--- a/src/HOL/SizeChange/Kleene_Algebras.thy Fri Jul 10 10:45:30 2009 -0400
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,469 +0,0 @@
-(* Title: HOL/Library/Kleene_Algebras.thy
- ID: $Id$
- Author: Alexander Krauss, TU Muenchen
-*)
-
-header "Kleene Algebras"
-
-theory Kleene_Algebras
-imports Main
-begin
-
-text {* A type class of kleene algebras *}
-
-class star =
- fixes star :: "'a \<Rightarrow> 'a"
-
-class idem_add = ab_semigroup_add +
- assumes add_idem [simp]: "x + x = x"
-
-lemma add_idem2[simp]: "(x::'a::idem_add) + (x + y) = x + y"
- unfolding add_assoc[symmetric]
- by simp
-
-class order_by_add = idem_add + ord +
- assumes order_def: "a \<le> b \<longleftrightarrow> a + b = b"
- assumes strict_order_def: "a < b \<longleftrightarrow> a \<le> b \<and> \<not> b \<le> a"
-begin
-
-lemma ord_simp1[simp]: "x \<le> y \<Longrightarrow> x + y = y"
- unfolding order_def .
-
-lemma ord_simp2[simp]: "x \<le> y \<Longrightarrow> y + x = y"
- unfolding order_def add_commute .
-
-lemma ord_intro: "x + y = y \<Longrightarrow> x \<le> y"
- unfolding order_def .
-
-subclass order proof
- fix x y z :: 'a
- show "x \<le> x" unfolding order_def by simp
- show "x \<le> y \<Longrightarrow> y \<le> z \<Longrightarrow> x \<le> z"
- proof (rule ord_intro)
- assume "x \<le> y" "y \<le> z"
- have "x + z = x + y + z" by (simp add:`y \<le> z` add_assoc)
- also have "\<dots> = y + z" by (simp add:`x \<le> y`)
- also have "\<dots> = z" by (simp add:`y \<le> z`)
- finally show "x + z = z" .
- qed
- show "x \<le> y \<Longrightarrow> y \<le> x \<Longrightarrow> x = y" unfolding order_def
- by (simp add: add_commute)
- show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x" by (fact strict_order_def)
-qed
-
-lemma plus_leI:
- "x \<le> z \<Longrightarrow> y \<le> z \<Longrightarrow> x + y \<le> z"
- unfolding order_def by (simp add: add_assoc)
-
-end
-
-class pre_kleene = semiring_1 + order_by_add
-begin
-
-subclass pordered_semiring proof
- fix x y z :: 'a
-
- assume "x \<le> y"
-
- show "z + x \<le> z + y"
- proof (rule ord_intro)
- have "z + x + (z + y) = x + y + z" by (simp add:add_ac)
- also have "\<dots> = z + y" by (simp add:`x \<le> y` add_ac)
- finally show "z + x + (z + y) = z + y" .
