Theory Sum

(*  Title:      ZF/Sum.thy
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1993  University of Cambridge

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 :: "ii" where
     "Inl(a)  0,a"

definition Inr :: "ii" where
     "Inr(b)  1,b"

definition "case" :: "[ii, ii, i]i" where
     "case(c,d)  (λy,z. cond(y, d(z), c(z)))"

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

subsection‹Rules for the termPart Primitive›

lemma Part_iff:
    "a  Part(A,h)  a  A  (y. a=h(y))"
  unfolding Part_def
apply (rule separation)

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
apply (unfold Part_def, blast)

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

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
by (unfold sum_defs, blast)

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

lemma Inl_iff [iff]: "Inl(a)=Inl(b)  a=b"
by (simp add: sum_defs)

lemma Inr_iff [iff]: "Inr(a)=Inr(b)  a=b"
by (simp add: sum_defs)

lemma Inl_Inr_iff [simp]: "Inl(a)=Inr(b)  False"
by (simp add: sum_defs)

lemma Inr_Inl_iff [simp]: "Inr(b)=Inl(a)  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  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"
by (simp add: sum_def, blast)

subsection‹The Eliminator: termcase

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  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 
        ((xA. u = Inl(x)  R(c(x))) 
        (yB. 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 termPart(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"
by (simp add: Part_def)

lemma Part_id: "Part(A,λx. x) = A"
by blast

lemma Part_Inr2: "Part(A+B, λx. Inr(h(x))) = {Inr(y). y  Part(B,h)}"
by blast

lemma Part_sum_equality: "C  A+B  Part(C,Inl)  Part(C,Inr) = C"
by blast