author | wenzelm |
Thu, 01 Sep 2016 16:05:22 +0200 | |
changeset 63750 | 9c8a366778e1 |
parent 63689 | 61171cbeedde |
child 63793 | e68a0b651eb5 |
permissions | -rw-r--r-- |
10249 | 1 |
(* Title: HOL/Library/Multiset.thy |
15072 | 2 |
Author: Tobias Nipkow, Markus Wenzel, Lawrence C Paulson, Norbert Voelker |
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3 |
Author: Andrei Popescu, TU Muenchen |
59813 | 4 |
Author: Jasmin Blanchette, Inria, LORIA, MPII |
5 |
Author: Dmitriy Traytel, TU Muenchen |
|
6 |
Author: Mathias Fleury, MPII |
|
10249 | 7 |
*) |
8 |
||
60500 | 9 |
section \<open>(Finite) multisets\<close> |
10249 | 10 |
|
15131 | 11 |
theory Multiset |
51599
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optionalized very specific code setup for multisets
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parents:
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diff
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|
12 |
imports Main |
15131 | 13 |
begin |
10249 | 14 |
|
60500 | 15 |
subsection \<open>The type of multisets\<close> |
10249 | 16 |
|
60606 | 17 |
definition "multiset = {f :: 'a \<Rightarrow> nat. finite {x. f x > 0}}" |
18 |
||
19 |
typedef 'a multiset = "multiset :: ('a \<Rightarrow> nat) set" |
|
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cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
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diff
changeset
|
20 |
morphisms count Abs_multiset |
45694
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prefer typedef without extra definition and alternative name;
wenzelm
parents:
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diff
changeset
|
21 |
unfolding multiset_def |
10249 | 22 |
proof |
45694
4a8743618257
prefer typedef without extra definition and alternative name;
wenzelm
parents:
45608
diff
changeset
|
23 |
show "(\<lambda>x. 0::nat) \<in> {f. finite {x. f x > 0}}" by simp |
10249 | 24 |
qed |
25 |
||
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ec64d94cbf9c
multiset operations are defined with lift_definitions;
bulwahn
parents:
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diff
changeset
|
26 |
setup_lifting type_definition_multiset |
19086 | 27 |
|
60606 | 28 |
lemma multiset_eq_iff: "M = N \<longleftrightarrow> (\<forall>a. count M a = count N a)" |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
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diff
changeset
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29 |
by (simp only: count_inject [symmetric] fun_eq_iff) |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
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diff
changeset
|
30 |
|
60606 | 31 |
lemma multiset_eqI: "(\<And>x. count A x = count B x) \<Longrightarrow> A = B" |
39302
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renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39301
diff
changeset
|
32 |
using multiset_eq_iff by auto |
34943
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cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
33 |
|
60606 | 34 |
text \<open>Preservation of the representing set @{term multiset}.\<close> |
35 |
||
36 |
lemma const0_in_multiset: "(\<lambda>a. 0) \<in> multiset" |
|
34943
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cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
37 |
by (simp add: multiset_def) |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
38 |
|
60606 | 39 |
lemma only1_in_multiset: "(\<lambda>b. if b = a then n else 0) \<in> multiset" |
34943
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cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
40 |
by (simp add: multiset_def) |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
41 |
|
60606 | 42 |
lemma union_preserves_multiset: "M \<in> multiset \<Longrightarrow> N \<in> multiset \<Longrightarrow> (\<lambda>a. M a + N a) \<in> multiset" |
34943
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cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
43 |
by (simp add: multiset_def) |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
44 |
|
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
45 |
lemma diff_preserves_multiset: |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
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diff
changeset
|
46 |
assumes "M \<in> multiset" |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
47 |
shows "(\<lambda>a. M a - N a) \<in> multiset" |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
48 |
proof - |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
49 |
have "{x. N x < M x} \<subseteq> {x. 0 < M x}" |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
50 |
by auto |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
51 |
with assms show ?thesis |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
52 |
by (auto simp add: multiset_def intro: finite_subset) |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
53 |
qed |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
54 |
|
41069
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
55 |
lemma filter_preserves_multiset: |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
56 |
assumes "M \<in> multiset" |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
57 |
shows "(\<lambda>x. if P x then M x else 0) \<in> multiset" |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
58 |
proof - |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
59 |
have "{x. (P x \<longrightarrow> 0 < M x) \<and> P x} \<subseteq> {x. 0 < M x}" |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
60 |
by auto |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
61 |
with assms show ?thesis |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
62 |
by (auto simp add: multiset_def intro: finite_subset) |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
63 |
qed |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
64 |
|
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
65 |
lemmas in_multiset = const0_in_multiset only1_in_multiset |
41069
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
66 |
union_preserves_multiset diff_preserves_multiset filter_preserves_multiset |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
67 |
|
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
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68 |
|
60500 | 69 |
subsection \<open>Representing multisets\<close> |
70 |
||
71 |
text \<open>Multiset enumeration\<close> |
|
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parents:
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72 |
|
48008 | 73 |
instantiation multiset :: (type) cancel_comm_monoid_add |
25571
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instantiation target rather than legacy instance
haftmann
parents:
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74 |
begin |
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25507
diff
changeset
|
75 |
|
47429
ec64d94cbf9c
multiset operations are defined with lift_definitions;
bulwahn
parents:
47308
diff
changeset
|
76 |
lift_definition zero_multiset :: "'a multiset" is "\<lambda>a. 0" |
ec64d94cbf9c
multiset operations are defined with lift_definitions;
bulwahn
parents:
47308
diff
changeset
|
77 |
by (rule const0_in_multiset) |
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
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diff
changeset
|
78 |
|
34943
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cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
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diff
changeset
|
79 |
abbreviation Mempty :: "'a multiset" ("{#}") where |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
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diff
changeset
|
80 |
"Mempty \<equiv> 0" |
25571
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instantiation target rather than legacy instance
haftmann
parents:
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diff
changeset
|
81 |
|
60606 | 82 |
lift_definition plus_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" is "\<lambda>M N. (\<lambda>a. M a + N a)" |
47429
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bulwahn
parents:
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diff
changeset
|
83 |
by (rule union_preserves_multiset) |
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25507
diff
changeset
|
84 |
|
60606 | 85 |
lift_definition minus_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" is "\<lambda> M N. \<lambda>a. M a - N a" |
59815
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59813
diff
changeset
|
86 |
by (rule diff_preserves_multiset) |
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59813
diff
changeset
|
87 |
|
48008 | 88 |
instance |
60678 | 89 |
by (standard; transfer; simp add: fun_eq_iff) |
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25507
diff
changeset
|
90 |
|
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
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diff
changeset
|
91 |
end |
10249 | 92 |
|
63195 | 93 |
context |
94 |
begin |
|
95 |
||
96 |
qualified definition is_empty :: "'a multiset \<Rightarrow> bool" where |
|
97 |
[code_abbrev]: "is_empty A \<longleftrightarrow> A = {#}" |
|
98 |
||
99 |
end |
|
100 |
||
101 |
||
60606 | 102 |
lift_definition single :: "'a \<Rightarrow> 'a multiset" is "\<lambda>a b. if b = a then 1 else 0" |
47429
ec64d94cbf9c
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bulwahn
parents:
47308
diff
changeset
|
103 |
by (rule only1_in_multiset) |
15869 | 104 |
|
26145 | 105 |
syntax |
60606 | 106 |
"_multiset" :: "args \<Rightarrow> 'a multiset" ("{#(_)#}") |
25507 | 107 |
translations |
108 |
"{#x, xs#}" == "{#x#} + {#xs#}" |
|
109 |
"{#x#}" == "CONST single x" |
|
110 |
||
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
111 |
lemma count_empty [simp]: "count {#} a = 0" |
47429
ec64d94cbf9c
multiset operations are defined with lift_definitions;
bulwahn
parents:
47308
diff
changeset
|
112 |
by (simp add: zero_multiset.rep_eq) |
10249 | 113 |
|
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
114 |
lemma count_single [simp]: "count {#b#} a = (if b = a then 1 else 0)" |
47429
ec64d94cbf9c
multiset operations are defined with lift_definitions;
bulwahn
parents:
47308
diff
changeset
|
115 |
by (simp add: single.rep_eq) |
29901 | 116 |
|
10249 | 117 |
|
60500 | 118 |
subsection \<open>Basic operations\<close> |
119 |
||
62430
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haftmann
parents:
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diff
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|
120 |
subsubsection \<open>Conversion to set and membership\<close> |
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more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
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|
121 |
|
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122 |
definition set_mset :: "'a multiset \<Rightarrow> 'a set" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
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parents:
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123 |
where "set_mset M = {x. count M x > 0}" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
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|
124 |
|
62537
7a9aa69f9b38
syntax for multiset membership modelled after syntax for set membership
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diff
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|
125 |
abbreviation Melem :: "'a \<Rightarrow> 'a multiset \<Rightarrow> bool" |
7a9aa69f9b38
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haftmann
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|
126 |
where "Melem a M \<equiv> a \<in> set_mset M" |
7a9aa69f9b38
syntax for multiset membership modelled after syntax for set membership
haftmann
parents:
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|
127 |
|
7a9aa69f9b38
syntax for multiset membership modelled after syntax for set membership
haftmann
parents:
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|
128 |
notation |
7a9aa69f9b38
syntax for multiset membership modelled after syntax for set membership
haftmann
parents:
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diff
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|
129 |
Melem ("op \<in>#") and |
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haftmann
parents:
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|
130 |
Melem ("(_/ \<in># _)" [51, 51] 50) |
7a9aa69f9b38
syntax for multiset membership modelled after syntax for set membership
haftmann
parents:
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diff
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|
131 |
|
7a9aa69f9b38
syntax for multiset membership modelled after syntax for set membership
haftmann
parents:
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diff
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|
132 |
notation (ASCII) |
7a9aa69f9b38
syntax for multiset membership modelled after syntax for set membership
haftmann
parents:
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diff
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|
133 |
Melem ("op :#") and |
7a9aa69f9b38
syntax for multiset membership modelled after syntax for set membership
haftmann
parents:
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diff
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|
134 |
Melem ("(_/ :# _)" [51, 51] 50) |
7a9aa69f9b38
syntax for multiset membership modelled after syntax for set membership
haftmann
parents:
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diff
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|
135 |
|
7a9aa69f9b38
syntax for multiset membership modelled after syntax for set membership
haftmann
parents:
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diff
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|
136 |
abbreviation not_Melem :: "'a \<Rightarrow> 'a multiset \<Rightarrow> bool" |
7a9aa69f9b38
syntax for multiset membership modelled after syntax for set membership
haftmann
parents:
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diff
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|
137 |
where "not_Melem a M \<equiv> a \<notin> set_mset M" |
7a9aa69f9b38
syntax for multiset membership modelled after syntax for set membership
haftmann
parents:
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diff
changeset
|
138 |
|
7a9aa69f9b38
syntax for multiset membership modelled after syntax for set membership
haftmann
parents:
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diff
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|
139 |
notation |
7a9aa69f9b38
syntax for multiset membership modelled after syntax for set membership
haftmann
parents:
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diff
changeset
|
140 |
not_Melem ("op \<notin>#") and |
7a9aa69f9b38
syntax for multiset membership modelled after syntax for set membership
haftmann
parents:
62430
diff
changeset
|
141 |
not_Melem ("(_/ \<notin># _)" [51, 51] 50) |
7a9aa69f9b38
syntax for multiset membership modelled after syntax for set membership
haftmann
parents:
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diff
changeset
|
142 |
|
7a9aa69f9b38
syntax for multiset membership modelled after syntax for set membership
haftmann
parents:
62430
diff
changeset
|
143 |
notation (ASCII) |
7a9aa69f9b38
syntax for multiset membership modelled after syntax for set membership
haftmann
parents:
62430
diff
changeset
|
144 |
not_Melem ("op ~:#") and |
7a9aa69f9b38
syntax for multiset membership modelled after syntax for set membership
haftmann
parents:
62430
diff
changeset
|
145 |
not_Melem ("(_/ ~:# _)" [51, 51] 50) |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
146 |
|
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
147 |
context |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
148 |
begin |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
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|
149 |
|
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
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|
150 |
qualified abbreviation Ball :: "'a multiset \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" |
9527ff088c15
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haftmann
parents:
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|
151 |
where "Ball M \<equiv> Set.Ball (set_mset M)" |
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|
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qualified abbreviation Bex :: "'a multiset \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" |
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154 |
where "Bex M \<equiv> Set.Bex (set_mset M)" |
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155 |
|
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end |
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|
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158 |
syntax |
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159 |
"_MBall" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool" ("(3\<forall>_\<in>#_./ _)" [0, 0, 10] 10) |
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160 |
"_MBex" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool" ("(3\<exists>_\<in>#_./ _)" [0, 0, 10] 10) |
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|
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syntax for multiset membership modelled after syntax for set membership
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162 |
syntax (ASCII) |
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"_MBall" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool" ("(3\<forall>_:#_./ _)" [0, 0, 10] 10) |
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164 |
"_MBex" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool" ("(3\<exists>_:#_./ _)" [0, 0, 10] 10) |
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|
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166 |
translations |
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"\<forall>x\<in>#A. P" \<rightleftharpoons> "CONST Multiset.Ball A (\<lambda>x. P)" |
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168 |
"\<exists>x\<in>#A. P" \<rightleftharpoons> "CONST Multiset.Bex A (\<lambda>x. P)" |
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169 |
|
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170 |
lemma count_eq_zero_iff: |
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171 |
"count M x = 0 \<longleftrightarrow> x \<notin># M" |
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172 |
by (auto simp add: set_mset_def) |
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173 |
|
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174 |
lemma not_in_iff: |
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175 |
"x \<notin># M \<longleftrightarrow> count M x = 0" |
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176 |
by (auto simp add: count_eq_zero_iff) |
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177 |
|
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178 |
lemma count_greater_zero_iff [simp]: |
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179 |
"count M x > 0 \<longleftrightarrow> x \<in># M" |
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180 |
by (auto simp add: set_mset_def) |
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181 |
|
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182 |
lemma count_inI: |
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183 |
assumes "count M x = 0 \<Longrightarrow> False" |
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184 |
shows "x \<in># M" |
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185 |
proof (rule ccontr) |
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186 |
assume "x \<notin># M" |
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187 |
with assms show False by (simp add: not_in_iff) |
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188 |
qed |
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189 |
|
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190 |
lemma in_countE: |
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191 |
assumes "x \<in># M" |
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192 |
obtains n where "count M x = Suc n" |
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193 |
proof - |
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194 |
from assms have "count M x > 0" by simp |
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195 |
then obtain n where "count M x = Suc n" |
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196 |
using gr0_conv_Suc by blast |
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197 |
with that show thesis . |
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198 |
qed |
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199 |
|
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200 |
lemma count_greater_eq_Suc_zero_iff [simp]: |
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201 |
"count M x \<ge> Suc 0 \<longleftrightarrow> x \<in># M" |
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202 |
by (simp add: Suc_le_eq) |
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203 |
|
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204 |
lemma count_greater_eq_one_iff [simp]: |
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205 |
"count M x \<ge> 1 \<longleftrightarrow> x \<in># M" |
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206 |
by simp |
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207 |
|
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208 |
lemma set_mset_empty [simp]: |
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209 |
"set_mset {#} = {}" |
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210 |
by (simp add: set_mset_def) |
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211 |
|
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212 |
lemma set_mset_single [simp]: |
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213 |
"set_mset {#b#} = {b}" |
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214 |
by (simp add: set_mset_def) |
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215 |
|
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216 |
lemma set_mset_eq_empty_iff [simp]: |
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217 |
"set_mset M = {} \<longleftrightarrow> M = {#}" |
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218 |
by (auto simp add: multiset_eq_iff count_eq_zero_iff) |
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219 |
|
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220 |
lemma finite_set_mset [iff]: |
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221 |
"finite (set_mset M)" |
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222 |
using count [of M] by (simp add: multiset_def) |
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223 |
|
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224 |
|
60500 | 225 |
subsubsection \<open>Union\<close> |
10249 | 226 |
|
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227 |
lemma count_union [simp]: |
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228 |
"count (M + N) a = count M a + count N a" |
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229 |
by (simp add: plus_multiset.rep_eq) |
10249 | 230 |
|
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231 |
lemma set_mset_union [simp]: |
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232 |
"set_mset (M + N) = set_mset M \<union> set_mset N" |
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233 |
by (simp only: set_eq_iff count_greater_zero_iff [symmetric] count_union) simp |
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234 |
|
10249 | 235 |
|
60500 | 236 |
subsubsection \<open>Difference\<close> |
10249 | 237 |
|
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238 |
instance multiset :: (type) comm_monoid_diff |
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239 |
by standard (transfer; simp add: fun_eq_iff) |
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240 |
|
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241 |
lemma count_diff [simp]: |
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242 |
"count (M - N) a = count M a - count N a" |
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|
243 |
by (simp add: minus_multiset.rep_eq) |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
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|
244 |
|
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245 |
lemma in_diff_count: |
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|
246 |
"a \<in># M - N \<longleftrightarrow> count N a < count M a" |
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247 |
by (simp add: set_mset_def) |
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|
248 |
|
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|
249 |
lemma count_in_diffI: |
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|
250 |
assumes "\<And>n. count N x = n + count M x \<Longrightarrow> False" |
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251 |
shows "x \<in># M - N" |
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|
252 |
proof (rule ccontr) |
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|
253 |
assume "x \<notin># M - N" |
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254 |
then have "count N x = (count N x - count M x) + count M x" |
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255 |
by (simp add: in_diff_count not_less) |
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|
256 |
with assms show False by auto |
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|
257 |
qed |
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|
258 |
|
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259 |
lemma in_diff_countE: |
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260 |
assumes "x \<in># M - N" |
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|
261 |
obtains n where "count M x = Suc n + count N x" |
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|
262 |
proof - |
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|
263 |
from assms have "count M x - count N x > 0" by (simp add: in_diff_count) |
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|
264 |
then have "count M x > count N x" by simp |
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|
265 |
then obtain n where "count M x = Suc n + count N x" |
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|
266 |
using less_iff_Suc_add by auto |
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|
267 |
with that show thesis . |
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|
268 |
qed |
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|
269 |
|
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270 |
lemma in_diffD: |
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271 |
assumes "a \<in># M - N" |
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|
272 |
shows "a \<in># M" |
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|
273 |
proof - |
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|
274 |
have "0 \<le> count N a" by simp |
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275 |
also from assms have "count N a < count M a" |
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|
276 |
by (simp add: in_diff_count) |
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|
277 |
finally show ?thesis by simp |
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haftmann
parents:
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diff
changeset
|
278 |
qed |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
279 |
|
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
280 |
lemma set_mset_diff: |
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haftmann
parents:
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diff
changeset
|
281 |
"set_mset (M - N) = {a. count N a < count M a}" |
9527ff088c15
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parents:
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diff
changeset
|
282 |
by (simp add: set_mset_def) |
9527ff088c15
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haftmann
parents:
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diff
changeset
|
283 |
|
17161 | 284 |
lemma diff_empty [simp]: "M - {#} = M \<and> {#} - M = {#}" |
52289 | 285 |
by rule (fact Groups.diff_zero, fact Groups.zero_diff) |
36903 | 286 |
|
62430
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parents:
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diff
changeset
|
287 |
lemma diff_cancel [simp]: "A - A = {#}" |
52289 | 288 |
by (fact Groups.diff_cancel) |
10249 | 289 |
|
36903 | 290 |
lemma diff_union_cancelR [simp]: "M + N - N = (M::'a multiset)" |
52289 | 291 |
by (fact add_diff_cancel_right') |
10249 | 292 |
|
36903 | 293 |
lemma diff_union_cancelL [simp]: "N + M - N = (M::'a multiset)" |
52289 | 294 |
by (fact add_diff_cancel_left') |
34943
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cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
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diff
changeset
|
295 |
|
52289 | 296 |
lemma diff_right_commute: |
60606 | 297 |
fixes M N Q :: "'a multiset" |
298 |
shows "M - N - Q = M - Q - N" |
|
52289 | 299 |
by (fact diff_right_commute) |
300 |
||
301 |
lemma diff_add: |
|
60606 | 302 |
fixes M N Q :: "'a multiset" |
303 |
shows "M - (N + Q) = M - N - Q" |
|
52289 | 304 |
by (rule sym) (fact diff_diff_add) |
58425 | 305 |
|
60606 | 306 |
lemma insert_DiffM: "x \<in># M \<Longrightarrow> {#x#} + (M - {#x#}) = M" |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39301
diff
changeset
|
307 |
by (clarsimp simp: multiset_eq_iff) |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
308 |
|
60606 | 309 |
lemma insert_DiffM2 [simp]: "x \<in># M \<Longrightarrow> M - {#x#} + {#x#} = M" |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39301
diff
changeset
|
310 |
by (clarsimp simp: multiset_eq_iff) |
34943
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cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
311 |
|
60606 | 312 |
lemma diff_union_swap: "a \<noteq> b \<Longrightarrow> M - {#a#} + {#b#} = M + {#b#} - {#a#}" |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39301
diff
changeset
|
313 |
by (auto simp add: multiset_eq_iff) |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
314 |
|
62430
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more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
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diff
changeset
|
315 |
lemma diff_union_single_conv: |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
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diff
changeset
|
316 |
"a \<in># J \<Longrightarrow> I + J - {#a#} = I + (J - {#a#})" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
317 |
by (simp add: multiset_eq_iff Suc_le_eq) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
318 |
|
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
319 |
lemma mset_add [elim?]: |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
320 |
assumes "a \<in># A" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
321 |
obtains B where "A = B + {#a#}" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
322 |
proof - |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
323 |
from assms have "A = (A - {#a#}) + {#a#}" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
324 |
by simp |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
325 |
with that show thesis . |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
326 |
qed |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
327 |
|
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
328 |
lemma union_iff: |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
329 |
"a \<in># A + B \<longleftrightarrow> a \<in># A \<or> a \<in># B" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
330 |
by auto |
26143
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
331 |
|
10249 | 332 |
|
60500 | 333 |
subsubsection \<open>Equality of multisets\<close> |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
334 |
|
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
335 |
lemma single_not_empty [simp]: "{#a#} \<noteq> {#} \<and> {#} \<noteq> {#a#}" |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39301
diff
changeset
|
336 |
by (simp add: multiset_eq_iff) |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
337 |
|
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
338 |
lemma single_eq_single [simp]: "{#a#} = {#b#} \<longleftrightarrow> a = b" |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39301
diff
changeset
|
339 |
by (auto simp add: multiset_eq_iff) |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
340 |
|
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
341 |
lemma union_eq_empty [iff]: "M + N = {#} \<longleftrightarrow> M = {#} \<and> N = {#}" |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39301
diff
changeset
|
342 |
by (auto simp add: multiset_eq_iff) |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
343 |
|
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
344 |
lemma empty_eq_union [iff]: "{#} = M + N \<longleftrightarrow> M = {#} \<and> N = {#}" |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39301
diff
changeset
|
345 |
by (auto simp add: multiset_eq_iff) |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
346 |
|
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
347 |
lemma multi_self_add_other_not_self [simp]: "M = M + {#x#} \<longleftrightarrow> False" |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39301
diff
changeset
|
348 |
by (auto simp add: multiset_eq_iff) |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
349 |
|
60606 | 350 |
lemma diff_single_trivial: "\<not> x \<in># M \<Longrightarrow> M - {#x#} = M" |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
351 |
by (auto simp add: multiset_eq_iff not_in_iff) |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
352 |
|
60606 | 353 |
lemma diff_single_eq_union: "x \<in># M \<Longrightarrow> M - {#x#} = N \<longleftrightarrow> M = N + {#x#}" |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
354 |
by auto |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
355 |
|
60606 | 356 |
lemma union_single_eq_diff: "M + {#x#} = N \<Longrightarrow> M = N - {#x#}" |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
357 |
by (auto dest: sym) |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
358 |
|
60606 | 359 |
lemma union_single_eq_member: "M + {#x#} = N \<Longrightarrow> x \<in># N" |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
360 |
by auto |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
361 |
|
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
362 |
lemma union_is_single: |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
363 |
"M + N = {#a#} \<longleftrightarrow> M = {#a#} \<and> N = {#} \<or> M = {#} \<and> N = {#a#}" |
60606 | 364 |
(is "?lhs = ?rhs") |
46730 | 365 |
proof |
60606 | 366 |
show ?lhs if ?rhs using that by auto |
367 |
show ?rhs if ?lhs |
|
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
368 |
by (metis Multiset.diff_cancel add.commute add_diff_cancel_left' diff_add_zero diff_single_trivial insert_DiffM that) |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
369 |
qed |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
370 |
|
60606 | 371 |
lemma single_is_union: "{#a#} = M + N \<longleftrightarrow> {#a#} = M \<and> N = {#} \<or> M = {#} \<and> {#a#} = N" |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
372 |
by (auto simp add: eq_commute [of "{#a#}" "M + N"] union_is_single) |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
373 |
|
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
374 |
lemma add_eq_conv_diff: |
60606 | 375 |
"M + {#a#} = N + {#b#} \<longleftrightarrow> M = N \<and> a = b \<or> M = N - {#a#} + {#b#} \<and> N = M - {#b#} + {#a#}" |
376 |
(is "?lhs \<longleftrightarrow> ?rhs") |
|
44890
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
44339
diff
changeset
|
377 |
(* shorter: by (simp add: multiset_eq_iff) fastforce *) |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
378 |
proof |
60606 | 379 |
show ?lhs if ?rhs |
380 |
using that |
|
381 |
by (auto simp add: add.assoc add.commute [of "{#b#}"]) |
|
382 |
(drule sym, simp add: add.assoc [symmetric]) |
|
383 |
show ?rhs if ?lhs |
|
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
384 |
proof (cases "a = b") |
60500 | 385 |
case True with \<open>?lhs\<close> show ?thesis by simp |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
386 |
next |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
387 |
case False |
60500 | 388 |
from \<open>?lhs\<close> have "a \<in># N + {#b#}" by (rule union_single_eq_member) |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
389 |
with False have "a \<in># N" by auto |
60500 | 390 |
moreover from \<open>?lhs\<close> have "M = N + {#b#} - {#a#}" by (rule union_single_eq_diff) |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
391 |
moreover note False |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
392 |
ultimately show ?thesis by (auto simp add: diff_right_commute [of _ "{#a#}"] diff_union_swap) |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
393 |
qed |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
394 |
qed |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
395 |
|
58425 | 396 |
lemma insert_noteq_member: |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
397 |
assumes BC: "B + {#b#} = C + {#c#}" |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
398 |
and bnotc: "b \<noteq> c" |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
399 |
shows "c \<in># B" |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
400 |
proof - |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
401 |
have "c \<in># C + {#c#}" by simp |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
402 |
have nc: "\<not> c \<in># {#b#}" using bnotc by simp |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
403 |
then have "c \<in># B + {#b#}" using BC by simp |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
404 |
then show "c \<in># B" using nc by simp |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
405 |
qed |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
406 |
|
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
407 |
lemma add_eq_conv_ex: |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
408 |
"(M + {#a#} = N + {#b#}) = |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
409 |
(M = N \<and> a = b \<or> (\<exists>K. M = K + {#b#} \<and> N = K + {#a#}))" |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
410 |
by (auto simp add: add_eq_conv_diff) |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
411 |
|
60606 | 412 |
lemma multi_member_split: "x \<in># M \<Longrightarrow> \<exists>A. M = A + {#x#}" |
60678 | 413 |
by (rule exI [where x = "M - {#x#}"]) simp |
51600
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
414 |
|
58425 | 415 |
lemma multiset_add_sub_el_shuffle: |
60606 | 416 |
assumes "c \<in># B" |
417 |
and "b \<noteq> c" |
|
58098 | 418 |
shows "B - {#c#} + {#b#} = B + {#b#} - {#c#}" |
419 |
proof - |
|
60500 | 420 |
from \<open>c \<in># B\<close> obtain A where B: "B = A + {#c#}" |
58098 | 421 |
by (blast dest: multi_member_split) |
422 |
have "A + {#b#} = A + {#b#} + {#c#} - {#c#}" by simp |
|
58425 | 423 |
then have "A + {#b#} = A + {#c#} + {#b#} - {#c#}" |
58098 | 424 |
by (simp add: ac_simps) |
425 |
then show ?thesis using B by simp |
|
426 |
qed |
|
427 |
||
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
428 |
|
60500 | 429 |
subsubsection \<open>Pointwise ordering induced by count\<close> |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
430 |
|
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61890
diff
changeset
|
431 |
definition subseteq_mset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "\<subseteq>#" 50) |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61890
diff
changeset
|
432 |
where "A \<subseteq># B = (\<forall>a. count A a \<le> count B a)" |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61890
diff
changeset
|
433 |
|
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61890
diff
changeset
|
434 |
definition subset_mset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "\<subset>#" 50) |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61890
diff
changeset
|
435 |
where "A \<subset># B = (A \<subseteq># B \<and> A \<noteq> B)" |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61890
diff
changeset
|
436 |
|
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
437 |
abbreviation (input) supseteq_mset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "\<supseteq>#" 50) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
438 |
where "supseteq_mset A B \<equiv> B \<subseteq># A" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
439 |
|
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
440 |
abbreviation (input) supset_mset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "\<supset>#" 50) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
441 |
where "supset_mset A B \<equiv> B \<subset># A" |
62208
ad43b3ab06e4
added 'supset' variants for new '<#' etc. symbols on multisets
blanchet
parents:
62082
diff
changeset
|
442 |
|
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61890
diff
changeset
|
443 |
notation (input) |
62208
ad43b3ab06e4
added 'supset' variants for new '<#' etc. symbols on multisets
blanchet
parents:
62082
diff
changeset
|
444 |
subseteq_mset (infix "\<le>#" 50) and |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
445 |
supseteq_mset (infix "\<ge>#" 50) |
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61890
diff
changeset
|
446 |
|
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61890
diff
changeset
|
447 |
notation (ASCII) |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61890
diff
changeset
|
448 |
subseteq_mset (infix "<=#" 50) and |
62208
ad43b3ab06e4
added 'supset' variants for new '<#' etc. symbols on multisets
blanchet
parents:
62082
diff
changeset
|
449 |
subset_mset (infix "<#" 50) and |
ad43b3ab06e4
added 'supset' variants for new '<#' etc. symbols on multisets
blanchet
parents:
62082
diff
changeset
|
450 |
supseteq_mset (infix ">=#" 50) and |
ad43b3ab06e4
added 'supset' variants for new '<#' etc. symbols on multisets
blanchet
parents:
62082
diff
changeset
|
451 |
supset_mset (infix ">#" 50) |
60397
f8a513fedb31
Renaming multiset operators < ~> <#,...
Mathias Fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
59999
diff
changeset
|
452 |
|
60606 | 453 |
interpretation subset_mset: ordered_ab_semigroup_add_imp_le "op +" "op -" "op \<subseteq>#" "op \<subset>#" |
60678 | 454 |
by standard (auto simp add: subset_mset_def subseteq_mset_def multiset_eq_iff intro: order_trans antisym) |
62837 | 455 |
\<comment> \<open>FIXME: avoid junk stemming from type class interpretation\<close> |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
456 |
|
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
457 |
lemma mset_subset_eqI: |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
458 |
"(\<And>a. count A a \<le> count B a) \<Longrightarrow> A \<subseteq># B" |
60397
f8a513fedb31
Renaming multiset operators < ~> <#,...