- qed
-
- show "z * x \<le> z * y"
- proof (rule ord_intro)
- from `x \<le> y` have "z * (x + y) = z * y" by simp
- thus "z * x + z * y = z * y" by (simp add:right_distrib)
- qed
-
- show "x * z \<le> y * z"
- proof (rule ord_intro)
- from `x \<le> y` have "(x + y) * z = y * z" by simp
- thus "x * z + y * z = y * z" by (simp add:left_distrib)
- qed
-qed
-
-lemma zero_minimum [simp]: "0 \<le> x"
- unfolding order_def by simp
-
-end
-
-class kleene = pre_kleene + star +
- assumes star1: "1 + a * star a \<le> star a"
- and star2: "1 + star a * a \<le> star a"
- and star3: "a * x \<le> x \<Longrightarrow> star a * x \<le> x"
- and star4: "x * a \<le> x \<Longrightarrow> x * star a \<le> x"
-
-class kleene_by_complete_lattice = pre_kleene
- + complete_lattice + power + star +
- assumes star_cont: "a * star b * c = SUPR UNIV (\<lambda>n. a * b ^ n * c)"
-begin
-
-lemma (in complete_lattice) le_SUPI':
- assumes "l \<le> M i"
- shows "l \<le> (SUP i. M i)"
- using assms by (rule order_trans) (rule le_SUPI [OF UNIV_I])
-
-end
-
-instance kleene_by_complete_lattice < kleene
-proof
-
- fix a x :: 'a
-
- have [simp]: "1 \<le> star a"
- unfolding star_cont[of 1 a 1, simplified]
- by (subst power_0[symmetric]) (rule le_SUPI [OF UNIV_I])
-
- show "1 + a * star a \<le> star a"
- apply (rule plus_leI, simp)
- apply (simp add:star_cont[of a a 1, simplified])
- apply (simp add:star_cont[of 1 a 1, simplified])
- apply (subst power_Suc[symmetric])
- by (intro SUP_leI le_SUPI UNIV_I)
-
- show "1 + star a * a \<le> star a"
- apply (rule plus_leI, simp)
- apply (simp add:star_cont[of 1 a a, simplified])
- apply (simp add:star_cont[of 1 a 1, simplified])
- by (auto intro: SUP_leI le_SUPI simp add: power_Suc[symmetric] power_commutes simp del: power_Suc)
-
- show "a * x \<le> x \<Longrightarrow> star a * x \<le> x"
- proof -
- assume a: "a * x \<le> x"
-
- {
- fix n
- have "a ^ (Suc n) * x \<le> a ^ n * x"
- proof (induct n)
- case 0 thus ?case by (simp add: a)
- next
- case (Suc n)
- hence "a * (a ^ Suc n * x) \<le> a * (a ^ n * x)"
- by (auto intro: mult_mono)
- thus ?case
- by (simp add: mult_assoc)
- qed
- }
- note a = this
-
- {
- fix n have "a ^ n * x \<le> x"
- proof (induct n)
- case 0 show ?case by simp
- next
- case (Suc n) with a[of n]
- show ?case by simp
- qed
- }
- note b = this
-
- show "star a * x \<le> x"
- unfolding star_cont[of 1 a x, simplified]
- by (rule SUP_leI) (rule b)
- qed
-
- show "x * a \<le> x \<Longrightarrow> x * star a \<le> x" (* symmetric *)
- proof -
- assume a: "x * a \<le> x"
-
- {
- fix n
- have "x * a ^ (Suc n) \<le> x * a ^ n"
- proof (induct n)
- case 0 thus ?case by (simp add: a)
- next
- case (Suc n)
- hence "(x * a ^ Suc n) * a \<le> (x * a ^ n) * a"
- by (auto intro: mult_mono)
- thus ?case
- by (simp add: power_commutes mult_assoc)
- qed
- }
- note a = this
-
- {
- fix n have "x * a ^ n \<le> x"
- proof (induct n)
- case 0 show ?case by simp
- next
- case (Suc n) with a[of n]
- show ?case by simp
- qed
- }
- note b = this
-
- show "x * star a \<le> x"
- unfolding star_cont[of x a 1, simplified]
- by (rule SUP_leI) (rule b)
- qed
-qed
-
-lemma less_add[simp]:
- fixes a b :: "'a :: order_by_add"
- shows "a \<le> a + b"
- and "b \<le> a + b"
- unfolding order_def
- by (auto simp:add_ac)
-
-lemma add_est1:
- fixes a b c :: "'a :: order_by_add"
- assumes a: "a + b \<le> c"
- shows "a \<le> c"
- using less_add(1) a
- by (rule order_trans)
-
-lemma add_est2:
- fixes a b c :: "'a :: order_by_add"
- assumes a: "a + b \<le> c"
- shows "b \<le> c"
- using less_add(2) a
- by (rule order_trans)
-
-
-lemma star3':
- fixes a b x :: "'a :: kleene"
- assumes a: "b + a * x \<le> x"
- shows "star a * b \<le> x"
-proof (rule order_trans)
- from a have "b \<le> x" by (rule add_est1)
- show "star a * b \<le> star a * x"
- by (rule mult_mono) (auto simp:`b \<le> x`)
-
- from a have "a * x \<le> x" by (rule add_est2)
- with star3 show "star a * x \<le> x" .
