(* Title: ZF/Coind/Static.thy ID: $Id: Static.thy,v 1.9 2005/06/17 14:15:10 haftmann Exp $ 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. ***) consts isof :: "[i,i] => o" axioms isof_app: "[|isof(c1,t_fun(t1,t2)); isof(c2,t1)|] ==> isof(c_app(c1,c2),t2)" (*Its extension to environments*) constdefs isofenv :: "[i,i] => o" "isofenv(ve,te) == ve_dom(ve) = te_dom(te) & (∀x ∈ ve_dom(ve). ∃c ∈ Const. ve_app(ve,x) = v_const(c) & isof(c,te_app(te,x)))" (*** Elaboration ***) consts ElabRel :: i inductive domains "ElabRel" <= "TyEnv * Exp * Ty" intros constI [intro!]: "[| te ∈ TyEnv; c ∈ Const; t ∈ Ty; isof(c,t) |] ==> <te,e_const(c),t> ∈ ElabRel" varI [intro!]: "[| te ∈ TyEnv; x ∈ ExVar; x ∈ te_dom(te) |] ==> <te,e_var(x),te_app(te,x)> ∈ ElabRel" fnI [intro!]: "[| te ∈ TyEnv; x ∈ ExVar; e ∈ Exp; t1 ∈ Ty; t2 ∈ Ty; <te_owr(te,x,t1),e,t2> ∈ ElabRel |] ==> <te,e_fn(x,e),t_fun(t1,t2)> ∈ ElabRel" fixI [intro!]: "[| te ∈ TyEnv; f ∈ ExVar; x ∈ ExVar; t1 ∈ Ty; t2 ∈ Ty; <te_owr(te_owr(te,f,t_fun(t1,t2)),x,t1),e,t2> ∈ ElabRel |] ==> <te,e_fix(f,x,e),t_fun(t1,t2)> ∈ ElabRel" appI [intro]: "[| te ∈ TyEnv; e1 ∈ Exp; e2 ∈ Exp; t1 ∈ Ty; t2 ∈ Ty; <te,e1,t_fun(t1,t2)> ∈ ElabRel; <te,e2,t1> ∈ ElabRel |] ==> <te,e_app(e1,e2),t2> ∈ ElabRel" type_intros te_appI Exp.intros Ty.intros inductive_cases elab_constE [elim!]: "<te,e_const(c),t> ∈ ElabRel" and elab_varE [elim!]: "<te,e_var(x),t> ∈ ElabRel" and elab_fnE [elim]: "<te,e_fn(x,e),t> ∈ ElabRel" and elab_fixE [elim!]: "<te,e_fix(f,x,e),t> ∈ ElabRel" and elab_appE [elim]: "<te,e_app(e1,e2),t> ∈ ElabRel" declare ElabRel.dom_subset [THEN subsetD, dest] end
lemmas elab_constE:
[| 〈te, e_const(c), t〉 ∈ ElabRel; [| te ∈ TyEnv; c ∈ Const; t ∈ Ty; isof(c, t) |] ==> Q |] ==> Q
and elab_varE:
[| 〈te, e_var(x), t〉 ∈ ElabRel; [| te ∈ TyEnv; x ∈ ExVar; x ∈ te_dom(te); t = te_app(te, x) |] ==> Q |] ==> Q
and elab_fnE:
[| 〈te, e_fn(x, e), t〉 ∈ ElabRel; !!t1 t2. [| te ∈ TyEnv; x ∈ ExVar; e ∈ Exp; t1 ∈ Ty; t2 ∈ Ty; 〈te_owr(te, x, t1), e, t2〉 ∈ ElabRel; t = t_fun(t1, t2) |] ==> Q |] ==> Q
and elab_fixE:
[| 〈te, e_fix(f, x, e), t〉 ∈ ElabRel; !!t1 t2. [| te ∈ TyEnv; f ∈ ExVar; x ∈ ExVar; t1 ∈ Ty; t2 ∈ Ty; 〈te_owr(te_owr(te, f, t_fun(t1, t2)), x, t1), e, t2〉 ∈ ElabRel; t = t_fun(t1, t2) |] ==> Q |] ==> Q
and elab_appE:
[| 〈te, e_app(e1.0, e2.0), t〉 ∈ ElabRel; !!t1. [| te ∈ TyEnv; e1.0 ∈ Exp; e2.0 ∈ Exp; t1 ∈ Ty; t ∈ Ty; 〈te, e1.0, t_fun(t1, t)〉 ∈ ElabRel; 〈te, e2.0, t1〉 ∈ ElabRel |] ==> Q |] ==> Q