--- a/src/HOL/Library/Kleene_Algebra.thy Fri May 14 14:14:22 2010 +0200
+++ b/src/HOL/Library/Kleene_Algebra.thy Sun May 23 10:37:43 2010 +0100
@@ -1,106 +1,164 @@
(* Title: HOL/Library/Kleene_Algebra.thy
Author: Alexander Krauss, TU Muenchen
+ Author: Tjark Weber, University of Cambridge
*)
-header "Kleene Algebra"
+header {* Kleene Algebra *}
theory Kleene_Algebra
imports Main
begin
-text {* WARNING: This is work in progress. Expect changes in the future *}
+text {* WARNING: This is work in progress. Expect changes in the future. *}
-text {* A type class of Kleene algebras *}
+text {* Various lemmas correspond to entries in a database of theorems
+ about Kleene algebras and related structures maintained by Peter
+ H\"ofner: see
+ \url{http://www.informatik.uni-augsburg.de/~hoefnepe/kleene_db/lemmas/index.html}. *}
+
+subsection {* Preliminaries *}
-class star =
- fixes star :: "'a \<Rightarrow> 'a"
+text {* A class where addition is idempotent. *}
-class idem_add = ab_semigroup_add +
+class idem_add = plus +
assumes add_idem [simp]: "x + x = x"
+
+text {* A class of idempotent abelian semigroups (written additively). *}
+
+class idem_ab_semigroup_add = ab_semigroup_add + idem_add
begin
-lemma add_idem2[simp]: "(x::'a) + (x + y) = x + y"
+lemma add_idem2 [simp]: "x + (x + y) = x + y"
unfolding add_assoc[symmetric] by simp
+lemma add_idem3 [simp]: "x + (y + x) = x + y"
+by (simp add: add_commute)
+
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"
+text {* A class where order is defined in terms of addition. *}
+
+class order_by_add = plus + ord +
+ assumes order_def: "x \<le> y \<longleftrightarrow> x + y = y"
+ assumes strict_order_def: "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
begin
-lemma ord_simp1[simp]: "x \<le> y \<Longrightarrow> x + y = y"
+lemma ord_simp [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 .
+end
+
+text {* A class of idempotent abelian semigroups (written additively)
+ where order is defined in terms of addition. *}
+
+class ordered_idem_ab_semigroup_add = idem_ab_semigroup_add + order_by_add
+begin
+
+lemma ord_simp2 [simp]: "x \<le> y \<Longrightarrow> y + x = y"
+ unfolding order_def add_commute .
+
subclass order proof
fix x y z :: 'a
- show "x \<le> x" unfolding order_def by simp
+ 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)
+ unfolding order_def by (metis add_assoc)
+ 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:
+subclass ordered_ab_semigroup_add proof
+ fix a b c :: 'a
+ assume "a \<le> b" show "c + a \<le> c + b"
+ proof (rule ord_intro)
+ have "c + a + (c + b) = a + b + c" by (simp add: add_ac)
+ also have "\<dots> = c + b" by (simp add: `a \<le> b` add_ac)
+ finally show "c + a + (c + b) = c + b" .
+ qed
+qed
+
+lemma plus_leI [simp]:
"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 less_add [simp]: "x \<le> x + y" "y \<le> x + y"
+unfolding order_def by (auto simp: add_ac)
-lemma add_est1: "a + b \<le> c \<Longrightarrow> a \<le> c"
+lemma add_est1 [elim]: "x + y \<le> z \<Longrightarrow> x \<le> z"
using less_add(1) by (rule order_trans)
-lemma add_est2: "a + b \<le> c \<Longrightarrow> b \<le> c"
+lemma add_est2 [elim]: "x + y \<le> z \<Longrightarrow> y \<le> z"
using less_add(2) by (rule order_trans)
+lemma add_supremum: "(x + y \<le> z) = (x \<le> z \<and> y \<le> z)"
+by auto
+
end
-class pre_kleene = semiring_1 + order_by_add
+text {* A class of commutative monoids (written additively) where
+ order is defined in terms of addition. *}
+
+class ordered_comm_monoid_add = comm_monoid_add + order_by_add
+begin
+
+lemma zero_minimum [simp]: "0 \<le> x"
+unfolding order_def by simp
+
+end
+
+text {* A class of idempotent commutative monoids (written additively)
+ where order is defined in terms of addition. *}
+
+class ordered_idem_comm_monoid_add = ordered_comm_monoid_add + idem_add
begin
+subclass ordered_idem_ab_semigroup_add ..
