(* Author: Florian Haftmann, TU Muenchen *)
section \Lexical order on functions\
theory Fun_Lexorder
imports Main
begin
definition less_fun :: "('a::linorder \ 'b::linorder) \ ('a \ 'b) \ bool"
where
"less_fun f g \ (\k. f k < g k \ (\k' < k. f k' = g k'))"
lemma less_funI:
assumes "\k. f k < g k \ (\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 "\k'. k' < k \ f k' = g k'"
using assms unfolding less_fun_def by blast
lemma less_fun_asym:
assumes "less_fun f g"
shows "\ less_fun g f"
proof
from assms obtain k1 where k1: "f k1 < g k1" "\k'. k' < k1 \ f k' = g k'"
by (blast elim!: less_funE)
assume "less_fun g f" then obtain k2 where k2: "g k2 < f k2" "\k'. k' < k2 \ 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:
"\ 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" "\k'. k' < k1 \ f k' = g k'"
by (blast elim!: less_funE)
from \less_fun g h\ obtain k2 where k2: "g k2 < h k2" "\k'. k' < k2 \ g k' = h k'"
by (blast elim!: less_funE)
show "\k. f k < h k \ (\k'k'. k' < k1 \ 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" "\k'. k' < k2 \ 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 (\f g. less_fun f g \ 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 \ g k}"
shows "less_fun f g \ f = g \ less_fun g f"
proof -
{ def K \ "{k. f k \ g k}"
assume "f \ g"
then obtain k' where "f k' \ g k'" by auto
then have [simp]: "K \ {}" by (auto simp add: K_def)
with assms have [simp]: "finite K" by (simp add: K_def)
def q \ "Min K"
then have "q \ K" and "\k. k \ K \ k \ q" by auto
then have "\k. \ k \ q \ k \ K" by blast
then have *: "\k. k < q \ f k = g k" by (simp add: K_def)
from \q \ K\ have "f q \ g q" by (simp add: K_def)
then have "f q < g q \ f q > g q" by auto
with * have "less_fun f g \ less_fun g f"
by (auto intro!: less_funI)
} then show ?thesis by blast
qed
end