--- a/src/HOL/Nat.thy Thu Jul 02 16:14:20 2015 +0200
+++ b/src/HOL/Nat.thy Fri Jul 03 08:26:34 2015 +0200
@@ -1416,6 +1416,42 @@
using Kleene_iter_lpfp[OF assms(1)] lfp_unfold[OF assms(1)] by simp
qed
+lemma mono_pow:
+ fixes f :: "'a \<Rightarrow> 'a::complete_lattice"
+ shows "mono f \<Longrightarrow> mono (f ^^ n)"
+ by (induction n) (auto simp: mono_def)
+
+lemma lfp_funpow:
+ assumes f: "mono f" shows "lfp (f ^^ Suc n) = lfp f"
+proof (rule antisym)
+ show "lfp f \<le> lfp (f ^^ Suc n)"
+ proof (rule lfp_lowerbound)
+ have "f (lfp (f ^^ Suc n)) = lfp (\<lambda>x. f ((f ^^ n) x))"
+ unfolding funpow_Suc_right by (simp add: lfp_rolling f mono_pow comp_def)
+ then show "f (lfp (f ^^ Suc n)) \<le> lfp (f ^^ Suc n)"
+ by (simp add: comp_def)
+ qed
+ have "(f^^n) (lfp f) = lfp f" for n
+ by (induction n) (auto intro: f lfp_unfold[symmetric])
+ then show "lfp (f^^Suc n) \<le> lfp f"
+ by (intro lfp_lowerbound) (simp del: funpow.simps)
+qed
+
+lemma gfp_funpow:
+ assumes f: "mono f" shows "gfp (f ^^ Suc n) = gfp f"
+proof (rule antisym)
+ show "gfp f \<ge> gfp (f ^^ Suc n)"
+ proof (rule gfp_upperbound)
+ have "f (gfp (f ^^ Suc n)) = gfp (\<lambda>x. f ((f ^^ n) x))"
+ unfolding funpow_Suc_right by (simp add: gfp_rolling f mono_pow comp_def)
+ then show "f (gfp (f ^^ Suc n)) \<ge> gfp (f ^^ Suc n)"
+ by (simp add: comp_def)
+ qed
+ have "(f^^n) (gfp f) = gfp f" for n
+ by (induction n) (auto intro: f gfp_unfold[symmetric])
+ then show "gfp (f^^Suc n) \<ge> gfp f"
+ by (intro gfp_upperbound) (simp del: funpow.simps)
+qed
subsection {* Embedding of the naturals into any @{text semiring_1}: @{term of_nat} *}