author | haftmann |
Fri, 25 Jul 2008 12:03:34 +0200 | |
changeset 27682 | 25aceefd4786 |
parent 25314 | 5eaf3e8b50a4 |
child 28004 | c8642f498aa3 |
permissions | -rw-r--r-- |
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(* Title: HOL/Library/Kleene_Algebras.thy |
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ID: $Id$ |
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Author: Alexander Krauss, TU Muenchen |
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*) |
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header "Kleene Algebras" |
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theory Kleene_Algebras |
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imports Main |
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begin |
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text {* A type class of kleene algebras *} |
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class star = type + |
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fixes star :: "'a \<Rightarrow> 'a" |
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class idem_add = ab_semigroup_add + |
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assumes add_idem [simp]: "x + x = x" |
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lemma add_idem2[simp]: "(x::'a::idem_add) + (x + y) = x + y" |
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unfolding add_assoc[symmetric] |
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by simp |
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class order_by_add = idem_add + ord + |
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assumes order_def: "a \<le> b \<longleftrightarrow> a + b = b" |
27682 | 26 |
assumes strict_order_def: "a < b \<longleftrightarrow> a \<le> b \<and> \<not> b \<le> a" |
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begin |
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27682 | 29 |
lemma ord_simp1[simp]: "x \<le> y \<Longrightarrow> x + y = y" |
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unfolding order_def . |
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27682 | 32 |
lemma ord_simp2[simp]: "x \<le> y \<Longrightarrow> y + x = y" |
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unfolding order_def add_commute . |
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lemma ord_intro: "x + y = y \<Longrightarrow> x \<le> y" |
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unfolding order_def . |
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subclass order proof |
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fix x y z :: 'a |
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show "x \<le> x" unfolding order_def by simp |
27682 | 41 |
show "x \<le> y \<Longrightarrow> y \<le> z \<Longrightarrow> x \<le> z" |
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proof (rule ord_intro) |
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assume "x \<le> y" "y \<le> z" |
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have "x + z = x + y + z" by (simp add:`y \<le> z` add_assoc) |
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also have "\<dots> = y + z" by (simp add:`x \<le> y`) |
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also have "\<dots> = z" by (simp add:`y \<le> z`) |
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finally show "x + z = z" . |
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qed |
27682 | 49 |
show "x \<le> y \<Longrightarrow> y \<le> x \<Longrightarrow> x = y" unfolding order_def |
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by (simp add: add_commute) |
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show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x" by (fact strict_order_def) |
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qed |
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27682 | 54 |
lemma plus_leI: |
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"x \<le> z \<Longrightarrow> y \<le> z \<Longrightarrow> x + y \<le> z" |
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unfolding order_def by (simp add: add_assoc) |
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58 |
end |
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class pre_kleene = semiring_1 + order_by_add |
27682 | 61 |
begin |
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27682 | 63 |
subclass pordered_semiring proof |
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fix x y z :: 'a |
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65 |
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assume "x \<le> y" |
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67 |
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show "z + x \<le> z + y" |
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proof (rule ord_intro) |
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have "z + x + (z + y) = x + y + z" by (simp add:add_ac) |
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also have "\<dots> = z + y" by (simp add:`x \<le> y` add_ac) |
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finally show "z + x + (z + y) = z + y" . |
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qed |
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74 |
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show "z * x \<le> z * y" |
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proof (rule ord_intro) |
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from `x \<le> y` have "z * (x + y) = z * y" by simp |
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thus "z * x + z * y = z * y" by (simp add:right_distrib) |
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qed |
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80 |
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show "x * z \<le> y * z" |
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proof (rule ord_intro) |
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from `x \<le> y` have "(x + y) * z = y * z" by simp |
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thus "x * z + y * z = y * z" by (simp add:left_distrib) |
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qed |
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qed |
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lemma zero_minimum [simp]: "0 \<le> x" |
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unfolding order_def by simp |
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90 |
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91 |
end |
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||
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class kleene = pre_kleene + star + |
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assumes star1: "1 + a * star a \<le> star a" |
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and star2: "1 + star a * a \<le> star a" |
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and star3: "a * x \<le> x \<Longrightarrow> star a * x \<le> x" |
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and star4: "x * a \<le> x \<Longrightarrow> x * star a \<le> x" |
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98 |
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class kleene_by_complete_lattice = pre_kleene |
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+ complete_lattice + recpower + star + |
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assumes star_cont: "a * star b * c = SUPR UNIV (\<lambda>n. a * b ^ n * c)" |
27682 | 102 |
begin |
