Theory TypeInf

Up to index of Isabelle/HOL/MicroJava

theory TypeInf
imports WellType
begin

(*  Title:      HOL/MicroJava/Comp/TypeInf.thy
    ID:         $Id: TypeInf.thy,v 1.5 2005/10/07 18:41:11 nipkow Exp $
    Author:     Martin Strecker
*)

(* Exact position in theory hierarchy still to be determined *)
theory TypeInf
imports "../J/WellType"
begin



(**********************************************************************)
;

(*** Inversion of typing rules -- to be moved into WellType.thy
     Also modify the wtpd_expr_… proofs in CorrComp.thy ***)

lemma NewC_invers: "E\<turnstile>NewC C::T 
  ==> T = Class C ∧ is_class (prg E) C"
by (erule ty_expr.cases, auto)

lemma Cast_invers: "E\<turnstile>Cast D e::T
  ==> ∃ C. T = Class D ∧ E\<turnstile>e::C ∧ is_class (prg E) D ∧ prg E\<turnstile>C\<preceq>? Class D"
by (erule ty_expr.cases, auto)

lemma Lit_invers: "E\<turnstile>Lit x::T
  ==> typeof (λv. None) x = Some T"
by (erule ty_expr.cases, auto)

lemma LAcc_invers: "E\<turnstile>LAcc v::T
  ==> localT E v = Some T ∧ is_type (prg E) T"
by (erule ty_expr.cases, auto)

lemma BinOp_invers: "E\<turnstile>BinOp bop e1 e2::T'
  ==> ∃ T. E\<turnstile>e1::T ∧ E\<turnstile>e2::T ∧ 
            (if bop = Eq then T' = PrimT Boolean
                        else T' = T ∧ T = PrimT Integer)"
by (erule ty_expr.cases, auto)

lemma LAss_invers: "E\<turnstile>v::=e::T'
  ==> ∃ T. v ~= This ∧ E\<turnstile>LAcc v::T ∧ E\<turnstile>e::T' ∧ prg E\<turnstile>T'\<preceq>T"
by (erule ty_expr.cases, auto)

lemma FAcc_invers: "E\<turnstile>{fd}a..fn::fT
  ==> ∃ C. E\<turnstile>a::Class C ∧ field (prg E,C) fn = Some (fd,fT)"
by (erule ty_expr.cases, auto)

lemma FAss_invers: "E\<turnstile>{fd}a..fn:=v::T' 
==> ∃ T. E\<turnstile>{fd}a..fn::T ∧ E\<turnstile>v ::T' ∧ prg E\<turnstile>T'\<preceq>T"
by (erule ty_expr.cases, auto)

lemma Call_invers: "E\<turnstile>{C}a..mn({pTs'}ps)::rT
  ==> ∃ pTs md. 
  E\<turnstile>a::Class C ∧ E\<turnstile>ps[::]pTs ∧ max_spec (prg E) C (mn, pTs) = {((md,rT),pTs')}"
by (erule ty_expr.cases, auto)


lemma Nil_invers: "E\<turnstile>[] [::] Ts ==> Ts = []"
by (erule ty_exprs.cases, auto)

lemma Cons_invers: "E\<turnstile>e#es[::]Ts ==> 
  ∃ T Ts'. Ts = T#Ts' ∧ E \<turnstile>e::T ∧ E \<turnstile>es[::]Ts'"
by (erule ty_exprs.cases, auto)


lemma Expr_invers: "E\<turnstile>Expr e\<surd> ==> ∃ T. E\<turnstile>e::T"
by (erule wt_stmt.cases, auto)

lemma Comp_invers: "E\<turnstile>s1;; s2\<surd> ==> E\<turnstile>s1\<surd> ∧ E\<turnstile>s2\<surd>"
by (erule wt_stmt.cases, auto)

lemma Cond_invers: "E\<turnstile>If(e) s1 Else s2\<surd> 
  ==> E\<turnstile>e::PrimT Boolean ∧ E\<turnstile>s1\<surd> ∧ E\<turnstile>s2\<surd>"
by (erule wt_stmt.cases, auto)

lemma Loop_invers: "E\<turnstile>While(e) s\<surd>
  ==> E\<turnstile>e::PrimT Boolean ∧ E\<turnstile>s\<surd>"
by (erule wt_stmt.cases, auto)


