src/HOL/Nat.thy
 changeset 30971 7fbebf75b3ef parent 30966 55104c664185 child 30975 b2fa60d56735
```     1.1 --- a/src/HOL/Nat.thy	Fri Apr 24 08:24:54 2009 +0200
1.2 +++ b/src/HOL/Nat.thy	Fri Apr 24 17:45:15 2009 +0200
1.3 @@ -1166,31 +1166,58 @@
1.4
1.5  subsection {* Natural operation of natural numbers on functions *}
1.6
1.7 -text {* @{text "f o^ n = f o ... o f"}, the n-fold composition of @{text f} *}
1.8 +text {*
1.9 +  We use the same logical constant for the power operations on
1.10 +  functions and relations, in order to share the same syntax.
1.11 +*}
1.12 +
1.13 +consts compow :: "nat \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)"
1.14 +
1.15 +abbreviation compower :: "('a \<Rightarrow> 'b) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'b" (infixr "^^" 80) where
1.16 +  "f ^^ n \<equiv> compow n f"
1.17 +
1.18 +notation (latex output)
1.19 +  compower ("(_\<^bsup>_\<^esup>)" [1000] 1000)
1.20 +
1.21 +notation (HTML output)
1.22 +  compower ("(_\<^bsup>_\<^esup>)" [1000] 1000)
1.23 +
1.24 +text {* @{text "f ^^ n = f o ... o f"}, the n-fold composition of @{text f} *}
1.25 +
1.27 +  funpow == "compow :: nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a)"
1.28 +begin
1.29
1.30  primrec funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where
1.31      "funpow 0 f = id"
1.32    | "funpow (Suc n) f = f o funpow n f"
1.33
1.34 -abbreviation funpower :: "('a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a" (infixr "o^" 80) where
1.35 -  "f o^ n \<equiv> funpow n f"
1.36 +end
1.37 +
1.38 +text {* for code generation *}
1.39 +
1.40 +definition funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where
1.41 +  funpow_code_def [code post]: "funpow = compow"
1.42
1.43 -notation (latex output)
1.44 -  funpower ("(_\<^bsup>_\<^esup>)" [1000] 1000)
1.45 +lemmas [code inline] = funpow_code_def [symmetric]
1.46
1.47 -notation (HTML output)
1.48 -  funpower ("(_\<^bsup>_\<^esup>)" [1000] 1000)
1.49 +lemma [code]:
1.50 +  "funpow 0 f = id"
1.51 +  "funpow (Suc n) f = f o funpow n f"
1.52 +  unfolding funpow_code_def by simp_all
1.53 +
1.54 +definition "foo = id ^^ (1 + 1)"
1.55
1.57 -  "f o^ (m + n) = f o^ m \<circ> f o^ n"
1.58 +  "f ^^ (m + n) = f ^^ m \<circ> f ^^ n"
1.59    by (induct m) simp_all
1.60
1.61  lemma funpow_swap1:
1.62 -  "f ((f o^ n) x) = (f o^ n) (f x)"
1.63 +  "f ((f ^^ n) x) = (f ^^ n) (f x)"
1.64  proof -
1.65 -  have "f ((f o^ n) x) = (f o^ (n + 1)) x" by simp
1.66 -  also have "\<dots>  = (f o^ n o f o^ 1) x" by (simp only: funpow_add)
1.67 -  also have "\<dots> = (f o^ n) (f x)" by simp
1.68 +  have "f ((f ^^ n) x) = (f ^^ (n + 1)) x" by simp
1.69 +  also have "\<dots>  = (f ^^ n o f ^^ 1) x" by (simp only: funpow_add)
1.70 +  also have "\<dots> = (f ^^ n) (f x)" by simp
1.71    finally show ?thesis .
1.72  qed
1.73
```