# Theory Sum

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theory Sum
imports Bool equalities
begin

    (*  Title:      ZF/sum.thy
        ID:         $Id: Sum.thy,v 1.17 2005/06/17 14:15:09 haftmann Exp $
        Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
        Copyright   1993  University of Cambridge

    *)

    header{*Disjoint Sums*}

    theory Sum imports Bool equalities begin

    text{*And the "Part" primitive for simultaneous recursive type definitions*}

    global

    constdefs
      sum     :: "[i,i]=>i"                     (infixr "+" 65)
         "A+B == {0}*A Un {1}*B"

      Inl     :: "i=>i"
         "Inl(a) == <0,a>"

      Inr     :: "i=>i"
         "Inr(b) == <1,b>"

      "case"  :: "[i=>i, i=>i, i]=>i"
         "case(c,d) == (%<y,z>. cond(y, d(z), c(z)))"

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

    local

    subsection{*Rules for the @{term Part} Primitive*}

    lemma Part_iff:
        "a : Part(A,h) <-> a:A & (EX y. a=h(y))"
    apply (unfold 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"
    apply (unfold 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"
    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, standard]
    lemmas Inr_inject = Inr_iff [THEN iffD1, standard]
    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 <-> (EX x. x:A & u=Inl(x)) | (EX 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: @{term case}*}

    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 ==>
            R(case(c,d,u)) <->
            ((ALL x:A. u = Inl(x) --> R(c(x))) &
            (ALL 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(%x. Inl(c'(x)), %y. Inr(d'(y)), z)) =
            case(%x. c(c'(x)), %y. 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, standard]

    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,%x. x) = A"
    by blast

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

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

    ML
    {*
    val sum_def = thm "sum_def";
    val Inl_def = thm "Inl_def";
    val Inr_def = thm "Inr_def";
    val sum_defs = thms "sum_defs";

    val Part_iff = thm "Part_iff";
    val Part_eqI = thm "Part_eqI";
    val PartI = thm "PartI";
    val PartE = thm "PartE";
    val Part_subset = thm "Part_subset";
    val Sigma_bool = thm "Sigma_bool";
    val InlI = thm "InlI";
    val InrI = thm "InrI";
    val sumE = thm "sumE";
    val Inl_iff = thm "Inl_iff";
    val Inr_iff = thm "Inr_iff";
    val Inl_Inr_iff = thm "Inl_Inr_iff";
    val Inr_Inl_iff = thm "Inr_Inl_iff";
    val sum_empty = thm "sum_empty";
    val Inl_inject = thm "Inl_inject";
    val Inr_inject = thm "Inr_inject";
    val Inl_neq_Inr = thm "Inl_neq_Inr";
    val Inr_neq_Inl = thm "Inr_neq_Inl";
    val InlD = thm "InlD";
    val InrD = thm "InrD";
    val sum_iff = thm "sum_iff";
    val sum_subset_iff = thm "sum_subset_iff";
    val sum_equal_iff = thm "sum_equal_iff";
    val sum_eq_2_times = thm "sum_eq_2_times";
    val case_Inl = thm "case_Inl";
    val case_Inr = thm "case_Inr";
    val case_type = thm "case_type";
    val expand_case = thm "expand_case";
    val case_cong = thm "case_cong";
    val case_case = thm "case_case";
    val Part_mono = thm "Part_mono";
    val Part_Collect = thm "Part_Collect";
    val Part_CollectE = thm "Part_CollectE";
    val Part_Inl = thm "Part_Inl";
    val Part_Inr = thm "Part_Inr";
    val PartD1 = thm "PartD1";
    val Part_id = thm "Part_id";
    val Part_Inr2 = thm "Part_Inr2";
    val Part_sum_equality = thm "Part_sum_equality";

    *}

    end

### Rules for the @{term Part} Primitive

lemma Part_iff:

      a ∈ Part(A, h) <-> a ∈ A ∧ (∃y. a = h(y))

lemma Part_eqI:

      [| a ∈ A; a = h(b) |] ==> a ∈ Part(A, h)

lemmas PartI:

      h(b) ∈ A ==> h(b) ∈ Part(A, h)

lemmas PartI:

      h(b) ∈ A ==> h(b) ∈ Part(A, h)

lemma PartE:

      [| a ∈ Part(A, h); !!z. [| a ∈ A; a = h(z) |] ==> P |] ==> P

lemma Part_subset:

