(* Title: HOL/MicroJava/J/TypeRel.thy ID: $Id: TypeRel.thy,v 1.27 2005/06/17 14:13:09 haftmann Exp $ Author: David von Oheimb Copyright 1999 Technische Universitaet Muenchen *) header {* \isaheader{Relations between Java Types} *} theory TypeRel imports Decl begin consts subcls1 :: "'c prog => (cname × cname) set" -- "subclass" widen :: "'c prog => (ty × ty ) set" -- "widening" cast :: "'c prog => (ty × ty ) set" -- "casting" syntax (xsymbols) subcls1 :: "'c prog => [cname, cname] => bool" ("_ \<turnstile> _ \<prec>C1 _" [71,71,71] 70) subcls :: "'c prog => [cname, cname] => bool" ("_ \<turnstile> _ \<preceq>C _" [71,71,71] 70) widen :: "'c prog => [ty , ty ] => bool" ("_ \<turnstile> _ \<preceq> _" [71,71,71] 70) cast :: "'c prog => [ty , ty ] => bool" ("_ \<turnstile> _ \<preceq>? _" [71,71,71] 70) syntax subcls1 :: "'c prog => [cname, cname] => bool" ("_ |- _ <=C1 _" [71,71,71] 70) subcls :: "'c prog => [cname, cname] => bool" ("_ |- _ <=C _" [71,71,71] 70) widen :: "'c prog => [ty , ty ] => bool" ("_ |- _ <= _" [71,71,71] 70) cast :: "'c prog => [ty , ty ] => bool" ("_ |- _ <=? _" [71,71,71] 70) translations "G\<turnstile>C \<prec>C1 D" == "(C,D) ∈ subcls1 G" "G\<turnstile>C \<preceq>C D" == "(C,D) ∈ (subcls1 G)^*" "G\<turnstile>S \<preceq> T" == "(S,T) ∈ widen G" "G\<turnstile>C \<preceq>? D" == "(C,D) ∈ cast G" -- "direct subclass, cf. 8.1.3" inductive "subcls1 G" intros subcls1I: "[|class G C = Some (D,rest); C ≠ Object|] ==> G\<turnstile>C\<prec>C1D" lemma subcls1D: "G\<turnstile>C\<prec>C1D ==> C ≠ Object ∧ (∃fs ms. class G C = Some (D,fs,ms))" apply (erule subcls1.elims) apply auto done lemma subcls1_def2: "subcls1 G = (SIGMA C: {C. is_class G C} . {D. C≠Object ∧ fst (the (class G C))=D})" by (auto simp add: is_class_def dest: subcls1D intro: subcls1I) lemma finite_subcls1: "finite (subcls1 G)" apply(subst subcls1_def2) apply(rule finite_SigmaI [OF finite_is_class]) apply(rule_tac B = "{fst (the (class G C))}" in finite_subset) apply auto done lemma subcls_is_class: "(C,D) ∈ (subcls1 G)^+ ==> is_class G C" apply (unfold is_class_def) apply(erule trancl_trans_induct) apply (auto dest!: subcls1D) done lemma subcls_is_class2 [rule_format (no_asm)]: "G\<turnstile>C\<preceq>C D ==> is_class G D --> is_class G C" apply (unfold is_class_def) apply (erule rtrancl_induct) apply (drule_tac [2] subcls1D) apply auto done constdefs class_rec :: "'c prog => cname => 'a => (cname => fdecl list => 'c mdecl list => 'a => 'a) => 'a" "class_rec G == wfrec ((subcls1 G)^-1) (λr C t f. case class G C of None => arbitrary | Some (D,fs,ms) => f C fs ms (if C = Object then t else r D t f))" lemma class_rec_lemma: "wf ((subcls1 G)^-1) ==> class G C = Some (D,fs,ms) ==> class_rec G C t f = f C fs ms (if C=Object then t else class_rec G D t f)" by (simp add: class_rec_def