haftmann@58196: (* Author: Florian Haftmann, TU Muenchen *)
haftmann@58196: 
haftmann@58196: header \<open>Lexical order on functions\<close>
haftmann@58196: 
haftmann@58196: theory Fun_Lexorder
haftmann@58196: imports Main
haftmann@58196: begin
haftmann@58196: 
haftmann@58196: definition less_fun :: "('a::linorder \<Rightarrow> 'b::linorder) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"
haftmann@58196: where
haftmann@58196:   "less_fun f g \<longleftrightarrow> (\<exists>k. f k < g k \<and> (\<forall>k' < k. f k' = g k'))"
haftmann@58196: 
haftmann@58196: lemma less_funI:
haftmann@58196:   assumes "\<exists>k. f k < g k \<and> (\<forall>k' < k. f k' = g k')"
haftmann@58196:   shows "less_fun f g"
haftmann@58196:   using assms by (simp add: less_fun_def)
haftmann@58196: 
haftmann@58196: lemma less_funE:
haftmann@58196:   assumes "less_fun f g"
haftmann@58196:   obtains k where "f k < g k" and "\<And>k'. k' < k \<Longrightarrow> f k' = g k'"
haftmann@58196:   using assms unfolding less_fun_def by blast
haftmann@58196: 
haftmann@58196: lemma less_fun_asym:
haftmann@58196:   assumes "less_fun f g"
haftmann@58196:   shows "\<not> less_fun g f"
haftmann@58196: proof
haftmann@58196:   from assms obtain k1 where k1: "f k1 < g k1" "\<And>k'. k' < k1 \<Longrightarrow> f k' = g k'"
haftmann@58196:     by (blast elim!: less_funE) 
haftmann@58196:   assume "less_fun g f" then obtain k2 where k2: "g k2 < f k2" "\<And>k'. k' < k2 \<Longrightarrow> g k' = f k'"
haftmann@58196:     by (blast elim!: less_funE) 
haftmann@58196:   show False proof (cases k1 k2 rule: linorder_cases)
haftmann@58196:     case equal with k1 k2 show False by simp
haftmann@58196:   next
haftmann@58196:     case less with k2 have "g k1 = f k1" by simp
haftmann@58196:     with k1 show False by simp
haftmann@58196:   next
haftmann@58196:     case greater with k1 have "f k2 = g k2" by simp
haftmann@58196:     with k2 show False by simp
haftmann@58196:   qed
haftmann@58196: qed
haftmann@58196: 
haftmann@58196: lemma less_fun_irrefl:
haftmann@58196:   "\<not> less_fun f f"
haftmann@58196: proof
haftmann@58196:   assume "less_fun f f"
haftmann@58196:   then obtain k where k: "f k < f k"
haftmann@58196:     by (blast elim!: less_funE)
haftmann@58196:   then show False by simp
haftmann@58196: qed
haftmann@58196: 
haftmann@58196: lemma less_fun_trans:
haftmann@58196:   assumes "less_fun f g" and "less_fun g h"
haftmann@58196:   shows "less_fun f h"
haftmann@58196: proof (rule less_funI)
haftmann@58196:   from `less_fun f g` obtain k1 where k1: "f k1 < g k1" "\<And>k'. k' < k1 \<Longrightarrow> f k' = g k'"
haftmann@58196:     by (blast elim!: less_funE) 
haftmann@58196:   from `less_fun g h` obtain k2 where k2: "g k2 < h k2" "\<And>k'. k' < k2 \<Longrightarrow> g k' = h k'"
haftmann@58196:     by (blast elim!: less_funE) 
haftmann@58196:   show "\<exists>k. f k < h k \<and> (\<forall>k'<k. f k' = h k')"
haftmann@58196:   proof (cases k1 k2 rule: linorder_cases)
haftmann@58196:     case equal with k1 k2 show ?thesis by (auto simp add: exI [of _ k2])
haftmann@58196:   next
haftmann@58196:     case less with k2 have "g k1 = h k1" "\<And>k'. k' < k1 \<Longrightarrow> g k' = h k'" by simp_all
haftmann@58196:     with k1 show ?thesis by (auto intro: exI [of _ k1])
haftmann@58196:   next
haftmann@58196:     case greater with k1 have "f k2 = g k2" "\<And>k'. k' < k2 \<Longrightarrow> f k' = g k'" by simp_all
haftmann@58196:     with k2 show ?thesis by (auto intro: exI [of _ k2])
haftmann@58196:   qed
haftmann@58196: qed
haftmann@58196: 
haftmann@58196: lemma order_less_fun:
haftmann@58196:   "class.order (\<lambda>f g. less_fun f g \<or> f = g) less_fun"
haftmann@58196:   by (rule order_strictI) (auto intro: less_fun_trans intro!: less_fun_irrefl less_fun_asym)
haftmann@58196: 
haftmann@58196: lemma less_fun_trichotomy:
haftmann@58196:   assumes "finite {k. f k \<noteq> g k}"
haftmann@58196:   shows "less_fun f g \<or> f = g \<or> less_fun g f"
haftmann@58196: proof -
haftmann@58196:   { def K \<equiv> "{k. f k \<noteq> g k}"
haftmann@58196:     assume "f \<noteq> g"
haftmann@58196:     then obtain k' where "f k' \<noteq> g k'" by auto
haftmann@58196:     then have [simp]: "K \<noteq> {}" by (auto simp add: K_def)
haftmann@58196:     with assms have [simp]: "finite K" by (simp add: K_def)
haftmann@58196:     def q \<equiv> "Min K"
haftmann@58196:     then have "q \<in> K" and "\<And>k. k \<in> K \<Longrightarrow> k \<ge> q" by auto
haftmann@58196:     then have "\<And>k. \<not> k \<ge> q \<Longrightarrow> k \<notin> K" by blast
haftmann@58196:     then have *: "\<And>k. k < q \<Longrightarrow> f k = g k" by (simp add: K_def)
haftmann@58196:     from `q \<in> K` have "f q \<noteq> g q" by (simp add: K_def)
haftmann@58196:     then have "f q < g q \<or> f q > g q" by auto
haftmann@58196:     with * have "less_fun f g \<or> less_fun g f"
haftmann@58196:       by (auto intro!: less_funI)
haftmann@58196:   } then show ?thesis by blast
haftmann@58196: qed
haftmann@58196: 
haftmann@58196: end