Mathias Fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
59999
diff
changeset
|
459 |
by (simp add: subseteq_mset_def) |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
460 |
|
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
461 |
lemma mset_subset_eq_count: |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
462 |
"A \<subseteq># B \<Longrightarrow> count A a \<le> count B a" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
463 |
by (simp add: subseteq_mset_def) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
464 |
|
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
465 |
lemma mset_subset_eq_exists_conv: "(A::'a multiset) \<subseteq># B \<longleftrightarrow> (\<exists>C. B = A + C)" |
60678 | 466 |
unfolding subseteq_mset_def |
467 |
apply (rule iffI) |
|
468 |
apply (rule exI [where x = "B - A"]) |
|
469 |
apply (auto intro: multiset_eq_iff [THEN iffD2]) |
|
470 |
done |
|
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
471 |
|
63560
3e3097ac37d1
more instantiations for multiset
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63547
diff
changeset
|
472 |
interpretation subset_mset: ordered_cancel_comm_monoid_diff "op +" 0 "op \<le>#" "op <#" "op -" |
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
473 |
by standard (simp, fact mset_subset_eq_exists_conv) |
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
474 |
|
63560
3e3097ac37d1
more instantiations for multiset
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63547
diff
changeset
|
475 |
interpretation subset_mset: ordered_ab_semigroup_monoid_add_imp_le "op +" 0 "op -" "op \<le>#" "op <#" |
3e3097ac37d1
more instantiations for multiset
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63547
diff
changeset
|
476 |
by standard |
3e3097ac37d1
more instantiations for multiset
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63547
diff
changeset
|
477 |
|
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
478 |
lemma mset_subset_eq_mono_add_right_cancel [simp]: "(A::'a multiset) + C \<subseteq># B + C \<longleftrightarrow> A \<subseteq># B" |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
479 |
by (fact subset_mset.add_le_cancel_right) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
480 |
|
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
481 |
lemma mset_subset_eq_mono_add_left_cancel [simp]: "C + (A::'a multiset) \<subseteq># C + B \<longleftrightarrow> A \<subseteq># B" |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
482 |
by (fact subset_mset.add_le_cancel_left) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
483 |
|
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
484 |
lemma mset_subset_eq_mono_add: "(A::'a multiset) \<subseteq># B \<Longrightarrow> C \<subseteq># D \<Longrightarrow> A + C \<subseteq># B + D" |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
485 |
by (fact subset_mset.add_mono) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
486 |
|
63560
3e3097ac37d1
more instantiations for multiset
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63547
diff
changeset
|
487 |
lemma mset_subset_eq_add_left: "(A::'a multiset) \<subseteq># A + B" |
3e3097ac37d1
more instantiations for multiset
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63547
diff
changeset
|
488 |
by simp |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
489 |
|
63560
3e3097ac37d1
more instantiations for multiset
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63547
diff
changeset
|
490 |
lemma mset_subset_eq_add_right: "B \<subseteq># (A::'a multiset) + B" |
3e3097ac37d1
more instantiations for multiset
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63547
diff
changeset
|
491 |
by simp |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
492 |
|
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
493 |
lemma single_subset_iff [simp]: |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
494 |
"{#a#} \<subseteq># M \<longleftrightarrow> a \<in># M" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
495 |
by (auto simp add: subseteq_mset_def Suc_le_eq) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
496 |
|
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
497 |
lemma mset_subset_eq_single: "a \<in># B \<Longrightarrow> {#a#} \<subseteq># B" |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
498 |
by (simp add: subseteq_mset_def Suc_le_eq) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
499 |
|
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
500 |
lemma multiset_diff_union_assoc: |
60606 | 501 |
fixes A B C D :: "'a multiset" |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
502 |
shows "C \<subseteq># B \<Longrightarrow> A + B - C = A + (B - C)" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
503 |
by (fact subset_mset.diff_add_assoc) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
504 |
|
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
505 |
lemma mset_subset_eq_multiset_union_diff_commute: |
60606 | 506 |
fixes A B C D :: "'a multiset" |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
507 |
shows "B \<subseteq># A \<Longrightarrow> A - B + C = A + C - B" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
508 |
by (fact subset_mset.add_diff_assoc2) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
509 |
|
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
510 |
lemma diff_subset_eq_self[simp]: |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
511 |
"(M::'a multiset) - N \<subseteq># M" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
512 |
by (simp add: subseteq_mset_def) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
513 |
|
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
514 |
lemma mset_subset_eqD: |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
515 |
assumes "A \<subseteq># B" and "x \<in># A" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
516 |
shows "x \<in># B" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
517 |
proof - |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
518 |
from \<open>x \<in># A\<close> have "count A x > 0" by simp |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
519 |
also from \<open>A \<subseteq># B\<close> have "count A x \<le> count B x" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
520 |
by (simp add: subseteq_mset_def) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
521 |
finally show ?thesis by simp |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
522 |
qed |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
523 |
|
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
524 |
lemma mset_subsetD: |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
525 |
"A \<subset># B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B" |
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
526 |
by (auto intro: mset_subset_eqD [of A]) |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
527 |
|
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
528 |
lemma set_mset_mono: |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
529 |
"A \<subseteq># B \<Longrightarrow> set_mset A \<subseteq> set_mset B" |
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
530 |
by (metis mset_subset_eqD subsetI) |
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
531 |
|
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
532 |
lemma mset_subset_eq_insertD: |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
533 |
"A + {#x#} \<subseteq># B \<Longrightarrow> x \<in># B \<and> A \<subset># B" |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
534 |
apply (rule conjI) |
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
535 |
apply (simp add: mset_subset_eqD) |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
536 |
apply (clarsimp simp: subset_mset_def subseteq_mset_def) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
537 |
apply safe |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
538 |
apply (erule_tac x = a in allE) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
539 |
apply (auto split: if_split_asm) |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
540 |
done |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
541 |
|
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
542 |
lemma mset_subset_insertD: |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
543 |
"A + {#x#} \<subset># B \<Longrightarrow> x \<in># B \<and> A \<subset># B" |
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
544 |
by (rule mset_subset_eq_insertD) simp |
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
545 |
|
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
546 |
lemma mset_subset_of_empty[simp]: "A \<subset># {#} \<longleftrightarrow> False" |
60397
f8a513fedb31
Renaming multiset operators < ~> <#,...
Mathias Fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
59999
diff
changeset
|
547 |
by (auto simp add: subseteq_mset_def subset_mset_def multiset_eq_iff) |
f8a513fedb31
Renaming multiset operators < ~> <#,...
Mathias Fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
59999
diff
changeset
|
548 |
|
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
549 |
lemma empty_le [simp]: "{#} \<subseteq># A" |
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
550 |
unfolding mset_subset_eq_exists_conv by auto |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
551 |
|
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
552 |
lemma insert_subset_eq_iff: |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
553 |
"{#a#} + A \<subseteq># B \<longleftrightarrow> a \<in># B \<and> A \<subseteq># B - {#a#}" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
554 |
using le_diff_conv2 [of "Suc 0" "count B a" "count A a"] |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
555 |
apply (auto simp add: subseteq_mset_def not_in_iff Suc_le_eq) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
556 |
apply (rule ccontr) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
557 |
apply (auto simp add: not_in_iff) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
558 |
done |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
559 |
|
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
560 |
lemma insert_union_subset_iff: |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
561 |
"{#a#} + A \<subset># B \<longleftrightarrow> a \<in># B \<and> A \<subset># B - {#a#}" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
562 |
by (auto simp add: insert_subset_eq_iff subset_mset_def insert_DiffM) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
563 |
|
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
564 |
lemma subset_eq_diff_conv: |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
565 |
"A - C \<subseteq># B \<longleftrightarrow> A \<subseteq># B + C" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
566 |
by (simp add: subseteq_mset_def le_diff_conv) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
567 |
|
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
568 |
lemma subset_eq_empty [simp]: "M \<subseteq># {#} \<longleftrightarrow> M = {#}" |
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
569 |
unfolding mset_subset_eq_exists_conv by auto |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
570 |
|
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
571 |
lemma multi_psub_of_add_self[simp]: "A \<subset># A + {#x#}" |
60397
f8a513fedb31
Renaming multiset operators < ~> <#,...
Mathias Fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
59999
diff
changeset
|
572 |
by (auto simp: subset_mset_def subseteq_mset_def) |
f8a513fedb31
Renaming multiset operators < ~> <#,...
Mathias Fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
59999
diff
changeset
|
573 |
|
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
574 |
lemma multi_psub_self[simp]: "(A::'a multiset) \<subset># A = False" |
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
575 |
by simp |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
576 |
|
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
577 |
lemma mset_subset_add_bothsides: "N + {#x#} \<subset># M + {#x#} \<Longrightarrow> N \<subset># M" |
60397
f8a513fedb31
Renaming multiset operators < ~> <#,...
Mathias Fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
59999
diff
changeset
|
578 |
by (fact subset_mset.add_less_imp_less_right) |
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
579 |
|
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
580 |
lemma mset_subset_empty_nonempty: "{#} \<subset># S \<longleftrightarrow> S \<noteq> {#}" |
62378
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62376
diff
changeset
|
581 |
by (fact subset_mset.zero_less_iff_neq_zero) |
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
582 |
|
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
583 |
lemma mset_subset_diff_self: "c \<in># B \<Longrightarrow> B - {#c#} \<subset># B" |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
584 |
by (auto simp: subset_mset_def elim: mset_add) |
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
585 |
|
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
586 |
|
60500 | 587 |
subsubsection \<open>Intersection\<close> |
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
588 |
|
60397
f8a513fedb31
Renaming multiset operators < ~> <#,...
Mathias Fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
59999
diff
changeset
|
589 |
definition inf_subset_mset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" (infixl "#\<inter>" 70) where |
f8a513fedb31
Renaming multiset operators < ~> <#,...
Mathias Fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
59999
diff
changeset
|
590 |
multiset_inter_def: "inf_subset_mset A B = A - (A - B)" |
f8a513fedb31
Renaming multiset operators < ~> <#,...
Mathias Fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
59999
diff
changeset
|
591 |
|
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
592 |
interpretation subset_mset: semilattice_inf inf_subset_mset "op \<subseteq>#" "op \<subset>#" |
46921 | 593 |
proof - |
60678 | 594 |
have [simp]: "m \<le> n \<Longrightarrow> m \<le> q \<Longrightarrow> m \<le> n - (n - q)" for m n q :: nat |
595 |
by arith |
|
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
596 |
show "class.semilattice_inf op #\<inter> op \<subseteq># op \<subset>#" |
60678 | 597 |
by standard (auto simp add: multiset_inter_def subseteq_mset_def) |
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
598 |
qed |
62837 | 599 |
\<comment> \<open>FIXME: avoid junk stemming from type class interpretation\<close> |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
600 |
|
41069
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
601 |
lemma multiset_inter_count [simp]: |
60606 | 602 |
fixes A B :: "'a multiset" |
603 |
shows "count (A #\<inter> B) x = min (count A x) (count B x)" |
|
47429
ec64d94cbf9c
multiset operations are defined with lift_definitions;
bulwahn
parents:
47308
diff
changeset
|
604 |
by (simp add: multiset_inter_def) |
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
605 |
|
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
606 |
lemma set_mset_inter [simp]: |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
607 |
"set_mset (A #\<inter> B) = set_mset A \<inter> set_mset B" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
608 |
by (simp only: set_eq_iff count_greater_zero_iff [symmetric] multiset_inter_count) simp |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
609 |
|
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
610 |
lemma diff_intersect_left_idem [simp]: |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
611 |
"M - M #\<inter> N = M - N" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
612 |
by (simp add: multiset_eq_iff min_def) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
613 |
|
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
614 |
lemma diff_intersect_right_idem [simp]: |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
615 |
"M - N #\<inter> M = M - N" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
616 |
by (simp add: multiset_eq_iff min_def) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
617 |
|
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
618 |
lemma multiset_inter_single: "a \<noteq> b \<Longrightarrow> {#a#} #\<inter> {#b#} = {#}" |
46730 | 619 |
by (rule multiset_eqI) auto |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
620 |
|
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
621 |
lemma multiset_union_diff_commute: |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
622 |
assumes "B #\<inter> C = {#}" |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
623 |
shows "A + B - C = A - C + B" |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39301
diff
changeset
|
624 |
proof (rule multiset_eqI) |
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
625 |
fix x |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
626 |
from assms have "min (count B x) (count C x) = 0" |
46730 | 627 |
by (auto simp add: multiset_eq_iff) |
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
628 |
then have "count B x = 0 \<or> count C x = 0" |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
629 |
unfolding min_def by (auto split: if_splits) |
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
630 |
then show "count (A + B - C) x = count (A - C + B) x" |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
631 |
by auto |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
632 |
qed |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
633 |
|
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
634 |
lemma disjunct_not_in: |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
635 |
"A #\<inter> B = {#} \<longleftrightarrow> (\<forall>a. a \<notin># A \<or> a \<notin># B)" (is "?P \<longleftrightarrow> ?Q") |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
636 |
proof |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
637 |
assume ?P |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
638 |
show ?Q |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
639 |
proof |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
640 |
fix a |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
641 |
from \<open>?P\<close> have "min (count A a) (count B a) = 0" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
642 |
by (simp add: multiset_eq_iff) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
643 |
then have "count A a = 0 \<or> count B a = 0" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
644 |
by (cases "count A a \<le> count B a") (simp_all add: min_def) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
645 |
then show "a \<notin># A \<or> a \<notin># B" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
646 |
by (simp add: not_in_iff) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
647 |
qed |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
648 |
next |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
649 |
assume ?Q |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
650 |
show ?P |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
651 |
proof (rule multiset_eqI) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
652 |
fix a |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
653 |
from \<open>?Q\<close> have "count A a = 0 \<or> count B a = 0" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
654 |
by (auto simp add: not_in_iff) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
655 |
then show "count (A #\<inter> B) a = count {#} a" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
656 |
by auto |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
657 |
qed |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
658 |
qed |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
659 |
|
60606 | 660 |
lemma empty_inter [simp]: "{#} #\<inter> M = {#}" |
51600
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
661 |
by (simp add: multiset_eq_iff) |
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
662 |
|
60606 | 663 |
lemma inter_empty [simp]: "M #\<inter> {#} = {#}" |
51600
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
664 |
by (simp add: multiset_eq_iff) |
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
665 |
|
60606 | 666 |
lemma inter_add_left1: "\<not> x \<in># N \<Longrightarrow> (M + {#x#}) #\<inter> N = M #\<inter> N" |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
667 |
by (simp add: multiset_eq_iff not_in_iff) |
51600
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
668 |
|
60606 | 669 |
lemma inter_add_left2: "x \<in># N \<Longrightarrow> (M + {#x#}) #\<inter> N = (M #\<inter> (N - {#x#})) + {#x#}" |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
670 |
by (auto simp add: multiset_eq_iff elim: mset_add) |
51600
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
671 |
|
60606 | 672 |
lemma inter_add_right1: "\<not> x \<in># N \<Longrightarrow> N #\<inter> (M + {#x#}) = N #\<inter> M" |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
673 |
by (simp add: multiset_eq_iff not_in_iff) |
51600
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
674 |
|
60606 | 675 |
lemma inter_add_right2: "x \<in># N \<Longrightarrow> N #\<inter> (M + {#x#}) = ((N - {#x#}) #\<inter> M) + {#x#}" |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
676 |
by (auto simp add: multiset_eq_iff elim: mset_add) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
677 |
|
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
678 |
lemma disjunct_set_mset_diff: |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
679 |
assumes "M #\<inter> N = {#}" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
680 |
shows "set_mset (M - N) = set_mset M" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
681 |
proof (rule set_eqI) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
682 |
fix a |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
683 |
from assms have "a \<notin># M \<or> a \<notin># N" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
684 |
by (simp add: disjunct_not_in) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
685 |
then show "a \<in># M - N \<longleftrightarrow> a \<in># M" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
686 |
by (auto dest: in_diffD) (simp add: in_diff_count not_in_iff) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
687 |
qed |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
688 |
|
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
689 |
lemma at_most_one_mset_mset_diff: |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
690 |
assumes "a \<notin># M - {#a#}" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
691 |
shows "set_mset (M - {#a#}) = set_mset M - {a}" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
692 |
using assms by (auto simp add: not_in_iff in_diff_count set_eq_iff) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
693 |
|
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
694 |
lemma more_than_one_mset_mset_diff: |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
695 |
assumes "a \<in># M - {#a#}" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
696 |
shows "set_mset (M - {#a#}) = set_mset M" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
697 |
proof (rule set_eqI) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
698 |
fix b |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
699 |
have "Suc 0 < count M b \<Longrightarrow> count M b > 0" by arith |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
700 |
then show "b \<in># M - {#a#} \<longleftrightarrow> b \<in># M" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
701 |
using assms by (auto simp add: in_diff_count) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
702 |
qed |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
703 |
|
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
704 |
lemma inter_iff: |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
705 |
"a \<in># A #\<inter> B \<longleftrightarrow> a \<in># A \<and> a \<in># B" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
706 |
by simp |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
707 |
|
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
708 |
lemma inter_union_distrib_left: |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
709 |
"A #\<inter> B + C = (A + C) #\<inter> (B + C)" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
710 |
by (simp add: multiset_eq_iff min_add_distrib_left) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
711 |
|
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
712 |
lemma inter_union_distrib_right: |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
713 |
"C + A #\<inter> B = (C + A) #\<inter> (C + B)" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
714 |
using inter_union_distrib_left [of A B C] by (simp add: ac_simps) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
715 |
|
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
716 |
lemma inter_subset_eq_union: |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
717 |
"A #\<inter> B \<subseteq># A + B" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
718 |
by (auto simp add: subseteq_mset_def) |
51600
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
719 |
|
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
720 |
|
60500 | 721 |
subsubsection \<open>Bounded union\<close> |
60678 | 722 |
|
723 |
definition sup_subset_mset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset"(infixl "#\<union>" 70) |
|
62837 | 724 |
where "sup_subset_mset A B = A + (B - A)" \<comment> \<open>FIXME irregular fact name\<close> |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
725 |
|
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
726 |
interpretation subset_mset: semilattice_sup sup_subset_mset "op \<subseteq>#" "op \<subset>#" |
51623 | 727 |
proof - |
60678 | 728 |
have [simp]: "m \<le> n \<Longrightarrow> q \<le> n \<Longrightarrow> m + (q - m) \<le> n" for m n q :: nat |
729 |
by arith |
|
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
730 |
show "class.semilattice_sup op #\<union> op \<subseteq># op \<subset>#" |
60678 | 731 |
by standard (auto simp add: sup_subset_mset_def subseteq_mset_def) |
51623 | 732 |
qed |
62837 | 733 |
\<comment> \<open>FIXME: avoid junk stemming from type class interpretation\<close> |
734 |
||
735 |
lemma sup_subset_mset_count [simp]: \<comment> \<open>FIXME irregular fact name\<close> |
|
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
736 |
"count (A #\<union> B) x = max (count A x) (count B x)" |
60397
f8a513fedb31
Renaming multiset operators < ~> <#,...
Mathias Fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
59999
diff
changeset
|
737 |
by (simp add: sup_subset_mset_def) |
51623 | 738 |
|
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
739 |
lemma set_mset_sup [simp]: |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
740 |
"set_mset (A #\<union> B) = set_mset A \<union> set_mset B" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
741 |
by (simp only: set_eq_iff count_greater_zero_iff [symmetric] sup_subset_mset_count) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
742 |
(auto simp add: not_in_iff elim: mset_add) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
743 |
|
60606 | 744 |
lemma empty_sup [simp]: "{#} #\<union> M = M" |
51623 | 745 |
by (simp add: multiset_eq_iff) |
746 |
||
60606 | 747 |
lemma sup_empty [simp]: "M #\<union> {#} = M" |
51623 | 748 |
by (simp add: multiset_eq_iff) |
749 |
||
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
750 |
lemma sup_union_left1: "\<not> x \<in># N \<Longrightarrow> (M + {#x#}) #\<union> N = (M #\<union> N) + {#x#}" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
751 |
by (simp add: multiset_eq_iff not_in_iff) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
752 |
|
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
753 |
lemma sup_union_left2: "x \<in># N \<Longrightarrow> (M + {#x#}) #\<union> N = (M #\<union> (N - {#x#})) + {#x#}" |
51623 | 754 |
by (simp add: multiset_eq_iff) |
755 |
||
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
756 |
lemma sup_union_right1: "\<not> x \<in># N \<Longrightarrow> N #\<union> (M + {#x#}) = (N #\<union> M) + {#x#}" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
757 |
by (simp add: multiset_eq_iff not_in_iff) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
758 |
|
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
759 |
lemma sup_union_right2: "x \<in># N \<Longrightarrow> N #\<union> (M + {#x#}) = ((N - {#x#}) #\<union> M) + {#x#}" |
51623 | 760 |
by (simp add: multiset_eq_iff) |
761 |
||
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
762 |
lemma sup_union_distrib_left: |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
763 |
"A #\<union> B + C = (A + C) #\<union> (B + C)" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
764 |
by (simp add: multiset_eq_iff max_add_distrib_left) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
765 |
|
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
766 |
lemma union_sup_distrib_right: |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
767 |
"C + A #\<union> B = (C + A) #\<union> (C + B)" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
768 |
using sup_union_distrib_left [of A B C] by (simp add: ac_simps) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
769 |
|
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
770 |
lemma union_diff_inter_eq_sup: |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
771 |
"A + B - A #\<inter> B = A #\<union> B" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
772 |
by (auto simp add: multiset_eq_iff) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
773 |
|
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
774 |
lemma union_diff_sup_eq_inter: |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
775 |
"A + B - A #\<union> B = A #\<inter> B" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
776 |
by (auto simp add: multiset_eq_iff) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
777 |
|
51623 | 778 |
|
60500 | 779 |
subsubsection \<open>Subset is an order\<close> |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
780 |
|
60397
f8a513fedb31
Renaming multiset operators < ~> <#,...
