src/HOL/Library/SCT_Misc.thy

+theory SCT_Misc
+imports Main
+begin
+
+
+subsection {* Searching in lists *}
+
+fun index_of :: "'a list \<Rightarrow> 'a \<Rightarrow> nat"
+where
+  "index_of [] c = 0"
+| "index_of (x#xs) c = (if x = c then 0 else Suc (index_of xs c))"
+
+lemma index_of_member: 
+  "(x \<in> set l) \<Longrightarrow> (l ! index_of l x = x)"
+  by (induct l) auto
+
+lemma index_of_length:
+  "(x \<in> set l) = (index_of l x < length l)"
+  by (induct l) auto
+
+
+
+subsection {* Some reasoning tools *}
+
+lemma inc_induct[consumes 1]:
+  assumes less: "i \<le> j"
+  assumes base: "P j"
+  assumes step: "\<And>i. \<lbrakk>i < j; P (Suc i)\<rbrakk> \<Longrightarrow> P i"
+  shows "P i"
+  using less
+proof (induct d\<equiv>"j - i" arbitrary: i)
+  case (0 i)
+  with `i \<le> j` have "i = j" by simp
+  with base show ?case by simp
+next
+  case (Suc d i)
+  hence "i < j" "P (Suc i)"
+    by simp_all
+  thus "P i" by (rule step)
+qed
+
+lemma strict_inc_induct[consumes 1]:
+  assumes less: "i < j"
+  assumes base: "\<And>i. j = Suc i \<Longrightarrow> P i"
+  assumes step: "\<And>i. \<lbrakk>i < j; P (Suc i)\<rbrakk> \<Longrightarrow> P i"
+  shows "P i"
+  using less
+proof (induct d\<equiv>"j - i - 1" arbitrary: i)
+  case (0 i)
+  with `i < j` have "j = Suc i" by simp
+  with base show ?case by simp
+next
+  case (Suc d i)
+  hence "i < j" "P (Suc i)"
+    by simp_all
+  thus "P i" by (rule step)
+qed
+
+
+lemma three_cases:
+  assumes "a1 \<Longrightarrow> thesis"
+  assumes "a2 \<Longrightarrow> thesis"
+  assumes "a3 \<Longrightarrow> thesis"
+  assumes "\<And>R. \<lbrakk>a1 \<Longrightarrow> R; a2 \<Longrightarrow> R; a3 \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R"
+  shows "thesis"
+  using prems
+  by auto
+
+
+section {* Sequences *}
+
+types
+  'a sequence = "nat \<Rightarrow> 'a"
+
+subsection {* Increasing sequences *}
+
+definition increasing :: "(nat \<Rightarrow> nat) \<Rightarrow> bool"
+where
+  "increasing s = (\<forall>i j. i < j \<longrightarrow> s i < s j)"
+
+lemma increasing_strict:
+  assumes "increasing s"
+  assumes "i < j"
+  shows "s i < s j"
+  using prems
+  unfolding increasing_def by simp
+
+lemma increasing_weak:
+  assumes "increasing s"
+  assumes "i \<le> j"
+  shows "s i \<le> s j"
+  using prems increasing_strict[of s i j]
+  by (cases "i<j") auto
+
+lemma increasing_inc:
+  assumes [simp]: "increasing s"
+  shows "n \<le> s n"
+proof (induct n)
+  case (Suc n)
+  with increasing_strict[of s n "Suc n"]
+  show ?case by auto
+qed auto
+
+
+lemma increasing_bij:
+  assumes [simp]: "increasing s"
+  shows "(s i < s j) = (i < j)"
+proof
+  assume "s i < s j"
+  show "i < j"
+  proof (rule classical)
+    assume "\<not> ?thesis"
+    hence "j \<le> i" by arith
+    with increasing_weak have "s j \<le> s i" by simp
+    with `s i < s j` show ?thesis by simp
+  qed
+qed (simp add:increasing_strict)
+
+
+
+
+subsection {* Sections induced by an increasing sequence *}
+
+abbreviation
+  "section s i == {s i ..< s (Suc i)}"
+
+definition
+  "section_of s n = (LEAST i. n < s (Suc i))"
+
+
+lemma section_help:
+  assumes [intro, simp]: "increasing s"
+  shows "\<exists>i. n < s (Suc i)" 
+proof -
+  from increasing_inc have "n \<le> s n" .
+  also from increasing_strict have "\<dots> < s (Suc n)" by simp
+  finally show ?thesis ..
+qed
+
+lemma section_of2:
+  assumes "increasing s"
+  shows "n < s (Suc (section_of s n))"
+  unfolding section_of_def
+  by (rule LeastI_ex) (rule section_help)
+
+
+lemma section_of1:
+  assumes [simp, intro]: "increasing s"
+  assumes "s i \<le> n"
+  shows "s (section_of s n) \<le> n"
+proof (rule classical)
+  let ?m = "section_of s n"
+
+  assume "\<not> ?thesis"
+  hence a: "n < s ?m" by simp
+  
+  have nonzero: "?m \<noteq> 0"
+  proof
+    assume "?m = 0"
+    from increasing_weak have "s 0 \<le> s i" by simp
+    also note `\<dots> \<le> n`
+    finally show False using `?m = 0` `n < s ?m` by simp 
+  qed
+  with a have "n < s (Suc (?m - 1))" by simp
+  with Least_le have "?m \<le> ?m - 1"
+    unfolding section_of_def .
+  with nonzero show ?thesis by simp
+qed
+
+lemma section_of_known: 
+  assumes [simp]: "increasing s"
+  assumes in_sect: "k \<in> section s i"
+  shows "section_of s k = i" (is "?s = i")
+proof (rule classical)
+  assume "\<not> ?thesis"
+
+  hence "?s < i \<or> ?s > i" by arith
+  thus ?thesis
+  proof
+    assume "?s < i"
+    hence "Suc ?s \<le> i" by simp
+    with increasing_weak have "s (Suc ?s) \<le> s i" by simp
+    moreover have "k < s (Suc ?s)" using section_of2 by simp
+    moreover from in_sect have "s i \<le> k" by simp
+    ultimately show ?thesis by simp 
+  next
+    assume "i < ?s" hence "Suc i \<le> ?s" by simp
+    with increasing_weak have "s (Suc i) \<le> s ?s" by simp
+    moreover 
+    from in_sect have "s i \<le> k" by simp
+    with section_of1 have "s ?s \<le> k" by simp
+    moreover from in_sect have "k < s (Suc i)" by simp
+    ultimately show ?thesis by simp
+  qed
+qed 
+
+  
+lemma in_section_of: 
+  assumes [simp, intro]: "increasing s"
+  assumes "s i \<le> k"
+  shows "k \<in> section s (section_of s k)"
+  using prems
+  by (auto intro:section_of1 section_of2)
+
+
+end