--- a/CONTRIBUTORS Sat Sep 06 20:12:32 2014 +0200
+++ b/CONTRIBUTORS Sat Sep 06 20:12:34 2014 +0200
@@ -6,6 +6,10 @@
Contributions to this Isabelle version
--------------------------------------
+* September 2014: Florian Haftmann, TUM
+ Lexicographic order on functions and
+ sum/product over function bodies.
+
* August 2014: Manuel Eberl, TUM
Generic euclidean algorithms for gcd et al.
--- /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
--- a/src/HOL/Library/Library.thy Sat Sep 06 20:12:32 2014 +0200
+++ b/src/HOL/Library/Library.thy Sat Sep 06 20:12:34 2014 +0200
@@ -26,6 +26,7 @@
Function_Division
Function_Growth
Fundamental_Theorem_Algebra
+ Fun_Lexorder
Indicator_Function
Infinite_Set
Inner_Product