author haftmann Wed Jan 30 10:57:44 2008 +0100 (2008-01-30) changeset 26013 8764a1f1253b parent 26012 f6917792f8a4 child 26014 00c2c3525bef
Theorem Inductive.lfp_ordinal_induct generalized to complete lattices
 NEWS file | annotate | diff | revisions src/HOL/Inductive.thy file | annotate | diff | revisions
1.1 --- a/NEWS	Tue Jan 29 18:00:12 2008 +0100
1.2 +++ b/NEWS	Wed Jan 30 10:57:44 2008 +0100
1.3 @@ -35,6 +35,9 @@
1.5  *** HOL ***
1.7 +* Theorem Inductive.lfp_ordinal_induct generalized to complete lattices.  The
1.8 +form set-specific version is available as Inductive.lfp_ordinal_induct_set.
1.9 +
1.10  * Theorems "power.simps" renamed to "power_int.simps".
1.12  * New class semiring_div provides basic abstract properties of semirings
2.1 --- a/src/HOL/Inductive.thy	Tue Jan 29 18:00:12 2008 +0100
2.2 +++ b/src/HOL/Inductive.thy	Wed Jan 30 10:57:44 2008 +0100
2.3 @@ -24,12 +24,15 @@
2.5  subsection {* Least and greatest fixed points *}
2.7 +context complete_lattice
2.8 +begin
2.9 +
2.10  definition
2.11 -  lfp :: "('a\<Colon>complete_lattice \<Rightarrow> 'a) \<Rightarrow> 'a" where
2.12 +  lfp :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a" where
2.13    "lfp f = Inf {u. f u \<le> u}"    --{*least fixed point*}
2.15  definition
2.16 -  gfp :: "('a\<Colon>complete_lattice \<Rightarrow> 'a) \<Rightarrow> 'a" where
2.17 +  gfp :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a" where
2.18    "gfp f = Sup {u. u \<le> f u}"    --{*greatest fixed point*}
2.21 @@ -44,6 +47,8 @@
2.22  lemma lfp_greatest: "(!!u. f u \<le> u ==> A \<le> u) ==> A \<le> lfp f"
2.23    by (auto simp add: lfp_def intro: Inf_greatest)
2.25 +end
2.26 +
2.27  lemma lfp_lemma2: "mono f ==> f (lfp f) \<le> lfp f"
2.28    by (iprover intro: lfp_greatest order_trans monoD lfp_lowerbound)
2.30 @@ -81,25 +86,34 @@
2.31    by (rule lfp_induct [THEN subsetD, THEN CollectD, OF mono _ lfp])
2.32      (auto simp: inf_set_eq intro: indhyp)
2.34 -lemma lfp_ordinal_induct:
2.35 +lemma lfp_ordinal_induct:
2.36 +  fixes f :: "'a\<Colon>complete_lattice \<Rightarrow> 'a"
2.37 +  assumes mono: "mono f"
2.38 +  and P_f: "\<And>S. P S \<Longrightarrow> P (f S)"
2.39 +  and P_Union: "\<And>M. \<forall>S\<in>M. P S \<Longrightarrow> P (Sup M)"
2.40 +  shows "P (lfp f)"
2.41 +proof -
2.42 +  let ?M = "{S. S \<le> lfp f \<and> P S}"
2.43 +  have "P (Sup ?M)" using P_Union by simp
2.44 +  also have "Sup ?M = lfp f"
2.45 +  proof (rule antisym)
2.46 +    show "Sup ?M \<le> lfp f" by (blast intro: Sup_least)
2.47 +    hence "f (Sup ?M) \<le> f (lfp f)" by (rule mono [THEN monoD])
2.48 +    hence "f (Sup ?M) \<le> lfp f" using mono [THEN lfp_unfold] by simp
2.49 +    hence "f (Sup ?M) \<in> ?M" using P_f P_Union by simp
2.50 +    hence "f (Sup ?M) \<le> Sup ?M" by (rule Sup_upper)
2.51 +    thus "lfp f \<le> Sup ?M" by (rule lfp_lowerbound)
2.52 +  qed
2.53 +  finally show ?thesis .
2.54 +qed
2.55 +
2.56 +lemma lfp_ordinal_induct_set:
2.57    assumes mono: "mono f"
2.58    and P_f: "!!S. P S ==> P(f S)"
2.59    and P_Union: "!!M. !S:M. P S ==> P(Union M)"
2.60    shows "P(lfp f)"
2.61 -proof -
2.62 -  let ?M = "{S. S \<subseteq> lfp f & P S}"
2.63 -  have "P (Union ?M)" using P_Union by simp
2.64 -  also have "Union ?M = lfp f"
2.65 -  proof
2.66 -    show "Union ?M \<subseteq> lfp f" by blast
2.67 -    hence "f (Union ?M) \<subseteq> f (lfp f)" by (rule mono [THEN monoD])
2.68 -    hence "f (Union ?M) \<subseteq> lfp f" using mono [THEN lfp_unfold] by simp
2.69 -    hence "f (Union ?M) \<in> ?M" using P_f P_Union by simp
2.70 -    hence "f (Union ?M) \<subseteq> Union ?M" by (rule Union_upper)
2.71 -    thus "lfp f \<subseteq> Union ?M" by (rule lfp_lowerbound)
2.72 -  qed
2.73 -  finally show ?thesis .
2.74 -qed
2.75 +  using assms unfolding Sup_set_def [symmetric]
2.76 +  by (rule lfp_ordinal_induct)
2.79  text{*Definition forms of @{text lfp_unfold} and @{text lfp_induct},