author | wenzelm |
Sat, 20 Aug 2011 23:36:18 +0200 | |
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parent 42871 | 1c0b99f950d9 |
child 44890 | 22f665a2e91c |
permissions | -rw-r--r-- |
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(* Title: HOL/Library/Multiset.thy |
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Author: Tobias Nipkow, Markus Wenzel, Lawrence C Paulson, Norbert Voelker |
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*) |
4 |
||
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5 |
header {* (Finite) multisets *} |
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|
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theory Multiset |
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8 |
imports Main |
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begin |
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|
11 |
subsection {* The type of multisets *} |
|
12 |
||
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typedef 'a multiset = "{f :: 'a => nat. finite {x. f x > 0}}" |
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14 |
morphisms count Abs_multiset |
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proof |
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show "(\<lambda>x. 0::nat) \<in> ?multiset" by simp |
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qed |
18 |
||
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lemmas multiset_typedef = Abs_multiset_inverse count_inverse count |
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|
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abbreviation Melem :: "'a => 'a multiset => bool" ("(_/ :# _)" [50, 51] 50) where |
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"a :# M == 0 < count M a" |
23 |
||
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notation (xsymbols) |
25 |
Melem (infix "\<in>#" 50) |
|
10249 | 26 |
|
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lemma multiset_eq_iff: |
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"M = N \<longleftrightarrow> (\<forall>a. count M a = count N a)" |
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29 |
by (simp only: count_inject [symmetric] fun_eq_iff) |
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30 |
|
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31 |
lemma multiset_eqI: |
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"(\<And>x. count A x = count B x) \<Longrightarrow> A = B" |
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33 |
using multiset_eq_iff by auto |
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34 |
|
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35 |
text {* |
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36 |
\medskip Preservation of the representing set @{term multiset}. |
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37 |
*} |
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38 |
|
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39 |
lemma const0_in_multiset: |
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40 |
"(\<lambda>a. 0) \<in> multiset" |
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41 |
by (simp add: multiset_def) |
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42 |
|
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43 |
lemma only1_in_multiset: |
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44 |
"(\<lambda>b. if b = a then n else 0) \<in> multiset" |
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45 |
by (simp add: multiset_def) |
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46 |
|
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47 |
lemma union_preserves_multiset: |
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48 |
"M \<in> multiset \<Longrightarrow> N \<in> multiset \<Longrightarrow> (\<lambda>a. M a + N a) \<in> multiset" |
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49 |
by (simp add: multiset_def) |
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50 |
|
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51 |
lemma diff_preserves_multiset: |
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52 |
assumes "M \<in> multiset" |
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53 |
shows "(\<lambda>a. M a - N a) \<in> multiset" |
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54 |
proof - |
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55 |
have "{x. N x < M x} \<subseteq> {x. 0 < M x}" |
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56 |
by auto |
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57 |
with assms show ?thesis |
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58 |
by (auto simp add: multiset_def intro: finite_subset) |
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59 |
qed |
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60 |
|
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61 |
lemma filter_preserves_multiset: |
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62 |
assumes "M \<in> multiset" |
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63 |
shows "(\<lambda>x. if P x then M x else 0) \<in> multiset" |
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64 |
proof - |
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65 |
have "{x. (P x \<longrightarrow> 0 < M x) \<and> P x} \<subseteq> {x. 0 < M x}" |
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66 |
by auto |
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67 |
with assms show ?thesis |
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68 |
by (auto simp add: multiset_def intro: finite_subset) |
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69 |
qed |
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70 |
|
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71 |
lemmas in_multiset = const0_in_multiset only1_in_multiset |
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72 |
union_preserves_multiset diff_preserves_multiset filter_preserves_multiset |
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73 |
|
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74 |
|
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75 |
subsection {* Representing multisets *} |
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76 |
|
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77 |
text {* Multiset enumeration *} |
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78 |
|
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79 |
instantiation multiset :: (type) "{zero, plus}" |
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80 |
begin |
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81 |
|
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82 |
definition Mempty_def: |
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83 |
"0 = Abs_multiset (\<lambda>a. 0)" |
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84 |
|
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85 |
abbreviation Mempty :: "'a multiset" ("{#}") where |
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86 |
"Mempty \<equiv> 0" |
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87 |
|
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88 |
definition union_def: |
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89 |
"M + N = Abs_multiset (\<lambda>a. count M a + count N a)" |
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90 |
|
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91 |
instance .. |
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92 |
|
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93 |
end |
10249 | 94 |
|
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95 |
definition single :: "'a => 'a multiset" where |
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96 |
"single a = Abs_multiset (\<lambda>b. if b = a then 1 else 0)" |
15869 | 97 |
|
26145 | 98 |
syntax |
26176 | 99 |
"_multiset" :: "args => 'a multiset" ("{#(_)#}") |
25507 | 100 |
translations |
101 |
"{#x, xs#}" == "{#x#} + {#xs#}" |
|
102 |
"{#x#}" == "CONST single x" |
|
103 |
||
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104 |
lemma count_empty [simp]: "count {#} a = 0" |
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105 |
by (simp add: Mempty_def in_multiset multiset_typedef) |
10249 | 106 |
|
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107 |
lemma count_single [simp]: "count {#b#} a = (if b = a then 1 else 0)" |
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108 |
by (simp add: single_def in_multiset multiset_typedef) |
29901 | 109 |
|
10249 | 110 |
|
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111 |
subsection {* Basic operations *} |
10249 | 112 |
|
113 |
subsubsection {* Union *} |
|
114 |
||
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115 |
lemma count_union [simp]: "count (M + N) a = count M a + count N a" |
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116 |
by (simp add: union_def in_multiset multiset_typedef) |
10249 | 117 |
|
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118 |
instance multiset :: (type) cancel_comm_monoid_add proof |
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119 |
qed (simp_all add: multiset_eq_iff) |
10277 | 120 |
|
10249 | 121 |
|
122 |
subsubsection {* Difference *} |
|
123 |
||
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124 |
instantiation multiset :: (type) minus |
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125 |
begin |
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126 |
|
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127 |
definition diff_def: |
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128 |
"M - N = Abs_multiset (\<lambda>a. count M a - count N a)" |
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129 |
|
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130 |
instance .. |
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131 |
|
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132 |
end |
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133 |
|
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134 |
lemma count_diff [simp]: "count (M - N) a = count M a - count N a" |
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135 |
by (simp add: diff_def in_multiset multiset_typedef) |
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136 |
|
17161 | 137 |
lemma diff_empty [simp]: "M - {#} = M \<and> {#} - M = {#}" |
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138 |
by(simp add: multiset_eq_iff) |
36903 | 139 |
|
140 |
lemma diff_cancel[simp]: "A - A = {#}" |
|
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141 |
by (rule multiset_eqI) simp |
10249 | 142 |
|
36903 | 143 |
lemma diff_union_cancelR [simp]: "M + N - N = (M::'a multiset)" |
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144 |
by(simp add: multiset_eq_iff) |
10249 | 145 |
|
36903 | 146 |
lemma diff_union_cancelL [simp]: "N + M - N = (M::'a multiset)" |
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147 |
by(simp add: multiset_eq_iff) |
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148 |
|
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149 |
lemma insert_DiffM: |
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150 |
"x \<in># M \<Longrightarrow> {#x#} + (M - {#x#}) = M" |
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151 |
by (clarsimp simp: multiset_eq_iff) |
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152 |
|
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153 |
lemma insert_DiffM2 [simp]: |
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154 |
"x \<in># M \<Longrightarrow> M - {#x#} + {#x#} = M" |
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155 |
by (clarsimp simp: multiset_eq_iff) |
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156 |
|
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157 |
lemma diff_right_commute: |
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158 |
"(M::'a multiset) - N - Q = M - Q - N" |
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159 |
by (auto simp add: multiset_eq_iff) |
36903 | 160 |
|
161 |
lemma diff_add: |
|
162 |
"(M::'a multiset) - (N + Q) = M - N - Q" |
|
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163 |
by (simp add: multiset_eq_iff) |
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164 |
|
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165 |
lemma diff_union_swap: |
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166 |
"a \<noteq> b \<Longrightarrow> M - {#a#} + {#b#} = M + {#b#} - {#a#}" |
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167 |
by (auto simp add: multiset_eq_iff) |
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168 |
|
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169 |
lemma diff_union_single_conv: |
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170 |
"a \<in># J \<Longrightarrow> I + J - {#a#} = I + (J - {#a#})" |
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171 |
by (simp add: multiset_eq_iff) |
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172 |
|
10249 | 173 |
|
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174 |
subsubsection {* Equality of multisets *} |
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175 |
|
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176 |
lemma single_not_empty [simp]: "{#a#} \<noteq> {#} \<and> {#} \<noteq> {#a#}" |
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177 |
by (simp add: multiset_eq_iff) |
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178 |
|
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179 |
lemma single_eq_single [simp]: "{#a#} = {#b#} \<longleftrightarrow> a = b" |
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180 |
by (auto simp add: multiset_eq_iff) |
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181 |
|
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182 |
lemma union_eq_empty [iff]: "M + N = {#} \<longleftrightarrow> M = {#} \<and> N = {#}" |
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183 |
by (auto simp add: multiset_eq_iff) |
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184 |
|
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185 |
lemma empty_eq_union [iff]: "{#} = M + N \<longleftrightarrow> M = {#} \<and> N = {#}" |
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186 |
by (auto simp add: multiset_eq_iff) |
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187 |
|
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188 |
lemma multi_self_add_other_not_self [simp]: "M = M + {#x#} \<longleftrightarrow> False" |
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189 |
by (auto simp add: multiset_eq_iff) |
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190 |
|
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191 |
lemma diff_single_trivial: |
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192 |
"\<not> x \<in># M \<Longrightarrow> M - {#x#} = M" |
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|
193 |
by (auto simp add: multiset_eq_iff) |
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194 |
|
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195 |
lemma diff_single_eq_union: |
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196 |
"x \<in># M \<Longrightarrow> M - {#x#} = N \<longleftrightarrow> M = N + {#x#}" |
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|
197 |
by auto |
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198 |
|
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199 |
lemma union_single_eq_diff: |
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200 |
"M + {#x#} = N \<Longrightarrow> M = N - {#x#}" |
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201 |
by (auto dest: sym) |
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202 |
|
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203 |
lemma union_single_eq_member: |
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204 |
"M + {#x#} = N \<Longrightarrow> x \<in># N" |
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205 |
by auto |
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206 |
|
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207 |
lemma union_is_single: |
36903 | 208 |
"M + N = {#a#} \<longleftrightarrow> M = {#a#} \<and> N={#} \<or> M = {#} \<and> N = {#a#}" (is "?lhs = ?rhs")proof |
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209 |
assume ?rhs then show ?lhs by auto |
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210 |
next |
36903 | 211 |
assume ?lhs thus ?rhs |
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212 |
by(simp add: multiset_eq_iff split:if_splits) (metis add_is_1) |
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213 |
qed |
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214 |
|
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215 |
lemma single_is_union: |
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216 |
"{#a#} = M + N \<longleftrightarrow> {#a#} = M \<and> N = {#} \<or> M = {#} \<and> {#a#} = N" |
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217 |
by (auto simp add: eq_commute [of "{#a#}" "M + N"] union_is_single) |
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218 |
|
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219 |
lemma add_eq_conv_diff: |
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220 |
"M + {#a#} = N + {#b#} \<longleftrightarrow> M = N \<and> a = b \<or> M = N - {#a#} + {#b#} \<and> N = M - {#b#} + {#a#}" (is "?lhs = ?rhs") |
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221 |
(* shorter: by (simp add: multiset_eq_iff) fastsimp *) |
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222 |
proof |
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223 |
assume ?rhs then show ?lhs |
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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
|
224 |
by (auto simp add: add_assoc add_commute [of "{#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
|
225 |
(drule sym, simp add: add_assoc [symmetric]) |
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
|
226 |
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:
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diff
changeset
|
227 |
assume ?lhs |
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
|
228 |
show ?rhs |
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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changeset
|
229 |
proof (cases "a = 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:
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diff
changeset
|
230 |
