src/HOL/Library/Kleene_Algebras.thy
author chaieb
Mon Jun 11 11:06:04 2007 +0200 (2007-06-11)
changeset 23315 df3a7e9ebadb
parent 22665 cf152ff55d16
child 23394 474ff28210c0
permissions -rw-r--r--
tuned Proof
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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 ""
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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 \<^loc>+ 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 \<sqsubseteq> b \<longleftrightarrow> a \<^loc>+ b = b"
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  assumes strict_order_def: "a \<sqsubset> b \<longleftrightarrow> a \<sqsubseteq> b \<and> a \<noteq> b"
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lemma ord_simp1[simp]: "(x::'a::order_by_add) \<le> y \<Longrightarrow> x + y = y"
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  unfolding order_def .
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lemma ord_simp2[simp]: "(x::'a::order_by_add) \<le> y \<Longrightarrow> y + x = y"
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  unfolding order_def add_commute .
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lemma ord_intro: "(x::'a::order_by_add) + y = y \<Longrightarrow> x \<le> y"
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  unfolding order_def .
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instance order_by_add \<subseteq> order
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proof
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  fix x y z :: 'a
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  show "x \<le> x" unfolding order_def by simp
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  show "\<lbrakk>x \<le> y; y \<le> z\<rbrakk> \<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
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  show "\<lbrakk>x \<le> y; y \<le> x\<rbrakk> \<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> x \<noteq> y" by (fact strict_order_def)
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qed
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class pre_kleene = semiring_1 + order_by_add
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instance pre_kleene \<subseteq> pordered_semiring
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proof
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  fix x y z :: 'a
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  assume "x \<le> y"
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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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  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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  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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class kleene = pre_kleene + star +
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  assumes star1: "\<^loc>1 \<^loc>+ a \<^loc>* star a \<sqsubseteq> star a"
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  and star2: "\<^loc>1 \<^loc>+ star a \<^loc>* a \<sqsubseteq> star a"
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  and star3: "a \<^loc>* x \<sqsubseteq> x \<Longrightarrow> star a \<^loc>* x \<sqsubseteq> x"
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  and star4: "x \<^loc>* a \<sqsubseteq> x \<Longrightarrow> x \<^loc>* star a \<sqsubseteq> x"
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class kleene_by_complete_lattice =
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  pre_kleene + complete_lattice + recpower + star +
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  assumes star_cont: "a \<^loc>* star b \<^loc>* c = (SUP n. a \<^loc>* b ^ n \<^loc>* c)"
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lemma plus_leI: 
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  fixes x :: "'a :: order_by_add"
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  shows "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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lemma le_SUPI':
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  fixes l :: "'a :: complete_lattice"
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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 prems
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  by (rule order_trans) (rule le_SUPI [OF UNIV_I])
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lemma zero_minimum[simp]: "(0::'a::pre_kleene) \<le> x"
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  unfolding order_def by simp
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instance kleene_by_complete_lattice \<subseteq> kleene
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proof
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  fix a x :: 'a
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  have [simp]: "1 \<le> star a"
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    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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  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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    by (intro SUP_leI le_SUPI UNIV_I)
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  show "1 + star a * a \<le> star a" 
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    apply (rule plus_leI, simp)
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    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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  show "a * x \<le> x \<Longrightarrow> star a * x \<le> x"
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  proof -
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    assume a: "a * x \<le> x"
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    {
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      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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      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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        thus ?case
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          by (simp add:power_Suc mult_assoc)
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      qed
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    }
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    note a = this
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    {
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      fix n have "a ^ n * x \<le> x"
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      proof (induct n)
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        case 0 show ?case by simp
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      next
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        case (Suc n) with a[of n]
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        show ?case by simp
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      qed
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    }
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    note b = this
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    show "star a * x \<le> x"
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      unfolding star_cont[of 1 a x, simplified]
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      by (rule SUP_leI) (rule b)
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  qed
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  show "x * a \<le> x \<Longrightarrow> x * star a \<le> x" (* symmetric *)
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  proof -
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    assume a: "x * a \<le> x"
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    {
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      fix n
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      have "x * a ^ (Suc n) \<le> x * a ^ n"
