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+++ b/src/HOL/Library/Kleene_Algebra.thy Fri Jul 10 09:24:50 2009 +0200
@@ -0,0 +1,495 @@
+(* Title: HOL/Library/Kleene_Algebra.thy
+ Author: Alexander Krauss, TU Muenchen
+*)
+
+header "Kleene Algebra"
+
+theory Kleene_Algebra
+imports Main
+begin
+
+text {* WARNING: This is work in progress. Expect changes in the future *}
+
+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"
+begin
+
+lemma add_idem2[simp]: "(x::'a) + (x + y) = x + y"
+unfolding add_assoc[symmetric] by simp
+
+end
+
+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)
+
+lemma less_add[simp]: "a \<le> a + b" "b \<le> a + b"
+unfolding order_def by (auto simp:add_ac)
+
+lemma add_est1: "a + b \<le> c \<Longrightarrow> a \<le> c"
+using less_add(1) by (rule order_trans)
+
+lemma add_est2: "a + b \<le> c \<Longrightarrow> b \<le> c"
+using less_add(2) by (rule order_trans)
+
+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"
+begin
+
+lemma star3':
+ 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':
+ 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_unfold_left:
+ shows "1 + a * star a = star a"
+proof (rule 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: "1 + star a * a = star a"
+proof (rule 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]: "star 0 = 1"
+by (fact star_unfold_left[of 0, simplified, symmetric])
+
+lemma star_one[simp]: "star 1 = 1"
+by (metis add_idem2 eq_iff mult_1_right ord_simp2 star3 star_unfold_left)
+
+lemma one_less_star: "1 \<le> star x"
+by (metis less_add(1) star_unfold_left)
+
+lemma ka1: "x * star x \<le> star x"
+by (metis less_add(2) star_unfold_left)
+
+lemma star_mult_idem[simp]: "star x * star x = star x"
+by (metis add_commute add_est1 eq_iff mult_1_right right_distrib star3 star_unfold_left)
+
+lemma less_star: "x \<le> star x"
+by (metis less_add(2) mult_1_right mult_left_mono one_less_star order_trans star_unfold_left zero_minimum)
+
+lemma star_simulation:
+ assumes a: "a * x = x * b"
+ shows "star a * x = x * star b"
+proof (rule 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_slide2[simp]: "star x * x = x * star x"
+by (metis star_simulation)
+
+lemma star_idemp[simp]: "star (star x) = star x"
+by (metis add_idem2 eq_iff less_star mult_1_right star3' star_mult_idem star_unfold_left)
+
+lemma star_slide[simp]: "star (x * y) * x = x * star (y * x)"
+by (auto simp: mult_assoc star_simulation)
+
+lemma star_one':
+ assumes "p * p' = 1" "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)
+ also have "\<dots> = star (p' * a * p)"
+ by (simp add: assms)
+ finally show ?thesis .
+qed
+
+lemma x_less_star[simp]: "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
+
+lemma star_mono: "x \<le> y \<Longrightarrow> star x \<le> star y"
+by (metis add_commute eq_iff less_star ord_simp2 order_trans star3 star4' star_idemp star_mult_idem x_less_star)
+
+lemma star_sub: "x \<le> 1 \<Longrightarrow> star x = 1"
+by (metis add_commute ord_simp1 star_idemp star_mono star_mult_idem star_one star_unfold_left)
+
+lemma star_unfold2: "star x * y = y + x * star x * y"
+by (subst star_unfold_right[symmetric]) (simp add: mult_assoc left_distrib)
+
+lemma star_absorb_one[simp]: "star (x + 1) = star x"
+by (metis add_commute eq_iff left_distrib less_add(1) less_add(2) mult_1_left mult_assoc star3 star_mono star_mult_idem star_unfold2 x_less_star)
+
+lemma star_absorb_one'[simp]: "star (1 + x) = star x"
+by (subst add_commute) (fact star_absorb_one)
+
+lemma ka16: "(y * star x) * star (y * star x) \<le> star x * star (y * star x)"
+by (metis ka1 less_add(1) mult_assoc order_trans star_unfold2)
+
+lemma ka16': "(star x * y) * star (star x * y) \<le> star (star x * y) * star x"
+by (metis ka1 mult_assoc order_trans star_slide x_less_star)
+
+lemma ka17: "(x * star x) * star (y * star x) \<le> star x * star (y * star x)"
+by (metis ka1 mult_assoc mult_right_mono zero_minimum)
+
+lemma ka18: "(x * star x) * star (y * star x) + (y * star x) * star (y * star x)
+ \<le> star x * star (y * star x)"
+by (metis ka16 ka17 left_distrib mult_assoc plus_leI)
+
+lemma kleene_church_rosser:
+ "star y * star x \<le> star x * star y \<Longrightarrow> star (x + y) \<le> star x * star y"
+oops
+
+lemma star_decomp: "star (a + b) = star a * star (b * star a)"
+oops
+
+lemma ka22: "y * star x \<le> star x * star y \<Longrightarrow> star y * star x \<le> star x * star y"
+by (metis mult_assoc mult_right_mono plus_leI star3' star_mult_idem x_less_star zero_minimum)
+
+lemma ka23: "star y * star x \<le> star x * star y \<Longrightarrow> y * star x \<le> star x * star y"
+by (metis less_star mult_right_mono order_trans zero_minimum)
+
+lemma ka24: "star (x + y) \<le> star (star x * star y)"
+by (metis add_est1 add_est2 less_add(1) mult_assoc order_def plus_leI star_absorb_one star_mono star_slide2 star_unfold2 star_unfold_left x_less_star)
+
+lemma ka25: "star y * star x \<le> star x * star y \<Longrightarrow> star (star y * star x) \<le> star x * star y"
+oops
+
+lemma kleene_bubblesort: "y * x \<le> x * y \<Longrightarrow> star (x + y) \<le> star x * star y"
+oops
+
+end
+
+subsection {* Complete lattices are Kleene algebras *}
+
+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])
+
+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
+
+subclass 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
+
+end
+
+
+subsection {* Transitive Closure *}
+
+context kleene
+begin
+
+definition
+ tcl_def: "tcl x = star x * x"
+
+lemma tcl_zero: "tcl 0 = 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 mult_assoc)
+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
+
+end
+
+
+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 ..
+
+context kleene
+begin
+
+lemma mk_tcl_lemma1:
+ "(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_assoc)
+qed
+
+lemma mk_tcl_lemma2:
+ shows "X * A \<le> X \<Longrightarrow> X * star A = X"
+ by (rule antisym) (auto simp:star4)
+
+end
+
+lemma mk_tcl_correctness:
+ fixes 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
+
+end