(fact_10_is__heap_Osimps_I2_J,axiom,
! [V: a] : ( is_heap_a @ ( t_a @ V @ e_a @ e_a ) ) ).
% is_heap.simps(2)
thf(fact_11_siftDown__Node,axiom,
! [T: tree_a,V: a,L: tree_a,R: tree_a] :
( ( T
= ( t_a @ V @ L @ R ) )
=> ? [L2: tree_a,V3: a,R2: tree_a] :
( ( ( heapIm1091024090Down_a @ T )
= ( t_a @ V3 @ L2 @ R2 ) )
& ( ord_less_eq_a @ V @ V3 ) ) ) ).
% siftDown_Node
thf(fact_12__C4_Ohyps_C,axiom,
! [L: tree_a,R: tree_a,V: a] :
( ~ ( ord_less_eq_a @ ( val_a @ ( t_a @ v1 @ l1 @ r1 ) ) @ v2 )
=> ( ( is_heap_a @ L )
=> ( ( is_heap_a @ R )
=> ( ( ( t_a @ v2 @ ( heapIm1140443833left_a @ ( t_a @ v1 @ l1 @ r1 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ v1 @ l1 @ r1 ) ) )
= ( t_a @ V @ L @ R ) )
=> ( is_heap_a @ ( heapIm1091024090Down_a @ ( t_a @ v2 @ ( heapIm1140443833left_a @ ( t_a @ v1 @ l1 @ r1 ) ) @ ( heapIm1257206334ight_a @ ( t_a @ v1 @ l1 @ r1 ) ) ) ) ) ) ) ) ) ).
% "4.hyps"
thf(fact_13_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_14_is__heap_Osimps_I1_J,axiom,
is_heap_a @ e_a ).
% is_heap.simps(1)
thf(fact_15_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_16_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_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_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_19_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_20_left_Osimps,axiom,
! [V: a,L: tree_a,R: tree_a] :
( ( heapIm1140443833left_a @ ( t_a @ V @ L @ R ) )
= L ) ).
% left.simps
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_val_Osimps,axiom,
! [V: a,Uu: tree_a,Uv: tree_a] :
( ( val_a @ ( t_a @ V @ Uu @ Uv ) )
= V ) ).
% val.simps
thf(fact_23_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_24_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_25_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_26_in__tree_Osimps_I1_J,axiom,
! [V: a] :
~ ( in_tree_a @ V @ e_a ) ).
% in_tree.simps(1)
thf(fact_27_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_28_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_29_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_30_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_31_siftDown__in__tree,axiom,
! [T: tree_a] :
( ( T != e_a )
=> ( in_tree_a @ ( val_a @ ( heapIm1091024090Down_a @ T ) ) @ T ) ) ).
% siftDown_in_tree
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_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_34_right_Osimps,axiom,
! [V: a,L: tree_a,R: tree_a] :
( ( heapIm1257206334ight_a @ ( t_a @ V @ L @ R ) )
= R ) ).
% right.simps
thf(fact_35_order__refl,axiom,
! [X: $o > a] : ( ord_less_eq_o_a @ X @ X ) ).
% order_refl
thf(fact_36_order__refl,axiom,
! [X: a] : ( ord_less_eq_a @ X @ X ) ).
% order_refl
thf(fact_37_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_38_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_39_le__funI,axiom,
! [F: $o > a,G: $o > a] :
( ! [X4: $o] : ( ord_less_eq_a @ ( F @ X4 ) @ ( G @ X4 ) )
=> ( ord_less_eq_o_a @ F @ G ) ) ).
% le_funI
thf(fact_40_le__fun__def,axiom,
( ord_less_eq_o_a
= ( ^ [F2: $o > a,G2: $o > a] :
! [X5: $o] : ( ord_less_eq_a @ ( F2 @ X5 ) @ ( G2 @ X5 ) ) ) ) ).