case True with `?lhs` 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:
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diff
changeset
|
231 |
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:
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diff
changeset
|
232 |
case 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:
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diff
changeset
|
233 |
from `?lhs` have "a \<in># N + {#b#}" by (rule union_single_eq_member) |
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
|
234 |
with False have "a \<in># N" 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:
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diff
changeset
|
235 |
moreover from `?lhs` have "M = N + {#b#} - {#a#}" by (rule union_single_eq_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:
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diff
changeset
|
236 |
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:
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diff
changeset
|
237 |
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
|
238 |
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:
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diff
changeset
|
239 |
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
|
240 |
|
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
|
241 |
lemma insert_noteq_member: |
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
|
242 |
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:
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diff
changeset
|
243 |
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
|
244 |
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:
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diff
changeset
|
245 |
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
|
246 |
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
|
247 |
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:
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diff
changeset
|
248 |
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
|
249 |
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
|
250 |
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
|
251 |
|
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
|
252 |
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
|
253 |
"(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:
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diff
changeset
|
254 |
(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
|
255 |
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
|
256 |
|
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
|
257 |
|
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
|
258 |
subsubsection {* Pointwise ordering induced by count *} |
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
|
259 |
|
35268
04673275441a
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haftmann
parents:
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diff
changeset
|
260 |
instantiation multiset :: (type) ordered_ab_semigroup_add_imp_le |
04673275441a
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haftmann
parents:
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diff
changeset
|
261 |
begin |
04673275441a
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haftmann
parents:
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diff
changeset
|
262 |
|
04673275441a
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haftmann
parents:
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diff
changeset
|
263 |
definition less_eq_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" where |
04673275441a
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haftmann
parents:
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diff
changeset
|
264 |
mset_le_def: "A \<le> B \<longleftrightarrow> (\<forall>a. count A a \<le> count B 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:
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diff
changeset
|
265 |
|
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
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diff
changeset
|
266 |
definition less_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" where |
04673275441a
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haftmann
parents:
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diff
changeset
|
267 |
mset_less_def: "(A::'a multiset) < B \<longleftrightarrow> A \<le> B \<and> A \<noteq> 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
|
268 |
|
35268
04673275441a
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haftmann
parents:
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diff
changeset
|
269 |
instance proof |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39301
diff
changeset
|
270 |
qed (auto simp add: mset_le_def mset_less_def multiset_eq_iff intro: order_trans antisym) |
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
271 |
|
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
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diff
changeset
|
272 |
end |
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
|
273 |
|
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
|
274 |
lemma mset_less_eqI: |
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
275 |
"(\<And>x. count A x \<le> count B x) \<Longrightarrow> A \<le> 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
|
276 |
by (simp add: mset_le_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:
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diff
changeset
|
277 |
|
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
278 |
lemma mset_le_exists_conv: |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
279 |
"(A::'a multiset) \<le> B \<longleftrightarrow> (\<exists>C. B = A + C)" |
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
|
280 |
apply (unfold mset_le_def, rule iffI, rule_tac x = "B - A" in exI) |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39301
diff
changeset
|
281 |
apply (auto intro: multiset_eq_iff [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
|
282 |
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
|
283 |
|
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
284 |
lemma mset_le_mono_add_right_cancel [simp]: |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
285 |
"(A::'a multiset) + C \<le> B + C \<longleftrightarrow> A \<le> B" |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
286 |
by (fact add_le_cancel_right) |
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
|
287 |
|
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
288 |
lemma mset_le_mono_add_left_cancel [simp]: |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
289 |
"C + (A::'a multiset) \<le> C + B \<longleftrightarrow> A \<le> B" |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
290 |
by (fact add_le_cancel_left) |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
291 |
|
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
292 |
lemma mset_le_mono_add: |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
293 |
"(A::'a multiset) \<le> B \<Longrightarrow> C \<le> D \<Longrightarrow> A + C \<le> B + D" |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
294 |
by (fact add_mono) |
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
|
295 |
|
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
296 |
lemma mset_le_add_left [simp]: |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
297 |
"(A::'a multiset) \<le> A + B" |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
298 |
unfolding mset_le_def by auto |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
299 |
|
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
300 |
lemma mset_le_add_right [simp]: |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
301 |
"B \<le> (A::'a multiset) + B" |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
302 |
unfolding mset_le_def by 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
|
303 |
|
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
304 |
lemma mset_le_single: |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
305 |
"a :# B \<Longrightarrow> {#a#} \<le> B" |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
306 |
by (simp add: mset_le_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
|
307 |
|
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
308 |
lemma multiset_diff_union_assoc: |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
309 |
"C \<le> B \<Longrightarrow> (A::'a multiset) + B - C = A + (B - C)" |
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 (simp add: multiset_eq_iff mset_le_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
|
311 |
|
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
|
312 |
lemma mset_le_multiset_union_diff_commute: |
36867 | 313 |
"B \<le> A \<Longrightarrow> (A::'a multiset) - 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
|
314 |
by (simp add: multiset_eq_iff mset_le_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
|
315 |
|
39301 | 316 |
lemma diff_le_self[simp]: "(M::'a multiset) - N \<le> M" |
317 |
by(simp add: mset_le_def) |
|
318 |
||
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
319 |
lemma mset_lessD: "A < B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># 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
|
320 |
apply (clarsimp simp: mset_le_def mset_less_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
|
321 |
apply (erule_tac x=x in allE) |
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
|
322 |
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
|
323 |
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
|
324 |
|
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
325 |
lemma mset_leD: "A \<le> B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B" |
34943
e97b22500a5c
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parents:
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changeset
|
326 |
apply (clarsimp simp: mset_le_def mset_less_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:
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diff
changeset
|
327 |
apply (erule_tac x = x in allE) |
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
|
328 |
apply auto |
e97b22500a5c
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parents:
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diff
changeset
|
329 |
done |
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parents:
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changeset
|
330 |
|
35268
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|
331 |
lemma mset_less_insertD: "(A + {#x#} < B) \<Longrightarrow> (x \<in># B \<and> A < B)" |
34943
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parents:
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changeset
|
332 |
apply (rule conjI) |
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
|
333 |
apply (simp add: mset_lessD) |
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
|
334 |
apply (clarsimp simp: mset_le_def mset_less_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
|
335 |
apply safe |
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
|
336 |
apply (erule_tac x = a in allE) |
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 |
apply (auto split: split_if_asm) |
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
|
338 |
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
|
339 |
|
35268
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parents:
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changeset
|
340 |
lemma mset_le_insertD: "(A + {#x#} \<le> B) \<Longrightarrow> (x \<in># B \<and> A \<le> B)" |
34943
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parents:
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diff
changeset
|
341 |
apply (rule conjI) |
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
|
342 |
apply (simp add: mset_leD) |
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 |
apply (force simp: mset_le_def mset_less_def split: split_if_asm) |
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
|
344 |
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
|
345 |
|
35268
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haftmann
parents:
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diff
changeset
|
346 |
lemma mset_less_of_empty[simp]: "A < {#} \<longleftrightarrow> False" |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
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parents:
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diff
changeset
|
347 |
by (auto simp add: mset_less_def mset_le_def 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:
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diff
changeset
|
348 |
|
35268
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haftmann
parents:
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changeset
|
349 |
lemma multi_psub_of_add_self[simp]: "A < A + {#x#}" |
04673275441a
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haftmann
parents:
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diff
changeset
|
350 |
by (auto simp: mset_le_def mset_less_def) |
34943
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parents:
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changeset
|
351 |
|
35268
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parents:
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|
352 |
lemma multi_psub_self[simp]: "(A::'a multiset) < A = False" |
04673275441a
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parents:
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changeset
|
353 |
by simp |
34943
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parents:
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diff
changeset
|
354 |
|
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 |
lemma mset_less_add_bothsides: |
35268
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haftmann
parents:
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diff
changeset
|
356 |
"T + {#x#} < S + {#x#} \<Longrightarrow> T < S" |
04673275441a
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haftmann
parents:
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diff
changeset
|
357 |
by (fact add_less_imp_less_right) |
04673275441a
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haftmann
parents:
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diff
changeset
|
358 |
|
04673275441a
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haftmann
parents:
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diff
changeset
|
359 |
lemma mset_less_empty_nonempty: |
04673275441a
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haftmann
parents:
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diff
changeset
|
360 |
"{#} < S \<longleftrightarrow> S \<noteq> {#}" |
04673275441a
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haftmann
parents:
35028
diff
changeset
|
361 |
by (auto simp: mset_le_def mset_less_def) |
04673275441a
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haftmann
parents:
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diff
changeset
|
362 |
|
04673275441a
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haftmann
parents:
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diff
changeset
|
363 |
lemma mset_less_diff_self: |
04673275441a
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haftmann
parents:
35028
diff
changeset
|
364 |
"c \<in># B \<Longrightarrow> B - {#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
|
365 |
by (auto simp: mset_le_def mset_less_def multiset_eq_iff) |
35268
04673275441a
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haftmann
parents:
35028
diff
changeset
|
366 |
|
04673275441a
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haftmann
parents:
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diff
changeset
|
367 |
|
04673275441a
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haftmann
parents:
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changeset
|
368 |
subsubsection {* Intersection *} |
04673275441a
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haftmann
parents:
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diff
changeset
|
369 |
|
04673275441a
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parents:
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|
370 |
instantiation multiset :: (type) semilattice_inf |
04673275441a
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haftmann
parents:
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|
371 |
begin |
04673275441a
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parents:
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changeset
|
372 |
|
04673275441a
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parents:
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|
373 |
definition inf_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" where |
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haftmann
parents:
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|
374 |
multiset_inter_def: "inf_multiset A B = A - (A - B)" |
04673275441a
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haftmann
parents:
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diff
changeset
|
375 |
|
04673275441a
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haftmann
parents:
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diff
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|
376 |
instance proof - |
04673275441a
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haftmann
parents:
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diff
changeset
|
377 |
have aux: "\<And>m n q :: nat. m \<le> n \<Longrightarrow> m \<le> q \<Longrightarrow> m \<le> n - (n - q)" by arith |
04673275441a
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haftmann
parents:
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changeset
|
378 |
show "OFCLASS('a multiset, semilattice_inf_class)" proof |
04673275441a
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haftmann
parents:
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diff
changeset
|
379 |
qed (auto simp add: multiset_inter_def mset_le_def aux) |
04673275441a
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haftmann
parents:
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diff
changeset
|
380 |
qed |
04673275441a
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haftmann
parents:
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diff
changeset
|
381 |
|
04673275441a
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haftmann
