--- a/src/HOL/Nat.thy Fri Apr 24 08:24:54 2009 +0200
+++ b/src/HOL/Nat.thy Fri Apr 24 17:45:15 2009 +0200
@@ -1166,31 +1166,58 @@
subsection {* Natural operation of natural numbers on functions *}
-text {* @{text "f o^ n = f o ... o f"}, the n-fold composition of @{text f} *}
+text {*
+ We use the same logical constant for the power operations on
+ functions and relations, in order to share the same syntax.
+*}
+
+consts compow :: "nat \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)"
+
+abbreviation compower :: "('a \<Rightarrow> 'b) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'b" (infixr "^^" 80) where
+ "f ^^ n \<equiv> compow n f"
+
+notation (latex output)
+ compower ("(_\<^bsup>_\<^esup>)" [1000] 1000)
+
+notation (HTML output)
+ compower ("(_\<^bsup>_\<^esup>)" [1000] 1000)
+
+text {* @{text "f ^^ n = f o ... o f"}, the n-fold composition of @{text f} *}
+
+overloading
+ funpow == "compow :: nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a)"
+begin
primrec funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where
"funpow 0 f = id"
| "funpow (Suc n) f = f o funpow n f"
-abbreviation funpower :: "('a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a" (infixr "o^" 80) where
- "f o^ n \<equiv> funpow n f"
+end
+
+text {* for code generation *}
+
+definition funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where
+ funpow_code_def [code post]: "funpow = compow"
-notation (latex output)
- funpower ("(_\<^bsup>_\<^esup>)" [1000] 1000)
+lemmas [code inline] = funpow_code_def [symmetric]
-notation (HTML output)
- funpower ("(_\<^bsup>_\<^esup>)" [1000] 1000)
+lemma [code]:
+ "funpow 0 f = id"
+ "funpow (Suc n) f = f o funpow n f"
+ unfolding funpow_code_def by simp_all
+
+definition "foo = id ^^ (1 + 1)"
lemma funpow_add:
- "f o^ (m + n) = f o^ m \<circ> f o^ n"
+ "f ^^ (m + n) = f ^^ m \<circ> f ^^ n"
by (induct m) simp_all
lemma funpow_swap1:
- "f ((f o^ n) x) = (f o^ n) (f x)"
+ "f ((f ^^ n) x) = (f ^^ n) (f x)"
proof -
- have "f ((f o^ n) x) = (f o^ (n + 1)) x" by simp
- also have "\<dots> = (f o^ n o f o^ 1) x" by (simp only: funpow_add)
- also have "\<dots> = (f o^ n) (f x)" by simp
+ have "f ((f ^^ n) x) = (f ^^ (n + 1)) x" by simp
+ also have "\<dots> = (f ^^ n o f ^^ 1) x" by (simp only: funpow_add)
+ also have "\<dots> = (f ^^ n) (f x)" by simp
finally show ?thesis .
qed