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(* Title: ZF/Coind/ECR.thy
Author: Jacob Frost, Cambridge University Computer Laboratory
Copyright 1995 University of Cambridge
*)
theory ECR imports Static Dynamic begin
(* The extended correspondence relation *)
consts
HasTyRel :: i
coinductive
domains "HasTyRel" \<subseteq> "Val * Ty"
intros
htr_constI [intro!]:
"[| c \ Const; t \ Ty; isof(c,t) |] ==> \ HasTyRel"
htr_closI [intro]:
"[| x \ ExVar; e \ Exp; t \ Ty; ve \ ValEnv; te \ TyEnv;
<te,e_fn(x,e),t> \<in> ElabRel;
ve_dom(ve) = te_dom(te);
{<ve_app(ve,y),te_app(te,y)>.y \<in> ve_dom(ve)} \<in> Pow(HasTyRel) |]
==> <v_clos(x,e,ve),t> \<in> HasTyRel"
monos Pow_mono
type_intros Val_ValEnv.intros
(* Pointwise extension to environments *)
definition
hastyenv :: "[i,i] => o" where
"hastyenv(ve,te) ==
ve_dom(ve) = te_dom(te) &
(\<forall>x \<in> ve_dom(ve). <ve_app(ve,x),te_app(te,x)> \<in> HasTyRel)"
(* Specialised co-induction rule *)
lemma htr_closCI [intro]:
"[| x \ ExVar; e \ Exp; t \ Ty; ve \ ValEnv; te \ TyEnv;
<te, e_fn(x, e), t> \<in> ElabRel; ve_dom(ve) = te_dom(te);
{<ve_app(ve,y),te_app(te,y)>.y \<in> ve_dom(ve)} \<in>
Pow({<v_clos(x,e,ve),t>} \<union> HasTyRel) |]
==> <v_clos(x, e, ve),t> \<in> HasTyRel"
apply (rule singletonI [THEN HasTyRel.coinduct], auto)
done
(* Specialised elimination rules *)
inductive_cases
htr_constE [elim!]: " \ HasTyRel"
and htr_closE [elim]: " \ HasTyRel"
(* Properties of the pointwise extension to environments *)
lemmas HasTyRel_non_zero =
HasTyRel.dom_subset [THEN subsetD, THEN SigmaD1, THEN ValNEE]
lemma hastyenv_owr:
"[| ve \ ValEnv; te \ TyEnv; hastyenv(ve,te); \ HasTyRel |]
==> hastyenv(ve_owr(ve,x,v),te_owr(te,x,t))"
by (auto simp add: hastyenv_def ve_app_owr HasTyRel_non_zero)
lemma basic_consistency_lem:
"[| ve \ ValEnv; te \ TyEnv; isofenv(ve,te) |] ==> hastyenv(ve,te)"
apply (unfold isofenv_def hastyenv_def)
apply (force intro: te_appI ve_domI)
done
(* ############################################################ *)
(* The Consistency theorem *)
(* ############################################################ *)
lemma consistency_const:
"[| c \ Const; hastyenv(ve,te); \ ElabRel |]
==> <v_const(c), t> \<in> HasTyRel"
by blast
lemma consistency_var:
"[| x \ ve_dom(ve); hastyenv(ve,te); \ ElabRel |] ==>
<ve_app(ve,x),t> \<in> HasTyRel"
by (unfold hastyenv_def, blast)
lemma consistency_fn:
"[| ve \ ValEnv; x \ ExVar; e \ Exp; hastyenv(ve,te);
<te,e_fn(x,e),t> \<in> ElabRel
|] ==> <v_clos(x, e, ve), t> \<in> HasTyRel"
by (unfold hastyenv_def, blast)
declare ElabRel.dom_subset [THEN subsetD, dest]
declare Ty.intros [simp, intro!]
declare TyEnv.intros [simp, intro!]
declare Val_ValEnv.intros [simp, intro!]