parents:
35028
diff
changeset
|
382 |
end |
04673275441a
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haftmann
parents:
35028
diff
changeset
|
383 |
|
04673275441a
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haftmann
parents:
35028
diff
changeset
|
384 |
abbreviation multiset_inter :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" (infixl "#\<inter>" 70) where |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
385 |
"multiset_inter \<equiv> inf" |
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 |
|
41069
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
387 |
lemma multiset_inter_count [simp]: |
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
388 |
"count (A #\<inter> B) x = min (count A x) (count B x)" |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
389 |
by (simp add: multiset_inter_def multiset_typedef) |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
390 |
|
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
391 |
lemma multiset_inter_single: "a \<noteq> b \<Longrightarrow> {#a#} #\<inter> {#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
|
392 |
by (rule multiset_eqI) (auto simp add: multiset_inter_count) |
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
|
393 |
|
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
394 |
lemma multiset_union_diff_commute: |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
395 |
assumes "B #\<inter> C = {#}" |
04673275441a
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haftmann
parents:
35028
diff
changeset
|
396 |
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
|
397 |
proof (rule multiset_eqI) |
35268
04673275441a
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haftmann
parents:
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diff
changeset
|
398 |
fix x |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
399 |
from assms have "min (count B x) (count C x) = 0" |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39301
diff
changeset
|
400 |
by (auto simp add: multiset_inter_count multiset_eq_iff) |
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
401 |
then have "count B x = 0 \<or> count C x = 0" |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
402 |
by auto |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
403 |
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
|
404 |
by auto |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
405 |
qed |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
406 |
|
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
407 |
|
41069
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
408 |
subsubsection {* Filter (with comprehension syntax) *} |
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
409 |
|
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
410 |
text {* Multiset comprehension *} |
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
411 |
|
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
412 |
definition filter :: "('a \<Rightarrow> bool) \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" where |
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
413 |
"filter P M = Abs_multiset (\<lambda>x. if P x then count M x else 0)" |
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
414 |
|
41069
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
415 |
hide_const (open) filter |
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
416 |
|
41069
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
417 |
lemma count_filter [simp]: |
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
418 |
"count (Multiset.filter P M) a = (if P a then count M a else 0)" |
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
419 |
by (simp add: filter_def in_multiset multiset_typedef) |
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
420 |
|
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
421 |
lemma filter_empty [simp]: |
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
422 |
"Multiset.filter P {#} = {#}" |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39301
diff
changeset
|
423 |
by (rule multiset_eqI) simp |
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
424 |
|
41069
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
425 |
lemma filter_single [simp]: |
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
426 |
"Multiset.filter P {#x#} = (if P x then {#x#} else {#})" |
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
427 |
by (rule multiset_eqI) simp |
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
428 |
|
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
429 |
lemma filter_union [simp]: |
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
430 |
"Multiset.filter P (M + N) = Multiset.filter P M + Multiset.filter 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
|
431 |
by (rule multiset_eqI) simp |
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
432 |
|
41069
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
433 |
lemma filter_diff [simp]: |
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
434 |
"Multiset.filter P (M - N) = Multiset.filter P M - Multiset.filter P N" |
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
435 |
by (rule multiset_eqI) simp |
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
436 |
|
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
437 |
lemma filter_inter [simp]: |
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
438 |
"Multiset.filter P (M #\<inter> N) = Multiset.filter P M #\<inter> Multiset.filter 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
|
439 |
by (rule multiset_eqI) simp |
10249 | 440 |
|
41069
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
441 |
syntax |
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
442 |
"_MCollect" :: "pttrn \<Rightarrow> 'a multiset \<Rightarrow> bool \<Rightarrow> 'a multiset" ("(1{# _ :# _./ _#})") |
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
443 |
syntax (xsymbol) |
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
444 |
"_MCollect" :: "pttrn \<Rightarrow> 'a multiset \<Rightarrow> bool \<Rightarrow> 'a multiset" ("(1{# _ \<in># _./ _#})") |
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
445 |
translations |
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
446 |
"{#x \<in># M. P#}" == "CONST Multiset.filter (\<lambda>x. P) M" |
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
447 |
|
10249 | 448 |
|
449 |
subsubsection {* Set of elements *} |
|
450 |
||
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
|
451 |
definition set_of :: "'a multiset => 'a set" 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:
33102
diff
changeset
|
452 |
"set_of M = {x. x :# 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
|
453 |
|
17161 | 454 |
lemma set_of_empty [simp]: "set_of {#} = {}" |
26178 | 455 |
by (simp add: set_of_def) |
10249 | 456 |
|
17161 | 457 |
lemma set_of_single [simp]: "set_of {#b#} = {b}" |
26178 | 458 |
by (simp add: set_of_def) |
10249 | 459 |
|
17161 | 460 |
lemma set_of_union [simp]: "set_of (M + N) = set_of M \<union> set_of N" |
26178 | 461 |
by (auto simp add: set_of_def) |
10249 | 462 |
|
17161 | 463 |
lemma set_of_eq_empty_iff [simp]: "(set_of M = {}) = (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
|
464 |
by (auto simp add: set_of_def multiset_eq_iff) |
10249 | 465 |
|
17161 | 466 |
lemma mem_set_of_iff [simp]: "(x \<in> set_of M) = (x :# M)" |
26178 | 467 |
by (auto simp add: set_of_def) |
26016 | 468 |
|
41069
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
469 |
lemma set_of_filter [simp]: "set_of {# x:#M. P x #} = set_of M \<inter> {x. P x}" |
26178 | 470 |
by (auto simp add: set_of_def) |
10249 | 471 |
|
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
|
472 |
lemma finite_set_of [iff]: "finite (set_of 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
|
473 |
using count [of M] by (simp add: multiset_def set_of_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
|
474 |
|
10249 | 475 |
|
476 |
subsubsection {* Size *} |
|
477 |
||
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
|
478 |
instantiation multiset :: (type) size |
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
|
479 |
begin |
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
|
480 |
|
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
|
481 |
definition size_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
|
482 |
"size M = setsum (count M) (set_of 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
|
483 |
|
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
|
484 |
instance .. |
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
|
485 |
|
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
|
486 |
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
|
487 |
|
28708
a1a436f09ec6
explicit check for pattern discipline before code translation
haftmann
parents:
28562
diff
changeset
|
488 |
lemma size_empty [simp]: "size {#} = 0" |
26178 | 489 |
by (simp add: size_def) |
10249 | 490 |
|
28708
a1a436f09ec6
explicit check for pattern discipline before code translation
haftmann
parents:
28562
diff
changeset
|
491 |
lemma size_single [simp]: "size {#b#} = 1" |
26178 | 492 |
by (simp add: size_def) |
10249 | 493 |
|
17161 | 494 |
lemma setsum_count_Int: |
26178 | 495 |
"finite A ==> setsum (count N) (A \<inter> set_of N) = setsum (count N) A" |
496 |
apply (induct rule: finite_induct) |
|
497 |
apply simp |
|
498 |
apply (simp add: Int_insert_left set_of_def) |
|
499 |
done |
|
10249 | 500 |
|
28708
a1a436f09ec6
explicit check for pattern discipline before code translation
haftmann
parents:
28562
diff
changeset
|
501 |
lemma size_union [simp]: "size (M + N::'a multiset) = size M + size N" |
26178 | 502 |
apply (unfold size_def) |
503 |
apply (subgoal_tac "count (M + N) = (\<lambda>a. count M a + count N a)") |
|
504 |
prefer 2 |
|
505 |
apply (rule ext, simp) |
|
506 |
apply (simp (no_asm_simp) add: setsum_Un_nat setsum_addf setsum_count_Int) |
|
507 |
apply (subst Int_commute) |
|
508 |
apply (simp (no_asm_simp) add: setsum_count_Int) |
|
509 |
done |
|
10249 | 510 |
|
17161 | 511 |
lemma size_eq_0_iff_empty [iff]: "(size M = 0) = (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
|
512 |
by (auto simp add: size_def multiset_eq_iff) |
26016 | 513 |
|
514 |
lemma nonempty_has_size: "(S \<noteq> {#}) = (0 < size S)" |
|
26178 | 515 |
by (metis gr0I gr_implies_not0 size_empty size_eq_0_iff_empty) |
10249 | 516 |
|
17161 | 517 |
lemma size_eq_Suc_imp_elem: "size M = Suc n ==> \<exists>a. a :# M" |
26178 | 518 |
apply (unfold size_def) |
519 |
apply (drule setsum_SucD) |
|
520 |
apply auto |
|
521 |
done |
|
10249 | 522 |
|
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
|
523 |
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
|
524 |
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
|
525 |
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
|
526 |
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
|
527 |
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
|
528 |
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
|
529 |
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
|
530 |
then show ?thesis by blast |
23611 | 531 |
qed |
15869 | 532 |
|
26016 | 533 |
|
534 |
subsection {* Induction and case splits *} |
|
10249 | 535 |
|
536 |
lemma setsum_decr: |
|
11701
3d51fbf81c17
sane numerals (stage 1): added generic 1, removed 1' and 2 on nat,
wenzelm
parents:
11655
diff
changeset
|
537 |
"finite F ==> (0::nat) < f a ==> |
15072 | 538 |
setsum (f (a := f a - 1)) F = (if a\<in>F then setsum f F - 1 else setsum f F)" |
26178 | 539 |
apply (induct rule: finite_induct) |
540 |
apply auto |
|
541 |
apply (drule_tac a = a in mk_disjoint_insert, auto) |
|
542 |
done |
|
10249 | 543 |
|
10313 | 544 |
lemma rep_multiset_induct_aux: |
26178 | 545 |
assumes 1: "P (\<lambda>a. (0::nat))" |
546 |
and 2: "!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))" |
|
547 |
shows "\<forall>f. f \<in> multiset --> setsum f {x. f x \<noteq> 0} = n --> P f" |
|
548 |
apply (unfold multiset_def) |
|
549 |
apply (induct_tac n, simp, clarify) |
|
550 |
apply (subgoal_tac "f = (\<lambda>a.0)") |
|
551 |
apply simp |
|
552 |
apply (rule 1) |
|
553 |
apply (rule ext, force, clarify) |
|
554 |
apply (frule setsum_SucD, clarify) |
|
555 |
apply (rename_tac a) |
|
556 |
apply (subgoal_tac "finite {x. (f (a := f a - 1)) x > 0}") |
|
557 |
prefer 2 |
|
558 |
apply (rule finite_subset) |
|
559 |
prefer 2 |
|
560 |
apply assumption |
|
561 |
apply simp |
|
562 |
apply blast |
|
563 |
apply (subgoal_tac "f = (f (a := f a - 1))(a := (f (a := f a - 1)) a + 1)") |
|
564 |
prefer 2 |
|
565 |
apply (rule ext) |
|
566 |
apply (simp (no_asm_simp)) |
|
567 |
apply (erule ssubst, rule 2 [unfolded multiset_def], blast) |
|
568 |
apply (erule allE, erule impE, erule_tac [2] mp, blast) |
|
569 |
apply (simp (no_asm_simp) add: setsum_decr del: fun_upd_apply One_nat_def) |
|
570 |
apply (subgoal_tac "{x. x \<noteq> a --> f x \<noteq> 0} = {x. f x \<noteq> 0}") |
|
571 |
prefer 2 |
|
572 |
apply blast |
|
573 |
apply (subgoal_tac "{x. x \<noteq> a \<and> f x \<noteq> 0} = {x. f x \<noteq> 0} - {a}") |
|
574 |
prefer 2 |
|
575 |
apply blast |
|
576 |
apply (simp add: le_imp_diff_is_add setsum_diff1_nat cong: conj_cong) |
|
577 |
done |
|
10249 | 578 |
|
10313 | 579 |
theorem rep_multiset_induct: |
11464 | 580 |
"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
|
581 |
(!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))) ==> P f" |
26178 | 582 |
using rep_multiset_induct_aux by blast |
10249 | 583 |
|
18258 | 584 |
theorem multiset_induct [case_names empty add, induct type: multiset]: |
26178 | 585 |
assumes empty: "P {#}" |
586 |
and add: "!!M x. P M ==> P (M + {#x#})" |
|
587 |
shows "P M" |
|
10249 | 588 |
proof - |
589 |
note defns = union_def single_def Mempty_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
|
590 |
note add' = add [unfolded defns, simplified] |
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
|
591 |
have aux: "\<And>a::'a. count (Abs_multiset (\<lambda>b. if b = a then 1 else 0)) = |
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
|
592 |
(\<lambda>b. if b = a then 1 else 0)" by (simp add: Abs_multiset_inverse in_multiset) |
10249 | 593 |
show ?thesis |
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
|
594 |
apply (rule count_inverse [THEN subst]) |
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
|
595 |
apply (rule count [THEN rep_multiset_induct]) |
18258 | 596 |
apply (rule empty [unfolded defns]) |
15072 | 597 |
apply (subgoal_tac "f(b := f b + 1) = (\<lambda>a. f a + (if a=b then 1 else 0))") |
10249 | 598 |
prefer 2 |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39301
diff
changeset
|
599 |
apply (simp add: fun_eq_iff) |
10249 | 600 |
apply (erule ssubst) |
17200 | 601 |
apply (erule Abs_multiset_inverse [THEN subst]) |
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
|
602 |
apply (drule add') |
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
|
603 |
apply (simp add: aux) |
10249 | 604 |
done |
605 |
qed |
|
606 |
||
25610 | 607 |
lemma multi_nonempty_split: "M \<noteq> {#} \<Longrightarrow> \<exists>A a. M = A + {#a#}" |
26178 | 608 |
by (induct M) auto |
25610 | 609 |
|
610 |
lemma multiset_cases [cases type, case_names empty add]: |
|
26178 | 611 |
assumes em: "M = {#} \<Longrightarrow> P" |
612 |
assumes add: "\<And>N x. M = N + {#x#} \<Longrightarrow> P" |
|
613 |
shows "P" |
|
25610 | 614 |
proof (cases "M = {#}") |
26145 | 615 |
assume "M = {#}" then show ?thesis using em by simp |
25610 | 616 |
next |
617 |
assume "M \<noteq> {#}" |
|
618 |
then obtain M' m where "M = M' + {#m#}" |
|
619 |
by (blast dest: multi_nonempty_split) |
|
26145 | 620 |
then show ?thesis using add by simp |
25610 | 621 |
qed |
622 |
||
623 |
lemma multi_member_split: "x \<in># M \<Longrightarrow> \<exists>A. M = A + {#x#}" |
|
26178 | 624 |
apply (cases M) |
625 |
apply simp |
|
626 |
apply (rule_tac x="M - {#x#}" in exI, simp) |
|
627 |
done |
|
25610 | 628 |
|
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
|
629 |
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
|
630 |
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
|
631 |
|
26033 | 632 |
lemma multiset_partition: "M = {# x:#M. P x #} + {# x:#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
|
633 |
apply (subst multiset_eq_iff) |
26178 | 634 |
apply auto |
635 |
done |
|
10249 | 636 |
|
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
637 |
lemma mset_less_size: "(A::'a multiset) < 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
|
638 |
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
|
639 |
case (empty 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
|
640 |
then have "M \<noteq> {#}" by (simp add: mset_less_empty_nonempty) |
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
|
641 |
then obtain M' x where "M = 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
|
642 |
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
|
643 |
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
|
644 |
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
|
645 |
case (add S x T) |
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
646 |
have IH: "\<And>B. S < B \<Longrightarrow> size S < size B" by fact |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
647 |
have SxsubT: "S + {#x#} < T" by fact |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
648 |
then have "x \<in># T" and "S < T" by (auto dest: mset_less_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
|
649 |
then obtain T' where T: "T = T' + {#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
|
650 |
by (blast dest: multi_member_split) |
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
651 |
then have "S < T'" using SxsubT |
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
|
652 |
by (blast intro: mset_less_add_bothsides) |
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
|
653 |
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
|
654 |
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
|
655 |
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
|
656 |
|
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
|
657 |
|
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
|
658 |
subsubsection {* Strong induction and subset induction for multisets *} |
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
|
659 |
|
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
|
660 |
text {* Well-foundedness of proper subset operator: *} |
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
|
661 |
|
