author | bulwahn |
Tue, 26 Feb 2008 11:18:43 +0100 | |
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parent 26033 | 278025d5282d |
child 26145 | 95670b6e1fa3 |
permissions | -rw-r--r-- |
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(* Title: HOL/Library/Multiset.thy |
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ID: $Id$ |
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Author: Tobias Nipkow, Markus Wenzel, Lawrence C Paulson, Norbert Voelker |
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*) |
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header {* Multisets *} |
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theory Multiset |
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imports List |
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begin |
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subsection {* The type of multisets *} |
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typedef 'a multiset = "{f::'a => nat. finite {x . f x > 0}}" |
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proof |
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show "(\<lambda>x. 0::nat) \<in> ?multiset" by simp |
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qed |
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lemmas multiset_typedef [simp] = |
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Abs_multiset_inverse Rep_multiset_inverse Rep_multiset |
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and [simp] = Rep_multiset_inject [symmetric] |
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definition |
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Mempty :: "'a multiset" ("{#}") where |
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"{#} = Abs_multiset (\<lambda>a. 0)" |
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definition |
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single :: "'a => 'a multiset" where |
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"single a = Abs_multiset (\<lambda>b. if b = a then 1 else 0)" |
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declare |
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Mempty_def[code func del] single_def[code func del] |
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definition |
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count :: "'a multiset => 'a => nat" where |
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"count = Rep_multiset" |
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definition |
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MCollect :: "'a multiset => ('a => bool) => 'a multiset" where |
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"MCollect M P = Abs_multiset (\<lambda>x. if P x then Rep_multiset M x else 0)" |
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abbreviation |
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Melem :: "'a => 'a multiset => bool" ("(_/ :# _)" [50, 51] 50) where |
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"a :# M == 0 < count M a" |
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notation (xsymbols) Melem (infix "\<in>#" 50) |
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syntax |
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"_MCollect" :: "pttrn => 'a multiset => bool => 'a multiset" ("(1{# _ :# _./ _#})") |
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translations |
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"{#x :# M. P#}" == "CONST MCollect M (\<lambda>x. P)" |
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definition |
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set_of :: "'a multiset => 'a set" where |
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"set_of M = {x. x :# M}" |
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instantiation multiset :: (type) "{plus, minus, zero, size}" |
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begin |
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definition |
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union_def[code func del]: |
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"M + N == Abs_multiset (\<lambda>a. Rep_multiset M a + Rep_multiset N a)" |
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definition |
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diff_def: "M - N == Abs_multiset (\<lambda>a. Rep_multiset M a - Rep_multiset N a)" |
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definition |
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Zero_multiset_def [simp]: "0 == {#}" |
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definition |
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size_def[code func del]: "size M == setsum (count M) (set_of M)" |
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instance .. |
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end |
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definition |
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multiset_inter :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" (infixl "#\<inter>" 70) where |
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"multiset_inter A B = A - (A - B)" |
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syntax -- "Multiset Enumeration" |
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"@multiset" :: "args => 'a multiset" ("{#(_)#}") |
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translations |
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"{#x, xs#}" == "{#x#} + {#xs#}" |
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"{#x#}" == "CONST single x" |
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text {* |
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\medskip Preservation of the representing set @{term multiset}. |
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*} |
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lemma const0_in_multiset: "(\<lambda>a. 0) \<in> multiset" |
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by (simp add: multiset_def) |
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lemma only1_in_multiset: "(\<lambda>b. if b = a then 1 else 0) \<in> multiset" |
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by (simp add: multiset_def) |
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lemma union_preserves_multiset: |
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"M \<in> multiset ==> N \<in> multiset ==> (\<lambda>a. M a + N a) \<in> multiset" |
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apply (simp add: multiset_def) |
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apply (drule (1) finite_UnI) |
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apply (simp del: finite_Un add: Un_def) |
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done |
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lemma diff_preserves_multiset: |
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"M \<in> multiset ==> (\<lambda>a. M a - N a) \<in> multiset" |
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apply (simp add: multiset_def) |
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apply (rule finite_subset) |
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apply auto |
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done |
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lemma MCollect_preserves_multiset: |
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"M \<in> multiset ==> (\<lambda>x. if P x then M x else 0) \<in> multiset" |
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apply (simp add: multiset_def) |
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apply (rule finite_subset, auto) |
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done |
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lemmas in_multiset = const0_in_multiset only1_in_multiset |
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union_preserves_multiset diff_preserves_multiset MCollect_preserves_multiset |
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subsection {* Algebraic properties *} |
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subsubsection {* Union *} |
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lemma union_empty [simp]: "M + {#} = M \<and> {#} + M = M" |
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by (simp add: union_def Mempty_def in_multiset) |
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lemma union_commute: "M + N = N + (M::'a multiset)" |
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by (simp add: union_def add_ac in_multiset) |
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lemma union_assoc: "(M + N) + K = M + (N + (K::'a multiset))" |
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by (simp add: union_def add_ac in_multiset) |
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lemma union_lcomm: "M + (N + K) = N + (M + (K::'a multiset))" |
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proof - |
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have "M + (N + K) = (N + K) + M" |
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by (rule union_commute) |
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also have "\<dots> = N + (K + M)" |
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by (rule union_assoc) |
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also have "K + M = M + K" |
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by (rule union_commute) |
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finally show ?thesis . |
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qed |
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lemmas union_ac = union_assoc union_commute union_lcomm |
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instance multiset :: (type) comm_monoid_add |
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proof |
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fix a b c :: "'a multiset" |