% le_fun_def
thf(fact_41_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 )
=> ( ! [X4: $o > a,Y2: $o > a] :
( ( ord_less_eq_o_a @ X4 @ Y2 )
=> ( ord_less_eq_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_42_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 )
=> ( ! [X4: a,Y2: a] :
( ( ord_less_eq_a @ X4 @ Y2 )
=> ( ord_less_eq_o_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_o_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_43_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 )
=> ( ! [X4: $o > a,Y2: $o > a] :
( ( ord_less_eq_o_a @ X4 @ Y2 )
=> ( ord_less_eq_o_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_o_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_44_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 )
=> ( ! [X4: a,Y2: a] :
( ( ord_less_eq_a @ X4 @ Y2 )
=> ( ord_less_eq_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_45_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 )
=> ( ! [X4: a,Y2: a] :
( ( ord_less_eq_a @ X4 @ Y2 )
=> ( ord_less_eq_o_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_o_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_46_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 )
=> ( ! [X4: $o > a,Y2: $o > a] :
( ( ord_less_eq_o_a @ X4 @ Y2 )
=> ( ord_less_eq_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_47_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 )
=> ( ! [X4: $o > a,Y2: $o > a] :
( ( ord_less_eq_o_a @ X4 @ Y2 )
=> ( ord_less_eq_o_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_o_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_48_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 )
=> ( ! [X4: a,Y2: a] :
( ( ord_less_eq_a @ X4 @ Y2 )
=> ( ord_less_eq_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_49_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_50_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 )
=> ( ! [X4: a,Y2: a] :
( ( ord_less_eq_a @ X4 @ Y2 )
=> ( ord_less_eq_o_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_o_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_51_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 )
=> ( ! [X4: $o > a,Y2: $o > a] :
( ( ord_less_eq_o_a @ X4 @ Y2 )
=> ( ord_less_eq_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_52_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 )
=> ( ! [X4: $o > a,Y2: $o > a] :
( ( ord_less_eq_o_a @ X4 @ Y2 )
=> ( ord_less_eq_o_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_o_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_53_ord__eq__le__subst,axiom,
! [A: a,F: a > a,B: a,C: a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_a @ B @ C )
=> ( ! [X4: a,Y2: a] :
( ( ord_less_eq_a @ X4 @ Y2 )
=> ( ord_less_eq_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_54_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 )
=> ( ! [X4: a,Y2: a] :
( ( ord_less_eq_a @ X4 @ Y2 )
=> ( ord_less_eq_o_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_o_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_55_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 )
=> ( ! [X4: $o > a,Y2: $o > a] :
( ( ord_less_eq_o_a @ X4 @ Y2 )
=> ( ord_less_eq_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_56_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 )
=> ( ! [X4: $o > a,Y2: $o > a] :
( ( ord_less_eq_o_a @ X4 @ Y2 )
=> ( ord_less_eq_o_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_o_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_57_ord__le__eq__subst,axiom,
! [A: a,B: a,F: a > a,C: a] :
( ( ord_less_eq_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: a,Y2: a] :
( ( ord_less_eq_a @ X4 @ Y2 )
=> ( ord_less_eq_a @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_58_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_59_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X5: a] : ( member_a @ X5 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_60_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_61_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_62_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_63_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_64_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_65_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_66_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_67_dual__order_Orefl,axiom,
! [A: $o > a] : ( ord_less_eq_o_a @ A @ A ) ).
% dual_order.refl
thf(fact_68_dual__order_Orefl,axiom,
! [A: a] : ( ord_less_eq_a @ A @ A ) ).
% dual_order.refl
thf(fact_69_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_70_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_71_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_72_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_73_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_74_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_75_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_76_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_77_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_78_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_79_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_80_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_81_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_82_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_83_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_84_le__cases,axiom,
! [X: a,Y: a] :
( ~ ( ord_less_eq_a @ X @ Y )
=> ( ord_less_eq_a @ Y @ X ) ) ).
% le_cases
thf(fact_85_eq__refl,axiom,
! [X: $o > a,Y: $o > a] :
( ( X = Y )
=> ( ord_less_eq_o_a @ X @ Y ) ) ).
% eq_refl
thf(fact_86_eq__refl,axiom,
! [X: a,Y: a] :
( ( X = Y )
=> ( ord_less_eq_a @ X @ Y ) ) ).
% eq_refl
thf(fact_87_linear,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
| ( ord_less_eq_a @ Y @ X ) ) ).
% linear
thf(fact_88_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_89_antisym,axiom,
! [X: a,Y: a] :
( ( ord_less_eq_a @ X @ Y )
=> ( ( ord_less_eq_a @ Y @ X )
=> ( X = Y ) ) ) ).
% antisym
thf(fact_90_eq__iff,axiom,
( ( ^ [Y3: $o > a,Z: $o > a] : ( Y3 = Z ) )
= ( ^ [X5: $o > a,Y4: $o > a] :
( ( ord_less_eq_o_a @ X5 @ Y4 )
& ( ord_less_eq_o_a @ Y4 @ X5 ) ) ) ) ).