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103 |
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27682 | 104 |
lemma (in complete_lattice) le_SUPI': |
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assumes "l \<le> M i" |
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shows "l \<le> (SUP i. M i)" |
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using assms by (rule order_trans) (rule le_SUPI [OF UNIV_I]) |
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108 |
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27682 | 109 |
end |
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instance kleene_by_complete_lattice < kleene |
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proof |
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113 |
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fix a x :: 'a |
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115 |
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have [simp]: "1 \<le> star a" |
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117 |
unfolding star_cont[of 1 a 1, simplified] |
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by (subst power_0[symmetric]) (rule le_SUPI [OF UNIV_I]) |
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119 |
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120 |
show "1 + a * star a \<le> star a" |
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apply (rule plus_leI, simp) |
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apply (simp add:star_cont[of a a 1, simplified]) |
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apply (simp add:star_cont[of 1 a 1, simplified]) |
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apply (subst power_Suc[symmetric]) |
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125 |
by (intro SUP_leI le_SUPI UNIV_I) |
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126 |
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show "1 + star a * a \<le> star a" |
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128 |
apply (rule plus_leI, simp) |
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129 |
apply (simp add:star_cont[of 1 a a, simplified]) |
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apply (simp add:star_cont[of 1 a 1, simplified]) |
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by (auto intro: SUP_leI le_SUPI UNIV_I simp add: power_Suc[symmetric] power_commutes) |
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132 |
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133 |
show "a * x \<le> x \<Longrightarrow> star a * x \<le> x" |
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proof - |
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135 |
assume a: "a * x \<le> x" |
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136 |
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137 |
{ |
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138 |
fix n |
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have "a ^ (Suc n) * x \<le> a ^ n * x" |
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proof (induct n) |
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case 0 thus ?case by (simp add:a power_Suc) |
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142 |
next |
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case (Suc n) |
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hence "a * (a ^ Suc n * x) \<le> a * (a ^ n * x)" |
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by (auto intro: mult_mono) |
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146 |
thus ?case |
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147 |
by (simp add:power_Suc mult_assoc) |
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qed |
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149 |
} |
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150 |
note a = this |
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151 |
|
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152 |
{ |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
153 |
fix n have "a ^ n * x \<le> x" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
154 |
proof (induct n) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
155 |
case 0 show ?case by simp |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
156 |
next |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
157 |
case (Suc n) with a[of n] |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
158 |
show ?case by simp |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
159 |
qed |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
160 |
} |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
161 |
note b = this |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
162 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
163 |
show "star a * x \<le> x" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
164 |
unfolding star_cont[of 1 a x, simplified] |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
165 |
by (rule SUP_leI) (rule b) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
166 |
qed |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
167 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
168 |
show "x * a \<le> x \<Longrightarrow> x * star a \<le> x" (* symmetric *) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
169 |
proof - |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
170 |
assume a: "x * a \<le> x" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
171 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
172 |
{ |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
173 |
fix n |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
174 |
have "x * a ^ (Suc n) \<le> x * a ^ n" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
175 |
proof (induct n) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
176 |
case 0 thus ?case by (simp add:a power_Suc) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
177 |
next |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
178 |
case (Suc n) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
179 |
hence "(x * a ^ Suc n) * a \<le> (x * a ^ n) * a" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
180 |
by (auto intro: mult_mono) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
181 |
thus ?case |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
182 |
by (simp add:power_Suc power_commutes mult_assoc) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
183 |
qed |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
184 |
} |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
185 |
note a = this |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
186 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
187 |
{ |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
188 |
fix n have "x * a ^ n \<le> x" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
189 |
proof (induct n) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
190 |
case 0 show ?case by simp |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
191 |
next |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
192 |
case (Suc n) with a[of n] |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
193 |
show ?case by simp |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
194 |
qed |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
195 |
} |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
196 |
note b = this |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
197 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
198 |