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
|
662 |
text {* proper multiset 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
|
663 |
|
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
|
664 |
definition |
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
|
665 |
mset_less_rel :: "('a multiset * 'a multiset) set" where |
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
666 |
"mset_less_rel = {(A,B). A < B}" |
10249 | 667 |
|
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
|
668 |
lemma multiset_add_sub_el_shuffle: |
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
|
669 |
assumes "c \<in># B" and "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
|
670 |
shows "B - {#c#} + {#b#} = B + {#b#} - {#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
|
671 |
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
|
672 |
from `c \<in># B` obtain A where B: "B = A + {#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
|
673 |
by (blast 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
|
674 |
have "A + {#b#} = A + {#b#} + {#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
|
675 |
then have "A + {#b#} = A + {#c#} + {#b#} - {#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
|
676 |
by (simp add: add_ac) |
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
|
677 |
then show ?thesis using B 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
|
678 |
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
|
679 |
|
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
|
680 |
lemma wf_mset_less_rel: "wf mset_less_rel" |
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
|
681 |
apply (unfold mset_less_rel_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
|
682 |
apply (rule wf_measure [THEN wf_subset, where f1=size]) |
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
|
683 |
apply (clarsimp simp: measure_def inv_image_def mset_less_size) |
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
|
684 |
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
|
685 |
|
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
|
686 |
text {* The induction rules: *} |
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
|
687 |
|
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
|
688 |
lemma full_multiset_induct [case_names less]: |
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
689 |
assumes ih: "\<And>B. \<forall>(A::'a multiset). A < 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
|
690 |
shows "P 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
|
691 |
apply (rule wf_mset_less_rel [THEN wf_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
|
692 |
apply (rule ih, auto simp: mset_less_rel_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
|
693 |
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
|
694 |
|
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
|
695 |
lemma multi_subset_induct [consumes 2, case_names empty add]: |
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
696 |
assumes "F \<le> 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
|
697 |
and empty: "P {#}" |
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
|
698 |
and insert: "\<And>a F. a \<in># A \<Longrightarrow> P F \<Longrightarrow> P (F + {#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
|
699 |
shows "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
|
700 |
proof - |
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
701 |
from `F \<le> 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
|
702 |
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
|
703 |
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
|
704 |
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
|
705 |
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
|
706 |
fix x F |
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
707 |
assume P: "F \<le> A \<Longrightarrow> P F" and i: "F + {#x#} \<le> 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
|
708 |
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
|
709 |
proof (rule insert) |
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
|
710 |
from i show "x \<in># A" by (auto dest: mset_le_insertD) |
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
711 |
from i have "F \<le> A" by (auto dest: mset_le_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
|
712 |
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
|
713 |
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
|
714 |
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
|
715 |
qed |
26145 | 716 |
|
17161 | 717 |
|
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
|
718 |
subsection {* Alternative representations *} |
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
|
719 |
|
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
|
720 |
subsubsection {* Lists *} |
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
|
721 |
|
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
|
722 |
primrec multiset_of :: "'a list \<Rightarrow> '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:
33102
diff
changeset
|
723 |
"multiset_of [] = {#}" | |
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
|
724 |
"multiset_of (a # x) = multiset_of x + {# 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
|
725 |
|
37107 | 726 |
lemma in_multiset_in_set: |
727 |
"x \<in># multiset_of xs \<longleftrightarrow> x \<in> set xs" |
|
728 |
by (induct xs) simp_all |
|
729 |
||
730 |
lemma count_multiset_of: |
|
731 |
"count (multiset_of xs) x = length (filter (\<lambda>y. x = y) xs)" |
|
732 |
by (induct xs) simp_all |
|
733 |
||
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
|
734 |
lemma multiset_of_zero_iff[simp]: "(multiset_of x = {#}) = (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
|
735 |
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
|
736 |
|
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
|
737 |
lemma multiset_of_zero_iff_right[simp]: "({#} = multiset_of x) = (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
|
738 |
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
|
739 |
|
40950 | 740 |
lemma set_of_multiset_of[simp]: "set_of (multiset_of 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
|
741 |
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
|
742 |
|
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
|
743 |
lemma mem_set_multiset_eq: "x \<in> set xs = (x :# multiset_of 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
|
744 |
by (induct xs) 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
|
745 |
|
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
|
746 |
lemma multiset_of_append [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
|
747 |
"multiset_of (xs @ ys) = multiset_of xs + multiset_of ys" |
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
|
748 |
by (induct xs arbitrary: ys) (auto simp: add_ac) |
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
|
749 |
|
40303
2d507370e879
lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents:
40250
diff
changeset
|
750 |
lemma multiset_of_filter: |
2d507370e879
lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents:
40250
diff
changeset
|
751 |
"multiset_of (filter P xs) = {#x :# multiset_of xs. P x #}" |
2d507370e879
lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents:
40250
diff
changeset
|
752 |
by (induct xs) simp_all |
2d507370e879
lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents:
40250
diff
changeset
|
753 |
|
40950 | 754 |
lemma multiset_of_rev [simp]: |
755 |
"multiset_of (rev xs) = multiset_of xs" |
|
756 |
by (induct xs) simp_all |
|
757 |
||
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
|
758 |
lemma surj_multiset_of: "surj multiset_of" |
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
|
759 |
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
|
760 |
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
|
761 |
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
|
762 |
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
|
763 |
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
|
764 |
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
|
765 |
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
|
766 |
|
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
|
767 |
lemma set_count_greater_0: "set x = {a. count (multiset_of x) a > 0}" |
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
|
768 |
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
|
769 |
|
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
|
770 |
lemma distinct_count_atmost_1: |
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
|
771 |
"distinct x = (! a. count (multiset_of x) a = (if a \<in> set x then 1 else 0))" |
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
|
772 |
apply (induct x, simp, rule iffI, simp_all) |
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
|
773 |
apply (rule conjI) |
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
|
774 |
apply (simp_all add: set_of_multiset_of [THEN sym] del: set_of_multiset_of) |
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
|
775 |
apply (erule_tac x = a in allE, simp, clarify) |
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
|
776 |
apply (erule_tac x = aa in allE, 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
|
777 |
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
|
778 |
|
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
|
779 |
lemma multiset_of_eq_setD: |
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
|
780 |
"multiset_of xs = multiset_of ys \<Longrightarrow> set xs = set ys" |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39301
diff
changeset
|
781 |
by (rule) (auto simp add:multiset_eq_iff set_count_greater_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
|
782 |
|
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
|
783 |
lemma set_eq_iff_multiset_of_eq_distinct: |
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
|
784 |
"distinct x \<Longrightarrow> distinct y \<Longrightarrow> |
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
|
785 |
(set x = set y) = (multiset_of x = multiset_of 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
|
786 |
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
|
787 |
|
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
|
788 |
lemma set_eq_iff_multiset_of_remdups_eq: |
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
|
789 |
"(set x = set y) = (multiset_of (remdups x) = multiset_of (remdups y))" |
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
|
790 |
apply (rule iffI) |
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
|
791 |
apply (simp add: set_eq_iff_multiset_of_eq_distinct[THEN iffD1]) |
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
|
792 |
apply (drule distinct_remdups [THEN distinct_remdups |
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
|
793 |
[THEN set_eq_iff_multiset_of_eq_distinct [THEN iffD2]]]) |
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
|
794 |
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
|
795 |
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
|
796 |
|
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
|
797 |
lemma multiset_of_compl_union [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
|
798 |
"multiset_of [x\<leftarrow>xs. P x] + multiset_of [x\<leftarrow>xs. \<not>P x] = multiset_of 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
|
799 |
by (induct xs) (auto simp: add_ac) |
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
|
800 |
|
41069
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
801 |
lemma count_multiset_of_length_filter: |
39533
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
802 |
"count (multiset_of xs) x = length (filter (\<lambda>y. x = y) xs)" |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
803 |
by (induct xs) 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
|
804 |
|
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
|
805 |
lemma nth_mem_multiset_of: "i < length ls \<Longrightarrow> (ls ! i) :# multiset_of ls" |
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
|
806 |
apply (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
|
807 |
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
|
808 |
apply (case_tac 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
|
809 |
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
|
810 |
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
|
811 |
|
36903 | 812 |
lemma multiset_of_remove1[simp]: |
813 |
"multiset_of (remove1 a xs) = multiset_of xs - {#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
|
814 |
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
|
815 |
|
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
|
816 |
lemma multiset_of_eq_length: |
37107 | 817 |
assumes "multiset_of xs = multiset_of ys" |
818 |
shows "length xs = length ys" |
|
819 |
using assms proof (induct xs arbitrary: ys) |
|
820 |
case Nil then show ?case by simp |
|
821 |
next |
|
822 |
case (Cons x xs) |
|
823 |
then have "x \<in># multiset_of ys" by (simp add: union_single_eq_member) |
|
824 |
then have "x \<in> set ys" by (simp add: in_multiset_in_set) |
|
825 |
from Cons.prems [symmetric] have "multiset_of xs = multiset_of (remove1 x ys)" |
|
826 |
by simp |
|
827 |
with Cons.hyps have "length xs = length (remove1 x ys)" . |
|
828 |
with `x \<in> set ys` show ?case |
|
829 |
by (auto simp add: length_remove1 dest: length_pos_if_in_set) |
|
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
|
830 |
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
|
831 |
|
39533
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
832 |
lemma multiset_of_eq_length_filter: |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
833 |
assumes "multiset_of xs = multiset_of ys" |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
834 |
shows "length (filter (\<lambda>x. z = x) xs) = length (filter (\<lambda>y. z = y) ys)" |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
835 |
proof (cases "z \<in># multiset_of xs") |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
836 |
case False |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
837 |
moreover have "\<not> z \<in># multiset_of ys" using assms False by simp |
41069
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
838 |
ultimately show ?thesis by (simp add: count_multiset_of_length_filter) |
39533
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
839 |
next |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
840 |
case True |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
841 |
moreover have "z \<in># multiset_of ys" using assms True by simp |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
842 |
show ?thesis using assms proof (induct xs arbitrary: ys) |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
843 |
case Nil then show ?case by simp |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
844 |
next |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
845 |
case (Cons x xs) |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
846 |
from `multiset_of (x # xs) = multiset_of ys` [symmetric] |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
847 |
have *: "multiset_of xs = multiset_of (remove1 x ys)" |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
848 |
and "x \<in> set ys" |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
849 |
by (auto simp add: mem_set_multiset_eq) |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
850 |
from * have "length (filter (\<lambda>x. z = x) xs) = length (filter (\<lambda>y. z = y) (remove1 x ys))" by (rule Cons.hyps) |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
851 |
moreover from `x \<in> set ys` have "length (filter (\<lambda>y. x = y) ys) > 0" by (simp add: filter_empty_conv) |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
852 |
ultimately show ?case using `x \<in> set ys` |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
853 |
by (simp add: filter_remove1) (auto simp add: length_remove1) |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
854 |
qed |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
855 |
qed |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
856 |
|
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
857 |
context linorder |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
858 |
begin |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
859 |
|
40210
aee7ef725330
sorting: avoid _key suffix if lemma applies both to simple and generalized variant; generalized insort_insert to insort_insert_key; additional lemmas
haftmann
parents:
39533
diff
changeset
|
860 |
lemma multiset_of_insort [simp]: |
39533
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
861 |
"multiset_of (insort_key k x xs) = {#x#} + multiset_of xs" |
37107 | 862 |
by (induct xs) (simp_all add: ac_simps) |
39533
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
863 |
|
40210
aee7ef725330
sorting: avoid _key suffix if lemma applies both to simple and generalized variant; generalized insort_insert to insort_insert_key; additional lemmas
haftmann
parents:
39533
diff
changeset
|
864 |
lemma multiset_of_sort [simp]: |
39533
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
865 |
"multiset_of (sort_key k xs) = multiset_of xs" |
37107 | 866 |
by (induct xs) (simp_all add: ac_simps) |
867 |
||
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
|
868 |
text {* |
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
|
869 |
This lemma shows which properties suffice to show that a function |
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
|
870 |
@{text "f"} with @{text "f xs = ys"} behaves like sort. |
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
|
871 |
*} |
37074 | 872 |
|
39533
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
873 |
lemma properties_for_sort_key: |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
874 |
assumes "multiset_of ys = multiset_of xs" |
40305
41833242cc42
tuned lemma proposition of properties_for_sort_key
haftmann
parents:
40303
diff
changeset
|
875 |
and "\<And>k. k \<in> set ys \<Longrightarrow> filter (\<lambda>x. f k = f 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
|
876 |
and "sorted (map f ys)" |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
877 |
shows "sort_key f xs = ys" |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
878 |
using assms 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
|
879 |
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
|
880 |
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
|
881 |
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
|
882 |