-qed
-
-
-lemma star4':
- fixes a b x :: "'a :: kleene"
- assumes a: "b + x * a \<le> x"
- shows "b * star a \<le> x"
-proof (rule order_trans)
- from a have "b \<le> x" by (rule add_est1)
- show "b * star a \<le> x * star a"
- by (rule mult_mono) (auto simp:`b \<le> x`)
-
- from a have "x * a \<le> x" by (rule add_est2)
- with star4 show "x * star a \<le> x" .
-qed
-
-
-lemma star_idemp:
- fixes x :: "'a :: kleene"
- shows "star (star x) = star x"
- oops
-
-lemma star_unfold_left:
- fixes a :: "'a :: kleene"
- shows "1 + a * star a = star a"
-proof (rule order_antisym, rule star1)
-
- have "1 + a * (1 + a * star a) \<le> 1 + a * star a"
- apply (rule add_mono, rule)
- apply (rule mult_mono, auto)
- apply (rule star1)
- done
-
- with star3' have "star a * 1 \<le> 1 + a * star a" .
- thus "star a \<le> 1 + a * star a" by simp
-qed
-
-
-lemma star_unfold_right:
- fixes a :: "'a :: kleene"
- shows "1 + star a * a = star a"
-proof (rule order_antisym, rule star2)
-
- have "1 + (1 + star a * a) * a \<le> 1 + star a * a"
- apply (rule add_mono, rule)
- apply (rule mult_mono, auto)
- apply (rule star2)
- done
-
- with star4' have "1 * star a \<le> 1 + star a * a" .
- thus "star a \<le> 1 + star a * a" by simp
-qed
-
-lemma star_zero[simp]:
- shows "star (0::'a::kleene) = 1"
- by (rule star_unfold_left[of 0, simplified])
-
-lemma star_commute:
- fixes a b x :: "'a :: kleene"
- assumes a: "a * x = x * b"
- shows "star a * x = x * star b"
-proof (rule order_antisym)
-
- show "star a * x \<le> x * star b"
- proof (rule star3', rule order_trans)
-
- from a have "a * x \<le> x * b" by simp
- hence "a * x * star b \<le> x * b * star b"
- by (rule mult_mono) auto
- thus "x + a * (x * star b) \<le> x + x * b * star b"
- using add_mono by (auto simp: mult_assoc)
-
- show "\<dots> \<le> x * star b"
- proof -
- have "x * (1 + b * star b) \<le> x * star b"
- by (rule mult_mono[OF _ star1]) auto
- thus ?thesis
- by (simp add:right_distrib mult_assoc)
- qed
- qed
-
- show "x * star b \<le> star a * x"
- proof (rule star4', rule order_trans)
-
- from a have b: "x * b \<le> a * x" by simp
- have "star a * x * b \<le> star a * a * x"
- unfolding mult_assoc
- by (rule mult_mono[OF _ b]) auto
- thus "x + star a * x * b \<le> x + star a * a * x"
- using add_mono by auto
-
- show "\<dots> \<le> star a * x"
- proof -
- have "(1 + star a * a) * x \<le> star a * x"
- by (rule mult_mono[OF star2]) auto
- thus ?thesis
- by (simp add:left_distrib mult_assoc)
- qed
- qed
-qed
-
-lemma star_assoc:
- fixes c d :: "'a :: kleene"
- shows "star (c * d) * c = c * star (d * c)"
- by (auto simp:mult_assoc star_commute)
-
-lemma star_dist:
- fixes a b :: "'a :: kleene"
- shows "star (a + b) = star a * star (b * star a)"
- oops
-
-lemma star_one:
- fixes a p p' :: "'a :: kleene"
- assumes "p * p' = 1" and "p' * p = 1"
- shows "p' * star a * p = star (p' * a * p)"
-proof -
- from assms
- have "p' * star a * p = p' * star (p * p' * a) * p"
- by simp
- also have "\<dots> = p' * p * star (p' * a * p)"
- by (simp add: mult_assoc star_assoc)
- also have "\<dots> = star (p' * a * p)"
- by (simp add: assms)
- finally show ?thesis .