wfrec cut_apply [OF converseI [OF subcls1I]]) consts method :: "'c prog × cname => ( sig \<rightharpoonup> cname × ty × 'c)" (* ###curry *) field :: "'c prog × cname => ( vname \<rightharpoonup> cname × ty )" (* ###curry *) fields :: "'c prog × cname => ((vname × cname) × ty) list" (* ###curry *) -- "methods of a class, with inheritance, overriding and hiding, cf. 8.4.6" defs method_def: "method ≡ λ(G,C). class_rec G C empty (λC fs ms ts. ts ++ map_of (map (λ(s,m). (s,(C,m))) ms))" lemma method_rec_lemma: "[|class G C = Some (D,fs,ms); wf ((subcls1 G)^-1)|] ==> method (G,C) = (if C = Object then empty else method (G,D)) ++ map_of (map (λ(s,m). (s,(C,m))) ms)" apply (unfold method_def) apply (simp split del: split_if) apply (erule (1) class_rec_lemma [THEN trans]); apply auto done -- "list of fields of a class, including inherited and hidden ones" defs fields_def: "fields ≡ λ(G,C). class_rec G C [] (λC fs ms ts. map (λ(fn,ft). ((fn,C),ft)) fs @ ts)" lemma fields_rec_lemma: "[|class G C = Some (D,fs,ms); wf ((subcls1 G)^-1)|] ==> fields (G,C) = map (λ(fn,ft). ((fn,C),ft)) fs @ (if C = Object then [] else fields (G,D))" apply (unfold fields_def) apply (simp split del: split_if) apply (erule (1) class_rec_lemma [THEN trans]); apply auto done defs field_def: "field == map_of o (map (λ((fn,fd),ft). (fn,(fd,ft)))) o fields" lemma field_fields: "field (G,C) fn = Some (fd, fT) ==> map_of (fields (G,C)) (fn, fd) = Some fT" apply (unfold field_def) apply (rule table_of_remap_SomeD) apply simp done -- "widening, viz. method invocation conversion,cf. 5.3 i.e. sort of syntactic subtyping" inductive "widen G" intros refl [intro!, simp]: "G\<turnstile> T \<preceq> T" -- "identity conv., cf. 5.1.1" subcls : "G\<turnstile>C\<preceq>C D ==> G\<turnstile>Class C \<preceq> Class D" null [intro!]: "G\<turnstile> NT \<preceq> RefT R" -- "casting conversion, cf. 5.5 / 5.1.5" -- "left out casts on primitve types" inductive "cast G" intros widen: "G\<turnstile> C\<preceq> D ==> G\<turnstile>C \<preceq>? D" subcls: "G\<turnstile> D\<preceq>C C ==> G\<turnstile>Class C \<preceq>? Class D" lemma widen_PrimT_RefT [iff]: "(G\<turnstile>PrimT pT\<preceq>RefT rT) = False" apply (rule iffI) apply (erule widen.elims) apply auto done lemma widen_RefT: "G\<turnstile>RefT R\<preceq>T ==> ∃t. T=RefT t" apply (ind_cases "G\<turnstile>S\<preceq>T") apply auto done lemma widen_RefT2: "G\<turnstile>S\<preceq>RefT R ==> ∃t. S=RefT t" apply (ind_cases "G\<turnstile>S\<preceq>T") apply auto done lemma widen_Class: "G\<turnstile>Class C\<preceq>T ==> ∃D. T=Class D" apply (ind_cases "G\<turnstile>S\<preceq>T") apply auto done lemma widen_Class_NullT [iff]: "(G\<turnstile>Class C\<preceq>NT) = False" apply (rule iffI) apply (ind_cases "G\<turnstile>S\<preceq>T") apply auto done lemma