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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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      next
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        case (Suc n)
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        hence "(x * a ^ Suc n) * a  \<le> (x * a ^ n) * a"
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          by (auto intro: mult_mono)
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        thus ?case
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          by (simp add:power_Suc power_commutes mult_assoc)
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      qed
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    }
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    note a = this
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    {
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      fix n have "x * a ^ n \<le> x"
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      proof (induct n)
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        case 0 show ?case by simp
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      next
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        case (Suc n) with a[of n]
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        show ?case by simp
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      qed
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    }
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    note b = this
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    show "x * star a \<le> x"
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      unfolding star_cont[of x a 1, simplified]
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      by (rule SUP_leI) (rule b)
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  qed
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qed
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lemma less_add[simp]:  
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  fixes a b :: "'a :: order_by_add"
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  shows "a \<le> a + b"
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  and "b \<le> a + b"
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  unfolding order_def 
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  by (auto simp:add_ac)
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lemma add_est1:
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  fixes a b c :: "'a :: order_by_add"
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  assumes a: "a + b \<le> c"
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  shows "a \<le> c"
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  using less_add(1) a
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  by (rule order_trans)
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lemma add_est2:
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  fixes a b c :: "'a :: order_by_add"
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  assumes a: "a + b \<le> c"
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  shows "b \<le> c"
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  using less_add(2) a
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  by (rule order_trans)
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lemma star3':
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  fixes a b x :: "'a :: kleene"
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  assumes a: "b + a * x \<le> x"
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  shows "star a * b \<le> x"
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proof (rule order_trans)
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  from a have "b \<le> x" by (rule add_est1)
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  show "star a * b \<le> star a * x"
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    by (rule mult_mono) (auto simp:`b \<le> x`)
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  from a have "a * x \<le> x" by (rule add_est2)
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  with star3 show "star a * x \<le> x" .
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qed
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lemma star4':
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  fixes a b x :: "'a :: kleene"
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  assumes a: "b + x * a \<le> x"
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  shows "b * star a \<le> x"
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proof (rule order_trans)
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  from a have "b \<le> x" by (rule add_est1)
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  show "b * star a \<le> x * star a"
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    by (rule mult_mono) (auto simp:`b \<le> x`)
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  from a have "x * a \<le> x" by (rule add_est2)
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  with star4 show "x * star a \<le> x" .
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qed
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lemma star_mono:
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  fixes x y :: "'a :: kleene"
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  assumes "x \<le> y"
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  shows "star x \<le> star y"
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  oops
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lemma star_idemp:
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  fixes x :: "'a :: kleene"
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  shows "star (star x) = star x"
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  oops
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lemma zero_star[simp]:
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  shows "star (0::'a::kleene) = 1"
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  oops
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lemma star_unfold_left:
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  fixes a :: "'a :: kleene"
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  shows "1 + a * star a = star a"
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proof (rule order_antisym, rule star1)
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  have "1 + a * (1 + a * star a) \<le> 1 + a * star a"
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    apply (rule add_mono, rule)
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    apply (rule mult_mono, auto)
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    apply (rule star1)
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    done
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  with star3' have "star a * 1 \<le> 1 + a * star a" .
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  thus "star a \<le> 1 + a * star a" by simp
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qed
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lemma star_unfold_right:  
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  fixes a :: "'a :: kleene"
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  shows "1 + star a * a = star a"
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proof (rule order_antisym, rule star2)
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  have "1 + (1 + star a * a) * a \<le> 1 + star a * a"
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    apply (rule add_mono, rule)
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    apply (rule mult_mono, auto)
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    apply (rule star2)
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    done
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  with star4' have "1 * star a \<le> 1 + star a * a" .
