# Theory Sum_Type

(*  Title:      HOL/Sum_Type.thy
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright   1992  University of Cambridge
*)

section ‹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  'a  'b  bool  bool"
where "Inl_Rep a x y p  x = a  p"

definition Inr_Rep :: "'b  'a  'b  bool  bool"
where "Inr_Rep b x y p  y = b  ¬ p"

definition "sum = {f. (a. f = Inl_Rep (a::'a))  (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  sum"
by (auto simp add: sum_def)

lemma Inr_RepI: "Inr_Rep b  sum"
by (auto simp add: sum_def)

lemma inj_on_Abs_sum: "A  sum  inj_on Abs_sum A"
by (rule inj_on_inverseI, rule Abs_sum_inverse) auto

lemma Inl_Rep_inject:
proof (rule inj_onI)
show "a c. Inl_Rep a = Inl_Rep c  a = c"
by (auto simp add: Inl_Rep_def fun_eq_iff)
qed

lemma Inr_Rep_inject:
proof (rule inj_onI)
show "b d. Inr_Rep b = Inr_Rep d  b = d"
by (auto simp add: Inr_Rep_def fun_eq_iff)
qed

lemma Inl_Rep_not_Inr_Rep: "Inl_Rep a  Inr_Rep b"
by (auto simp add: Inl_Rep_def Inr_Rep_def fun_eq_iff)

definition Inl :: "'a  'a + 'b"
where "Inl = Abs_sum  Inl_Rep"

definition Inr :: "'b  'a + 'b"
where "Inr = Abs_sum  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  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  x = y"
using inj_Inr by (rule injD)

lemma Inl_not_Inr: "Inl a  Inr b"
proof -
have "{Inl_Rep a, Inr_Rep b}  sum"
using Inl_RepI [of a] Inr_RepI [of b] 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)  Abs_sum (Inr_Rep b)"
by auto
then show ?thesis
by (simp add: Inl_def Inr_def)
qed

lemma Inr_not_Inl: "Inr b  Inl a"
using Inl_not_Inr by (rule not_sym)

lemma sumE:
assumes "x::'a. s = Inl x  P"
and "y::'b. s = Inr y  P"
shows P
proof (rule Abs_sum_cases [of s])
fix f
assume "s = Abs_sum f" and "f  sum"
with assms show P
by (auto simp add: sum_def Inl_def Inr_def)
qed

free_constructors case_sum for
isl: Inl projl
| Inr projr
by (erule sumE, assumption) (auto dest: Inl_inject Inr_inject simp add: Inl_not_Inr)

text ‹Avoid name clashes by prefixing the output of old_rep_datatype› with old›.›

setup Sign.mandatory_path "old"

old_rep_datatype Inl Inr
proof -
fix P
fix s :: "'a + 'b"
assume x: "x::'a. P (Inl x)" and y: "y::'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)

setup Sign.parent_path

text ‹But erase the prefix for properties that are not generated by free_constructors›.›

setup Sign.mandatory_path "sum"

declare
old.sum.inject[iff del]
old.sum.distinct(1)[simp del, induct_simp del]

lemmas induct = old.sum.induct
lemmas inducts = old.sum.inducts
lemmas rec = old.sum.rec
lemmas simps = sum.inject sum.distinct sum.case sum.rec

setup Sign.parent_path

primrec map_sum :: "('a  'c)  ('b  'd)  'a + 'b  'c + 'd"
where
"map_sum f1 f2 (Inl a) = Inl (f1 a)"
| "map_sum f1 f2 (Inr a) = Inr (f2 a)"

functor map_sum: map_sum
proof -
show "map_sum f g  map_sum h i = map_sum (f  h) (g  i)" for f g h i
proof
show "(map_sum f g  map_sum h i) s = map_sum (f  h) (g  i) s" for s
by (cases s) simp_all
qed
show
proof
show "map_sum id id s = id s" for s
by (cases s) simp_all
qed
qed

lemma split_sum_all: "(x. P x)  (x. P (Inl x))  (x. P (Inr x))"
by (auto intro: sum.induct)

lemma split_sum_ex: "(x. P x)  (x. P (Inl x))  (x. P (Inr x))"
using split_sum_all[of "λx. ¬P x"] by blast

subsection ‹Projections›

lemma case_sum_KK [simp]: "case_sum (λx. a) (λx. a) = (λx. a)"
by (rule ext) (simp split: sum.split)

lemma surjective_sum: "case_sum (λx::'a. f (Inl x)) (λy::'b. f (Inr y)) = f"
proof
fix s :: "'a + 'b"
show "(case s of Inl (x::'a)  f (Inl x) | Inr (y::'b)  f (Inr y)) = f s"
by (cases s) simp_all
qed

lemma case_sum_inject:
assumes a: "case_sum f1 f2 = case_sum g1 g2"
and r: "f1 = g1  f2 = g2  P"
shows P
proof (rule r)
show "f1 = g1"
proof
fix x :: 'a
from a have "case_sum f1 f2 (Inl x) = case_sum 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 "case_sum f1 f2 (Inr y) = case_sum g1 g2 (Inr y)" by simp
then show "f2 y = g2 y" by simp
qed
qed

primrec Suml :: "('a  'c)  'a + 'b  'c"
where "Suml f (Inl x) = f x"

primrec Sumr :: "('b  'c)  'a + 'b  '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 :: '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 :: '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  'b set  ('a + 'b) set"  (infixr "<+>" 65)
where "A <+> B = Inl ` A  Inr ` B"

hide_const (open) Plus ― ‹Valuable identifier›

lemma InlI [intro!]: "a  A  Inl a  A <+> B"
by (simp add: Plus_def)

lemma InrI [intro!]: "b  B  Inr b  A <+> B"
by (simp add: Plus_def)

text ‹Exhaustion rule for sums, a degenerate form of induction›

lemma PlusE [elim!]:
"u  A <+> B  (x. x  A  u = Inl x  P)  (y. y  B  u = Inr y  P)  P"
by (auto simp add: Plus_def)

lemma Plus_eq_empty_conv [simp]: "A <+> B = {}  A = {}  B = {}"
by auto

lemma UNIV_Plus_UNIV [simp]:
proof (rule set_eqI)
fix u :: "'a + 'b"
show  by (cases u) auto
qed

lemma UNIV_sum:
proof -
have "x  range Inl" if "x  range Inr" for x :: "'a + 'b"
using that by (cases x) simp_all
then show ?thesis by auto
qed

hide_const (open) Suml Sumr sum

end