(* Title: ZF/Coind/Static.thy
Author: Jacob Frost, Cambridge University Computer Laboratory
Copyright 1995 University of Cambridge
*)
theory Static imports Values Types begin
(*** Basic correspondence relation -- not completely specified, as it is a
parameter of the proof. A concrete version could be defined inductively.
***)
axiomatization isof :: "[i,i] => o"
where isof_app: "[|isof(c1,t_fun(t1,t2)); isof(c2,t1)|] ==> isof(c_app(c1,c2),t2)"
(*Its extension to environments*)
definition
isofenv :: "[i,i] => o" where
"isofenv(ve,te) ==
ve_dom(ve) = te_dom(te) &
(\<forall>x \<in> ve_dom(ve).
\<exists>c \<in> Const. ve_app(ve,x) = v_const(c) & isof(c,te_app(te,x)))"
(*** Elaboration ***)
consts ElabRel :: i
inductive
domains "ElabRel" \<subseteq> "TyEnv * Exp * Ty"
intros
constI [intro!]:
"[| te \<in> TyEnv; c \<in> Const; t \<in> Ty; isof(c,t) |] ==>
<te,e_const(c),t> \<in> ElabRel"
varI [intro!]:
"[| te \<in> TyEnv; x \<in> ExVar; x \<in> te_dom(te) |] ==>
<te,e_var(x),te_app(te,x)> \<in> ElabRel"
fnI [intro!]:
"[| te \<in> TyEnv; x \<in> ExVar; e \<in> Exp; t1 \<in> Ty; t2 \<in> Ty;
<te_owr(te,x,t1),e,t2> \<in> ElabRel |] ==>
<te,e_fn(x,e),t_fun(t1,t2)> \<in> ElabRel"
fixI [intro!]:
"[| te \<in> TyEnv; f \<in> ExVar; x \<in> ExVar; t1 \<in> Ty; t2 \<in> Ty;
<te_owr(te_owr(te,f,t_fun(t1,t2)),x,t1),e,t2> \<in> ElabRel |] ==>
<te,e_fix(f,x,e),t_fun(t1,t2)> \<in> ElabRel"
appI [intro]:
"[| te \<in> TyEnv; e1 \<in> Exp; e2 \<in> Exp; t1 \<in> Ty; t2 \<in> Ty;
<te,e1,t_fun(t1,t2)> \<in> ElabRel;
<te,e2,t1> \<in> ElabRel |] ==> <te,e_app(e1,e2),t2> \<in> ElabRel"
type_intros te_appI Exp.intros Ty.intros
inductive_cases
elab_constE [elim!]: "<te,e_const(c),t> \<in> ElabRel"
and elab_varE [elim!]: "<te,e_var(x),t> \<in> ElabRel"
and elab_fnE [elim]: "<te,e_fn(x,e),t> \<in> ElabRel"
and elab_fixE [elim!]: "<te,e_fix(f,x,e),t> \<in> ElabRel"
and elab_appE [elim]: "<te,e_app(e1,e2),t> \<in> ElabRel"
declare ElabRel.dom_subset [THEN subsetD, dest]
end