Theory Focus_ex

(* 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. fst (f$(x,fix$(LAM k. snd (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 "HOLCF-Library.Stream"
begin

typedecl ('a, 'b) tc
axiomatization where tc_arity: "OFCLASS(('a::pcpo, 'b::pcpo) tc, pcop_class)"
instance tc :: (pcpo, pcpo) pcpo by (rule tc_arity)

axiomatization
  Rf :: "('b stream * ('b,'c) tc stream * 'c stream * ('b,'c) tc stream) ⇒ bool"

definition
  is_f :: "('b stream * ('b,'c) tc stream → 'c stream * ('b,'c) tc stream) ⇒ bool" where
  "is_f f ⟷ (∀i1 i2 o1 o2. f⋅(i1, i2) = (o1, o2) ⟶ Rf (i1, i2, o1, o2))"

definition
  is_net_g :: "('b stream * ('b,'c) tc stream → 'c stream * ('b,'c) tc stream) ⇒
    'b stream ⇒ 'c stream ⇒ bool" where
  "is_net_g f x y ≡ (∃z.
                        (y, z) = f⋅(x,z) ∧
                        (∀oy hz. (oy, hz) = f⋅(x, hz) ⟶ z << hz))"

definition
  is_g :: "('b stream → 'c stream) ⇒ bool" where
  "is_g g ≡ (∃f. is_f f ∧ (∀x y. g⋅x = y ⟶ is_net_g f x y))"

definition
  def_g :: "('b stream → 'c stream) => bool" where
  "def_g g ≡ (∃f. is_f f ∧ g = (LAM x. fst (f⋅(x, fix⋅(LAM  k. snd (f⋅(x, k)))))))"


(* first some logical trading *)

lemma lemma1:
  "is_g g ⟷
    (∃f. is_f(f) ∧ (∀x.(∃z. (g⋅x,z) = f⋅(x,z) ∧ (∀w y. (y, w) = f⋅(x, w) ⟶ z << w))))"
apply (simp add: is_g_def is_net_g_def)
apply fast
done

lemma lemma2:
  "(∃f. is_f f ∧ (∀x. (∃z. (g⋅x, z) = f⋅(x, z) ∧ (∀w y. (y, w) = f⋅(x,w) ⟶ z << w)))) ⟷
  (∃f. is_f f ∧ (∀x. ∃z.
        g⋅x = fst (f⋅(x, z)) ∧
          z = snd (f⋅(x, z)) ∧
        (∀w y. (y, w) = f⋅(x, w) ⟶ z << w)))"
apply (rule iffI)
apply (erule exE)
apply (rule_tac x = "f" in exI)
apply (erule conjE)+
apply (erule conjI)
apply (intro strip)
apply (erule allE)
apply (erule exE)
apply (rule_tac x = "z" in exI)
apply (erule conjE)+
apply (rule conjI)
apply (rule_tac [2] conjI)
prefer 3 apply (assumption)
apply (drule sym)
apply (simp)
apply (drule sym)
apply (simp)
apply (erule exE)
apply (rule_tac x = "f" in exI)
apply (erule conjE)+
apply (erule conjI)
apply (intro strip)
apply (erule allE)
apply (erule exE)
apply (rule_tac x = "z" in exI)
apply (erule conjE)+
apply (rule conjI)
prefer 2 apply (assumption)
apply (rule prod_eqI)
apply simp
apply simp
done

lemma lemma3: "def_g g ⟶ is_g g"
apply (tactic ‹simp_tac (put_simpset HOL_ss \<^context>
  addsimps [@{thm def_g_def}, @{thm lemma1}, @{thm lemma2}]) 1›)
apply (rule impI)
apply (erule exE)
apply (rule_tac x = "f" in exI)
apply (erule conjE)+
apply (erule conjI)
apply (intro strip)
apply (rule_tac x = "fix⋅(LAM k. snd (f⋅(x, k)))" in exI)
apply (rule conjI)
 apply (simp)
 apply (rule prod_eqI, simp, simp)
 apply (rule trans)
  apply (rule fix_eq)
 apply (simp (no_asm))
apply (intro strip)
apply (rule fix_least)
apply (simp (no_asm))
apply (erule exE)
apply (drule sym)
back
apply simp
done

lemma lemma4: "is_g g ⟶ def_g g"
apply (tactic ‹simp_tac (put_simpset HOL_ss \<^context>
  delsimps (@{thms HOL.ex_simps} @ @{thms HOL.all_simps})
  addsimps [@{thm lemma1}, @{thm lemma2}, @{thm def_g_def}]) 1›)
apply (rule impI)
apply (erule exE)
apply (rule_tac x = "f" in exI)
apply (erule conjE)+
apply (erule conjI)
apply (rule cfun_eqI)
apply (erule_tac x = "x" in allE)
apply (erule exE)
apply (erule conjE)+
apply (subgoal_tac "fix⋅(LAM k. snd (f⋅(x, k))) = z")
 apply simp
apply (subgoal_tac "∀w y. f⋅(x, w) = (y, w) ⟶ z << w")
apply (rule fix_eqI)
apply simp
apply (subgoal_tac "f⋅(x, za) = (fst (f⋅(x, za)), za)")
apply fast
apply (rule prod_eqI, simp, simp)
apply (intro strip)
apply (erule allE)+
apply (erule mp)
apply (erule sym)
done

(* now we assemble the result *)

lemma loopback_eq: "def_g = is_g"
apply (rule ext)
apply (rule iffI)
apply (erule lemma3 [THEN mp])
apply (erule lemma4 [THEN mp])
done

lemma L2:
  "(∃f. is_f (f::'b stream * ('b,'c) tc stream → 'c stream * ('b,'c) tc stream)) ⟶
    (∃g. def_g (g::'b stream → 'c stream))"
apply (simp add: def_g_def)
done

theorem conservative_loopback:
  "(∃f. is_f (f::'b stream * ('b,'c) tc stream → 'c stream * ('b,'c) tc stream)) ⟶
    (∃g. is_g (g::'b stream → 'c stream))"
apply (rule loopback_eq [THEN subst])
apply (rule L2)
done

end