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show "(a + b) + c = a + (b + c)" by (rule union_assoc) |
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show "a + b = b + a" by (rule union_commute) |
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show "0 + a = a" by simp |
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qed |
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subsubsection {* Difference *} |
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lemma diff_empty [simp]: "M - {#} = M \<and> {#} - M = {#}" |
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by (simp add: Mempty_def diff_def in_multiset) |
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lemma diff_union_inverse2 [simp]: "M + {#a#} - {#a#} = M" |
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by (simp add: union_def diff_def in_multiset) |
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lemma diff_cancel: "A - A = {#}" |
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by (simp add: diff_def Mempty_def) |
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subsubsection {* Count of elements *} |
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lemma count_empty [simp]: "count {#} a = 0" |
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by (simp add: count_def Mempty_def in_multiset) |
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lemma count_single [simp]: "count {#b#} a = (if b = a then 1 else 0)" |
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by (simp add: count_def single_def in_multiset) |
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lemma count_union [simp]: "count (M + N) a = count M a + count N a" |
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by (simp add: count_def union_def in_multiset) |
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lemma count_diff [simp]: "count (M - N) a = count M a - count N a" |
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by (simp add: count_def diff_def in_multiset) |
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lemma count_MCollect [simp]: |
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"count {# x:#M. P x #} a = (if P a then count M a else 0)" |
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by (simp add: count_def MCollect_def in_multiset) |
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subsubsection {* Set of elements *} |
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lemma set_of_empty [simp]: "set_of {#} = {}" |
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by (simp add: set_of_def) |
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lemma set_of_single [simp]: "set_of {#b#} = {b}" |
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by (simp add: set_of_def) |
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lemma set_of_union [simp]: "set_of (M + N) = set_of M \<union> set_of N" |
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by (auto simp add: set_of_def) |
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lemma set_of_eq_empty_iff [simp]: "(set_of M = {}) = (M = {#})" |
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by(auto simp: set_of_def Mempty_def in_multiset count_def expand_fun_eq) |
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lemma mem_set_of_iff [simp]: "(x \<in> set_of M) = (x :# M)" |
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by (auto simp add: set_of_def) |
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lemma set_of_MCollect [simp]: "set_of {# x:#M. P x #} = set_of M \<inter> {x. P x}" |
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by (auto simp add: set_of_def) |
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subsubsection {* Size *} |
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lemma size_empty [simp,code func]: "size {#} = 0" |
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by (simp add: size_def) |
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lemma size_single [simp,code func]: "size {#b#} = 1" |
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by (simp add: size_def) |
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lemma finite_set_of [iff]: "finite (set_of M)" |
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using Rep_multiset [of M] |
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by (simp add: multiset_def set_of_def count_def) |
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lemma setsum_count_Int: |
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"finite A ==> setsum (count N) (A \<inter> set_of N) = setsum (count N) A" |
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apply (induct rule: finite_induct) |
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apply simp |
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apply (simp add: Int_insert_left set_of_def) |
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done |
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lemma size_union[simp,code func]: "size (M + N::'a multiset) = size M + size N" |
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apply (unfold size_def) |
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apply (subgoal_tac "count (M + N) = (\<lambda>a. count M a + count N a)") |
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prefer 2 |
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apply (rule ext, simp) |
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apply (simp (no_asm_simp) add: setsum_Un_nat setsum_addf setsum_count_Int) |
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apply (subst Int_commute) |
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apply (simp (no_asm_simp) add: setsum_count_Int) |
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done |
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lemma size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})" |
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apply (unfold size_def Mempty_def count_def, auto simp: in_multiset) |
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apply (simp add: set_of_def count_def in_multiset expand_fun_eq) |
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done |
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lemma nonempty_has_size: "(S \<noteq> {#}) = (0 < size S)" |
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by(metis gr0I gr_implies_not0 size_empty size_eq_0_iff_empty) |
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lemma size_eq_Suc_imp_elem: "size M = Suc n ==> \<exists>a. a :# M" |
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apply (unfold size_def) |
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apply (drule setsum_SucD, auto) |
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done |
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subsubsection {* Equality of multisets *} |
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lemma multiset_eq_conv_count_eq: "(M = N) = (\<forall>a. count M a = count N a)" |
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by (simp add: count_def expand_fun_eq) |
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lemma single_not_empty [simp]: "{#a#} \<noteq> {#} \<and> {#} \<noteq> {#a#}" |
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by (simp add: single_def Mempty_def in_multiset expand_fun_eq) |
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lemma single_eq_single [simp]: "({#a#} = {#b#}) = (a = b)" |
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by (auto simp add: single_def in_multiset expand_fun_eq) |
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lemma union_eq_empty [iff]: "(M + N = {#}) = (M = {#} \<and> N = {#})" |
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by (auto simp add: union_def Mempty_def in_multiset expand_fun_eq) |
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lemma empty_eq_union [iff]: "({#} = M + N) = (M = {#} \<and> N = {#})" |
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by (auto simp add: union_def Mempty_def in_multiset expand_fun_eq) |
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lemma union_right_cancel [simp]: "(M + K = N + K) = (M = (N::'a multiset))" |
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by (simp add: union_def in_multiset expand_fun_eq) |
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lemma union_left_cancel [simp]: "(K + M = K + N) = (M = (N::'a multiset))" |
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by (simp add: union_def in_multiset expand_fun_eq) |
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lemma union_is_single: |
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"(M + N = {#a#}) = (M = {#a#} \<and> N={#} \<or> M = {#} \<and> N = {#a#})" |
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apply (simp add: Mempty_def single_def union_def in_multiset add_is_1 expand_fun_eq) |
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apply blast |
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done |
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lemma single_is_union: |
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"({#a#} = M + N) = ({#a#} = M \<and> N = {#} \<or> M = {#} \<and> {#a#} = N)" |
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apply (unfold Mempty_def single_def union_def) |
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apply (simp add: add_is_1 one_is_add in_multiset expand_fun_eq) |
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apply (blast dest: sym) |
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done |
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lemma add_eq_conv_diff: |
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"(M + {#a#} = N + {#b#}) = |
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(M = N \<and> a = b \<or> M = N - {#a#} + {#b#} \<and> N = M - {#b#} + {#a#})" |
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using [[simproc del: neq]] |
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apply (unfold single_def union_def diff_def) |
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apply (simp (no_asm) add: in_multiset expand_fun_eq) |