from Cons.prems(2) have |
40305
41833242cc42
tuned lemma proposition of properties_for_sort_key
haftmann
parents:
40303
diff
changeset
|
883 |
"\<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
|
884 |
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
|
885 |
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
|
886 |
by (auto intro!: Cons.hyps simp add: sorted_map_remove1) |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
887 |
moreover from Cons.prems have "x \<in> set ys" |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
888 |
by (auto simp add: mem_set_multiset_eq intro!: ccontr) |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
889 |
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
|
890 |
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
|
891 |
|
39533
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
892 |
lemma properties_for_sort: |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
893 |
assumes multiset: "multiset_of ys = multiset_of xs" |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
894 |
and "sorted ys" |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
895 |
shows "sort xs = ys" |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
896 |
proof (rule properties_for_sort_key) |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
897 |
from multiset show "multiset_of ys = multiset_of xs" . |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
898 |
from `sorted ys` show "sorted (map (\<lambda>x. x) ys)" by simp |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
899 |
from multiset have "\<And>k. length (filter (\<lambda>y. k = y) ys) = length (filter (\<lambda>x. k = x) xs)" |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
900 |
by (rule multiset_of_eq_length_filter) |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
901 |
then have "\<And>k. replicate (length (filter (\<lambda>y. k = y) ys)) k = replicate (length (filter (\<lambda>x. k = x) xs)) k" |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
902 |
by simp |
40305
41833242cc42
tuned lemma proposition of properties_for_sort_key
haftmann
parents:
40303
diff
changeset
|
903 |
then show "\<And>k. k \<in> set ys \<Longrightarrow> filter (\<lambda>y. k = y) ys = filter (\<lambda>x. k = x) xs" |
39533
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
904 |
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
|
905 |
qed |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
906 |
|
40303
2d507370e879
lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents:
40250
diff
changeset
|
907 |
lemma sort_key_by_quicksort: |
2d507370e879
lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents:
40250
diff
changeset
|
908 |
"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
|
909 |
@ [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
|
910 |
@ 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
|
911 |
proof (rule properties_for_sort_key) |
2d507370e879
lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents:
40250
diff
changeset
|
912 |
show "multiset_of ?rhs = multiset_of ?lhs" |
2d507370e879
lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents:
40250
diff
changeset
|
913 |
by (rule multiset_eqI) (auto simp add: multiset_of_filter) |
2d507370e879
lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents:
40250
diff
changeset
|
914 |
next |
2d507370e879
lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents:
40250
diff
changeset
|
915 |
show "sorted (map f ?rhs)" |
2d507370e879
lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents:
40250
diff
changeset
|
916 |
by (auto simp add: sorted_append intro: sorted_map_same) |
2d507370e879
lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents:
40250
diff
changeset
|
917 |
next |
40305
41833242cc42
tuned lemma proposition of properties_for_sort_key
haftmann
parents:
40303
diff
changeset
|
918 |
fix l |
41833242cc42
tuned lemma proposition of properties_for_sort_key
haftmann
parents:
40303
diff
changeset
|
919 |
assume "l \<in> set ?rhs" |
40346 | 920 |
let ?pivot = "f (xs ! (length xs div 2))" |
921 |
have *: "\<And>x. f l = f x \<longleftrightarrow> f x = f l" by auto |
|
40306 | 922 |
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
|
923 |
unfolding filter_sort by (rule properties_for_sort_key) (auto intro: sorted_map_same) |
40346 | 924 |
with * have **: "[x \<leftarrow> sort_key f xs . f l = f x] = [x \<leftarrow> xs. f l = f x]" by simp |
925 |
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 |
|
926 |
then have "\<And>P. [x \<leftarrow> sort_key f xs . P (f x) ?pivot \<and> f l = f x] = |
|
927 |
[x \<leftarrow> sort_key f xs. P (f l) ?pivot \<and> f l = f x]" by simp |
|
928 |
note *** = this [of "op <"] this [of "op >"] this [of "op ="] |
|
40306 | 929 |
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
|
930 |
proof (cases "f l" ?pivot rule: linorder_cases) |
40307 | 931 |
case less then moreover have "f l \<noteq> ?pivot" and "\<not> f l > ?pivot" by auto |
932 |
ultimately show ?thesis |
|
40346 | 933 |
by (simp add: filter_sort [symmetric] ** ***) |
40305
41833242cc42
tuned lemma proposition of properties_for_sort_key
haftmann
parents:
40303
diff
changeset
|
934 |
next |
40306 | 935 |
case equal then show ?thesis |
40346 | 936 |
by (simp add: * less_le) |
40305
41833242cc42
tuned lemma proposition of properties_for_sort_key
haftmann
parents:
40303
diff
changeset
|
937 |
next |
40307 | 938 |
case greater then moreover have "f l \<noteq> ?pivot" and "\<not> f l < ?pivot" by auto |
939 |
ultimately show ?thesis |
|
40346 | 940 |
by (simp add: filter_sort [symmetric] ** ***) |
40306 | 941 |
qed |
40303
2d507370e879
lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents:
40250
diff
changeset
|
942 |
qed |
2d507370e879
lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents:
40250
diff
changeset
|
943 |
|
2d507370e879
lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents:
40250
diff
changeset
|
944 |
lemma sort_by_quicksort: |
2d507370e879
lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents:
40250
diff
changeset
|
945 |
"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
|
946 |
@ [x\<leftarrow>xs. x = xs ! (length xs div 2)] |
2d507370e879
lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents:
40250
diff
changeset
|
947 |
@ 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
|
948 |
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
|
949 |
|
40347
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
950 |
text {* A stable parametrized quicksort *} |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
951 |
|
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
952 |
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
|
953 |
"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
|
954 |
|
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
955 |
lemma part_code [code]: |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
956 |
"part f pivot [] = ([], [], [])" |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
957 |
"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
|
958 |
if x' < pivot then (x # lts, eqs, gts) |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
959 |
else if x' > pivot then (lts, eqs, x # gts) |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
960 |
else (lts, x # eqs, gts))" |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
961 |
by (auto simp add: part_def Let_def split_def) |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
962 |
|
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
963 |
lemma sort_key_by_quicksort_code [code]: |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
964 |
"sort_key f xs = (case xs of [] \<Rightarrow> [] |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
965 |
| [x] \<Rightarrow> xs |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
966 |
| [x, y] \<Rightarrow> (if f x \<le> f y then xs else [y, x]) |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
967 |
| _ \<Rightarrow> (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
|
968 |
in sort_key f lts @ eqs @ sort_key f gts))" |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
969 |
proof (cases xs) |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
970 |
case Nil then show ?thesis by simp |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
971 |
next |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
972 |
case (Cons _ ys) note hyps = Cons show ?thesis proof (cases ys) |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
973 |
case Nil with hyps show ?thesis by simp |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
974 |
next |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
975 |
case (Cons _ zs) note hyps = hyps Cons show ?thesis proof (cases zs) |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
976 |
case Nil with hyps show ?thesis by auto |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
977 |
next |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
978 |
case Cons |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
979 |
from sort_key_by_quicksort [of f xs] |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
980 |
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
|
981 |
in sort_key f lts @ eqs @ sort_key f gts)" |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
982 |
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
|
983 |
with hyps Cons show ?thesis by (simp only: list.cases) |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
984 |
qed |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
985 |
qed |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
986 |
qed |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
987 |
|
39533
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
988 |
end |
91a0ff0ff237
generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents:
39314
diff
changeset
|
989 |
|
40347
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
990 |
hide_const (open) part |
429bf4388b2f
added code lemmas for stable parametrized quicksort
haftmann
parents:
40346
diff
changeset
|
991 |
|
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
992 |
lemma multiset_of_remdups_le: "multiset_of (remdups xs) \<le> multiset_of xs" |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
993 |
by (induct xs) (auto intro: 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
|
994 |
|
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
|
995 |
lemma multiset_of_update: |
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
|
996 |
"i < length ls \<Longrightarrow> multiset_of (ls[i := v]) = multiset_of ls - {#ls ! i#} + {#v#}" |
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
|
997 |
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
|
998 |
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
|
999 |
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
|
1000 |
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
|
1001 |
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
|
1002 |
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
|
1003 |
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
|
1004 |
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
|
1005 |
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
|
1006 |
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
|
1007 |
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
|
1008 |
apply (subst add_assoc) |
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
|
1009 |
apply (subst add_commute [of "{#v#}" "{#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
|
1010 |
apply (subst add_assoc [symmetric]) |
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
|
1011 |
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
|
1012 |
apply (rule mset_le_multiset_union_diff_commute) |
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
|
1013 |
apply (simp add: mset_le_single nth_mem_multiset_of) |
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
|
1014 |
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
|
1015 |
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
|
1016 |
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
|
1017 |
|
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
|
1018 |
lemma multiset_of_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
|
1019 |
"i < length ls \<Longrightarrow> j < length ls \<Longrightarrow> |
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
|
1020 |
multiset_of (ls[j := ls ! i, i := ls ! j]) = multiset_of ls" |
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
|
1021 |
by (cases "i = j") (simp_all add: multiset_of_update nth_mem_multiset_of) |
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
|
1022 |
|
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
|
1023 |
|
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
|
1024 |
subsubsection {* Association lists -- including rudimentary code generation *} |
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
|
1025 |
|
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
|
1026 |
definition count_of :: "('a \<times> nat) list \<Rightarrow> 'a \<Rightarrow> nat" 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:
33102
diff
changeset
|
1027 |
"count_of xs x = (case map_of xs x of None \<Rightarrow> 0 | Some n \<Rightarrow> 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
|
1028 |
|
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
|
1029 |
lemma count_of_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
|
1030 |
"count_of xs \<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
|
1031 |
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
|
1032 |
let ?A = "{x::'a. 0 < (case map_of xs x of None \<Rightarrow> 0\<Colon>nat | Some (n\<Colon>nat) \<Rightarrow> 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
|
1033 |
have "?A \<subseteq> dom (map_of 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
|
1034 |
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
|
1035 |
fix 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
|
1036 |
assume "x \<in> ?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
|
1037 |
then have "0 < (case map_of xs x of None \<Rightarrow> 0\<Colon>nat | Some (n\<Colon>nat) \<Rightarrow> n)" 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
|
1038 |
then have "map_of xs x \<noteq> None" by (cases "map_of xs 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
|
1039 |
then show "x \<in> dom (map_of xs)" 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
|
1040 |
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
|
1041 |
with finite_dom_map_of [of xs] have "finite ?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
|
1042 |
by (auto 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
|
1043 |
then show ?thesis |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39301
diff
changeset
|
1044 |
by (simp add: count_of_def fun_eq_iff multiset_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
|
1045 |
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
|
1046 |
|
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
|
1047 |
lemma count_simps [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
|
1048 |
"count_of [] = (\<lambda>_. 0)" |
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
|
1049 |
"count_of ((x, n) # xs) = (\<lambda>y. if x = y then n else count_of xs 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
|
1050 |
by (simp_all add: count_of_def 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
|
1051 |
|
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
|
1052 |
lemma count_of_empty: |
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
|
1053 |
"x \<notin> fst ` set xs \<Longrightarrow> count_of xs x = 0" |
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
|
1054 |
by (induct xs) (simp_all add: count_of_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
|
1055 |
|
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
|
1056 |
lemma count_of_filter: |
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
|
1057 |
"count_of (filter (P \<circ> fst) xs) x = (if P x then count_of xs x else 0)" |
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
|
1058 |
by (induct xs) 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
|
1059 |
|
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
|
1060 |
definition Bag :: "('a \<times> nat) list \<Rightarrow> '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:
33102
diff
changeset
|
1061 |
"Bag xs = Abs_multiset (count_of 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
|
1062 |
|
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
|
1063 |
code_datatype Bag |
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
|
1064 |
|
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
|
1065 |
lemma count_Bag [simp, code]: |
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
|
1066 |
"count (Bag xs) = count_of 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
|
1067 |
by (simp add: Bag_def count_of_multiset Abs_multiset_inverse) |
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
|
1068 |
|
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
|
1069 |
lemma Mempty_Bag [code]: |
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
|
1070 |
"{#} = Bag []" |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39301
diff
changeset
|
1071 |
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
|
1072 |
|
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
|
1073 |
lemma single_Bag [code]: |
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
|
1074 |
"{#x#} = Bag [(x, 1)]" |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39301
diff
changeset
|
1075 |
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
|
1076 |
|
41069
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
1077 |
lemma filter_Bag [code]: |
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
1078 |
"Multiset.filter P (Bag xs) = Bag (filter (P \<circ> fst) xs)" |
6fabc0414055
name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents:
40968
diff
changeset
|
1079 |
by (rule multiset_eqI) (simp add: count_of_filter) |
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
|
1080 |
|
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
|
1081 |
lemma mset_less_eq_Bag [code]: |
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1082 |
"Bag xs \<le> A \<longleftrightarrow> (\<forall>(x, n) \<in> set xs. count_of xs x \<le> count A 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
|
1083 |
(is "?lhs \<longleftrightarrow> ?rhs") |
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
|
1084 |
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
|
1085 |
assume ?lhs then show ?rhs |
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