% eq_iff
thf(fact_91_eq__iff,axiom,
( ( ^ [Y3: a,Z: a] : ( Y3 = Z ) )
= ( ^ [X5: a,Y4: a] :
( ( ord_less_eq_a @ X5 @ Y4 )
& ( ord_less_eq_a @ Y4 @ X5 ) ) ) ) ).
% eq_iff
thf(fact_92_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_93_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_94_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 ) )
=> ( ! [X4: $o > a] :
( ( P @ X4 )
=> ( ! [Y5: $o > a] :
( ( P @ Y5 )
=> ( ord_less_eq_o_a @ Y5 @ X4 ) )
=> ( Q @ X4 ) ) )
=> ( Q @ ( order_Greatest_o_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_95_GreatestI2__order,axiom,
! [P: a > $o,X: a,Q: a > $o] :
( ( P @ X )
=> ( ! [Y2: a] :
( ( P @ Y2 )
=> ( ord_less_eq_a @ Y2 @ X ) )
=> ( ! [X4: a] :
( ( P @ X4 )
=> ( ! [Y5: a] :
( ( P @ Y5 )
=> ( ord_less_eq_a @ Y5 @ X4 ) )
=> ( Q @ X4 ) ) )
=> ( Q @ ( order_Greatest_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_96_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_97_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_98_Tree_Orel__induct,axiom,
! [R3: a > a > $o,X: tree_a,Y: tree_a,Q: tree_a > tree_a > $o] :
( ( rel_Tree_a_a @ R3 @ X @ Y )
=> ( ( Q @ e_a @ e_a )
=> ( ! [A21: a,A22: tree_a,A23: tree_a,B21: a,B22: tree_a,B23: tree_a] :
( ( R3 @ A21 @ B21 )
=> ( ( Q @ A22 @ B22 )
=> ( ( Q @ A23 @ B23 )
=> ( Q @ ( t_a @ A21 @ A22 @ A23 ) @ ( t_a @ B21 @ B22 @ B23 ) ) ) ) )
=> ( Q @ X @ Y ) ) ) ) ).
% Tree.rel_induct
thf(fact_99_Tree_Orel__mono,axiom,
! [R3: a > a > $o,Ra: a > a > $o] :
( ( ord_less_eq_a_a_o @ R3 @ Ra )
=> ( ord_le1530450702ee_a_o @ ( rel_Tree_a_a @ R3 ) @ ( rel_Tree_a_a @ Ra ) ) ) ).
% Tree.rel_mono
thf(fact_100_Tree_Orel__eq,axiom,
( ( rel_Tree_a_a
@ ^ [Y3: a,Z: a] : ( Y3 = Z ) )
= ( ^ [Y3: tree_a,Z: tree_a] : ( Y3 = Z ) ) ) ).
% Tree.rel_eq
thf(fact_101_Tree_Orel__refl,axiom,
! [Ra: a > a > $o,X: tree_a] :
( ! [X4: a] : ( Ra @ X4 @ X4 )
=> ( rel_Tree_a_a @ Ra @ X @ X ) ) ).
% Tree.rel_refl
thf(fact_102_Tree_Orel__inject_I2_J,axiom,
! [R3: a > a > $o,X21: a,X22: tree_a,X23: tree_a,Y21: a,Y22: tree_a,Y23: tree_a] :
( ( rel_Tree_a_a @ R3 @ ( t_a @ X21 @ X22 @ X23 ) @ ( t_a @ Y21 @ Y22 @ Y23 ) )
= ( ( R3 @ X21 @ Y21 )
& ( rel_Tree_a_a @ R3 @ X22 @ Y22 )
& ( rel_Tree_a_a @ R3 @ X23 @ Y23 ) ) ) ).
% Tree.rel_inject(2)
thf(fact_103_Tree_Orel__intros_I2_J,axiom,
! [R3: a > a > $o,X21: a,Y21: a,X22: tree_a,Y22: tree_a,X23: tree_a,Y23: tree_a] :
( ( R3 @ X21 @ Y21 )
=> ( ( rel_Tree_a_a @ R3 @ X22 @ Y22 )
=> ( ( rel_Tree_a_a @ R3 @ X23 @ Y23 )
=> ( rel_Tree_a_a @ R3 @ ( t_a @ X21 @ X22 @ X23 ) @ ( t_a @ Y21 @ Y22 @ Y23 ) ) ) ) ) ).
