Theory Buffer

(*  Title:      HOLCF/FOCUS/Buffer.thy
    ID:         $Id: Buffer.thy,v 1.7 2005/09/26 06:41:24 haftmann Exp $
    Author:     David von Oheimb, TU Muenchen

Formalization of section 4 of

@inproceedings {broy_mod94,
    author = {Manfred Broy},
    title = {{Specification and Refinement of a Buffer of Length One}},
    booktitle = {Deductive Program Design},
    year = {1994},
    editor = {Manfred Broy},
    volume = {152},
    series = {ASI Series, Series F: Computer and System Sciences},
    pages = {273 -- 304},
    publisher = {Springer}
}

Slides available from http://ddvo.net/talks/1-Buffer.ps.gz

*)

theory Buffer
imports FOCUS
begin

typedecl D

datatype

  M     = Md D | Mreq ("•")

datatype

  State = Sd D | Snil ("¤")

types

  SPF11         = "M fstream -> D fstream"
  SPEC11        = "SPF11 set"
  SPSF11        = "State => SPF11"
  SPECS11       = "SPSF11 set"

consts

  BufEq_F       :: "SPEC11 => SPEC11"
  BufEq         :: "SPEC11"
  BufEq_alt     :: "SPEC11"

  BufAC_Asm_F   :: " (M fstream set) => (M fstream set)"
  BufAC_Asm     :: " (M fstream set)"
  BufAC_Cmt_F   :: "((M fstream * D fstream) set) =>
                    ((M fstream * D fstream) set)"
  BufAC_Cmt     :: "((M fstream * D fstream) set)"
  BufAC         :: "SPEC11"

  BufSt_F       :: "SPECS11 => SPECS11"
  BufSt_P       :: "SPECS11"
  BufSt         :: "SPEC11"

defs

  BufEq_F_def:   "BufEq_F B ≡ {f. ∀d. f·(Md d\<leadsto><>) = <> ∧
                                 (∀x. ∃ff∈B. f·(Md d\<leadsto>•\<leadsto>x) = d\<leadsto>ff·x)}"
  BufEq_def:     "BufEq     ≡ gfp BufEq_F"
  BufEq_alt_def: "BufEq_alt ≡ gfp (λB. {f. ∀d. f·(Md d\<leadsto><> ) = <> ∧
                                 (∃ff∈B. (∀x. f·(Md d\<leadsto>•\<leadsto>x) = d\<leadsto>ff·x))})"
  BufAC_Asm_F_def: "BufAC_Asm_F A ≡ {s. s = <> ∨
                  (∃d x. s = Md d\<leadsto>x ∧ (x = <> ∨ (ft·x = Def • ∧ (rt·x)∈A)))}"
  BufAC_Asm_def: "BufAC_Asm ≡ gfp BufAC_Asm_F"

  BufAC_Cmt_F_def: "BufAC_Cmt_F C ≡ {(s,t). ∀d x.
                           (s = <>         -->     t = <>                 ) ∧
                           (s = Md d\<leadsto><>   -->     t = <>                 ) ∧
                           (s = Md d\<leadsto>•\<leadsto>x --> (ft·t = Def d ∧ (x,rt·t)∈C))}"
  BufAC_Cmt_def: "BufAC_Cmt ≡ gfp BufAC_Cmt_F"
  BufAC_def:     "BufAC ≡ {f. ∀x. x∈BufAC_Asm --> (x,f·x)∈BufAC_Cmt}"

  BufSt_F_def:   "BufSt_F H ≡ {h. ∀s  . h s      ·<>        = <>         ∧
                                 (∀d x. h ¤     ·(Md d\<leadsto>x) = h (Sd d)·x ∧
                                (∃hh∈H. h (Sd d)·(•   \<leadsto>x) = d\<leadsto>(hh ¤·x)))}"
  BufSt_P_def:   "BufSt_P ≡ gfp BufSt_F"
  BufSt_def:     "BufSt ≡ {f. ∃h∈BufSt_P. f = h ¤}"

ML {* use_legacy_bindings (the_context ()) *}

end