Theory Stream_Processor

theory Stream_Processor
imports Stream BNF_Corec
(*  Title:      HOL/Datatype_Examples/Stream_Processor.thy
Stream processors---a syntactic representation of continuous functions on streams.
*)

section {* Stream Processors---A Syntactic Representation of Continuous Functions on Streams *}

theory Stream_Processor
imports "~~/src/HOL/Library/Stream" "../../BNF_Corec"
begin

datatype (discs_sels) ('a, 'b, 'c) spμ = Get "'a ⇒ ('a, 'b, 'c) spμ" | Put "'b" "'c"
codatatype ('a, 'b) spν = In (out: "('a, 'b, ('a, 'b) spν) spμ")

declare spμ.map_transfer[transfer_rule]

definition "sub ≡ {(f a, Get f) | a f. True}"

lemma subI[intro]: "(f a, Get f) ∈ sub"
  unfolding sub_def by blast

lemma wf_sub[simp, intro]: "wf sub"
proof (rule wfUNIVI)
  fix P  :: "('a, 'b, 'c) spμ ⇒ bool" and x
  assume "∀x. (∀y. (y, x) ∈ sub ⟶ P y) ⟶ P x"
  hence I: "⋀x. (∀y. (∃a f. y = f a ∧ x = Get f) ⟶ P y) ⟹ P x" unfolding sub_def by blast
  show "P x" by (induct x) (auto intro: I)
qed

corecursive run :: "('a, 'b) spν ⇒ 'a stream ⇒ 'b stream" where
  "run sp s = (case out sp of
     Get f ⇒ run (In (f (shd s))) (stl s)
   | Put b sp ⇒ b ## run sp s)"
  by (relation "map_prod In In ` sub <*lex*> {}")
    (auto simp: inj_on_def out_def split: spν.splits intro: wf_map_prod_image)

corec copy where
  "copy = In (Get (λa. Put a copy))"

corec (friend) get where
  "get f = In (Get (λa. out (f a)))"

lemma run_simps [simp]:
  "run (In (Get f)) s = run (In (f (shd s))) (stl s)"
  "run (In (Put b sp)) s = b ## run sp s"
by(subst run.code; simp; fail)+

lemma copy_sel[simp]: "out copy = Get (λa. Put a copy)"
  by (subst copy.code; simp)

corecursive sp_comp (infixl "oo" 65) where
  "sp oo sp' = (case (out sp, out sp') of
      (Put b sp, _) ⇒ In (Put b (sp oo sp'))
    | (Get f, Put b sp) ⇒ In (f b) oo sp
    | (_, Get g) ⇒ get (λa. (sp oo In (g a))))"
  by (relation "map_prod In In ` sub <*lex*> map_prod In In ` sub")
    (auto simp: inj_on_def out_def split: spν.splits intro: wf_map_prod_image)

lemma run_copy_unique: 
    "(⋀s. h s = shd s ## h (stl s)) ⟹ h = run copy"
apply corec_unique
apply(rule ext)
apply(subst copy.code)
apply simp
done

export_code run copy sp_comp in Haskell module_name Stream_Processor file code

end