|
1086 |
by (auto simp add: mset_le_def count_Bag) |
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
|
1087 |
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
|
1088 |
assume ?rhs |
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
|
1089 |
show ?lhs |
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
|
1090 |
proof (rule mset_less_eqI) |
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
|
1091 |
fix 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
|
1092 |
from `?rhs` have "count_of xs x \<le> count A 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
|
1093 |
by (cases "x \<in> fst ` set xs") (auto simp add: count_of_empty) |
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
|
1094 |
then show "count (Bag xs) x \<le> count A 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
|
1095 |
by (simp add: mset_le_def count_Bag) |
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
|
1096 |
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
|
1097 |
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
|
1098 |
|
38857
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
38287
diff
changeset
|
1099 |
instantiation multiset :: (equal) equal |
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
|
1100 |
begin |
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
|
1101 |
|
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
|
1102 |
definition |
38857
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
38287
diff
changeset
|
1103 |
"HOL.equal A B \<longleftrightarrow> (A::'a multiset) \<le> B \<and> B \<le> 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
|
1104 |
|
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
|
1105 |
instance proof |
38857
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
38287
diff
changeset
|
1106 |
qed (simp add: equal_multiset_def 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
|
1107 |
|
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
|
1108 |
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
|
1109 |
|
38857
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
38287
diff
changeset
|
1110 |
lemma [code nbe]: |
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
38287
diff
changeset
|
1111 |
"HOL.equal (A :: 'a::equal multiset) A \<longleftrightarrow> True" |
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
38287
diff
changeset
|
1112 |
by (fact equal_refl) |
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
38287
diff
changeset
|
1113 |
|
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
|
1114 |
definition (in term_syntax) |
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
|
1115 |
bagify :: "('a\<Colon>typerep \<times> nat) list \<times> (unit \<Rightarrow> Code_Evaluation.term) |
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 |
\<Rightarrow> 'a multiset \<times> (unit \<Rightarrow> Code_Evaluation.term)" 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:
33102
diff
changeset
|
1117 |
[code_unfold]: "bagify xs = Code_Evaluation.valtermify Bag {\<cdot>} 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
|
1118 |
|
37751 | 1119 |
notation fcomp (infixl "\<circ>>" 60) |
1120 |
notation scomp (infixl "\<circ>\<rightarrow>" 60) |
|
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
|
1121 |
|
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
|
1122 |
instantiation multiset :: (random) random |
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
|
1123 |
begin |
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
|
1124 |
|
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
|
1125 |
definition |
37751 | 1126 |
"Quickcheck.random i = Quickcheck.random i \<circ>\<rightarrow> (\<lambda>xs. Pair (bagify xs))" |
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
|
1127 |
|
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
|
1128 |
instance .. |
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
|
1129 |
|
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
|
1130 |
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
|
1131 |
|
37751 | 1132 |
no_notation fcomp (infixl "\<circ>>" 60) |
1133 |
no_notation scomp (infixl "\<circ>\<rightarrow>" 60) |
|
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
|
1134 |
|
36176
3fe7e97ccca8
replaced generic 'hide' command by more conventional 'hide_class', 'hide_type', 'hide_const', 'hide_fact' -- frees some popular keywords;
wenzelm
parents:
35712
diff
changeset
|
1135 |
hide_const (open) bagify |
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
|
1136 |
|
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
|
1137 |
|
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
|
1138 |
subsection {* The multiset order *} |
10249 | 1139 |
|
1140 |
subsubsection {* Well-foundedness *} |
|
1141 |
||
28708
a1a436f09ec6
explicit check for pattern discipline before code translation
haftmann
parents:
28562
diff
changeset
|
1142 |
definition mult1 :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where |
37765 | 1143 |
"mult1 r = {(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> |
23751 | 1144 |
(\<forall>b. b :# K --> (b, a) \<in> r)}" |
10249 | 1145 |
|
28708
a1a436f09ec6
explicit check for pattern discipline before code translation
haftmann
parents:
28562
diff
changeset
|
1146 |
definition mult :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where |
37765 | 1147 |
"mult r = (mult1 r)\<^sup>+" |
10249 | 1148 |
|
23751 | 1149 |
lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r" |
26178 | 1150 |
by (simp add: mult1_def) |
10249 | 1151 |
|
23751 | 1152 |
lemma less_add: "(N, M0 + {#a#}) \<in> mult1 r ==> |
1153 |
(\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or> |
|
1154 |
(\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K)" |
|
19582 | 1155 |
(is "_ \<Longrightarrow> ?case1 (mult1 r) \<or> ?case2") |
10249 | 1156 |
proof (unfold mult1_def) |
23751 | 1157 |
let ?r = "\<lambda>K a. \<forall>b. b :# K --> (b, a) \<in> r" |
11464 | 1158 |
let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a" |
23751 | 1159 |
let ?case1 = "?case1 {(N, M). ?R N M}" |
10249 | 1160 |
|
23751 | 1161 |
assume "(N, M0 + {#a#}) \<in> {(N, M). ?R N M}" |
18258 | 1162 |
then have "\<exists>a' M0' K. |
11464 | 1163 |
M0 + {#a#} = M0' + {#a'#} \<and> N = M0' + K \<and> ?r K a'" by simp |
18258 | 1164 |
then show "?case1 \<or> ?case2" |
10249 | 1165 |
proof (elim exE conjE) |
1166 |
fix a' M0' K |
|
1167 |
assume N: "N = M0' + K" and r: "?r K a'" |
|
1168 |
assume "M0 + {#a#} = M0' + {#a'#}" |
|
18258 | 1169 |
then have "M0 = M0' \<and> a = a' \<or> |
11464 | 1170 |
(\<exists>K'. M0 = K' + {#a'#} \<and> M0' = K' + {#a#})" |
10249 | 1171 |
by (simp only: add_eq_conv_ex) |
18258 | 1172 |
then show ?thesis |
10249 | 1173 |
proof (elim disjE conjE exE) |
1174 |
assume "M0 = M0'" "a = a'" |
|
11464 | 1175 |
with N r have "?r K a \<and> N = M0 + K" by simp |
18258 | 1176 |
then have ?case2 .. then show ?thesis .. |
10249 | 1177 |
next |
1178 |
fix K' |
|
1179 |
assume "M0' = K' + {#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
|
1180 |
with N have n: "N = K' + K + {#a#}" by (simp add: add_ac) |
10249 | 1181 |
|
1182 |
assume "M0 = K' + {#a'#}" |
|
1183 |
with r have "?R (K' + K) M0" by blast |
|
18258 | 1184 |
with n have ?case1 by simp then show ?thesis .. |
10249 | 1185 |
qed |
1186 |
qed |
|
1187 |
qed |
|
1188 |
||
23751 | 1189 |
lemma all_accessible: "wf r ==> \<forall>M. M \<in> acc (mult1 r)" |
10249 | 1190 |
proof |
1191 |
let ?R = "mult1 r" |
|
1192 |
let ?W = "acc ?R" |
|
1193 |
{ |
|
1194 |
fix M M0 a |
|
23751 | 1195 |
assume M0: "M0 \<in> ?W" |
1196 |
and wf_hyp: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)" |
|
1197 |
and acc_hyp: "\<forall>M. (M, M0) \<in> ?R --> M + {#a#} \<in> ?W" |
|
1198 |
have "M0 + {#a#} \<in> ?W" |
|
1199 |
proof (rule accI [of "M0 + {#a#}"]) |
|
10249 | 1200 |
fix N |
23751 | 1201 |
assume "(N, M0 + {#a#}) \<in> ?R" |
1202 |
then have "((\<exists>M. (M, M0) \<in> ?R \<and> N = M + {#a#}) \<or> |
|
1203 |
(\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K))" |
|
10249 | 1204 |
by (rule less_add) |
23751 | 1205 |
then show "N \<in> ?W" |
10249 | 1206 |
proof (elim exE disjE conjE) |
23751 | 1207 |
fix M assume "(M, M0) \<in> ?R" and N: "N = M + {#a#}" |
1208 |
from acc_hyp have "(M, M0) \<in> ?R --> M + {#a#} \<in> ?W" .. |
|
1209 |
from this and `(M, M0) \<in> ?R` have "M + {#a#} \<in> ?W" .. |
|
1210 |
then show "N \<in> ?W" by (simp only: N) |
|
10249 | 1211 |
next |
1212 |
fix K |
|
1213 |
assume N: "N = M0 + K" |
|
23751 | 1214 |
assume "\<forall>b. b :# K --> (b, a) \<in> r" |
1215 |
then have "M0 + K \<in> ?W" |
|
10249 | 1216 |
proof (induct K) |
18730 | 1217 |
case empty |
23751 | 1218 |
from M0 show "M0 + {#} \<in> ?W" by simp |
18730 | 1219 |
next |
1220 |
case (add K x) |
|
23751 | 1221 |
from add.prems have "(x, a) \<in> r" by simp |
1222 |
with wf_hyp have "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast |
|
1223 |
moreover from add have "M0 + K \<in> ?W" by simp |
|
1224 |
ultimately have "(M0 + K) + {#x#} \<in> ?W" .. |
|
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
|
1225 |
then show "M0 + (K + {#x#}) \<in> ?W" by (simp only: add_assoc) |
10249 | 1226 |
qed |
23751 | 1227 |
then show "N \<in> ?W" by (simp only: N) |
10249 | 1228 |
qed |
1229 |
qed |
|
1230 |
} note tedious_reasoning = this |
|
1231 |
||
23751 | 1232 |
assume wf: "wf r" |
10249 | 1233 |
fix M |
23751 | 1234 |
show "M \<in> ?W" |
10249 | 1235 |
proof (induct M) |
23751 | 1236 |
show "{#} \<in> ?W" |
10249 | 1237 |
proof (rule accI) |
23751 | 1238 |
fix b assume "(b, {#}) \<in> ?R" |
1239 |
with not_less_empty show "b \<in> ?W" by contradiction |
|
10249 | 1240 |
qed |
1241 |
||
23751 | 1242 |
fix M a assume "M \<in> ?W" |
1243 |
from wf have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W" |
|
10249 | 1244 |
proof induct |
1245 |
fix a |
|
23751 | 1246 |
assume r: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)" |
1247 |
show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W" |
|
10249 | 1248 |
proof |
23751 | 1249 |
fix M assume "M \<in> ?W" |
1250 |
then show "M + {#a#} \<in> ?W" |
|
23373 | 1251 |
by (rule acc_induct) (rule tedious_reasoning [OF _ r]) |
10249 | 1252 |
qed |
1253 |
qed |
|
23751 | 1254 |
from this and `M \<in> ?W` show "M + {#a#} \<in> ?W" .. |
10249 | 1255 |
qed |
1256 |
qed |
|
1257 |
||
23751 | 1258 |
theorem wf_mult1: "wf r ==> wf (mult1 r)" |
26178 | 1259 |
by (rule acc_wfI) (rule all_accessible) |
10249 | 1260 |
|
23751 | 1261 |
theorem wf_mult: "wf r ==> wf (mult r)" |
26178 | 1262 |
unfolding mult_def by (rule wf_trancl) (rule wf_mult1) |
10249 | 1263 |
|
1264 |
||
1265 |
subsubsection {* Closure-free presentation *} |
|
1266 |
||
1267 |
text {* One direction. *} |
|
1268 |
||
1269 |
lemma mult_implies_one_step: |
|
23751 | 1270 |
"trans r ==> (M, N) \<in> mult r ==> |
11464 | 1271 |
\<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and> |
23751 | 1272 |
(\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r)" |
26178 | 1273 |
apply (unfold mult_def mult1_def set_of_def) |
1274 |
apply (erule converse_trancl_induct, clarify) |
|
1275 |
apply (rule_tac x = M0 in exI, simp, clarify) |
|
1276 |
apply (case_tac "a :# K") |
|
1277 |
apply (rule_tac x = I in exI) |
|
1278 |
apply (simp (no_asm)) |
|
1279 |
apply (rule_tac x = "(K - {#a#}) + Ka" in exI) |
|
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 |
apply (simp (no_asm_simp) add: add_assoc [symmetric]) |
26178 | 1281 |
apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong) |
1282 |
apply (simp add: diff_union_single_conv) |
|
1283 |
apply (simp (no_asm_use) add: trans_def) |
|
1284 |
apply blast |
|
1285 |
apply (subgoal_tac "a :# I") |
|
1286 |
apply (rule_tac x = "I - {#a#}" in exI) |
|
1287 |
apply (rule_tac x = "J + {#a#}" in exI) |
|
1288 |
apply (rule_tac x = "K + Ka" in exI) |
|
1289 |
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
|
1290 |
apply (simp add: multiset_eq_iff split: nat_diff_split) |
26178 | 1291 |
apply (rule conjI) |
1292 |
apply (drule_tac f = "\<lambda>M. M - {#a#}" 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
|
1293 |
apply (simp add: multiset_eq_iff split: nat_diff_split) |
26178 | 1294 |
apply (simp (no_asm_use) add: trans_def) |
1295 |
apply blast |
|
1296 |
apply (subgoal_tac "a :# (M0 + {#a#})") |
|
1297 |
apply simp |
|
1298 |
apply (simp (no_asm)) |
|
1299 |
done |
|
10249 | 1300 |
|
1301 |
lemma one_step_implies_mult_aux: |
|
23751 | 1302 |
"trans r ==> |
1303 |
\<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)) |
|
1304 |
--> (I + K, I + J) \<in> mult r" |
|
26178 | 1305 |
apply (induct_tac n, auto) |
1306 |
apply (frule size_eq_Suc_imp_eq_union, clarify) |
|
1307 |
apply (rename_tac "J'", simp) |
|
1308 |
apply (erule notE, auto) |
|
1309 |
apply (case_tac "J' = {#}") |
|
1310 |
apply (simp add: mult_def) |
|
1311 |
apply (rule r_into_trancl) |
|
1312 |
apply (simp add: mult1_def set_of_def, blast) |
|
1313 |
txt {* Now we know @{term "J' \<noteq> {#}"}. *} |
|
1314 |
apply (cut_tac M = K and P = "\<lambda>x. (x, a) \<in> r" in multiset_partition) |
|
1315 |
apply (erule_tac P = "\<forall>k \<in> set_of K. ?P k" in rev_mp) |
|
1316 |
apply (erule ssubst) |
|
1317 |
apply (simp add: Ball_def, auto) |
|
1318 |
apply (subgoal_tac |
|
1319 |
"((I + {# x :# K. (x, a) \<in> r #}) + {# x :# K. (x, a) \<notin> r #}, |
|
1320 |
(I + {# x :# K. (x, a) \<in> r #}) + J') \<in> mult r") |
|
1321 |
prefer 2 |
|
1322 |
apply force |
|
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
|
1323 |
apply (simp (no_asm_use) add: add_assoc [symmetric] mult_def) |
26178 | 1324 |
apply (erule trancl_trans) |
1325 |
apply (rule r_into_trancl) |
|
1326 |
apply (simp add: mult1_def set_of_def) |
|
1327 |
apply (rule_tac x = a in exI) |
|
1328 |
apply (rule_tac x = "I + J'" in exI) |
|
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
|
1329 |
apply (simp add: add_ac) |
26178 | 1330 |
done |
10249 | 1331 |
|
17161 | 1332 |
lemma one_step_implies_mult: |
23751 | 1333 |
"trans r ==> J \<noteq> {#} ==> \<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r |
1334 |
==> (I + K, I + J) \<in> mult r" |
|
26178 | 1335 |
using one_step_implies_mult_aux by blast |
10249 | 1336 |
|
1337 |
||
1338 |
subsubsection {* Partial-order properties *} |
|
1339 |
||
35273 | 1340 |
definition less_multiset :: "'a\<Colon>order multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "<#" 50) where |
1341 |
"M' <# M \<longleftrightarrow> (M', M) \<in> mult {(x', x). x' < x}" |
|
10249 | 1342 |
|
35273 | 1343 |
definition le_multiset :: "'a\<Colon>order multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "<=#" 50) where |
1344 |
"M' <=# M \<longleftrightarrow> M' <# M \<or> M' = M" |
|
1345 |
||
35308 | 1346 |
notation (xsymbols) less_multiset (infix "\<subset>#" 50) |
1347 |
notation (xsymbols) le_multiset (infix "\<subseteq>#" 50) |
|
10249 | 1348 |
|
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1349 |
interpretation multiset_order: order le_multiset less_multiset |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1350 |
proof - |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1351 |
have irrefl: "\<And>M :: 'a multiset. \<not> M \<subset># M" |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1352 |
proof |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1353 |
fix M :: "'a multiset" |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1354 |
assume "M \<subset># M" |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1355 |
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
|
1356 |
have "trans {(x'::'a, x). x' < x}" |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1357 |
by (rule transI) simp |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1358 |
moreover note MM |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1359 |
ultimately have "\<exists>I J K. M = I + J \<and> M = I + K |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1360 |
\<and> J \<noteq> {#} \<and> (\<forall>k\<in>set_of K. \<exists>j\<in>set_of J. (k, j) \<in> {(x, y). x < y})" |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1361 |
by (rule mult_implies_one_step) |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1362 |
then obtain I J K where "M = I + J" and "M = I + K" |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1363 |
and "J \<noteq> {#}" and "(\<forall>k\<in>set_of K. \<exists>j\<in>set_of J. (k, j) \<in> {(x, y). x < y})" by blast |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1364 |
then have aux1: "K \<noteq> {#}" and aux2: "\<forall>k\<in>set_of K. \<exists>j\<in>set_of K. k < j" by auto |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1365 |
have "finite (set_of K)" by simp |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1366 |
moreover note aux2 |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1367 |
ultimately have "set_of K = {}" |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1368 |
by (induct rule: finite_induct) (auto intro: order_less_trans) |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1369 |
with aux1 show False by simp |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1370 |
qed |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1371 |
have trans: "\<And>K M N :: 'a multiset. K \<subset># M \<Longrightarrow> M \<subset># N \<Longrightarrow> K \<subset># N" |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1372 |
unfolding less_multiset_def mult_def by (blast intro: trancl_trans) |
36635
080b755377c0
locale predicates of classes carry a mandatory "class" prefix
haftmann
parents:
36176
diff
changeset
|
1373 |
show "class.order (le_multiset :: 'a multiset \<Rightarrow> _) less_multiset" proof |
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1374 |
qed (auto simp add: le_multiset_def irrefl dest: trans) |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1375 |
qed |
10249 | 1376 |
|
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1377 |
lemma mult_less_irrefl [elim!]: |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1378 |
"M \<subset># (M::'a::order multiset) ==> R" |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1379 |
by (simp add: multiset_order.less_irrefl) |
26567
7bcebb8c2d33
instantiation replacing primitive instance plus overloaded defs; more conservative type arities
haftmann
parents:
26178
diff
changeset
|
1380 |
|
10249 | 1381 |
|
1382 |
subsubsection {* Monotonicity of multiset union *} |
|
1383 |
||
17161 | 1384 |
lemma mult1_union: |
40249
cd404ecb9107
Remove unnecessary premise of mult1_union
Lars Noschinski <noschinl@in.tum.de>
parents:
39533
diff
changeset
|
1385 |