widen_Class_Class [iff]: "(G\<turnstile>Class C\<preceq> Class D) = (G\<turnstile>C\<preceq>C D)" apply (rule iffI) apply (ind_cases "G\<turnstile>S\<preceq>T") apply (auto elim: widen.subcls) done lemma widen_NT_Class [simp]: "G \<turnstile> T \<preceq> NT ==> G \<turnstile> T \<preceq> Class D" by (ind_cases "G \<turnstile> T \<preceq> NT", auto) lemma cast_PrimT_RefT [iff]: "(G\<turnstile>PrimT pT\<preceq>? RefT rT) = False" apply (rule iffI) apply (erule cast.elims) apply auto done lemma cast_RefT: "G \<turnstile> C \<preceq>? Class D ==> ∃ rT. C = RefT rT" apply (erule cast.cases) apply simp apply (erule widen.cases) apply auto done theorem widen_trans[trans]: "[|G\<turnstile>S\<preceq>U; G\<turnstile>U\<preceq>T|] ==> G\<turnstile>S\<preceq>T" proof - assume "G\<turnstile>S\<preceq>U" thus "!!T. G\<turnstile>U\<preceq>T ==> G\<turnstile>S\<preceq>T" proof induct case (refl T T') thus "G\<turnstile>T\<preceq>T'" . next case (subcls C D T) then obtain E where "T = Class E" by (blast dest: widen_Class) with subcls show "G\<turnstile>Class C\<preceq>T" by (auto elim: rtrancl_trans) next case (null R RT) then obtain rt where "RT = RefT rt" by (blast dest: widen_RefT) thus "G\<turnstile>NT\<preceq>RT" by auto qed qed end
lemma subcls1D:
G |- C <=C1 D ==> C ≠ Object ∧ (∃fs ms. class G C = Some (D, fs, ms))
lemma subcls1_def2:
subcls1 G = (SIGMA C:{C. is_class G C}. {D. C ≠ Object ∧ fst (the (class G C)) = D})
lemma finite_subcls1:
finite (subcls1 G)
lemma subcls_is_class:
(C, D) ∈ (subcls1 G)+ ==> is_class G C
lemma subcls_is_class2:
[| G |- C <=C D; is_class G D |] ==> is_class G C
lemma class_rec_lemma:
[| wf ((subcls1 G)^-1); class G C = Some (D, fs, ms) |] ==> class_rec G C t f = f C fs ms (if C = Object then t else class_rec G D t f)
lemma method_rec_lemma:
[| class G C = Some (D, fs, ms); wf ((subcls1 G)^-1) |] ==> method (G, C) = (if C = Object then empty else method (G, D)) ++ map_of (map (%(s, m). (s, C, m)) ms)
lemma fields_rec_lemma:
[| class G C = Some (D, fs, ms); wf ((subcls1 G)^-1) |] ==> fields (G, C) = map (%(fn, ft). ((fn, C), ft)) fs @ (if C = Object then [] else fields (G, D))
lemma field_fields:
field (G, C) fn = Some (fd, fT) ==> map_of (fields (G, C)) (fn, fd) = Some fT
lemma widen_PrimT_RefT:
G |- PrimT pT <= RefT rT = False
lemma widen_RefT:
G |- RefT R <= T ==> ∃t. T = RefT t
lemma widen_RefT2:
G |- S <= RefT R ==> ∃t. S = RefT t
lemma widen_Class:
G |- Class C <= T ==> ∃D. T = Class D
lemma widen_Class_NullT:
G |- Class C <= NT = False
lemma widen_Class_Class:
G |- Class C <= Class D = G |- C <=C D
lemma widen_NT_Class:
G |- T <= NT ==> G |- T <= Class D
lemma cast_PrimT_RefT:
G |- PrimT pT <=? RefT rT = False
lemma cast_RefT:
G |- C <=? Class D ==> ∃rT. C = RefT rT
theorem widen_trans:
[| G |- S <= U; G |- U <= T |] ==> G |- S <= T