-qed
-
-lemma star_mono:
- fixes x y :: "'a :: kleene"
- assumes "x \<le> y"
- shows "star x \<le> star y"
- oops
-
-
-
-(* Own lemmas *)
-
-
-lemma x_less_star[simp]:
- fixes x :: "'a :: kleene"
- shows "x \<le> x * star a"
-proof -
- have "x \<le> x * (1 + a * star a)" by (simp add:right_distrib)
- also have "\<dots> = x * star a" by (simp only: star_unfold_left)
- finally show ?thesis .
-qed
-
-subsection {* Transitive Closure *}
-
-definition
- "tcl (x::'a::kleene) = star x * x"
-
-lemma tcl_zero:
- "tcl (0::'a::kleene) = 0"
- unfolding tcl_def by simp
-
-lemma tcl_unfold_right: "tcl a = a + tcl a * a"
-proof -
- from star_unfold_right[of a]
- have "a * (1 + star a * a) = a * star a" by simp
- from this[simplified right_distrib, simplified]
- show ?thesis
- by (simp add:tcl_def star_commute mult_ac)
-qed
-
-lemma less_tcl: "a \<le> tcl a"
-proof -
- have "a \<le> a + tcl a * a" by simp
- also have "\<dots> = tcl a" by (rule tcl_unfold_right[symmetric])
- finally show ?thesis .
-qed
-
-subsection {* Naive Algorithm to generate the transitive closure *}
-
-function (default "\<lambda>x. 0", tailrec, domintros)
- mk_tcl :: "('a::{plus,times,ord,zero}) \<Rightarrow> 'a \<Rightarrow> 'a"
-where
- "mk_tcl A X = (if X * A \<le> X then X else mk_tcl A (X + X * A))"
- by pat_completeness simp
-
-declare mk_tcl.simps[simp del] (* loops *)
-
-lemma mk_tcl_code[code]:
- "mk_tcl A X =
- (let XA = X * A
- in if XA \<le> X then X else mk_tcl A (X + XA))"
- unfolding mk_tcl.simps[of A X] Let_def ..
-
-lemma mk_tcl_lemma1:
- fixes X :: "'a :: kleene"
- shows "(X + X * A) * star A = X * star A"
-proof -
- have "A * star A \<le> 1 + A * star A" by simp
- also have "\<dots> = star A" by (simp add:star_unfold_left)
- finally have "star A + A * star A = star A" by simp
- hence "X * (star A + A * star A) = X * star A" by simp
- thus ?thesis by (simp add:left_distrib right_distrib mult_ac)
-qed
-
-lemma mk_tcl_lemma2:
- fixes X :: "'a :: kleene"
- shows "X * A \<le> X \<Longrightarrow> X * star A = X"
- by (rule order_antisym) (auto simp:star4)
-
-
-
-
-lemma mk_tcl_correctness:
- fixes A X :: "'a :: {kleene}"
- assumes "mk_tcl_dom (A, X)"
- shows "mk_tcl A X = X * star A"
- using assms
- by induct (auto simp:mk_tcl_lemma1 mk_tcl_lemma2)
-
-lemma graph_implies_dom: "mk_tcl_graph x y \<Longrightarrow> mk_tcl_dom x"
- by (rule mk_tcl_graph.induct) (auto intro:accp.accI elim:mk_tcl_rel.cases)
-
-lemma mk_tcl_default: "\<not> mk_tcl_dom (a,x) \<Longrightarrow> mk_tcl a x = 0"
- unfolding mk_tcl_def
- by (rule fundef_default_value[OF mk_tcl_sumC_def graph_implies_dom])
-
-
-text {* We can replace the dom-Condition of the correctness theorem
- with something executable *}
-
-lemma mk_tcl_correctness2:
- fixes A X :: "'a :: {kleene}"
- assumes "mk_tcl A A \<noteq> 0"
- shows "mk_tcl A A = tcl A"
- using assms mk_tcl_default mk_tcl_correctness
- unfolding tcl_def
- by (auto simp:star_commute)
-
-end