--- a/src/HOL/Relation_Power.thy Thu May 28 22:54:57 2009 -0700
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,154 +0,0 @@
-(* Title: HOL/Relation_Power.thy
- Author: Tobias Nipkow
- Copyright 1996 TU Muenchen
-*)
-
-header{*Powers of Relations and Functions*}
-
-theory Relation_Power
-imports Power Transitive_Closure Plain
-begin
-
-consts funpower :: "('a \<Rightarrow> 'b) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'b" (infixr "^^" 80)
-
-overloading
- relpow \<equiv> "funpower \<Colon> ('a \<times> 'a) set \<Rightarrow> nat \<Rightarrow> ('a \<times> 'a) set"
-begin
-
-text {* @{text "R ^^ n = R O ... O R"}, the n-fold composition of @{text R} *}
-
-primrec relpow where
- "(R \<Colon> ('a \<times> 'a) set) ^^ 0 = Id"
- | "(R \<Colon> ('a \<times> 'a) set) ^^ Suc n = R O (R ^^ n)"
-
-end
-
-overloading
- funpow \<equiv> "funpower \<Colon> ('a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"
-begin
-
-text {* @{text "f ^^ n = f o ... o f"}, the n-fold composition of @{text f} *}
-
-primrec funpow where
- "(f \<Colon> 'a \<Rightarrow> 'a) ^^ 0 = id"
- | "(f \<Colon> 'a \<Rightarrow> 'a) ^^ Suc n = f o (f ^^ n)"
-
-end
-
-primrec fun_pow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where
- "fun_pow 0 f = id"
- | "fun_pow (Suc n) f = f o fun_pow n f"
-
-lemma funpow_fun_pow [code unfold]:
- "f ^^ n = fun_pow n f"
- unfolding funpow_def fun_pow_def ..
-
-lemma funpow_add:
- "f ^^ (m + n) = f ^^ m o f ^^ n"
- by (induct m) simp_all
-
-lemma funpow_swap1:
- "f ((f ^^ n) x) = (f ^^ n) (f x)"
-proof -
- have "f ((f ^^ n) x) = (f ^^ (n+1)) x" unfolding One_nat_def by simp
- also have "\<dots> = (f ^^ n o f ^^ 1) x" by (simp only: funpow_add)
- also have "\<dots> = (f ^^ n) (f x)" unfolding One_nat_def by simp
- finally show ?thesis .
-qed
-
-lemma rel_pow_1 [simp]:
- fixes R :: "('a * 'a) set"
- shows "R ^^ 1 = R"
- by simp
-
-lemma rel_pow_0_I:
- "(x, x) \<in> R ^^ 0"
- by simp
-
-lemma rel_pow_Suc_I:
- "(x, y) \<in> R ^^ n \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> (x, z) \<in> R ^^ Suc n"
- by auto
-
-lemma rel_pow_Suc_I2:
- "(x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ n \<Longrightarrow> (x, z) \<in> R ^^ Suc n"
- by (induct n arbitrary: z) (simp, fastsimp)
-
-lemma rel_pow_0_E:
- "(x, y) \<in> R ^^ 0 \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> P"
- by simp
-
-lemma rel_pow_Suc_E:
- "(x, z) \<in> R ^^ Suc n \<Longrightarrow> (\<And>y. (x, y) \<in> R ^^ n \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> P) \<Longrightarrow> P"
- by auto
-
-lemma rel_pow_E:
- "(x, z) \<in> R ^^ n \<Longrightarrow> (n = 0 \<Longrightarrow> x = z \<Longrightarrow> P)
- \<Longrightarrow> (\<And>y m. n = Suc m \<Longrightarrow> (x, y) \<in> R ^^ m \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> P)
- \<Longrightarrow> P"
- by (cases n) auto
-
-lemma rel_pow_Suc_D2:
- "(x, z) \<in> R ^^ Suc n \<Longrightarrow> (\<exists>y. (x, y) \<in> R \<and> (y, z) \<in> R ^^ n)"
- apply (induct n arbitrary: x z)
- apply (blast intro: rel_pow_0_I elim: rel_pow_0_E rel_pow_Suc_E)
- apply (blast intro: rel_pow_Suc_I elim: rel_pow_0_E rel_pow_Suc_E)
- done
-
-lemma rel_pow_Suc_D2':
- "\<forall>x y z. (x, y) \<in> R ^^ n \<and> (y, z) \<in> R \<longrightarrow> (\<exists>w. (x, w) \<in> R \<and> (w, z) \<in> R ^^ n)"
- by (induct n) (simp_all, blast)
-
-lemma rel_pow_E2:
- "(x, z) \<in> R ^^ n \<Longrightarrow> (n = 0 \<Longrightarrow> x = z \<Longrightarrow> P)
- \<Longrightarrow> (\<And>y m. n = Suc m \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ m \<Longrightarrow> P)
- \<Longrightarrow> P"
- apply (cases n, simp)
- apply (cut_tac n=nat and R=R in rel_pow_Suc_D2', simp, blast)
- done
-
-lemma rtrancl_imp_UN_rel_pow:
- "p \<in> R^* \<Longrightarrow> p \<in> (\<Union>n. R ^^ n)"
- apply (cases p) apply (simp only:)
- apply (erule rtrancl_induct)
- apply (blast intro: rel_pow_0_I rel_pow_Suc_I)+
- done
-
-lemma rel_pow_imp_rtrancl:
- "p \<in> R ^^ n \<Longrightarrow> p \<in> R^*"
- apply (induct n arbitrary: p)
- apply (simp_all only: split_tupled_all)
- apply (blast intro: rtrancl_refl elim: rel_pow_0_E)
- apply (blast elim: rel_pow_Suc_E intro: rtrancl_into_rtrancl)
- done
-
-lemma rtrancl_is_UN_rel_pow:
- "R^* = (UN n. R ^^ n)"
- by (blast intro: rtrancl_imp_UN_rel_pow rel_pow_imp_rtrancl)
-
-lemma trancl_power:
- "x \<in> r^+ = (\<exists>n > 0. x \<in> r ^^ n)"
- apply (cases x)
- apply simp
- apply (rule iffI)
- apply (drule tranclD2)
- apply (clarsimp simp: rtrancl_is_UN_rel_pow)
- apply (rule_tac x="Suc x" in exI)
- apply (clarsimp simp: rel_comp_def)
- apply fastsimp
- apply clarsimp
- apply (case_tac n, simp)
- apply clarsimp
- apply (drule rel_pow_imp_rtrancl)
- apply fastsimp
- done
-
-lemma single_valued_rel_pow:
- fixes R :: "('a * 'a) set"
- shows "single_valued R \<Longrightarrow> single_valued (R ^^ n)"
- apply (induct n arbitrary: R)
- apply simp_all
- apply (rule single_valuedI)
- apply (fast dest: single_valuedD elim: rel_pow_Suc_E)
- done
-
-end