theory OG_Syntax
imports OG_Tactics Quote_Antiquote
begin

text‹Syntax for commands and for assertions and boolean expressions in
 commands ‹com› and annotated commands ‹ann_com›.›

abbreviation Skip :: "'a com"  (‹SKIP› 63)
  where "SKIP ≡ Basic id"

abbreviation AnnSkip :: "'a assn → 'a ann_com"  (‹_//SKIP› [90] 63)
  where "r SKIP ≡ AnnBasic r id"

notation
  Seq  (‹_,,/ _› [55, 56] 55) and
  AnnSeq  (‹_;;/ _› [60,61] 60)

syntax
  "_Assign"      :: "idt → 'b → 'a com"    (‹(🚫_ :=/ _)› [70, 65] 61)
  "_AnnAssign"   :: "'a assn → idt → 'b → 'a com"    (‹(_ 🚫_ :=/ _)› [90,70,65] 61)

translations
  "🚫x := a" ⇀ "CONST Basic «🚫(_update_name x (λ_. a))¬"
  "r 🚫x := a" ⇀ "CONST AnnBasic r «🚫(_update_name x (λ_. a))¬"

syntax
  "_AnnCond1"    :: "'a assn → 'a bexp → 'a ann_com → 'a ann_com → 'a ann_com"
                    (‹_ //IF _ /THEN _ /ELSE _ /FI›  [90,0,0,0] 61)
  "_AnnCond2"    :: "'a assn → 'a bexp → 'a ann_com → 'a ann_com"
                    (‹_ //IF _ /THEN _ /FI›  [90,0,0] 61)
  "_AnnWhile"    :: "'a assn → 'a bexp → 'a assn → 'a ann_com → 'a ann_com"
                    (‹_ //WHILE _ /INV _ //DO _//OD›  [90,0,0,0] 61)
  "_AnnAwait"    :: "'a assn → 'a bexp → 'a com → 'a ann_com"
                    (‹_ //AWAIT _ /THEN /_ /END›  [90,0,0] 61)
  "_AnnAtom"     :: "'a assn → 'a com → 'a ann_com"   (‹_//⟨_⟩› [90,0] 61)
  "_AnnWait"     :: "'a assn → 'a bexp → 'a ann_com"   (‹_//WAIT _ END› [90,0] 61)

  "_Cond"        :: "'a bexp → 'a com → 'a com → 'a com"
                                  (‹(0IF _/ THEN _/ ELSE _/ FI)› [0, 0, 0] 61)
  "_Cond2"       :: "'a bexp → 'a com → 'a com"   (‹IF _ THEN _ FI› [0,0] 56)
  "_While_inv"   :: "'a bexp → 'a assn → 'a com → 'a com"
                    (‹(0WHILE _/ INV _ //DO _ /OD)›  [0, 0, 0] 61)
  "_While"       :: "'a bexp → 'a com → 'a com"
                    (‹(0WHILE _ //DO _ /OD)›  [0, 0] 61)

translations
  "IF b THEN c1 ELSE c2 FI" ⇀ "CONST Cond {b} c1 c2"
  "IF b THEN c FI" ⇌ "IF b THEN c ELSE SKIP FI"
  "WHILE b INV i DO c OD" ⇀ "CONST While {b} i c"
  "WHILE b DO c OD" ⇌ "WHILE b INV CONST undefined DO c OD"

  "r IF b THEN c1 ELSE c2 FI" ⇀ "CONST AnnCond1 r {b} c1 c2"
  "r IF b THEN c FI" ⇀ "CONST AnnCond2 r {b} c"
  "r WHILE b INV i DO c OD" ⇀ "CONST AnnWhile r {b} i c"
  "r AWAIT b THEN c END" ⇀ "CONST AnnAwait r {b} c"
  "r ⟨c⟩" ⇌ "r AWAIT CONST True THEN c END"
  "r WAIT b END" ⇌ "r AWAIT b THEN SKIP END"

nonterminal prgs

syntax
  "_PAR" :: "prgs → 'a"              (‹COBEGIN//_//COEND› [57] 56)
  "_prg" :: "['a, 'a] → prgs"        (‹_//_› [60, 90] 57)
  "_prgs" :: "['a, 'a, prgs] → prgs"  (‹_//_//∥//_› [60,90,57] 57)

