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
 . Null is zero. Method succ adds an object in front, method add
  as many objects in front as the parameter specifies.

 , 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
  into lists of references based on given lists of variable and field
  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