--- a/src/HOL/Enum.thy Thu Oct 13 23:02:59 2011 +0200
+++ b/src/HOL/Enum.thy Thu Oct 13 23:27:46 2011 +0200
@@ -758,69 +758,13 @@
unfolding enum_the_def by (auto split: list.split)
qed
+
subsection {* An executable card operator on finite types *}
-lemma
- [code]: "card R = length (filter R enum)"
-by (simp add: distinct_length_filter[OF enum_distinct] enum_UNIV Collect_def)
-
-subsection {* An executable (reflexive) transitive closure on finite relations *}
-
-text {* Definitions could be moved to Transitive Closure theory if they are of more general use. *}
-
-definition ntrancl :: "('a * 'a => bool) => nat => ('a * 'a => bool)"
-where
- [code del]: "ntrancl R n = (UN i : {i. 0 < i & i <= (Suc n)}. R ^^ i)"
-
-lemma [code]:
- "ntrancl (R :: 'a * 'a => bool) 0 = R"
-proof
- show "R <= ntrancl R 0"
- unfolding ntrancl_def by fastforce
-next
- {
- fix i have "(0 < i & i <= Suc 0) = (i = 1)" by auto
- }
- from this show "ntrancl R 0 <= R"
- unfolding ntrancl_def by auto
-qed
-
lemma [code]:
- "ntrancl (R :: 'a * 'a => bool) (Suc n) = (ntrancl R n) O (Id Un R)"
-proof
- {
- fix a b
- assume "(a, b) : ntrancl R (Suc n)"
- from this obtain i where "0 < i" "i <= Suc (Suc n)" "(a, b) : R ^^ i"
- unfolding ntrancl_def by auto
- have "(a, b) : ntrancl R n O (Id Un R)"
- proof (cases "i = 1")
- case True
- from this `(a, b) : R ^^ i` show ?thesis
- unfolding ntrancl_def by auto
- next
- case False
- from this `0 < i` obtain j where j: "i = Suc j" "0 < j"
- by (cases i) auto
- from this `(a, b) : R ^^ i` obtain c where c1: "(a, c) : R ^^ j" and c2:"(c, b) : R"
- by auto
- from c1 j `i <= Suc (Suc n)` have "(a, c): ntrancl R n"
- unfolding ntrancl_def by fastforce
- from this c2 show ?thesis by fastforce
- qed
- }
- from this show "ntrancl R (Suc n) <= ntrancl R n O (Id Un R)" by auto
-next
- show "ntrancl R n O (Id Un R) <= ntrancl R (Suc n)"
- unfolding ntrancl_def by fastforce
-qed
+ "card R = length (filter R enum)"
+ by (simp add: distinct_length_filter [OF enum_distinct] enum_UNIV Collect_def)
-lemma [code]: "trancl (R :: ('a :: finite) * 'a => bool) = ntrancl R (card R - 1)"
-by (cases "card R") (auto simp add: trancl_finite_eq_rel_pow rel_pow_empty ntrancl_def)
-
-(* a copy of Nitpick.rtrancl_unfold, should be moved to Transitive_Closure *)
-lemma [code]: "r^* = (r^+)^="
-by simp
subsection {* Closing up *}
--- a/src/HOL/Nitpick.thy Thu Oct 13 23:02:59 2011 +0200
+++ b/src/HOL/Nitpick.thy Thu Oct 13 23:27:46 2011 +0200
@@ -63,7 +63,7 @@
by (auto simp: mem_def)
lemma rtrancl_unfold [nitpick_unfold, no_atp]: "r\<^sup>* \<equiv> (r\<^sup>+)\<^sup>="
-by simp
+ by (simp only: rtrancl_trancl_reflcl)
lemma rtranclp_unfold [nitpick_unfold, no_atp]:
"rtranclp r a b \<equiv> (a = b \<or> tranclp r a b)"
--- a/src/HOL/Transitive_Closure.thy Thu Oct 13 23:02:59 2011 +0200
+++ b/src/HOL/Transitive_Closure.thy Thu Oct 13 23:27:46 2011 +0200
@@ -576,6 +576,9 @@
apply (erule rtrancl_reflcl [THEN equalityD2, THEN subsetD], fast)
done
+lemma rtrancl_trancl_reflcl [code]: "r^* = (r^+)^="
+ by simp
+
lemma trancl_empty [simp]: "{}^+ = {}"
by (auto elim: trancl_induct)
@@ -980,6 +983,65 @@
apply (fast dest: single_valuedD elim: rel_pow_Suc_E)
done
+
+subsection {* Bounded transitive closure *}
+
+definition ntrancl :: "nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"
+where
+ "ntrancl n R = (\<Union>i\<in>{i. 0 < i \<and> i \<le> Suc n}. R ^^ i)"
+
+lemma ntrancl_Zero [simp, code]:
+ "ntrancl 0 R = R"
+proof
+ show "R \<subseteq> ntrancl 0 R"
+ unfolding ntrancl_def by fastforce
+next
+ {
+ fix i have "0 < i \<and> i \<le> Suc 0 \<longleftrightarrow> i = 1" by auto
+ }
+ from this show "ntrancl 0 R \<le> R"
+ unfolding ntrancl_def by auto
+qed
+
+lemma ntrancl_Suc [simp, code]:
+ "ntrancl (Suc n) R = ntrancl n R O (Id \<union> R)"
+proof
+ {
+ fix a b
+ assume "(a, b) \<in> ntrancl (Suc n) R"
+ from this obtain i where "0 < i" "i \<le> Suc (Suc n)" "(a, b) \<in> R ^^ i"
+ unfolding ntrancl_def by auto
+ have "(a, b) \<in> ntrancl n R O (Id \<union> R)"
+ proof (cases "i = 1")
+ case True
+ from this `(a, b) \<in> R ^^ i` show ?thesis
+ unfolding ntrancl_def by auto
+ next
+ case False
+ from this `0 < i` obtain j where j: "i = Suc j" "0 < j"
+ by (cases i) auto
+ from this `(a, b) \<in> R ^^ i` obtain c where c1: "(a, c) \<in> R ^^ j" and c2:"(c, b) \<in> R"
+ by auto
+ from c1 j `i \<le> Suc (Suc n)` have "(a, c) \<in> ntrancl n R"
+ unfolding ntrancl_def by fastforce
+ from this c2 show ?thesis by fastforce
+ qed
+ }
+ from this show "ntrancl (Suc n) R \<subseteq> ntrancl n R O (Id \<union> R)"
+ by auto
+next
+ show "ntrancl n R O (Id \<union> R) \<subseteq> ntrancl (Suc n) R"
+ unfolding ntrancl_def by fastforce
+qed
+
+lemma finnite_trancl_ntranl:
+ "finite R \<Longrightarrow> trancl R = ntrancl (card R - 1) R"
+ by (cases "card R") (auto simp add: trancl_finite_eq_rel_pow rel_pow_empty ntrancl_def)
+
+lemma [code]: "trancl (R :: (('a :: finite) \<times> 'a) set) = ntrancl (card R - 1) R"
+ by (simp add: finnite_trancl_ntranl)
+
+
subsection {* Acyclic relations *}
definition acyclic :: "('a * 'a) set => bool" where