Mathias Fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
59999
diff
changeset
|
781 |
interpretation subset_mset: order "op \<le>#" "op <#" by unfold_locales auto |
51623 | 782 |
|
63409
3f3223b90239
moved lemmas and locales around (with minor incompatibilities)
blanchet
parents:
63388
diff
changeset
|
783 |
|
63358
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
784 |
subsubsection \<open>Conditionally complete lattice\<close> |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
785 |
|
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
786 |
instantiation multiset :: (type) Inf |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
787 |
begin |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
788 |
|
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
789 |
lift_definition Inf_multiset :: "'a multiset set \<Rightarrow> 'a multiset" is |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
790 |
"\<lambda>A i. if A = {} then 0 else Inf ((\<lambda>f. f i) ` A)" |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
791 |
proof - |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
792 |
fix A :: "('a \<Rightarrow> nat) set" assume *: "\<And>x. x \<in> A \<Longrightarrow> x \<in> multiset" |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
793 |
have "finite {i. (if A = {} then 0 else Inf ((\<lambda>f. f i) ` A)) > 0}" unfolding multiset_def |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
794 |
proof (cases "A = {}") |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
795 |
case False |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
796 |
then obtain f where "f \<in> A" by blast |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
797 |
hence "{i. Inf ((\<lambda>f. f i) ` A) > 0} \<subseteq> {i. f i > 0}" |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
798 |
by (auto intro: less_le_trans[OF _ cInf_lower]) |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
799 |
moreover from \<open>f \<in> A\<close> * have "finite \<dots>" by (simp add: multiset_def) |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
800 |
ultimately have "finite {i. Inf ((\<lambda>f. f i) ` A) > 0}" by (rule finite_subset) |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
801 |
with False show ?thesis by simp |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
802 |
qed simp_all |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
803 |
thus "(\<lambda>i. if A = {} then 0 else INF f:A. f i) \<in> multiset" by (simp add: multiset_def) |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
804 |
qed |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
805 |
|
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
806 |
instance .. |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
807 |
|
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
808 |
end |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
809 |
|
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
810 |
lemma Inf_multiset_empty: "Inf {} = {#}" |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
811 |
by transfer simp_all |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
812 |
|
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
813 |
lemma count_Inf_multiset_nonempty: "A \<noteq> {} \<Longrightarrow> count (Inf A) x = Inf ((\<lambda>X. count X x) ` A)" |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
814 |
by transfer simp_all |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
815 |
|
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
816 |
|
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
817 |
instantiation multiset :: (type) Sup |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
818 |
begin |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
819 |
|
63360 | 820 |
definition Sup_multiset :: "'a multiset set \<Rightarrow> 'a multiset" where |
821 |
"Sup_multiset A = (if A \<noteq> {} \<and> subset_mset.bdd_above A then |
|
822 |
Abs_multiset (\<lambda>i. Sup ((\<lambda>X. count X i) ` A)) else {#})" |
|
823 |
||
824 |
lemma Sup_multiset_empty: "Sup {} = {#}" |
|
825 |
by (simp add: Sup_multiset_def) |
|
826 |
||
827 |
lemma Sup_multiset_unbounded: "\<not>subset_mset.bdd_above A \<Longrightarrow> Sup A = {#}" |
|
828 |
by (simp add: Sup_multiset_def) |
|
63358
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
829 |
|
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
830 |
instance .. |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
831 |
|
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
832 |
end |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
833 |
|
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
834 |
lemma bdd_below_multiset [simp]: "subset_mset.bdd_below A" |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
835 |
by (intro subset_mset.bdd_belowI[of _ "{#}"]) simp_all |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
836 |
|
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
837 |
lemma bdd_above_multiset_imp_bdd_above_count: |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
838 |
assumes "subset_mset.bdd_above (A :: 'a multiset set)" |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
839 |
shows "bdd_above ((\<lambda>X. count X x) ` A)" |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
840 |
proof - |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
841 |
from assms obtain Y where Y: "\<forall>X\<in>A. X \<subseteq># Y" |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
842 |
by (auto simp: subset_mset.bdd_above_def) |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
843 |
hence "count X x \<le> count Y x" if "X \<in> A" for X |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
844 |
using that by (auto intro: mset_subset_eq_count) |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
845 |
thus ?thesis by (intro bdd_aboveI[of _ "count Y x"]) auto |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
846 |
qed |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
847 |
|
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
848 |
lemma bdd_above_multiset_imp_finite_support: |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
849 |
assumes "A \<noteq> {}" "subset_mset.bdd_above (A :: 'a multiset set)" |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
850 |
shows "finite (\<Union>X\<in>A. {x. count X x > 0})" |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
851 |
proof - |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
852 |
from assms obtain Y where Y: "\<forall>X\<in>A. X \<subseteq># Y" |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
853 |
by (auto simp: subset_mset.bdd_above_def) |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
854 |
hence "count X x \<le> count Y x" if "X \<in> A" for X x |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
855 |
using that by (auto intro: mset_subset_eq_count) |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
856 |
hence "(\<Union>X\<in>A. {x. count X x > 0}) \<subseteq> {x. count Y x > 0}" |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
857 |
by safe (erule less_le_trans) |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
858 |
moreover have "finite \<dots>" by simp |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
859 |
ultimately show ?thesis by (rule finite_subset) |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
860 |
qed |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
861 |
|
63360 | 862 |
lemma Sup_multiset_in_multiset: |
863 |
assumes "A \<noteq> {}" "subset_mset.bdd_above A" |
|
864 |
shows "(\<lambda>i. SUP X:A. count X i) \<in> multiset" |
|
865 |
unfolding multiset_def |
|
866 |
proof |
|
867 |
have "{i. Sup ((\<lambda>X. count X i) ` A) > 0} \<subseteq> (\<Union>X\<in>A. {i. 0 < count X i})" |
|
868 |
proof safe |
|
869 |
fix i assume pos: "(SUP X:A. count X i) > 0" |
|
870 |
show "i \<in> (\<Union>X\<in>A. {i. 0 < count X i})" |
|
871 |
proof (rule ccontr) |
|
872 |
assume "i \<notin> (\<Union>X\<in>A. {i. 0 < count X i})" |
|
873 |
hence "\<forall>X\<in>A. count X i \<le> 0" by (auto simp: count_eq_zero_iff) |
|
874 |
with assms have "(SUP X:A. count X i) \<le> 0" |
|
875 |
by (intro cSup_least bdd_above_multiset_imp_bdd_above_count) auto |
|
876 |
with pos show False by simp |
|
877 |
qed |
|
878 |
qed |
|
879 |
moreover from assms have "finite \<dots>" by (rule bdd_above_multiset_imp_finite_support) |
|
880 |
ultimately show "finite {i. Sup ((\<lambda>X. count X i) ` A) > 0}" by (rule finite_subset) |
|
881 |
qed |
|
882 |
||
63358
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
883 |
lemma count_Sup_multiset_nonempty: |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
884 |
assumes "A \<noteq> {}" "subset_mset.bdd_above A" |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
885 |
shows "count (Sup A) x = (SUP X:A. count X x)" |
63360 | 886 |
using assms by (simp add: Sup_multiset_def Abs_multiset_inverse Sup_multiset_in_multiset) |
63358
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
887 |
|
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
888 |
|
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
889 |
interpretation subset_mset: conditionally_complete_lattice Inf Sup "op #\<inter>" "op \<subseteq>#" "op \<subset>#" "op #\<union>" |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
890 |
proof |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
891 |
fix X :: "'a multiset" and A |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
892 |
assume "X \<in> A" |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
893 |
show "Inf A \<subseteq># X" |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
894 |
proof (rule mset_subset_eqI) |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
895 |
fix x |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
896 |
from \<open>X \<in> A\<close> have "A \<noteq> {}" by auto |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
897 |
hence "count (Inf A) x = (INF X:A. count X x)" |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
898 |
by (simp add: count_Inf_multiset_nonempty) |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
899 |
also from \<open>X \<in> A\<close> have "\<dots> \<le> count X x" |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
900 |
by (intro cInf_lower) simp_all |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
901 |
finally show "count (Inf A) x \<le> count X x" . |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
902 |
qed |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
903 |
next |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
904 |
fix X :: "'a multiset" and A |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
905 |
assume nonempty: "A \<noteq> {}" and le: "\<And>Y. Y \<in> A \<Longrightarrow> X \<subseteq># Y" |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
906 |
show "X \<subseteq># Inf A" |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
907 |
proof (rule mset_subset_eqI) |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
908 |
fix x |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
909 |
from nonempty have "count X x \<le> (INF X:A. count X x)" |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
910 |
by (intro cInf_greatest) (auto intro: mset_subset_eq_count le) |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
911 |
also from nonempty have "\<dots> = count (Inf A) x" by (simp add: count_Inf_multiset_nonempty) |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
912 |
finally show "count X x \<le> count (Inf A) x" . |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
913 |
qed |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
914 |
next |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
915 |
fix X :: "'a multiset" and A |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
916 |
assume X: "X \<in> A" and bdd: "subset_mset.bdd_above A" |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
917 |
show "X \<subseteq># Sup A" |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
918 |
proof (rule mset_subset_eqI) |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
919 |
fix x |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
920 |
from X have "A \<noteq> {}" by auto |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
921 |
have "count X x \<le> (SUP X:A. count X x)" |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
922 |
by (intro cSUP_upper X bdd_above_multiset_imp_bdd_above_count bdd) |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
923 |
also from count_Sup_multiset_nonempty[OF \<open>A \<noteq> {}\<close> bdd] |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
924 |
have "(SUP X:A. count X x) = count (Sup A) x" by simp |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
925 |
finally show "count X x \<le> count (Sup A) x" . |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
926 |
qed |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
927 |
next |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
928 |
fix X :: "'a multiset" and A |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
929 |
assume nonempty: "A \<noteq> {}" and ge: "\<And>Y. Y \<in> A \<Longrightarrow> Y \<subseteq># X" |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
930 |
from ge have bdd: "subset_mset.bdd_above A" by (rule subset_mset.bdd_aboveI[of _ X]) |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
931 |
show "Sup A \<subseteq># X" |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
932 |
proof (rule mset_subset_eqI) |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
933 |
fix x |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
934 |
from count_Sup_multiset_nonempty[OF \<open>A \<noteq> {}\<close> bdd] |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
935 |
have "count (Sup A) x = (SUP X:A. count X x)" . |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
936 |
also from nonempty have "\<dots> \<le> count X x" |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
937 |
by (intro cSup_least) (auto intro: mset_subset_eq_count ge) |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
938 |
finally show "count (Sup A) x \<le> count X x" . |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
939 |
qed |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
940 |
qed |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
941 |
|
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
942 |
lemma set_mset_Inf: |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
943 |
assumes "A \<noteq> {}" |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
944 |
shows "set_mset (Inf A) = (\<Inter>X\<in>A. set_mset X)" |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
945 |
proof safe |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
946 |
fix x X assume "x \<in># Inf A" "X \<in> A" |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
947 |
hence nonempty: "A \<noteq> {}" by (auto simp: Inf_multiset_empty) |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
948 |
from \<open>x \<in># Inf A\<close> have "{#x#} \<subseteq># Inf A" by auto |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
949 |
also from \<open>X \<in> A\<close> have "\<dots> \<subseteq># X" by (rule subset_mset.cInf_lower) simp_all |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
950 |
finally show "x \<in># X" by simp |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
951 |
next |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
952 |
fix x assume x: "x \<in> (\<Inter>X\<in>A. set_mset X)" |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
953 |
hence "{#x#} \<subseteq># X" if "X \<in> A" for X using that by auto |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
954 |
from assms and this have "{#x#} \<subseteq># Inf A" by (rule subset_mset.cInf_greatest) |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
955 |
thus "x \<in># Inf A" by simp |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
956 |
qed |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
957 |
|
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
958 |
lemma in_Inf_multiset_iff: |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
959 |
assumes "A \<noteq> {}" |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
960 |
shows "x \<in># Inf A \<longleftrightarrow> (\<forall>X\<in>A. x \<in># X)" |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
961 |
proof - |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
962 |
from assms have "set_mset (Inf A) = (\<Inter>X\<in>A. set_mset X)" by (rule set_mset_Inf) |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
963 |
also have "x \<in> \<dots> \<longleftrightarrow> (\<forall>X\<in>A. x \<in># X)" by simp |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
964 |
finally show ?thesis . |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
965 |
qed |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
966 |
|
63360 | 967 |
lemma in_Inf_multisetD: "x \<in># Inf A \<Longrightarrow> X \<in> A \<Longrightarrow> x \<in># X" |
968 |
by (subst (asm) in_Inf_multiset_iff) auto |
|
969 |
||
63358
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
970 |
lemma set_mset_Sup: |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
971 |
assumes "subset_mset.bdd_above A" |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
972 |
shows "set_mset (Sup A) = (\<Union>X\<in>A. set_mset X)" |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
973 |
proof safe |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
974 |
fix x assume "x \<in># Sup A" |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
975 |
hence nonempty: "A \<noteq> {}" by (auto simp: Sup_multiset_empty) |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
976 |
show "x \<in> (\<Union>X\<in>A. set_mset X)" |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
977 |
proof (rule ccontr) |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
978 |
assume x: "x \<notin> (\<Union>X\<in>A. set_mset X)" |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
979 |
have "count X x \<le> count (Sup A) x" if "X \<in> A" for X x |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
980 |
using that by (intro mset_subset_eq_count subset_mset.cSup_upper assms) |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
981 |
with x have "X \<subseteq># Sup A - {#x#}" if "X \<in> A" for X |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
982 |
using that by (auto simp: subseteq_mset_def algebra_simps not_in_iff) |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
983 |
hence "Sup A \<subseteq># Sup A - {#x#}" by (intro subset_mset.cSup_least nonempty) |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
984 |
with \<open>x \<in># Sup A\<close> show False |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
985 |
by (auto simp: subseteq_mset_def count_greater_zero_iff [symmetric] |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
986 |
simp del: count_greater_zero_iff dest!: spec[of _ x]) |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
987 |
qed |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
988 |
next |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
989 |
fix x X assume "x \<in> set_mset X" "X \<in> A" |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
990 |
hence "{#x#} \<subseteq># X" by auto |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
991 |
also have "X \<subseteq># Sup A" by (intro subset_mset.cSup_upper \<open>X \<in> A\<close> assms) |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
992 |
finally show "x \<in> set_mset (Sup A)" by simp |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
993 |
qed |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
994 |
|
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
995 |
lemma in_Sup_multiset_iff: |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
996 |
assumes "subset_mset.bdd_above A" |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
997 |
shows "x \<in># Sup A \<longleftrightarrow> (\<exists>X\<in>A. x \<in># X)" |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
998 |
proof - |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
999 |
from assms have "set_mset (Sup A) = (\<Union>X\<in>A. set_mset X)" by (rule set_mset_Sup) |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
1000 |
also have "x \<in> \<dots> \<longleftrightarrow> (\<exists>X\<in>A. x \<in># X)" by simp |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
1001 |
finally show ?thesis . |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
1002 |
qed |
a500677d4cec
Conditionally complete lattice of multisets
Manuel Eberl <eberlm@in.tum.de>
parents:
63310
diff
changeset
|
1003 |
|
63360 | 1004 |
lemma in_Sup_multisetD: |
1005 |
assumes "x \<in># Sup A" |
|
1006 |
shows "\<exists>X\<in>A. x \<in># X" |
|
1007 |
proof - |
|
1008 |
have "subset_mset.bdd_above A" |
|
1009 |
by (rule ccontr) (insert assms, simp_all add: Sup_multiset_unbounded) |
|
1010 |
with assms show ?thesis by (simp add: in_Sup_multiset_iff) |
|
63534
523b488b15c9
Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents:
63524
diff
changeset
|
1011 |
qed |
523b488b15c9
Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents:
63524
diff
changeset
|
1012 |
|
523b488b15c9
Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents:
63524
diff
changeset
|
1013 |
interpretation subset_mset: distrib_lattice "op #\<inter>" "op \<subseteq>#" "op \<subset>#" "op #\<union>" |
523b488b15c9
Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents:
63524
diff
changeset
|
1014 |
proof |
523b488b15c9
Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents:
63524
diff
changeset
|
1015 |
fix A B C :: "'a multiset" |
523b488b15c9
Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents:
63524
diff
changeset
|
1016 |
show "A #\<union> (B #\<inter> C) = A #\<union> B #\<inter> (A #\<union> C)" |
523b488b15c9
Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents:
63524
diff
changeset
|
1017 |
by (intro multiset_eqI) simp_all |
523b488b15c9
Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents:
63524
diff
changeset
|
1018 |
qed |
63360 | 1019 |
|
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1020 |
|
60500 | 1021 |
subsubsection \<open>Filter (with comprehension syntax)\<close> |
1022 |
||
1023 |
text \<open>Multiset comprehension\<close> |
|
41069
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
1024 |
|
59998
c54d36be22ef
renamed Multiset.fold -> fold_mset, Multiset.filter -> filter_mset
nipkow
parents:
59986
diff
changeset
|
1025 |
lift_definition filter_mset :: "('a \<Rightarrow> bool) \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" |
c54d36be22ef
renamed Multiset.fold -> fold_mset, Multiset.filter -> filter_mset
nipkow
parents:
59986
diff
changeset
|
1026 |
is "\<lambda>P M. \<lambda>x. if P x then M x else 0" |
47429
ec64d94cbf9c
multiset operations are defined with lift_definitions;
bulwahn
parents:
47308
diff
changeset
|
1027 |
by (rule filter_preserves_multiset) |
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1028 |
|
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1029 |
syntax (ASCII) |
63689 | 1030 |
"_MCollect" :: "pttrn \<Rightarrow> 'a multiset \<Rightarrow> bool \<Rightarrow> 'a multiset" ("(1{#_ :# _./ _#})") |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1031 |
syntax |
63689 | 1032 |
"_MCollect" :: "pttrn \<Rightarrow> 'a multiset \<Rightarrow> bool \<Rightarrow> 'a multiset" ("(1{#_ \<in># _./ _#})") |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1033 |
translations |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1034 |
"{#x \<in># M. P#}" == "CONST filter_mset (\<lambda>x. P) M" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1035 |
|
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1036 |
lemma count_filter_mset [simp]: |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1037 |
"count (filter_mset P M) a = (if P a then count M a else 0)" |
59998
c54d36be22ef
renamed Multiset.fold -> fold_mset, Multiset.filter -> filter_mset
nipkow
parents:
59986
diff
changeset
|
1038 |
by (simp add: filter_mset.rep_eq) |
c54d36be22ef
renamed Multiset.fold -> fold_mset, Multiset.filter -> filter_mset
nipkow
parents:
59986
diff
changeset
|
1039 |
|
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1040 |
lemma set_mset_filter [simp]: |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1041 |
"set_mset (filter_mset P M) = {a \<in> set_mset M. P a}" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1042 |
by (simp only: set_eq_iff count_greater_zero_iff [symmetric] count_filter_mset) simp |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1043 |
|
60606 | 1044 |
lemma filter_empty_mset [simp]: "filter_mset P {#} = {#}" |
59998
c54d36be22ef
renamed Multiset.fold -> fold_mset, Multiset.filter -> filter_mset
nipkow
parents:
59986
diff
changeset
|
1045 |
by (rule multiset_eqI) simp |
c54d36be22ef
renamed Multiset.fold -> fold_mset, Multiset.filter -> filter_mset
nipkow
parents:
59986
diff
changeset
|
1046 |
|
60606 | 1047 |
lemma filter_single_mset [simp]: "filter_mset P {#x#} = (if P x then {#x#} else {#})" |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39301
diff
changeset
|
1048 |
by (rule multiset_eqI) simp |
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1049 |
|
60606 | 1050 |
lemma filter_union_mset [simp]: "filter_mset P (M + N) = filter_mset P M + filter_mset P N" |
41069
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
1051 |
by (rule multiset_eqI) simp |
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
1052 |
|
60606 | 1053 |
lemma filter_diff_mset [simp]: "filter_mset P (M - N) = filter_mset P M - filter_mset P N" |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39301
diff
changeset
|
1054 |
by (rule multiset_eqI) simp |
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1055 |
|
60606 | 1056 |
lemma filter_inter_mset [simp]: "filter_mset P (M #\<inter> N) = filter_mset P M #\<inter> filter_mset P N" |
41069
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
1057 |
by (rule multiset_eqI) simp |
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
1058 |
|
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1059 |
lemma multiset_filter_subset[simp]: "filter_mset f M \<subseteq># M" |
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
1060 |
by (simp add: mset_subset_eqI) |
60397
f8a513fedb31
Renaming multiset operators < ~> <#,...
Mathias Fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
59999
diff
changeset
|
1061 |
|
60606 | 1062 |
lemma multiset_filter_mono: |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1063 |
assumes "A \<subseteq># B" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1064 |
shows "filter_mset f A \<subseteq># filter_mset f B" |
58035 | 1065 |
proof - |
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
1066 |
from assms[unfolded mset_subset_eq_exists_conv] |
58035 | 1067 |
obtain C where B: "B = A + C" by auto |
1068 |
show ?thesis unfolding B by auto |
|
1069 |
qed |
|
1070 |
||
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1071 |
lemma filter_mset_eq_conv: |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1072 |
"filter_mset P M = N \<longleftrightarrow> N \<subseteq># M \<and> (\<forall>b\<in>#N. P b) \<and> (\<forall>a\<in>#M - N. \<not> P a)" (is "?P \<longleftrightarrow> ?Q") |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1073 |
proof |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1074 |
assume ?P then show ?Q by auto (simp add: multiset_eq_iff in_diff_count) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1075 |
next |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1076 |
assume ?Q |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1077 |
then obtain Q where M: "M = N + Q" |
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
1078 |
by (auto simp add: mset_subset_eq_exists_conv) |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1079 |
then have MN: "M - N = Q" by simp |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1080 |
show ?P |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1081 |
proof (rule multiset_eqI) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1082 |
fix a |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1083 |
from \<open>?Q\<close> MN have *: "\<not> P a \<Longrightarrow> a \<notin># N" "P a \<Longrightarrow> a \<notin># Q" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1084 |
by auto |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1085 |
show "count (filter_mset P M) a = count N a" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1086 |
proof (cases "a \<in># M") |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1087 |
case True |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1088 |
with * show ?thesis |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1089 |
by (simp add: not_in_iff M) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1090 |
next |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1091 |
case False then have "count M a = 0" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1092 |
by (simp add: not_in_iff) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1093 |
with M show ?thesis by simp |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1094 |
qed |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1095 |
qed |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1096 |
qed |
59813 | 1097 |
|
1098 |
||
60500 | 1099 |
subsubsection \<open>Size\<close> |
10249 | 1100 |
|
56656 | 1101 |
definition wcount where "wcount f M = (\<lambda>x. count M x * Suc (f x))" |
1102 |
||
1103 |
lemma wcount_union: "wcount f (M + N) a = wcount f M a + wcount f N a" |
|
1104 |
by (auto simp: wcount_def add_mult_distrib) |
|
1105 |
||
1106 |
definition size_multiset :: "('a \<Rightarrow> nat) \<Rightarrow> 'a multiset \<Rightarrow> nat" where |
|
60495 | 1107 |
"size_multiset f M = setsum (wcount f M) (set_mset M)" |
56656 | 1108 |
|
1109 |
lemmas size_multiset_eq = size_multiset_def[unfolded wcount_def] |
|
1110 |
||
60606 | 1111 |
instantiation multiset :: (type) size |
1112 |
begin |
|
1113 |
||
56656 | 1114 |
definition size_multiset where |
1115 |
size_multiset_overloaded_def: "size_multiset = Multiset.size_multiset (\<lambda>_. 0)" |
|
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1116 |
instance .. |
60606 | 1117 |
|
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1118 |
end |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1119 |
|
56656 | 1120 |
lemmas size_multiset_overloaded_eq = |
1121 |
size_multiset_overloaded_def[THEN fun_cong, unfolded size_multiset_eq, simplified] |
|
1122 |
||
1123 |
lemma size_multiset_empty [simp]: "size_multiset f {#} = 0" |
|
1124 |
by (simp add: size_multiset_def) |
|
1125 |
||
28708
a1a436f09ec6
explicit check for pattern discipline before code translation
haftmann
parents:
28562
diff
changeset
|
1126 |
lemma size_empty [simp]: "size {#} = 0" |
56656 | 1127 |
by (simp add: size_multiset_overloaded_def) |
1128 |
||
1129 |
lemma size_multiset_single [simp]: "size_multiset f {#b#} = Suc (f b)" |
|
1130 |
by (simp add: size_multiset_eq) |
|
10249 | 1131 |
|
28708
a1a436f09ec6
explicit check for pattern discipline before code translation
haftmann
parents:
28562
diff
changeset
|
1132 |
lemma size_single [simp]: "size {#b#} = 1" |
56656 | 1133 |
by (simp add: size_multiset_overloaded_def) |
1134 |
||
1135 |
lemma setsum_wcount_Int: |
|
60495 | 1136 |
"finite A \<Longrightarrow> setsum (wcount f N) (A \<inter> set_mset N) = setsum (wcount f N) A" |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1137 |
by (induct rule: finite_induct) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1138 |
(simp_all add: Int_insert_left wcount_def count_eq_zero_iff) |
56656 | 1139 |
|
1140 |
lemma size_multiset_union [simp]: |
|
1141 |
"size_multiset f (M + N::'a multiset) = size_multiset f M + size_multiset f N" |
|
57418 | 1142 |
apply (simp add: size_multiset_def setsum_Un_nat setsum.distrib setsum_wcount_Int wcount_union) |
56656 | 1143 |
apply (subst Int_commute) |
1144 |
apply (simp add: setsum_wcount_Int) |
|
26178 | 1145 |
done |
10249 | 1146 |
|
28708
a1a436f09ec6
explicit check for pattern discipline before code translation
haftmann
parents:
28562
diff
changeset
|
1147 |
lemma size_union [simp]: "size (M + N::'a multiset) = size M + size N" |
56656 | 1148 |
by (auto simp add: size_multiset_overloaded_def) |
1149 |
||
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1150 |
lemma size_multiset_eq_0_iff_empty [iff]: |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1151 |
"size_multiset f M = 0 \<longleftrightarrow> M = {#}" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1152 |
by (auto simp add: size_multiset_eq count_eq_zero_iff) |
10249 | 1153 |
|
17161 | 1154 |
lemma size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})" |
56656 | 1155 |
by (auto simp add: size_multiset_overloaded_def) |
26016 | 1156 |
|
1157 |
lemma nonempty_has_size: "(S \<noteq> {#}) = (0 < size S)" |
|
26178 | 1158 |
by (metis gr0I gr_implies_not0 size_empty size_eq_0_iff_empty) |
10249 | 1159 |
|
60607 | 1160 |
lemma size_eq_Suc_imp_elem: "size M = Suc n \<Longrightarrow> \<exists>a. a \<in># M" |
56656 | 1161 |
apply (unfold size_multiset_overloaded_eq) |
26178 | 1162 |
apply (drule setsum_SucD) |
1163 |
apply auto |
|
1164 |
done |
|
10249 | 1165 |
|
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1166 |
lemma size_eq_Suc_imp_eq_union: |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1167 |
assumes "size M = Suc n" |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1168 |
shows "\<exists>a N. M = N + {#a#}" |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1169 |
proof - |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1170 |
from assms obtain a where "a \<in># M" |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1171 |
by (erule size_eq_Suc_imp_elem [THEN exE]) |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1172 |
then have "M = M - {#a#} + {#a#}" by simp |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1173 |
then show ?thesis by blast |
23611 | 1174 |
qed |
15869 | 1175 |
|
60606 | 1176 |
lemma size_mset_mono: |
1177 |
fixes A B :: "'a multiset" |
|
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1178 |
assumes "A \<subseteq># B" |
60606 | 1179 |
shows "size A \<le> size B" |
59949 | 1180 |
proof - |
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
1181 |
from assms[unfolded mset_subset_eq_exists_conv] |
59949 | 1182 |
obtain C where B: "B = A + C" by auto |
60606 | 1183 |
show ?thesis unfolding B by (induct C) auto |
59949 | 1184 |
qed |
1185 |
||
59998
c54d36be22ef
renamed Multiset.fold -> fold_mset, Multiset.filter -> filter_mset
nipkow
parents:
59986
diff
changeset
|
1186 |
lemma size_filter_mset_lesseq[simp]: "size (filter_mset f M) \<le> size M" |
59949 | 1187 |
by (rule size_mset_mono[OF multiset_filter_subset]) |
1188 |
||
1189 |
lemma size_Diff_submset: |
|
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1190 |
"M \<subseteq># M' \<Longrightarrow> size (M' - M) = size M' - size(M::'a multiset)" |
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
1191 |
by (metis add_diff_cancel_left' size_union mset_subset_eq_exists_conv) |
26016 | 1192 |
|
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1193 |
|
60500 | 1194 |
subsection \<open>Induction and case splits\<close> |
10249 | 1195 |
|
18258 | 1196 |
theorem multiset_induct [case_names empty add, induct type: multiset]: |
48009 | 1197 |
assumes empty: "P {#}" |
1198 |
assumes add: "\<And>M x. P M \<Longrightarrow> P (M + {#x#})" |
|
1199 |
shows "P M" |
|
1200 |
proof (induct n \<equiv> "size M" arbitrary: M) |
|
1201 |
case 0 thus "P M" by (simp add: empty) |
|
1202 |
next |
|
1203 |
case (Suc k) |
|
1204 |
obtain N x where "M = N + {#x#}" |
|
60500 | 1205 |
using \<open>Suc k = size M\<close> [symmetric] |
48009 | 1206 |
using size_eq_Suc_imp_eq_union by fast |
1207 |
with Suc add show "P M" by simp |
|
10249 | 1208 |
qed |
1209 |
||
25610 | 1210 |
lemma multi_nonempty_split: "M \<noteq> {#} \<Longrightarrow> \<exists>A a. M = A + {#a#}" |
26178 | 1211 |
by (induct M) auto |
25610 | 1212 |
|
55913 | 1213 |
lemma multiset_cases [cases type]: |
1214 |
obtains (empty) "M = {#}" |
|
1215 |
| (add) N x where "M = N + {#x#}" |
|
63092 | 1216 |
by (induct M) simp_all |
25610 | 1217 |
|
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1218 |
lemma multi_drop_mem_not_eq: "c \<in># B \<Longrightarrow> B - {#c#} \<noteq> B" |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1219 |
by (cases "B = {#}") (auto dest: multi_member_split) |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1220 |
|
60607 | 1221 |
lemma multiset_partition: "M = {# x\<in>#M. P x #} + {# x\<in>#M. \<not> P x #}" |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39301
diff
changeset
|
1222 |
apply (subst multiset_eq_iff) |
26178 | 1223 |
apply auto |
1224 |
done |
|
10249 | 1225 |
|
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
1226 |
lemma mset_subset_size: "(A::'a multiset) \<subset># B \<Longrightarrow> size A < size B" |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1227 |
proof (induct A arbitrary: B) |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1228 |
case (empty M) |
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
1229 |
then have "M \<noteq> {#}" by (simp add: mset_subset_empty_nonempty) |
58425 | 1230 |
then obtain M' x where "M = M' + {#x#}" |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1231 |
by (blast dest: multi_nonempty_split) |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1232 |
then show ?case by simp |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1233 |
next |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1234 |
case (add S x T) |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1235 |
have IH: "\<And>B. S \<subset># B \<Longrightarrow> size S < size B" by fact |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1236 |
have SxsubT: "S + {#x#} \<subset># T" by fact |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1237 |
then have "x \<in># T" and "S \<subset># T" |
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
1238 |
by (auto dest: mset_subset_insertD) |
58425 | 1239 |
then obtain T' where T: "T = T' + {#x#}" |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1240 |
by (blast dest: multi_member_split) |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1241 |
then have "S \<subset># T'" using SxsubT |
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
1242 |
by (blast intro: mset_subset_add_bothsides) |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1243 |
then have "size S < size T'" using IH by simp |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1244 |
then show ?case using T by simp |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1245 |
qed |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1246 |
|
59949 | 1247 |
lemma size_1_singleton_mset: "size M = 1 \<Longrightarrow> \<exists>a. M = {#a#}" |
1248 |
by (cases M) auto |
|
1249 |
||
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1250 |
|
60500 | 1251 |
subsubsection \<open>Strong induction and subset induction for multisets\<close> |
1252 |
||
1253 |
text \<open>Well-foundedness of strict subset relation\<close> |
|
58098 | 1254 |
|
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
1255 |
lemma wf_subset_mset_rel: "wf {(M, N :: 'a multiset). M \<subset># N}" |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1256 |
apply (rule wf_measure [THEN wf_subset, where f1=size]) |
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
1257 |
apply (clarsimp simp: measure_def inv_image_def mset_subset_size) |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1258 |
done |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1259 |
|
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1260 |
lemma full_multiset_induct [case_names less]: |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1261 |
assumes ih: "\<And>B. \<forall>(A::'a multiset). A \<subset># B \<longrightarrow> P A \<Longrightarrow> P B" |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1262 |
shows "P B" |
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
1263 |
apply (rule wf_subset_mset_rel [THEN wf_induct]) |
58098 | 1264 |
apply (rule ih, auto) |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1265 |
done |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1266 |
|
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1267 |
lemma multi_subset_induct [consumes 2, case_names empty add]: |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1268 |
assumes "F \<subseteq># A" |
60606 | 1269 |
and empty: "P {#}" |
1270 |
and insert: "\<And>a F. a \<in># A \<Longrightarrow> P F \<Longrightarrow> P (F + {#a#})" |
|
1271 |
shows "P F" |
|
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1272 |
proof - |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1273 |
from \<open>F \<subseteq># A\<close> |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1274 |
show ?thesis |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1275 |
proof (induct F) |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1276 |
show "P {#}" by fact |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1277 |
next |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1278 |
fix x F |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1279 |
assume P: "F \<subseteq># A \<Longrightarrow> P F" and i: "F + {#x#} \<subseteq># A" |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1280 |
show "P (F + {#x#})" |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1281 |
proof (rule insert) |
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
1282 |
from i show "x \<in># A" by (auto dest: mset_subset_eq_insertD) |
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
1283 |
from i have "F \<subseteq># A" by (auto dest: mset_subset_eq_insertD) |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1284 |
with P show "P F" . |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1285 |
qed |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1286 |
qed |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1287 |
qed |
26145 | 1288 |
|
17161 | 1289 |
|
60500 | 1290 |
subsection \<open>The fold combinator\<close> |
48023 | 1291 |
|
59998
c54d36be22ef
renamed Multiset.fold -> fold_mset, Multiset.filter -> filter_mset
nipkow
parents:
59986
diff
changeset
|
1292 |
definition fold_mset :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a multiset \<Rightarrow> 'b" |
48023 | 1293 |
where |
60495 | 1294 |
"fold_mset f s M = Finite_Set.fold (\<lambda>x. f x ^^ count M x) s (set_mset M)" |
48023 | 1295 |
|
60606 | 1296 |
lemma fold_mset_empty [simp]: "fold_mset f s {#} = s" |
59998
c54d36be22ef
renamed Multiset.fold -> fold_mset, Multiset.filter -> filter_mset
nipkow
parents:
59986
diff
changeset
|
1297 |
by (simp add: fold_mset_def) |
48023 | 1298 |
|
1299 |
context comp_fun_commute |
|
1300 |
begin |
|
1301 |
||
60606 | 1302 |
lemma fold_mset_insert: "fold_mset f s (M + {#x#}) = f x (fold_mset f s M)" |
49822
0cfc1651be25
simplified construction of fold combinator on multisets;
haftmann
parents:
49717
diff
changeset
|
1303 |
proof - |
0cfc1651be25
simplified construction of fold combinator on multisets;
haftmann
parents:
