src/HOL/Library/Fun_Lexorder.thy
changeset 58196 1b3fbfb85980
child 58881 b9556a055632
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Fun_Lexorder.thy	Sat Sep 06 20:12:34 2014 +0200
@@ -0,0 +1,95 @@
+(* Author: Florian Haftmann, TU Muenchen *)
+
+header \<open>Lexical order on functions\<close>
+
+theory Fun_Lexorder
+imports Main
+begin
+
+definition less_fun :: "('a::linorder \<Rightarrow> 'b::linorder) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"
+where
+  "less_fun f g \<longleftrightarrow> (\<exists>k. f k < g k \<and> (\<forall>k' < k. f k' = g k'))"
+
+lemma less_funI:
+  assumes "\<exists>k. f k < g k \<and> (\<forall>k' < k. f k' = g k')"
+  shows "less_fun f g"
+  using assms by (simp add: less_fun_def)
+
+lemma less_funE:
+  assumes "less_fun f g"
+  obtains k where "f k < g k" and "\<And>k'. k' < k \<Longrightarrow> f k' = g k'"
+  using assms unfolding less_fun_def by blast
+
+lemma less_fun_asym:
+  assumes "less_fun f g"
+  shows "\<not> less_fun g f"
+proof
+  from assms obtain k1 where k1: "f k1 < g k1" "\<And>k'. k' < k1 \<Longrightarrow> f k' = g k'"
+    by (blast elim!: less_funE) 
+  assume "less_fun g f" then obtain k2 where k2: "g k2 < f k2" "\<And>k'. k' < k2 \<Longrightarrow> g k' = f k'"
+    by (blast elim!: less_funE) 
+  show False proof (cases k1 k2 rule: linorder_cases)
+    case equal with k1 k2 show False by simp
+  next
+    case less with k2 have "g k1 = f k1" by simp
+    with k1 show False by simp
+  next
+    case greater with k1 have "f k2 = g k2" by simp
+    with k2 show False by simp
+  qed
+qed
+
+lemma less_fun_irrefl:
+  "\<not> less_fun f f"
+proof
+  assume "less_fun f f"
+  then obtain k where k: "f k < f k"
+    by (blast elim!: less_funE)
+  then show False by simp
+qed
+
+lemma less_fun_trans:
+  assumes "less_fun f g" and "less_fun g h"
+  shows "less_fun f h"
+proof (rule less_funI)
+  from `less_fun f g` obtain k1 where k1: "f k1 < g k1" "\<And>k'. k' < k1 \<Longrightarrow> f k' = g k'"
+    by (blast elim!: less_funE) 
+  from `less_fun g h` obtain k2 where k2: "g k2 < h k2" "\<And>k'. k' < k2 \<Longrightarrow> g k' = h k'"
+    by (blast elim!: less_funE) 
+  show "\<exists>k. f k < h k \<and> (\<forall>k'<k. f k' = h k')"
+  proof (cases k1 k2 rule: linorder_cases)
+    case equal with k1 k2 show ?thesis by (auto simp add: exI [of _ k2])
+  next
+    case less with k2 have "g k1 = h k1" "\<And>k'. k' < k1 \<Longrightarrow> g k' = h k'" by simp_all
+    with k1 show ?thesis by (auto intro: exI [of _ k1])
+  next
+    case greater with k1 have "f k2 = g k2" "\<And>k'. k' < k2 \<Longrightarrow> f k' = g k'" by simp_all
+    with k2 show ?thesis by (auto intro: exI [of _ k2])
+  qed
+qed
+
+lemma order_less_fun:
+  "class.order (\<lambda>f g. less_fun f g \<or> f = g) less_fun"
+  by (rule order_strictI) (auto intro: less_fun_trans intro!: less_fun_irrefl less_fun_asym)
+
+lemma less_fun_trichotomy:
+  assumes "finite {k. f k \<noteq> g k}"
+  shows "less_fun f g \<or> f = g \<or> less_fun g f"
+proof -
+  { def K \<equiv> "{k. f k \<noteq> g k}"
+    assume "f \<noteq> g"
+    then obtain k' where "f k' \<noteq> g k'" by auto
+    then have [simp]: "K \<noteq> {}" by (auto simp add: K_def)
+    with assms have [simp]: "finite K" by (simp add: K_def)
+    def q \<equiv> "Min K"
+    then have "q \<in> K" and "\<And>k. k \<in> K \<Longrightarrow> k \<ge> q" by auto
+    then have "\<And>k. \<not> k \<ge> q \<Longrightarrow> k \<notin> K" by blast
+    then have *: "\<And>k. k < q \<Longrightarrow> f k = g k" by (simp add: K_def)
+    from `q \<in> K` have "f q \<noteq> g q" by (simp add: K_def)
+    then have "f q < g q \<or> f q > g q" by auto
+    with * have "less_fun f g \<or> less_fun g f"
+      by (auto intro!: less_funI)
+  } then show ?thesis by blast
+qed
+
+end