% Tree.rel_intros(2)
thf(fact_104_Tree_Octr__transfer_I1_J,axiom,
! [R3: a > a > $o] : ( rel_Tree_a_a @ R3 @ e_a @ e_a ) ).
% Tree.ctr_transfer(1)
thf(fact_105_Tree_Orel__distinct_I2_J,axiom,
! [R3: a > a > $o,Y21: a,Y22: tree_a,Y23: tree_a] :
~ ( rel_Tree_a_a @ R3 @ ( t_a @ Y21 @ Y22 @ Y23 ) @ e_a ) ).
% Tree.rel_distinct(2)
thf(fact_106_Tree_Orel__distinct_I1_J,axiom,
! [R3: a > a > $o,Y21: a,Y22: tree_a,Y23: tree_a] :
~ ( rel_Tree_a_a @ R3 @ e_a @ ( t_a @ Y21 @ Y22 @ Y23 ) ) ).
% Tree.rel_distinct(1)
thf(fact_107_Tree_Orel__cases,axiom,
! [R3: a > a > $o,A: tree_a,B: tree_a] :
( ( rel_Tree_a_a @ R3 @ A @ B )
=> ( ( ( A = e_a )
=> ( B != e_a ) )
=> ~ ! [X1: a,X2: tree_a,X3: tree_a] :
( ( A
= ( t_a @ X1 @ X2 @ X3 ) )
=> ! [Y1: a,Y24: tree_a,Y32: tree_a] :
( ( B
= ( t_a @ Y1 @ Y24 @ Y32 ) )
=> ( ( R3 @ X1 @ Y1 )
=> ( ( rel_Tree_a_a @ R3 @ X2 @ Y24 )
=> ~ ( rel_Tree_a_a @ R3 @ X3 @ Y32 ) ) ) ) ) ) ) ).
% Tree.rel_cases
thf(fact_108_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_109_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_110_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_111_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_112_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_113_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_114_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_115_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_116_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_117_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_118_multiset__induct__max,axiom,
! [P: multiset_a > $o,M: multiset_a] :
( ( P @ zero_zero_multiset_a )
=> ( ! [X4: a,M2: multiset_a] :
( ( P @ M2 )
=> ( ! [Xa: a] :
( ( member_a @ Xa @ ( set_mset_a @ M2 ) )
=> ( ord_less_eq_a @ Xa @ X4 ) )
=> ( P @ ( add_mset_a @ X4 @ M2 ) ) ) )
=> ( P @ M ) ) ) ).
% multiset_induct_max
thf(fact_119_multiset__induct__min,axiom,
! [P: multiset_a > $o,M: multiset_a] :
( ( P @ zero_zero_multiset_a )
=> ( ! [X4: a,M2: multiset_a] :
( ( P @ M2 )
=> ( ! [Xa: a] :
( ( member_a @ Xa @ ( set_mset_a @ M2 ) )
=> ( ord_less_eq_a @ X4 @ Xa ) )
=> ( P @ ( add_mset_a @ X4 @ M2 ) ) ) )
=> ( P @ M ) ) ) ).
% multiset_induct_min
thf(fact_120_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_121_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_122_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_123_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_124_multiset__cases,axiom,
! [M: multiset_a] :
( ( M != zero_zero_multiset_a )
=> ~ ! [X4: a,N2: multiset_a] :
( M
!= ( add_mset_a @ X4 @ N2 ) ) ) ).
% multiset_cases
thf(fact_125_multiset__induct,axiom,
! [P: multiset_a > $o,M: multiset_a] :
( ( P @ zero_zero_multiset_a )
=> ( ! [X4: a,M2: multiset_a] :
( ( P @ M2 )
=> ( P @ ( add_mset_a @ X4 @ M2 ) ) )
=> ( P @ M ) ) ) ).
% multiset_induct
thf(fact_126_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_127_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_128_multiset__nonemptyE,axiom,
! [A2: multiset_a] :
( ( A2 != zero_zero_multiset_a )
=> ~ ! [X4: a] :
~ ( member_a @ X4 @ ( set_mset_a @ A2 ) ) ) ).
% multiset_nonemptyE
thf(fact_129_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_130_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_131_multiset_Osimps_I1_J,axiom,
( ( multiset_a2 @ e_a )
= zero_zero_multiset_a ) ).