      Part(A, h) ⊆ A

### Rules for Disjoint Sums

lemmas sum_defs:

      A + B == {0} × A ∪ {1} × B

      Inl(a) == ⟨0, a⟩

      Inr(b) == ⟨1, b⟩

      case(c, d) == %⟨y,z⟩. cond(y, d(z), c(z))

lemmas sum_defs:

      A + B == {0} × A ∪ {1} × B

      Inl(a) == ⟨0, a⟩

      Inr(b) == ⟨1, b⟩

      case(c, d) == %⟨y,z⟩. cond(y, d(z), c(z))

lemma Sigma_bool:

      Sigma(bool, C) = C(0) + C(1)

lemma InlI:

      a ∈ A ==> Inl(a) ∈ A + B

lemma InrI:

      b ∈ B ==> Inr(b) ∈ A + B

lemma sumE:

      [| u ∈ A + B; !!x. [| x ∈ A; u = Inl(x) |] ==> P;
         !!y. [| y ∈ B; u = Inr(y) |] ==> P |]
      ==> P

lemma Inl_iff:

      Inl(a) = Inl(b) <-> a = b

lemma Inr_iff:

      Inr(a) = Inr(b) <-> a = b

lemma Inl_Inr_iff:

      Inl(a) = Inr(b) <-> False

lemma Inr_Inl_iff:

      Inr(b) = Inl(a) <-> False

lemma sum_empty:

      0 + 0 = 0

lemmas Inl_inject:

      Inl(a) = Inl(b) ==> a = b

lemmas Inl_inject:

      Inl(a) = Inl(b) ==> a = b

lemmas Inr_inject:

      Inr(a) = Inr(b) ==> a = b

lemmas Inr_inject:

      Inr(a) = Inr(b) ==> a = b

lemmas Inl_neq_Inr:

      Inl(a2) = Inr(b2) ==> P

lemmas Inl_neq_Inr:

      Inl(a2) = Inr(b2) ==> P

lemmas Inr_neq_Inl:

      Inr(b2) = Inl(a2) ==> P

lemmas Inr_neq_Inl:

      Inr(b2) = Inl(a2) ==> P

lemma InlD:

      Inl(a) ∈ A + B ==> a ∈ A

lemma InrD:

      Inr(b) ∈ A + B ==> b ∈ B

lemma sum_iff:

      u ∈ A + B <-> (∃x. x ∈ A ∧ u = Inl(x)) ∨ (∃y. y ∈ B ∧ u = Inr(y))

lemma Inl_in_sum_iff:

      Inl(x) ∈ A + B <-> x ∈ A

lemma Inr_in_sum_iff:

      Inr(y) ∈ A + B <-> y ∈ B

lemma sum_subset_iff:

      A + B ⊆ C + D <-> A ⊆ C ∧ B ⊆ D

lemma sum_equal_iff:

      A + B = C + D <-> A = C ∧ B = D

lemma sum_eq_2_times:

      A + A = 2 × A

### The Eliminator: @{term case}

lemma case_Inl:

      case(c, d, Inl(a)) = c(a)

lemma case_Inr:

      case(c, d, Inr(b)) = d(b)

lemma case_type:

      [| 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)

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)))

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)

lemma case_case:

      z ∈ 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)

### More Rules for @{term "Part(A,h)"}

lemma Part_mono:

      A ⊆ B ==> Part(A, h) ⊆ Part(B, h)

lemma Part_Collect:

      Part(Collect(A, P), h) = Collect(Part(A, h), P)

lemmas Part_CollectE:

      [| a ∈ Part({x ∈ A . P(x)}, h); [| a ∈ Part(A, h); P(a) |] ==> R |] ==> R

lemmas Part_CollectE:

      [| a ∈ Part({x ∈ A . P(x)}, h); [| a ∈ Part(A, h); P(a) |] ==> R |] ==> R

lemma Part_Inl:

      Part(A + B, Inl) = {Inl(x) . x ∈ A}

lemma Part_Inr:

      Part(A + B, Inr) = {Inr(y) . y ∈ B}

lemma PartD1:

      a ∈ Part(A, h) ==> a ∈ A

lemma Part_id:

      Part(A, %x. x) = A

lemma Part_Inr2:

      Part(A + B, %x. Inr(h(x))) = {Inr(y) . y ∈ Part(B, h)}

lemma Part_sum_equality:

      C ⊆ A + B ==> Part(C, Inl) ∪ Part(C, Inr) = C