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  thus "star a \<le> 1 + star a * a" by simp
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qed
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lemma star_commute:
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  fixes a b x :: "'a :: kleene"
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  assumes a: "a * x = x * b"
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  shows "star a * x = x * star b"
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proof (rule order_antisym)
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  show "star a * x \<le> x * star b"
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  proof (rule star3', rule order_trans)
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    from a have "a * x \<le> x * b" by simp
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    hence "a * x * star b \<le> x * b * star b"
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      by (rule mult_mono) auto
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    thus "x + a * (x * star b) \<le> x + x * b * star b"
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      using add_mono by (auto simp: mult_assoc)
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    show "\<dots> \<le> x * star b"
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    proof -
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      have "x * (1 + b * star b) \<le> x * star b"
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        by (rule mult_mono[OF _ star1]) auto
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      thus ?thesis
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        by (simp add:right_distrib mult_assoc)
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    qed
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  qed
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  show "x * star b \<le> star a * x"
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  proof (rule star4', rule order_trans)
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    from a have b: "x * b \<le> a * x" by simp
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    have "star a * x * b \<le> star a * a * x"
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      unfolding mult_assoc
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      by (rule mult_mono[OF _ b]) auto
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    thus "x + star a * x * b \<le> x + star a * a * x"
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      using add_mono by auto
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    show "\<dots> \<le> star a * x"
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    proof -
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      have "(1 + star a * a) * x \<le> star a * x"
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        by (rule mult_mono[OF star2]) auto
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      thus ?thesis
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        by (simp add:left_distrib mult_assoc)
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    qed
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  qed
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qed      
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lemma star_assoc:
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  fixes c d :: "'a :: kleene"
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  shows "star (c * d) * c = c * star (d * c)"
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  oops
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lemma star_dist:
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  fixes a b :: "'a :: kleene"
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  shows "star (a + b) = star a * star (b * star a)"
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  oops
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lemma star_one:
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  fixes a p p' :: "'a :: kleene"
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  assumes "p * p' = 1" and "p' * p = 1"
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  shows "p' * star a * p = star (p' * a * p)"
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  oops
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lemma star_zero: 
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  "star (0::'a::kleene) = 1"
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  by (rule star_unfold_left[of 0, simplified])
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(* Own lemmas *)
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lemma x_less_star[simp]: 
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  fixes x :: "'a :: kleene"
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  shows "x \<le> x * star a"
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proof -
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  have "x \<le> x * (1 + a * star a)" by (simp add:right_distrib)
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  also have "\<dots> = x * star a" by (simp only: star_unfold_left)
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  finally show ?thesis .
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qed
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subsection {* Transitive Closure *}
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definition 
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  "tcl (x::'a::kleene) = star x * x"
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lemma tcl_zero: 
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  "tcl (0::'a::kleene) = 0"
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  unfolding tcl_def by simp
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subsection {* Naive Algorithm to generate the transitive closure *}
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function (default "\<lambda>x. 0", tailrec)
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  mk_tcl :: "('a::{plus,times,ord,zero}) \<Rightarrow> 'a \<Rightarrow> 'a"
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where
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  "mk_tcl A X = (if X * A \<le> X then X else mk_tcl A (X + X * A))"
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  by pat_completeness simp
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declare mk_tcl.simps[simp del] (* loops *)
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lemma mk_tcl_code[code]:
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  "mk_tcl A X = 
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  (let XA = X * A 
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  in if XA \<le> X then X else mk_tcl A (X + XA))"
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  unfolding mk_tcl.simps[of A X] Let_def ..
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lemma mk_tcl_lemma1:
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  fixes X :: "'a :: kleene"
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  shows "(X + X * A) * star A = X * star A"
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proof -
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  have "A * star A \<le> 1 + A * star A" by simp
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  also have "\<dots> = star A" by (simp add:star_unfold_left)
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  finally have "star A + A * star A = star A" by simp
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  hence "X * (star A + A * star A) = X * star A" by simp
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  thus ?thesis by (simp add:left_distrib right_distrib mult_ac)
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qed
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lemma mk_tcl_lemma2:
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  fixes X :: "'a :: kleene"
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  shows "X * A \<le> X \<Longrightarrow> X * star A = X"
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  by (rule order_antisym) (auto simp:star4)
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lemma mk_tcl_correctness:
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  fixes A X :: "'a :: {kleene}"
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  assumes "mk_tcl_dom (A, X)"
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  shows "mk_tcl A X = X * star A"
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  using prems
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  by induct (auto simp:mk_tcl_lemma1 mk_tcl_lemma2)
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lemma graph_implies_dom: "mk_tcl_graph x y \<Longrightarrow> mk_tcl_dom x"
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  by (rule mk_tcl_graph.induct) (auto intro:accI elim:mk_tcl_rel.cases)
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lemma mk_tcl_default: "\<not> mk_tcl_dom (a,x) \<Longrightarrow> mk_tcl a x = 0"
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  unfolding mk_tcl_def
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  by (rule fundef_default_value[OF mk_tcl_sum_def graph_implies_dom])
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text {* We can replace the dom-Condition of the correctness theorem 
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  with something executable *}
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lemma mk_tcl_correctness2:
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  fixes A X :: "'a :: {kleene}"
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  assumes "mk_tcl A A \<noteq> 0"
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  shows "mk_tcl A A = tcl A"
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  using prems mk_tcl_default mk_tcl_correctness
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  unfolding tcl_def 
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  by (auto simp:star_commute)
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   453
end