--- a/src/HOL/SizeChange/Correctness.thy Fri Jul 10 00:47:17 2009 +0200
+++ b/src/HOL/SizeChange/Correctness.thy Fri Jul 10 00:49:32 2009 +0200
@@ -250,7 +250,7 @@
have "tcl A = A * star A"
unfolding tcl_def
- by (simp add: star_commute[of A A A, simplified])
+ by (simp add: star_simulation[of A A A, simplified])
with edge
have "has_edge (A * star A) x G y" by simp
@@ -272,7 +272,7 @@
have "has_edge (star A * A) x G y" by (simp add:tcl_def)
then obtain H H' z
where AH': "has_edge A z H' y" and G: "G = H * H'"
- by (auto simp:in_grcomp)
+ by (auto simp:in_grcomp simp del: star_slide2)
from B
obtain m' e' where "has_edge H' m' e' n"
by (auto simp:G in_grcomp)
--- a/src/HOL/SizeChange/Implementation.thy Fri Jul 10 00:47:17 2009 +0200
+++ b/src/HOL/SizeChange/Implementation.thy Fri Jul 10 00:49:32 2009 +0200
@@ -100,7 +100,7 @@
assumes fA: "finite_acg A"
shows "mk_tcl A A = tcl A"
using mk_tcl_finite_terminates[OF fA]
- by (simp only: tcl_def mk_tcl_correctness star_commute)
+ by (simp only: tcl_def mk_tcl_correctness star_simulation)
definition test_SCT :: "nat acg \<Rightarrow> bool"
where
--- a/src/HOL/SizeChange/Kleene_Algebras.thy Fri Jul 10 00:47:17 2009 +0200
+++ b/src/HOL/SizeChange/Kleene_Algebras.thy Fri Jul 10 00:49:32 2009 +0200
@@ -9,17 +9,19 @@
imports Main
begin
-text {* A type class of kleene algebras *}
+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::idem_add) + (x + y) = x + y"
- unfolding add_assoc[symmetric]
- by simp
+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"
@@ -55,6 +57,15 @@
"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
@@ -95,22 +106,206 @@
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
-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
+subclass kleene
proof
-
fix a x :: 'a
have [simp]: "1 \<le> star a"
@@ -201,193 +396,19 @@
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
+end
-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"
+context kleene
+begin
-lemma tcl_zero:
- "tcl (0::'a::kleene) = 0"
- unfolding tcl_def by simp
+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 -
@@ -395,7 +416,7 @@
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)
+ by (simp add:tcl_def mult_assoc)
qed
lemma less_tcl: "a \<le> tcl a"
@@ -405,6 +426,9 @@
finally show ?thesis .
qed
+end
+
+
subsection {* Naive Algorithm to generate the transitive closure *}
function (default "\<lambda>x. 0", tailrec, domintros)
@@ -421,31 +445,32 @@
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:
- fixes X :: "'a :: kleene"
- shows "(X + X * A) * star A = X * star A"
+ "(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)
+ thus ?thesis by (simp add:left_distrib right_distrib mult_assoc)
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)
+ by (rule antisym) (auto simp:star4)
-
-
+end
lemma mk_tcl_correctness:
- fixes A X :: "'a :: {kleene}"
+ 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)
+ 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)
@@ -464,6 +489,6 @@
shows "mk_tcl A A = tcl A"
using assms mk_tcl_default mk_tcl_correctness
unfolding tcl_def
- by (auto simp:star_commute)
+ by auto
end