# Theory Sum

```(*  Title:      ZF/Sum.thy
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
*)

section‹Disjoint Sums›

theory Sum imports Bool equalities begin

text‹And the "Part" primitive for simultaneous recursive type definitions›

definition sum :: "[i,i]⇒i" (infixr ‹+› 65) where
"A+B ≡ {0}*A ∪ {1}*B"

definition Inl :: "i⇒i" where
"Inl(a) ≡ ⟨0,a⟩"

definition Inr :: "i⇒i" where
"Inr(b) ≡ ⟨1,b⟩"

definition "case" :: "[i⇒i, i⇒i, i]⇒i" where
"case(c,d) ≡ (λ⟨y,z⟩. cond(y, d(z), c(z)))"

(*operator for selecting out the various summands*)
definition Part :: "[i,i⇒i] ⇒ i" where
"Part(A,h) ≡ {x ∈ A. ∃z. x = h(z)}"

subsection‹Rules for the \<^term>‹Part› Primitive›

lemma Part_iff:
"a ∈ Part(A,h) ⟷ a ∈ A ∧ (∃y. a=h(y))"
unfolding Part_def
apply (rule separation)
done

lemma Part_eqI [intro]:
"⟦a ∈ A;  a=h(b)⟧ ⟹ a ∈ Part(A,h)"
by (unfold Part_def, blast)

lemmas PartI = refl [THEN [2] Part_eqI]

lemma PartE [elim!]:
"⟦a ∈ Part(A,h);  ⋀z. ⟦a ∈ A;  a=h(z)⟧ ⟹ P
⟧ ⟹ P"
apply (unfold Part_def, blast)
done

lemma Part_subset: "Part(A,h) ⊆ A"
unfolding Part_def
apply (rule Collect_subset)
done

subsection‹Rules for Disjoint Sums›

lemmas sum_defs = sum_def Inl_def Inr_def case_def

lemma Sigma_bool: "Sigma(bool,C) = C(0) + C(1)"
by (unfold bool_def sum_def, blast)

(** Introduction rules for the injections **)

lemma InlI [intro!,simp,TC]: "a ∈ A ⟹ Inl(a) ∈ A+B"
by (unfold sum_defs, blast)

lemma InrI [intro!,simp,TC]: "b ∈ B ⟹ Inr(b) ∈ A+B"
by (unfold sum_defs, blast)

(** Elimination rules **)

lemma sumE [elim!]:
"⟦u ∈ A+B;
⋀x. ⟦x ∈ A;  u=Inl(x)⟧ ⟹ P;
⋀y. ⟦y ∈ B;  u=Inr(y)⟧ ⟹ P
⟧ ⟹ P"
by (unfold sum_defs, blast)

(** Injection and freeness equivalences, for rewriting **)

lemma Inl_iff [iff]: "Inl(a)=Inl(b) ⟷ a=b"

lemma Inr_iff [iff]: "Inr(a)=Inr(b) ⟷ a=b"

lemma Inl_Inr_iff [simp]: "Inl(a)=Inr(b) ⟷ False"

lemma Inr_Inl_iff [simp]: "Inr(b)=Inl(a) ⟷ False"

lemma sum_empty [simp]: "0+0 = 0"

(*Injection and freeness rules*)

lemmas Inl_inject = Inl_iff [THEN iffD1]
lemmas Inr_inject = Inr_iff [THEN iffD1]
lemmas Inl_neq_Inr = Inl_Inr_iff [THEN iffD1, THEN FalseE, elim!]
lemmas Inr_neq_Inl = Inr_Inl_iff [THEN iffD1, THEN FalseE, elim!]

lemma InlD: "Inl(a): A+B ⟹ a ∈ A"
by blast

lemma InrD: "Inr(b): A+B ⟹ b ∈ B"
by blast

lemma sum_iff: "u ∈ A+B ⟷ (∃x. x ∈ A ∧ u=Inl(x)) | (∃y. y ∈ B ∧ u=Inr(y))"
by blast

lemma Inl_in_sum_iff [simp]: "(Inl(x) ∈ A+B) ⟷ (x ∈ A)"
by auto

lemma Inr_in_sum_iff [simp]: "(Inr(y) ∈ A+B) ⟷ (y ∈ B)"
by auto

lemma sum_subset_iff: "A+B ⊆ C+D ⟷ A<=C ∧ B<=D"
by blast

lemma sum_equal_iff: "A+B = C+D ⟷ A=C ∧ B=D"
by (simp add: extension sum_subset_iff, blast)

lemma sum_eq_2_times: "A+A = 2*A"

subsection‹The Eliminator: \<^term>‹case››

lemma case_Inl [simp]: "case(c, d, Inl(a)) = c(a)"

lemma case_Inr [simp]: "case(c, d, Inr(b)) = d(b)"

lemma case_type [TC]:
"⟦u ∈ A+B;
⋀x. x ∈ A ⟹ c(x): C(Inl(x));
⋀y. y ∈ B ⟹ d(y): C(Inr(y))
⟧ ⟹ case(c,d,u) ∈ C(u)"
by auto

lemma expand_case: "u ∈ A+B ⟹
R(case(c,d,u)) ⟷
((∀x∈A. u = Inl(x) ⟶ R(c(x))) ∧
(∀y∈B. u = Inr(y) ⟶ R(d(y))))"
by auto

lemma case_cong:
"⟦z ∈ A+B;
⋀x. x ∈ A ⟹ c(x)=c'(x);
⋀y. y ∈ B ⟹ d(y)=d'(y)
⟧ ⟹ case(c,d,z) = case(c',d',z)"
by auto

lemma case_case: "z ∈ A+B ⟹
case(c, d, case(λx. Inl(c'(x)), λy. Inr(d'(y)), z)) =
case(λx. c(c'(x)), λy. d(d'(y)), z)"
by auto

subsection‹More Rules for \<^term>‹Part(A,h)››

lemma Part_mono: "A<=B ⟹ Part(A,h)<=Part(B,h)"
by blast

lemma Part_Collect: "Part(Collect(A,P), h) = Collect(Part(A,h), P)"
by blast

lemmas Part_CollectE =
Part_Collect [THEN equalityD1, THEN subsetD, THEN CollectE]

lemma Part_Inl: "Part(A+B,Inl) = {Inl(x). x ∈ A}"
by blast

lemma Part_Inr: "Part(A+B,Inr) = {Inr(y). y ∈ B}"
by blast

lemma PartD1: "a ∈ Part(A,h) ⟹ a ∈ A"