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apply (rule conjI, force, safe, simp_all) |
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apply (simp add: eq_sym_conv) |
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done |
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declare Rep_multiset_inject [symmetric, simp del] |
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instance multiset :: (type) cancel_ab_semigroup_add |
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proof |
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fix a b c :: "'a multiset" |
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show "a + b = a + c \<Longrightarrow> b = c" by simp |
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qed |
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lemma insert_DiffM: |
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"x \<in># M \<Longrightarrow> {#x#} + (M - {#x#}) = M" |
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by (clarsimp simp: multiset_eq_conv_count_eq) |
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lemma insert_DiffM2[simp]: |
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"x \<in># M \<Longrightarrow> M - {#x#} + {#x#} = M" |
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by (clarsimp simp: multiset_eq_conv_count_eq) |
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lemma multi_union_self_other_eq: |
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"(A::'a multiset) + X = A + Y \<Longrightarrow> X = Y" |
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by (induct A arbitrary: X Y, auto) |
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lemma multi_self_add_other_not_self[simp]: "(A = A + {#x#}) = False" |
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by (metis single_not_empty union_empty union_left_cancel) |
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lemma insert_noteq_member: |
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321 |
assumes BC: "B + {#b#} = C + {#c#}" |
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and bnotc: "b \<noteq> c" |
|
323 |
shows "c \<in># B" |
|
324 |
proof - |
|
325 |
have "c \<in># C + {#c#}" by simp |
|
326 |
have nc: "\<not> c \<in># {#b#}" using bnotc by simp |
|
327 |
hence "c \<in># B + {#b#}" using BC by simp |
|
328 |
thus "c \<in># B" using nc by simp |
|
329 |
qed |
|
330 |
||
331 |
||
26016 | 332 |
lemma add_eq_conv_ex: |
333 |
"(M + {#a#} = N + {#b#}) = |
|
334 |
(M = N \<and> a = b \<or> (\<exists>K. M = K + {#b#} \<and> N = K + {#a#}))" |
|
335 |
by (auto simp add: add_eq_conv_diff) |
|
336 |
||
337 |
||
338 |
lemma empty_multiset_count: |
|
339 |
"(\<forall>x. count A x = 0) = (A = {#})" |
|
340 |
by(metis count_empty multiset_eq_conv_count_eq) |
|
341 |
||
342 |
||
15869 | 343 |
subsubsection {* Intersection *} |
344 |
||
345 |
lemma multiset_inter_count: |
|
26016 | 346 |
"count (A #\<inter> B) x = min (count A x) (count B x)" |
347 |
by (simp add: multiset_inter_def min_def) |
|
15869 | 348 |
|
349 |
lemma multiset_inter_commute: "A #\<inter> B = B #\<inter> A" |
|
26016 | 350 |
by (simp add: multiset_eq_conv_count_eq multiset_inter_count |
21214
a91bab12b2bd
adjusted two lemma names due to name change in interpretation
haftmann
parents:
20770
diff
changeset
|
351 |
min_max.inf_commute) |
15869 | 352 |
|
353 |
lemma multiset_inter_assoc: "A #\<inter> (B #\<inter> C) = A #\<inter> B #\<inter> C" |
|
26016 | 354 |
by (simp add: multiset_eq_conv_count_eq multiset_inter_count |
21214
a91bab12b2bd
adjusted two lemma names due to name change in interpretation
haftmann
parents:
20770
diff
changeset
|
355 |
min_max.inf_assoc) |
15869 | 356 |
|
357 |
lemma multiset_inter_left_commute: "A #\<inter> (B #\<inter> C) = B #\<inter> (A #\<inter> C)" |
|
26016 | 358 |
by (simp add: multiset_eq_conv_count_eq multiset_inter_count min_def) |
15869 | 359 |
|
17161 | 360 |
lemmas multiset_inter_ac = |
361 |
multiset_inter_commute |
|
362 |
multiset_inter_assoc |
|
363 |
multiset_inter_left_commute |
|
15869 | 364 |
|
26143
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
365 |
lemma multiset_inter_single: "a \<noteq> b \<Longrightarrow> {#a#} #\<inter> {#b#} = {#}" |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
366 |
by (simp add: multiset_eq_conv_count_eq multiset_inter_count) |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
367 |
|
15869 | 368 |
lemma multiset_union_diff_commute: "B #\<inter> C = {#} \<Longrightarrow> A + B - C = A - C + B" |
17200 | 369 |
apply (simp add: multiset_eq_conv_count_eq multiset_inter_count min_def |
17161 | 370 |
split: split_if_asm) |
15869 | 371 |
apply clarsimp |
17161 | 372 |
apply (erule_tac x = a in allE) |
15869 | 373 |
apply auto |
374 |
done |
|
375 |
||
10249 | 376 |
|
26016 | 377 |
subsubsection {* Comprehension (filter) *} |
378 |
||
379 |
lemma MCollect_empty[simp, code func]: "MCollect {#} P = {#}" |
|
380 |
by(simp add:MCollect_def Mempty_def Abs_multiset_inject in_multiset expand_fun_eq) |
|
381 |
||
382 |
lemma MCollect_single[simp, code func]: |
|
383 |
"MCollect {#x#} P = (if P x then {#x#} else {#})" |
|
384 |
by(simp add:MCollect_def Mempty_def single_def Abs_multiset_inject in_multiset expand_fun_eq) |
|
385 |
||
386 |
lemma MCollect_union[simp, code func]: |
|
387 |
"MCollect (M+N) f = MCollect M f + MCollect N f" |
|
388 |
by(simp add:MCollect_def union_def Abs_multiset_inject in_multiset expand_fun_eq) |
|
389 |
||
390 |
||
391 |
subsection {* Induction and case splits *} |
|
10249 | 392 |
|
393 |
lemma setsum_decr: |
|
11701
3d51fbf81c17
sane numerals (stage 1): added generic 1, removed 1' and 2 on nat,
wenzelm
parents:
11655
diff
changeset
|
394 |
"finite F ==> (0::nat) < f a ==> |
15072 | 395 |
setsum (f (a := f a - 1)) F = (if a\<in>F then setsum f F - 1 else setsum f F)" |
18258 | 396 |
apply (induct rule: finite_induct) |
397 |
apply auto |
|
15072 | 398 |
apply (drule_tac a = a in mk_disjoint_insert, auto) |
10249 | 399 |
done |
400 |
||
10313 | 401 |
lemma rep_multiset_induct_aux: |
18730 | 402 |
assumes 1: "P (\<lambda>a. (0::nat))" |
403 |
and 2: "!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))" |
|
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
24035
diff
changeset
|
404 |
shows "\<forall>f. f \<in> multiset --> setsum f {x. f x \<noteq> 0} = n --> P f" |
18730 | 405 |
apply (unfold multiset_def) |
406 |
apply (induct_tac n, simp, clarify) |
|
407 |
apply (subgoal_tac "f = (\<lambda>a.0)") |
|
408 |
apply simp |
|
409 |
apply (rule 1) |
|
410 |
apply (rule ext, force, clarify) |
|
411 |
apply (frule setsum_SucD, clarify) |
|
412 |
apply (rename_tac a) |
|
25162 | 413 |
apply (subgoal_tac "finite {x. (f (a := f a - 1)) x > 0}") |
18730 | 414 |
prefer 2 |
415 |
apply (rule finite_subset) |
|
416 |
prefer 2 |
|
417 |
apply assumption |
|
418 |
apply simp |
|
419 |
apply blast |
|
420 |
apply (subgoal_tac "f = (f (a := f a - 1))(a := (f (a := f a - 1)) a + 1)") |
|
421 |
prefer 2 |
|
422 |
apply (rule ext) |
|
423 |
apply (simp (no_asm_simp)) |
|
424 |
apply (erule ssubst, rule 2 [unfolded multiset_def], blast) |
|
425 |
apply (erule allE, erule impE, erule_tac [2] mp, blast) |
|
426 |
apply (simp (no_asm_simp) add: setsum_decr del: fun_upd_apply One_nat_def) |
|
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
24035
diff
changeset
|
427 |
apply (subgoal_tac "{x. x \<noteq> a --> f x \<noteq> 0} = {x. f x \<noteq> 0}") |
18730 | 428 |
prefer 2 |
429 |
apply blast |
|
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
24035
diff
changeset
|
430 |
apply (subgoal_tac "{x. x \<noteq> a \<and> f x \<noteq> 0} = {x. f x \<noteq> 0} - {a}") |
18730 | 431 |
prefer 2 |
432 |
apply blast |
|
433 |
apply (simp add: le_imp_diff_is_add setsum_diff1_nat cong: conj_cong) |
|
434 |
done |
|
10249 | 435 |
|
10313 | 436 |
theorem rep_multiset_induct: |
11464 | 437 |
"f \<in> multiset ==> P (\<lambda>a. 0) ==> |
11701
3d51fbf81c17
sane numerals (stage 1): added generic 1, removed 1' and 2 on nat,
wenzelm
parents:
11655
diff
changeset
|
438 |
(!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))) ==> P f" |
17161 | 439 |
using rep_multiset_induct_aux by blast |
10249 | 440 |
|
18258 | 441 |
theorem multiset_induct [case_names empty add, induct type: multiset]: |
442 |
assumes empty: "P {#}" |
|
443 |
and add: "!!M x. P M ==> P (M + {#x#})" |
|
17161 | 444 |
shows "P M" |
10249 | 445 |
proof - |
446 |
note defns = union_def single_def Mempty_def |
|
447 |
show ?thesis |
|
448 |
apply (rule Rep_multiset_inverse [THEN subst]) |
|
10313 | 449 |
apply (rule Rep_multiset [THEN rep_multiset_induct]) |
18258 | 450 |
apply (rule empty [unfolded defns]) |
15072 | 451 |
apply (subgoal_tac "f(b := f b + 1) = (\<lambda>a. f a + (if a=b then 1 else 0))") |
10249 | 452 |
prefer 2 |
453 |
apply (simp add: expand_fun_eq) |
|
454 |
apply (erule ssubst) |
|
17200 | 455 |
apply (erule Abs_multiset_inverse [THEN subst]) |
26016 | 456 |
apply (drule add [unfolded defns, simplified]) |
457 |
apply(simp add:in_multiset) |
|
10249 | 458 |
done |
459 |
qed |
|
460 |
||
25610 | 461 |
lemma multi_nonempty_split: "M \<noteq> {#} \<Longrightarrow> \<exists>A a. M = A + {#a#}" |
462 |
by (induct M, auto) |
|
463 |
||
464 |
lemma multiset_cases [cases type, case_names empty add]: |
|
465 |
assumes em: "M = {#} \<Longrightarrow> P" |
|
466 |
assumes add: "\<And>N x. M = N + {#x#} \<Longrightarrow> P" |
|
467 |
shows "P" |
|
468 |
proof (cases "M = {#}") |
|
469 |
assume "M = {#}" thus ?thesis using em by simp |
|
470 |
next |
|
471 |
assume "M \<noteq> {#}" |
|
472 |
then obtain M' m where "M = M' + {#m#}" |
|
473 |
by (blast dest: multi_nonempty_split) |
|
474 |
thus ?thesis using add by simp |
|
475 |
qed |
|
476 |
||
477 |
lemma multi_member_split: "x \<in># M \<Longrightarrow> \<exists>A. M = A + {#x#}" |
|
478 |
apply (cases M, simp) |
|
479 |
apply (rule_tac x="M - {#x#}" in exI, simp) |
|
480 |
done |
|
481 |
||
26033 | 482 |
lemma multiset_partition: "M = {# x:#M. P x #} + {# x:#M. \<not> P x #}" |
17161 | 483 |
