Up to index of Isabelle/HOLCF/ex
theory Focus_ex(* $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