1(* Author: Tobias Nipkow, Daniel St��we *)
2
3section \<open>Three-Way Comparison\<close>
4
5theory Cmp
6imports Main
7begin
8
9datatype cmp_val = LT | EQ | GT
10
11definition cmp :: "'a:: linorder \<Rightarrow> 'a \<Rightarrow> cmp_val" where
12"cmp x y = (if x < y then LT else if x=y then EQ else GT)"
13
14lemma 
15    LT[simp]: "cmp x y = LT \<longleftrightarrow> x < y"
16and EQ[simp]: "cmp x y = EQ \<longleftrightarrow> x = y"
17and GT[simp]: "cmp x y = GT \<longleftrightarrow> x > y"
18by (auto simp: cmp_def)
19
20lemma case_cmp_if[simp]: "(case c of EQ \<Rightarrow> e | LT \<Rightarrow> l | GT \<Rightarrow> g) =
21  (if c = LT then l else if c = GT then g else e)"
22by(simp split: cmp_val.split)
23
24end
25