by (subst multiset_eq_conv_count_eq, auto) |
10249 | 484 |
|
15869 | 485 |
declare multiset_typedef [simp del] |
10249 | 486 |
|
25610 | 487 |
lemma multi_drop_mem_not_eq: "c \<in># B \<Longrightarrow> B - {#c#} \<noteq> B" |
488 |
by (cases "B={#}", auto dest: multi_member_split) |
|
17161 | 489 |
|
26016 | 490 |
subsection {* Orderings *} |
10249 | 491 |
|
492 |
subsubsection {* Well-foundedness *} |
|
493 |
||
19086 | 494 |
definition |
23751 | 495 |
mult1 :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where |
19086 | 496 |
"mult1 r = |
23751 | 497 |
{(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> |
498 |
(\<forall>b. b :# K --> (b, a) \<in> r)}" |
|
10249 | 499 |
|
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21214
diff
changeset
|
500 |
definition |
23751 | 501 |
mult :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where |
502 |
"mult r = (mult1 r)\<^sup>+" |
|
10249 | 503 |
|
23751 | 504 |
lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r" |
10277 | 505 |
by (simp add: mult1_def) |
10249 | 506 |
|
23751 | 507 |
lemma less_add: "(N, M0 + {#a#}) \<in> mult1 r ==> |
508 |
(\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or> |
|
509 |
(\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K)" |
|
19582 | 510 |
(is "_ \<Longrightarrow> ?case1 (mult1 r) \<or> ?case2") |
10249 | 511 |
proof (unfold mult1_def) |
23751 | 512 |
let ?r = "\<lambda>K a. \<forall>b. b :# K --> (b, a) \<in> r" |
11464 | 513 |
let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a" |
23751 | 514 |
let ?case1 = "?case1 {(N, M). ?R N M}" |
10249 | 515 |
|
23751 | 516 |
assume "(N, M0 + {#a#}) \<in> {(N, M). ?R N M}" |
18258 | 517 |
then have "\<exists>a' M0' K. |
11464 | 518 |
M0 + {#a#} = M0' + {#a'#} \<and> N = M0' + K \<and> ?r K a'" by simp |
18258 | 519 |
then show "?case1 \<or> ?case2" |
10249 | 520 |
proof (elim exE conjE) |
521 |
fix a' M0' K |
|
522 |
assume N: "N = M0' + K" and r: "?r K a'" |
|
523 |
assume "M0 + {#a#} = M0' + {#a'#}" |
|
18258 | 524 |
then have "M0 = M0' \<and> a = a' \<or> |
11464 | 525 |
(\<exists>K'. M0 = K' + {#a'#} \<and> M0' = K' + {#a#})" |
10249 | 526 |
by (simp only: add_eq_conv_ex) |
18258 | 527 |
then show ?thesis |
10249 | 528 |
proof (elim disjE conjE exE) |
529 |
assume "M0 = M0'" "a = a'" |
|
11464 | 530 |
with N r have "?r K a \<and> N = M0 + K" by simp |
18258 | 531 |
then have ?case2 .. then show ?thesis .. |
10249 | 532 |
next |
533 |
fix K' |
|
534 |
assume "M0' = K' + {#a#}" |
|
535 |
with N have n: "N = K' + K + {#a#}" by (simp add: union_ac) |
|
536 |
||
537 |
assume "M0 = K' + {#a'#}" |
|
538 |
with r have "?R (K' + K) M0" by blast |
|
18258 | 539 |
with n have ?case1 by simp then show ?thesis .. |
10249 | 540 |
qed |
541 |
qed |
|
542 |
qed |
|
543 |
||
23751 | 544 |
lemma all_accessible: "wf r ==> \<forall>M. M \<in> acc (mult1 r)" |
10249 | 545 |
proof |
546 |
let ?R = "mult1 r" |
|
547 |
let ?W = "acc ?R" |
|
548 |
{ |
|
549 |
fix M M0 a |
|
23751 | 550 |
assume M0: "M0 \<in> ?W" |
551 |
and wf_hyp: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)" |
|
552 |
and acc_hyp: "\<forall>M. (M, M0) \<in> ?R --> M + {#a#} \<in> ?W" |
|
553 |
have "M0 + {#a#} \<in> ?W" |
|
554 |
proof (rule accI [of "M0 + {#a#}"]) |
|
10249 | 555 |
fix N |
23751 | 556 |
assume "(N, M0 + {#a#}) \<in> ?R" |
557 |
then have "((\<exists>M. (M, M0) \<in> ?R \<and> N = M + {#a#}) \<or> |
|
558 |
(\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K))" |
|
10249 | 559 |
by (rule less_add) |
23751 | 560 |
then show "N \<in> ?W" |
10249 | 561 |
proof (elim exE disjE conjE) |
23751 | 562 |
fix M assume "(M, M0) \<in> ?R" and N: "N = M + {#a#}" |
563 |
from acc_hyp have "(M, M0) \<in> ?R --> M + {#a#} \<in> ?W" .. |
|
564 |
from this and `(M, M0) \<in> ?R` have "M + {#a#} \<in> ?W" .. |
|
565 |
then show "N \<in> ?W" by (simp only: N) |
|
10249 | 566 |
next |
567 |
fix K |
|
568 |
assume N: "N = M0 + K" |
|
23751 | 569 |
assume "\<forall>b. b :# K --> (b, a) \<in> r" |
570 |
then have "M0 + K \<in> ?W" |
|
10249 | 571 |
proof (induct K) |
18730 | 572 |
case empty |
23751 | 573 |
from M0 show "M0 + {#} \<in> ?W" by simp |
18730 | 574 |
next |
575 |
case (add K x) |
|
23751 | 576 |
from add.prems have "(x, a) \<in> r" by simp |
577 |
with wf_hyp have "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast |
|
578 |
moreover from add have "M0 + K \<in> ?W" by simp |
|
579 |
ultimately have "(M0 + K) + {#x#} \<in> ?W" .. |
|
580 |
then show "M0 + (K + {#x#}) \<in> ?W" by (simp only: union_assoc) |
|
10249 | 581 |
qed |
23751 | 582 |
then show "N \<in> ?W" by (simp only: N) |
10249 | 583 |
qed |
584 |
qed |
|
585 |
} note tedious_reasoning = this |
|
586 |
||
23751 | 587 |
assume wf: "wf r" |
10249 | 588 |
fix M |
23751 | 589 |
show "M \<in> ?W" |
10249 | 590 |
proof (induct M) |
23751 | 591 |
show "{#} \<in> ?W" |
10249 | 592 |
proof (rule accI) |
23751 | 593 |
fix b assume "(b, {#}) \<in> ?R" |
594 |
with not_less_empty show "b \<in> ?W" by contradiction |
|
10249 | 595 |
qed |
596 |
||
23751 | 597 |
fix M a assume "M \<in> ?W" |
598 |
from wf have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W" |
|
10249 | 599 |
proof induct |
600 |
fix a |
|
23751 | 601 |
assume r: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)" |
602 |
show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W" |
|
10249 | 603 |
proof |
23751 | 604 |
fix M assume "M \<in> ?W" |
605 |
then show "M + {#a#} \<in> ?W" |
|
23373 | 606 |
by (rule acc_induct) (rule tedious_reasoning [OF _ r]) |
10249 | 607 |
qed |
608 |
qed |
|
23751 | 609 |
from this and `M \<in> ?W` show "M + {#a#} \<in> ?W" .. |
10249 | 610 |
qed |
611 |
qed |
|
612 |
||
23751 | 613 |
theorem wf_mult1: "wf r ==> wf (mult1 r)" |
23373 | 614 |
by (rule acc_wfI) (rule all_accessible) |
10249 | 615 |
|
23751 | 616 |
theorem wf_mult: "wf r ==> wf (mult r)" |
617 |
unfolding mult_def by (rule wf_trancl) (rule wf_mult1) |
|
10249 | 618 |
|
619 |
||
620 |
subsubsection {* Closure-free presentation *} |
|
621 |
||
622 |
(*Badly needed: a linear arithmetic procedure for multisets*) |
|
623 |
||
624 |
lemma diff_union_single_conv: "a :# J ==> I + J - {#a#} = I + (J - {#a#})" |
|
23373 | 625 |
by (simp add: multiset_eq_conv_count_eq) |
10249 | 626 |
|
627 |
text {* One direction. *} |
|
628 |
||
629 |
lemma mult_implies_one_step: |
|
23751 | 630 |
"trans r ==> (M, N) \<in> mult r ==> |
11464 | 631 |
\<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and> |
23751 | 632 |
(\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r)" |
10249 | 633 |
apply (unfold mult_def mult1_def set_of_def) |
23751 | 634 |
apply (erule converse_trancl_induct, clarify) |
15072 | 635 |
apply (rule_tac x = M0 in exI, simp, clarify) |
23751 | 636 |
apply (case_tac "a :# K") |
10249 | 637 |
apply (rule_tac x = I in exI) |
638 |
apply (simp (no_asm)) |
|
23751 | 639 |
apply (rule_tac x = "(K - {#a#}) + Ka" in exI) |
10249 | 640 |
apply (simp (no_asm_simp) add: union_assoc [symmetric]) |
11464 | 641 |
apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong) |
10249 | 642 |
apply (simp add: diff_union_single_conv) |
643 |
apply (simp (no_asm_use) add: trans_def) |
|
644 |
apply blast |
|
645 |
apply (subgoal_tac "a :# I") |
|
646 |
apply (rule_tac x = "I - {#a#}" in exI) |
|
647 |
apply (rule_tac x = "J + {#a#}" in exI) |
|
648 |
apply (rule_tac x = "K + Ka" in exI) |
|
649 |
apply (rule conjI) |
|
650 |
apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split) |
|
651 |
apply (rule conjI) |
|
15072 | 652 |
apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong, simp) |
10249 | 653 |
apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split) |
654 |
apply (simp (no_asm_use) add: trans_def) |
|
655 |
apply blast |
|
10277 | 656 |
apply (subgoal_tac "a :# (M0 + {#a#})") |
10249 | 657 |
apply simp |
658 |
apply (simp (no_asm)) |
|
659 |
done |
|
660 |
||
661 |
lemma elem_imp_eq_diff_union: "a :# M ==> M = M - {#a#} + {#a#}" |
|
23373 | 662 |
by (simp add: multiset_eq_conv_count_eq) |
10249 | 663 |
|
11464 | 664 |
lemma size_eq_Suc_imp_eq_union: "size M = Suc n ==> \<exists>a N. M = N + {#a#}" |
10249 | 665 |
apply (erule size_eq_Suc_imp_elem [THEN exE]) |
15072 | 666 |
apply (drule elem_imp_eq_diff_union, auto) |
10249 | 667 |
done |
668 |
||
669 |
lemma one_step_implies_mult_aux: |
|
23751 | 670 |
"trans r ==> |
671 |
\<forall>I J K. (size J = n \<and> J \<noteq> {#} \<and> (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r)) |
|
672 |
--> (I + K, I + J) \<in> mult r" |
|
15072 | 673 |
apply (induct_tac n, auto) |
674 |
apply (frule size_eq_Suc_imp_eq_union, clarify) |
|
675 |
apply (rename_tac "J'", simp) |
|
676 |
apply (erule notE, auto) |
|
10249 | 677 |
apply (case_tac "J' = {#}") |
678 |
apply (simp add: mult_def) |
|
23751 | 679 |
apply (rule r_into_trancl) |
15072 | 680 |
apply (simp add: mult1_def set_of_def, blast) |
11464 | 681 |
txt {* Now we know @{term "J' \<noteq> {#}"}. *} |
23751 | 682 |
apply (cut_tac M = K and P = "\<lambda>x. (x, a) \<in> r" in multiset_partition) |
11464 | 683 |
apply (erule_tac P = "\<forall>k \<in> set_of K. ?P k" in rev_mp) |
10249 | 684 |
apply (erule ssubst) |
15072 | 685 |
apply (simp add: Ball_def, auto) |
10249 | 686 |
apply (subgoal_tac |
26033 | 687 |
"((I + {# x :# K. (x, a) \<in> r #}) + {# x :# K. (x, a) \<notin> r #}, |
688 |
(I + {# x :# K. (x, a) \<in> r #}) + J') \<in> mult r") |
|
10249 | 689 |
prefer 2 |
690 |
apply force |
|
691 |
apply (simp (no_asm_use) add: union_assoc [symmetric] mult_def) |
|
23751 | 692 |
apply (erule trancl_trans) |
693 |
apply (rule r_into_trancl) |
|
10249 | 694 |
apply (simp add: mult1_def set_of_def) |
695 |
apply (rule_tac x = a in exI) |
|
696 |
apply (rule_tac x = "I + J'" in exI) |
|
697 |
apply (simp add: union_ac) |
|
698 |
done |
|
699 |
||
17161 | 700 |
lemma one_step_implies_mult: |
23751 | 701 |
"trans r ==> J \<noteq> {#} ==> \<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r |
702 |
==> (I + K, I + J) \<in> mult r" |
|
23373 | 703 |
using one_step_implies_mult_aux by blast |
10249 | 704 |
|
705 |
||
706 |
subsubsection {* Partial-order properties *} |
