(* Title: HOL/Sum_Type.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1992 University of Cambridge
*)
header{*The Disjoint Sum of Two Types*}
theory Sum_Type
imports Typedef Inductive Fun
begin
subsection {* Construction of the sum type and its basic abstract operations *}
definition Inl_Rep :: "'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool \<Rightarrow> bool" where
"Inl_Rep a x y p \<longleftrightarrow> x = a \<and> p"
definition Inr_Rep :: "'b \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool \<Rightarrow> bool" where
"Inr_Rep b x y p \<longleftrightarrow> y = b \<and> \<not> p"
definition "sum = {f. (\<exists>a. f = Inl_Rep (a::'a)) \<or> (\<exists>b. f = Inr_Rep (b::'b))}"
typedef ('a, 'b) sum (infixr "+" 10) = "sum :: ('a => 'b => bool => bool) set"
unfolding sum_def by auto
lemma Inl_RepI: "Inl_Rep a \<in> sum"
by (auto simp add: sum_def)
lemma Inr_RepI: "Inr_Rep b \<in> sum"
by (auto simp add: sum_def)
lemma inj_on_Abs_sum: "A \<subseteq> sum \<Longrightarrow> inj_on Abs_sum A"
by (rule inj_on_inverseI, rule Abs_sum_inverse) auto
lemma Inl_Rep_inject: "inj_on Inl_Rep A"
proof (rule inj_onI)
show "\<And>a c. Inl_Rep a = Inl_Rep c \<Longrightarrow> a = c"
by (auto simp add: Inl_Rep_def fun_eq_iff)
qed
lemma Inr_Rep_inject: "inj_on Inr_Rep A"
proof (rule inj_onI)
show "\<And>b d. Inr_Rep b = Inr_Rep d \<Longrightarrow> b = d"
by (auto simp add: Inr_Rep_def fun_eq_iff)
qed
lemma Inl_Rep_not_Inr_Rep: "Inl_Rep a \<noteq> Inr_Rep b"
by (auto simp add: Inl_Rep_def Inr_Rep_def fun_eq_iff)
definition Inl :: "'a \<Rightarrow> 'a + 'b" where
"Inl = Abs_sum \<circ> Inl_Rep"
definition Inr :: "'b \<Rightarrow> 'a + 'b" where
"Inr = Abs_sum \<circ> Inr_Rep"
lemma inj_Inl [simp]: "inj_on Inl A"
by (auto simp add: Inl_def intro!: comp_inj_on Inl_Rep_inject inj_on_Abs_sum Inl_RepI)
lemma Inl_inject: "Inl x = Inl y \<Longrightarrow> x = y"
using inj_Inl by (rule injD)
lemma inj_Inr [simp]: "inj_on Inr A"
by (auto simp add: Inr_def intro!: comp_inj_on Inr_Rep_inject inj_on_Abs_sum Inr_RepI)
lemma Inr_inject: "Inr x = Inr y \<Longrightarrow> x = y"
using inj_Inr by (rule injD)
lemma Inl_not_Inr: "Inl a \<noteq> Inr b"
proof -
from Inl_RepI [of a] Inr_RepI [of b] have "{Inl_Rep a, Inr_Rep b} \<subseteq> sum" by auto
with inj_on_Abs_sum have "inj_on Abs_sum {Inl_Rep a, Inr_Rep b}" .
