Theory Hered

(*  Title:      CCL/Hered.thy
    ID:         $Id: Hered.thy,v 1.5 2005/09/17 15:35:27 wenzelm Exp $
    Author:     Martin Coen
    Copyright   1993  University of Cambridge
*)

header {* Hereditary Termination -- cf. Martin Lo\"f *}

theory Hered
imports Type
uses "coinduction.ML"
begin

text {*
  Note that this is based on an untyped equality and so @{text "lam
  x. b(x)"} is only hereditarily terminating if @{text "ALL x. b(x)"}
  is.  Not so useful for functions!
*}

consts
      (*** Predicates ***)
  HTTgen     ::       "i set => i set"
  HTT        ::       "i set"

axioms
  (*** Definitions of Hereditary Termination ***)

  HTTgen_def:
  "HTTgen(R) == {t. t=true | t=false | (EX a b. t=<a,b>      & a : R & b : R) |
                                      (EX f.  t=lam x. f(x) & (ALL x. f(x) : R))}"
  HTT_def:       "HTT == gfp(HTTgen)"

ML {* use_legacy_bindings (the_context ()) *}

end

theorem lfpI:

  [| mono(f); a : f(lfp(f)) |] ==> a : lfp(f)

theorem ssubst_single:

  [| a = a'; a' : A |] ==> a : A

theorem ssubst_pair:

  [| a = a'; b = b'; <a',b'> : A |] ==> <a,b> : A

theorem PO_refl:

  <a,a> : PO

theorem EQ_refl:

  <a,a> : EQ

theorem HTTgen_mono:

  mono(HTTgen)

theorem HTTgenXH:

  t : HTTgen(A) <->
  t = true ∨
  t = false ∨
  (∃a b. t = <a,b> ∧ a : A ∧ b : A) ∨ (∃f. t = lam x. f(x) ∧ (∀x. f(x) : A))

theorem HTTXH:

  t : HTT <->
  t = true ∨
  t = false ∨
  (∃a b. t = <a,b> ∧ a : HTT ∧ b : HTT) ∨ (∃f. t = lam x. f(x) ∧ (∀x. f(x) : HTT))

theorem HTT_bot:

  ¬ bot : HTT

theorem HTT_true:

  true : HTT

theorem HTT_false:

  false : HTT

theorem HTT_pair:

  <a,b> : HTT <-> a : HTT ∧ b : HTT

theorem HTT_lam:

  lam x. f(x) : HTT <-> (∀x. f(x) : HTT)

theorem HTT_coinduct:

  [| t : R; R <= HTTgen(R) |] ==> t : HTT

theorem HTT_coinduct3:

  [| t : R; R <= HTTgen(lfp(%x. HTTgen(x) Un R Un HTT)) |] ==> t : HTT

theorem UnitF:

  Unit <= HTT

theorem BoolF:

  Bool <= HTT

theorem PlusF:

  [| A <= HTT; B <= HTT |] ==> A + B <= HTT

theorem SigmaF:

  [| A <= HTT; !!x. x : A ==> B(x) <= HTT |] ==> SUM x:A. B(x) <= HTT

theorem NatF:

  Nat <= HTT

theorem ListF:

  A <= HTT ==> List(A) <= HTT

theorem ListsF:

  A <= HTT ==> Lists(A) <= HTT

theorem IListsF:

  A <= HTT ==> ILists(A) <= HTT