--- a/src/HOLCF/Eventual.thy Tue Oct 05 17:53:00 2010 -0700
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,153 +0,0 @@
-(* Title: HOLCF/Eventual.thy
- Author: Brian Huffman
-*)
-
-header {* Eventually-constant sequences *}
-
-theory Eventual
-imports Infinite_Set
-begin
-
-subsection {* Lemmas about MOST *}
-
-lemma MOST_INFM:
- assumes inf: "infinite (UNIV::'a set)"
- shows "MOST x::'a. P x \<Longrightarrow> INFM x::'a. P x"
- unfolding Alm_all_def Inf_many_def
- apply (auto simp add: Collect_neg_eq)
- apply (drule (1) finite_UnI)
- apply (simp add: Compl_partition2 inf)
- done
-
-lemma MOST_SucI: "MOST n. P n \<Longrightarrow> MOST n. P (Suc n)"
-by (rule MOST_inj [OF _ inj_Suc])
-
-lemma MOST_SucD: "MOST n. P (Suc n) \<Longrightarrow> MOST n. P n"
-unfolding MOST_nat
-apply (clarify, rule_tac x="Suc m" in exI, clarify)
-apply (erule Suc_lessE, simp)
-done
-
-lemma MOST_Suc_iff: "(MOST n. P (Suc n)) \<longleftrightarrow> (MOST n. P n)"
-by (rule iffI [OF MOST_SucD MOST_SucI])
-
-lemma INFM_finite_Bex_distrib:
- "finite A \<Longrightarrow> (INFM y. \<exists>x\<in>A. P x y) \<longleftrightarrow> (\<exists>x\<in>A. INFM y. P x y)"
-by (induct set: finite, simp, simp add: INFM_disj_distrib)
-
-lemma MOST_finite_Ball_distrib:
- "finite A \<Longrightarrow> (MOST y. \<forall>x\<in>A. P x y) \<longleftrightarrow> (\<forall>x\<in>A. MOST y. P x y)"
-by (induct set: finite, simp, simp add: MOST_conj_distrib)
-
-lemma MOST_ge_nat: "MOST n::nat. m \<le> n"
-unfolding MOST_nat_le by fast
-
-subsection {* Eventually constant sequences *}
-
-definition
- eventually_constant :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"
-where
- "eventually_constant S = (\<exists>x. MOST i. S i = x)"
-
-lemma eventually_constant_MOST_MOST:
- "eventually_constant S \<longleftrightarrow> (MOST m. MOST n. S n = S m)"
-unfolding eventually_constant_def MOST_nat
-apply safe
-apply (rule_tac x=m in exI, clarify)
-apply (rule_tac x=m in exI, clarify)
-apply simp
-apply fast
-done
-
-lemma eventually_constantI: "MOST i. S i = x \<Longrightarrow> eventually_constant S"
-unfolding eventually_constant_def by fast
-
-lemma eventually_constant_comp:
- "eventually_constant (\<lambda>i. S i) \<Longrightarrow> eventually_constant (\<lambda>i. f (S i))"
-unfolding eventually_constant_def
-apply (erule exE, rule_tac x="f x" in exI)
-apply (erule MOST_mono, simp)
-done
-
-lemma eventually_constant_Suc_iff:
- "eventually_constant (\<lambda>i. S (Suc i)) \<longleftrightarrow> eventually_constant (\<lambda>i. S i)"
-unfolding eventually_constant_def
-by (subst MOST_Suc_iff, rule refl)
-
-lemma eventually_constant_SucD:
- "eventually_constant (\<lambda>i. S (Suc i)) \<Longrightarrow> eventually_constant (\<lambda>i. S i)"
-by (rule eventually_constant_Suc_iff [THEN iffD1])
-
-subsection {* Limits of eventually constant sequences *}
-
-definition
- eventual :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a" where
- "eventual S = (THE x. MOST i. S i = x)"
-
-lemma eventual_eqI: "MOST i. S i = x \<Longrightarrow> eventual S = x"
-unfolding eventual_def
-apply (rule the_equality, assumption)
-apply (rename_tac y)
-apply (subgoal_tac "MOST i::nat. y = x", simp)
-apply (erule MOST_rev_mp)
-apply (erule MOST_rev_mp)
-apply simp
-done
-
-lemma MOST_eq_eventual:
- "eventually_constant S \<Longrightarrow> MOST i. S i = eventual S"
-unfolding eventually_constant_def
-by (erule exE, simp add: eventual_eqI)
-
-lemma eventual_mem_range:
- "eventually_constant S \<Longrightarrow> eventual S \<in> range S"
-apply (drule MOST_eq_eventual)
-apply (simp only: MOST_nat_le, clarify)
-apply (drule spec, drule mp, rule order_refl)
-apply (erule range_eqI [OF sym])
-done
-
-lemma eventually_constant_MOST_iff:
- assumes S: "eventually_constant S"
- shows "(MOST n. P (S n)) \<longleftrightarrow> P (eventual S)"
-apply (subgoal_tac "(MOST n. P (S n)) \<longleftrightarrow> (MOST n::nat. P (eventual S))")
-apply simp
-apply (rule iffI)
-apply (rule MOST_rev_mp [OF MOST_eq_eventual [OF S]])
-apply (erule MOST_mono, force)
-apply (rule MOST_rev_mp [OF MOST_eq_eventual [OF S]])
-apply (erule MOST_mono, simp)
-done
-
-lemma MOST_eventual:
- "\<lbrakk>eventually_constant S; MOST n. P (S n)\<rbrakk> \<Longrightarrow> P (eventual S)"
-proof -
- assume "eventually_constant S"
- hence "MOST n. S n = eventual S"
- by (rule MOST_eq_eventual)
- moreover assume "MOST n. P (S n)"
- ultimately have "MOST n. S n = eventual S \<and> P (S n)"
- by (rule MOST_conj_distrib [THEN iffD2, OF conjI])
- hence "MOST n::nat. P (eventual S)"
- by (rule MOST_mono) auto
- thus ?thesis by simp
-qed
-
-lemma eventually_constant_MOST_Suc_eq:
- "eventually_constant S \<Longrightarrow> MOST n. S (Suc n) = S n"
-apply (drule MOST_eq_eventual)
-apply (frule MOST_Suc_iff [THEN iffD2])
-apply (erule MOST_rev_mp)
-apply (erule MOST_rev_mp)
-apply simp
-done
-
-lemma eventual_comp:
- "eventually_constant S \<Longrightarrow> eventual (\<lambda>i. f (S i)) = f (eventual (\<lambda>i. S i))"
-apply (rule eventual_eqI)
-apply (rule MOST_mono)
-apply (erule MOST_eq_eventual)
-apply simp
-done
-
-end