src/HOLCF/explicit_domains/Stream.thy

repaired some proofs

(* 
    ID:         $Id$
    Author:     Franz Regensburger
    Copyright   1993 Technische Universitaet Muenchen

Theory for streams without defined empty stream 
  'a stream = 'a ** ('a stream)u

The type is axiomatized as the least solution of the domain equation above.
The functor term that specifies the domain equation is: 

  FT = <**,K_{'a},U>

For details see chapter 5 of:

[Franz Regensburger] HOLCF: Eine konservative Erweiterung von HOL um LCF,
                     Dissertation, Technische Universit"at M"unchen, 1994
*)

Stream = Dnat2 +

types stream 1

(* ----------------------------------------------------------------------- *)
(* arity axiom is validated by semantic reasoning                          *)
(* partial ordering is implicit in the isomorphism axioms and their cont.  *)

arities stream::(pcpo)pcpo

consts

(* ----------------------------------------------------------------------- *)
(* essential constants                                                     *)

stream_rep      :: "('a stream) -> ('a ** ('a stream)u)"
stream_abs      :: "('a ** ('a stream)u) -> ('a stream)"

(* ----------------------------------------------------------------------- *)
(* abstract constants and auxiliary constants                              *)

stream_copy     :: "('a stream -> 'a stream) ->'a stream -> 'a stream"

scons           :: "'a -> 'a stream -> 'a stream"
stream_when     :: "('a -> 'a stream -> 'b) -> 'a stream -> 'b"
is_scons        :: "'a stream -> tr"
shd             :: "'a stream -> 'a"
stl             :: "'a stream -> 'a stream"
stream_take     :: "nat => 'a stream -> 'a stream"
stream_finite   :: "'a stream => bool"
stream_bisim    :: "('a stream => 'a stream => bool) => bool"

rules

(* ----------------------------------------------------------------------- *)
(* axiomatization of recursive type 'a stream                              *)
(* ----------------------------------------------------------------------- *)
(* ('a stream,stream_abs) is the initial F-algebra where                   *)
(* F is the locally continuous functor determined by functor term FT.      *)
(* domain equation: 'a stream = 'a ** ('a stream)u                         *)
(* functor term:    FT = <**,K_{'a},U>                                     *)
(* ----------------------------------------------------------------------- *)
(* stream_abs is an isomorphism with inverse stream_rep                    *)
(* identity is the least endomorphism on 'a stream                         *)

stream_abs_iso  "stream_rep`(stream_abs`x) = x"
stream_rep_iso  "stream_abs`(stream_rep`x) = x"
stream_copy_def "stream_copy == (LAM f. stream_abs oo 
                (ssplit`(LAM x y. (|x , (fup`(up oo f))`y|) )) oo stream_rep)"
stream_reach    "(fix`stream_copy)`x = x"

defs
(* ----------------------------------------------------------------------- *)
(* properties of additional constants                                      *)
(* ----------------------------------------------------------------------- *)
(* constructors                                                            *)

scons_def       "scons == (LAM x l. stream_abs`(| x, up`l |))"

(* ----------------------------------------------------------------------- *)
(* discriminator functional                                                *)

stream_when_def 
"stream_when == (LAM f l.ssplit `(LAM x l.f`x`(fup`ID`l)) `(stream_rep`l))"

(* ----------------------------------------------------------------------- *)
(* discriminators and selectors                                            *)

is_scons_def    "is_scons == stream_when`(LAM x l.TT)"
shd_def         "shd == stream_when`(LAM x l.x)"
stl_def         "stl == stream_when`(LAM x l.l)"

(* ----------------------------------------------------------------------- *)
(* the taker for streams                                                   *)

stream_take_def "stream_take == (%n.iterate n stream_copy UU)"

(* ----------------------------------------------------------------------- *)

stream_finite_def       "stream_finite == (%s.? n.stream_take n `s=s)"

(* ----------------------------------------------------------------------- *)
(* definition of bisimulation is determined by domain equation             *)
(* simplification and rewriting for abstract constants yields def below    *)

stream_bisim_def "stream_bisim ==
(%R.!s1 s2.
        R s1 s2 -->
  ((s1=UU & s2=UU) |
  (? x s11 s21. x~=UU & s1=scons`x`s11 & s2 = scons`x`s21 & R s11 s21)))"

end