show "x * star a \<le> x" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
199 |
unfolding star_cont[of x a 1, simplified] |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
200 |
by (rule SUP_leI) (rule b) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
201 |
qed |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
202 |
qed |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
203 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
204 |
lemma less_add[simp]: |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
205 |
fixes a b :: "'a :: order_by_add" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
206 |
shows "a \<le> a + b" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
207 |
and "b \<le> a + b" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
208 |
unfolding order_def |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
209 |
by (auto simp:add_ac) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
210 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
211 |
lemma add_est1: |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
212 |
fixes a b c :: "'a :: order_by_add" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
213 |
assumes a: "a + b \<le> c" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
214 |
shows "a \<le> c" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
215 |
using less_add(1) a |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
216 |
by (rule order_trans) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
217 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
218 |
lemma add_est2: |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
219 |
fixes a b c :: "'a :: order_by_add" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
220 |
assumes a: "a + b \<le> c" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
221 |
shows "b \<le> c" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
222 |
using less_add(2) a |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
223 |
by (rule order_trans) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
224 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
225 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
226 |
lemma star3': |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
227 |
fixes a b x :: "'a :: kleene" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
228 |
assumes a: "b + a * x \<le> x" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
229 |
shows "star a * b \<le> x" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
230 |
proof (rule order_trans) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
231 |
from a have "b \<le> x" by (rule add_est1) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
232 |
show "star a * b \<le> star a * x" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
233 |
by (rule mult_mono) (auto simp:`b \<le> x`) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
234 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
235 |
from a have "a * x \<le> x" by (rule add_est2) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
236 |
with star3 show "star a * x \<le> x" . |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
237 |
qed |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
238 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
239 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
240 |
lemma star4': |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
241 |
fixes a b x :: "'a :: kleene" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
242 |
assumes a: "b + x * a \<le> x" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
243 |
shows "b * star a \<le> x" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
244 |
proof (rule order_trans) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
245 |
from a have "b \<le> x" by (rule add_est1) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
246 |
show "b * star a \<le> x * star a" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
247 |
by (rule mult_mono) (auto simp:`b \<le> x`) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
248 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
249 |
from a have "x * a \<le> x" by (rule add_est2) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
250 |
with star4 show "x * star a \<le> x" . |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
251 |
qed |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
252 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
253 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
254 |
lemma star_idemp: |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
255 |
fixes x :: "'a :: kleene" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
256 |
shows "star (star x) = star x" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
257 |
oops |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
258 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
259 |
lemma star_unfold_left: |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
260 |
fixes a :: "'a :: kleene" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
261 |
shows "1 + a * star a = star a" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
262 |
proof (rule order_antisym, rule star1) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
263 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
264 |
have "1 + a * (1 + a * star a) \<le> 1 + a * star a" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
265 |
apply (rule add_mono, rule) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
266 |
apply (rule mult_mono, auto) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
267 |
apply (rule star1) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
268 |
done |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
269 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
270 |
with star3' have "star a * 1 \<le> 1 + a * star a" . |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
271 |
thus "star a \<le> 1 + a * star a" by simp |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
272 |
qed |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
273 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
274 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
275 |
lemma star_unfold_right: |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
276 |
fixes a :: "'a :: kleene" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
277 |
shows "1 + star a * a = star a" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
278 |
proof (rule order_antisym, rule star2) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
279 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
280 |
have "1 + (1 + star a * a) * a \<le> 1 + star a * a" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
281 |
apply (rule add_mono, rule) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
282 |
apply (rule mult_mono, auto) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
283 |
apply (rule star2) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
284 |
done |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
285 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
286 |
with star4' have "1 * star a \<le> 1 + star a * a" . |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
287 |
thus "star a \<le> 1 + star a * a" by simp |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
288 |
qed |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
289 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
290 |
lemma star_zero[simp]: |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
291 |
shows "star (0::'a::kleene) = 1" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