|
707 |
||
12338
de0f4a63baa5
renamed class "term" to "type" (actually "HOL.type");
wenzelm
parents:
11868
diff
changeset
|
708 |
instance multiset :: (type) ord .. |
10249 | 709 |
|
710 |
defs (overloaded) |
|
23751 | 711 |
less_multiset_def: "M' < M == (M', M) \<in> mult {(x', x). x' < x}" |
11464 | 712 |
le_multiset_def: "M' <= M == M' = M \<or> M' < (M::'a multiset)" |
10249 | 713 |
|
23751 | 714 |
lemma trans_base_order: "trans {(x', x). x' < (x::'a::order)}" |
18730 | 715 |
unfolding trans_def by (blast intro: order_less_trans) |
10249 | 716 |
|
717 |
text {* |
|
718 |
\medskip Irreflexivity. |
|
719 |
*} |
|
720 |
||
721 |
lemma mult_irrefl_aux: |
|
18258 | 722 |
"finite A ==> (\<forall>x \<in> A. \<exists>y \<in> A. x < (y::'a::order)) \<Longrightarrow> A = {}" |
23373 | 723 |
by (induct rule: finite_induct) (auto intro: order_less_trans) |
10249 | 724 |
|
17161 | 725 |
lemma mult_less_not_refl: "\<not> M < (M::'a::order multiset)" |
15072 | 726 |
apply (unfold less_multiset_def, auto) |
727 |
apply (drule trans_base_order [THEN mult_implies_one_step], auto) |
|
10249 | 728 |
apply (drule finite_set_of [THEN mult_irrefl_aux [rule_format (no_asm)]]) |
729 |
apply (simp add: set_of_eq_empty_iff) |
|
730 |
done |
|
731 |
||
732 |
lemma mult_less_irrefl [elim!]: "M < (M::'a::order multiset) ==> R" |
|
23373 | 733 |
using insert mult_less_not_refl by fast |
10249 | 734 |
|
735 |
||
736 |
text {* Transitivity. *} |
|
737 |
||
738 |
theorem mult_less_trans: "K < M ==> M < N ==> K < (N::'a::order multiset)" |
|
23751 | 739 |
unfolding less_multiset_def mult_def by (blast intro: trancl_trans) |
10249 | 740 |
|
741 |
text {* Asymmetry. *} |
|
742 |
||
11464 | 743 |
theorem mult_less_not_sym: "M < N ==> \<not> N < (M::'a::order multiset)" |
10249 | 744 |
apply auto |
745 |
apply (rule mult_less_not_refl [THEN notE]) |
|
15072 | 746 |
apply (erule mult_less_trans, assumption) |
10249 | 747 |
done |
748 |
||
749 |
theorem mult_less_asym: |
|
11464 | 750 |
"M < N ==> (\<not> P ==> N < (M::'a::order multiset)) ==> P" |
15072 | 751 |
by (insert mult_less_not_sym, blast) |
10249 | 752 |
|
753 |
theorem mult_le_refl [iff]: "M <= (M::'a::order multiset)" |
|
18730 | 754 |
unfolding le_multiset_def by auto |
10249 | 755 |
|
756 |
text {* Anti-symmetry. *} |
|
757 |
||
758 |
theorem mult_le_antisym: |
|
759 |
"M <= N ==> N <= M ==> M = (N::'a::order multiset)" |
|
18730 | 760 |
unfolding le_multiset_def by (blast dest: mult_less_not_sym) |
10249 | 761 |
|
762 |
text {* Transitivity. *} |
|
763 |
||
764 |
theorem mult_le_trans: |
|
765 |
"K <= M ==> M <= N ==> K <= (N::'a::order multiset)" |
|
18730 | 766 |
unfolding le_multiset_def by (blast intro: mult_less_trans) |
10249 | 767 |
|
11655 | 768 |
theorem mult_less_le: "(M < N) = (M <= N \<and> M \<noteq> (N::'a::order multiset))" |
18730 | 769 |
unfolding le_multiset_def by auto |
10249 | 770 |
|
10277 | 771 |
text {* Partial order. *} |
772 |
||
773 |
instance multiset :: (order) order |
|
774 |
apply intro_classes |
|
23751 | 775 |
apply (rule mult_less_le) |
776 |
apply (rule mult_le_refl) |
|
777 |
apply (erule mult_le_trans, assumption) |
|
778 |
apply (erule mult_le_antisym, assumption) |
|
10277 | 779 |
done |
780 |
||
10249 | 781 |
|
782 |
subsubsection {* Monotonicity of multiset union *} |
|
783 |
||
17161 | 784 |
lemma mult1_union: |
23751 | 785 |
"(B, D) \<in> mult1 r ==> trans r ==> (C + B, C + D) \<in> mult1 r" |
15072 | 786 |
apply (unfold mult1_def, auto) |
10249 | 787 |
apply (rule_tac x = a in exI) |
788 |
apply (rule_tac x = "C + M0" in exI) |
|
789 |
apply (simp add: union_assoc) |
|
790 |
done |
|
791 |
||
792 |
lemma union_less_mono2: "B < D ==> C + B < C + (D::'a::order multiset)" |
|
793 |
apply (unfold less_multiset_def mult_def) |
|
23751 | 794 |
apply (erule trancl_induct) |
795 |
apply (blast intro: mult1_union transI order_less_trans r_into_trancl) |
|
796 |
apply (blast intro: mult1_union transI order_less_trans r_into_trancl trancl_trans) |
|
10249 | 797 |
done |
798 |
||
799 |
lemma union_less_mono1: "B < D ==> B + C < D + (C::'a::order multiset)" |
|
800 |
apply (subst union_commute [of B C]) |
|
801 |
apply (subst union_commute [of D C]) |
|
802 |
apply (erule union_less_mono2) |
|
803 |
done |
|
804 |
||
17161 | 805 |
lemma union_less_mono: |
10249 | 806 |
"A < C ==> B < D ==> A + B < C + (D::'a::order multiset)" |
807 |
apply (blast intro!: union_less_mono1 union_less_mono2 mult_less_trans) |
|
808 |
done |
|
809 |
||
17161 | 810 |
lemma union_le_mono: |
10249 | 811 |
"A <= C ==> B <= D ==> A + B <= C + (D::'a::order multiset)" |
18730 | 812 |
unfolding le_multiset_def |
813 |
by (blast intro: union_less_mono union_less_mono1 union_less_mono2) |
|
10249 | 814 |
|
17161 | 815 |
lemma empty_leI [iff]: "{#} <= (M::'a::order multiset)" |
10249 | 816 |
apply (unfold le_multiset_def less_multiset_def) |
817 |
apply (case_tac "M = {#}") |
|
818 |
prefer 2 |
|
23751 | 819 |
apply (subgoal_tac "({#} + {#}, {#} + M) \<in> mult (Collect (split op <))") |
10249 | 820 |
prefer 2 |
821 |
apply (rule one_step_implies_mult) |
|
23751 | 822 |
apply (simp only: trans_def, auto) |
10249 | 823 |
done |
824 |
||
17161 | 825 |
lemma union_upper1: "A <= A + (B::'a::order multiset)" |
15072 | 826 |
proof - |
17200 | 827 |
have "A + {#} <= A + B" by (blast intro: union_le_mono) |
18258 | 828 |
then show ?thesis by simp |
15072 | 829 |
qed |
830 |
||
17161 | 831 |
lemma union_upper2: "B <= A + (B::'a::order multiset)" |
18258 | 832 |
by (subst union_commute) (rule union_upper1) |
15072 | 833 |
|
23611 | 834 |
instance multiset :: (order) pordered_ab_semigroup_add |
835 |
apply intro_classes |
|
836 |
apply(erule union_le_mono[OF mult_le_refl]) |
|
837 |
done |
|
15072 | 838 |
|
17200 | 839 |
subsection {* Link with lists *} |
15072 | 840 |
|
26016 | 841 |
primrec multiset_of :: "'a list \<Rightarrow> 'a multiset" where |
842 |
"multiset_of [] = {#}" | |
|
843 |
"multiset_of (a # x) = multiset_of x + {# a #}" |
|
15072 | 844 |
|
845 |
lemma multiset_of_zero_iff[simp]: "(multiset_of x = {#}) = (x = [])" |
|
18258 | 846 |
by (induct x) auto |
15072 | 847 |
|
848 |
lemma multiset_of_zero_iff_right[simp]: "({#} = multiset_of x) = (x = [])" |
|
18258 | 849 |
by (induct x) auto |
15072 | 850 |
|
851 |
lemma set_of_multiset_of[simp]: "set_of(multiset_of x) = set x" |
|
18258 | 852 |
by (induct x) auto |
15867 | 853 |
|
854 |
lemma mem_set_multiset_eq: "x \<in> set xs = (x :# multiset_of xs)" |
|
855 |
by (induct xs) auto |
|
15072 | 856 |
|
18258 | 857 |
lemma multiset_of_append [simp]: |
858 |
"multiset_of (xs @ ys) = multiset_of xs + multiset_of ys" |
|
20503 | 859 |
by (induct xs arbitrary: ys) (auto simp: union_ac) |
18730 | 860 |
|
15072 | 861 |
lemma surj_multiset_of: "surj multiset_of" |
17200 | 862 |
apply (unfold surj_def, rule allI) |
863 |
apply (rule_tac M=y in multiset_induct, auto) |
|
864 |
apply (rule_tac x = "x # xa" in exI, auto) |
|
10249 | 865 |
done |
866 |
||
25162 | 867 |
lemma set_count_greater_0: "set x = {a. count (multiset_of x) a > 0}" |
18258 | 868 |
by (induct x) auto |
15072 | 869 |
|
17200 | 870 |
lemma distinct_count_atmost_1: |
15072 | 871 |
"distinct x = (! a. count (multiset_of x) a = (if a \<in> set x then 1 else 0))" |
18258 | 872 |
apply (induct x, simp, rule iffI, simp_all) |
17200 | 873 |
apply (rule conjI) |
874 |
apply (simp_all add: set_of_multiset_of [THEN sym] del: set_of_multiset_of) |
|
15072 | 875 |
apply (erule_tac x=a in allE, simp, clarify) |
17200 | 876 |
apply (erule_tac x=aa in allE, simp) |
15072 | 877 |
done |
878 |
||
17200 | 879 |
lemma multiset_of_eq_setD: |
15867 | 880 |
"multiset_of xs = multiset_of ys \<Longrightarrow> set xs = set ys" |
881 |
by (rule) (auto simp add:multiset_eq_conv_count_eq set_count_greater_0) |
|
882 |
||
17200 | 883 |
lemma set_eq_iff_multiset_of_eq_distinct: |
884 |
"\<lbrakk>distinct x; distinct y\<rbrakk> |
|
15072 | 885 |
\<Longrightarrow> (set x = set y) = (multiset_of x = multiset_of y)" |
17200 | 886 |
by (auto simp: multiset_eq_conv_count_eq distinct_count_atmost_1) |
15072 | 887 |
|
17200 | 888 |
lemma set_eq_iff_multiset_of_remdups_eq: |
15072 | 889 |
"(set x = set y) = (multiset_of (remdups x) = multiset_of (remdups y))" |
17200 | 890 |
apply (rule iffI) |
891 |
apply (simp add: set_eq_iff_multiset_of_eq_distinct[THEN iffD1]) |
|
892 |
apply (drule distinct_remdups[THEN distinct_remdups |
|
893 |
[THEN set_eq_iff_multiset_of_eq_distinct[THEN iffD2]]]) |
|
15072 | 894 |
apply simp |
10249 | 895 |
done |
896 |
||
18258 | 897 |
lemma multiset_of_compl_union [simp]: |
23281 | 898 |
"multiset_of [x\<leftarrow>xs. P x] + multiset_of [x\<leftarrow>xs. \<not>P x] = multiset_of xs" |
15630 | 899 |
by (induct xs) (auto simp: union_ac) |
15072 | 900 |
|
17200 | 901 |
lemma count_filter: |
23281 | 902 |
"count (multiset_of xs) x = length [y \<leftarrow> xs. y = x]" |
18258 | 903 |
by (induct xs) auto |
15867 | 904 |
|
26143
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
905 |
lemma nth_mem_multiset_of: "i < length ls \<Longrightarrow> (ls ! i) :# multiset_of ls" |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
906 |
by (induct ls arbitrary: i, simp, case_tac i, auto) |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
907 |
|
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
908 |
lemma multiset_of_remove1: "multiset_of (remove1 a xs) = multiset_of xs - {#a#}" |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
909 |
by (induct xs, auto simp add: multiset_eq_conv_count_eq) |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
910 |
|
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
911 |
lemma multiset_of_eq_length: |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
912 |
assumes "multiset_of xs = multiset_of ys" |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
913 |
shows "List.length xs = List.length ys" |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
914 |
using assms |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
915 |
proof (induct arbitrary: ys rule: length_induct) |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