"(B, D) \<in> mult1 r ==> (C + B, C + D) \<in> mult1 r" |
26178 | 1386 |
apply (unfold mult1_def) |
1387 |
apply auto |
|
1388 |
apply (rule_tac x = a in exI) |
|
1389 |
apply (rule_tac x = "C + M0" in exI) |
|
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
|
1390 |
apply (simp add: add_assoc) |
26178 | 1391 |
done |
10249 | 1392 |
|
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1393 |
lemma union_less_mono2: "B \<subset># D ==> C + B \<subset># C + (D::'a::order multiset)" |
26178 | 1394 |
apply (unfold less_multiset_def mult_def) |
1395 |
apply (erule trancl_induct) |
|
40249
cd404ecb9107
Remove unnecessary premise of mult1_union
Lars Noschinski <noschinl@in.tum.de>
parents:
39533
diff
changeset
|
1396 |
apply (blast intro: mult1_union) |
cd404ecb9107
Remove unnecessary premise of mult1_union
Lars Noschinski <noschinl@in.tum.de>
parents:
39533
diff
changeset
|
1397 |
apply (blast intro: mult1_union trancl_trans) |
26178 | 1398 |
done |
10249 | 1399 |
|
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1400 |
lemma union_less_mono1: "B \<subset># D ==> B + C \<subset># D + (C::'a::order 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
|
1401 |
apply (subst add_commute [of B 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
|
1402 |
apply (subst add_commute [of D C]) |
26178 | 1403 |
apply (erule union_less_mono2) |
1404 |
done |
|
10249 | 1405 |
|
17161 | 1406 |
lemma union_less_mono: |
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1407 |
"A \<subset># C ==> B \<subset># D ==> A + B \<subset># C + (D::'a::order multiset)" |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1408 |
by (blast intro!: union_less_mono1 union_less_mono2 multiset_order.less_trans) |
10249 | 1409 |
|
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1410 |
interpretation multiset_order: ordered_ab_semigroup_add plus le_multiset less_multiset |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1411 |
proof |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1412 |
qed (auto simp add: le_multiset_def intro: union_less_mono2) |
26145 | 1413 |
|
15072 | 1414 |
|
25610 | 1415 |
subsection {* The fold combinator *} |
1416 |
||
26145 | 1417 |
text {* |
1418 |
The intended behaviour is |
|
1419 |
@{text "fold_mset f z {#x\<^isub>1, ..., x\<^isub>n#} = f x\<^isub>1 (\<dots> (f x\<^isub>n z)\<dots>)"} |
|
1420 |
if @{text f} is associative-commutative. |
|
25610 | 1421 |
*} |
1422 |
||
26145 | 1423 |
text {* |
1424 |
The graph of @{text "fold_mset"}, @{text "z"}: the start element, |
|
1425 |
@{text "f"}: folding function, @{text "A"}: the multiset, @{text |
|
1426 |
"y"}: the result. |
|
1427 |
*} |
|
25610 | 1428 |
inductive |
25759 | 1429 |
fold_msetG :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a multiset \<Rightarrow> 'b \<Rightarrow> bool" |
25610 | 1430 |
for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" |
1431 |
and z :: 'b |
|
1432 |
where |
|
25759 | 1433 |
emptyI [intro]: "fold_msetG f z {#} z" |
1434 |
| insertI [intro]: "fold_msetG f z A y \<Longrightarrow> fold_msetG f z (A + {#x#}) (f x y)" |
|
25610 | 1435 |
|
25759 | 1436 |
inductive_cases empty_fold_msetGE [elim!]: "fold_msetG f z {#} x" |
1437 |
inductive_cases insert_fold_msetGE: "fold_msetG f z (A + {#}) y" |
|
25610 | 1438 |
|
1439 |
definition |
|
26145 | 1440 |
fold_mset :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a multiset \<Rightarrow> 'b" where |
1441 |
"fold_mset f z A = (THE x. fold_msetG f z A x)" |
|
25610 | 1442 |
|
25759 | 1443 |
lemma Diff1_fold_msetG: |
26145 | 1444 |
"fold_msetG f z (A - {#x#}) y \<Longrightarrow> x \<in># A \<Longrightarrow> fold_msetG f z A (f x y)" |
26178 | 1445 |
apply (frule_tac x = x in fold_msetG.insertI) |
1446 |
apply auto |
|
1447 |
done |
|
25610 | 1448 |
|
25759 | 1449 |
lemma fold_msetG_nonempty: "\<exists>x. fold_msetG f z A x" |
26178 | 1450 |
apply (induct A) |
1451 |
apply blast |
|
1452 |
apply clarsimp |
|
1453 |
apply (drule_tac x = x in fold_msetG.insertI) |
|
1454 |
apply auto |
|
1455 |
done |
|
25610 | 1456 |
|
25759 | 1457 |
lemma fold_mset_empty[simp]: "fold_mset f z {#} = z" |
26178 | 1458 |
unfolding fold_mset_def by blast |
25610 | 1459 |
|
42871
1c0b99f950d9
names of fold_set locales resemble name of characteristic property more closely
haftmann
parents:
42809
diff
changeset
|
1460 |
context comp_fun_commute |
26145 | 1461 |
begin |
25610 | 1462 |
|
26145 | 1463 |
lemma fold_msetG_determ: |
1464 |
"fold_msetG f z A x \<Longrightarrow> fold_msetG f z A y \<Longrightarrow> y = x" |
|
25610 | 1465 |
proof (induct arbitrary: x y z rule: full_multiset_induct) |
1466 |
case (less M x\<^isub>1 x\<^isub>2 Z) |
|
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1467 |
have IH: "\<forall>A. A < M \<longrightarrow> |
25759 | 1468 |
(\<forall>x x' x''. fold_msetG f x'' A x \<longrightarrow> fold_msetG f x'' A x' |
25610 | 1469 |
\<longrightarrow> x' = x)" by fact |
25759 | 1470 |
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 | 1471 |
show ?case |
25759 | 1472 |
proof (rule fold_msetG.cases [OF Mfoldx\<^isub>1]) |
25610 | 1473 |
assume "M = {#}" and "x\<^isub>1 = Z" |
26145 | 1474 |
then show ?case using Mfoldx\<^isub>2 by auto |
25610 | 1475 |
next |
1476 |
fix B b u |
|
25759 | 1477 |
assume "M = B + {#b#}" and "x\<^isub>1 = f b u" and Bu: "fold_msetG f Z B u" |
26145 | 1478 |
then have MBb: "M = B + {#b#}" and x\<^isub>1: "x\<^isub>1 = f b u" by auto |
25610 | 1479 |
show ?case |
25759 | 1480 |
proof (rule fold_msetG.cases [OF Mfoldx\<^isub>2]) |
25610 | 1481 |
assume "M = {#}" "x\<^isub>2 = Z" |
26145 | 1482 |
then show ?case using Mfoldx\<^isub>1 by auto |
25610 | 1483 |
next |
1484 |
fix C c v |
|
25759 | 1485 |
assume "M = C + {#c#}" and "x\<^isub>2 = f c v" and Cv: "fold_msetG f Z C v" |
26145 | 1486 |
then have MCc: "M = C + {#c#}" and x\<^isub>2: "x\<^isub>2 = f c v" by auto |
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1487 |
then have CsubM: "C < M" by simp |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1488 |
from MBb have BsubM: "B < M" by simp |
25610 | 1489 |
show ?case |
1490 |
proof cases |
|
1491 |
assume "b=c" |
|
1492 |
then moreover have "B = C" using MBb MCc by auto |
|
1493 |
ultimately show ?thesis using Bu Cv x\<^isub>1 x\<^isub>2 CsubM IH by auto |
|
1494 |
next |
|
1495 |
assume diff: "b \<noteq> c" |
|
1496 |
let ?D = "B - {#c#}" |
|
1497 |
have cinB: "c \<in># B" and binC: "b \<in># C" using MBb MCc diff |
|
1498 |
by (auto intro: insert_noteq_member dest: sym) |
|
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1499 |
have "B - {#c#} < B" using cinB by (rule mset_less_diff_self) |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1500 |
then have DsubM: "?D < M" using BsubM by (blast intro: order_less_trans) |
25610 | 1501 |
from MBb MCc have "B + {#b#} = C + {#c#}" by blast |
26145 | 1502 |
then have [simp]: "B + {#b#} - {#c#} = C" |
25610 | 1503 |
using MBb MCc binC cinB by auto |
1504 |
have B: "B = ?D + {#c#}" and C: "C = ?D + {#b#}" |
|
1505 |
using MBb MCc diff binC cinB |
|
1506 |
by (auto simp: multiset_add_sub_el_shuffle) |
|
25759 | 1507 |
then obtain d where Dfoldd: "fold_msetG f Z ?D d" |
1508 |
using fold_msetG_nonempty by iprover |
|
26145 | 1509 |
then have "fold_msetG f Z B (f c d)" using cinB |
25759 | 1510 |
by (rule Diff1_fold_msetG) |
26145 | 1511 |
then have "f c d = u" using IH BsubM Bu by blast |
25610 | 1512 |
moreover |
25759 | 1513 |
have "fold_msetG f Z C (f b d)" using binC cinB diff Dfoldd |
25610 | 1514 |
by (auto simp: multiset_add_sub_el_shuffle |
25759 | 1515 |
dest: fold_msetG.insertI [where x=b]) |
26145 | 1516 |
then have "f b d = v" using IH CsubM Cv by blast |
25610 | 1517 |
ultimately show ?thesis using x\<^isub>1 x\<^isub>2 |
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
|
1518 |
by (auto simp: fun_left_comm) |
25610 | 1519 |
qed |
1520 |
qed |
|
1521 |
qed |
|
1522 |
qed |
|
1523 |
||
26145 | 1524 |
lemma fold_mset_insert_aux: |
1525 |
"(fold_msetG f z (A + {#x#}) v) = |
|
25759 | 1526 |
(\<exists>y. fold_msetG f z A y \<and> v = f x y)" |
26178 | 1527 |
apply (rule iffI) |
1528 |
prefer 2 |
|
1529 |
apply blast |
|
1530 |
apply (rule_tac A=A and f=f in fold_msetG_nonempty [THEN exE, standard]) |
|
1531 |
apply (blast intro: fold_msetG_determ) |
|
1532 |
done |
|
25610 | 1533 |
|
26145 | 1534 |
lemma fold_mset_equality: "fold_msetG f z A y \<Longrightarrow> fold_mset f z A = y" |
26178 | 1535 |
unfolding fold_mset_def by (blast intro: fold_msetG_determ) |
25610 | 1536 |
|
26145 | 1537 |
lemma fold_mset_insert: |
26178 | 1538 |
"fold_mset f z (A + {#x#}) = f x (fold_mset f z A)" |
1539 |
apply (simp add: fold_mset_def fold_mset_insert_aux) |
|
1540 |
apply (rule the_equality) |
|
1541 |
apply (auto cong add: conj_cong |
|
26145 | 1542 |
simp add: fold_mset_def [symmetric] fold_mset_equality fold_msetG_nonempty) |
26178 | 1543 |
done |
25610 | 1544 |
|
26145 | 1545 |
lemma fold_mset_commute: "f x (fold_mset f z A) = fold_mset f (f x z) 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
|
1546 |
by (induct A) (auto simp: fold_mset_insert fun_left_comm [of x]) |
26178 | 1547 |
|
26145 | 1548 |
lemma fold_mset_single [simp]: "fold_mset f z {#x#} = f x z" |
26178 | 1549 |
using fold_mset_insert [of z "{#}"] by simp |
25610 | 1550 |
|
26145 | 1551 |
lemma fold_mset_union [simp]: |
1552 |
"fold_mset f z (A+B) = fold_mset f (fold_mset f z A) B" |
|
25759 | 1553 |
proof (induct A) |
26145 | 1554 |
case empty then show ?case by simp |
25759 | 1555 |
next |
26145 | 1556 |
case (add A 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
|
1557 |
have "A + {#x#} + B = (A+B) + {#x#}" by (simp add: add_ac) |
26145 | 1558 |
then have "fold_mset f z (A + {#x#} + B) = f x (fold_mset f z (A + B))" |
1559 |
by (simp add: fold_mset_insert) |
|
1560 |
also have "\<dots> = fold_mset f (fold_mset f z (A + {#x#})) B" |
|
1561 |
by (simp add: fold_mset_commute[of x,symmetric] add fold_mset_insert) |
|
1562 |
finally show ?case . |
|
25759 | 1563 |
qed |
1564 |
||
26145 | 1565 |
lemma fold_mset_fusion: |
42871
1c0b99f950d9
names of fold_set locales resemble name of characteristic property more closely
haftmann
parents:
42809
diff
changeset
|
1566 |
assumes "comp_fun_commute g" |
27611 | 1567 |
shows "(\<And>x y. h (g x y) = f x (h y)) \<Longrightarrow> h (fold_mset g w A) = fold_mset f (h w) A" (is "PROP ?P") |
1568 |
proof - |
|
42871
1c0b99f950d9
names of fold_set locales resemble name of characteristic property more closely
haftmann
parents:
42809
diff
changeset
|
1569 |
interpret comp_fun_commute g by (fact assms) |
27611 | 1570 |
show "PROP ?P" by (induct A) auto |
1571 |
qed |
|
25610 | 1572 |
|
26145 | 1573 |
lemma fold_mset_rec: |
1574 |
assumes "a \<in># A" |
|
25759 | 1575 |
shows "fold_mset f z A = f a (fold_mset f z (A - {#a#}))" |
25610 | 1576 |
proof - |
26145 | 1577 |
from assms obtain A' where "A = A' + {#a#}" |
1578 |
by (blast dest: multi_member_split) |
|
1579 |
then show ?thesis by simp |
|
25610 | 1580 |
qed |
1581 |
||
26145 | 1582 |
end |
1583 |
||
1584 |
text {* |
|
1585 |
A note on code generation: When defining some function containing a |
|
1586 |
subterm @{term"fold_mset F"}, code generation is not automatic. When |
|
1587 |
interpreting locale @{text left_commutative} with @{text F}, the |
|
1588 |
would be code thms for @{const fold_mset} become thms like |
|
1589 |
@{term"fold_mset F z {#} = z"} where @{text F} is not a pattern but |
|
1590 |
contains defined symbols, i.e.\ is not a code thm. Hence a separate |
|
1591 |
constant with its own code thms needs to be introduced for @{text |
|
1592 |
F}. See the image operator below. |
|
1593 |
*} |
|
1594 |
||
26016 | 1595 |
|
1596 |
subsection {* Image *} |
|
1597 |
||
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
|
1598 |
definition image_mset :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a multiset \<Rightarrow> 'b 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:
33102
diff
changeset
|
1599 |
"image_mset f = fold_mset (op + o single o f) {#}" |
26016 | 1600 |
|
44339
eda6aef75939
odd workaround for odd problem of load order in HOL/ex/ROOT.ML (!??);
wenzelm
parents:
42871
diff
changeset
|
1601 |
interpretation image_fun_commute: comp_fun_commute "op + o single o f" for f |
42809
5b45125b15ba
use pointfree characterisation for fold_set locale
haftmann
parents:
41505
diff
changeset
|
1602 |
proof qed (simp add: add_ac fun_eq_iff) |
26016 | 1603 |
|
28708
a1a436f09ec6
explicit check for pattern discipline before code translation
haftmann
parents:
28562
diff
changeset
|
1604 |
lemma image_mset_empty [simp]: "image_mset f {#} = {#}" |
26178 | 1605 |
by (simp add: image_mset_def) |
26016 | 1606 |
|
28708
a1a436f09ec6
explicit check for pattern discipline before code translation
haftmann
parents:
28562
diff
changeset
|
1607 |
lemma image_mset_single [simp]: "image_mset f {#x#} = {#f x#}" |
26178 | 1608 |
by (simp add: image_mset_def) |
26016 | 1609 |
|
1610 |
lemma image_mset_insert: |
|
1611 |
"image_mset f (M + {#a#}) = image_mset f M + {#f a#}" |
|
26178 | 1612 |
by (simp add: image_mset_def add_ac) |
26016 | 1613 |
|
28708
a1a436f09ec6
explicit check for pattern discipline before code translation
haftmann
parents:
28562
diff
changeset
|
1614 |
lemma image_mset_union [simp]: |
26016 | 1615 |
"image_mset f (M+N) = image_mset f M + image_mset f N" |
26178 | 1616 |
apply (induct N) |
1617 |
apply 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
|
1618 |
apply (simp add: add_assoc [symmetric] image_mset_insert) |
26178 | 1619 |
done |
26016 | 1620 |
|
26145 | 1621 |
lemma size_image_mset [simp]: "size (image_mset f M) = size M" |
26178 | 1622 |
by (induct M) simp_all |
26016 | 1623 |
|
26145 | 1624 |
lemma image_mset_is_empty_iff [simp]: "image_mset f M = {#} \<longleftrightarrow> M = {#}" |
26178 | 1625 |
by (cases M) auto |
26016 | 1626 |
|
26145 | 1627 |
syntax |
35352 | 1628 |
"_comprehension1_mset" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> 'a multiset" |
26145 | 1629 |
("({#_/. _ :# _#})") |
1630 |
translations |
|
1631 |
"{#e. x:#M#}" == "CONST image_mset (%x. e) M" |
|
26016 | 1632 |
|
26145 | 1633 |
syntax |
35352 | 1634 |
"_comprehension2_mset" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> bool \<Rightarrow> 'a multiset" |
26145 | 1635 |
("({#_/ | _ :# _./ _#})") |
26016 | 1636 |
translations |
26033 | 1637 |
"{#e | x:#M. P#}" => "{#e. x :# {# x:#M. P#}#}" |
26016 | 1638 |
|
26145 | 1639 |
text {* |
1640 |
This allows to write not just filters like @{term "{#x:#M. x<c#}"} |
|
1641 |
but also images like @{term "{#x+x. x:#M #}"} and @{term [source] |
|
1642 |
"{#x+x|x:#M. x<c#}"}, where the latter is currently displayed as |
|
1643 |
@{term "{#x+x|x:#M. x<c#}"}. |
|
1644 |
*} |
|
26016 | 1645 |
|
41505
6d19301074cf
"enriched_type" replaces less specific "type_lifting"
haftmann
parents:
41372
diff
changeset
|
1646 |
enriched_type image_mset: image_mset proof - |
41372 | 1647 |
fix f g |
1648 |
show "image_mset f \<circ> image_mset g = image_mset (f \<circ> g)" |
|
1649 |
proof |
|
1650 |
fix A |
|
1651 |
show "(image_mset f \<circ> image_mset g) A = image_mset (f \<circ> g) A" |
|
1652 |
by (induct A) simp_all |
|
1653 |
qed |
|
40606 | 1654 |
next |
41372 | 1655 |
show "image_mset id = id" |
1656 |
proof |
|
1657 |
fix A |
|
1658 |
show "image_mset id A = id A" |
|
1659 |
by (induct A) simp_all |
|
1660 |
qed |
|
40606 | 1661 |
qed |
1662 |
||
29125
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1663 |
|
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1664 |
subsection {* Termination proofs with multiset orders *} |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1665 |
|
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1666 |
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
|
1667 |
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
|
1668 |
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
|
1669 |
by auto |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1670 |
|
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1671 |
definition "ms_strict = mult pair_less" |
37765 | 1672 |
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
|
1673 |
|
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1674 |
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
|
1675 |
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
|
1676 |
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
|
1677 |
|
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1678 |
lemma smsI: |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1679 |
"(set_of A, set_of B) \<in> max_strict \<Longrightarrow> (Z + A, Z + B) \<in> ms_strict" |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1680 |
unfolding ms_strict_def |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1681 |
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
|
1682 |
|
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1683 |
lemma wmsI: |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1684 |
"(set_of A, set_of B) \<in> max_strict \<or> A = {#} \<and> B = {#} |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1685 |
\<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
|
1686 |
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
|
1687 |
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
|
1688 |
|
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1689 |
inductive pw_leq |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1690 |
where |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1691 |
pw_leq_empty: "pw_leq {#} {#}" |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1692 |
| 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
|
1693 |
|
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1694 |
lemma pw_leq_lstep: |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1695 |
"(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
|
1696 |
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
|
1697 |
|
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1698 |
lemma pw_leq_split: |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1699 |
assumes "pw_leq X Y" |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1700 |
shows "\<exists>A B Z. X = A + Z \<and> Y = B + Z \<and> ((set_of A, set_of B) \<in> max_strict \<or> (B = {#} \<and> A = {#}))" |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1701 |
using assms |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1702 |
proof (induct) |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1703 |
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
|
1704 |
next |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1705 |
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
|
1706 |