292 |
by (rule star_unfold_left[of 0, simplified]) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
293 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
294 |
lemma star_commute: |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
295 |
fixes a b x :: "'a :: kleene" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
296 |
assumes a: "a * x = x * b" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
297 |
shows "star a * x = x * star b" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
298 |
proof (rule order_antisym) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
299 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
300 |
show "star a * x \<le> x * star b" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
301 |
proof (rule star3', rule order_trans) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
302 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
303 |
from a have "a * x \<le> x * b" by simp |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
304 |
hence "a * x * star b \<le> x * b * star b" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
305 |
by (rule mult_mono) auto |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
306 |
thus "x + a * (x * star b) \<le> x + x * b * star b" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
307 |
using add_mono by (auto simp: mult_assoc) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
308 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
309 |
show "\<dots> \<le> x * star b" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
310 |
proof - |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
311 |
have "x * (1 + b * star b) \<le> x * star b" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
312 |
by (rule mult_mono[OF _ star1]) auto |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
313 |
thus ?thesis |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
314 |
by (simp add:right_distrib mult_assoc) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
315 |
qed |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
316 |
qed |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
317 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
318 |
show "x * star b \<le> star a * x" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
319 |
proof (rule star4', rule order_trans) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
320 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
321 |
from a have b: "x * b \<le> a * x" by simp |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
322 |
have "star a * x * b \<le> star a * a * x" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
323 |
unfolding mult_assoc |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
324 |
by (rule mult_mono[OF _ b]) auto |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
325 |
thus "x + star a * x * b \<le> x + star a * a * x" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
326 |
using add_mono by auto |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
327 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
328 |
show "\<dots> \<le> star a * x" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
329 |
proof - |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
330 |
have "(1 + star a * a) * x \<le> star a * x" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
331 |
by (rule mult_mono[OF star2]) auto |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
332 |
thus ?thesis |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
333 |
by (simp add:left_distrib mult_assoc) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
334 |
qed |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
335 |
qed |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
336 |
qed |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
337 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
338 |
lemma star_assoc: |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
339 |
fixes c d :: "'a :: kleene" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
340 |
shows "star (c * d) * c = c * star (d * c)" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
341 |
by (auto simp:mult_assoc star_commute) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
342 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
343 |
lemma star_dist: |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
344 |
fixes a b :: "'a :: kleene" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
345 |
shows "star (a + b) = star a * star (b * star a)" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
346 |
oops |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
347 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
348 |
lemma star_one: |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
349 |
fixes a p p' :: "'a :: kleene" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
350 |
assumes "p * p' = 1" and "p' * p = 1" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
351 |
shows "p' * star a * p = star (p' * a * p)" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
352 |
proof - |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
353 |
from assms |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
354 |
have "p' * star a * p = p' * star (p * p' * a) * p" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
355 |
by simp |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
356 |
also have "\<dots> = p' * p * star (p' * a * p)" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
357 |
by (simp add: mult_assoc star_assoc) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
358 |
also have "\<dots> = star (p' * a * p)" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
359 |
by (simp add: assms) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
360 |
finally show ?thesis . |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
361 |
qed |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
362 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
363 |
lemma star_mono: |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
364 |
fixes x y :: "'a :: kleene" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
365 |
assumes "x \<le> y" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
366 |
shows "star x \<le> star y" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
367 |
oops |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
368 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
369 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
370 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
371 |
(* Own lemmas *) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
372 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
373 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
374 |
lemma x_less_star[simp]: |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
375 |
fixes x :: "'a :: kleene" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
376 |
shows "x \<le> x * star a" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
377 |
proof - |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
378 |
have "x \<le> x * (1 + a * star a)" by (simp add:right_distrib) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
379 |
also have "\<dots> = x * star a" by (simp only: star_unfold_left) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
380 |
finally show ?thesis . |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
381 |
qed |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
382 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
383 |
subsection {* Transitive Closure *} |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