  "_prg_scheme" :: "['a, 'a, 'a, 'a, 'a] → prgs"
                  (‹SCHEME [_ ≤ _ < _] _// _› [0,0,0,60, 90] 57)

translations
  "_prg c q" ⇌ "[(CONST Some c, q)]"
  "_prgs c q ps" ⇌ "(CONST Some c, q) # ps"
  "_PAR ps" ⇌ "CONST Parallel ps"

  "_prg_scheme j i k c q" ⇌ "CONST map (λi. (CONST Some c, q)) [j..<k]"

print_translation ‹
 let
 fun quote_tr' f (t :: ts) =
 Term.list_comb (f $ Syntax_Trans.quote_tr' syntax_const‹_antiquote› t, ts)
 | quote_tr' _ _ = raise Match;

 fun annquote_tr' f (r :: t :: ts) =
 Term.list_comb (f $ r $ Syntax_Trans.quote_tr' syntax_const‹_antiquote› t, ts)
 | annquote_tr' _ _ = raise Match;

 val assert_tr' = quote_tr' (Syntax.const syntax_const‹_Assert›);

 fun bexp_tr' name ((Const (const_syntax‹Collect›, _) $ t) :: ts) =
 quote_tr' (Syntax.const name) (t :: ts)
 | bexp_tr' _ _ = raise Match;

 fun annbexp_tr' name (r :: (Const (const_syntax‹Collect›, _) $ t) :: ts) =
 annquote_tr' (Syntax.const name) (r :: t :: ts)
 | annbexp_tr' _ _ = raise Match;

 fun assign_tr' (Abs (x, _, f $ k $ Bound 0) :: ts) =
 quote_tr' (Syntax.const syntax_const‹_Assign› $ Syntax_Trans.update_name_tr' f)
 (Abs (x, dummyT, Syntax_Trans.const_abs_tr' k) :: ts)
 | assign_tr' _ = raise Match;

 fun annassign_tr' (r :: Abs (x, _, f $ k $ Bound 0) :: ts) =
 quote_tr' (Syntax.const syntax_const‹_AnnAssign› $ r $ Syntax_Trans.update_name_tr' f)
 (Abs (x, dummyT, Syntax_Trans.const_abs_tr' k) :: ts)
 | annassign_tr' _ = raise Match;

 fun Parallel_PAR [(Const (const_syntax‹Cons›, _) $
 (Const (const_syntax‹Pair›, _) $ (Const (const_syntax‹Some›,_) $ t1 ) $ t2) $
 Const (const_syntax‹Nil›, _))] = Syntax.const syntax_const‹_prg› $ t1 $ t2
 | Parallel_PAR [(Const (const_syntax‹Cons›, _) $
 (Const (const_syntax‹Pair›, _) $ (Const (const_syntax‹Some›, _) $ t1) $ t2) $ ts)] =
 Syntax.const syntax_const‹_prgs› $ t1 $ t2 $ Parallel_PAR [ts]
 | Parallel_PAR _ = raise Match;

 fun Parallel_tr' ts = Syntax.const syntax_const‹_PAR› $ Parallel_PAR ts;
 in
 [(const_syntax‹Collect›, K assert_tr'),
 (const_syntax‹Basic›, K assign_tr'),
 (const_syntax‹Cond›, K (bexp_tr' syntax_const‹_Cond›)),
 (const_syntax‹While›, K (bexp_tr' syntax_const‹_While_inv›)),
 (const_syntax‹AnnBasic›, K annassign_tr'),
 (const_syntax‹AnnWhile›, K (annbexp_tr' syntax_const‹_AnnWhile›)),
 (const_syntax‹AnnAwait›, K (annbexp_tr' syntax_const‹_AnnAwait›)),
 (const_syntax‹AnnCond1›, K (annbexp_tr' syntax_const‹_AnnCond1›)),
 (const_syntax‹AnnCond2›, K (annbexp_tr' syntax_const‹_AnnCond2›))]
 end
 ›

end