then obtain A B Z where |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1707 |
[simp]: "X = A + Z" "Y = B + Z" |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1708 |
and 1[simp]: "(set_of A, set_of B) \<in> max_strict \<or> (B = {#} \<and> A = {#})" |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1709 |
by auto |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1710 |
from pw_leq_step have "x = y \<or> (x, y) \<in> pair_less" |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1711 |
unfolding pair_leq_def by auto |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1712 |
thus ?case |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1713 |
proof |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1714 |
assume [simp]: "x = y" |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1715 |
have |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1716 |
"{#x#} + X = A + ({#y#}+Z) |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1717 |
\<and> {#y#} + Y = B + ({#y#}+Z) |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1718 |
\<and> ((set_of A, set_of B) \<in> max_strict \<or> (B = {#} \<and> A = {#}))" |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1719 |
by (auto simp: add_ac) |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1720 |
thus ?case by (intro exI) |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1721 |
next |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1722 |
assume A: "(x, y) \<in> pair_less" |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1723 |
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
|
1724 |
have "{#x#} + X = ?A' + Z" |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1725 |
"{#y#} + Y = ?B' + Z" |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1726 |
by (auto simp add: add_ac) |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1727 |
moreover have |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1728 |
"(set_of ?A', set_of ?B') \<in> max_strict" |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1729 |
using 1 A unfolding max_strict_def |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1730 |
by (auto elim!: max_ext.cases) |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1731 |
ultimately show ?thesis by blast |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1732 |
qed |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1733 |
qed |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1734 |
|
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1735 |
lemma |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1736 |
assumes pwleq: "pw_leq Z Z'" |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1737 |
shows ms_strictI: "(set_of A, set_of B) \<in> max_strict \<Longrightarrow> (Z + A, Z' + B) \<in> ms_strict" |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1738 |
and ms_weakI1: "(set_of A, set_of B) \<in> max_strict \<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
|
1739 |
and ms_weakI2: "(Z + {#}, Z' + {#}) \<in> ms_weak" |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1740 |
proof - |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1741 |
from pw_leq_split[OF pwleq] |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1742 |
obtain A' B' Z'' |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1743 |
where [simp]: "Z = A' + Z''" "Z' = B' + Z''" |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1744 |
and mx_or_empty: "(set_of A', set_of B') \<in> max_strict \<or> (A' = {#} \<and> B' = {#})" |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1745 |
by blast |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1746 |
{ |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1747 |
assume max: "(set_of A, set_of B) \<in> max_strict" |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1748 |
from mx_or_empty |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1749 |
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
|
1750 |
proof |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1751 |
assume max': "(set_of A', set_of B') \<in> max_strict" |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1752 |
with max have "(set_of (A + A'), set_of (B + B')) \<in> max_strict" |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1753 |
by (auto simp: max_strict_def intro: max_ext_additive) |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1754 |
thus ?thesis by (rule smsI) |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1755 |
next |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1756 |
assume [simp]: "A' = {#} \<and> B' = {#}" |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1757 |
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
|
1758 |
qed |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1759 |
thus "(Z + A, Z' + B) \<in> ms_strict" by (simp add:add_ac) |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1760 |
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
|
1761 |
} |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1762 |
from mx_or_empty |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1763 |
have "(Z'' + A', Z'' + B') \<in> ms_weak" by (rule wmsI) |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1764 |
thus "(Z + {#}, Z' + {#}) \<in> ms_weak" by (simp add:add_ac) |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1765 |
qed |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1766 |
|
39301 | 1767 |
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
|
1768 |
and nonempty_plus: "{# x #} + rs \<noteq> {#}" |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1769 |
and nonempty_single: "{# x #} \<noteq> {#}" |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1770 |
by auto |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1771 |
|
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1772 |
setup {* |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1773 |
let |
35402 | 1774 |
fun msetT T = Type (@{type_name multiset}, [T]); |
29125
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1775 |
|
35402 | 1776 |
fun mk_mset T [] = Const (@{const_abbrev Mempty}, msetT T) |
29125
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1777 |
| mk_mset T [x] = Const (@{const_name single}, T --> msetT T) $ x |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1778 |
| mk_mset T (x :: xs) = |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1779 |
Const (@{const_name plus}, msetT T --> msetT T --> msetT T) $ |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1780 |
mk_mset T [x] $ mk_mset T xs |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1781 |
|
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1782 |
fun mset_member_tac m i = |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1783 |
(if m <= 0 then |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1784 |
rtac @{thm multi_member_this} i ORELSE rtac @{thm multi_member_last} i |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1785 |
else |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1786 |
rtac @{thm multi_member_skip} i THEN mset_member_tac (m - 1) i) |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1787 |
|
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1788 |
val mset_nonempty_tac = |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1789 |
rtac @{thm nonempty_plus} ORELSE' rtac @{thm nonempty_single} |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1790 |
|
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1791 |
val regroup_munion_conv = |
35402 | 1792 |
Function_Lib.regroup_conv @{const_abbrev Mempty} @{const_name plus} |
39301 | 1793 |
(map (fn t => t RS eq_reflection) (@{thms add_ac} @ @{thms empty_neutral})) |
29125
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1794 |
|
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1795 |
fun unfold_pwleq_tac i = |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1796 |
(rtac @{thm pw_leq_step} i THEN (fn st => unfold_pwleq_tac (i + 1) st)) |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1797 |
ORELSE (rtac @{thm pw_leq_lstep} i) |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1798 |
ORELSE (rtac @{thm pw_leq_empty} i) |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1799 |
|
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1800 |
val set_of_simps = [@{thm set_of_empty}, @{thm set_of_single}, @{thm set_of_union}, |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1801 |
@{thm Un_insert_left}, @{thm Un_empty_left}] |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1802 |
in |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1803 |
ScnpReconstruct.multiset_setup (ScnpReconstruct.Multiset |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1804 |
{ |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1805 |
msetT=msetT, mk_mset=mk_mset, mset_regroup_conv=regroup_munion_conv, |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1806 |
mset_member_tac=mset_member_tac, mset_nonempty_tac=mset_nonempty_tac, |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1807 |
mset_pwleq_tac=unfold_pwleq_tac, set_of_simps=set_of_simps, |
30595
c87a3350f5a9
proper spacing before ML antiquotations -- note that @ may be part of symbolic ML identifiers;
wenzelm
parents:
30428
diff
changeset
|
1808 |
smsI'= @{thm ms_strictI}, wmsI2''= @{thm ms_weakI2}, wmsI1= @{thm ms_weakI1}, |
c87a3350f5a9
proper spacing before ML antiquotations -- note that @ may be part of symbolic ML identifiers;
wenzelm
parents:
30428
diff
changeset
|
1809 |
reduction_pair= @{thm ms_reduction_pair} |
29125
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1810 |
}) |
10249 | 1811 |
end |
29125
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1812 |
*} |
d41182a8135c
method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents:
28823
diff
changeset
|
1813 |
|
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
|
1814 |
|
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
|
1815 |
subsection {* Legacy theorem bindings *} |
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
|
1816 |
|
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39301
diff
changeset
|
1817 |
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
|
1818 |
|
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
|
1819 |
lemma union_commute: "M + N = N + (M::'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
|
1820 |
by (fact add_commute) |
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
|
1821 |
|
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
|
1822 |
lemma union_assoc: "(M + N) + K = M + (N + (K::'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
|
1823 |
by (fact add_assoc) |
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
|
1824 |
|
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
|
1825 |
lemma union_lcomm: "M + (N + K) = N + (M + (K::'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
|
1826 |
by (fact add_left_commute) |
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
|
1827 |
|
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
|
1828 |
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
|
1829 |
|
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
|
1830 |
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
|
1831 |
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
|
1832 |
|
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
|
1833 |
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
|
1834 |
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
|
1835 |
|
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
|
1836 |
lemma multi_union_self_other_eq: "(A::'a multiset) + X = A + Y \<Longrightarrow> X = Y" |
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
|
1837 |
by (fact add_imp_eq) |
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
|
1838 |
|
35268
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1839 |
lemma mset_less_trans: "(M::'a multiset) < K \<Longrightarrow> K < N \<Longrightarrow> M < N" |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1840 |
by (fact order_less_trans) |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1841 |
|
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1842 |
lemma multiset_inter_commute: "A #\<inter> B = B #\<inter> A" |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1843 |
by (fact inf.commute) |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1844 |
|
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1845 |
lemma multiset_inter_assoc: "A #\<inter> (B #\<inter> C) = A #\<inter> B #\<inter> C" |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1846 |
by (fact inf.assoc [symmetric]) |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1847 |
|
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1848 |
lemma multiset_inter_left_commute: "A #\<inter> (B #\<inter> C) = B #\<inter> (A #\<inter> C)" |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1849 |
by (fact inf.left_commute) |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1850 |
|
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1851 |
lemmas multiset_inter_ac = |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1852 |
multiset_inter_commute |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1853 |
multiset_inter_assoc |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1854 |
multiset_inter_left_commute |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1855 |
|
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1856 |
lemma mult_less_not_refl: |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1857 |
"\<not> M \<subset># (M::'a::order multiset)" |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1858 |
by (fact multiset_order.less_irrefl) |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1859 |
|
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1860 |
lemma mult_less_trans: |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1861 |
"K \<subset># M ==> M \<subset># N ==> K \<subset># (N::'a::order multiset)" |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1862 |
by (fact multiset_order.less_trans) |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1863 |
|
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1864 |
lemma mult_less_not_sym: |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1865 |
"M \<subset># N ==> \<not> N \<subset># (M::'a::order multiset)" |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1866 |
by (fact multiset_order.less_not_sym) |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1867 |
|
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1868 |
lemma mult_less_asym: |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1869 |
"M \<subset># N ==> (\<not> P ==> N \<subset># (M::'a::order multiset)) ==> P" |
04673275441a
switched notations for pointwise and multiset order
haftmann
parents:
35028
diff
changeset
|
1870 |
by (fact multiset_order.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
|
1871 |
|
35712
77aa29bf14ee
added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents:
35402
diff
changeset
|
1872 |
ML {* |
77aa29bf14ee
added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents:
35402
diff
changeset
|
1873 |
fun multiset_postproc _ maybe_name all_values (T as Type (_, [elem_T])) |
77aa29bf14ee
added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents:
35402
diff
changeset
|
1874 |
(Const _ $ t') = |
77aa29bf14ee
added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents:
35402
diff
changeset
|
1875 |
let |
77aa29bf14ee
added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents:
35402
diff
changeset
|
1876 |
val (maybe_opt, ps) = |
77aa29bf14ee
added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents:
35402
diff
changeset
|
1877 |
Nitpick_Model.dest_plain_fun t' ||> op ~~ |
77aa29bf14ee
added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents:
35402
diff
changeset
|
1878 |
||> map (apsnd (snd o HOLogic.dest_number)) |
77aa29bf14ee
added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents:
35402
diff
changeset
|
1879 |
fun elems_for t = |
77aa29bf14ee
added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents:
35402
diff
changeset
|
1880 |
case AList.lookup (op =) ps t of |
77aa29bf14ee
added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents:
35402
diff
changeset
|
1881 |
SOME n => replicate n t |
77aa29bf14ee
added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents:
35402
diff
changeset
|
1882 |
| NONE => [Const (maybe_name, elem_T --> elem_T) $ t] |
77aa29bf14ee
added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents:
35402
diff
changeset
|
1883 |
in |
77aa29bf14ee
added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents:
35402
diff
changeset
|
1884 |
case maps elems_for (all_values elem_T) @ |
37261 | 1885 |
(if maybe_opt then [Const (Nitpick_Model.unrep (), elem_T)] |
1886 |
else []) of |
|
35712
77aa29bf14ee
added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents:
35402
diff
changeset
|
1887 |
[] => Const (@{const_name zero_class.zero}, T) |
77aa29bf14ee
added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents:
35402
diff
changeset
|
1888 |
| ts => foldl1 (fn (t1, t2) => |
77aa29bf14ee
added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents:
35402
diff
changeset
|
1889 |
Const (@{const_name plus_class.plus}, T --> T --> T) |
77aa29bf14ee
added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents:
35402
diff
changeset
|
1890 |
$ t1 $ t2) |
77aa29bf14ee
added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents:
35402
diff
changeset
|
1891 |
(map (curry (op $) (Const (@{const_name single}, |
77aa29bf14ee
added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents:
35402
diff
changeset
|
1892 |
elem_T --> T))) ts) |
77aa29bf14ee
added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents:
35402
diff
changeset
|
1893 |
end |
77aa29bf14ee
added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents:
35402
diff
changeset
|
1894 |
| multiset_postproc _ _ _ _ t = t |
77aa29bf14ee
added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents:
35402
diff
changeset
|
1895 |
*} |
77aa29bf14ee
added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents:
35402
diff
changeset
|
1896 |
|
38287 | 1897 |
declaration {* |
1898 |
Nitpick_Model.register_term_postprocessor @{typ "'a multiset"} |
|
38242 | 1899 |
multiset_postproc |
35712
77aa29bf14ee
added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents:
35402
diff
changeset
|
1900 |
*} |
77aa29bf14ee
added a mechanism to Nitpick to support custom rendering of terms, and used it for multisets
blanchet
parents:
35402
diff
changeset
|
1901 |
|
37169
f69efa106feb
make Nitpick "show_all" option behave less surprisingly
blanchet
parents:
37107
diff
changeset
|
1902 |
end |