(* Title: ZF/Sum.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
*)
section\<open>Disjoint Sums\<close>
theory Sum imports Bool equalities begin
text\<open>And the "Part" primitive for simultaneous recursive type definitions\<close>
definition sum :: "[i,i]=>i" (infixr \<open>+\<close> 65) where
"A+B == {0}*A \<union> {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) == (%<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 \<in> A. \<exists>z. x = h(z)}"
subsection\<open>Rules for the \<^term>\<open>Part\<close> Primitive\<close>
lemma Part_iff:
"a \<in> Part(A,h) \<longleftrightarrow> a \<in> A & (\<exists>y. a=h(y))"
apply (unfold Part_def)
apply (rule separation)
done
lemma Part_eqI [intro]:
"[| a \<in> A; a=h(b) |] ==> a \<in> Part(A,h)"
by (unfold Part_def, blast)
lemmas PartI = refl [THEN [2] Part_eqI]
lemma PartE [elim!]:
"[| a \<in> Part(A,h); !!z. [| a \<in> A; a=h(z) |] ==> P
|] ==> P"
apply (unfold Part_def, blast)
done
lemma Part_subset: "Part(A,h) \<subseteq> A"
apply (unfold Part_def)
apply (rule Collect_subset)
done
subsection\<open>Rules for Disjoint Sums\<close>
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 \<in> A ==> Inl(a) \<in> A+B"
by (unfold sum_defs, blast)
lemma InrI [intro!,simp,TC]: "b \<in> B ==> Inr(b) \<in> A+B"
by (unfold sum_defs, blast)
(** Elimination rules **)
lemma sumE [elim!]:
"[| u \<in> A+B;
!!x. [| x \<in> A; u=Inl(x) |] ==> P;
!!y. [| y \<in> 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) \<longleftrightarrow> a=b"
by (simp add: sum_defs)
lemma Inr_iff [iff]: "Inr(a)=Inr(b) \<longleftrightarrow> a=b"
by (simp add: sum_defs)
lemma Inl_Inr_iff [simp]: "Inl(a)=Inr(b) \<longleftrightarrow> False"
by (simp add: sum_defs)
lemma Inr_Inl_iff [simp]: "Inr(b)=Inl(a) \<longleftrightarrow> False"
by (simp add: sum_defs)
lemma sum_empty [simp]: "0+0 = 0"
by (simp add: sum_defs)
(*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 \<in> A"
by blast
lemma InrD: "Inr(b): A+B ==> b \<in> B"
by blast
lemma sum_iff: "u \<in> A+B \<longleftrightarrow> (\<exists>x. x \<in> A & u=Inl(x)) | (\<exists>y. y \<in> B & u=Inr(y))"
by blast
lemma Inl_in_sum_iff [simp]: "(Inl(x) \<in> A+B) \<longleftrightarrow> (x \<in> A)"
by auto
lemma Inr_in_sum_iff [simp]: "(Inr(y) \<in> A+B) \<longleftrightarrow> (y \<in> B)"
by auto
lemma sum_subset_iff: "A+B \<subseteq> C+D \<longleftrightarrow> A<=C & B<=D"
by blast
lemma sum_equal_iff: "A+B = C+D \<longleftrightarrow> A=C & B=D"
by (simp add: extension sum_subset_iff, blast)
lemma sum_eq_2_times: "A+A = 2*A"
by (simp add: sum_def, blast)
subsection\<open>The Eliminator: \<^term>\<open>case\<close>\<close>
lemma case_Inl [simp]: "case(c, d, Inl(a)) = c(a)"
by (simp add: sum_defs)
lemma case_Inr [simp]: "case(c, d, Inr(b)) = d(b)"
by (simp add: sum_defs)
lemma case_type [TC]:
"[| u \<in> A+B;
!!x. x \<in> A ==> c(x): C(Inl(x));
!!y. y \<in> B ==> d(y): C(Inr(y))
|] ==> case(c,d,u) \<in> C(u)"
by auto
lemma expand_case: "u \<in> A+B ==>
R(case(c,d,u)) \<longleftrightarrow>
((\<forall>x\<in>A. u = Inl(x) \<longrightarrow> R(c(x))) &
(\<forall>y\<in>B. u = Inr(y) \<longrightarrow> R(d(y))))"
by auto
lemma case_cong:
"[| z \<in> A+B;
!!x. x \<in> A ==> c(x)=c'(x);
!!y. y \<in> B ==> d(y)=d'(y)
|] ==> case(c,d,z) = case(c',d',z)"
by auto
lemma case_case: "z \<in> A+B ==>
case(c, d, case(%x. Inl(c'(x)), %y. Inr(d'(y)), z)) =
case(%x. c(c'(x)), %y. d(d'(y)), z)"
by auto
subsection\<open>More Rules for \<^term>\<open>Part(A,h)\<close>\<close>
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 \<in> A}"
by blast
lemma Part_Inr: "Part(A+B,Inr) = {Inr(y). y \<in> B}"
by blast
lemma PartD1: "a \<in> Part(A,h) ==> a \<in> A"
by (simp add: Part_def)
lemma Part_id: "Part(A,%x. x) = A"
by blast
lemma Part_Inr2: "Part(A+B, %x. Inr(h(x))) = {Inr(y). y \<in> Part(B,h)}"
by blast
lemma Part_sum_equality: "C \<subseteq> A+B ==> Part(C,Inl) \<union> Part(C,Inr) = C"
by blast
end