Theory OO

theory OO imports Main begin

subsection "Towards an OO Language: A Language of Records"

(* FIXME: move to HOL/Fun *)
abbreviation fun_upd2 :: "('a ⇒ 'b ⇒ 'c) ⇒ 'a ⇒ 'b ⇒ 'c ⇒ 'a ⇒ 'b ⇒ 'c"
  ("_/'((2_,_ :=/ _)')" [1000,0,0,0] 900)
where "f(x,y := z) == f(x := (f x)(y := z))"

type_synonym addr = nat
datatype ref = null | Ref addr

type_synonym obj = "string ⇒ ref"
type_synonym venv = "string ⇒ ref"
type_synonym store = "addr ⇒ obj"

datatype exp =
  Null |
  New |
  V string |
  Faccess exp string       ("_∙/_" [63,1000] 63) |
  Vassign string exp       ("(_ ::=/ _)" [1000,61] 62) |
  Fassign exp string exp   ("(_∙_ ::=/ _)" [63,0,62] 62) |
  Mcall exp string exp     ("(_∙/_<_>)" [63,0,0] 63) |
  Seq exp exp              ("_;/ _" [61,60] 60) |
  If bexp exp exp          ("IF _/ THEN (2_)/ ELSE (2_)" [0,0,61] 61)
and bexp = B bool | Not bexp | And bexp bexp | Eq exp exp

type_synonym menv = "string ⇒ exp"
type_synonym config = "venv × store × addr"

inductive
  big_step :: "menv ⇒ exp × config ⇒ ref × config ⇒ bool"
    ("(_ ⊢/ (_/ ⇒ _))" [60,0,60] 55) and
  bval ::  "menv ⇒ bexp × config ⇒ bool × config ⇒ bool"
    ("_ ⊢ _ → _" [60,0,60] 55)
where
Null:
"me ⊢ (Null,c) ⇒ (null,c)" |
New:
"me ⊢ (New,ve,s,n) ⇒ (Ref n,ve,s(n := (λf. null)),n+1)" |
Vaccess:
"me ⊢ (V x,ve,sn) ⇒ (ve x,ve,sn)" |
Faccess:
"me ⊢ (e,c) ⇒ (Ref a,ve',s',n') ⟹
 me ⊢ (e∙f,c) ⇒ (s' a f,ve',s',n')" |
Vassign:
"me ⊢ (e,c) ⇒ (r,ve',sn') ⟹
 me ⊢ (x ::= e,c) ⇒ (r,ve'(x:=r),sn')" |
Fassign:
"⟦ me ⊢ (oe,c1) ⇒ (Ref a,c2);  me ⊢ (e,c2) ⇒ (r,ve3,s3,n3) ⟧ ⟹
 me ⊢ (oe∙f ::= e,c1) ⇒ (r,ve3,s3(a,f := r),n3)" |
Mcall:
"⟦ me ⊢ (oe,c1) ⇒ (or,c2);  me ⊢ (pe,c2) ⇒ (pr,ve3,sn3);
   ve = (λx. null)(''this'' := or, ''param'' := pr);
   me ⊢ (me m,ve,sn3) ⇒ (r,ve',sn4) ⟧
  ⟹
 me ⊢ (oe∙m<pe>,c1) ⇒ (r,ve3,sn4)" for or |
Seq:
"⟦ me ⊢ (e1,c1) ⇒ (r,c2);  me ⊢ (e2,c2) ⇒ c3 ⟧ ⟹
 me ⊢ (e1; e2,c1) ⇒ c3" |
IfTrue:
"⟦ me ⊢ (b,c1) → (True,c2);  me ⊢ (e1,c2) ⇒ c3 ⟧ ⟹
 me ⊢ (IF b THEN e1 ELSE e2,c1) ⇒ c3" |
IfFalse:
"⟦ me ⊢ (b,c1) → (False,c2);  me ⊢ (e2,c2) ⇒ c3 ⟧ ⟹
 me ⊢ (IF b THEN e1 ELSE e2,c1) ⇒ c3" |

"me ⊢ (B bv,c) → (bv,c)" |

"me ⊢ (b,c1) → (bv,c2) ⟹ me ⊢ (Not b,c1) → (¬bv,c2)" |

"⟦ me ⊢ (b1,c1) → (bv1,c2);  me ⊢ (b2,c2) → (bv2,c3) ⟧ ⟹
 me ⊢ (And b1 b2,c1) → (bv1∧bv2,c3)" |

"⟦ me ⊢ (e1,c1) ⇒ (r1,c2);  me ⊢ (e2,c2) ⇒ (r2,c3) ⟧ ⟹
 me ⊢ (Eq e1 e2,c1) → (r1=r2,c3)"


code_pred (modes: i => i => o => bool) big_step .

text‹Example: natural numbers encoded as objects with a predecessor
field. Null is zero. Method succ adds an object in front, method add
adds as many objects in front as the parameter specifies.

First, the method bodies:›

definition
"m_succ  =  (''s'' ::= New)∙''pred'' ::= V ''this''; V ''s''"

definition "m_add =
  IF Eq (V ''param'') Null
  THEN V ''this''
  ELSE V ''this''∙''succ''<Null>∙''add''<V ''param''∙''pred''>"

text‹The method environment:›
definition
"menv = (λm. Null)(''succ'' := m_succ, ''add'' := m_add)"

text‹The main code, adding 1 and 2:›
definition "main =
  ''1'' ::= Null∙''succ''<Null>;
  ''2'' ::= V ''1''∙''succ''<Null>;
  V ''2'' ∙ ''add'' <V ''1''>"

text‹Execution of semantics. The final variable environment and store are
converted into lists of references based on given lists of variable and field
names to extract.›

values
 "{(r, map ve' [''1'',''2''], map (λn. map (s' n)[''pred'']) [0..<n])|
    r ve' s' n. menv ⊢ (main, λx. null, nth[], 0) ⇒ (r,ve',s',n)}"

end