lemma consistency_fix:
"[| ve \ ValEnv; x \ ExVar; e \ Exp; f \ ExVar; cl \ Val;
v_clos(x,e,ve_owr(ve,f,cl)) = cl;
hastyenv(ve,te); <te,e_fix(f,x,e),t> \<in> ElabRel |] ==>
<cl,t> \<in> HasTyRel"
apply (unfold hastyenv_def)
apply (erule elab_fixE, safe)
apply hypsubst_thin
apply (rule subst, assumption)
apply (rule_tac te="te_owr(te, f, t_fun(t1, t2))" in htr_closCI)
apply simp_all
apply (blast intro: ve_owrI)
apply (rule ElabRel.fnI)
apply (simp_all add: ValNEE, force)
done
lemma consistency_app1:
"[| ve \ ValEnv; e1 \ Exp; e2 \ Exp; c1 \ Const; c2 \ Const;
<ve,e1,v_const(c1)> \<in> EvalRel;
\<forall>t te.
hastyenv(ve,te) \<longrightarrow> <te,e1,t> \<in> ElabRel \<longrightarrow> <v_const(c1),t> \<in> HasTyRel;
<ve, e2, v_const(c2)> \<in> EvalRel;
\<forall>t te.
hastyenv(ve,te) \<longrightarrow> <te,e2,t> \<in> ElabRel \<longrightarrow> <v_const(c2),t> \<in> HasTyRel;
hastyenv(ve, te);
<te,e_app(e1,e2),t> \<in> ElabRel |]
==> <v_const(c_app(c1, c2)),t> \<in> HasTyRel"
by (blast intro!: c_appI intro: isof_app)
lemma consistency_app2:
"[| ve \ ValEnv; vem \ ValEnv; e1 \ Exp; e2 \ Exp; em \ Exp; xm \ ExVar;
v \<in> Val;
<ve,e1,v_clos(xm,em,vem)> \<in> EvalRel;
\<forall>t te. hastyenv(ve,te) \<longrightarrow>
<te,e1,t> \<in> ElabRel \<longrightarrow> <v_clos(xm,em,vem),t> \<in> HasTyRel;
<ve,e2,v2> \<in> EvalRel;
\<forall>t te. hastyenv(ve,te) \<longrightarrow> <te,e2,t> \<in> ElabRel \<longrightarrow> <v2,t> \<in> HasTyRel;
<ve_owr(vem,xm,v2),em,v> \<in> EvalRel;
\<forall>t te. hastyenv(ve_owr(vem,xm,v2),te) \<longrightarrow>
<te,em,t> \<in> ElabRel \<longrightarrow> <v,t> \<in> HasTyRel;
hastyenv(ve,te); <te,e_app(e1,e2),t> \<in> ElabRel |]
==> <v,t> \<in> HasTyRel"
apply (erule elab_appE)
apply (drule spec [THEN spec, THEN mp, THEN mp], assumption+)
apply (drule spec [THEN spec, THEN mp, THEN mp], assumption+)
apply (erule htr_closE)
apply (erule elab_fnE, simp)
apply clarify
apply (drule spec [THEN spec, THEN mp, THEN mp])
prefer 2 apply assumption+
apply (rule hastyenv_owr, assumption)
apply assumption
apply (simp add: hastyenv_def, blast+)
done
lemma consistency [rule_format]:
" \ EvalRel
==> (\<forall>t te. hastyenv(ve,te) \<longrightarrow> <te,e,t> \<in> ElabRel \<longrightarrow> <v,t> \<in> HasTyRel)"
apply (erule EvalRel.induct)
apply (simp_all add: consistency_const consistency_var consistency_fn
consistency_fix consistency_app1)
apply (blast intro: consistency_app2)
done
lemma basic_consistency:
"[| ve \ ValEnv; te \ TyEnv; isofenv(ve,te);
<ve,e,v_const(c)> \<in> EvalRel; <te,e,t> \<in> ElabRel |]
==> isof(c,t)"
by (blast dest: consistency intro!: basic_consistency_lem)
end
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