Theory Types

theory Types
imports Language
(*  Title:      ZF/Coind/Types.thy
Author: Jacob Frost, Cambridge University Computer Laboratory
Copyright 1995 University of Cambridge
*)


theory Types imports Language begin

consts
Ty :: i (* Datatype of types *)
TyConst :: i (* Abstract type of type constants *)

datatype
"Ty" = t_const ("tc ∈ TyConst")
| t_fun ("t1 ∈ Ty","t2 ∈ Ty")


(* Definition of type environments and associated operators *)

consts
TyEnv :: i

datatype
"TyEnv" = te_emp
| te_owr ("te ∈ TyEnv","x ∈ ExVar","t ∈ Ty")

consts
te_dom :: "i => i"
te_app :: "[i,i] => i"


primrec (*domain of the type environment*)
"te_dom (te_emp) = 0"
"te_dom (te_owr(te,x,v)) = te_dom(te) ∪ {x}"

primrec (*lookup up identifiers in the type environment*)
"te_app (te_emp,x) = 0"
"te_app (te_owr(te,y,t),x) = (if x=y then t else te_app(te,x))"

inductive_cases te_owrE [elim!]: "te_owr(te,f,t) ∈ TyEnv"

(*redundant??*)
lemma te_app_owr1: "te_app(te_owr(te,x,t),x) = t"
by simp

(*redundant??*)
lemma te_app_owr2: "x ≠ y ==> te_app(te_owr(te,x,t),y) = te_app(te,y)"
by auto

lemma te_app_owr [simp]:
"te_app(te_owr(te,x,t),y) = (if x=y then t else te_app(te,y))"
by auto

lemma te_appI:
"[| te ∈ TyEnv; x ∈ ExVar; x ∈ te_dom(te) |] ==> te_app(te,x) ∈ Ty"
apply (erule_tac P = "x ∈ te_dom (te) " in rev_mp)
apply (erule TyEnv.induct, auto)
done


















end