916 |
case (1 xs ys) |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
917 |
show ?case |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
918 |
proof (cases xs) |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
919 |
case Nil with 1(2) show ?thesis by simp |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
920 |
next |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
921 |
case (Cons x xs') |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
922 |
note xCons = Cons |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
923 |
show ?thesis |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
924 |
proof (cases ys) |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
925 |
case Nil |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
926 |
with 1(2) Cons show ?thesis by simp |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
927 |
next |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
928 |
case (Cons y ys') |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
929 |
have x_in_ys: "x = y \<or> x \<in> set ys'" |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
930 |
proof (cases "x = y") |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
931 |
case True thus ?thesis .. |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
932 |
next |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
933 |
case False |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
934 |
from 1(2)[symmetric] xCons Cons have "x :# multiset_of ys' + {#y#}" by simp |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
935 |
with False show ?thesis by (simp add: mem_set_multiset_eq) |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
936 |
qed |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
937 |
from 1(1) have IH: "List.length xs' < List.length xs \<longrightarrow> |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
938 |
(\<forall>x. multiset_of xs' = multiset_of x \<longrightarrow> List.length xs' = List.length x)" by blast |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
939 |
from 1(2) x_in_ys Cons xCons have "multiset_of xs' = multiset_of (remove1 x (y#ys'))" |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
940 |
apply - |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
941 |
apply (simp add: multiset_of_remove1, simp only: add_eq_conv_diff) |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
942 |
apply fastsimp |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
943 |
done |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
944 |
with IH xCons have IH': "List.length xs' = List.length (remove1 x (y#ys'))" by fastsimp |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
945 |
from x_in_ys have "x \<noteq> y \<Longrightarrow> List.length ys' > 0" by auto |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
946 |
with Cons xCons x_in_ys IH' show ?thesis by (auto simp add: length_remove1) |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
947 |
qed |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
948 |
qed |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
949 |
qed |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
950 |
|
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
951 |
text {* This lemma shows which properties suffice to show that |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
952 |
a function f with f xs = ys behaves like sort. *} |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
953 |
lemma properties_for_sort: "\<lbrakk> multiset_of ys = multiset_of xs; sorted ys\<rbrakk> \<Longrightarrow> sort xs = ys" |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
954 |
proof (induct xs arbitrary: ys) |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
955 |
case Nil thus ?case by simp |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
956 |
next |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
957 |
case (Cons x xs) |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
958 |
hence "x \<in> set ys" by (auto simp add: mem_set_multiset_eq intro!: ccontr) |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
959 |
with Cons.prems Cons.hyps [of "remove1 x ys"] show ?case |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
960 |
by (simp add: sorted_remove1 multiset_of_remove1 insort_remove1) |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
961 |
qed |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
962 |
|
15867 | 963 |
|
15072 | 964 |
subsection {* Pointwise ordering induced by count *} |
965 |
||
19086 | 966 |
definition |
25610 | 967 |
mset_le :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "\<le>#" 50) where |
968 |
"(A \<le># B) = (\<forall>a. count A a \<le> count B a)" |
|
23611 | 969 |
definition |
25610 | 970 |
mset_less :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "<#" 50) where |
971 |
"(A <# B) = (A \<le># B \<and> A \<noteq> B)" |
|
972 |
||
973 |
notation mset_le (infix "\<subseteq>#" 50) |
|
974 |
notation mset_less (infix "\<subset>#" 50) |
|
15072 | 975 |
|
23611 | 976 |
lemma mset_le_refl[simp]: "A \<le># A" |
18730 | 977 |
unfolding mset_le_def by auto |
15072 | 978 |
|
23611 | 979 |
lemma mset_le_trans: "\<lbrakk> A \<le># B; B \<le># C \<rbrakk> \<Longrightarrow> A \<le># C" |
18730 | 980 |
unfolding mset_le_def by (fast intro: order_trans) |
15072 | 981 |
|
23611 | 982 |
lemma mset_le_antisym: "\<lbrakk> A \<le># B; B \<le># A \<rbrakk> \<Longrightarrow> A = B" |
17200 | 983 |
apply (unfold mset_le_def) |
984 |
apply (rule multiset_eq_conv_count_eq[THEN iffD2]) |
|
15072 | 985 |
apply (blast intro: order_antisym) |
986 |
done |
|
987 |
||
17200 | 988 |
lemma mset_le_exists_conv: |
23611 | 989 |
"(A \<le># B) = (\<exists>C. B = A + C)" |
990 |
apply (unfold mset_le_def, rule iffI, rule_tac x = "B - A" in exI) |
|
15072 | 991 |
apply (auto intro: multiset_eq_conv_count_eq [THEN iffD2]) |
992 |
done |
|
993 |
||
23611 | 994 |
lemma mset_le_mono_add_right_cancel[simp]: "(A + C \<le># B + C) = (A \<le># B)" |
18730 | 995 |
unfolding mset_le_def by auto |
15072 | 996 |
|
23611 | 997 |
lemma mset_le_mono_add_left_cancel[simp]: "(C + A \<le># C + B) = (A \<le># B)" |
18730 | 998 |
unfolding mset_le_def by auto |
15072 | 999 |
|
23611 | 1000 |
lemma mset_le_mono_add: "\<lbrakk> A \<le># B; C \<le># D \<rbrakk> \<Longrightarrow> A + C \<le># B + D" |
17200 | 1001 |
apply (unfold mset_le_def) |
1002 |
apply auto |
|
15072 | 1003 |
apply (erule_tac x=a in allE)+ |
1004 |
apply auto |
|
1005 |
done |
|
1006 |
||
23611 | 1007 |
lemma mset_le_add_left[simp]: "A \<le># A + B" |
18730 | 1008 |
unfolding mset_le_def by auto |
15072 | 1009 |
|
23611 | 1010 |
lemma mset_le_add_right[simp]: "B \<le># A + B" |
18730 | 1011 |
unfolding mset_le_def by auto |
15072 | 1012 |
|
26143
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1013 |
lemma mset_le_single: "a :# B \<Longrightarrow> {#a#} \<le># B" |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1014 |
by (simp add: mset_le_def) |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1015 |
|
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1016 |
lemma multiset_diff_union_assoc: "C \<le># B \<Longrightarrow> A + B - C = A + (B - C)" |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1017 |
by (simp add: multiset_eq_conv_count_eq mset_le_def) |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1018 |
|
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1019 |
lemma mset_le_multiset_union_diff_commute: |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1020 |
assumes "B \<le># A" |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1021 |
shows "A - B + C = A + C - B" |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1022 |
proof - |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1023 |
from mset_le_exists_conv [of "B" "A"] assms have "\<exists>D. A = B + D" .. |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1024 |
from this obtain D where "A = B + D" .. |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1025 |
thus ?thesis |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1026 |
apply - |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1027 |
apply simp |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1028 |
apply (subst union_commute) |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1029 |
apply (subst multiset_diff_union_assoc) |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1030 |
apply simp |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1031 |
apply (simp add: diff_cancel) |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1032 |
apply (subst union_assoc) |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1033 |
apply (subst union_commute[of "B" _]) |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1034 |
apply (subst multiset_diff_union_assoc) |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1035 |
apply simp |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1036 |
apply (simp add: diff_cancel) |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1037 |
done |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1038 |
qed |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1039 |
|
23611 | 1040 |
lemma multiset_of_remdups_le: "multiset_of (remdups xs) \<le># multiset_of xs" |
1041 |
apply (induct xs) |
|
1042 |
apply auto |
|
1043 |
apply (rule mset_le_trans) |
|
1044 |
apply auto |
|
1045 |
done |
|
1046 |
||
26143
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1047 |
lemma multiset_of_update: "i < length ls \<Longrightarrow> multiset_of (ls[i := v]) = multiset_of ls - {#ls ! i#} + {#v#}" |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1048 |
proof (induct ls arbitrary: i) |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1049 |
case Nil thus ?case by simp |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1050 |
next |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1051 |
case (Cons x xs) |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1052 |
show ?case |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1053 |
proof (cases i) |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1054 |
