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(* Title: ZF/Coind/Values.thy
Author: Jacob Frost, Cambridge University Computer Laboratory
Copyright 1995 University of Cambridge
*)
theory Values imports Language Map begin
(* Values, values environments and associated operators *)
consts
Val :: i
ValEnv :: i
Val_ValEnv :: i
codatatype
"Val" = v_const ("c \ Const")
| v_clos ("x \ ExVar","e \ Exp","ve \ ValEnv")
and
"ValEnv" = ve_mk ("m \ PMap(ExVar,Val)")
monos PMap_mono
type_intros A_into_univ mapQU
consts
ve_owr :: "[i,i,i] => i"
ve_dom :: "i=>i"
ve_app :: "[i,i] => i"
primrec "ve_owr(ve_mk(m), x, v) = ve_mk(map_owr(m,x,v))"
primrec "ve_dom(ve_mk(m)) = domain(m)"
primrec "ve_app(ve_mk(m), a) = map_app(m,a)"
definition
ve_emp :: i where
"ve_emp == ve_mk(map_emp)"
(* Elimination rules *)
lemma ValEnvE:
"[| ve \ ValEnv; !!m.[| ve=ve_mk(m); m \ PMap(ExVar,Val) |] ==> Q |] ==> Q"
apply (unfold Part_def Val_def ValEnv_def, clarify)
apply (erule Val_ValEnv.cases)
apply (auto simp add: Val_def Part_def Val_ValEnv.con_defs)
done
lemma ValE:
"[| v \ Val;
!!c. [| v = v_const(c); c \<in> Const |] ==> Q;
!!e ve x.
[| v = v_clos(x,e,ve); x \<in> ExVar; e \<in> Exp; ve \<in> ValEnv |] ==> Q
|] ==>
Q"
apply (unfold Part_def Val_def ValEnv_def, clarify)
apply (erule Val_ValEnv.cases)
apply (auto simp add: ValEnv_def Part_def Val_ValEnv.con_defs)
done
(* Nonempty sets *)
lemma v_closNE [simp]: "v_clos(x,e,ve) \ 0"
by (unfold QPair_def QInl_def QInr_def Val_ValEnv.con_defs, blast)
declare v_closNE [THEN notE, elim!]
lemma v_constNE [simp]: "c \ Const ==> v_const(c) \ 0"
apply (unfold QPair_def QInl_def QInr_def Val_ValEnv.con_defs)
apply (drule constNEE, auto)
done
(* Proving that the empty set is not a value *)
lemma ValNEE: "v \ Val ==> v \ 0"
by (erule ValE, auto)
(* Equalities for value environments *)
lemma ve_dom_owr [simp]:
"[| ve \ ValEnv; v \0 |] ==> ve_dom(ve_owr(ve,x,v)) = ve_dom(ve) \ {x}"
apply (erule ValEnvE)
apply (auto simp add: map_domain_owr)
done
lemma ve_app_owr [simp]:
"ve \ ValEnv
==> ve_app(ve_owr(ve,y,v),x) = (if x=y then v else ve_app(ve,x))"
by (erule ValEnvE, simp add: map_app_owr)
(* Introduction rules for operators on value environments *)
lemma ve_appI: "[| ve \ ValEnv; x \ ve_dom(ve) |] ==> ve_app(ve,x):Val"
by (erule ValEnvE, simp add: pmap_appI)
lemma ve_domI: "[| ve \ ValEnv; x \ ve_dom(ve) |] ==> x \ ExVar"
apply (erule ValEnvE, simp)
apply (blast dest: pmap_domainD)
done
lemma ve_empI: "ve_emp \ ValEnv"
apply (unfold ve_emp_def)
apply (rule Val_ValEnv.intros)
apply (rule pmap_empI)
done
lemma ve_owrI:
"[|ve \ ValEnv; x \ ExVar; v \ Val |] ==> ve_owr(ve,x,v):ValEnv"
apply (erule ValEnvE, simp)
apply (blast intro: pmap_owrI Val_ValEnv.intros)
done
end
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