with Inl_Rep_not_Inr_Rep [of a b] inj_on_contraD have "Abs_sum (Inl_Rep a) \<noteq> Abs_sum (Inr_Rep b)" by auto
then show ?thesis by (simp add: Inl_def Inr_def)
qed
lemma Inr_not_Inl: "Inr b \<noteq> Inl a"
using Inl_not_Inr by (rule not_sym)
lemma sumE:
assumes "\<And>x::'a. s = Inl x \<Longrightarrow> P"
and "\<And>y::'b. s = Inr y \<Longrightarrow> P"
shows P
proof (rule Abs_sum_cases [of s])
fix f
assume "s = Abs_sum f" and "f \<in> sum"
with assms show P by (auto simp add: sum_def Inl_def Inr_def)
qed
rep_datatype Inl Inr
proof -
fix P
fix s :: "'a + 'b"
assume x: "\<And>x\<Colon>'a. P (Inl x)" and y: "\<And>y\<Colon>'b. P (Inr y)"
then show "P s" by (auto intro: sumE [of s])
qed (auto dest: Inl_inject Inr_inject simp add: Inl_not_Inr)
primrec sum_map :: "('a \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'a + 'b \<Rightarrow> 'c + 'd" where
"sum_map f1 f2 (Inl a) = Inl (f1 a)"
| "sum_map f1 f2 (Inr a) = Inr (f2 a)"
enriched_type sum_map: sum_map proof -
fix f g h i
show "sum_map f g \<circ> sum_map h i = sum_map (f \<circ> h) (g \<circ> i)"
proof
fix s
show "(sum_map f g \<circ> sum_map h i) s = sum_map (f \<circ> h) (g \<circ> i) s"
by (cases s) simp_all
qed
next
fix s
show "sum_map id id = id"
proof
fix s
show "sum_map id id s = id s"
by (cases s) simp_all
qed
qed
subsection {* Projections *}
lemma sum_case_KK [simp]: "sum_case (\<lambda>x. a) (\<lambda>x. a) = (\<lambda>x. a)"
by (rule ext) (simp split: sum.split)
lemma surjective_sum: "sum_case (\<lambda>x::'a. f (Inl x)) (\<lambda>y::'b. f (Inr y)) = f"
proof
fix s :: "'a + 'b"
show "(case s of Inl (x\<Colon>'a) \<Rightarrow> f (Inl x) | Inr (y\<Colon>'b) \<Rightarrow> f (Inr y)) = f s"
by (cases s) simp_all
qed
lemma sum_case_inject:
assumes a: "sum_case f1 f2 = sum_case g1 g2"
assumes r: "f1 = g1 \<Longrightarrow> f2 = g2 \<Longrightarrow> P"
shows P
proof (rule r)
show "f1 = g1" proof
fix x :: 'a
from a have "sum_case f1 f2 (Inl x) = sum_case g1 g2 (Inl x)" by simp
then show "f1 x = g1 x" by simp
qed
show "f2 = g2" proof
fix y :: 'b
from a have "sum_case f1 f2 (Inr y) = sum_case g1 g2 (Inr y)" by simp
then show "f2 y = g2 y" by simp
qed
qed
lemma sum_case_weak_cong:
"s = t \<Longrightarrow> sum_case f g s = sum_case f g t"
-- {* Prevents simplification of @{text f} and @{text g}: much faster. *}
by simp
primrec Projl :: "'a + 'b \<Rightarrow> 'a" where
Projl_Inl: "Projl (Inl x) = x"
primrec Projr :: "'a + 'b \<Rightarrow> 'b" where
Projr_Inr: "Projr (Inr x) = x"
primrec Suml :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a + 'b \<Rightarrow> 'c" where
"Suml f (Inl x) = f x"
primrec Sumr :: "('b \<Rightarrow> 'c) \<Rightarrow> 'a + 'b \<Rightarrow> 'c" where
"Sumr f (Inr x) = f x"
lemma Suml_inject:
assumes "Suml f = Suml g" shows "f = g"
proof
fix x :: 'a
let ?s = "Inl x \<Colon> 'a + 'b"
from assms have "Suml f ?s = Suml g ?s" by simp
then show "f x = g x" by simp
qed
lemma Sumr_inject:
assumes "Sumr f = Sumr g" shows "f = g"
proof
fix x :: 'b
let ?s = "Inr x \<Colon> 'a + 'b"
from assms have "Sumr f ?s = Sumr g ?s" by simp
then show "f x = g x" by simp
qed
subsection {* The Disjoint Sum of Sets *}
definition Plus :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a + 'b) set" (infixr "<+>" 65) where
"A <+> B = Inl ` A \<union> Inr ` B"
hide_const (open) Plus --"Valuable identifier"
lemma InlI [intro!]: "a \<in> A \<Longrightarrow> Inl a \<in> A <+> B"
by (simp add: Plus_def)
lemma InrI [intro!]: "b \<in> B \<Longrightarrow> Inr b \<in> A <+> B"
by (simp add: Plus_def)
text {* Exhaustion rule for sums, a degenerate form of induction *}
lemma PlusE [elim!]:
"u \<in> A <+> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> u = Inl x \<Longrightarrow> P) \<Longrightarrow> (\<And>y. y \<in> B \<Longrightarrow> u = Inr y \<Longrightarrow> P) \<Longrightarrow> P"
by (auto simp add: Plus_def)
lemma Plus_eq_empty_conv [simp]: "A <+> B = {} \<longleftrightarrow> A = {} \<and> B = {}"
by auto
lemma UNIV_Plus_UNIV [simp]: "UNIV <+> UNIV = UNIV"
proof (rule set_eqI)
fix u :: "'a + 'b"
show "u \<in> UNIV <+> UNIV \<longleftrightarrow> u \<in> UNIV" by (cases u) auto
qed
hide_const (open) Suml Sumr Projl Projr
hide_const (open) sum
end