Theory TypeRel

theory TypeRel
imports Decl Wfrec
(*  Title:      HOL/MicroJava/J/TypeRel.thy
Author: David von Oheimb, Technische Universitaet Muenchen
*)


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

theory TypeRel
imports Decl "~~/src/HOL/Library/Wfrec"
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 *}

code_pred
(modes: i => i => o => bool, i => i => i => bool)
subcls1p
.

declare subcls1_def [code_pred_def]

code_pred
(modes: i => i × o => bool, i => i × i => bool)
[inductify]
subcls1
.

definition subcls' where "subcls' G = (subcls1p G)^**"

code_pred
(modes: i => i => i => bool, i => i => o => bool)
[inductify]
subcls'
.

lemma subcls_conv_subcls' [code_unfold]:
"(subcls1 G)^* = {(C, D). subcls' G C D}"
by(simp add: subcls'_def subcls1_def rtrancl_def)

lemma class_rec_code [code]:
"class_rec G C t f =
(if wf_class G then
(case class G C of
None => class_rec G C t f
| Some (D, fs, ms) =>
if C = Object then f Object fs ms t else f C fs ms (class_rec G D t f))
else class_rec G C t f)"

apply(cases "wf_class G")
apply(unfold class_rec_def wf_class_def)
apply(subst wfrec, assumption)
apply(cases "class G C")
apply(simp add: wfrec)
apply clarsimp
apply(rename_tac D fs ms)
apply(rule_tac f="f C fs ms" in arg_cong)
apply(clarsimp simp add: cut_def)
apply(blast intro: subcls1I)
apply simp
done

lemma wf_class_code [code]:
"wf_class G <-> (∀(C, rest) ∈ set G. C ≠ Object --> ¬ G \<turnstile> fst (the (class G C)) \<preceq>C C)"
proof
assume "wf_class G"
hence wf: "wf (((subcls1 G)^+)^-1)" unfolding wf_class_def by(rule wf_converse_trancl)
hence acyc: "acyclic ((subcls1 G)^+)" by(auto dest: wf_acyclic)
show "∀(C, rest) ∈ set G. C ≠ Object --> ¬ G \<turnstile> fst (the (class G C)) \<preceq>C C"
proof(safe)
fix C D fs ms
assume "(C, D, fs, ms) ∈ set G"
and "C ≠ Object"
and subcls: "G \<turnstile> fst (the (class G C)) \<preceq>C C"
from `(C, D, fs, ms) ∈ set G` obtain D' fs' ms'
where "class": "class G C = Some (D', fs', ms')"
unfolding class_def by(auto dest!: weak_map_of_SomeI)
hence "G \<turnstile> C \<prec>C1 D'" using `C ≠ Object` ..
hence *: "(C, D') ∈ (subcls1 G)^+" ..
also from * acyc have "C ≠ D'" by(auto simp add: acyclic_def)
with subcls "class" have "(D', C) ∈ (subcls1 G)^+" by(auto dest: rtranclD)
finally show False using acyc by(auto simp add: acyclic_def)
qed
next
assume rhs[rule_format]: "∀(C, rest) ∈ set G. C ≠ Object --> ¬ G \<turnstile> fst (the (class G C)) \<preceq>C C"
have "acyclic (subcls1 G)"
proof(intro acyclicI strip notI)
fix C
assume "(C, C) ∈ (subcls1 G)+"
thus False
proof(cases)
case base
then obtain rest where "class G C = Some (C, rest)"
and "C ≠ Object" by cases
from `class G C = Some (C, rest)` have "(C, C, rest) ∈ set G"
unfolding class_def by(rule map_of_SomeD)
with `C ≠ Object` `class G C = Some (C, rest)`
have "¬ G \<turnstile> C \<preceq>C C" by(auto dest: rhs)
thus False by simp
next
case (step D)
from `G \<turnstile> D \<prec>C1 C` obtain rest where "class G D = Some (C, rest)"
and "D ≠ Object" by cases
from `class G D = Some (C, rest)` have "(D, C, rest) ∈ set G"
unfolding class_def by(rule map_of_SomeD)
with `D ≠ Object` `class G D = Some (C, rest)`
have "¬ G \<turnstile> C \<preceq>C D" by(auto dest: rhs)
moreover from `(C, D) ∈ (subcls1 G)+`
have "G \<turnstile> C \<preceq>C D" by(rule trancl_into_rtrancl)
ultimately show False by contradiction
qed
qed
thus "wf_class G" unfolding wf_class_def
by(rule finite_acyclic_wf_converse[OF finite_subcls1])
qed

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 [code]: "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 [code]: "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 [code]: "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"

code_pred widen .

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