(**********************************************************************)


declare split_paired_All [simp del]
declare split_paired_Ex [simp del]

(* Uniqueness of types property *)

lemma uniqueness_of_types: "
  (∀ (E::'a prog × (vname => ty option)) T1 T2. 
  E\<turnstile>e :: T1 --> E\<turnstile>e :: T2 --> T1 = T2) ∧
  (∀ (E::'a prog × (vname => ty option)) Ts1 Ts2. 
  E\<turnstile>es [::] Ts1 --> E\<turnstile>es [::] Ts2 --> Ts1 = Ts2)"
apply (rule expr.induct)

(* NewC *)
apply (intro strip) 
apply (erule ty_expr.cases) apply simp+
apply (erule ty_expr.cases) apply simp+

(* Cast *)
apply (intro strip) 
apply (erule ty_expr.cases) apply simp+
apply (erule ty_expr.cases) apply simp+

(* Lit *)
apply (intro strip) 
apply (erule ty_expr.cases) apply simp+
apply (erule ty_expr.cases) apply simp+

(* BinOp *)
apply (intro strip)
apply (case_tac binop)
(* Eq *)
apply (erule ty_expr.cases) apply simp+
apply (erule ty_expr.cases) apply simp+
(* Add *)
apply (erule ty_expr.cases) apply simp+
apply (erule ty_expr.cases) apply simp+

(* LAcc *)
apply (intro strip) 
apply (erule ty_expr.cases) apply simp+
apply (erule ty_expr.cases) apply simp+

(* LAss *)
apply (intro strip) 
apply (erule ty_expr.cases) apply simp+
apply (erule ty_expr.cases) apply simp+


(* FAcc *)
apply (intro strip)
apply (drule FAcc_invers)+ apply (erule exE)+ 
  apply (subgoal_tac "C = Ca", simp) apply blast


(* FAss *)
apply (intro strip)
apply (drule FAss_invers)+ apply (erule exE)+ apply (erule conjE)+
apply (drule FAcc_invers)+ apply (erule exE)+ apply blast 


(* Call *)
apply (intro strip)
apply (drule Call_invers)+ apply (erule exE)+ apply (erule conjE)+
apply (subgoal_tac "pTs = pTsa", simp) apply blast

(* expression lists *)
apply (intro strip)
apply (erule ty_exprs.cases)+ apply simp+

apply (intro strip)
apply (erule ty_exprs.cases, simp)
apply (erule ty_exprs.cases, simp)
apply (subgoal_tac "e = ea", simp) apply simp
done


lemma uniqueness_of_types_expr [rule_format (no_asm)]: "
  (∀ E T1 T2. E\<turnstile>e :: T1 --> E\<turnstile>e :: T2 --> T1 = T2)"
by (rule uniqueness_of_types [THEN conjunct1])

lemma uniqueness_of_types_exprs [rule_format (no_asm)]: "
  (∀ E Ts1 Ts2. E\<turnstile>es [::] Ts1 --> E\<turnstile>es [::] Ts2 --> Ts1 = Ts2)"
by (rule uniqueness_of_types [THEN conjunct2])


  

constdefs 
  inferred_tp  :: "[java_mb env, expr] => ty"
  "inferred_tp E e == (SOME T. E\<turnstile>e :: T)"
  inferred_tps :: "[java_mb env, expr list] => ty list"
  "inferred_tps E es == (SOME Ts. E\<turnstile>es [::] Ts)"

(* get inferred type(s) for well-typed term *)
lemma inferred_tp_wt: "E\<turnstile>e :: T ==> (inferred_tp E e) = T"
by (auto simp: inferred_tp_def intro: uniqueness_of_types_expr)

lemma inferred_tps_wt: "E\<turnstile>es [::] Ts ==> (inferred_tps E es) = Ts"
by (auto simp: inferred_tps_def intro: uniqueness_of_types_exprs)


end


lemma NewC_invers:

