(* Title: HOL/MicroJava/Comp/TypeInf.thy ID: $Id: TypeInf.thy,v 1.4 2005/06/17 14:13:09 haftmann Exp $ Author: Martin Strecker *) (* Exact position in theory hierarchy still to be determined *) theory TypeInf imports 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 |- NewC C :: T ==> T = Class C ∧ is_class (fst E) C
lemma Cast_invers:
E |- Cast D e :: T ==> ∃C. T = Class D ∧ E |- e :: C ∧ is_class (fst E) D ∧ fst E |- C <=? Class D
lemma Lit_invers:
E |- Lit x :: T ==> typeof empty x = Some T
lemma LAcc_invers:
E |- LAcc v :: T ==> snd E v = Some T ∧ is_type (fst E) T
lemma BinOp_invers:
E |- BinOp bop e1.0 e2.0 :: T' ==> ∃T. E |- e1.0 :: T ∧ E |- e2.0 :: T ∧ (if bop = Eq then T' = PrimT Boolean else T' = T ∧ T = PrimT Integer)
lemma LAss_invers:
E |- v::=e :: T' ==> ∃T. v ≠ This ∧ E |- LAcc v :: T ∧ E |- e :: T' ∧ fst E |- T' <= T
lemma FAcc_invers:
E |- {fd}a..fn :: fT ==> ∃C. E |- a :: Class C ∧ field (fst E, C) fn = Some (fd, fT)
lemma FAss_invers:
E |- {fd}a..fn:=v :: T' ==> ∃T. E |- {fd}a..fn :: T ∧ E |- v :: T' ∧ fst E |- T' <= T
lemma Call_invers:
E |- {C}a..mn( {pTs'}ps) :: rT ==> ∃pTs md. E |- a :: Class C ∧ E |- ps [::] pTs ∧ max_spec (fst E) C (mn, pTs) = {((md, rT), pTs')}
lemma Nil_invers:
E |- [] [::] Ts ==> Ts = []
lemma Cons_invers:
E |- e # es [::] Ts ==> ∃T Ts'. Ts = T # Ts' ∧ E |- e :: T ∧ E |- es [::] Ts'
lemma Expr_invers:
E |- Expr e [ok] ==> ∃T. E |- e :: T
lemma Comp_invers:
E |- s1.0;; s2.0 [ok] ==> E |- s1.0 [ok] ∧ E |- s2.0 [ok]
lemma Cond_invers:
E |- If (e) s1.0 Else s2.0 [ok] ==> E |- e :: PrimT Boolean ∧ E |- s1.0 [ok] ∧ E |- s2.0 [ok]
lemma Loop_invers:
E |- While (e) s [ok] ==> E |- e :: PrimT Boolean ∧ E |- s [ok]
lemma uniqueness_of_types:
(∀E T1 T2. E |- e :: T1 --> E |- e :: T2 --> T1 = T2) ∧ (∀E Ts1 Ts2. E |- es [::] Ts1 --> E |- es [::] Ts2 --> Ts1 = Ts2)
lemma uniqueness_of_types_expr:
[| E |- e :: T1.0; E |- e :: T2.0 |] ==> T1.0 = T2.0
lemma uniqueness_of_types_exprs:
[| E |- es [::] Ts1.0; E |- es [::] Ts2.0 |] ==> Ts1.0 = Ts2.0
lemma inferred_tp_wt:
E |- e :: T ==> inferred_tp E e = T
lemma inferred_tps_wt:
E |- es [::] Ts ==> inferred_tps E es = Ts