49717
diff
changeset
|
1304 |
interpret mset: comp_fun_commute "\<lambda>y. f y ^^ count M y" |
0cfc1651be25
simplified construction of fold combinator on multisets;
haftmann
parents:
49717
diff
changeset
|
1305 |
by (fact comp_fun_commute_funpow) |
0cfc1651be25
simplified construction of fold combinator on multisets;
haftmann
parents:
49717
diff
changeset
|
1306 |
interpret mset_union: comp_fun_commute "\<lambda>y. f y ^^ count (M + {#x#}) y" |
0cfc1651be25
simplified construction of fold combinator on multisets;
haftmann
parents:
49717
diff
changeset
|
1307 |
by (fact comp_fun_commute_funpow) |
0cfc1651be25
simplified construction of fold combinator on multisets;
haftmann
parents:
49717
diff
changeset
|
1308 |
show ?thesis |
60495 | 1309 |
proof (cases "x \<in> set_mset M") |
49822
0cfc1651be25
simplified construction of fold combinator on multisets;
haftmann
parents:
49717
diff
changeset
|
1310 |
case False |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1311 |
then have *: "count (M + {#x#}) x = 1" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1312 |
by (simp add: not_in_iff) |
60495 | 1313 |
from False have "Finite_Set.fold (\<lambda>y. f y ^^ count (M + {#x#}) y) s (set_mset M) = |
1314 |
Finite_Set.fold (\<lambda>y. f y ^^ count M y) s (set_mset M)" |
|
49822
0cfc1651be25
simplified construction of fold combinator on multisets;
haftmann
parents:
49717
diff
changeset
|
1315 |
by (auto intro!: Finite_Set.fold_cong comp_fun_commute_funpow) |
0cfc1651be25
simplified construction of fold combinator on multisets;
haftmann
parents:
49717
diff
changeset
|
1316 |
with False * show ?thesis |
59998
c54d36be22ef
renamed Multiset.fold -> fold_mset, Multiset.filter -> filter_mset
nipkow
parents:
59986
diff
changeset
|
1317 |
by (simp add: fold_mset_def del: count_union) |
48023 | 1318 |
next |
49822
0cfc1651be25
simplified construction of fold combinator on multisets;
haftmann
parents:
49717
diff
changeset
|
1319 |
case True |
63040 | 1320 |
define N where "N = set_mset M - {x}" |
60495 | 1321 |
from N_def True have *: "set_mset M = insert x N" "x \<notin> N" "finite N" by auto |
49822
0cfc1651be25
simplified construction of fold combinator on multisets;
haftmann
parents:
49717
diff
changeset
|
1322 |
then have "Finite_Set.fold (\<lambda>y. f y ^^ count (M + {#x#}) y) s N = |
0cfc1651be25
simplified construction of fold combinator on multisets;
haftmann
parents:
49717
diff
changeset
|
1323 |
Finite_Set.fold (\<lambda>y. f y ^^ count M y) s N" |
0cfc1651be25
simplified construction of fold combinator on multisets;
haftmann
parents:
49717
diff
changeset
|
1324 |
by (auto intro!: Finite_Set.fold_cong comp_fun_commute_funpow) |
59998
c54d36be22ef
renamed Multiset.fold -> fold_mset, Multiset.filter -> filter_mset
nipkow
parents:
59986
diff
changeset
|
1325 |
with * show ?thesis by (simp add: fold_mset_def del: count_union) simp |
48023 | 1326 |
qed |
1327 |
qed |
|
1328 |
||
60606 | 1329 |
corollary fold_mset_single [simp]: "fold_mset f s {#x#} = f x s" |
49822
0cfc1651be25
simplified construction of fold combinator on multisets;
haftmann
parents:
49717
diff
changeset
|
1330 |
proof - |
59998
c54d36be22ef
renamed Multiset.fold -> fold_mset, Multiset.filter -> filter_mset
nipkow
parents:
59986
diff
changeset
|
1331 |
have "fold_mset f s ({#} + {#x#}) = f x s" by (simp only: fold_mset_insert) simp |
49822
0cfc1651be25
simplified construction of fold combinator on multisets;
haftmann
parents:
49717
diff
changeset
|
1332 |
then show ?thesis by simp |
0cfc1651be25
simplified construction of fold combinator on multisets;
haftmann
parents:
49717
diff
changeset
|
1333 |
qed |
48023 | 1334 |
|
60606 | 1335 |
lemma fold_mset_fun_left_comm: "f x (fold_mset f s M) = fold_mset f (f x s) M" |
49822
0cfc1651be25
simplified construction of fold combinator on multisets;
haftmann
parents:
49717
diff
changeset
|
1336 |
by (induct M) (simp_all add: fold_mset_insert fun_left_comm) |
48023 | 1337 |
|
60606 | 1338 |
lemma fold_mset_union [simp]: "fold_mset f s (M + N) = fold_mset f (fold_mset f s M) N" |
49822
0cfc1651be25
simplified construction of fold combinator on multisets;
haftmann
parents:
49717
diff
changeset
|
1339 |
proof (induct M) |
48023 | 1340 |
case empty then show ?case by simp |
1341 |
next |
|
49822
0cfc1651be25
simplified construction of fold combinator on multisets;
haftmann
parents:
49717
diff
changeset
|
1342 |
case (add M x) |
0cfc1651be25
simplified construction of fold combinator on multisets;
haftmann
parents:
49717
diff
changeset
|
1343 |
have "M + {#x#} + N = (M + N) + {#x#}" |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
1344 |
by (simp add: ac_simps) |
51548
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1345 |
with add show ?case by (simp add: fold_mset_insert fold_mset_fun_left_comm) |
48023 | 1346 |
qed |
1347 |
||
1348 |
lemma fold_mset_fusion: |
|
1349 |
assumes "comp_fun_commute g" |
|
60606 | 1350 |
and *: "\<And>x y. h (g x y) = f x (h y)" |
1351 |
shows "h (fold_mset g w A) = fold_mset f (h w) A" |
|
48023 | 1352 |
proof - |
1353 |
interpret comp_fun_commute g by (fact assms) |
|
60606 | 1354 |
from * show ?thesis by (induct A) auto |
48023 | 1355 |
qed |
1356 |
||
1357 |
end |
|
1358 |
||
60500 | 1359 |
text \<open> |
48023 | 1360 |
A note on code generation: When defining some function containing a |
59998
c54d36be22ef
renamed Multiset.fold -> fold_mset, Multiset.filter -> filter_mset
nipkow
parents:
59986
diff
changeset
|
1361 |
subterm @{term "fold_mset F"}, code generation is not automatic. When |
61585 | 1362 |
interpreting locale \<open>left_commutative\<close> with \<open>F\<close>, the |
59998
c54d36be22ef
renamed Multiset.fold -> fold_mset, Multiset.filter -> filter_mset
nipkow
parents:
59986
diff
changeset
|
1363 |
would be code thms for @{const fold_mset} become thms like |
61585 | 1364 |
@{term "fold_mset F z {#} = z"} where \<open>F\<close> is not a pattern but |
48023 | 1365 |
contains defined symbols, i.e.\ is not a code thm. Hence a separate |
61585 | 1366 |
constant with its own code thms needs to be introduced for \<open>F\<close>. See the image operator below. |
60500 | 1367 |
\<close> |
1368 |
||
1369 |
||
1370 |
subsection \<open>Image\<close> |
|
48023 | 1371 |
|
1372 |
definition image_mset :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a multiset \<Rightarrow> 'b multiset" where |
|
60607 | 1373 |
"image_mset f = fold_mset (plus \<circ> single \<circ> f) {#}" |
1374 |
||
1375 |
lemma comp_fun_commute_mset_image: "comp_fun_commute (plus \<circ> single \<circ> f)" |
|
49823 | 1376 |
proof |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
1377 |
qed (simp add: ac_simps fun_eq_iff) |
48023 | 1378 |
|
1379 |
lemma image_mset_empty [simp]: "image_mset f {#} = {#}" |
|
49823 | 1380 |
by (simp add: image_mset_def) |
48023 | 1381 |
|
1382 |
lemma image_mset_single [simp]: "image_mset f {#x#} = {#f x#}" |
|
49823 | 1383 |
proof - |
60607 | 1384 |
interpret comp_fun_commute "plus \<circ> single \<circ> f" |
49823 | 1385 |
by (fact comp_fun_commute_mset_image) |
1386 |
show ?thesis by (simp add: image_mset_def) |
|
1387 |
qed |
|
48023 | 1388 |
|
60606 | 1389 |
lemma image_mset_union [simp]: "image_mset f (M + N) = image_mset f M + image_mset f N" |
49823 | 1390 |
proof - |
60607 | 1391 |
interpret comp_fun_commute "plus \<circ> single \<circ> f" |
49823 | 1392 |
by (fact comp_fun_commute_mset_image) |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
1393 |
show ?thesis by (induct N) (simp_all add: image_mset_def ac_simps) |
49823 | 1394 |
qed |
1395 |
||
60606 | 1396 |
corollary image_mset_insert: "image_mset f (M + {#a#}) = image_mset f M + {#f a#}" |
49823 | 1397 |
by simp |
48023 | 1398 |
|
60606 | 1399 |
lemma set_image_mset [simp]: "set_mset (image_mset f M) = image f (set_mset M)" |
49823 | 1400 |
by (induct M) simp_all |
48040 | 1401 |
|
60606 | 1402 |
lemma size_image_mset [simp]: "size (image_mset f M) = size M" |
49823 | 1403 |
by (induct M) simp_all |
48023 | 1404 |
|
60606 | 1405 |
lemma image_mset_is_empty_iff [simp]: "image_mset f M = {#} \<longleftrightarrow> M = {#}" |
49823 | 1406 |
by (cases M) auto |
48023 | 1407 |
|
63099
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1408 |
lemma image_mset_If: |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1409 |
"image_mset (\<lambda>x. if P x then f x else g x) A = |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1410 |
image_mset f (filter_mset P A) + image_mset g (filter_mset (\<lambda>x. \<not>P x) A)" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1411 |
by (induction A) (auto simp: add_ac) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1412 |
|
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1413 |
lemma image_mset_Diff: |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1414 |
assumes "B \<subseteq># A" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1415 |
shows "image_mset f (A - B) = image_mset f A - image_mset f B" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1416 |
proof - |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1417 |
have "image_mset f (A - B + B) = image_mset f (A - B) + image_mset f B" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1418 |
by simp |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1419 |
also from assms have "A - B + B = A" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1420 |
by (simp add: subset_mset.diff_add) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1421 |
finally show ?thesis by simp |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1422 |
qed |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1423 |
|
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1424 |
lemma count_image_mset: |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1425 |
"count (image_mset f A) x = (\<Sum>y\<in>f -` {x} \<inter> set_mset A. count A y)" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1426 |
by (induction A) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1427 |
(auto simp: setsum.distrib setsum.delta' intro!: setsum.mono_neutral_left) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1428 |
|
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1429 |
|
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61890
diff
changeset
|
1430 |
syntax (ASCII) |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61890
diff
changeset
|
1431 |
"_comprehension_mset" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> 'a multiset" ("({#_/. _ :# _#})") |
48023 | 1432 |
syntax |
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61890
diff
changeset
|
1433 |
"_comprehension_mset" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> 'a multiset" ("({#_/. _ \<in># _#})") |
59813 | 1434 |
translations |
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61890
diff
changeset
|
1435 |
"{#e. x \<in># M#}" \<rightleftharpoons> "CONST image_mset (\<lambda>x. e) M" |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61890
diff
changeset
|
1436 |
|
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61890
diff
changeset
|
1437 |
syntax (ASCII) |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61890
diff
changeset
|
1438 |
"_comprehension_mset'" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> bool \<Rightarrow> 'a multiset" ("({#_/ | _ :# _./ _#})") |
48023 | 1439 |
syntax |
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61890
diff
changeset
|
1440 |
"_comprehension_mset'" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> bool \<Rightarrow> 'a multiset" ("({#_/ | _ \<in># _./ _#})") |
59813 | 1441 |
translations |
60606 | 1442 |
"{#e | x\<in>#M. P#}" \<rightharpoonup> "{#e. x \<in># {# x\<in>#M. P#}#}" |
59813 | 1443 |
|
60500 | 1444 |
text \<open> |
60607 | 1445 |
This allows to write not just filters like @{term "{#x\<in>#M. x<c#}"} |
1446 |
but also images like @{term "{#x+x. x\<in>#M #}"} and @{term [source] |
|
1447 |
"{#x+x|x\<in>#M. x<c#}"}, where the latter is currently displayed as |
|
1448 |
@{term "{#x+x|x\<in>#M. x<c#}"}. |
|
60500 | 1449 |
\<close> |
48023 | 1450 |
|
60495 | 1451 |
lemma in_image_mset: "y \<in># {#f x. x \<in># M#} \<longleftrightarrow> y \<in> f ` set_mset M" |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1452 |
by (metis set_image_mset) |
59813 | 1453 |
|
55467
a5c9002bc54d
renamed 'enriched_type' to more informative 'functor' (following the renaming of enriched type constructors to bounded natural functors)
blanchet
parents:
55417
diff
changeset
|
1454 |
functor image_mset: image_mset |
48023 | 1455 |
proof - |
1456 |
fix f g show "image_mset f \<circ> image_mset g = image_mset (f \<circ> g)" |
|
1457 |
proof |
|
1458 |
fix A |
|
1459 |
show "(image_mset f \<circ> image_mset g) A = image_mset (f \<circ> g) A" |
|
1460 |
by (induct A) simp_all |
|
1461 |
qed |
|
1462 |
show "image_mset id = id" |
|
1463 |
proof |
|
1464 |
fix A |
|
1465 |
show "image_mset id A = id A" |
|
1466 |
by (induct A) simp_all |
|
1467 |
qed |
|
1468 |
qed |
|
1469 |
||
59813 | 1470 |
declare |
1471 |
image_mset.id [simp] |
|
1472 |
image_mset.identity [simp] |
|
1473 |
||
1474 |
lemma image_mset_id[simp]: "image_mset id x = x" |
|
1475 |
unfolding id_def by auto |
|
1476 |
||
1477 |
lemma image_mset_cong: "(\<And>x. x \<in># M \<Longrightarrow> f x = g x) \<Longrightarrow> {#f x. x \<in># M#} = {#g x. x \<in># M#}" |
|
1478 |
by (induct M) auto |
|
1479 |
||
1480 |
lemma image_mset_cong_pair: |
|
1481 |
"(\<forall>x y. (x, y) \<in># M \<longrightarrow> f x y = g x y) \<Longrightarrow> {#f x y. (x, y) \<in># M#} = {#g x y. (x, y) \<in># M#}" |
|
1482 |
by (metis image_mset_cong split_cong) |
|
49717 | 1483 |
|
48023 | 1484 |
|
60500 | 1485 |
subsection \<open>Further conversions\<close> |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1486 |
|
60515 | 1487 |
primrec mset :: "'a list \<Rightarrow> 'a multiset" where |
1488 |
"mset [] = {#}" | |
|
1489 |
"mset (a # x) = mset x + {# a #}" |
|
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1490 |
|
37107 | 1491 |
lemma in_multiset_in_set: |
60515 | 1492 |
"x \<in># mset xs \<longleftrightarrow> x \<in> set xs" |
37107 | 1493 |
by (induct xs) simp_all |
1494 |
||
60515 | 1495 |
lemma count_mset: |
1496 |
"count (mset xs) x = length (filter (\<lambda>y. x = y) xs)" |
|
37107 | 1497 |
by (induct xs) simp_all |
1498 |
||
60515 | 1499 |
lemma mset_zero_iff[simp]: "(mset x = {#}) = (x = [])" |
59813 | 1500 |
by (induct x) auto |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1501 |
|
60515 | 1502 |
lemma mset_zero_iff_right[simp]: "({#} = mset x) = (x = [])" |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1503 |
by (induct x) auto |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1504 |
|
60515 | 1505 |
lemma set_mset_mset[simp]: "set_mset (mset x) = set x" |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1506 |
by (induct x) auto |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1507 |
|
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1508 |
lemma set_mset_comp_mset [simp]: "set_mset \<circ> mset = set" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1509 |
by (simp add: fun_eq_iff) |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1510 |
|
60515 | 1511 |
lemma size_mset [simp]: "size (mset xs) = length xs" |
48012 | 1512 |
by (induct xs) simp_all |
1513 |
||
60606 | 1514 |
lemma mset_append [simp]: "mset (xs @ ys) = mset xs + mset ys" |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
1515 |
by (induct xs arbitrary: ys) (auto simp: ac_simps) |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1516 |
|
60607 | 1517 |
lemma mset_filter: "mset (filter P xs) = {#x \<in># mset xs. P x #}" |
40303
2d507370e879
lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents:
40250
diff
changeset
|
1518 |
by (induct xs) simp_all |
2d507370e879
lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents:
40250
diff
changeset
|
1519 |
|
60515 | 1520 |
lemma mset_rev [simp]: |
1521 |
"mset (rev xs) = mset xs" |
|
40950 | 1522 |
by (induct xs) simp_all |
1523 |
||
60515 | 1524 |
lemma surj_mset: "surj mset" |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1525 |
apply (unfold surj_def) |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1526 |
apply (rule allI) |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1527 |
apply (rule_tac M = y in multiset_induct) |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1528 |
apply auto |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1529 |
apply (rule_tac x = "x # xa" in exI) |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1530 |
apply auto |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1531 |
done |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1532 |
|
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1533 |
lemma distinct_count_atmost_1: |
60606 | 1534 |
"distinct x = (\<forall>a. count (mset x) a = (if a \<in> set x then 1 else 0))" |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1535 |
proof (induct x) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1536 |
case Nil then show ?case by simp |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1537 |
next |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1538 |
case (Cons x xs) show ?case (is "?lhs \<longleftrightarrow> ?rhs") |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1539 |
proof |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1540 |
assume ?lhs then show ?rhs using Cons by simp |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1541 |
next |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1542 |
assume ?rhs then have "x \<notin> set xs" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1543 |
by (simp split: if_splits) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1544 |
moreover from \<open>?rhs\<close> have "(\<forall>a. count (mset xs) a = |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1545 |
(if a \<in> set xs then 1 else 0))" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1546 |
by (auto split: if_splits simp add: count_eq_zero_iff) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1547 |
ultimately show ?lhs using Cons by simp |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1548 |
qed |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1549 |
qed |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1550 |
|
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1551 |
lemma mset_eq_setD: |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1552 |
assumes "mset xs = mset ys" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1553 |
shows "set xs = set ys" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1554 |
proof - |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1555 |
from assms have "set_mset (mset xs) = set_mset (mset ys)" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1556 |
by simp |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1557 |
then show ?thesis by simp |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1558 |
qed |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1559 |
|
60515 | 1560 |
lemma set_eq_iff_mset_eq_distinct: |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1561 |
"distinct x \<Longrightarrow> distinct y \<Longrightarrow> |
60515 | 1562 |
(set x = set y) = (mset x = mset y)" |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39301
diff
changeset
|
1563 |
by (auto simp: multiset_eq_iff distinct_count_atmost_1) |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1564 |
|
60515 | 1565 |
lemma set_eq_iff_mset_remdups_eq: |
1566 |
"(set x = set y) = (mset (remdups x) = mset (remdups y))" |
|
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1567 |
apply (rule iffI) |
60515 | 1568 |
apply (simp add: set_eq_iff_mset_eq_distinct[THEN iffD1]) |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1569 |
apply (drule distinct_remdups [THEN distinct_remdups |
60515 | 1570 |
[THEN set_eq_iff_mset_eq_distinct [THEN iffD2]]]) |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1571 |
apply simp |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1572 |
done |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1573 |
|
60606 | 1574 |
lemma mset_compl_union [simp]: "mset [x\<leftarrow>xs. P x] + mset [x\<leftarrow>xs. \<not>P x] = mset xs" |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
1575 |
by (induct xs) (auto simp: ac_simps) |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1576 |
|
60607 | 1577 |
lemma nth_mem_mset: "i < length ls \<Longrightarrow> (ls ! i) \<in># mset ls" |
60678 | 1578 |
proof (induct ls arbitrary: i) |
1579 |
case Nil |
|
1580 |
then show ?case by simp |
|
1581 |
next |
|
1582 |
case Cons |
|
1583 |
then show ?case by (cases i) auto |
|
1584 |
qed |
|
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1585 |
|
60606 | 1586 |
lemma mset_remove1[simp]: "mset (remove1 a xs) = mset xs - {#a#}" |
60678 | 1587 |
by (induct xs) (auto simp add: multiset_eq_iff) |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1588 |
|
60515 | 1589 |
lemma mset_eq_length: |
1590 |
assumes "mset xs = mset ys" |
|
37107 | 1591 |
shows "length xs = length ys" |
60515 | 1592 |
using assms by (metis size_mset) |
1593 |
||
1594 |
lemma mset_eq_length_filter: |
|
1595 |
assumes "mset xs = mset ys" |
|
39533
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
1596 |
shows "length (filter (\<lambda>x. z = x) xs) = length (filter (\<lambda>y. z = y) ys)" |
60515 | 1597 |
using assms by (metis count_mset) |
39533
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
1598 |
|
45989
b39256df5f8a
moved theorem requiring multisets from More_List to Multiset
haftmann
parents:
45866
diff
changeset
|
1599 |
lemma fold_multiset_equiv: |
b39256df5f8a
moved theorem requiring multisets from More_List to Multiset
haftmann
parents:
45866
diff
changeset
|
1600 |
assumes f: "\<And>x y. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> f x \<circ> f y = f y \<circ> f x" |
60515 | 1601 |
and equiv: "mset xs = mset ys" |
49822
0cfc1651be25
simplified construction of fold combinator on multisets;
haftmann
parents:
49717
diff
changeset
|
1602 |
shows "List.fold f xs = List.fold f ys" |
60606 | 1603 |
using f equiv [symmetric] |
46921 | 1604 |
proof (induct xs arbitrary: ys) |
60678 | 1605 |
case Nil |
1606 |
then show ?case by simp |
|
45989
b39256df5f8a
moved theorem requiring multisets from More_List to Multiset
haftmann
parents:
45866
diff
changeset
|
1607 |
next |
b39256df5f8a
moved theorem requiring multisets from More_List to Multiset
haftmann
parents:
45866
diff
changeset
|
1608 |
case (Cons x xs) |
60678 | 1609 |
then have *: "set ys = set (x # xs)" |
1610 |
by (blast dest: mset_eq_setD) |
|
58425 | 1611 |
have "\<And>x y. x \<in> set ys \<Longrightarrow> y \<in> set ys \<Longrightarrow> f x \<circ> f y = f y \<circ> f x" |
45989
b39256df5f8a
moved theorem requiring multisets from More_List to Multiset
haftmann
parents:
45866
diff
changeset
|
1612 |
by (rule Cons.prems(1)) (simp_all add: *) |
60678 | 1613 |
moreover from * have "x \<in> set ys" |
1614 |
by simp |
|
1615 |
ultimately have "List.fold f ys = List.fold f (remove1 x ys) \<circ> f x" |
|
1616 |
by (fact fold_remove1_split) |
|
1617 |
moreover from Cons.prems have "List.fold f xs = List.fold f (remove1 x ys)" |
|
1618 |
by (auto intro: Cons.hyps) |
|
45989
b39256df5f8a
moved theorem requiring multisets from More_List to Multiset
haftmann
parents:
45866
diff
changeset
|
1619 |
ultimately show ?case by simp |
b39256df5f8a
moved theorem requiring multisets from More_List to Multiset
haftmann
parents:
45866
diff
changeset
|
1620 |
qed |
b39256df5f8a
moved theorem requiring multisets from More_List to Multiset
haftmann
parents:
45866
diff
changeset
|
1621 |
|
60606 | 1622 |
lemma mset_insort [simp]: "mset (insort x xs) = mset xs + {#x#}" |
51548
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1623 |
by (induct xs) (simp_all add: ac_simps) |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1624 |
|
63524
4ec755485732
adding mset_map to the simp rules
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63410
diff
changeset
|
1625 |
lemma mset_map[simp]: "mset (map f xs) = image_mset f (mset xs)" |
51600
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
1626 |
by (induct xs) simp_all |
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
1627 |
|
61890
f6ded81f5690
abandoned attempt to unify sublocale and interpretation into global theories
haftmann
parents:
61832
diff
changeset
|
1628 |
global_interpretation mset_set: folding "\<lambda>x M. {#x#} + M" "{#}" |
61832 | 1629 |
defines mset_set = "folding.F (\<lambda>x M. {#x#} + M) {#}" |
1630 |
by standard (simp add: fun_eq_iff ac_simps) |
|
51548
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1631 |
|
60513 | 1632 |
lemma count_mset_set [simp]: |
1633 |
"finite A \<Longrightarrow> x \<in> A \<Longrightarrow> count (mset_set A) x = 1" (is "PROP ?P") |
|
1634 |
"\<not> finite A \<Longrightarrow> count (mset_set A) x = 0" (is "PROP ?Q") |
|
1635 |
"x \<notin> A \<Longrightarrow> count (mset_set A) x = 0" (is "PROP ?R") |
|
51600
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
1636 |
proof - |
60606 | 1637 |
have *: "count (mset_set A) x = 0" if "x \<notin> A" for A |
1638 |
proof (cases "finite A") |
|
1639 |
case False then show ?thesis by simp |
|
1640 |
next |
|
1641 |
case True from True \<open>x \<notin> A\<close> show ?thesis by (induct A) auto |
|
1642 |
qed |
|
51600
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
1643 |
then show "PROP ?P" "PROP ?Q" "PROP ?R" |
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
1644 |
by (auto elim!: Set.set_insert) |
61585 | 1645 |
qed \<comment> \<open>TODO: maybe define @{const mset_set} also in terms of @{const Abs_multiset}\<close> |
60513 | 1646 |
|
1647 |
lemma elem_mset_set[simp, intro]: "finite A \<Longrightarrow> x \<in># mset_set A \<longleftrightarrow> x \<in> A" |
|
59813 | 1648 |
by (induct A rule: finite_induct) simp_all |
1649 |
||
63099
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1650 |
lemma mset_set_Union: |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1651 |
"finite A \<Longrightarrow> finite B \<Longrightarrow> A \<inter> B = {} \<Longrightarrow> mset_set (A \<union> B) = mset_set A + mset_set B" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1652 |
by (induction A rule: finite_induct) (auto simp: add_ac) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1653 |
|
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1654 |
lemma filter_mset_mset_set [simp]: |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1655 |
"finite A \<Longrightarrow> filter_mset P (mset_set A) = mset_set {x\<in>A. P x}" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1656 |
proof (induction A rule: finite_induct) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1657 |
case (insert x A) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1658 |
from insert.hyps have "filter_mset P (mset_set (insert x A)) = |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1659 |
filter_mset P (mset_set A) + mset_set (if P x then {x} else {})" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1660 |
by (simp add: add_ac) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1661 |
also have "filter_mset P (mset_set A) = mset_set {x\<in>A. P x}" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1662 |
by (rule insert.IH) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1663 |
also from insert.hyps |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1664 |
have "\<dots> + mset_set (if P x then {x} else {}) = |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1665 |
mset_set ({x \<in> A. P x} \<union> (if P x then {x} else {}))" (is "_ = mset_set ?A") |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1666 |
by (intro mset_set_Union [symmetric]) simp_all |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1667 |
also from insert.hyps have "?A = {y\<in>insert x A. P y}" by auto |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1668 |
finally show ?case . |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1669 |
qed simp_all |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1670 |
|
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1671 |
lemma mset_set_Diff: |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1672 |
assumes "finite A" "B \<subseteq> A" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1673 |
shows "mset_set (A - B) = mset_set A - mset_set B" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1674 |
proof - |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1675 |
from assms have "mset_set ((A - B) \<union> B) = mset_set (A - B) + mset_set B" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1676 |
by (intro mset_set_Union) (auto dest: finite_subset) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1677 |
also from assms have "A - B \<union> B = A" by blast |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1678 |
finally show ?thesis by simp |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1679 |
qed |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1680 |
|
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1681 |
lemma mset_set_set: "distinct xs \<Longrightarrow> mset_set (set xs) = mset xs" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1682 |
by (induction xs) (simp_all add: add_ac) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1683 |
|
51548
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1684 |
context linorder |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1685 |
begin |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1686 |
|
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1687 |
definition sorted_list_of_multiset :: "'a multiset \<Rightarrow> 'a list" |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1688 |
where |
59998
c54d36be22ef
renamed Multiset.fold -> fold_mset, Multiset.filter -> filter_mset
nipkow
parents:
59986
diff
changeset
|
1689 |
"sorted_list_of_multiset M = fold_mset insort [] M" |
51548
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1690 |
|
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1691 |
lemma sorted_list_of_multiset_empty [simp]: |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1692 |
"sorted_list_of_multiset {#} = []" |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1693 |
by (simp add: sorted_list_of_multiset_def) |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1694 |
|
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1695 |
lemma sorted_list_of_multiset_singleton [simp]: |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1696 |
"sorted_list_of_multiset {#x#} = [x]" |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1697 |
proof - |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1698 |
interpret comp_fun_commute insort by (fact comp_fun_commute_insort) |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1699 |
show ?thesis by (simp add: sorted_list_of_multiset_def) |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1700 |
qed |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1701 |
|
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1702 |
lemma sorted_list_of_multiset_insert [simp]: |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1703 |
"sorted_list_of_multiset (M + {#x#}) = List.insort x (sorted_list_of_multiset M)" |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1704 |
proof - |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1705 |
interpret comp_fun_commute insort by (fact comp_fun_commute_insort) |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1706 |
show ?thesis by (simp add: sorted_list_of_multiset_def) |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1707 |
qed |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1708 |
|
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1709 |
end |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1710 |
|
60515 | 1711 |
lemma mset_sorted_list_of_multiset [simp]: |
1712 |
"mset (sorted_list_of_multiset M) = M" |
|
60513 | 1713 |
by (induct M) simp_all |
51548
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1714 |
|
60515 | 1715 |
lemma sorted_list_of_multiset_mset [simp]: |
1716 |
"sorted_list_of_multiset (mset xs) = sort xs" |
|
60513 | 1717 |
by (induct xs) simp_all |
1718 |
||
1719 |
lemma finite_set_mset_mset_set[simp]: |
|
1720 |
"finite A \<Longrightarrow> set_mset (mset_set A) = A" |
|
1721 |
by (induct A rule: finite_induct) simp_all |
|
1722 |
||
63099
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1723 |
lemma mset_set_empty_iff: "mset_set A = {#} \<longleftrightarrow> A = {} \<or> infinite A" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1724 |
using finite_set_mset_mset_set by fastforce |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1725 |
|
60513 | 1726 |
lemma infinite_set_mset_mset_set: |
1727 |
"\<not> finite A \<Longrightarrow> set_mset (mset_set A) = {}" |
|
1728 |
by simp |
|
51548
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1729 |
|
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1730 |
lemma set_sorted_list_of_multiset [simp]: |
60495 | 1731 |
"set (sorted_list_of_multiset M) = set_mset M" |
60513 | 1732 |
by (induct M) (simp_all add: set_insort) |
1733 |
||
1734 |
lemma sorted_list_of_mset_set [simp]: |
|
1735 |
"sorted_list_of_multiset (mset_set A) = sorted_list_of_set A" |
|
1736 |
by (cases "finite A") (induct A rule: finite_induct, simp_all add: ac_simps) |
|
51548
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1737 |
|
63099
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1738 |
lemma mset_upt [simp]: "mset [m..<n] = mset_set {m..<n}" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1739 |
by (induction n) (simp_all add: atLeastLessThanSuc add_ac) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1740 |
|
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1741 |
lemma image_mset_map_of: |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1742 |
"distinct (map fst xs) \<Longrightarrow> {#the (map_of xs i). i \<in># mset (map fst xs)#} = mset (map snd xs)" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1743 |
proof (induction xs) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1744 |
case (Cons x xs) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1745 |
have "{#the (map_of (x # xs) i). i \<in># mset (map fst (x # xs))#} = |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1746 |
{#the (if i = fst x then Some (snd x) else map_of xs i). |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1747 |
i \<in># mset (map fst xs)#} + {#snd x#}" (is "_ = ?A + _") by simp |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1748 |
also from Cons.prems have "?A = {#the (map_of xs i). i :# mset (map fst xs)#}" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1749 |
by (cases x, intro image_mset_cong) (auto simp: in_multiset_in_set) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1750 |
also from Cons.prems have "\<dots> = mset (map snd xs)" by (intro Cons.IH) simp_all |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1751 |
finally show ?case by simp |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1752 |
qed simp_all |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1753 |
|
51548
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1754 |
|
60804 | 1755 |
subsection \<open>Replicate operation\<close> |
1756 |
||
1757 |
definition replicate_mset :: "nat \<Rightarrow> 'a \<Rightarrow> 'a multiset" where |
|
1758 |
"replicate_mset n x = ((op + {#x#}) ^^ n) {#}" |
|
1759 |
||
1760 |
lemma replicate_mset_0[simp]: "replicate_mset 0 x = {#}" |
|
1761 |
unfolding replicate_mset_def by simp |
|
1762 |
||
1763 |
lemma replicate_mset_Suc[simp]: "replicate_mset (Suc n) x = replicate_mset n x + {#x#}" |
|
1764 |
unfolding replicate_mset_def by (induct n) (auto intro: add.commute) |
|
1765 |
||
1766 |
lemma in_replicate_mset[simp]: "x \<in># replicate_mset n y \<longleftrightarrow> n > 0 \<and> x = y" |
|
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1767 |
unfolding replicate_mset_def by (induct n) auto |
60804 | 1768 |
|
1769 |
lemma count_replicate_mset[simp]: "count (replicate_mset n x) y = (if y = x then n else 0)" |
|
1770 |
unfolding replicate_mset_def by (induct n) simp_all |
|
1771 |
||
1772 |
lemma set_mset_replicate_mset_subset[simp]: "set_mset (replicate_mset n x) = (if n = 0 then {} else {x})" |
|
1773 |
by (auto split: if_splits) |
|
1774 |
||
1775 |
lemma size_replicate_mset[simp]: "size (replicate_mset n M) = n" |
|
1776 |
by (induct n, simp_all) |
|
1777 |
||
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
1778 |
lemma count_le_replicate_mset_subset_eq: "n \<le> count M x \<longleftrightarrow> replicate_mset n x \<subseteq># M" |
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
1779 |
by (auto simp add: mset_subset_eqI) (metis count_replicate_mset subseteq_mset_def) |
60804 | 1780 |
|
1781 |
lemma filter_eq_replicate_mset: "{#y \<in># D. y = x#} = replicate_mset (count D x) x" |
|
1782 |
by (induct D) simp_all |
|
1783 |
||
61031
162bd20dae23
more lemmas on sorting and multisets (due to Thomas Sewell)
haftmann
parents:
60804
diff
changeset
|
1784 |
lemma replicate_count_mset_eq_filter_eq: |
162bd20dae23
more lemmas on sorting and multisets (due to Thomas Sewell)
haftmann
parents:
60804
diff
changeset
|
1785 |
"replicate (count (mset xs) k) k = filter (HOL.eq k) xs" |
162bd20dae23
more lemmas on sorting and multisets (due to Thomas Sewell)
haftmann
parents:
60804
diff
changeset
|
1786 |
by (induct xs) auto |
162bd20dae23
more lemmas on sorting and multisets (due to Thomas Sewell)
haftmann
parents:
60804
diff
changeset
|
1787 |
|
62366 | 1788 |
lemma replicate_mset_eq_empty_iff [simp]: |
1789 |
"replicate_mset n a = {#} \<longleftrightarrow> n = 0" |
|
1790 |
by (induct n) simp_all |
|
1791 |
||
1792 |
lemma replicate_mset_eq_iff: |
|
1793 |
"replicate_mset m a = replicate_mset n b \<longleftrightarrow> |
|
1794 |
m = 0 \<and> n = 0 \<or> m = n \<and> a = b" |
|
1795 |
by (auto simp add: multiset_eq_iff) |
|
1796 |
||
60804 | 1797 |
|
60500 | 1798 |
subsection \<open>Big operators\<close> |
51548
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1799 |
|
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1800 |
locale comm_monoid_mset = comm_monoid |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1801 |
begin |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1802 |
|
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1803 |
definition F :: "'a multiset \<Rightarrow> 'a" |
63290
9ac558ab0906
boldify syntax in abstract algebraic structures, to avoid clashes with concrete syntax in corresponding type classes
haftmann
parents:
63195
diff
changeset
|
1804 |
where eq_fold: "F M = fold_mset f \<^bold>1 M" |
9ac558ab0906
boldify syntax in abstract algebraic structures, to avoid clashes with concrete syntax in corresponding type classes
haftmann
parents:
63195
diff
changeset
|
1805 |
|
9ac558ab0906
boldify syntax in abstract algebraic structures, to avoid clashes with concrete syntax in corresponding type classes
haftmann
parents:
63195
diff
changeset
|
1806 |
lemma empty [simp]: "F {#} = \<^bold>1" |
51548
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1807 |
by (simp add: eq_fold) |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1808 |
|
60678 | 1809 |
lemma singleton [simp]: "F {#x#} = x" |
51548
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1810 |
proof - |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1811 |
interpret comp_fun_commute |
60678 | 1812 |
by standard (simp add: fun_eq_iff left_commute) |
51548
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1813 |
show ?thesis by (simp add: eq_fold) |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1814 |
qed |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1815 |
|
63290
9ac558ab0906
boldify syntax in abstract algebraic structures, to avoid clashes with concrete syntax in corresponding type classes
haftmann
parents:
63195
diff
changeset
|
1816 |
lemma union [simp]: "F (M + N) = F M \<^bold>* F N" |
51548
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1817 |
proof - |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1818 |
interpret comp_fun_commute f |
60678 | 1819 |
by standard (simp add: fun_eq_iff left_commute) |
1820 |
show ?thesis |
|
1821 |
by (induct N) (simp_all add: left_commute eq_fold) |
|
51548
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1822 |
qed |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1823 |
|
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1824 |
end |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1825 |
|
61076 | 1826 |
lemma comp_fun_commute_plus_mset[simp]: "comp_fun_commute (op + :: 'a multiset \<Rightarrow> _ \<Rightarrow> _)" |
60678 | 1827 |
by standard (simp add: add_ac comp_def) |
59813 | 1828 |
|
1829 |
declare comp_fun_commute.fold_mset_insert[OF comp_fun_commute_plus_mset, simp] |
|
1830 |
||
59998
c54d36be22ef
renamed Multiset.fold -> fold_mset, Multiset.filter -> filter_mset
nipkow
parents:
59986
diff
changeset
|
1831 |
lemma in_mset_fold_plus_iff[iff]: "x \<in># fold_mset (op +) M NN \<longleftrightarrow> x \<in># M \<or> (\<exists>N. N \<in># NN \<and> x \<in># N)" |
59813 | 1832 |
by (induct NN) auto |
1833 |
||
54868 | 1834 |
context comm_monoid_add |
1835 |
begin |
|
1836 |
||
61605 | 1837 |
sublocale msetsum: comm_monoid_mset plus 0 |
61832 | 1838 |
defines msetsum = msetsum.F .. |
51548
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1839 |
|
60804 | 1840 |
lemma (in semiring_1) msetsum_replicate_mset [simp]: |
1841 |
"msetsum (replicate_mset n a) = of_nat n * a" |
|
1842 |
by (induct n) (simp_all add: algebra_simps) |
|
1843 |
||
51548
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1844 |
lemma setsum_unfold_msetsum: |
60513 | 1845 |
"setsum f A = msetsum (image_mset f (mset_set A))" |
51548
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1846 |
by (cases "finite A") (induct A rule: finite_induct, simp_all) |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1847 |
|
63534
523b488b15c9
Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents:
63524
diff
changeset
|
1848 |
lemma msetsum_delta: "msetsum (image_mset (\<lambda>x. if x = y then c else 0) A) = c * count A y" |
523b488b15c9
Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents:
63524
diff
changeset
|
1849 |
by (induction A) simp_all |
523b488b15c9
Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents:
63524
diff
changeset
|
1850 |
|
523b488b15c9
Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents:
63524
diff
changeset
|
1851 |
lemma msetsum_delta': "msetsum (image_mset (\<lambda>x. if y = x then c else 0) A) = c * count A y" |
523b488b15c9
Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents:
63524
diff
changeset
|
1852 |
by (induction A) simp_all |
523b488b15c9
Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents:
63524
diff
changeset
|
1853 |
|
51548
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1854 |
end |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1855 |
|
59813 | 1856 |
lemma msetsum_diff: |
61076 | 1857 |
fixes M N :: "('a :: ordered_cancel_comm_monoid_diff) multiset" |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1858 |
shows "N \<subseteq># M \<Longrightarrow> msetsum (M - N) = msetsum M - msetsum N" |
60397
f8a513fedb31
Renaming multiset operators < ~> <#,...