+
+lemma sum_is_zero: "(x + y = 0) = (x = 0 \<and> y = 0)"
+by (simp add: add_supremum eq_iff)
+
+end
+
+section {* Kleene Algebras *}
+
+text {* Class @{text pre_kleene} provides all operations of Kleene
+ algebras except for the Kleene star. *}
+
+class pre_kleene = semiring_1 + idem_add + order_by_add
+begin
+
+subclass ordered_idem_comm_monoid_add ..
+
subclass ordered_semiring proof
- fix x y z :: 'a
+ fix a b c :: 'a
+ assume "a \<le> b"
- assume "x \<le> y"
-
- show "z + x \<le> z + y"
+ show "c * a \<le> c * b"
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" .
+ from `a \<le> b` have "c * (a + b) = c * b" by simp
+ thus "c * a + c * b = c * b" by (simp add: right_distrib)
qed
- show "z * x \<le> z * y"
+ show "a * c \<le> b * c"
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)
+ from `a \<le> b` have "(a + b) * c = b * c" by simp
+ thus "a * c + b * c = b * c" by (simp add: left_distrib)
qed
qed
-lemma zero_minimum [simp]: "0 \<le> x"
- unfolding order_def by simp
+end
+
+text {* A class that provides a star operator. *}
-end
+class star =
+ fixes star :: "'a \<Rightarrow> 'a"
+
+text {* Finally, a class of Kleene algebras. *}
class kleene = pre_kleene + star +
assumes star1: "1 + a * star a \<le> star a"
@@ -109,38 +167,20 @@
and star4: "x * a \<le> x \<Longrightarrow> x * star a \<le> x"
begin
-lemma star3':
+lemma star3' [simp]:
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`)
+by (metis assms less_add mult_left_mono order_trans star3 zero_minimum)
- from a have "a * x \<le> x" by (rule add_est2)
- with star3 show "star a * x \<le> x" .
-qed
-
-lemma star4':
+lemma star4' [simp]:
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`)
+by (metis assms less_add mult_right_mono order_trans star4 zero_minimum)
- 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"
+lemma star_unfold_left: "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
+ by (metis add_left_mono mult_left_mono star1 zero_minimum)
with star3' have "star a * 1 \<le> 1 + a * star a" .
thus "star a \<le> 1 + a * star a" by simp
qed
@@ -148,76 +188,66 @@
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
+ by (metis add_left_mono mult_right_mono star2 zero_minimum)
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"
+lemma star_zero [simp]: "star 0 = 1"
by (fact star_unfold_left[of 0, simplified, symmetric])
-lemma star_one[simp]: "star 1 = 1"
+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"
+lemma one_less_star [simp]: "1 \<le> star x"
by (metis less_add(1) star_unfold_left)
-lemma ka1: "x * star x \<le> star x"
+lemma ka1 [simp]: "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"
+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"
+lemma less_star [simp]: "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:
+lemma star_simulation_leq_1:
+ assumes a: "a * x \<le> x * b"
+ shows "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_right_mono) simp
+ thus "x + a * (x * star b) \<le> x + x * b * star b"
+ using add_left_mono by (auto simp: mult_assoc)
+ show "\<dots> \<le> x * star b"
+ by (metis add_supremum ka1 mult.right_neutral mult_assoc mult_left_mono one_less_star zero_minimum)
+qed
+
+lemma star_simulation_leq_2:
+ assumes a: "x * a \<le> b * x"
+ shows "x * star a \<le> star b * x"
+proof (rule star4', rule order_trans)
+ have "star b * x * a \<le> star b * b * x"
+ by (metis assms mult_assoc mult_mono order_refl zero_minimum)
+ thus "x + star b * x * a \<le> x + star b * b * x"
+ using add_mono by auto
+ show "\<dots> \<le> star b * x"
+ by (metis add_supremum left_distrib less_add mult.left_neutral mult_assoc mult_right_mono star_unfold_left star_unfold_right zero_minimum)
+qed
+
+lemma star_simulation [simp]:
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
+by (metis antisym assms order_refl star_simulation_leq_1 star_simulation_leq_2)
-lemma star_slide2[simp]: "star x * x = x * star x"
+lemma star_slide2 [simp]: "star x * x = x * star x"
by (metis star_simulation)
-lemma star_idemp[simp]: "star (star x) = star x"
+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)"
+lemma star_slide [simp]: "star (x * y) * x = x * star (y * x)"
by (auto simp: mult_assoc star_simulation)
lemma star_one':
@@ -234,26 +264,22 @@
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 x_less_star [simp]: "x \<le> x * star a"
+by (metis mult.right_neutral mult_left_mono one_less_star zero_minimum)
-lemma star_mono: "x \<le> y \<Longrightarrow> star x \<le> star y"