% multiset.simps(1)
thf(fact_132_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_133_single__subset__iff,axiom,
! [A: a,M: multiset_a] :
( ( subseteq_mset_a @ ( add_mset_a @ A @ zero_zero_multiset_a ) @ M )
= ( member_a @ A @ ( set_mset_a @ M ) ) ) ).
% single_subset_iff
thf(fact_134_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_135_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_136_add__mset__subseteq__single__iff,axiom,
! [A: a,M: multiset_a,B: a] :
( ( subseteq_mset_a @ ( add_mset_a @ A @ M ) @ ( add_mset_a @ B @ zero_zero_multiset_a ) )
= ( ( M = zero_zero_multiset_a )
& ( A = B ) ) ) ).
% add_mset_subseteq_single_iff
thf(fact_137_verit__sum__simplify,axiom,
! [A: multiset_a] :
( ( plus_plus_multiset_a @ A @ zero_zero_multiset_a )
= A ) ).
% verit_sum_simplify
thf(fact_138_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_139_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_140_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_141_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_142_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_143_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_144_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_145_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_146_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_147_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_148_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_149_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_150_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_151_multi__subset__induct,axiom,
! [F3: multiset_a,A2: multiset_a,P: multiset_a > $o] :
( ( subseteq_mset_a @ F3 @ A2 )
=> ( ( P @ zero_zero_multiset_a )
=> ( ! [A4: a,F4: multiset_a] :
( ( member_a @ A4 @ ( set_mset_a @ A2 ) )
=> ( ( P @ F4 )
=> ( P @ ( add_mset_a @ A4 @ F4 ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% multi_subset_induct
thf(fact_152_mset__subset__eq__single,axiom,
! [A: a,B4: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ B4 ) )
=> ( subseteq_mset_a @ ( add_mset_a @ A @ zero_zero_multiset_a ) @ B4 ) ) ).
% mset_subset_eq_single
thf(fact_153_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_154_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_155_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 ) )
=> ! [K3: multiset_a] :
( ( N
= ( plus_plus_multiset_a @ M0 @ K3 ) )
=> ~ ! [B6: a] :
( ( member_a @ B6 @ ( set_mset_a @ K3 ) )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ B6 @ A4 ) @ R ) ) ) ) ) ).
% mult1E
thf(fact_156_mult1I,axiom,
! [M: multiset_a,A: a,M02: multiset_a,N: multiset_a,K4: multiset_a,R: set_Product_prod_a_a] :
( ( M
= ( add_mset_a @ A @ M02 ) )
=> ( ( N
= ( plus_plus_multiset_a @ M02 @ K4 ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ ( set_mset_a @ K4 ) )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ B3 @ A ) @ R ) )
=> ( member340150864iset_a @ ( produc2037245207iset_a @ N @ M ) @ ( mult1_a @ R ) ) ) ) ) ).
% mult1I
thf(fact_157_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 ) ) )
| ? [K3: multiset_a] :
( ! [B6: a] :
( ( member_a @ B6 @ ( set_mset_a @ K3 ) )
=> ( member449909584od_a_a @ ( product_Pair_a_a @ B6 @ A ) @ R ) )
& ( N
= ( plus_plus_multiset_a @ M02 @ K3 ) ) ) ) ) ).
% less_add
thf(fact_158_subsetI,axiom,
! [A2: set_a,B4: set_a] :
( ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ( member_a @ X4 @ B4 ) )
=> ( ord_less_eq_set_a @ A2 @ B4 ) ) ).
% subsetI
thf(fact_159_in__mono,axiom,
! [A2: set_a,B4: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ B4 )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ X @ B4 ) ) ) ).
% in_mono
thf(fact_160_subsetD,axiom,
! [A2: set_a,B4: set_a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B4 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B4 ) ) ) ).
% subsetD
thf(fact_161_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B7: set_a] :
! [X5: a] :
( ( member_a @ X5 @ A6 )
=> ( member_a @ X5 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_162_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B7: set_a] :
! [T2: a] :
( ( member_a @ T2 @ A6 )
=> ( member_a @ T2 @ B7 ) ) ) ) ).
