# Theory Focus_ex

Up to index of Isabelle/HOLCF/ex

theory Focus_ex
imports Stream
uses [Focus_ex.ML]
begin

    (* $Id: Focus_ex.thy,v 1.4 2005/09/06 17:28:59 wenzelm Exp $ *)

    (* Specification of the following loop back device

              g
               --------------------
              |      -------       |
           x  |     |       |      |  y
        ------|---->|       |------| ----->
              |  z  |   f   | z    |
              |  -->|       |---   |
              | |   |       |   |  |
              | |    -------    |  |
              | |               |  |
              |  <--------------   |
              |                    |
               --------------------

    First step: Notation in Agent Network Description Language (ANDL)
    -----------------------------------------------------------------

    agent f
            input  channel i1:'b i2: ('b,'c) tc
            output channel o1:'c o2: ('b,'c) tc
    is
            Rf(i1,i2,o1,o2)  (left open in the example)
    end f

    agent g
            input  channel x:'b
            output channel y:'c
    is network
            <y,z> = f$<x,z>
    end network
    end g

    Remark: the type of the feedback depends at most on the types of the input and
            output of g. (No type miracles inside g)

    Second step: Translation of ANDL specification to HOLCF Specification
    ---------------------------------------------------------------------

    Specification of agent f ist translated to predicate is_f

    is_f :: ('b stream * ('b,'c) tc stream ->
                    'c stream * ('b,'c) tc stream) => bool

    is_f f  = !i1 i2 o1 o2.
            f$<i1,i2> = <o1,o2> --> Rf(i1,i2,o1,o2)

    Specification of agent g is translated to predicate is_g which uses
    predicate is_net_g

    is_net_g :: ('b stream * ('b,'c) tc stream -> 'c stream * ('b,'c) tc stream) =>
                'b stream => 'c stream => bool

    is_net_g f x y =
            ? z. <y,z> = f$<x,z> &
            !oy hz. <oy,hz> = f$<x,hz> --> z << hz

    is_g :: ('b stream -> 'c stream) => bool

    is_g g  = ? f. is_f f  & (!x y. g$x = y --> is_net_g f x y

    Third step: (show conservativity)
    -----------

    Suppose we have a model for the theory TH1 which contains the axiom

            ? f. is_f f

    In this case there is also a model for the theory TH2 that enriches TH1 by
    axiom

            ? g. is_g g

    The result is proved by showing that there is a definitional extension
    that extends TH1 by a definition of g.

    We define:

    def_g g  =
             (? f. is_f f  &
                  g = (LAM x. cfst$(f$<x,fix$(LAM k.csnd$(f$<x,k>))>)) )

    Now we prove:

            (? f. is_f f ) --> (? g. is_g g)

    using the theorems

    loopback_eq)    def_g = is_g                    (real work)

    L1)             (? f. is_f f ) --> (? g. def_g g)  (trivial)

    *)

    theory Focus_ex
    imports Stream
    begin

    typedecl ('a, 'b) tc
    arities tc:: (pcpo, pcpo) pcpo

    consts

    is_f     ::
     "('b stream * ('b,'c) tc stream -> 'c stream * ('b,'c) tc stream) => bool"
    is_net_g :: "('b stream *('b,'c) tc stream -> 'c stream * ('b,'c) tc stream) =>
                'b stream => 'c stream => bool"
    is_g     :: "('b stream -> 'c stream) => bool"
    def_g    :: "('b stream -> 'c stream) => bool"
    Rf       ::
    "('b stream * ('b,'c) tc stream * 'c stream * ('b,'c) tc stream) => bool"

    defs

    is_f:           "is_f f == (!i1 i2 o1 o2.
                            f$<i1,i2> = <o1,o2> --> Rf(i1,i2,o1,o2))"

    is_net_g:       "is_net_g f x y == (? z.
                            <y,z> = f$<x,z> &
                            (!oy hz. <oy,hz> = f$<x,hz> --> z << hz))"

    is_g:           "is_g g  == (? f.
                            is_f f  &
                            (!x y. g$x = y --> is_net_g f x y))"

    def_g:          "def_g g == (? f.
                            is_f f  &
                            g = (LAM x. cfst$(f$<x,fix$(LAM  k. csnd$(f$<x,k>))>)))"

    ML {* use_legacy_bindings (the_context ()) *}

    end