  E \<turnstile> NewC C :: T ==> T = Class C ∧ is_class (fst E) C

lemma Cast_invers:

  E \<turnstile> Cast D e :: T
  ==> ∃C. T = Class DE \<turnstile> e :: C ∧
          is_class (fst E) D ∧ fst E \<turnstile> C \<preceq>? Class D

lemma Lit_invers:

  E \<turnstile> Lit x :: T ==> typeof empty x = Some T

lemma LAcc_invers:

  E \<turnstile> LAcc v :: T ==> snd E v = Some T ∧ is_type (fst E) T

lemma BinOp_invers:

  E \<turnstile> BinOp bop e1.0 e2.0 :: T'
  ==> ∃T. E \<turnstile> e1.0 :: TE \<turnstile> e2.0 :: T ∧
          (if bop = Eq then T' = PrimT Boolean else T' = TT = PrimT Integer)

lemma LAss_invers:

  E \<turnstile> v::=e :: T'
  ==> ∃T. v  This ∧
          E \<turnstile> LAcc v :: TE \<turnstile> e :: T' ∧ fst E \<turnstile> T' \<preceq> T

lemma FAcc_invers:

  E \<turnstile> {fd}a..fn :: fT
  ==> ∃C. E \<turnstile> a :: Class C ∧
          TypeRel.field (fst E, C) fn = Some (fd, fT)

lemma FAss_invers:

  E \<turnstile> {fd}a..fn:=v :: T'
  ==> ∃T. E \<turnstile> {fd}a..fn :: TE \<turnstile> v :: T' ∧ fst E \<turnstile> T' \<preceq> T

lemma Call_invers:

  E \<turnstile> {C}a..mn( {pTs'}ps) :: rT
  ==> ∃pTs md.
         E \<turnstile> a :: Class CE \<turnstile> ps [::] pTs ∧
         max_spec (fst E) C (mn, pTs) = {((md, rT), pTs')}

lemma Nil_invers:

  E \<turnstile> [] [::] Ts ==> Ts = []

lemma Cons_invers:

  E \<turnstile> e # es [::] Ts
  ==> ∃T Ts'. Ts = T # Ts'E \<turnstile> e :: TE \<turnstile> es [::] Ts'

lemma Expr_invers:

  E \<turnstile> Expr e \<surd> ==> ∃T. E \<turnstile> e :: T

lemma Comp_invers:

  E \<turnstile> s1.0;; s2.0 \<surd>
  ==> E \<turnstile> s1.0 \<surd>E \<turnstile> s2.0 \<surd>

lemma Cond_invers:

  E \<turnstile> If (e) s1.0 Else s2.0 \<surd>
  ==> E \<turnstile> e :: PrimT Boolean ∧
      E \<turnstile> s1.0 \<surd>E \<turnstile> s2.0 \<surd>

lemma Loop_invers:

  E \<turnstile> While (e) s \<surd>
  ==> E \<turnstile> e :: PrimT Boolean ∧ E \<turnstile> s \<surd>

lemma uniqueness_of_types:

  (∀E T1 T2. E \<turnstile> e :: T1 --> E \<turnstile> e :: T2 --> T1 = T2) ∧
  (∀E Ts1 Ts2.
      E \<turnstile> es [::] Ts1 --> E \<turnstile> es [::] Ts2 --> Ts1 = Ts2)

lemma uniqueness_of_types_expr:

  [| E \<turnstile> e :: T1.0; E \<turnstile> e :: T2.0 |] ==> T1.0 = T2.0

lemma uniqueness_of_types_exprs:

  [| E \<turnstile> es [::] Ts1.0; E \<turnstile> es [::] Ts2.0 |]
  ==> Ts1.0 = Ts2.0

lemma inferred_tp_wt:

  E \<turnstile> e :: T ==> inferred_tp E e = T

lemma inferred_tps_wt:

  E \<turnstile> es [::] Ts ==> inferred_tps E es = Ts