Mathias Fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
59999
diff
changeset
|
1859 |
by (metis add_diff_cancel_right' msetsum.union subset_mset.diff_add) |
59813 | 1860 |
|
59949 | 1861 |
lemma size_eq_msetsum: "size M = msetsum (image_mset (\<lambda>_. 1) M)" |
1862 |
proof (induct M) |
|
1863 |
case empty then show ?case by simp |
|
1864 |
next |
|
1865 |
case (add M x) then show ?case |
|
60495 | 1866 |
by (cases "x \<in> set_mset M") |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
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diff
changeset
|
1867 |
(simp_all add: size_multiset_overloaded_eq setsum.distrib setsum.delta' insert_absorb not_in_iff) |
59949 | 1868 |
qed |
1869 |
||
63099
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1870 |
lemma size_mset_set [simp]: "size (mset_set A) = card A" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1871 |
by (simp only: size_eq_msetsum card_eq_setsum setsum_unfold_msetsum) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63092
diff
changeset
|
1872 |
|
62366 | 1873 |
syntax (ASCII) |
1874 |
"_msetsum_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add" ("(3SUM _:#_. _)" [0, 51, 10] 10) |
|
1875 |
syntax |
|
1876 |
"_msetsum_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add" ("(3\<Sum>_\<in>#_. _)" [0, 51, 10] 10) |
|
1877 |
translations |
|
1878 |
"\<Sum>i \<in># A. b" \<rightleftharpoons> "CONST msetsum (CONST image_mset (\<lambda>i. b) A)" |
|
59949 | 1879 |
|
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61890
diff
changeset
|
1880 |
abbreviation Union_mset :: "'a multiset multiset \<Rightarrow> 'a multiset" ("\<Union>#_" [900] 900) |
62837 | 1881 |
where "\<Union># MM \<equiv> msetsum MM" \<comment> \<open>FIXME ambiguous notation -- |
1882 |
could likewise refer to \<open>\<Squnion>#\<close>\<close> |
|
59813 | 1883 |
|
60495 | 1884 |
lemma set_mset_Union_mset[simp]: "set_mset (\<Union># MM) = (\<Union>M \<in> set_mset MM. set_mset M)" |
59813 | 1885 |
by (induct MM) auto |
1886 |
||
1887 |
lemma in_Union_mset_iff[iff]: "x \<in># \<Union># MM \<longleftrightarrow> (\<exists>M. M \<in># MM \<and> x \<in># M)" |
|
1888 |
by (induct MM) auto |
|
1889 |
||
62366 | 1890 |
lemma count_setsum: |
1891 |
"count (setsum f A) x = setsum (\<lambda>a. count (f a) x) A" |
|
1892 |
by (induct A rule: infinite_finite_induct) simp_all |
|
1893 |
||
1894 |
lemma setsum_eq_empty_iff: |
|
1895 |
assumes "finite A" |
|
1896 |
shows "setsum f A = {#} \<longleftrightarrow> (\<forall>a\<in>A. f a = {#})" |
|
1897 |
using assms by induct simp_all |
|
51548
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1898 |
|
54868 | 1899 |
context comm_monoid_mult |
1900 |
begin |
|
1901 |
||
61605 | 1902 |
sublocale msetprod: comm_monoid_mset times 1 |
61832 | 1903 |
defines msetprod = msetprod.F .. |
51548
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1904 |
|
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1905 |
lemma msetprod_empty: |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1906 |
"msetprod {#} = 1" |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1907 |
by (fact msetprod.empty) |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1908 |
|
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1909 |
lemma msetprod_singleton: |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1910 |
"msetprod {#x#} = x" |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1911 |
by (fact msetprod.singleton) |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1912 |
|
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1913 |
lemma msetprod_Un: |
58425 | 1914 |
"msetprod (A + B) = msetprod A * msetprod B" |
51548
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1915 |
by (fact msetprod.union) |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1916 |
|
60804 | 1917 |
lemma msetprod_replicate_mset [simp]: |
1918 |
"msetprod (replicate_mset n a) = a ^ n" |
|
1919 |
by (induct n) (simp_all add: ac_simps) |
|
1920 |
||
51548
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1921 |
lemma setprod_unfold_msetprod: |
60513 | 1922 |
"setprod f A = msetprod (image_mset f (mset_set A))" |
51548
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1923 |
by (cases "finite A") (induct A rule: finite_induct, simp_all) |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1924 |
|
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1925 |
lemma msetprod_multiplicity: |
60495 | 1926 |
"msetprod M = setprod (\<lambda>x. x ^ count M x) (set_mset M)" |
59998
c54d36be22ef
renamed Multiset.fold -> fold_mset, Multiset.filter -> filter_mset
nipkow
parents:
59986
diff
changeset
|
1927 |
by (simp add: fold_mset_def setprod.eq_fold msetprod.eq_fold funpow_times_power comp_def) |
51548
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1928 |
|
63534
523b488b15c9
Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents:
63524
diff
changeset
|
1929 |
lemma msetprod_delta: "msetprod (image_mset (\<lambda>x. if x = y then c else 1) A) = c ^ count A y" |
523b488b15c9
Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents:
63524
diff
changeset
|
1930 |
by (induction A) (simp_all add: mult_ac) |
523b488b15c9
Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents:
63524
diff
changeset
|
1931 |
|
523b488b15c9
Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents:
63524
diff
changeset
|
1932 |
lemma msetprod_delta': "msetprod (image_mset (\<lambda>x. if y = x then c else 1) A) = c ^ count A y" |
523b488b15c9
Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents:
63524
diff
changeset
|
1933 |
by (induction A) (simp_all add: mult_ac) |
523b488b15c9
Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents:
63524
diff
changeset
|
1934 |
|
51548
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1935 |
end |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1936 |
|
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61890
diff
changeset
|
1937 |
syntax (ASCII) |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61890
diff
changeset
|
1938 |
"_msetprod_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult" ("(3PROD _:#_. _)" [0, 51, 10] 10) |
51548
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1939 |
syntax |
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61890
diff
changeset
|
1940 |
"_msetprod_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult" ("(3\<Prod>_\<in>#_. _)" [0, 51, 10] 10) |
51548
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1941 |
translations |
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61890
diff
changeset
|
1942 |
"\<Prod>i \<in># A. b" \<rightleftharpoons> "CONST msetprod (CONST image_mset (\<lambda>i. b) A)" |
51548
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1943 |
|
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1944 |
lemma (in comm_semiring_1) dvd_msetprod: |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1945 |
assumes "x \<in># A" |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1946 |
shows "x dvd msetprod A" |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1947 |
proof - |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1948 |
from assms have "A = (A - {#x#}) + {#x#}" by simp |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1949 |
then obtain B where "A = B + {#x#}" .. |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1950 |
then show ?thesis by simp |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1951 |
qed |
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1952 |
|
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1953 |
lemma (in semidom) msetprod_zero_iff [iff]: |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1954 |
"msetprod A = 0 \<longleftrightarrow> 0 \<in># A" |
62366 | 1955 |
by (induct A) auto |
1956 |
||
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1957 |
lemma (in semidom_divide) msetprod_diff: |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1958 |
assumes "B \<subseteq># A" and "0 \<notin># B" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1959 |
shows "msetprod (A - B) = msetprod A div msetprod B" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1960 |
proof - |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1961 |
from assms obtain C where "A = B + C" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1962 |
by (metis subset_mset.add_diff_inverse) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1963 |
with assms show ?thesis by simp |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1964 |
qed |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1965 |
|
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1966 |
lemma (in semidom_divide) msetprod_minus: |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1967 |
assumes "a \<in># A" and "a \<noteq> 0" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1968 |
shows "msetprod (A - {#a#}) = msetprod A div a" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1969 |
using assms msetprod_diff [of "{#a#}" A] |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1970 |
by (auto simp add: single_subset_iff) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1971 |
|
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1972 |
lemma (in normalization_semidom) normalized_msetprodI: |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1973 |
assumes "\<And>a. a \<in># A \<Longrightarrow> normalize a = a" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1974 |
shows "normalize (msetprod A) = msetprod A" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1975 |
using assms by (induct A) (simp_all add: normalize_mult) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
1976 |
|
51548
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1977 |
|
60500 | 1978 |
subsection \<open>Alternative representations\<close> |
1979 |
||
1980 |
subsubsection \<open>Lists\<close> |
|
51548
757fa47af981
centralized various multiset operations in theory multiset;
haftmann
parents:
51161
diff
changeset
|
1981 |
|
39533
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
1982 |
context linorder |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
1983 |
begin |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
1984 |
|
60515 | 1985 |
lemma mset_insort [simp]: |
1986 |
"mset (insort_key k x xs) = {#x#} + mset xs" |
|
37107 | 1987 |
by (induct xs) (simp_all add: ac_simps) |
58425 | 1988 |
|
60515 | 1989 |
lemma mset_sort [simp]: |
1990 |
"mset (sort_key k xs) = mset xs" |
|
37107 | 1991 |
by (induct xs) (simp_all add: ac_simps) |
1992 |
||
60500 | 1993 |
text \<open> |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
1994 |
This lemma shows which properties suffice to show that a function |
61585 | 1995 |
\<open>f\<close> with \<open>f xs = ys\<close> behaves like sort. |
60500 | 1996 |
\<close> |
37074 | 1997 |
|
39533
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
1998 |
lemma properties_for_sort_key: |
60515 | 1999 |
assumes "mset ys = mset xs" |
60606 | 2000 |
and "\<And>k. k \<in> set ys \<Longrightarrow> filter (\<lambda>x. f k = f x) ys = filter (\<lambda>x. f k = f x) xs" |
2001 |
and "sorted (map f ys)" |
|
39533
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
2002 |
shows "sort_key f xs = ys" |
60606 | 2003 |
using assms |
46921 | 2004 |
proof (induct xs arbitrary: ys) |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
2005 |
case Nil then show ?case by simp |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
2006 |
next |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
2007 |
case (Cons x xs) |
39533
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
2008 |
from Cons.prems(2) have |
40305
41833242cc42
tuned lemma proposition of properties_for_sort_key
haftmann
parents:
40303
diff
changeset
|
2009 |
"\<forall>k \<in> set ys. filter (\<lambda>x. f k = f x) (remove1 x ys) = filter (\<lambda>x. f k = f x) xs" |
39533
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
2010 |
by (simp add: filter_remove1) |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
2011 |
with Cons.prems have "sort_key f xs = remove1 x ys" |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
2012 |
by (auto intro!: Cons.hyps simp add: sorted_map_remove1) |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
2013 |
moreover from Cons.prems have "x \<in># mset ys" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
2014 |
by auto |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
2015 |
then have "x \<in> set ys" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
2016 |
by simp |
39533
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
2017 |
ultimately show ?case using Cons.prems by (simp add: insort_key_remove1) |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
2018 |
qed |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
2019 |
|
39533
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
2020 |
lemma properties_for_sort: |
60515 | 2021 |
assumes multiset: "mset ys = mset xs" |
60606 | 2022 |
and "sorted ys" |
39533
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
2023 |
shows "sort xs = ys" |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
2024 |
proof (rule properties_for_sort_key) |
60515 | 2025 |
from multiset show "mset ys = mset xs" . |
60500 | 2026 |
from \<open>sorted ys\<close> show "sorted (map (\<lambda>x. x) ys)" by simp |
60678 | 2027 |
from multiset have "length (filter (\<lambda>y. k = y) ys) = length (filter (\<lambda>x. k = x) xs)" for k |
60515 | 2028 |
by (rule mset_eq_length_filter) |
60678 | 2029 |
then have "replicate (length (filter (\<lambda>y. k = y) ys)) k = |
2030 |
replicate (length (filter (\<lambda>x. k = x) xs)) k" for k |
|
39533
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
2031 |
by simp |
60678 | 2032 |
then show "k \<in> set ys \<Longrightarrow> filter (\<lambda>y. k = y) ys = filter (\<lambda>x. k = x) xs" for k |
39533
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
2033 |
by (simp add: replicate_length_filter) |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
2034 |
qed |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
2035 |
|
61031
162bd20dae23
more lemmas on sorting and multisets (due to Thomas Sewell)
haftmann
parents:
60804
diff
changeset
|
2036 |
lemma sort_key_inj_key_eq: |
162bd20dae23
more lemmas on sorting and multisets (due to Thomas Sewell)
haftmann
parents:
60804
diff
changeset
|
2037 |
assumes mset_equal: "mset xs = mset ys" |
162bd20dae23
more lemmas on sorting and multisets (due to Thomas Sewell)
haftmann
parents:
60804
diff
changeset
|
2038 |
and "inj_on f (set xs)" |
162bd20dae23
more lemmas on sorting and multisets (due to Thomas Sewell)
haftmann
parents:
60804
diff
changeset
|
2039 |
and "sorted (map f ys)" |
162bd20dae23
more lemmas on sorting and multisets (due to Thomas Sewell)
haftmann
parents:
60804
diff
changeset
|
2040 |
shows "sort_key f xs = ys" |
162bd20dae23
more lemmas on sorting and multisets (due to Thomas Sewell)
haftmann
parents:
60804
diff
changeset
|
2041 |
proof (rule properties_for_sort_key) |
162bd20dae23
more lemmas on sorting and multisets (due to Thomas Sewell)
haftmann
parents:
60804
diff
changeset
|
2042 |
from mset_equal |
162bd20dae23
more lemmas on sorting and multisets (due to Thomas Sewell)
haftmann
parents:
60804
diff
changeset
|
2043 |
show "mset ys = mset xs" by simp |
61188 | 2044 |
from \<open>sorted (map f ys)\<close> |
61031
162bd20dae23
more lemmas on sorting and multisets (due to Thomas Sewell)
haftmann
parents:
60804
diff
changeset
|
2045 |
show "sorted (map f ys)" . |
162bd20dae23
more lemmas on sorting and multisets (due to Thomas Sewell)
haftmann
parents:
60804
diff
changeset
|
2046 |
show "[x\<leftarrow>ys . f k = f x] = [x\<leftarrow>xs . f k = f x]" if "k \<in> set ys" for k |
162bd20dae23
more lemmas on sorting and multisets (due to Thomas Sewell)
haftmann
parents:
60804
diff
changeset
|
2047 |
proof - |
162bd20dae23
more lemmas on sorting and multisets (due to Thomas Sewell)
haftmann
parents:
60804
diff
changeset
|
2048 |
from mset_equal |
162bd20dae23
more lemmas on sorting and multisets (due to Thomas Sewell)
haftmann
parents:
60804
diff
changeset
|
2049 |
have set_equal: "set xs = set ys" by (rule mset_eq_setD) |
162bd20dae23
more lemmas on sorting and multisets (due to Thomas Sewell)
haftmann
parents:
60804
diff
changeset
|
2050 |
with that have "insert k (set ys) = set ys" by auto |
61188 | 2051 |
with \<open>inj_on f (set xs)\<close> have inj: "inj_on f (insert k (set ys))" |
61031
162bd20dae23
more lemmas on sorting and multisets (due to Thomas Sewell)
haftmann
parents:
60804
diff
changeset
|
2052 |
by (simp add: set_equal) |
162bd20dae23
more lemmas on sorting and multisets (due to Thomas Sewell)
haftmann
parents:
60804
diff
changeset
|
2053 |
from inj have "[x\<leftarrow>ys . f k = f x] = filter (HOL.eq k) ys" |
162bd20dae23
more lemmas on sorting and multisets (due to Thomas Sewell)
haftmann
parents:
60804
diff
changeset
|
2054 |
by (auto intro!: inj_on_filter_key_eq) |
162bd20dae23
more lemmas on sorting and multisets (due to Thomas Sewell)
haftmann
parents:
60804
diff
changeset
|
2055 |
also have "\<dots> = replicate (count (mset ys) k) k" |
162bd20dae23
more lemmas on sorting and multisets (due to Thomas Sewell)
haftmann
parents:
60804
diff
changeset
|
2056 |
by (simp add: replicate_count_mset_eq_filter_eq) |
162bd20dae23
more lemmas on sorting and multisets (due to Thomas Sewell)
haftmann
parents:
60804
diff
changeset
|
2057 |
also have "\<dots> = replicate (count (mset xs) k) k" |
162bd20dae23
more lemmas on sorting and multisets (due to Thomas Sewell)
haftmann
parents:
60804
diff
changeset
|
2058 |
using mset_equal by simp |
162bd20dae23
more lemmas on sorting and multisets (due to Thomas Sewell)
haftmann
parents:
60804
diff
changeset
|
2059 |
also have "\<dots> = filter (HOL.eq k) xs" |
162bd20dae23
more lemmas on sorting and multisets (due to Thomas Sewell)
haftmann
parents:
60804
diff
changeset
|
2060 |
by (simp add: replicate_count_mset_eq_filter_eq) |
162bd20dae23
more lemmas on sorting and multisets (due to Thomas Sewell)
haftmann
parents:
60804
diff
changeset
|
2061 |
also have "\<dots> = [x\<leftarrow>xs . f k = f x]" |
162bd20dae23
more lemmas on sorting and multisets (due to Thomas Sewell)
haftmann
parents:
60804
diff
changeset
|
2062 |
using inj by (auto intro!: inj_on_filter_key_eq [symmetric] simp add: set_equal) |
162bd20dae23
more lemmas on sorting and multisets (due to Thomas Sewell)
haftmann
parents:
60804
diff
changeset
|
2063 |
finally show ?thesis . |
162bd20dae23
more lemmas on sorting and multisets (due to Thomas Sewell)
haftmann
parents:
60804
diff
changeset
|
2064 |
qed |
162bd20dae23
more lemmas on sorting and multisets (due to Thomas Sewell)
haftmann
parents:
60804
diff
changeset
|
2065 |
qed |
162bd20dae23
more lemmas on sorting and multisets (due to Thomas Sewell)
haftmann
parents:
60804
diff
changeset
|
2066 |
|
162bd20dae23
more lemmas on sorting and multisets (due to Thomas Sewell)
haftmann
parents:
60804
diff
changeset
|
2067 |
lemma sort_key_eq_sort_key: |
162bd20dae23
more lemmas on sorting and multisets (due to Thomas Sewell)
haftmann
parents:
60804
diff
changeset
|
2068 |
assumes "mset xs = mset ys" |
162bd20dae23
more lemmas on sorting and multisets (due to Thomas Sewell)
haftmann
parents:
60804
diff
changeset
|
2069 |
and "inj_on f (set xs)" |
162bd20dae23
more lemmas on sorting and multisets (due to Thomas Sewell)
haftmann
parents:
60804
diff
changeset
|
2070 |
shows "sort_key f xs = sort_key f ys" |
162bd20dae23
more lemmas on sorting and multisets (due to Thomas Sewell)
haftmann
parents:
60804
diff
changeset
|
2071 |
by (rule sort_key_inj_key_eq) (simp_all add: assms) |
162bd20dae23
more lemmas on sorting and multisets (due to Thomas Sewell)
haftmann
parents:
60804
diff
changeset
|
2072 |
|
40303
2d507370e879
lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents:
40250
diff
changeset
|
2073 |
lemma sort_key_by_quicksort: |
2d507370e879
lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents:
40250
diff
changeset
|
2074 |
"sort_key f xs = sort_key f [x\<leftarrow>xs. f x < f (xs ! (length xs div 2))] |
2d507370e879
lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents:
40250
diff
changeset
|
2075 |
@ [x\<leftarrow>xs. f x = f (xs ! (length xs div 2))] |
2d507370e879
lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents:
40250
diff
changeset
|
2076 |
@ sort_key f [x\<leftarrow>xs. f x > f (xs ! (length xs div 2))]" (is "sort_key f ?lhs = ?rhs") |
2d507370e879
lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents:
40250
diff
changeset
|
2077 |
proof (rule properties_for_sort_key) |
60515 | 2078 |
show "mset ?rhs = mset ?lhs" |
2079 |
by (rule multiset_eqI) (auto simp add: mset_filter) |
|
40303
2d507370e879
lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents:
40250
diff
changeset
|
2080 |
show "sorted (map f ?rhs)" |
2d507370e879
lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents:
40250
diff
changeset
|
2081 |
by (auto simp add: sorted_append intro: sorted_map_same) |
2d507370e879
lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents:
40250
diff
changeset
|
2082 |
next |
40305
41833242cc42
tuned lemma proposition of properties_for_sort_key
haftmann
parents:
40303
diff
changeset
|
2083 |
fix l |
41833242cc42
tuned lemma proposition of properties_for_sort_key
haftmann
parents:
40303
diff
changeset
|
2084 |
assume "l \<in> set ?rhs" |
40346 | 2085 |
let ?pivot = "f (xs ! (length xs div 2))" |
2086 |
have *: "\<And>x. f l = f x \<longleftrightarrow> f x = f l" by auto |
|
40306 | 2087 |
have "[x \<leftarrow> sort_key f xs . f x = f l] = [x \<leftarrow> xs. f x = f l]" |
40305
41833242cc42
tuned lemma proposition of properties_for_sort_key
haftmann
parents:
40303
diff
changeset
|
2088 |
unfolding filter_sort by (rule properties_for_sort_key) (auto intro: sorted_map_same) |
40346 | 2089 |
with * have **: "[x \<leftarrow> sort_key f xs . f l = f x] = [x \<leftarrow> xs. f l = f x]" by simp |
2090 |
have "\<And>x P. P (f x) ?pivot \<and> f l = f x \<longleftrightarrow> P (f l) ?pivot \<and> f l = f x" by auto |
|
2091 |
then have "\<And>P. [x \<leftarrow> sort_key f xs . P (f x) ?pivot \<and> f l = f x] = |
|
2092 |
[x \<leftarrow> sort_key f xs. P (f l) ?pivot \<and> f l = f x]" by simp |
|
2093 |
note *** = this [of "op <"] this [of "op >"] this [of "op ="] |
|
40306 | 2094 |
show "[x \<leftarrow> ?rhs. f l = f x] = [x \<leftarrow> ?lhs. f l = f x]" |
40305
41833242cc42
tuned lemma proposition of properties_for_sort_key
haftmann
parents:
40303
diff
changeset
|
2095 |
proof (cases "f l" ?pivot rule: linorder_cases) |
46730 | 2096 |
case less |
2097 |
then have "f l \<noteq> ?pivot" and "\<not> f l > ?pivot" by auto |
|
2098 |
with less show ?thesis |
|
40346 | 2099 |
by (simp add: filter_sort [symmetric] ** ***) |
40305
41833242cc42
tuned lemma proposition of properties_for_sort_key
haftmann
parents:
40303
diff
changeset
|
2100 |
next |
40306 | 2101 |
case equal then show ?thesis |
40346 | 2102 |
by (simp add: * less_le) |
40305
41833242cc42
tuned lemma proposition of properties_for_sort_key
haftmann
parents:
40303
diff
changeset
|
2103 |
next |
46730 | 2104 |
case greater |
2105 |
then have "f l \<noteq> ?pivot" and "\<not> f l < ?pivot" by auto |
|
2106 |
with greater show ?thesis |
|
40346 | 2107 |
by (simp add: filter_sort [symmetric] ** ***) |
40306 | 2108 |
qed |
40303
2d507370e879
lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents:
40250
diff
changeset
|
2109 |
qed |
2d507370e879
lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents:
40250
diff
changeset
|
2110 |
|
2d507370e879
lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents:
40250
diff
changeset
|
2111 |
lemma sort_by_quicksort: |
2d507370e879
lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents:
40250
diff
changeset
|
2112 |
"sort xs = sort [x\<leftarrow>xs. x < xs ! (length xs div 2)] |
2d507370e879
lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents:
40250
diff
changeset
|
2113 |
@ [x\<leftarrow>xs. x = xs ! (length xs div 2)] |
2d507370e879
lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents:
40250
diff
changeset
|
2114 |
@ sort [x\<leftarrow>xs. x > xs ! (length xs div 2)]" (is "sort ?lhs = ?rhs") |
2d507370e879
lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents:
40250
diff
changeset
|
2115 |
using sort_key_by_quicksort [of "\<lambda>x. x", symmetric] by simp |
2d507370e879
lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents:
40250
diff
changeset
|
2116 |
|
60500 | 2117 |
text \<open>A stable parametrized quicksort\<close> |
40347
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
2118 |
|
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
2119 |
definition part :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b list \<Rightarrow> 'b list \<times> 'b list \<times> 'b list" where |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
2120 |
"part f pivot xs = ([x \<leftarrow> xs. f x < pivot], [x \<leftarrow> xs. f x = pivot], [x \<leftarrow> xs. pivot < f x])" |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
2121 |
|
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
2122 |
lemma part_code [code]: |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
2123 |
"part f pivot [] = ([], [], [])" |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
2124 |
"part f pivot (x # xs) = (let (lts, eqs, gts) = part f pivot xs; x' = f x in |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
2125 |
if x' < pivot then (x # lts, eqs, gts) |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
2126 |
else if x' > pivot then (lts, eqs, x # gts) |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
2127 |
else (lts, x # eqs, gts))" |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
2128 |
by (auto simp add: part_def Let_def split_def) |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
2129 |
|
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
2130 |
lemma sort_key_by_quicksort_code [code]: |
60606 | 2131 |
"sort_key f xs = |
2132 |
(case xs of |
|
2133 |
[] \<Rightarrow> [] |
|
40347
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
2134 |
| [x] \<Rightarrow> xs |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
2135 |
| [x, y] \<Rightarrow> (if f x \<le> f y then xs else [y, x]) |
60606 | 2136 |
| _ \<Rightarrow> |
2137 |
let (lts, eqs, gts) = part f (f (xs ! (length xs div 2))) xs |
|
2138 |
in sort_key f lts @ eqs @ sort_key f gts)" |
|
40347
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
2139 |
proof (cases xs) |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
2140 |
case Nil then show ?thesis by simp |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
2141 |
next |
46921 | 2142 |
case (Cons _ ys) note hyps = Cons show ?thesis |
2143 |
proof (cases ys) |
|
40347
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
2144 |
case Nil with hyps show ?thesis by simp |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
2145 |
next |
46921 | 2146 |
case (Cons _ zs) note hyps = hyps Cons show ?thesis |
2147 |
proof (cases zs) |
|
40347
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
2148 |
case Nil with hyps show ?thesis by auto |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
2149 |
next |
58425 | 2150 |
case Cons |
40347
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
2151 |
from sort_key_by_quicksort [of f xs] |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
2152 |
have "sort_key f xs = (let (lts, eqs, gts) = part f (f (xs ! (length xs div 2))) xs |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
2153 |
in sort_key f lts @ eqs @ sort_key f gts)" |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
2154 |
by (simp only: split_def Let_def part_def fst_conv snd_conv) |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
2155 |
with hyps Cons show ?thesis by (simp only: list.cases) |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
2156 |
qed |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
2157 |
qed |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
2158 |
qed |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
2159 |
|
39533
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
2160 |
end |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
2161 |
|
40347
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
2162 |
hide_const (open) part |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
2163 |
|
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
2164 |
lemma mset_remdups_subset_eq: "mset (remdups xs) \<subseteq># mset xs" |
60397
f8a513fedb31
Renaming multiset operators < ~> <#,...
