Theory Sum

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theory Sum
imports Bool equalities
begin

(*  Title:      ZF/sum.thy
    ID:         $Id: Sum.thy,v 1.18 2007/10/07 19:19:32 wenzelm 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

end

Rules for the @{term Part} Primitive

lemma Part_iff:

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

lemma Part_eqI:

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

lemma PartI:

  h(b) ∈ A ==> h(b) ∈ Part(A, h)

lemma PartE:

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

lemma Part_subset:

  Part(A, h) ⊆ A

Rules for Disjoint Sums

lemma sum_defs:

  A + B == {0} × A ∪ {1} × B
  Inl(a) == ⟨0, a
  Inr(b) == ⟨1, b
  case(c, d) == λ⟨y,z⟩. cond(y, d(z), c(z))

lemma Sigma_bool:

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

lemma InlI:

  aA ==> Inl(a) ∈ A + B

lemma InrI:

  bB ==> Inr(b) ∈ A + B

lemma sumE:

  [| uA + B; !!x. [| xA; u = Inl(x) |] ==> P;
     !!y. [| yB; 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

lemma Inl_inject:

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

lemma Inr_inject:

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

lemma Inl_neq_Inr:

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

lemma Inr_neq_Inl:

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

lemma InlD:

  Inl(a) ∈ A + B ==> aA

lemma InrD:

  Inr(b) ∈ A + B ==> bB

lemma sum_iff:

  uA + B <-> (∃x. xAu = Inl(x)) ∨ (∃y. yBu = Inr(y))

lemma Inl_in_sum_iff:

  Inl(x) ∈ A + B <-> xA

lemma Inr_in_sum_iff:

  Inr(y) ∈ A + B <-> yB

lemma sum_subset_iff:

  A + BC + D <-> ACBD

lemma sum_equal_iff:

  A + B = C + D <-> A = CB = 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:

  [| uA + B; !!x. xA ==> c(x) ∈ C(Inl(x));
     !!y. yB ==> d(y) ∈ C(Inr(y)) |]
  ==> case(c, d, u) ∈ C(u)

lemma expand_case:

  uA + B
  ==> R(case(c, d, u)) <->
      (∀xA. u = Inl(x) --> R(c(x))) ∧ (∀yB. u = Inr(y) --> R(d(y)))

lemma case_cong:

  [| zA + B; !!x. xA ==> c(x) = c'(x); !!y. yB ==> d(y) = d'(y) |]
  ==> case(c, d, z) = case(c', d', z)

lemma case_case:

  zA + B
  ==> case(c, d, case(λx. Inl(c'(x)), λy. Inr(d'(y)), z)) =
      case(λx. c(c'(x)), λy. d(d'(y)), z)

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

lemma Part_mono:

  AB ==> Part(A, h) ⊆ Part(B, h)

lemma Part_Collect:

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

lemma Part_CollectE:

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

lemma Part_Inl:

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

lemma Part_Inr:

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

lemma PartD1:

  a ∈ Part(A, h) ==> aA

lemma Part_id:

  Part(A, λx. x) = A

lemma Part_Inr2:

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

lemma Part_sum_equality:

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