case 0 thus ?thesis by simp |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1055 |
next |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1056 |
case (Suc i') |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1057 |
with Cons show ?thesis |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1058 |
apply - |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1059 |
apply simp |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1060 |
apply (subst union_assoc) |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1061 |
apply (subst union_commute[where M="{#v#}" and N="{#x#}"]) |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1062 |
apply (subst union_assoc[symmetric]) |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1063 |
apply simp |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1064 |
apply (rule mset_le_multiset_union_diff_commute) |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1065 |
apply (simp add: mset_le_single nth_mem_multiset_of) |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1066 |
done |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1067 |
qed |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1068 |
qed |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1069 |
|
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1070 |
lemma multiset_of_swap: "\<lbrakk> i < length ls; j < length ls \<rbrakk> \<Longrightarrow> multiset_of (ls[j := ls ! i, i := ls ! j]) = multiset_of ls" |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1071 |
apply (case_tac "i=j") |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1072 |
apply simp |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1073 |
apply (simp add: multiset_of_update) |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1074 |
apply (subst elem_imp_eq_diff_union[symmetric]) |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1075 |
apply (simp add: nth_mem_multiset_of) |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1076 |
apply simp |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1077 |
done |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26033
diff
changeset
|
1078 |
|
25208 | 1079 |
interpretation mset_order: |
1080 |
order ["op \<le>#" "op <#"] |
|
1081 |
by (auto intro: order.intro mset_le_refl mset_le_antisym |
|
1082 |
mset_le_trans simp: mset_less_def) |
|
23611 | 1083 |
|
1084 |
interpretation mset_order_cancel_semigroup: |
|
25622 | 1085 |
pordered_cancel_ab_semigroup_add ["op +" "op \<le>#" "op <#"] |
25208 | 1086 |
by unfold_locales (erule mset_le_mono_add [OF mset_le_refl]) |
23611 | 1087 |
|
1088 |
interpretation mset_order_semigroup_cancel: |
|
25622 | 1089 |
pordered_ab_semigroup_add_imp_le ["op +" "op \<le>#" "op <#"] |
25208 | 1090 |
by (unfold_locales) simp |
15072 | 1091 |
|
25610 | 1092 |
|
1093 |
lemma mset_lessD: |
|
1094 |
"\<lbrakk> A \<subset># B ; x \<in># A \<rbrakk> \<Longrightarrow> x \<in># B" |
|
1095 |
apply (clarsimp simp: mset_le_def mset_less_def) |
|
1096 |
apply (erule_tac x=x in allE) |
|
1097 |
apply auto |
|
1098 |
done |
|
1099 |
||
1100 |
lemma mset_leD: |
|
1101 |
"\<lbrakk> A \<subseteq># B ; x \<in># A \<rbrakk> \<Longrightarrow> x \<in># B" |
|
1102 |
apply (clarsimp simp: mset_le_def mset_less_def) |
|
1103 |
apply (erule_tac x=x in allE) |
|
1104 |
apply auto |
|
1105 |
done |
|
1106 |
||
1107 |
lemma mset_less_insertD: |
|
1108 |
"(A + {#x#} \<subset># B) \<Longrightarrow> (x \<in># B \<and> A \<subset># B)" |
|
1109 |
apply (rule conjI) |
|
1110 |
apply (simp add: mset_lessD) |
|
1111 |
apply (clarsimp simp: mset_le_def mset_less_def) |
|
1112 |
apply safe |
|
1113 |
apply (erule_tac x=a in allE) |
|
1114 |
apply (auto split: split_if_asm) |
|
1115 |
done |
|
1116 |
||
1117 |
lemma mset_le_insertD: |
|
1118 |
"(A + {#x#} \<subseteq># B) \<Longrightarrow> (x \<in># B \<and> A \<subseteq># B)" |
|
1119 |
apply (rule conjI) |
|
1120 |
apply (simp add: mset_leD) |
|
1121 |
apply (force simp: mset_le_def mset_less_def split: split_if_asm) |
|
1122 |
done |
|
1123 |
||
1124 |
lemma mset_less_of_empty[simp]: "A \<subset># {#} = False" |
|
1125 |
by (induct A, auto simp: mset_le_def mset_less_def) |
|
1126 |
||
1127 |
lemma multi_psub_of_add_self[simp]: "A \<subset># A + {#x#}" |
|
1128 |
by (clarsimp simp: mset_le_def mset_less_def) |
|
1129 |
||
1130 |
lemma multi_psub_self[simp]: "A \<subset># A = False" |
|
1131 |
by (clarsimp simp: mset_le_def mset_less_def) |
|
1132 |
||
1133 |
lemma mset_less_add_bothsides: |
|
1134 |
"T + {#x#} \<subset># S + {#x#} \<Longrightarrow> T \<subset># S" |
|
1135 |
by (clarsimp simp: mset_le_def mset_less_def) |
|
1136 |
||
1137 |
lemma mset_less_empty_nonempty: "({#} \<subset># S) = (S \<noteq> {#})" |
|
1138 |
by (auto simp: mset_le_def mset_less_def) |
|
1139 |
||
1140 |
lemma mset_less_size: "A \<subset># B \<Longrightarrow> size A < size B" |
|
1141 |
proof (induct A arbitrary: B) |
|
1142 |
case (empty M) |
|
1143 |
hence "M \<noteq> {#}" by (simp add: mset_less_empty_nonempty) |
|
1144 |
then obtain M' x where "M = M' + {#x#}" |
|
1145 |
by (blast dest: multi_nonempty_split) |
|
1146 |
thus ?case by simp |
|
1147 |
next |
|
1148 |
case (add S x T) |
|
1149 |
have IH: "\<And>B. S \<subset># B \<Longrightarrow> size S < size B" by fact |
|
1150 |
have SxsubT: "S + {#x#} \<subset># T" by fact |
|
1151 |
hence "x \<in># T" and "S \<subset># T" by (auto dest: mset_less_insertD) |
|
1152 |
then obtain T' where T: "T = T' + {#x#}" |
|
1153 |
by (blast dest: multi_member_split) |
|
1154 |
hence "S \<subset># T'" using SxsubT |
|
1155 |
by (blast intro: mset_less_add_bothsides) |
|
1156 |
hence "size S < size T'" using IH by simp |
|
1157 |
thus ?case using T by simp |
|
1158 |
qed |
|
1159 |
||
1160 |
lemmas mset_less_trans = mset_order.less_eq_less.less_trans |
|
1161 |
||
1162 |
lemma mset_less_diff_self: "c \<in># B \<Longrightarrow> B - {#c#} \<subset># B" |
|
1163 |
by (auto simp: mset_le_def mset_less_def multi_drop_mem_not_eq) |
|
1164 |
||
1165 |
subsection {* Strong induction and subset induction for multisets *} |
|
1166 |
||
26016 | 1167 |
text {* Well-foundedness of proper subset operator: *} |
25610 | 1168 |
|
1169 |
definition |
|
1170 |
mset_less_rel :: "('a multiset * 'a multiset) set" |
|
1171 |
where |
|
1172 |
--{* proper multiset subset *} |
|
1173 |
"mset_less_rel \<equiv> {(A,B). A \<subset># B}" |
|
1174 |
||
1175 |
lemma multiset_add_sub_el_shuffle: |
|
1176 |
assumes cinB: "c \<in># B" and bnotc: "b \<noteq> c" |
|
1177 |
shows "B - {#c#} + {#b#} = B + {#b#} - {#c#}" |
|
1178 |
proof - |
|
1179 |
from cinB obtain A where B: "B = A + {#c#}" |
|
1180 |
by (blast dest: multi_member_split) |
|
1181 |
have "A + {#b#} = A + {#b#} + {#c#} - {#c#}" by simp |
|
1182 |
hence "A + {#b#} = A + {#c#} + {#b#} - {#c#}" |
|
1183 |
by (simp add: union_ac) |
|
1184 |
thus ?thesis using B by simp |
|
1185 |
qed |
|
1186 |
||
1187 |
lemma wf_mset_less_rel: "wf mset_less_rel" |
|
1188 |
apply (unfold mset_less_rel_def) |
|
1189 |
apply (rule wf_measure [THEN wf_subset, where f1=size]) |
|
1190 |
apply (clarsimp simp: measure_def inv_image_def mset_less_size) |
|
1191 |
done |
|
1192 |
||
26016 | 1193 |
text {* The induction rules: *} |
25610 | 1194 |
|
1195 |
lemma full_multiset_induct [case_names less]: |
|
1196 |
assumes ih: "\<And>B. \<forall>A. A \<subset># B \<longrightarrow> P A \<Longrightarrow> P B" |
|
1197 |
shows "P B" |
|
1198 |
apply (rule wf_mset_less_rel [THEN wf_induct]) |
|
1199 |
apply (rule ih, auto simp: mset_less_rel_def) |
|
1200 |
done |
|
1201 |
||
1202 |
lemma multi_subset_induct [consumes 2, case_names empty add]: |
|
1203 |
assumes "F \<subseteq># A" |
|
1204 |
and empty: "P {#}" |
|
1205 |
and insert: "\<And>a F. a \<in># A \<Longrightarrow> P F \<Longrightarrow> P (F + {#a#})" |
|
1206 |
shows "P F" |
|
1207 |
proof - |
|
1208 |
from `F \<subseteq># A` |
|
1209 |
show ?thesis |
|
1210 |
proof (induct F) |
|
1211 |
show "P {#}" by fact |
|
1212 |
next |
|
1213 |
fix x F |
|
1214 |
assume P: "F \<subseteq># A \<Longrightarrow> P F" and i: "F + {#x#} \<subseteq># A" |
|
1215 |
show "P (F + {#x#})" |
|
1216 |
proof (rule insert) |
|
1217 |
from i show "x \<in># A" by (auto dest: mset_le_insertD) |
|
1218 |
from i have "F \<subseteq># A" by (auto simp: mset_le_insertD) |
|
1219 |
with P show "P F" . |
|
1220 |
qed |
|
1221 |
qed |
|
1222 |
qed |
|
1223 |
||
26016 | 1224 |
text{* A consequence: Extensionality. *} |
25610 | 1225 |
|
1226 |
lemma multi_count_eq: |
|
1227 |
"(\<forall>x. count A x = count B x) = (A = B)" |
|
1228 |
apply (rule iffI) |
|
1229 |
prefer 2 |
|
1230 |
apply clarsimp |
|
1231 |
apply (induct A arbitrary: B rule: full_multiset_induct) |
|
1232 |
apply (rename_tac C) |
|
1233 |
apply (case_tac B rule: multiset_cases) |
|
1234 |
apply (simp add: empty_multiset_count) |
|
1235 |
apply simp |
|
1236 |
apply (case_tac "x \<in># C") |
|
1237 |
apply (force dest: multi_member_split) |
|
1238 |
apply (erule_tac x=x in allE) |
|
1239 |
apply simp |
|
1240 |
done |
|
1241 |
||
1242 |
lemmas multi_count_ext = multi_count_eq [THEN iffD1, rule_format] |
|
1243 |
||
1244 |
subsection {* The fold combinator *} |
|
1245 |
||
1246 |
text {* The intended behaviour is |
|
25759 | 1247 |
@{text "fold_mset f z {#x\<^isub>1, ..., x\<^isub>n#} = f x\<^isub>1 (\<dots> (f x\<^isub>n z)\<dots>)"} |
25610 | 1248 |
if @{text f} is associative-commutative. |
1249 |
*} |
|
1250 |
||
25759 | 1251 |
(* the graph of fold_mset, z = the start element, f = folding function, |
25610 | 1252 |
A the multiset, y the result *) |
1253 |
inductive |
|
25759 | 1254 |
fold_msetG :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a multiset \<Rightarrow> 'b \<Rightarrow> bool" |
25610 | 1255 |
for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" |
1256 |
and z :: 'b |
|
1257 |
where |
|
25759 | 1258 |
emptyI [intro]: "fold_msetG f z {#} z" |
1259 |
| insertI [intro]: "fold_msetG f z A y \<Longrightarrow> fold_msetG f z (A + {#x#}) (f x y)" |
|
25610 | 1260 |
|
25759 | 1261 |
inductive_cases empty_fold_msetGE [elim!]: "fold_msetG f z {#} x" |
1262 |