Mathias Fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
59999
diff
changeset
|
2165 |
by (induct xs) (auto intro: subset_mset.order_trans) |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
2166 |
|
60515 | 2167 |
lemma mset_update: |
2168 |
"i < length ls \<Longrightarrow> mset (ls[i := v]) = mset ls - {#ls ! i#} + {#v#}" |
|
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
2169 |
proof (induct ls arbitrary: i) |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
2170 |
case Nil then show ?case by simp |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
2171 |
next |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
2172 |
case (Cons x xs) |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
2173 |
show ?case |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
2174 |
proof (cases i) |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
2175 |
case 0 then show ?thesis by simp |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
2176 |
next |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
2177 |
case (Suc i') |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
2178 |
with Cons show ?thesis |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
2179 |
apply simp |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
2180 |
apply (subst add.assoc) |
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
2181 |
apply (subst add.commute [of "{#v#}" "{#x#}"]) |
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
2182 |
apply (subst add.assoc [symmetric]) |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
2183 |
apply simp |
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
2184 |
apply (rule mset_subset_eq_multiset_union_diff_commute) |
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
2185 |
apply (simp add: mset_subset_eq_single nth_mem_mset) |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
2186 |
done |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
2187 |
qed |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
2188 |
qed |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
2189 |
|
60515 | 2190 |
lemma mset_swap: |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
2191 |
"i < length ls \<Longrightarrow> j < length ls \<Longrightarrow> |
60515 | 2192 |
mset (ls[j := ls ! i, i := ls ! j]) = mset ls" |
2193 |
by (cases "i = j") (simp_all add: mset_update nth_mem_mset) |
|
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
2194 |
|
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
2195 |
|
60500 | 2196 |
subsection \<open>The multiset order\<close> |
2197 |
||
2198 |
subsubsection \<open>Well-foundedness\<close> |
|
10249 | 2199 |
|
60606 | 2200 |
definition mult1 :: "('a \<times> 'a) set \<Rightarrow> ('a multiset \<times> 'a multiset) set" where |
37765 | 2201 |
"mult1 r = {(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> |
60607 | 2202 |
(\<forall>b. b \<in># K \<longrightarrow> (b, a) \<in> r)}" |
60606 | 2203 |
|
2204 |
definition mult :: "('a \<times> 'a) set \<Rightarrow> ('a multiset \<times> 'a multiset) set" where |
|
37765 | 2205 |
"mult r = (mult1 r)\<^sup>+" |
10249 | 2206 |
|
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
2207 |
lemma mult1I: |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
2208 |
assumes "M = M0 + {#a#}" and "N = M0 + K" and "\<And>b. b \<in># K \<Longrightarrow> (b, a) \<in> r" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
2209 |
shows "(N, M) \<in> mult1 r" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
2210 |
using assms unfolding mult1_def by blast |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
2211 |
|
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
2212 |
lemma mult1E: |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
2213 |
assumes "(N, M) \<in> mult1 r" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
2214 |
obtains a M0 K where "M = M0 + {#a#}" "N = M0 + K" "\<And>b. b \<in># K \<Longrightarrow> (b, a) \<in> r" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
2215 |
using assms unfolding mult1_def by blast |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
2216 |
|
63660
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2217 |
lemma mono_mult1: |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2218 |
assumes "r \<subseteq> r'" shows "mult1 r \<subseteq> mult1 r'" |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2219 |
unfolding mult1_def using assms by blast |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2220 |
|
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2221 |
lemma mono_mult: |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2222 |
assumes "r \<subseteq> r'" shows "mult r \<subseteq> mult r'" |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2223 |
unfolding mult_def using mono_mult1[OF assms] trancl_mono by blast |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2224 |
|
23751 | 2225 |
lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r" |
26178 | 2226 |
by (simp add: mult1_def) |
10249 | 2227 |
|
60608 | 2228 |
lemma less_add: |
2229 |
assumes mult1: "(N, M0 + {#a#}) \<in> mult1 r" |
|
2230 |
shows |
|
2231 |
"(\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or> |
|
2232 |
(\<exists>K. (\<forall>b. b \<in># K \<longrightarrow> (b, a) \<in> r) \<and> N = M0 + K)" |
|
2233 |
proof - |
|
60607 | 2234 |
let ?r = "\<lambda>K a. \<forall>b. b \<in># K \<longrightarrow> (b, a) \<in> r" |
11464 | 2235 |
let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a" |
60608 | 2236 |
obtain a' M0' K where M0: "M0 + {#a#} = M0' + {#a'#}" |
2237 |
and N: "N = M0' + K" |
|
2238 |
and r: "?r K a'" |
|
2239 |
using mult1 unfolding mult1_def by auto |
|
2240 |
show ?thesis (is "?case1 \<or> ?case2") |
|
60606 | 2241 |
proof - |
2242 |
from M0 consider "M0 = M0'" "a = a'" |
|
2243 |
| K' where "M0 = K' + {#a'#}" "M0' = K' + {#a#}" |
|
2244 |
by atomize_elim (simp only: add_eq_conv_ex) |
|
18258 | 2245 |
then show ?thesis |
60606 | 2246 |
proof cases |
2247 |
case 1 |
|
11464 | 2248 |
with N r have "?r K a \<and> N = M0 + K" by simp |
60606 | 2249 |
then have ?case2 .. |
2250 |
then show ?thesis .. |
|
10249 | 2251 |
next |
60606 | 2252 |
case 2 |
2253 |
from N 2(2) have n: "N = K' + K + {#a#}" by (simp add: ac_simps) |
|
2254 |
with r 2(1) have "?R (K' + K) M0" by blast |
|
60608 | 2255 |
with n have ?case1 by (simp add: mult1_def) |
60606 | 2256 |
then show ?thesis .. |
10249 | 2257 |
qed |
2258 |
qed |
|
2259 |
qed |
|
2260 |
||
60608 | 2261 |
lemma all_accessible: |
2262 |
assumes "wf r" |
|
2263 |
shows "\<forall>M. M \<in> Wellfounded.acc (mult1 r)" |
|
10249 | 2264 |
proof |
2265 |
let ?R = "mult1 r" |
|
54295 | 2266 |
let ?W = "Wellfounded.acc ?R" |
10249 | 2267 |
{ |
2268 |
fix M M0 a |
|
23751 | 2269 |
assume M0: "M0 \<in> ?W" |
60606 | 2270 |
and wf_hyp: "\<And>b. (b, a) \<in> r \<Longrightarrow> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)" |
2271 |
and acc_hyp: "\<forall>M. (M, M0) \<in> ?R \<longrightarrow> M + {#a#} \<in> ?W" |
|
23751 | 2272 |
have "M0 + {#a#} \<in> ?W" |
2273 |
proof (rule accI [of "M0 + {#a#}"]) |
|
10249 | 2274 |
fix N |
23751 | 2275 |
assume "(N, M0 + {#a#}) \<in> ?R" |
60608 | 2276 |
then consider M where "(M, M0) \<in> ?R" "N = M + {#a#}" |
2277 |
| K where "\<forall>b. b \<in># K \<longrightarrow> (b, a) \<in> r" "N = M0 + K" |
|
2278 |
by atomize_elim (rule less_add) |
|
23751 | 2279 |
then show "N \<in> ?W" |
60608 | 2280 |
proof cases |
2281 |
case 1 |
|
60606 | 2282 |
from acc_hyp have "(M, M0) \<in> ?R \<longrightarrow> M + {#a#} \<in> ?W" .. |
60500 | 2283 |
from this and \<open>(M, M0) \<in> ?R\<close> have "M + {#a#} \<in> ?W" .. |
60608 | 2284 |
then show "N \<in> ?W" by (simp only: \<open>N = M + {#a#}\<close>) |
10249 | 2285 |
next |
60608 | 2286 |
case 2 |
2287 |
from this(1) have "M0 + K \<in> ?W" |
|
10249 | 2288 |
proof (induct K) |
18730 | 2289 |
case empty |
23751 | 2290 |
from M0 show "M0 + {#} \<in> ?W" by simp |
18730 | 2291 |
next |
2292 |
case (add K x) |
|
23751 | 2293 |
from add.prems have "(x, a) \<in> r" by simp |
2294 |
with wf_hyp have "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast |
|
2295 |
moreover from add have "M0 + K \<in> ?W" by simp |
|
2296 |
ultimately have "(M0 + K) + {#x#} \<in> ?W" .. |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
2297 |
then show "M0 + (K + {#x#}) \<in> ?W" by (simp only: add.assoc) |
10249 | 2298 |
qed |
60608 | 2299 |
then show "N \<in> ?W" by (simp only: 2(2)) |
10249 | 2300 |
qed |
2301 |
qed |
|
2302 |
} note tedious_reasoning = this |
|
2303 |
||
60608 | 2304 |
show "M \<in> ?W" for M |
10249 | 2305 |
proof (induct M) |
23751 | 2306 |
show "{#} \<in> ?W" |
10249 | 2307 |
proof (rule accI) |
23751 | 2308 |
fix b assume "(b, {#}) \<in> ?R" |
2309 |
with not_less_empty show "b \<in> ?W" by contradiction |
|
10249 | 2310 |
qed |
2311 |
||
23751 | 2312 |
fix M a assume "M \<in> ?W" |
60608 | 2313 |
from \<open>wf r\<close> have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W" |
10249 | 2314 |
proof induct |
2315 |
fix a |
|
60606 | 2316 |
assume r: "\<And>b. (b, a) \<in> r \<Longrightarrow> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)" |
23751 | 2317 |
show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W" |
10249 | 2318 |
proof |
23751 | 2319 |
fix M assume "M \<in> ?W" |
2320 |
then show "M + {#a#} \<in> ?W" |
|
23373 | 2321 |
by (rule acc_induct) (rule tedious_reasoning [OF _ r]) |
10249 | 2322 |
qed |
2323 |
qed |
|
60500 | 2324 |
from this and \<open>M \<in> ?W\<close> show "M + {#a#} \<in> ?W" .. |
10249 | 2325 |
qed |
2326 |
qed |
|
2327 |
||
60606 | 2328 |
theorem wf_mult1: "wf r \<Longrightarrow> wf (mult1 r)" |
26178 | 2329 |
by (rule acc_wfI) (rule all_accessible) |
10249 | 2330 |
|
60606 | 2331 |
theorem wf_mult: "wf r \<Longrightarrow> wf (mult r)" |
26178 | 2332 |
unfolding mult_def by (rule wf_trancl) (rule wf_mult1) |
10249 | 2333 |
|
2334 |
||
60500 | 2335 |
subsubsection \<open>Closure-free presentation\<close> |
2336 |
||
2337 |
text \<open>One direction.\<close> |
|
10249 | 2338 |
|
2339 |
lemma mult_implies_one_step: |
|
60606 | 2340 |
"trans r \<Longrightarrow> (M, N) \<in> mult r \<Longrightarrow> |
11464 | 2341 |
\<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and> |
60495 | 2342 |
(\<forall>k \<in> set_mset K. \<exists>j \<in> set_mset J. (k, j) \<in> r)" |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
2343 |
apply (unfold mult_def mult1_def) |
26178 | 2344 |
apply (erule converse_trancl_induct, clarify) |
2345 |
apply (rule_tac x = M0 in exI, simp, clarify) |
|
60607 | 2346 |
apply (case_tac "a \<in># K") |
26178 | 2347 |
apply (rule_tac x = I in exI) |
2348 |
apply (simp (no_asm)) |
|
2349 |
apply (rule_tac x = "(K - {#a#}) + Ka" in exI) |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
2350 |
apply (simp (no_asm_simp) add: add.assoc [symmetric]) |
59807 | 2351 |
apply (drule_tac f = "\<lambda>M. M - {#a#}" and x="S + T" for S T in arg_cong) |
26178 | 2352 |
apply (simp add: diff_union_single_conv) |
2353 |
apply (simp (no_asm_use) add: trans_def) |
|
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
2354 |
apply (metis (no_types, hide_lams) Multiset.diff_right_commute Un_iff diff_single_trivial multi_drop_mem_not_eq) |
60607 | 2355 |
apply (subgoal_tac "a \<in># I") |
26178 | 2356 |
apply (rule_tac x = "I - {#a#}" in exI) |
2357 |
apply (rule_tac x = "J + {#a#}" in exI) |
|
2358 |
apply (rule_tac x = "K + Ka" in exI) |
|
2359 |
apply (rule conjI) |
|
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39301
diff
changeset
|
2360 |
apply (simp add: multiset_eq_iff split: nat_diff_split) |
26178 | 2361 |
apply (rule conjI) |
59807 | 2362 |
apply (drule_tac f = "\<lambda>M. M - {#a#}" and x="S + T" for S T in arg_cong, simp) |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39301
diff
changeset
|
2363 |
apply (simp add: multiset_eq_iff split: nat_diff_split) |
26178 | 2364 |
apply (simp (no_asm_use) add: trans_def) |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
2365 |
apply (subgoal_tac "a \<in># (M0 + {#a#})") |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
2366 |
apply (simp_all add: not_in_iff) |
26178 | 2367 |
apply blast |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
2368 |
apply (metis add.comm_neutral add_diff_cancel_right' count_eq_zero_iff diff_single_trivial multi_self_add_other_not_self plus_multiset.rep_eq) |
26178 | 2369 |
done |
10249 | 2370 |
|
2371 |
lemma one_step_implies_mult_aux: |
|
60678 | 2372 |
"\<forall>I J K. size J = n \<and> J \<noteq> {#} \<and> (\<forall>k \<in> set_mset K. \<exists>j \<in> set_mset J. (k, j) \<in> r) |
2373 |
\<longrightarrow> (I + K, I + J) \<in> mult r" |
|
2374 |
apply (induct n) |
|
2375 |
apply auto |
|
26178 | 2376 |
apply (frule size_eq_Suc_imp_eq_union, clarify) |
2377 |
apply (rename_tac "J'", simp) |
|
2378 |
apply (erule notE, auto) |
|
2379 |
apply (case_tac "J' = {#}") |
|
2380 |
apply (simp add: mult_def) |
|
2381 |
apply (rule r_into_trancl) |
|
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
2382 |
apply (simp add: mult1_def, blast) |
60500 | 2383 |
txt \<open>Now we know @{term "J' \<noteq> {#}"}.\<close> |
26178 | 2384 |
apply (cut_tac M = K and P = "\<lambda>x. (x, a) \<in> r" in multiset_partition) |
60495 | 2385 |
apply (erule_tac P = "\<forall>k \<in> set_mset K. P k" for P in rev_mp) |
26178 | 2386 |
apply (erule ssubst) |
2387 |
apply (simp add: Ball_def, auto) |
|
2388 |
apply (subgoal_tac |
|
60607 | 2389 |
"((I + {# x \<in># K. (x, a) \<in> r #}) + {# x \<in># K. (x, a) \<notin> r #}, |
2390 |
(I + {# x \<in># K. (x, a) \<in> r #}) + J') \<in> mult r") |
|
26178 | 2391 |
prefer 2 |
2392 |
apply force |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
2393 |
apply (simp (no_asm_use) add: add.assoc [symmetric] mult_def) |
26178 | 2394 |
apply (erule trancl_trans) |
2395 |
apply (rule r_into_trancl) |
|
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
2396 |
apply (simp add: mult1_def) |
26178 | 2397 |
apply (rule_tac x = a in exI) |
2398 |
apply (rule_tac x = "I + J'" in exI) |
|
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
2399 |
apply (simp add: ac_simps) |
26178 | 2400 |
done |
10249 | 2401 |
|
17161 | 2402 |
lemma one_step_implies_mult: |
62651 | 2403 |
"J \<noteq> {#} \<Longrightarrow> \<forall>k \<in> set_mset K. \<exists>j \<in> set_mset J. (k, j) \<in> r |
60606 | 2404 |
\<Longrightarrow> (I + K, I + J) \<in> mult r" |
26178 | 2405 |
using one_step_implies_mult_aux by blast |
10249 | 2406 |
|
63660
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2407 |
|
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2408 |
subsection \<open>The multiset extension is cancellative for multiset union\<close> |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2409 |
|
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2410 |
lemma mult_cancel: |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2411 |
assumes "trans s" and "irrefl s" |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2412 |
shows "(X + Z, Y + Z) \<in> mult s \<longleftrightarrow> (X, Y) \<in> mult s" (is "?L \<longleftrightarrow> ?R") |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2413 |
proof |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2414 |
assume ?L thus ?R |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2415 |
proof (induct Z) |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2416 |
case (add Z z) |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2417 |
obtain X' Y' Z' where *: "X + Z + {#z#} = Z' + X'" "Y + Z + {#z#} = Z' + Y'" "Y' \<noteq> {#}" |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2418 |
"\<forall>x \<in> set_mset X'. \<exists>y \<in> set_mset Y'. (x, y) \<in> s" |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2419 |
using mult_implies_one_step[OF `trans s` add(2)] unfolding add.assoc by blast |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2420 |
consider Z2 where "Z' = Z2 + {#z#}" | X2 Y2 where "X' = X2 + {#z#}" "Y' = Y2 + {#z#}" |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2421 |
using *(1,2) by (metis mset_add union_iff union_single_eq_member) |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2422 |
thus ?case |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2423 |
proof (cases) |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2424 |
case 1 thus ?thesis using * one_step_implies_mult[of Y' X' s Z2] |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2425 |
by (auto simp: add.commute[of _ "{#_#}"] add.assoc intro: add(1)) |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2426 |
next |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2427 |
case 2 then obtain y where "y \<in> set_mset Y2" "(z, y) \<in> s" using *(4) `irrefl s` |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2428 |
by (auto simp: irrefl_def) |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2429 |
moreover from this transD[OF `trans s` _ this(2)] |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2430 |
have "x' \<in> set_mset X2 \<Longrightarrow> \<exists>y \<in> set_mset Y2. (x', y) \<in> s" for x' |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2431 |
using 2 *(4)[rule_format, of x'] by auto |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2432 |
ultimately show ?thesis using * one_step_implies_mult[of Y2 X2 s Z'] 2 |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2433 |
by (force simp: add.commute[of "{#_#}"] add.assoc[symmetric] intro: add(1)) |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2434 |
qed |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2435 |
qed auto |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2436 |
next |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2437 |
assume ?R then obtain I J K |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2438 |
where "Y = I + J" "X = I + K" "J \<noteq> {#}" "\<forall>k \<in> set_mset K. \<exists>j \<in> set_mset J. (k, j) \<in> s" |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2439 |
using mult_implies_one_step[OF `trans s`] by blast |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2440 |
thus ?L using one_step_implies_mult[of J K s "I + Z"] by (auto simp: ac_simps) |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2441 |
qed |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2442 |
|
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2443 |
lemma mult_cancel_max: |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2444 |
assumes "trans s" and "irrefl s" |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2445 |
shows "(X, Y) \<in> mult s \<longleftrightarrow> (X - X #\<inter> Y, Y - X #\<inter> Y) \<in> mult s" (is "?L \<longleftrightarrow> ?R") |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2446 |
proof - |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2447 |
have "X - X #\<inter> Y + X #\<inter> Y = X" "Y - X #\<inter> Y + X #\<inter> Y = Y" by (auto simp: count_inject[symmetric]) |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2448 |
thus ?thesis using mult_cancel[OF assms, of "X - X #\<inter> Y" "X #\<inter> Y" "Y - X #\<inter> Y"] by auto |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2449 |
qed |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2450 |
|
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2451 |
|
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2452 |
subsection \<open>Quasi-executable version of the multiset extension\<close> |
63088
f2177f5d2aed
a quasi-recursive characterization of the multiset order (by Christian Sternagel)
haftmann
parents:
63060
diff
changeset
|
2453 |
|
f2177f5d2aed
a quasi-recursive characterization of the multiset order (by Christian Sternagel)
haftmann
parents:
63060
diff
changeset
|
2454 |
text \<open> |
63660
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2455 |
Predicate variants of \<open>mult\<close> and the reflexive closure of \<open>mult\<close>, which are |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2456 |
executable whenever the given predicate \<open>P\<close> is. Together with the standard |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2457 |
code equations for \<open>op #\<inter>\<close> and \<open>op -\<close> this should yield quadratic |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2458 |
(with respect to calls to \<open>P\<close>) implementations of \<open>multp\<close> and \<open>multeqp\<close>. |
63088
f2177f5d2aed
a quasi-recursive characterization of the multiset order (by Christian Sternagel)
haftmann
parents:
63060
diff
changeset
|
2459 |
\<close> |
63660
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2460 |
|
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2461 |
definition multp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" where |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2462 |
"multp P N M = |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2463 |
(let Z = M #\<inter> N; X = M - Z in |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2464 |
X \<noteq> {#} \<and> (let Y = N - Z in (\<forall>y \<in> set_mset Y. \<exists>x \<in> set_mset X. P y x)))" |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2465 |
|
63088
f2177f5d2aed
a quasi-recursive characterization of the multiset order (by Christian Sternagel)
haftmann
parents:
63060
diff
changeset
|
2466 |
definition multeqp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" where |
f2177f5d2aed
a quasi-recursive characterization of the multiset order (by Christian Sternagel)
haftmann
parents:
63060
diff
changeset
|
2467 |
"multeqp P N M = |
f2177f5d2aed
a quasi-recursive characterization of the multiset order (by Christian Sternagel)
haftmann
parents:
63060
diff
changeset
|
2468 |
(let Z = M #\<inter> N; X = M - Z; Y = N - Z in |
f2177f5d2aed
a quasi-recursive characterization of the multiset order (by Christian Sternagel)
haftmann
parents:
63060
diff
changeset
|
2469 |
(\<forall>y \<in> set_mset Y. \<exists>x \<in> set_mset X. P y x))" |
f2177f5d2aed
a quasi-recursive characterization of the multiset order (by Christian Sternagel)
haftmann
parents:
63060
diff
changeset
|
2470 |
|
63660
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2471 |
lemma multp_iff: |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2472 |
assumes "irrefl R" and "trans R" and [simp]: "\<And>x y. P x y \<longleftrightarrow> (x, y) \<in> R" |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2473 |
shows "multp P N M \<longleftrightarrow> (N, M) \<in> mult R" (is "?L \<longleftrightarrow> ?R") |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2474 |
proof - |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2475 |
have *: "M #\<inter> N + (N - M #\<inter> N) = N" "M #\<inter> N + (M - M #\<inter> N) = M" |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2476 |
"(M - M #\<inter> N) #\<inter> (N - M #\<inter> N) = {#}" by (auto simp: count_inject[symmetric]) |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2477 |
show ?thesis |
63088
f2177f5d2aed
a quasi-recursive characterization of the multiset order (by Christian Sternagel)
haftmann
parents:
63060
diff
changeset
|
2478 |
proof |
63660
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2479 |
assume ?L thus ?R |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2480 |
using one_step_implies_mult[of "M - M #\<inter> N" "N - M #\<inter> N" R "M #\<inter> N"] * |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2481 |
by (auto simp: multp_def Let_def) |
63088
f2177f5d2aed
a quasi-recursive characterization of the multiset order (by Christian Sternagel)
haftmann
parents:
63060
diff
changeset
|
2482 |
next |
63660
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2483 |
{ fix I J K :: "'a multiset" assume "(I + J) #\<inter> (I + K) = {#}" |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2484 |
then have "I = {#}" by (metis inter_union_distrib_right union_eq_empty) |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2485 |
} note [dest!] = this |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2486 |
assume ?R thus ?L |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2487 |
using mult_implies_one_step[OF assms(2), of "N - M #\<inter> N" "M - M #\<inter> N"] |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2488 |
mult_cancel_max[OF assms(2,1), of "N" "M"] * by (auto simp: multp_def ac_simps) |
63088
f2177f5d2aed
a quasi-recursive characterization of the multiset order (by Christian Sternagel)
haftmann
parents:
63060
diff
changeset
|
2489 |
qed |
63660
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2490 |
qed |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2491 |
|
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2492 |
lemma multeqp_iff: |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2493 |
assumes "irrefl R" and "trans R" and "\<And>x y. P x y \<longleftrightarrow> (x, y) \<in> R" |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2494 |
shows "multeqp P N M \<longleftrightarrow> (N, M) \<in> (mult R)\<^sup>=" |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2495 |
proof - |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2496 |
{ assume "N \<noteq> M" "M - M #\<inter> N = {#}" |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2497 |
then obtain y where "count N y \<noteq> count M y" by (auto simp: count_inject[symmetric]) |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2498 |
then have "\<exists>y. count M y < count N y" using `M - M #\<inter> N = {#}` |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2499 |
by (auto simp: count_inject[symmetric] dest!: le_neq_implies_less fun_cong[of _ _ y]) |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2500 |
} |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2501 |
then have "multeqp P N M \<longleftrightarrow> multp P N M \<or> N = M" |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2502 |
by (auto simp: multeqp_def multp_def Let_def in_diff_count) |
76302202a92d
add monotonicity propertyies of `mult1` and `mult`
Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at>
parents:
63560
diff
changeset
|
2503 |
thus ?thesis using multp_iff[OF assms] by simp |
63088
f2177f5d2aed
a quasi-recursive characterization of the multiset order (by Christian Sternagel)
haftmann
parents:
63060
diff
changeset
|
2504 |
qed |
f2177f5d2aed
a quasi-recursive characterization of the multiset order (by Christian Sternagel)
haftmann
parents:
63060
diff
changeset
|
2505 |
|
10249 | 2506 |
|
60500 | 2507 |
subsubsection \<open>Partial-order properties\<close> |
10249 | 2508 |
|
63410
9789ccc2a477
more instantiations for multiset
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63409
diff
changeset
|
2509 |
lemma (in preorder) mult1_lessE: |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
2510 |
assumes "(N, M) \<in> mult1 {(a, b). a < b}" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
2511 |
obtains a M0 K where "M = M0 + {#a#}" "N = M0 + K" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
2512 |
"a \<notin># K" "\<And>b. b \<in># K \<Longrightarrow> b < a" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
2513 |
proof - |
63539 | 2514 |
from assms obtain a M0 K where "M = M0 + {#a#}" "N = M0 + K" and |
2515 |
*: "b \<in># K \<Longrightarrow> b < a" for b by (blast elim: mult1E) |
|
2516 |
moreover from * [of a] have "a \<notin># K" by auto |
|
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
2517 |
ultimately show thesis by (auto intro: that) |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
2518 |
qed |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
2519 |
|
63410
9789ccc2a477
more instantiations for multiset
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63409
diff
changeset
|
2520 |
instantiation multiset :: (preorder) order |
63388
a095acd4cfbf
instantiate multiset with multiset ordering
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63360
diff
changeset
|
2521 |
begin |
a095acd4cfbf
instantiate multiset with multiset ordering
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63360
diff
changeset
|
2522 |
|
a095acd4cfbf
instantiate multiset with multiset ordering
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63360
diff
changeset
|
2523 |
definition less_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" |
a095acd4cfbf
instantiate multiset with multiset ordering
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63360
diff
changeset
|
2524 |
where "M' < M \<longleftrightarrow> (M', M) \<in> mult {(x', x). x' < x}" |
a095acd4cfbf
instantiate multiset with multiset ordering
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63360
diff
changeset
|
2525 |
|
a095acd4cfbf
instantiate multiset with multiset ordering
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63360
diff
changeset
|
2526 |
definition less_eq_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" |
a095acd4cfbf
instantiate multiset with multiset ordering
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63360
diff
changeset
|
2527 |
where "less_eq_multiset M' M \<longleftrightarrow> M' < M \<or> M' = M" |
a095acd4cfbf
instantiate multiset with multiset ordering
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63360
diff
changeset
|
2528 |
|
a095acd4cfbf
instantiate multiset with multiset ordering
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63360
diff
changeset
|
2529 |
instance |
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
2530 |
proof - |
63388
a095acd4cfbf
instantiate multiset with multiset ordering
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63360
diff
changeset
|
2531 |
have irrefl: "\<not> M < M" for M :: "'a multiset" |
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
2532 |
proof |
63388
a095acd4cfbf
instantiate multiset with multiset ordering
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63360
diff
changeset
|
2533 |
assume "M < M" |
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
2534 |
then have MM: "(M, M) \<in> mult {(x, y). x < y}" by (simp add: less_multiset_def) |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
2535 |
have "trans {(x'::'a, x). x' < x}" |
63410
9789ccc2a477
more instantiations for multiset
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63409
diff
changeset
|
2536 |
by (metis (mono_tags, lifting) case_prodD case_prodI less_trans mem_Collect_eq transI) |
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
2537 |
moreover note MM |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
2538 |
ultimately have "\<exists>I J K. M = I + J \<and> M = I + K |
60495 | 2539 |
\<and> J \<noteq> {#} \<and> (\<forall>k\<in>set_mset K. \<exists>j\<in>set_mset J. (k, j) \<in> {(x, y). x < y})" |
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
2540 |
by (rule mult_implies_one_step) |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
2541 |
then obtain I J K where "M = I + J" and "M = I + K" |
60495 | 2542 |
and "J \<noteq> {#}" and "(\<forall>k\<in>set_mset K. \<exists>j\<in>set_mset J. (k, j) \<in> {(x, y). x < y})" by blast |
60678 | 2543 |
then have *: "K \<noteq> {#}" and **: "\<forall>k\<in>set_mset K. \<exists>j\<in>set_mset K. k < j" by auto |
60495 | 2544 |
have "finite (set_mset K)" by simp |
60678 | 2545 |
moreover note ** |
60495 | 2546 |
ultimately have "set_mset K = {}" |
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
2547 |
by (induct rule: finite_induct) (auto intro: order_less_trans) |
60678 | 2548 |
with * show False by simp |
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
2549 |
qed |
63388
a095acd4cfbf
instantiate multiset with multiset ordering
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63360
diff
changeset
|
2550 |
have trans: "K < M \<Longrightarrow> M < N \<Longrightarrow> K < N" for K M N :: "'a multiset" |
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
2551 |
unfolding less_multiset_def mult_def by (blast intro: trancl_trans) |
63388
a095acd4cfbf
instantiate multiset with multiset ordering
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63360
diff
changeset
|
2552 |
show "OFCLASS('a multiset, order_class)" |
a095acd4cfbf
instantiate multiset with multiset ordering
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63360
diff
changeset
|
2553 |
by standard (auto simp add: less_eq_multiset_def irrefl dest: trans) |
a095acd4cfbf
instantiate multiset with multiset ordering
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63360
diff
changeset
|
2554 |
qed |
a095acd4cfbf
instantiate multiset with multiset ordering
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63360
diff
changeset
|
2555 |
end \<comment> \<open>FIXME avoid junk stemming from type class interpretation\<close> |
a095acd4cfbf
instantiate multiset with multiset ordering
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63360
diff
changeset
|
2556 |
|
a095acd4cfbf
instantiate multiset with multiset ordering
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63360
diff
changeset
|
2557 |
lemma mset_le_irrefl [elim!]: |
63410
9789ccc2a477
more instantiations for multiset
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63409
diff
changeset
|
2558 |
fixes M :: "'a::preorder multiset" |
63388
a095acd4cfbf
instantiate multiset with multiset ordering
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63360
diff
changeset
|
2559 |
shows "M < M \<Longrightarrow> R" |
46730 | 2560 |
by simp |
26567
7bcebb8c2d33
instantiation replacing primitive instance plus overloaded defs; more conservative type arities
haftmann
parents:
26178
diff
changeset
|
2561 |
|
10249 | 2562 |
|
60500 | 2563 |