+lemma star_mono [simp]: "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)
+by (metis add_commute ord_simp 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 (x + 1) = star x"
+by (metis add_commute eq_iff left_distrib less_add 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"
+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)"
@@ -269,21 +295,17 @@
\<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)"
+lemma star_decomp: "star (x + y) = star x * star (y * star x)"
proof (rule antisym)
- have "1 + (a + b) * star a * star (b * star a) \<le>
- 1 + a * star a * star (b * star a) + b * star a * star (b * star a)"
+ have "1 + (x + y) * star x * star (y * star x) \<le>
+ 1 + x * star x * star (y * star x) + y * star x * star (y * star x)"
by (metis add_commute add_left_commute eq_iff left_distrib mult_assoc)
- also have "\<dots> \<le> star a * star (b * star a)"
+ also have "\<dots> \<le> star x * star (y * star x)"
by (metis add_commute add_est1 add_left_commute ka18 plus_leI star_unfold_left x_less_star)
- finally show "star (a + b) \<le> star a * star (b * star a)"
+ finally show "star (x + y) \<le> star x * star (y * star x)"
by (metis mult_1_right mult_assoc star3')
next
- show "star a * star (b * star a) \<le> star (a + b)"
+ show "star x * star (y * star x) \<le> star (x + y)"
by (metis add_assoc add_est1 add_est2 add_left_commute less_star mult_mono'
star_absorb_one star_absorb_one' star_idemp star_mono star_mult_idem zero_minimum)
qed
@@ -298,14 +320,40 @@
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
+-- {* Takes several minutes on my computer. *}
+by (metis mult_assoc mult_right_mono order_trans star_idemp star_mult_idem star_simulation_leq_2 star_slide x_less_star zero_minimum)
+
+lemma church_rosser:
+ "star y * star x \<le> star x * star y \<Longrightarrow> star (x + y) \<le> star x * star y"
+by (metis add_commute ka24 ka25 order_trans)
lemma kleene_bubblesort: "y * x \<le> x * y \<Longrightarrow> star (x + y) \<le> star x * star y"
-oops
+by (metis church_rosser star_simulation_leq_1 star_simulation_leq_2)
+
+lemma ka27: "star (x + star y) = star (x + y)"
+by (metis add_commute star_decomp star_idemp)
+
+lemma ka28: "star (star x + star y) = star (x + y)"
+by (metis add_commute ka27)
+
+lemma ka29: "(y * (1 + x) \<le> (1 + x) * star y) = (y * x \<le> (1 + x) * star y)"
+by (metis add_supremum left_distrib less_add(1) less_star mult.left_neutral mult.right_neutral order_trans right_distrib)
+
+lemma ka30: "star x * star y \<le> star (x + y)"
+by (metis mult_left_mono star_decomp star_mono x_less_star zero_minimum)
+
+lemma simple_simulation: "x * y = 0 \<Longrightarrow> star x * y = y"
+by (metis mult.right_neutral mult_zero_right star_simulation star_zero)
+
+lemma ka32: "star (x * y) = 1 + x * star (y * x) * y"
+by (metis mult_assoc star_slide star_unfold_left)
+
+lemma ka33: "x * y + 1 \<le> y \<Longrightarrow> star x \<le> y"
+by (metis add_commute mult.right_neutral star3')
end
-subsection {* Complete lattices are Kleene algebras *}
+subsection {* Complete Lattices are Kleene Algebras *}
lemma (in complete_lattice) le_SUPI':
assumes "l \<le> M i"
@@ -325,7 +373,7 @@
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"
+ 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])
@@ -411,38 +459,26 @@
end
-
subsection {* Transitive Closure *}
context kleene
begin
-definition
- tcl_def: "tcl x = star x * x"
+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
+by (metis star_slide2 star_unfold2 tcl_def)
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
+by (metis star_slide2 tcl_def x_less_star)
end
-
-subsection {* Naive Algorithm to generate the transitive closure *}
+subsection {* A 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"
@@ -461,19 +497,11 @@
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_lemma1: "(X + X * A) * star A = X * star A"
+by (metis ka1 left_distrib mult_assoc mult_left_mono ord_simp2 zero_minimum)
-lemma mk_tcl_lemma2:
- shows "X * A \<le> X \<Longrightarrow> X * star A = X"
- by (rule antisym) (auto simp:star4)
+lemma mk_tcl_lemma2: "X * A \<le> X \<Longrightarrow> X * star A = X"
+by (rule antisym) (auto simp: star4)
end
@@ -484,7 +512,6 @@
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)
@@ -492,9 +519,8 @@
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 *}
+ with something executable: *}
lemma mk_tcl_correctness2:
fixes A X :: "'a :: {kleene}"