% subset_iff
thf(fact_163_one__step__implies__mult,axiom,
! [J2: multiset_a,K4: multiset_a,R: set_Product_prod_a_a,I2: multiset_a] :
( ( J2 != zero_zero_multiset_a )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( set_mset_a @ K4 ) )
=> ? [Xa: a] :
( ( member_a @ Xa @ ( set_mset_a @ J2 ) )
& ( member449909584od_a_a @ ( product_Pair_a_a @ X4 @ Xa ) @ R ) ) )
=> ( member340150864iset_a @ ( produc2037245207iset_a @ ( plus_plus_multiset_a @ I2 @ K4 ) @ ( plus_plus_multiset_a @ I2 @ J2 ) ) @ ( mult_a @ R ) ) ) ) ).
% one_step_implies_mult
thf(fact_164_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 ) )
& ? [K3: multiset_a] :
( ( M
= ( plus_plus_multiset_a @ I3 @ K3 ) )
& ( J3 != zero_zero_multiset_a )
& ! [X7: a] :
( ( member_a @ X7 @ ( set_mset_a @ K3 ) )
=> ? [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_165_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 )
= ( ! [X5: multiset_a] :
( ( member_multiset_a @ X5 @ ( set_mset_multiset_a @ M ) )
=> ( X5 = zero_zero_multiset_a ) ) ) ) ).
% subset_mset.sum_mset_0_iff
thf(fact_166_mult__cancel__add__mset,axiom,
! [S: set_Product_prod_a_a,Uu: a,X8: multiset_a,Y7: multiset_a] :
( ( trans_a @ S )
=> ( ( irrefl_a @ S )
=> ( ( member340150864iset_a @ ( produc2037245207iset_a @ ( add_mset_a @ Uu @ X8 ) @ ( add_mset_a @ Uu @ Y7 ) ) @ ( mult_a @ S ) )
= ( member340150864iset_a @ ( produc2037245207iset_a @ X8 @ Y7 ) @ ( mult_a @ S ) ) ) ) ) ).
% mult_cancel_add_mset
thf(fact_167_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_168_union__fold__mset__add__mset,axiom,
( plus_plus_multiset_a
= ( fold_m364285649iset_a @ add_mset_a ) ) ).
% union_fold_mset_add_mset
thf(fact_169_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_170_subset__mset_Osum__mset__mono,axiom,
! [K4: multiset_a,F: a > multiset_a,G: a > multiset_a] :
( ! [I4: a] :
( ( member_a @ I4 @ ( set_mset_a @ K4 ) )
=> ( subseteq_mset_a @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( subseteq_mset_a @ ( comm_m315775925iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ ( image_929116801iset_a @ F @ K4 ) ) @ ( comm_m315775925iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ ( image_929116801iset_a @ G @ K4 ) ) ) ) ).
% subset_mset.sum_mset_mono
thf(fact_171_image__mset__add__mset,axiom,
! [F: a > a,A: a,M: multiset_a] :
( ( image_mset_a_a @ F @ ( add_mset_a @ A @ M ) )
= ( add_mset_a @ ( F @ A ) @ ( image_mset_a_a @ F @ M ) ) ) ).
% image_mset_add_mset
thf(fact_172_msed__map__invR,axiom,
! [F: a > a,M: multiset_a,B: a,N: multiset_a] :
( ( ( image_mset_a_a @ F @ M )
= ( add_mset_a @ B @ N ) )
=> ? [M1: multiset_a,A4: a] :
( ( M
= ( add_mset_a @ A4 @ M1 ) )
& ( ( F @ A4 )
= B )
& ( ( image_mset_a_a @ F @ M1 )
= N ) ) ) ).
% msed_map_invR
thf(fact_173_msed__map__invL,axiom,
! [F: a > a,A: a,M: multiset_a,N: multiset_a] :
( ( ( image_mset_a_a @ F @ ( add_mset_a @ A @ M ) )
= N )
=> ? [N1: multiset_a] :
( ( N
= ( add_mset_a @ ( F @ A ) @ N1 ) )
& ( ( image_mset_a_a @ F @ M )
= N1 ) ) ) ).
% msed_map_invL
thf(fact_174_image__mset__single,axiom,
! [F: a > a,X: a] :
( ( image_mset_a_a @ F @ ( add_mset_a @ X @ zero_zero_multiset_a ) )
= ( add_mset_a @ ( F @ X ) @ zero_zero_multiset_a ) ) ).
% image_mset_single
% Conjectures (1)
thf(conj_0,conjecture,
is_heap_a @ ( heapIm1091024090Down_a @ ( t_a @ v2 @ e_a @ ( t_a @ v1 @ l1 @ r1 ) ) ) ).
%------------------------------------------------------------------------------