subsubsection \<open>Monotonicity of multiset union\<close> |
10249 | 2564 |
|
60606 | 2565 |
lemma mult1_union: "(B, D) \<in> mult1 r \<Longrightarrow> (C + B, C + D) \<in> mult1 r" |
26178 | 2566 |
apply (unfold mult1_def) |
2567 |
apply auto |
|
2568 |
apply (rule_tac x = a in exI) |
|
2569 |
apply (rule_tac x = "C + M0" in exI) |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
2570 |
apply (simp add: add.assoc) |
26178 | 2571 |
done |
10249 | 2572 |
|
63410
9789ccc2a477
more instantiations for multiset
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63409
diff
changeset
|
2573 |
lemma union_le_mono2: "B < D \<Longrightarrow> C + B < C + (D::'a::preorder multiset)" |
26178 | 2574 |
apply (unfold less_multiset_def mult_def) |
2575 |
apply (erule trancl_induct) |
|
40249
cd404ecb9107
Remove unnecessary premise of mult1_union
Lars Noschinski <noschinl@in.tum.de>
parents:
39533
diff
changeset
|
2576 |
apply (blast intro: mult1_union) |
cd404ecb9107
Remove unnecessary premise of mult1_union
Lars Noschinski <noschinl@in.tum.de>
parents:
39533
diff
changeset
|
2577 |
apply (blast intro: mult1_union trancl_trans) |
26178 | 2578 |
done |
10249 | 2579 |
|
63410
9789ccc2a477
more instantiations for multiset
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63409
diff
changeset
|
2580 |
lemma union_le_mono1: "B < D \<Longrightarrow> B + C < D + (C::'a::preorder multiset)" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
2581 |
apply (subst add.commute [of B C]) |
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
2582 |
apply (subst add.commute [of D C]) |
63388
a095acd4cfbf
instantiate multiset with multiset ordering
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63360
diff
changeset
|
2583 |
apply (erule union_le_mono2) |
26178 | 2584 |
done |
10249 | 2585 |
|
17161 | 2586 |
lemma union_less_mono: |
63410
9789ccc2a477
more instantiations for multiset
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63409
diff
changeset
|
2587 |
fixes A B C D :: "'a::preorder multiset" |
63388
a095acd4cfbf
instantiate multiset with multiset ordering
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63360
diff
changeset
|
2588 |
shows "A < C \<Longrightarrow> B < D \<Longrightarrow> A + B < C + D" |
a095acd4cfbf
instantiate multiset with multiset ordering
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63360
diff
changeset
|
2589 |
by (blast intro!: union_le_mono1 union_le_mono2 less_trans) |
a095acd4cfbf
instantiate multiset with multiset ordering
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63360
diff
changeset
|
2590 |
|
63410
9789ccc2a477
more instantiations for multiset
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63409
diff
changeset
|
2591 |
instantiation multiset :: (preorder) ordered_ab_semigroup_add |
63388
a095acd4cfbf
instantiate multiset with multiset ordering
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63360
diff
changeset
|
2592 |
begin |
a095acd4cfbf
instantiate multiset with multiset ordering
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63360
diff
changeset
|
2593 |
instance |
a095acd4cfbf
instantiate multiset with multiset ordering
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63360
diff
changeset
|
2594 |
by standard (auto simp add: less_eq_multiset_def intro: union_le_mono2) |
a095acd4cfbf
instantiate multiset with multiset ordering
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63360
diff
changeset
|
2595 |
end |
15072 | 2596 |
|
63409
3f3223b90239
moved lemmas and locales around (with minor incompatibilities)
blanchet
parents:
63388
diff
changeset
|
2597 |
|
60500 | 2598 |
subsubsection \<open>Termination proofs with multiset orders\<close> |
29125
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2599 |
|
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2600 |
lemma multi_member_skip: "x \<in># XS \<Longrightarrow> x \<in># {# y #} + XS" |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2601 |
and multi_member_this: "x \<in># {# x #} + XS" |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2602 |
and multi_member_last: "x \<in># {# x #}" |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2603 |
by auto |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2604 |
|
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2605 |
definition "ms_strict = mult pair_less" |
37765 | 2606 |
definition "ms_weak = ms_strict \<union> Id" |
29125
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2607 |
|
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2608 |
lemma ms_reduction_pair: "reduction_pair (ms_strict, ms_weak)" |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2609 |
unfolding reduction_pair_def ms_strict_def ms_weak_def pair_less_def |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2610 |
by (auto intro: wf_mult1 wf_trancl simp: mult_def) |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2611 |
|
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2612 |
lemma smsI: |
60495 | 2613 |
"(set_mset A, set_mset B) \<in> max_strict \<Longrightarrow> (Z + A, Z + B) \<in> ms_strict" |
29125
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2614 |
unfolding ms_strict_def |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2615 |
by (rule one_step_implies_mult) (auto simp add: max_strict_def pair_less_def elim!:max_ext.cases) |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2616 |
|
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2617 |
lemma wmsI: |
60495 | 2618 |
"(set_mset A, set_mset B) \<in> max_strict \<or> A = {#} \<and> B = {#} |
29125
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2619 |
\<Longrightarrow> (Z + A, Z + B) \<in> ms_weak" |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2620 |
unfolding ms_weak_def ms_strict_def |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2621 |
by (auto simp add: pair_less_def max_strict_def elim!:max_ext.cases intro: one_step_implies_mult) |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2622 |
|
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2623 |
inductive pw_leq |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2624 |
where |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2625 |
pw_leq_empty: "pw_leq {#} {#}" |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2626 |
| pw_leq_step: "\<lbrakk>(x,y) \<in> pair_leq; pw_leq X Y \<rbrakk> \<Longrightarrow> pw_leq ({#x#} + X) ({#y#} + Y)" |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2627 |
|
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2628 |
lemma pw_leq_lstep: |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2629 |
"(x, y) \<in> pair_leq \<Longrightarrow> pw_leq {#x#} {#y#}" |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2630 |
by (drule pw_leq_step) (rule pw_leq_empty, simp) |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2631 |
|
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2632 |
lemma pw_leq_split: |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2633 |
assumes "pw_leq X Y" |
60495 | 2634 |
shows "\<exists>A B Z. X = A + Z \<and> Y = B + Z \<and> ((set_mset A, set_mset B) \<in> max_strict \<or> (B = {#} \<and> A = {#}))" |
29125
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2635 |
using assms |
60606 | 2636 |
proof induct |
29125
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2637 |
case pw_leq_empty thus ?case by auto |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2638 |
next |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2639 |
case (pw_leq_step x y X Y) |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2640 |
then obtain A B Z where |
58425 | 2641 |
[simp]: "X = A + Z" "Y = B + Z" |
60495 | 2642 |
and 1[simp]: "(set_mset A, set_mset B) \<in> max_strict \<or> (B = {#} \<and> A = {#})" |
29125
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2643 |
by auto |
60606 | 2644 |
from pw_leq_step consider "x = y" | "(x, y) \<in> pair_less" |
29125
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2645 |
unfolding pair_leq_def by auto |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2646 |
thus ?case |
60606 | 2647 |
proof cases |
2648 |
case [simp]: 1 |
|
2649 |
have "{#x#} + X = A + ({#y#}+Z) \<and> {#y#} + Y = B + ({#y#}+Z) \<and> |
|
2650 |
((set_mset A, set_mset B) \<in> max_strict \<or> (B = {#} \<and> A = {#}))" |
|
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
2651 |
by (auto simp: ac_simps) |
60606 | 2652 |
thus ?thesis by blast |
29125
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2653 |
next |
60606 | 2654 |
case 2 |
29125
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2655 |
let ?A' = "{#x#} + A" and ?B' = "{#y#} + B" |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2656 |
have "{#x#} + X = ?A' + Z" |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2657 |
"{#y#} + Y = ?B' + Z" |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
2658 |
by (auto simp add: ac_simps) |
58425 | 2659 |
moreover have |
60495 | 2660 |
"(set_mset ?A', set_mset ?B') \<in> max_strict" |
60606 | 2661 |
using 1 2 unfolding max_strict_def |
29125
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2662 |
by (auto elim!: max_ext.cases) |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2663 |
ultimately show ?thesis by blast |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2664 |
qed |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2665 |
qed |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2666 |
|
58425 | 2667 |
lemma |
29125
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2668 |
assumes pwleq: "pw_leq Z Z'" |
60495 | 2669 |
shows ms_strictI: "(set_mset A, set_mset B) \<in> max_strict \<Longrightarrow> (Z + A, Z' + B) \<in> ms_strict" |
60606 | 2670 |
and ms_weakI1: "(set_mset A, set_mset B) \<in> max_strict \<Longrightarrow> (Z + A, Z' + B) \<in> ms_weak" |
2671 |
and ms_weakI2: "(Z + {#}, Z' + {#}) \<in> ms_weak" |
|
29125
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2672 |
proof - |
58425 | 2673 |
from pw_leq_split[OF pwleq] |
29125
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2674 |
obtain A' B' Z'' |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2675 |
where [simp]: "Z = A' + Z''" "Z' = B' + Z''" |
60495 | 2676 |
and mx_or_empty: "(set_mset A', set_mset B') \<in> max_strict \<or> (A' = {#} \<and> B' = {#})" |
29125
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2677 |
by blast |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2678 |
{ |
60495 | 2679 |
assume max: "(set_mset A, set_mset B) \<in> max_strict" |
29125
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2680 |
from mx_or_empty |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2681 |
have "(Z'' + (A + A'), Z'' + (B + B')) \<in> ms_strict" |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2682 |
proof |
60495 | 2683 |
assume max': "(set_mset A', set_mset B') \<in> max_strict" |
2684 |
with max have "(set_mset (A + A'), set_mset (B + B')) \<in> max_strict" |
|
29125
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2685 |
by (auto simp: max_strict_def intro: max_ext_additive) |
58425 | 2686 |
thus ?thesis by (rule smsI) |
29125
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2687 |
next |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2688 |
assume [simp]: "A' = {#} \<and> B' = {#}" |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2689 |
show ?thesis by (rule smsI) (auto intro: max) |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2690 |
qed |
60606 | 2691 |
thus "(Z + A, Z' + B) \<in> ms_strict" by (simp add: ac_simps) |
29125
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2692 |
thus "(Z + A, Z' + B) \<in> ms_weak" by (simp add: ms_weak_def) |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2693 |
} |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2694 |
from mx_or_empty |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2695 |
have "(Z'' + A', Z'' + B') \<in> ms_weak" by (rule wmsI) |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
2696 |
thus "(Z + {#}, Z' + {#}) \<in> ms_weak" by (simp add:ac_simps) |
29125
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2697 |
qed |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2698 |
|
39301 | 2699 |
lemma empty_neutral: "{#} + x = x" "x + {#} = x" |
29125
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2700 |
and nonempty_plus: "{# x #} + rs \<noteq> {#}" |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2701 |
and nonempty_single: "{# x #} \<noteq> {#}" |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2702 |
by auto |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2703 |
|
60500 | 2704 |
setup \<open> |
60606 | 2705 |
let |
2706 |
fun msetT T = Type (@{type_name multiset}, [T]); |
|
2707 |
||
2708 |
fun mk_mset T [] = Const (@{const_abbrev Mempty}, msetT T) |
|
2709 |
| mk_mset T [x] = Const (@{const_name single}, T --> msetT T) $ x |
|
2710 |
| mk_mset T (x :: xs) = |
|
2711 |
Const (@{const_name plus}, msetT T --> msetT T --> msetT T) $ |
|
2712 |
mk_mset T [x] $ mk_mset T xs |
|
2713 |
||
60752 | 2714 |
fun mset_member_tac ctxt m i = |
60606 | 2715 |
if m <= 0 then |
60752 | 2716 |
resolve_tac ctxt @{thms multi_member_this} i ORELSE |
2717 |
resolve_tac ctxt @{thms multi_member_last} i |
|
60606 | 2718 |
else |
60752 | 2719 |
resolve_tac ctxt @{thms multi_member_skip} i THEN mset_member_tac ctxt (m - 1) i |
2720 |
||
2721 |
fun mset_nonempty_tac ctxt = |
|
2722 |
resolve_tac ctxt @{thms nonempty_plus} ORELSE' |
|
2723 |
resolve_tac ctxt @{thms nonempty_single} |
|
29125
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
2724 |
|
60606 | 2725 |
fun regroup_munion_conv ctxt = |
2726 |
Function_Lib.regroup_conv ctxt @{const_abbrev Mempty} @{const_name plus} |
|
2727 |
(map (fn t => t RS eq_reflection) (@{thms ac_simps} @ @{thms empty_neutral})) |
|
2728 |
||
60752 | 2729 |
fun unfold_pwleq_tac ctxt i = |
2730 |
(resolve_tac ctxt @{thms pw_leq_step} i THEN (fn st => unfold_pwleq_tac ctxt (i + 1) st)) |
|
2731 |
ORELSE (resolve_tac ctxt @{thms pw_leq_lstep} i) |
|
2732 |
ORELSE (resolve_tac ctxt @{thms pw_leq_empty} i) |
|
60606 | 2733 |
|
2734 |
val set_mset_simps = [@{thm set_mset_empty}, @{thm set_mset_single}, @{thm set_mset_union}, |
|
2735 |
@{thm Un_insert_left}, @{thm Un_empty_left}] |
|
2736 |
in |
|
2737 |
ScnpReconstruct.multiset_setup (ScnpReconstruct.Multiset |
|
2738 |
{ |
|
2739 |
msetT=msetT, mk_mset=mk_mset, mset_regroup_conv=regroup_munion_conv, |
|
2740 |
mset_member_tac=mset_member_tac, mset_nonempty_tac=mset_nonempty_tac, |
|
2741 |
mset_pwleq_tac=unfold_pwleq_tac, set_of_simps=set_mset_simps, |
|
2742 |
smsI'= @{thm ms_strictI}, wmsI2''= @{thm ms_weakI2}, wmsI1= @{thm ms_weakI1}, |
|
60752 | 2743 |
reduction_pair = @{thm ms_reduction_pair} |
60606 | 2744 |
}) |
2745 |
end |
|
60500 | 2746 |
\<close> |
2747 |
||
2748 |
||
2749 |
subsection \<open>Legacy theorem bindings\<close> |
|
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
2750 |
|
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39301
diff
changeset
|
2751 |
lemmas multi_count_eq = multiset_eq_iff [symmetric] |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
2752 |
|
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
2753 |
lemma union_commute: "M + N = N + (M::'a multiset)" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
2754 |
by (fact add.commute) |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
2755 |
|
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
2756 |
lemma union_assoc: "(M + N) + K = M + (N + (K::'a multiset))" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
2757 |
by (fact add.assoc) |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
2758 |
|
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
2759 |
lemma union_lcomm: "M + (N + K) = N + (M + (K::'a multiset))" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
2760 |
by (fact add.left_commute) |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
2761 |
|
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
2762 |
lemmas union_ac = union_assoc union_commute union_lcomm |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
2763 |
|
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
2764 |
lemma union_right_cancel: "M + K = N + K \<longleftrightarrow> M = (N::'a multiset)" |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
2765 |
by (fact add_right_cancel) |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
2766 |
|
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
2767 |
lemma union_left_cancel: "K + M = K + N \<longleftrightarrow> M = (N::'a multiset)" |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
2768 |
by (fact add_left_cancel) |
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
2769 |
|
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
2770 |
lemma multi_union_self_other_eq: "(A::'a multiset) + X = A + Y \<Longrightarrow> X = Y" |
59557 | 2771 |
by (fact add_left_imp_eq) |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
2772 |
|
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
2773 |
lemma mset_subset_trans: "(M::'a multiset) \<subset># K \<Longrightarrow> K \<subset># N \<Longrightarrow> M \<subset># N" |
60397
f8a513fedb31
Renaming multiset operators < ~> <#,...
Mathias Fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
59999
diff
changeset
|
2774 |
by (fact subset_mset.less_trans) |
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
2775 |
|
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
2776 |
lemma multiset_inter_commute: "A #\<inter> B = B #\<inter> A" |
60397
f8a513fedb31
Renaming multiset operators < ~> <#,...
Mathias Fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
59999
diff
changeset
|
2777 |
by (fact subset_mset.inf.commute) |
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
2778 |
|
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
2779 |
lemma multiset_inter_assoc: "A #\<inter> (B #\<inter> C) = A #\<inter> B #\<inter> C" |
60397
f8a513fedb31
Renaming multiset operators < ~> <#,...
Mathias Fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
59999
diff
changeset
|
2780 |
by (fact subset_mset.inf.assoc [symmetric]) |
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
2781 |
|
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
2782 |
lemma multiset_inter_left_commute: "A #\<inter> (B #\<inter> C) = B #\<inter> (A #\<inter> C)" |
60397
f8a513fedb31
Renaming multiset operators < ~> <#,...
Mathias Fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
59999
diff
changeset
|
2783 |
by (fact subset_mset.inf.left_commute) |
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
2784 |
|
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
2785 |
lemmas multiset_inter_ac = |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
2786 |
multiset_inter_commute |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
2787 |
multiset_inter_assoc |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
2788 |
multiset_inter_left_commute |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
2789 |
|
63410
9789ccc2a477
more instantiations for multiset
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63409
diff
changeset
|
2790 |
lemma mset_le_not_refl: "\<not> M < (M::'a::preorder multiset)" |
63388
a095acd4cfbf
instantiate multiset with multiset ordering
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63360
diff
changeset
|
2791 |
by (fact less_irrefl) |
a095acd4cfbf
instantiate multiset with multiset ordering
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63360
diff
changeset
|
2792 |
|
63410
9789ccc2a477
more instantiations for multiset
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63409
diff
changeset
|
2793 |
lemma mset_le_trans: "K < M \<Longrightarrow> M < N \<Longrightarrow> K < (N::'a::preorder multiset)" |
63388
a095acd4cfbf
instantiate multiset with multiset ordering
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63360
diff
changeset
|
2794 |
by (fact less_trans) |
a095acd4cfbf
instantiate multiset with multiset ordering
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63360
diff
changeset
|
2795 |
|
63410
9789ccc2a477
more instantiations for multiset
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63409
diff
changeset
|
2796 |
lemma mset_le_not_sym: "M < N \<Longrightarrow> \<not> N < (M::'a::preorder multiset)" |
63388
a095acd4cfbf
instantiate multiset with multiset ordering
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63360
diff
changeset
|
2797 |
by (fact less_not_sym) |
a095acd4cfbf
instantiate multiset with multiset ordering
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63360
diff
changeset
|
2798 |
|
63410
9789ccc2a477
more instantiations for multiset
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63409
diff
changeset
|
2799 |
lemma mset_le_asym: "M < N \<Longrightarrow> (\<not> P \<Longrightarrow> N < (M::'a::preorder multiset)) \<Longrightarrow> P" |
63388
a095acd4cfbf
instantiate multiset with multiset ordering
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63360
diff
changeset
|
2800 |
by (fact less_asym) |
34943
e97b22500a5c
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents:
33102
diff
changeset
|
2801 |
|
60500 | 2802 |
declaration \<open> |
60606 | 2803 |
let |
2804 |
fun multiset_postproc _ maybe_name all_values (T as Type (_, [elem_T])) (Const _ $ t') = |
|
2805 |
let |
|
2806 |
val (maybe_opt, ps) = |
|
2807 |
Nitpick_Model.dest_plain_fun t' |
|
2808 |
||> op ~~ |
|
2809 |
||> map (apsnd (snd o HOLogic.dest_number)) |
|
2810 |
fun elems_for t = |
|
2811 |
(case AList.lookup (op =) ps t of |
|
2812 |
SOME n => replicate n t |
|
2813 |
| NONE => [Const (maybe_name, elem_T --> elem_T) $ t]) |
|
2814 |
in |
|
2815 |
(case maps elems_for (all_values elem_T) @ |
|
61333 | 2816 |
(if maybe_opt then [Const (Nitpick_Model.unrep_mixfix (), elem_T)] else []) of |
60606 | 2817 |
[] => Const (@{const_name zero_class.zero}, T) |
2818 |
| ts => |
|
2819 |
foldl1 (fn (t1, t2) => |
|
2820 |
Const (@{const_name plus_class.plus}, T --> T --> T) $ t1 $ t2) |
|
2821 |
(map (curry (op $) (Const (@{const_name single}, elem_T --> T))) ts)) |
|
2822 |
end |
|
2823 |
| multiset_postproc _ _ _ _ t = t |
|
2824 |
in Nitpick_Model.register_term_postprocessor @{typ "'a multiset"} multiset_postproc end |
|
60500 | 2825 |
\<close> |
2826 |
||
2827 |
||
2828 |
subsection \<open>Naive implementation using lists\<close> |
|
51600
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2829 |
|
60515 | 2830 |
code_datatype mset |
51600
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2831 |
|
60606 | 2832 |
lemma [code]: "{#} = mset []" |
51600
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2833 |
by simp |
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2834 |
|
63195 | 2835 |
lemma [code]: "Multiset.is_empty (mset xs) \<longleftrightarrow> List.null xs" |
2836 |
by (simp add: Multiset.is_empty_def List.null_def) |
|
2837 |
||
60606 | 2838 |
lemma [code]: "{#x#} = mset [x]" |
51600
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2839 |
by simp |
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2840 |
|
60606 | 2841 |
lemma union_code [code]: "mset xs + mset ys = mset (xs @ ys)" |
51600
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2842 |
by simp |
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2843 |
|
60606 | 2844 |
lemma [code]: "image_mset f (mset xs) = mset (map f xs)" |
60515 | 2845 |
by (simp add: mset_map) |
51600
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2846 |
|
60606 | 2847 |
lemma [code]: "filter_mset f (mset xs) = mset (filter f xs)" |
60515 | 2848 |
by (simp add: mset_filter) |
51600
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2849 |
|
60606 | 2850 |
lemma [code]: "mset xs - mset ys = mset (fold remove1 ys xs)" |
51600
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2851 |
by (rule sym, induct ys arbitrary: xs) (simp_all add: diff_add diff_right_commute) |
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2852 |
|
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2853 |
lemma [code]: |
60515 | 2854 |
"mset xs #\<inter> mset ys = |
2855 |
mset (snd (fold (\<lambda>x (ys, zs). |
|
51600
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2856 |
if x \<in> set ys then (remove1 x ys, x # zs) else (ys, zs)) xs (ys, [])))" |
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2857 |
proof - |
60515 | 2858 |
have "\<And>zs. mset (snd (fold (\<lambda>x (ys, zs). |
51600
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2859 |
if x \<in> set ys then (remove1 x ys, x # zs) else (ys, zs)) xs (ys, zs))) = |
60515 | 2860 |
(mset xs #\<inter> mset ys) + mset zs" |
51623 | 2861 |
by (induct xs arbitrary: ys) |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
2862 |
(auto simp add: inter_add_right1 inter_add_right2 ac_simps) |
51623 | 2863 |
then show ?thesis by simp |
2864 |
qed |
|
2865 |
||
2866 |
lemma [code]: |
|
60515 | 2867 |
"mset xs #\<union> mset ys = |
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61378
diff
changeset
|
2868 |
mset (case_prod append (fold (\<lambda>x (ys, zs). (remove1 x ys, x # zs)) xs (ys, [])))" |
51623 | 2869 |
proof - |
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61378
diff
changeset
|
2870 |
have "\<And>zs. mset (case_prod append (fold (\<lambda>x (ys, zs). (remove1 x ys, x # zs)) xs (ys, zs))) = |
60515 | 2871 |
(mset xs #\<union> mset ys) + mset zs" |
51623 | 2872 |
by (induct xs arbitrary: ys) (simp_all add: multiset_eq_iff) |
51600
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2873 |
then show ?thesis by simp |
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2874 |
qed |
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2875 |
|
59813 | 2876 |
declare in_multiset_in_set [code_unfold] |
51600
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2877 |
|
60606 | 2878 |
lemma [code]: "count (mset xs) x = fold (\<lambda>y. if x = y then Suc else id) xs 0" |
51600
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2879 |
proof - |
60515 | 2880 |
have "\<And>n. fold (\<lambda>y. if x = y then Suc else id) xs n = count (mset xs) x + n" |
51600
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2881 |
by (induct xs) simp_all |
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2882 |
then show ?thesis by simp |
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2883 |
qed |
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2884 |
|
60515 | 2885 |
declare set_mset_mset [code] |
2886 |
||
2887 |
declare sorted_list_of_multiset_mset [code] |
|
51600
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2888 |
|
61585 | 2889 |
lemma [code]: \<comment> \<open>not very efficient, but representation-ignorant!\<close> |
60515 | 2890 |
"mset_set A = mset (sorted_list_of_set A)" |
51600
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2891 |
apply (cases "finite A") |
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2892 |
apply simp_all |
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2893 |
apply (induct A rule: finite_induct) |
59813 | 2894 |
apply (simp_all add: add.commute) |
51600
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2895 |
done |
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2896 |
|
60515 | 2897 |
declare size_mset [code] |
51600
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2898 |
|
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
2899 |
fun subset_eq_mset_impl :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool option" where |
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
2900 |
"subset_eq_mset_impl [] ys = Some (ys \<noteq> [])" |
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
2901 |
| "subset_eq_mset_impl (Cons x xs) ys = (case List.extract (op = x) ys of |
55808
488c3e8282c8
added Rene Thiemann's patch for the nonterminating equality/subset test code for multisets
nipkow
parents:
55565
diff
changeset
|
2902 |
None \<Rightarrow> None |
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
2903 |
| Some (ys1,_,ys2) \<Rightarrow> subset_eq_mset_impl xs (ys1 @ ys2))" |
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
2904 |
|
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
2905 |
lemma subset_eq_mset_impl: "(subset_eq_mset_impl xs ys = None \<longleftrightarrow> \<not> mset xs \<subseteq># mset ys) \<and> |
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
2906 |
(subset_eq_mset_impl xs ys = Some True \<longleftrightarrow> mset xs \<subset># mset ys) \<and> |
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
2907 |
(subset_eq_mset_impl xs ys = Some False \<longrightarrow> mset xs = mset ys)" |
55808
488c3e8282c8
added Rene Thiemann's patch for the nonterminating equality/subset test code for multisets
nipkow
parents:
55565
diff
changeset
|
2908 |
proof (induct xs arbitrary: ys) |
488c3e8282c8
added Rene Thiemann's patch for the nonterminating equality/subset test code for multisets
nipkow
parents:
55565
diff
changeset
|
2909 |
case (Nil ys) |
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
2910 |
show ?case by (auto simp: mset_subset_empty_nonempty) |
55808
488c3e8282c8
added Rene Thiemann's patch for the nonterminating equality/subset test code for multisets
nipkow
parents:
55565
diff
changeset
|
2911 |
next |
488c3e8282c8
added Rene Thiemann's patch for the nonterminating equality/subset test code for multisets
nipkow
parents:
55565
diff
changeset
|
2912 |
case (Cons x xs ys) |
488c3e8282c8
added Rene Thiemann's patch for the nonterminating equality/subset test code for multisets
nipkow
parents:
55565
diff
changeset
|
2913 |
show ?case |
488c3e8282c8
added Rene Thiemann's patch for the nonterminating equality/subset test code for multisets
nipkow
parents:
55565
diff
changeset
|
2914 |
proof (cases "List.extract (op = x) ys") |
488c3e8282c8
added Rene Thiemann's patch for the nonterminating equality/subset test code for multisets
nipkow
parents:
55565
diff
changeset
|
2915 |
case None |
488c3e8282c8
added Rene Thiemann's patch for the nonterminating equality/subset test code for multisets
nipkow
parents:
55565
diff
changeset
|
2916 |
hence x: "x \<notin> set ys" by (simp add: extract_None_iff) |
488c3e8282c8
added Rene Thiemann's patch for the nonterminating equality/subset test code for multisets
nipkow
parents:
55565
diff
changeset
|
2917 |
{ |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
2918 |
assume "mset (x # xs) \<subseteq># mset ys" |
60495 | 2919 |
from set_mset_mono[OF this] x have False by simp |
55808
488c3e8282c8
added Rene Thiemann's patch for the nonterminating equality/subset test code for multisets
nipkow
parents:
55565
diff
changeset
|
2920 |
} note nle = this |
488c3e8282c8
added Rene Thiemann's patch for the nonterminating equality/subset test code for multisets
nipkow
parents:
55565
diff
changeset
|
2921 |
moreover |
488c3e8282c8
added Rene Thiemann's patch for the nonterminating equality/subset test code for multisets
nipkow
parents:
55565
diff
changeset
|
2922 |
{ |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
2923 |
assume "mset (x # xs) \<subset># mset ys" |
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
2924 |
hence "mset (x # xs) \<subseteq># mset ys" by auto |
55808
488c3e8282c8
added Rene Thiemann's patch for the nonterminating equality/subset test code for multisets
nipkow
parents:
55565
diff
changeset
|
2925 |
from nle[OF this] have False . |
488c3e8282c8
added Rene Thiemann's patch for the nonterminating equality/subset test code for multisets
nipkow
parents:
55565
diff
changeset
|
2926 |
} |
488c3e8282c8
added Rene Thiemann's patch for the nonterminating equality/subset test code for multisets
nipkow
parents:
55565
diff
changeset
|
2927 |
ultimately show ?thesis using None by auto |
488c3e8282c8
added Rene Thiemann's patch for the nonterminating equality/subset test code for multisets
nipkow
parents:
55565
diff
changeset
|
2928 |
next |
488c3e8282c8
added Rene Thiemann's patch for the nonterminating equality/subset test code for multisets
nipkow
parents:
55565
diff
changeset
|
2929 |
case (Some res) |
488c3e8282c8
added Rene Thiemann's patch for the nonterminating equality/subset test code for multisets
nipkow
parents:
55565
diff
changeset
|
2930 |
obtain ys1 y ys2 where res: "res = (ys1,y,ys2)" by (cases res, auto) |
488c3e8282c8
added Rene Thiemann's patch for the nonterminating equality/subset test code for multisets
nipkow
parents:
55565
diff
changeset
|
2931 |
note Some = Some[unfolded res] |
488c3e8282c8
added Rene Thiemann's patch for the nonterminating equality/subset test code for multisets
nipkow
parents:
55565
diff
changeset
|
2932 |
from extract_SomeE[OF Some] have "ys = ys1 @ x # ys2" by simp |
60515 | 2933 |
hence id: "mset ys = mset (ys1 @ ys2) + {#x#}" |
55808
488c3e8282c8
added Rene Thiemann's patch for the nonterminating equality/subset test code for multisets
nipkow
parents:
55565
diff
changeset
|
2934 |
by (auto simp: ac_simps) |
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
2935 |
show ?thesis unfolding subset_eq_mset_impl.simps |
55808
488c3e8282c8
added Rene Thiemann's patch for the nonterminating equality/subset test code for multisets
nipkow
parents:
55565
diff
changeset
|
2936 |
unfolding Some option.simps split |
488c3e8282c8
added Rene Thiemann's patch for the nonterminating equality/subset test code for multisets
nipkow
parents:
55565
diff
changeset
|
2937 |
unfolding id |
488c3e8282c8
added Rene Thiemann's patch for the nonterminating equality/subset test code for multisets
nipkow
parents:
55565
diff
changeset
|
2938 |
using Cons[of "ys1 @ ys2"] |
60397
f8a513fedb31
Renaming multiset operators < ~> <#,...
