Theory TypeRel

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theory TypeRel
imports Decl

(*  Title:      HOL/MicroJava/J/TypeRel.thy
ID: $Id$
Author: David von Oheimb
Copyright 1999 Technische Universitaet Muenchen
*)


header {* \isaheader{Relations between Java Types} *}

theory TypeRel imports Decl begin

-- "direct subclass, cf. 8.1.3"


inductive_set
subcls1 :: "'c prog => (cname × cname) set"
and subcls1' :: "'c prog => cname => cname => bool" ("_ \<turnstile> _ \<prec>C1 _" [71,71,71] 70)
for G :: "'c prog"
where
"G \<turnstile> C \<prec>C1 D ≡ (C, D) ∈ subcls1 G"
| subcls1I: "[|class G C = Some (D,rest); C ≠ Object|] ==> G \<turnstile> C \<prec>C1 D"


abbreviation
subcls :: "'c prog => cname => cname => bool" ("_ \<turnstile> _ \<preceq>C _" [71,71,71] 70)
where "G \<turnstile> C \<preceq>C D ≡ (C, D) ∈ (subcls1 G)^*"


lemma subcls1D:
"G\<turnstile>C\<prec>C1D ==> C ≠ Object ∧ (∃fs ms. class G C = Some (D,fs,ms))"

apply (erule subcls1.cases)
apply auto
done

lemma subcls1_def2:
"subcls1 P =
(SIGMA C:{C. is_class P C}. {D. C≠Object ∧ fst (the (class P C))=D})"

by (auto simp add: is_class_def dest: subcls1D intro: subcls1I)

lemma finite_subcls1: "finite (subcls1 G)"
apply(simp add: subcls1_def2 del: mem_Sigma_iff)
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

definition class_rec :: "'c prog => cname => 'a =>
(cname => fdecl list => 'c mdecl list => 'a => 'a) => 'a"
where
"class_rec G == wfrec ((subcls1 G)^-1)
(λr C t f. case class G C of
None => undefined
| Some (D,fs,ms) =>
f C fs ms (if C = Object then t else r D t f))"


lemma class_rec_lemma:
assumes wf: "wf ((subcls1 G)^-1)"
and cls: "class G C = Some (D, fs, ms)"
shows "class_rec G C t f = f C fs ms (if C=Object then t else class_rec G D t f)"

proof -
from wf have step: "!!H a. wfrec ((subcls1 G)¯) H a =
H (cut (wfrec ((subcls1 G)¯) H) ((subcls1 G)¯) a) a"

by (rule wfrec)
have cut: "!!f. C ≠ Object ==> cut f ((subcls1 G)¯) C D = f D"
by (rule cut_apply [where r="(subcls1 G)^-1", simplified, OF subcls1I, OF cls])
from cls show ?thesis by (simp add: step cut class_rec_def)
qed

definition
"wf_class G = wf ((subcls1 G)^-1)"



text {* Code generator setup (FIXME!) *}

consts_code
"wfrec" ("\<module>wfrec?")
attach {*
fun wfrec f x = f (wfrec f) x;
*}


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 :: "'c prog => [ty , ty ] => bool" ("_ \<turnstile> _ \<preceq> _" [71,71,71] 70)
for G :: "'c prog"
where
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"


lemmas refl = HOL.refl

-- "casting conversion, cf. 5.5 / 5.1.5"
-- "left out casts on primitve types"

inductive
cast :: "'c prog => [ty , ty ] => bool" ("_ \<turnstile> _ \<preceq>? _" [71,71,71] 70)
for G :: "'c prog"
where
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.cases)
apply auto
done

lemma widen_RefT: "G\<turnstile>RefT R\<preceq>T ==> ∃t. T=RefT t"
apply (ind_cases "G\<turnstile>RefT R\<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>RefT R")
apply auto
done

lemma widen_Class: "G\<turnstile>Class C\<preceq>T ==> ∃D. T=Class D"
apply (ind_cases "G\<turnstile>Class C\<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>Class C\<preceq>NT")
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>Class C \<preceq> Class D")
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.cases)
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
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