# Theory HOL-Library.Fun_Lexorder

```(* Author: Florian Haftmann, TU Muenchen *)

section ‹Lexicographic 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' < k1 ⟹ f k' = g k'" for k'
by (blast elim!: less_funE)
assume "less_fun g f" then obtain k2 where k2: "g k2 < f k2" "k' < k2 ⟹ g k' = f k'" for 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' < k1 ⟹ f k' = g k'" for k'
by (blast elim!: less_funE)
from ‹less_fun g h› obtain k2 where k2: "g k2 < h k2" "k' < k2 ⟹ g k' = h k'" for k'
by (blast elim!: less_funE)
show "∃k. f k < h k ∧ (∀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" "⋀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 (λf 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 -
{ define K where "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)
define q where "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
```