Mathias Fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
59999
diff
changeset
|
2939 |
unfolding subset_mset_def subseteq_mset_def by auto |
55808
488c3e8282c8
added Rene Thiemann's patch for the nonterminating equality/subset test code for multisets
nipkow
parents:
55565
diff
changeset
|
2940 |
qed |
488c3e8282c8
added Rene Thiemann's patch for the nonterminating equality/subset test code for multisets
nipkow
parents:
55565
diff
changeset
|
2941 |
qed |
488c3e8282c8
added Rene Thiemann's patch for the nonterminating equality/subset test code for multisets
nipkow
parents:
55565
diff
changeset
|
2942 |
|
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
2943 |
lemma [code]: "mset xs \<subseteq># mset ys \<longleftrightarrow> subset_eq_mset_impl xs ys \<noteq> None" |
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
2944 |
using subset_eq_mset_impl[of xs ys] by (cases "subset_eq_mset_impl xs ys", auto) |
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
2945 |
|
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
2946 |
lemma [code]: "mset xs \<subset># mset ys \<longleftrightarrow> subset_eq_mset_impl xs ys = Some True" |
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
2947 |
using subset_eq_mset_impl[of xs ys] by (cases "subset_eq_mset_impl xs ys", auto) |
51600
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2948 |
|
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2949 |
instantiation multiset :: (equal) equal |
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2950 |
begin |
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2951 |
|
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2952 |
definition |
55808
488c3e8282c8
added Rene Thiemann's patch for the nonterminating equality/subset test code for multisets
nipkow
parents:
55565
diff
changeset
|
2953 |
[code del]: "HOL.equal A (B :: 'a multiset) \<longleftrightarrow> A = B" |
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
2954 |
lemma [code]: "HOL.equal (mset xs) (mset ys) \<longleftrightarrow> subset_eq_mset_impl xs ys = Some False" |
55808
488c3e8282c8
added Rene Thiemann's patch for the nonterminating equality/subset test code for multisets
nipkow
parents:
55565
diff
changeset
|
2955 |
unfolding equal_multiset_def |
63310
caaacf37943f
normalising multiset theorem names
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63290
diff
changeset
|
2956 |
using subset_eq_mset_impl[of xs ys] by (cases "subset_eq_mset_impl xs ys", auto) |
51600
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2957 |
|
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2958 |
instance |
60678 | 2959 |
by standard (simp add: equal_multiset_def) |
2960 |
||
37169
f69efa106feb
make Nitpick "show_all" option behave less surprisingly
blanchet
parents:
37107
diff
changeset
|
2961 |
end |
49388 | 2962 |
|
60606 | 2963 |
lemma [code]: "msetsum (mset xs) = listsum xs" |
51600
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2964 |
by (induct xs) (simp_all add: add.commute) |
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2965 |
|
60606 | 2966 |
lemma [code]: "msetprod (mset xs) = fold times xs 1" |
51600
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2967 |
proof - |
60515 | 2968 |
have "\<And>x. fold times xs x = msetprod (mset xs) * x" |
51600
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2969 |
by (induct xs) (simp_all add: mult.assoc) |
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2970 |
then show ?thesis by simp |
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2971 |
qed |
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2972 |
|
60500 | 2973 |
text \<open> |
63388
a095acd4cfbf
instantiate multiset with multiset ordering
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63360
diff
changeset
|
2974 |
Exercise for the casual reader: add implementations for @{term "op \<le>"} |
a095acd4cfbf
instantiate multiset with multiset ordering
fleury <Mathias.Fleury@mpi-inf.mpg.de>
parents:
63360
diff
changeset
|
2975 |
and @{term "op <"} (multiset order). |
60500 | 2976 |
\<close> |
2977 |
||
2978 |
text \<open>Quickcheck generators\<close> |
|
51600
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2979 |
|
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2980 |
definition (in term_syntax) |
61076 | 2981 |
msetify :: "'a::typerep list \<times> (unit \<Rightarrow> Code_Evaluation.term) |
51600
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2982 |
\<Rightarrow> 'a multiset \<times> (unit \<Rightarrow> Code_Evaluation.term)" where |
60515 | 2983 |
[code_unfold]: "msetify xs = Code_Evaluation.valtermify mset {\<cdot>} xs" |
51600
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2984 |
|
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2985 |
notation fcomp (infixl "\<circ>>" 60) |
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2986 |
notation scomp (infixl "\<circ>\<rightarrow>" 60) |
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2987 |
|
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2988 |
instantiation multiset :: (random) random |
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2989 |
begin |
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2990 |
|
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2991 |
definition |
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2992 |
"Quickcheck_Random.random i = Quickcheck_Random.random i \<circ>\<rightarrow> (\<lambda>xs. Pair (msetify xs))" |
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2993 |
|
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2994 |
instance .. |
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2995 |
|
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2996 |
end |
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2997 |
|
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2998 |
no_notation fcomp (infixl "\<circ>>" 60) |
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
2999 |
no_notation scomp (infixl "\<circ>\<rightarrow>" 60) |
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
3000 |
|
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
3001 |
instantiation multiset :: (full_exhaustive) full_exhaustive |
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
3002 |
begin |
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
3003 |
|
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
3004 |
definition full_exhaustive_multiset :: "('a multiset \<times> (unit \<Rightarrow> term) \<Rightarrow> (bool \<times> term list) option) \<Rightarrow> natural \<Rightarrow> (bool \<times> term list) option" |
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
3005 |
where |
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
3006 |
"full_exhaustive_multiset f i = Quickcheck_Exhaustive.full_exhaustive (\<lambda>xs. f (msetify xs)) i" |
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
3007 |
|
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
3008 |
instance .. |
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
3009 |
|
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
3010 |
end |
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
3011 |
|
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
3012 |
hide_const (open) msetify |
197e25f13f0c
default implementation of multisets by list with reasonable coverage of operations on multisets
haftmann
parents:
51599
diff
changeset
|
3013 |
|
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3014 |
|
60500 | 3015 |
subsection \<open>BNF setup\<close> |
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3016 |
|
57966 | 3017 |
definition rel_mset where |
60515 | 3018 |
"rel_mset R X Y \<longleftrightarrow> (\<exists>xs ys. mset xs = X \<and> mset ys = Y \<and> list_all2 R xs ys)" |
3019 |
||
3020 |
lemma mset_zip_take_Cons_drop_twice: |
|
57966 | 3021 |
assumes "length xs = length ys" "j \<le> length xs" |
60515 | 3022 |
shows "mset (zip (take j xs @ x # drop j xs) (take j ys @ y # drop j ys)) = |
3023 |
mset (zip xs ys) + {#(x, y)#}" |
|
60606 | 3024 |
using assms |
57966 | 3025 |
proof (induct xs ys arbitrary: x y j rule: list_induct2) |
3026 |
case Nil |
|
3027 |
thus ?case |
|
3028 |
by simp |
|
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3029 |
next |
57966 | 3030 |
case (Cons x xs y ys) |
3031 |
thus ?case |
|
3032 |
proof (cases "j = 0") |
|
3033 |
case True |
|
3034 |
thus ?thesis |
|
3035 |
by simp |
|
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3036 |
next |
57966 | 3037 |
case False |
3038 |
then obtain k where k: "j = Suc k" |
|
60678 | 3039 |
by (cases j) simp |
57966 | 3040 |
hence "k \<le> length xs" |
3041 |
using Cons.prems by auto |
|
60515 | 3042 |
hence "mset (zip (take k xs @ x # drop k xs) (take k ys @ y # drop k ys)) = |
3043 |
mset (zip xs ys) + {#(x, y)#}" |
|
57966 | 3044 |
by (rule Cons.hyps(2)) |
3045 |
thus ?thesis |
|
3046 |
unfolding k by (auto simp: add.commute union_lcomm) |
|
58425 | 3047 |
qed |
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3048 |
qed |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3049 |
|
60515 | 3050 |
lemma ex_mset_zip_left: |
3051 |
assumes "length xs = length ys" "mset xs' = mset xs" |
|
3052 |
shows "\<exists>ys'. length ys' = length xs' \<and> mset (zip xs' ys') = mset (zip xs ys)" |
|
58425 | 3053 |
using assms |
57966 | 3054 |
proof (induct xs ys arbitrary: xs' rule: list_induct2) |
3055 |
case Nil |
|
3056 |
thus ?case |
|
3057 |
by auto |
|
3058 |
next |
|
3059 |
case (Cons x xs y ys xs') |
|
3060 |
obtain j where j_len: "j < length xs'" and nth_j: "xs' ! j = x" |
|
60515 | 3061 |
by (metis Cons.prems in_set_conv_nth list.set_intros(1) mset_eq_setD) |
58425 | 3062 |
|
63040 | 3063 |
define xsa where "xsa = take j xs' @ drop (Suc j) xs'" |
60515 | 3064 |
have "mset xs' = {#x#} + mset xsa" |
57966 | 3065 |
unfolding xsa_def using j_len nth_j |
58247
98d0f85d247f
enamed drop_Suc_conv_tl and nth_drop' to Cons_nth_drop_Suc
nipkow
parents:
58098
diff
changeset
|
3066 |
by (metis (no_types) ab_semigroup_add_class.add_ac(1) append_take_drop_id Cons_nth_drop_Suc |
60515 | 3067 |
mset.simps(2) union_code add.commute) |
3068 |
hence ms_x: "mset xsa = mset xs" |
|
3069 |
by (metis Cons.prems add.commute add_right_imp_eq mset.simps(2)) |
|
57966 | 3070 |
then obtain ysa where |
60515 | 3071 |
len_a: "length ysa = length xsa" and ms_a: "mset (zip xsa ysa) = mset (zip xs ys)" |
57966 | 3072 |
using Cons.hyps(2) by blast |
3073 |
||
63040 | 3074 |
define ys' where "ys' = take j ysa @ y # drop j ysa" |
57966 | 3075 |
have xs': "xs' = take j xsa @ x # drop j xsa" |
3076 |
using ms_x j_len nth_j Cons.prems xsa_def |
|
58247
98d0f85d247f
enamed drop_Suc_conv_tl and nth_drop' to Cons_nth_drop_Suc
nipkow
parents:
58098
diff
changeset
|
3077 |
by (metis append_eq_append_conv append_take_drop_id diff_Suc_Suc Cons_nth_drop_Suc length_Cons |
60515 | 3078 |
length_drop size_mset) |
57966 | 3079 |
have j_len': "j \<le> length xsa" |
3080 |
using j_len xs' xsa_def |
|
3081 |
by (metis add_Suc_right append_take_drop_id length_Cons length_append less_eq_Suc_le not_less) |
|
3082 |
have "length ys' = length xs'" |
|
3083 |
unfolding ys'_def using Cons.prems len_a ms_x |
|
60515 | 3084 |
by (metis add_Suc_right append_take_drop_id length_Cons length_append mset_eq_length) |
3085 |
moreover have "mset (zip xs' ys') = mset (zip (x # xs) (y # ys))" |
|
57966 | 3086 |
unfolding xs' ys'_def |
60515 | 3087 |
by (rule trans[OF mset_zip_take_Cons_drop_twice]) |
57966 | 3088 |
(auto simp: len_a ms_a j_len' add.commute) |
3089 |
ultimately show ?case |
|
3090 |
by blast |
|
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3091 |
qed |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3092 |
|
57966 | 3093 |
lemma list_all2_reorder_left_invariance: |
60515 | 3094 |
assumes rel: "list_all2 R xs ys" and ms_x: "mset xs' = mset xs" |
3095 |
shows "\<exists>ys'. list_all2 R xs' ys' \<and> mset ys' = mset ys" |
|
57966 | 3096 |
proof - |
3097 |
have len: "length xs = length ys" |
|
3098 |
using rel list_all2_conv_all_nth by auto |
|
3099 |
obtain ys' where |
|
60515 | 3100 |
len': "length xs' = length ys'" and ms_xy: "mset (zip xs' ys') = mset (zip xs ys)" |
3101 |
using len ms_x by (metis ex_mset_zip_left) |
|
57966 | 3102 |
have "list_all2 R xs' ys'" |
60515 | 3103 |
using assms(1) len' ms_xy unfolding list_all2_iff by (blast dest: mset_eq_setD) |
3104 |
moreover have "mset ys' = mset ys" |
|
3105 |
using len len' ms_xy map_snd_zip mset_map by metis |
|
57966 | 3106 |
ultimately show ?thesis |
3107 |
by blast |
|
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3108 |
qed |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3109 |
|
60515 | 3110 |
lemma ex_mset: "\<exists>xs. mset xs = X" |
3111 |
by (induct X) (simp, metis mset.simps(2)) |
|
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3112 |
|
62324 | 3113 |
inductive pred_mset :: "('a \<Rightarrow> bool) \<Rightarrow> 'a multiset \<Rightarrow> bool" |
3114 |
where |
|
3115 |
"pred_mset P {#}" |
|
3116 |
| "\<lbrakk>P a; pred_mset P M\<rbrakk> \<Longrightarrow> pred_mset P (M + {#a#})" |
|
3117 |
||
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3118 |
bnf "'a multiset" |
57966 | 3119 |
map: image_mset |
60495 | 3120 |
sets: set_mset |
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3121 |
bd: natLeq |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3122 |
wits: "{#}" |
57966 | 3123 |
rel: rel_mset |
62324 | 3124 |
pred: pred_mset |
57966 | 3125 |
proof - |
3126 |
show "image_mset id = id" |
|
3127 |
by (rule image_mset.id) |
|
60606 | 3128 |
show "image_mset (g \<circ> f) = image_mset g \<circ> image_mset f" for f g |
59813 | 3129 |
unfolding comp_def by (rule ext) (simp add: comp_def image_mset.compositionality) |
60606 | 3130 |
show "(\<And>z. z \<in> set_mset X \<Longrightarrow> f z = g z) \<Longrightarrow> image_mset f X = image_mset g X" for f g X |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
3131 |
by (induct X) simp_all |
60606 | 3132 |
show "set_mset \<circ> image_mset f = op ` f \<circ> set_mset" for f |
57966 | 3133 |
by auto |
3134 |
show "card_order natLeq" |
|
3135 |
by (rule natLeq_card_order) |
|
3136 |
show "BNF_Cardinal_Arithmetic.cinfinite natLeq" |
|
3137 |
by (rule natLeq_cinfinite) |
|
60606 | 3138 |
show "ordLeq3 (card_of (set_mset X)) natLeq" for X |
57966 | 3139 |
by transfer |
3140 |
(auto intro!: ordLess_imp_ordLeq simp: finite_iff_ordLess_natLeq[symmetric] multiset_def) |
|
60606 | 3141 |
show "rel_mset R OO rel_mset S \<le> rel_mset (R OO S)" for R S |
57966 | 3142 |
unfolding rel_mset_def[abs_def] OO_def |
3143 |
apply clarify |
|
60678 | 3144 |
subgoal for X Z Y xs ys' ys zs |
3145 |
apply (drule list_all2_reorder_left_invariance [where xs = ys' and ys = zs and xs' = ys]) |
|
3146 |
apply (auto intro: list_all2_trans) |
|
3147 |
done |
|
60606 | 3148 |
done |
3149 |
show "rel_mset R = |
|
62324 | 3150 |
(\<lambda>x y. \<exists>z. set_mset z \<subseteq> {(x, y). R x y} \<and> |
3151 |
image_mset fst z = x \<and> image_mset snd z = y)" for R |
|
3152 |
unfolding rel_mset_def[abs_def] |
|
57966 | 3153 |
apply (rule ext)+ |
62324 | 3154 |
apply safe |
3155 |
apply (rule_tac x = "mset (zip xs ys)" in exI; |
|
3156 |
auto simp: in_set_zip list_all2_iff mset_map[symmetric]) |
|
57966 | 3157 |
apply (rename_tac XY) |
60515 | 3158 |
apply (cut_tac X = XY in ex_mset) |
57966 | 3159 |
apply (erule exE) |
3160 |
apply (rename_tac xys) |
|
3161 |
apply (rule_tac x = "map fst xys" in exI) |
|
60515 | 3162 |
apply (auto simp: mset_map) |
57966 | 3163 |
apply (rule_tac x = "map snd xys" in exI) |
60515 | 3164 |
apply (auto simp: mset_map list_all2I subset_eq zip_map_fst_snd) |
59997 | 3165 |
done |
60606 | 3166 |
show "z \<in> set_mset {#} \<Longrightarrow> False" for z |
57966 | 3167 |
by auto |
62324 | 3168 |
show "pred_mset P = (\<lambda>x. Ball (set_mset x) P)" for P |
3169 |
proof (intro ext iffI) |
|
3170 |
fix x |
|
3171 |
assume "pred_mset P x" |
|
3172 |
then show "Ball (set_mset x) P" by (induct pred: pred_mset; simp) |
|
3173 |
next |
|
3174 |
fix x |
|
3175 |
assume "Ball (set_mset x) P" |
|
3176 |
then show "pred_mset P x" by (induct x; auto intro: pred_mset.intros) |
|
3177 |
qed |
|
57966 | 3178 |
qed |
3179 |
||
60606 | 3180 |
inductive rel_mset' |
3181 |
where |
|
57966 | 3182 |
Zero[intro]: "rel_mset' R {#} {#}" |
3183 |
| Plus[intro]: "\<lbrakk>R a b; rel_mset' R M N\<rbrakk> \<Longrightarrow> rel_mset' R (M + {#a#}) (N + {#b#})" |
|
3184 |
||
3185 |
lemma rel_mset_Zero: "rel_mset R {#} {#}" |
|
3186 |
unfolding rel_mset_def Grp_def by auto |
|
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3187 |
|
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3188 |
declare multiset.count[simp] |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3189 |
declare Abs_multiset_inverse[simp] |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3190 |
declare multiset.count_inverse[simp] |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3191 |
declare union_preserves_multiset[simp] |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3192 |
|
57966 | 3193 |
lemma rel_mset_Plus: |
60606 | 3194 |
assumes ab: "R a b" |
3195 |
and MN: "rel_mset R M N" |
|
3196 |
shows "rel_mset R (M + {#a#}) (N + {#b#})" |
|
3197 |
proof - |
|
3198 |
have "\<exists>ya. image_mset fst y + {#a#} = image_mset fst ya \<and> |
|
3199 |
image_mset snd y + {#b#} = image_mset snd ya \<and> |
|
3200 |
set_mset ya \<subseteq> {(x, y). R x y}" |
|
3201 |
if "R a b" and "set_mset y \<subseteq> {(x, y). R x y}" for y |
|
3202 |
using that by (intro exI[of _ "y + {#(a,b)#}"]) auto |
|
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3203 |
thus ?thesis |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3204 |
using assms |
57966 | 3205 |
unfolding multiset.rel_compp_Grp Grp_def by blast |
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3206 |
qed |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3207 |
|
60606 | 3208 |
lemma rel_mset'_imp_rel_mset: "rel_mset' R M N \<Longrightarrow> rel_mset R M N" |
60678 | 3209 |
by (induct rule: rel_mset'.induct) (auto simp: rel_mset_Zero rel_mset_Plus) |
57966 | 3210 |
|
60606 | 3211 |
lemma rel_mset_size: "rel_mset R M N \<Longrightarrow> size M = size N" |
60678 | 3212 |
unfolding multiset.rel_compp_Grp Grp_def by auto |
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3213 |
|
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3214 |
lemma multiset_induct2[case_names empty addL addR]: |
60678 | 3215 |
assumes empty: "P {#} {#}" |
3216 |
and addL: "\<And>M N a. P M N \<Longrightarrow> P (M + {#a#}) N" |
|
3217 |
and addR: "\<And>M N a. P M N \<Longrightarrow> P M (N + {#a#})" |
|
3218 |
shows "P M N" |
|
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3219 |
apply(induct N rule: multiset_induct) |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3220 |
apply(induct M rule: multiset_induct, rule empty, erule addL) |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3221 |
apply(induct M rule: multiset_induct, erule addR, erule addR) |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3222 |
done |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3223 |
|
59949 | 3224 |
lemma multiset_induct2_size[consumes 1, case_names empty add]: |
60606 | 3225 |
assumes c: "size M = size N" |
3226 |
and empty: "P {#} {#}" |
|
3227 |
and add: "\<And>M N a b. P M N \<Longrightarrow> P (M + {#a#}) (N + {#b#})" |
|
3228 |
shows "P M N" |
|
60678 | 3229 |
using c |
3230 |
proof (induct M arbitrary: N rule: measure_induct_rule[of size]) |
|
60606 | 3231 |
case (less M) |
3232 |
show ?case |
|
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3233 |
proof(cases "M = {#}") |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3234 |
case True hence "N = {#}" using less.prems by auto |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3235 |
thus ?thesis using True empty by auto |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3236 |
next |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3237 |
case False then obtain M1 a where M: "M = M1 + {#a#}" by (metis multi_nonempty_split) |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3238 |
have "N \<noteq> {#}" using False less.prems by auto |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3239 |
then obtain N1 b where N: "N = N1 + {#b#}" by (metis multi_nonempty_split) |
59949 | 3240 |
have "size M1 = size N1" using less.prems unfolding M N by auto |
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3241 |
thus ?thesis using M N less.hyps add by auto |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3242 |
qed |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3243 |
qed |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3244 |
|
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3245 |
lemma msed_map_invL: |
60606 | 3246 |
assumes "image_mset f (M + {#a#}) = N" |
3247 |
shows "\<exists>N1. N = N1 + {#f a#} \<and> image_mset f M = N1" |
|
3248 |
proof - |
|
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3249 |
have "f a \<in># N" |
60606 | 3250 |
using assms multiset.set_map[of f "M + {#a#}"] by auto |
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3251 |
then obtain N1 where N: "N = N1 + {#f a#}" using multi_member_split by metis |
57966 | 3252 |
have "image_mset f M = N1" using assms unfolding N by simp |
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3253 |
thus ?thesis using N by blast |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3254 |
qed |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3255 |
|
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3256 |
lemma msed_map_invR: |
60606 | 3257 |
assumes "image_mset f M = N + {#b#}" |
3258 |
shows "\<exists>M1 a. M = M1 + {#a#} \<and> f a = b \<and> image_mset f M1 = N" |
|
3259 |
proof - |
|
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54868
diff
changeset
|
3260 |
obtain a where a: "a \<in># M" and fa: "f a = b" |
60606 | 3261 |
using multiset.set_map[of f M] unfolding assms |
62430
9527ff088c15
more succint formulation of membership for multisets, similar to lists;
haftmann
parents:
62390
diff
changeset
|
3262 |
by (metis image_iff union_single_eq_member) |
55129
26bd1cba3ab5
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parents:
54868
diff
changeset
|
3263 |
then obtain M1 where M: "M = M1 + {#a#}" using multi_member_split by metis |
57966 | 3264 |
have "image_mset f M1 = N" using assms unfolding M fa[symmetric] by simp |
55129
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parents:
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diff
changeset
|
3265 |
thus ?thesis using M fa by blast |
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parents:
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diff
changeset
|
3266 |
qed |
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parents:
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diff
changeset
|
3267 |
|
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changeset
|
3268 |
lemma msed_rel_invL: |
60606 | 3269 |
assumes "rel_mset R (M + {#a#}) N" |
3270 |
shows "\<exists>N1 b. N = N1 + {#b#} \<and> R a b \<and> rel_mset R M N1" |
|
3271 |
proof - |
|
57966 | 3272 |
obtain K where KM: "image_mset fst K = M + {#a#}" |
60606 | 3273 |
and KN: "image_mset snd K = N" and sK: "set_mset K \<subseteq> {(a, b). R a b}" |
3274 |
using assms |
|
3275 |
unfolding multiset.rel_compp_Grp Grp_def by auto |
|
55129
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changeset
|
3276 |
obtain K1 ab where K: "K = K1 + {#ab#}" and a: "fst ab = a" |
60606 | 3277 |
and K1M: "image_mset fst K1 = M" using msed_map_invR[OF KM] by auto |
57966 | 3278 |
obtain N1 where N: "N = N1 + {#snd ab#}" and K1N1: "image_mset snd K1 = N1" |
60606 | 3279 |
using msed_map_invL[OF KN[unfolded K]] by auto |
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parents:
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diff
changeset
|
3280 |
have Rab: "R a (snd ab)" using sK a unfolding K by auto |
57966 | 3281 |
have "rel_mset R M N1" using sK K1M K1N1 |
60606 | 3282 |
unfolding K multiset.rel_compp_Grp Grp_def by auto |
55129
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parents:
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changeset
|
3283 |
thus ?thesis using N Rab by auto |
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blanchet
parents:
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diff
changeset
|
3284 |
qed |
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parents:
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diff
changeset
|
3285 |
|
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diff
changeset
|
3286 |
lemma msed_rel_invR: |
60606 | 3287 |
assumes "rel_mset R M (N + {#b#})" |
3288 |
shows "\<exists>M1 a. M = M1 + {#a#} \<and> R a b \<and> rel_mset R M1 N" |
|
3289 |
proof - |
|
57966 | 3290 |
obtain K where KN: "image_mset snd K = N + {#b#}" |
60606 | 3291 |
and KM: "image_mset fst K = M" and sK: "set_mset K \<subseteq> {(a, b). R a b}" |
3292 |
using assms |
|
3293 |
unfolding multiset.rel_compp_Grp Grp_def by auto |
|
55129
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parents:
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diff
changeset
|
3294 |
obtain K1 ab where K: "K = K1 + {#ab#}" and b: "snd ab = b" |
60606 | 3295 |
and K1N: "image_mset snd K1 = N" using msed_map_invR[OF KN] by auto |
57966 | 3296 |
obtain M1 where M: "M = M1 + {#fst ab#}" and K1M1: "image_mset fst K1 = M1" |
60606 | 3297 |
using msed_map_invL[OF KM[unfolded K]] by auto |
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parents:
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diff
changeset
|
3298 |
have Rab: "R (fst ab) b" using sK b unfolding K by auto |
57966 | 3299 |
have "rel_mset R M1 N" using sK K1N K1M1 |
60606 | 3300 |
unfolding K multiset.rel_compp_Grp Grp_def by auto |
55129
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blanchet
parents:
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diff
changeset
|
3301 |
thus ?thesis using M Rab by auto |
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blanchet
parents:
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diff
changeset
|
3302 |
qed |
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parents:
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diff
changeset
|
3303 |
|
57966 | 3304 |
lemma rel_mset_imp_rel_mset': |
60606 | 3305 |
assumes "rel_mset R M N" |
3306 |
shows "rel_mset' R M N" |
|
59949 | 3307 |
using assms proof(induct M arbitrary: N rule: measure_induct_rule[of size]) |
55129
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parents:
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diff
changeset
|
3308 |
case (less M) |
59949 | 3309 |
have c: "size M = size N" using rel_mset_size[OF less.prems] . |
55129
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diff
changeset
|
3310 |
show ?case |
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changeset
|
3311 |
proof(cases "M = {#}") |
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changeset
|
3312 |
case True hence "N = {#}" using c by simp |
57966 | 3313 |
thus ?thesis using True rel_mset'.Zero by auto |
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changeset
|
3314 |
next |
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parents:
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diff
changeset
|
3315 |
case False then obtain M1 a where M: "M = M1 + {#a#}" by (metis multi_nonempty_split) |
57966 | 3316 |
obtain N1 b where N: "N = N1 + {#b#}" and R: "R a b" and ms: "rel_mset R M1 N1" |
60606 | 3317 |
using msed_rel_invL[OF less.prems[unfolded M]] by auto |
57966 | 3318 |
have "rel_mset' R M1 N1" using less.hyps[of M1 N1] ms unfolding M by simp |
3319 |
thus ?thesis using rel_mset'.Plus[of R a b, OF R] unfolding M N by simp |
|
55129
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blanchet
parents:
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diff
changeset
|
3320 |
qed |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
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diff
changeset
|
3321 |
qed |
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parents:
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diff
changeset
|
3322 |
|
60606 | 3323 |
lemma rel_mset_rel_mset': "rel_mset R M N = rel_mset' R M N" |
60678 | 3324 |
using rel_mset_imp_rel_mset' rel_mset'_imp_rel_mset by auto |
57966 | 3325 |
|
60613 | 3326 |
text \<open>The main end product for @{const rel_mset}: inductive characterization:\<close> |
61337 | 3327 |
lemmas rel_mset_induct[case_names empty add, induct pred: rel_mset] = |
60606 | 3328 |
rel_mset'.induct[unfolded rel_mset_rel_mset'[symmetric]] |
55129
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parents:
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changeset
|
3329 |
|
56656 | 3330 |
|
60500 | 3331 |
subsection \<open>Size setup\<close> |
56656 | 3332 |
|
57966 | 3333 |
lemma multiset_size_o_map: "size_multiset g \<circ> image_mset f = size_multiset (g \<circ> f)" |
60678 | 3334 |
apply (rule ext) |
3335 |
subgoal for x by (induct x) auto |
|
3336 |
done |
|
56656 | 3337 |
|
60500 | 3338 |
setup \<open> |
60606 | 3339 |
BNF_LFP_Size.register_size_global @{type_name multiset} @{const_name size_multiset} |
62082 | 3340 |
@{thm size_multiset_overloaded_def} |
60606 | 3341 |
@{thms size_multiset_empty size_multiset_single size_multiset_union size_empty size_single |
3342 |
size_union} |
|
3343 |
@{thms multiset_size_o_map} |
|
60500 | 3344 |
\<close> |
56656 | 3345 |
|
3346 |
hide_const (open) wcount |
|
3347 |
||
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changeset
|
3348 |
end |