inductive_cases insert_fold_msetGE: "fold_msetG f z (A + {#}) y" |
|
25610 | 1263 |
|
1264 |
definition |
|
25759 | 1265 |
fold_mset :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a multiset \<Rightarrow> 'b" |
25610 | 1266 |
where |
25759 | 1267 |
"fold_mset f z A \<equiv> THE x. fold_msetG f z A x" |
25610 | 1268 |
|
25759 | 1269 |
lemma Diff1_fold_msetG: |
1270 |
"\<lbrakk> fold_msetG f z (A - {#x#}) y; x \<in># A \<rbrakk> \<Longrightarrow> fold_msetG f z A (f x y)" |
|
1271 |
by (frule_tac x=x in fold_msetG.insertI, auto) |
|
25610 | 1272 |
|
25759 | 1273 |
lemma fold_msetG_nonempty: "\<exists>x. fold_msetG f z A x" |
25610 | 1274 |
apply (induct A) |
1275 |
apply blast |
|
1276 |
apply clarsimp |
|
25759 | 1277 |
apply (drule_tac x=x in fold_msetG.insertI) |
25610 | 1278 |
apply auto |
1279 |
done |
|
1280 |
||
25759 | 1281 |
lemma fold_mset_empty[simp]: "fold_mset f z {#} = z" |
1282 |
by (unfold fold_mset_def, blast) |
|
25610 | 1283 |
|
1284 |
locale left_commutative = |
|
25623 | 1285 |
fixes f :: "'a => 'b => 'b" |
1286 |
assumes left_commute: "f x (f y z) = f y (f x z)" |
|
25610 | 1287 |
|
25759 | 1288 |
lemma (in left_commutative) fold_msetG_determ: |
1289 |
"\<lbrakk>fold_msetG f z A x; fold_msetG f z A y\<rbrakk> \<Longrightarrow> y = x" |
|
25610 | 1290 |
proof (induct arbitrary: x y z rule: full_multiset_induct) |
1291 |
case (less M x\<^isub>1 x\<^isub>2 Z) |
|
1292 |
have IH: "\<forall>A. A \<subset># M \<longrightarrow> |
|
25759 | 1293 |
(\<forall>x x' x''. fold_msetG f x'' A x \<longrightarrow> fold_msetG f x'' A x' |
25610 | 1294 |
\<longrightarrow> x' = x)" by fact |
25759 | 1295 |
have Mfoldx\<^isub>1: "fold_msetG f Z M x\<^isub>1" and Mfoldx\<^isub>2: "fold_msetG f Z M x\<^isub>2" by fact+ |
25610 | 1296 |
show ?case |
25759 | 1297 |
proof (rule fold_msetG.cases [OF Mfoldx\<^isub>1]) |
25610 | 1298 |
assume "M = {#}" and "x\<^isub>1 = Z" |
1299 |
thus ?case using Mfoldx\<^isub>2 by auto |
|
1300 |
next |
|
1301 |
fix B b u |
|
25759 | 1302 |
assume "M = B + {#b#}" and "x\<^isub>1 = f b u" and Bu: "fold_msetG f Z B u" |
25623 | 1303 |
hence MBb: "M = B + {#b#}" and x\<^isub>1: "x\<^isub>1 = f b u" by auto |
25610 | 1304 |
show ?case |
25759 | 1305 |
proof (rule fold_msetG.cases [OF Mfoldx\<^isub>2]) |
25610 | 1306 |
assume "M = {#}" "x\<^isub>2 = Z" |
1307 |
thus ?case using Mfoldx\<^isub>1 by auto |
|
1308 |
next |
|
1309 |
fix C c v |
|
25759 | 1310 |
assume "M = C + {#c#}" and "x\<^isub>2 = f c v" and Cv: "fold_msetG f Z C v" |
25623 | 1311 |
hence MCc: "M = C + {#c#}" and x\<^isub>2: "x\<^isub>2 = f c v" by auto |
25610 | 1312 |
hence CsubM: "C \<subset># M" by simp |
1313 |
from MBb have BsubM: "B \<subset># M" by simp |
|
1314 |
show ?case |
|
1315 |
proof cases |
|
1316 |
assume "b=c" |
|
1317 |
then moreover have "B = C" using MBb MCc by auto |
|
1318 |
ultimately show ?thesis using Bu Cv x\<^isub>1 x\<^isub>2 CsubM IH by auto |
|
1319 |
next |
|
1320 |
assume diff: "b \<noteq> c" |
|
1321 |
let ?D = "B - {#c#}" |
|
1322 |
have cinB: "c \<in># B" and binC: "b \<in># C" using MBb MCc diff |
|
1323 |
by (auto intro: insert_noteq_member dest: sym) |
|
1324 |
have "B - {#c#} \<subset># B" using cinB by (rule mset_less_diff_self) |
|
1325 |
hence DsubM: "?D \<subset># M" using BsubM by (blast intro: mset_less_trans) |
|
1326 |
from MBb MCc have "B + {#b#} = C + {#c#}" by blast |
|
1327 |
hence [simp]: "B + {#b#} - {#c#} = C" |
|
1328 |
using MBb MCc binC cinB by auto |
|
1329 |
have B: "B = ?D + {#c#}" and C: "C = ?D + {#b#}" |
|
1330 |
using MBb MCc diff binC cinB |
|
1331 |
by (auto simp: multiset_add_sub_el_shuffle) |
|
25759 | 1332 |
then obtain d where Dfoldd: "fold_msetG f Z ?D d" |
1333 |
using fold_msetG_nonempty by iprover |
|
1334 |
hence "fold_msetG f Z B (f c d)" using cinB |
|
1335 |
by (rule Diff1_fold_msetG) |
|
25623 | 1336 |
hence "f c d = u" using IH BsubM Bu by blast |
25610 | 1337 |
moreover |
25759 | 1338 |
have "fold_msetG f Z C (f b d)" using binC cinB diff Dfoldd |
25610 | 1339 |
by (auto simp: multiset_add_sub_el_shuffle |
25759 | 1340 |
dest: fold_msetG.insertI [where x=b]) |
25623 | 1341 |
hence "f b d = v" using IH CsubM Cv by blast |
25610 | 1342 |
ultimately show ?thesis using x\<^isub>1 x\<^isub>2 |
1343 |
by (auto simp: left_commute) |
|
1344 |
qed |
|
1345 |
qed |
|
1346 |
qed |
|
1347 |
qed |
|
1348 |
||
25759 | 1349 |
lemma (in left_commutative) fold_mset_insert_aux: " |
1350 |
(fold_msetG f z (A + {#x#}) v) = |
|
1351 |
(\<exists>y. fold_msetG f z A y \<and> v = f x y)" |
|
25610 | 1352 |
apply (rule iffI) |
1353 |
prefer 2 |
|
1354 |
apply blast |
|
25759 | 1355 |
apply (rule_tac A=A and f=f in fold_msetG_nonempty [THEN exE, standard]) |
1356 |
apply (blast intro: fold_msetG_determ) |
|
25610 | 1357 |
done |
1358 |
||
25759 | 1359 |
lemma (in left_commutative) fold_mset_equality: "fold_msetG f z A y \<Longrightarrow> fold_mset f z A = y" |
1360 |
by (unfold fold_mset_def) (blast intro: fold_msetG_determ) |
|
25610 | 1361 |
|
25759 | 1362 |
lemma (in left_commutative) fold_mset_insert: |
1363 |
"fold_mset f z (A + {#x#}) = f x (fold_mset f z A)" |
|
1364 |
apply (simp add: fold_mset_def fold_mset_insert_aux union_commute) |
|
25610 | 1365 |
apply (rule the_equality) |
1366 |
apply (auto cong add: conj_cong |
|
25759 | 1367 |
simp add: fold_mset_def [symmetric] fold_mset_equality fold_msetG_nonempty) |
1368 |
done |
|
1369 |
||
1370 |
lemma (in left_commutative) fold_mset_insert_idem: |
|
1371 |
shows "fold_mset f z (A + {#a#}) = f a (fold_mset f z A)" |
|
1372 |
apply (simp add: fold_mset_def fold_mset_insert_aux) |
|
1373 |
apply (rule the_equality) |
|
1374 |
apply (auto cong add: conj_cong |
|
1375 |
simp add: fold_mset_def [symmetric] fold_mset_equality fold_msetG_nonempty) |
|
25610 | 1376 |
done |
1377 |
||
25759 | 1378 |
lemma (in left_commutative) fold_mset_commute: |
1379 |
"f x (fold_mset f z A) = fold_mset f (f x z) A" |
|
1380 |
by (induct A, auto simp: fold_mset_insert left_commute [of x]) |
|
1381 |
||
1382 |
lemma (in left_commutative) fold_mset_single [simp]: |
|
1383 |
"fold_mset f z {#x#} = f x z" |
|
1384 |
using fold_mset_insert[of z "{#}"] by simp |
|
25610 | 1385 |
|
25759 | 1386 |
lemma (in left_commutative) fold_mset_union [simp]: |
1387 |
"fold_mset f z (A+B) = fold_mset f (fold_mset f z A) B" |
|
1388 |
proof (induct A) |
|
1389 |
case empty thus ?case by simp |
|
1390 |
next |
|
1391 |
case (add A x) |
|
1392 |
have "A + {#x#} + B = (A+B) + {#x#}" by(simp add:union_ac) |
|
1393 |
hence "fold_mset f z (A + {#x#} + B) = f x (fold_mset f z (A + B))" |
|
1394 |
by (simp add: fold_mset_insert) |
|
1395 |
also have "\<dots> = fold_mset f (fold_mset f z (A + {#x#})) B" |
|
1396 |
by (simp add: fold_mset_commute[of x,symmetric] add fold_mset_insert) |
|
1397 |
finally show ?case . |
|
1398 |
qed |
|
1399 |
||
1400 |
lemma (in left_commutative) fold_mset_fusion: |
|
25610 | 1401 |
includes left_commutative g |
25759 | 1402 |
shows "\<lbrakk>\<And>x y. h (g x y) = f x (h y) \<rbrakk> \<Longrightarrow> h (fold_mset g w A) = fold_mset f (h w) A" |
25610 | 1403 |
by (induct A, auto) |
1404 |
||
25759 | 1405 |
lemma (in left_commutative) fold_mset_rec: |
25610 | 1406 |
assumes a: "a \<in># A" |
25759 | 1407 |
shows "fold_mset f z A = f a (fold_mset f z (A - {#a#}))" |
25610 | 1408 |
proof - |
1409 |
from a obtain A' where "A = A' + {#a#}" by (blast dest: multi_member_split) |
|
1410 |
thus ?thesis by simp |
|
1411 |
qed |
|
1412 |
||
26016 | 1413 |
text{* A note on code generation: When defining some |
1414 |
function containing a subterm @{term"fold_mset F"}, code generation is |
|
1415 |
not automatic. When interpreting locale @{text left_commutative} with |
|
1416 |
@{text F}, the would be code thms for @{const fold_mset} become thms like |
|
1417 |
@{term"fold_mset F z {#} = z"} where @{text F} is not a pattern but contains |
|
1418 |
defined symbols, i.e.\ is not a code thm. Hence a separate constant with its |
|
1419 |
own code thms needs to be introduced for @{text F}. See the image operator |
|
1420 |
below. *} |
|
1421 |
||
1422 |
subsection {* Image *} |
|
1423 |
||
1424 |
definition [code func del]: "image_mset f == fold_mset (op + o single o f) {#}" |
|
1425 |
||
1426 |
interpretation image_left_comm: left_commutative["op + o single o f"] |
|
1427 |
by(unfold_locales)(simp add:union_ac) |
|
1428 |
||
1429 |
lemma image_mset_empty[simp,code func]: "image_mset f {#} = {#}" |
|
1430 |
by(simp add:image_mset_def) |
|
1431 |
||
1432 |
lemma image_mset_single[simp,code func]: "image_mset f {#x#} = {#f x#}" |
|
1433 |
by(simp add:image_mset_def) |
|
1434 |
||
1435 |
lemma image_mset_insert: |
|
1436 |
"image_mset f (M + {#a#}) = image_mset f M + {#f a#}" |
|
1437 |
by(simp add:image_mset_def add_ac) |
|
1438 |
||
1439 |
lemma image_mset_union[simp, code func]: |
|
1440 |
"image_mset f (M+N) = image_mset f M + image_mset f N" |
|
1441 |
apply(induct N) |
|
1442 |
apply simp |
|
1443 |
apply(simp add:union_assoc[symmetric] image_mset_insert) |
|
1444 |
done |
|
1445 |
||
1446 |
lemma size_image_mset[simp]: "size(image_mset f M) = size M" |
|
1447 |
by(induct M) simp_all |
|
1448 |
||
1449 |
lemma image_mset_is_empty_iff[simp]: "image_mset f M = {#} \<longleftrightarrow> M={#}" |
|
1450 |
by (cases M) auto |
|
1451 |
||
1452 |
||
1453 |
syntax comprehension1_mset :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> 'a multiset" |
|
1454 |
("({#_/. _ :# _#})") |
|
1455 |
translations "{#e. x:#M#}" == "CONST image_mset (%x. e) M" |
|
1456 |
||
1457 |
syntax comprehension2_mset :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> bool \<Rightarrow> 'a multiset" |
|
1458 |
("({#_/ | _ :# _./ _#})") |
|
1459 |
translations |
|
26033 | 1460 |
"{#e | x:#M. P#}" => "{#e. x :# {# x:#M. P#}#}" |
26016 | 1461 |
|
26033 | 1462 |
text{* This allows to write not just filters like @{term"{#x:#M. x<c#}"} |
26016 | 1463 |
but also images like @{term"{#x+x. x:#M #}"} |
1464 |
and @{term[source]"{#x+x|x:#M. x<c#}"}, where the latter is currently |
|
1465 |
displayed as @{term"{#x+x|x:#M. x<c#}"}. *} |
|
1466 |
||
10249 | 1467 |
end |