384 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
385 |
definition |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
386 |
"tcl (x::'a::kleene) = star x * x" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
387 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
388 |
lemma tcl_zero: |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
389 |
"tcl (0::'a::kleene) = 0" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
390 |
unfolding tcl_def by simp |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
391 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
392 |
lemma tcl_unfold_right: "tcl a = a + tcl a * a" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
393 |
proof - |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
394 |
from star_unfold_right[of a] |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
395 |
have "a * (1 + star a * a) = a * star a" by simp |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
396 |
from this[simplified right_distrib, simplified] |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
397 |
show ?thesis |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
398 |
by (simp add:tcl_def star_commute mult_ac) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
399 |
qed |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
400 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
401 |
lemma less_tcl: "a \<le> tcl a" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
402 |
proof - |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
403 |
have "a \<le> a + tcl a * a" by simp |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
404 |
also have "\<dots> = tcl a" by (rule tcl_unfold_right[symmetric]) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
405 |
finally show ?thesis . |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
406 |
qed |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
407 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
408 |
subsection {* Naive Algorithm to generate the transitive closure *} |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
409 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
410 |
function (default "\<lambda>x. 0", tailrec, domintros) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
411 |
mk_tcl :: "('a::{plus,times,ord,zero}) \<Rightarrow> 'a \<Rightarrow> 'a" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
412 |
where |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
413 |
"mk_tcl A X = (if X * A \<le> X then X else mk_tcl A (X + X * A))" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
414 |
by pat_completeness simp |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
415 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
416 |
declare mk_tcl.simps[simp del] (* loops *) |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
417 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
418 |
lemma mk_tcl_code[code]: |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
419 |
"mk_tcl A X = |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
420 |
(let XA = X * A |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
421 |
in if XA \<le> X then X else mk_tcl A (X + XA))" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
422 |
unfolding mk_tcl.simps[of A X] Let_def .. |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
423 |
|
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
424 |
lemma mk_tcl_lemma1: |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
425 |
fixes X :: "'a :: kleene" |
5eaf3e8b50a4
moved stuff about size change termination to its own session
krauss
parents:
diff
changeset
|
426 |
shows "(X + X * A) * star A = X * star A" |
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427 |
proof - |
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428 |
have "A * star A \<le> 1 + A * star A" by simp |
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429 |
also have "\<dots> = star A" by (simp add:star_unfold_left) |
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430 |
finally have "star A + A * star A = star A" by simp |
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431 |
hence "X * (star A + A * star A) = X * star A" by simp |
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432 |
thus ?thesis by (simp add:left_distrib right_distrib mult_ac) |
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433 |
qed |
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434 |
|
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435 |
lemma mk_tcl_lemma2: |
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436 |
fixes X :: "'a :: kleene" |
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437 |
shows "X * A \<le> X \<Longrightarrow> X * star A = X" |
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438 |
by (rule order_antisym) (auto simp:star4) |
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439 |
|
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440 |
|
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441 |
|
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442 |
|
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443 |
lemma mk_tcl_correctness: |
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444 |
fixes A X :: "'a :: {kleene}" |
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445 |
assumes "mk_tcl_dom (A, X)" |
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446 |
shows "mk_tcl A X = X * star A" |
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447 |
using assms |
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448 |
by induct (auto simp:mk_tcl_lemma1 mk_tcl_lemma2) |
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449 |
|
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450 |
lemma graph_implies_dom: "mk_tcl_graph x y \<Longrightarrow> mk_tcl_dom x" |
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451 |
by (rule mk_tcl_graph.induct) (auto intro:accp.accI elim:mk_tcl_rel.cases) |
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452 |
|
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453 |
lemma mk_tcl_default: "\<not> mk_tcl_dom (a,x) \<Longrightarrow> mk_tcl a x = 0" |
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454 |
unfolding mk_tcl_def |
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455 |
by (rule fundef_default_value[OF mk_tcl_sum_def graph_implies_dom]) |
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456 |
|
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457 |
|
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458 |
text {* We can replace the dom-Condition of the correctness theorem |
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459 |
with something executable *} |
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460 |
|
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461 |
lemma mk_tcl_correctness2: |
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462 |
fixes A X :: "'a :: {kleene}" |
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463 |
assumes "mk_tcl A A \<noteq> 0" |
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464 |
shows "mk_tcl A A = tcl A" |
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465 |
using assms mk_tcl_default mk_tcl_correctness |
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466 |
unfolding tcl_def |
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467 |
by (auto simp:star_commute) |
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468 |
|
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469 |
end |