Theory Type (Isabelle2005: October 2005)

(*  Title:      CCL/Type.thy
    ID:         $Id: Type.thy,v 1.7 2005/09/17 15:35:29 wenzelm Exp $
    Author:     Martin Coen
    Copyright   1993  University of Cambridge
*)

header {* Types in CCL are defined as sets of terms *}

theory Type
imports Term
begin

consts

  Subtype       :: "['a set, 'a => o] => 'a set"
  Bool          :: "i set"
  Unit          :: "i set"
  "+"           :: "[i set, i set] => i set"         (infixr 55)
  Pi            :: "[i set, i => i set] => i set"
  Sigma         :: "[i set, i => i set] => i set"
  Nat           :: "i set"
  List          :: "i set => i set"
  Lists         :: "i set => i set"
  ILists        :: "i set => i set"
  TAll          :: "(i set => i set) => i set"       (binder "TALL " 55)
  TEx           :: "(i set => i set) => i set"       (binder "TEX " 55)
  Lift          :: "i set => i set"                  ("(3[_])")

  SPLIT         :: "[i, [i, i] => i set] => i set"

syntax
  "@Pi"         :: "[idt, i set, i set] => i set"    ("(3PROD _:_./ _)"
                                [0,0,60] 60)

  "@Sigma"      :: "[idt, i set, i set] => i set"    ("(3SUM _:_./ _)"
                                [0,0,60] 60)

  "@->"         :: "[i set, i set] => i set"         ("(_ ->/ _)"  [54, 53] 53)
  "@*"          :: "[i set, i set] => i set"         ("(_ */ _)" [56, 55] 55)
  "@Subtype"    :: "[idt, 'a set, o] => 'a set"      ("(1{_: _ ./ _})")

translations
  "PROD x:A. B" => "Pi(A, %x. B)"
  "A -> B"      => "Pi(A, _K(B))"
  "SUM x:A. B"  => "Sigma(A, %x. B)"
  "A * B"       => "Sigma(A, _K(B))"
  "{x: A. B}"   == "Subtype(A, %x. B)"

print_translation {*
  [("Pi", dependent_tr' ("@Pi", "@->")),
   ("Sigma", dependent_tr' ("@Sigma", "@*"))] *}

axioms
  Subtype_def: "{x:A. P(x)} == {x. x:A & P(x)}"
  Unit_def:          "Unit == {x. x=one}"
  Bool_def:          "Bool == {x. x=true | x=false}"
  Plus_def:           "A+B == {x. (EX a:A. x=inl(a)) | (EX b:B. x=inr(b))}"
  Pi_def:         "Pi(A,B) == {x. EX b. x=lam x. b(x) & (ALL x:A. b(x):B(x))}"
  Sigma_def:   "Sigma(A,B) == {x. EX a:A. EX b:B(a).x=<a,b>}"
  Nat_def:            "Nat == lfp(% X. Unit + X)"
  List_def:       "List(A) == lfp(% X. Unit + A*X)"

  Lists_def:     "Lists(A) == gfp(% X. Unit + A*X)"
  ILists_def:   "ILists(A) == gfp(% X.{} + A*X)"

  Tall_def:   "TALL X. B(X) == Inter({X. EX Y. X=B(Y)})"
  Tex_def:     "TEX X. B(X) == Union({X. EX Y. X=B(Y)})"
  Lift_def:           "[A] == A Un {bot}"

  SPLIT_def:   "SPLIT(p,B) == Union({A. EX x y. p=<x,y> & A=B(x,y)})"

ML {* use_legacy_bindings (the_context ()) *}

end

theorem subsetXH:

  A <= B <-> (∀x. x : A --> x : B)

theorem SubtypeI:

  [| a : A; P(a) |] ==> a : Subtype(A, P)

theorem SubtypeE:

  [| a : Subtype(A, P); [| a : A; P(a) |] ==> Q |] ==> Q

theorem idM:

  mono(%X. X)

theorem constM:

  mono(%X. A)

theorem LiftM:

  mono(A) ==> mono(%X. [A(X)])

theorem SgM:

  [| mono(A); !!x X. x : A(X) ==> mono(%X. B(X, x)) |]
  ==> mono(%X. Sigma(A(X), B(X)))

theorem PiM:

  (!!x. x : A ==> mono(%X. B(X, x))) ==> mono(%X. Pi(A, B(X)))

theorem PlusM:

  [| mono(A); mono(B) |] ==> mono(%X. A(X) + B(X))

theorem NatM:

  mono(op +(Unit))

theorem def_NatB:

  Nat = Unit + Nat

theorem ListM:

  mono(%X. Unit + A * X)

theorem def_ListB:

  List(A) = Unit + A * List(A)

theorem def_ListsB:

  Lists(A) = Unit + A * Lists(A)

theorem IListsM:

  mono(%X. {} + A * X)

theorem def_IListsB:

  ILists(A) = {} + A * ILists(A)

theorem SgE2:

  [| <a,b> : Sigma(A, B); [| a : A; b : B(a) |] ==> P |] ==> P

theorem lfp_subset_gfp:

  mono(f) ==> lfp(f) <= gfp(f)

theorem gfpI:

  [| a : A; !!x X. [| x : A; ALL y:A. t(y) : X |] ==> t(x) : B(X) |]
  ==> t(a) : gfp(B)

theorem def_gfpI:

  [| C == gfp(B); a : A; !!x X. [| x : A; ALL y:A. t(y) : X |] ==> t(x) : B(X) |]
  ==> t(a) : C