Theory Static

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theory Static
imports Values Types
begin
(*  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