source: src/common/Identifiers.ma @ 1624

include "basics/types.ma".
include "ASM/String.ma".
include "utilities/binary/positive.ma".
include "common/Errors.ma".

(* identifiers and their generators are tagged to differentiate them, and to
   provide extra type checking. *)

(* in common/PreIdentifiers.ma, via Errors.ma.
inductive identifier (tag:String) : Type[0] ≝
  an_identifier : Pos → identifier tag.
*)

record universe (tag:String) : Type[0] ≝
{
  next_identifier : Pos
}.

definition new_universe : ∀tag:String. universe tag ≝
  λtag. mk_universe tag one.

(* Fresh identifier generation uses delayed overflow checking.  To make sure
   that the identifiers really were fresh, use the check_universe_ok function
   below afterwards. *)
definition fresh : ∀tag:String. universe tag → identifier tag × (universe tag) ≝
  λtag.
  λuniv: universe tag.
  let id ≝ next_identifier ? univ in
  〈an_identifier tag id, mk_universe tag (succ id)〉.

definition eq_identifier : ∀t. identifier t → identifier t → bool ≝
  λt,l,r.
  match l with
  [ an_identifier l' ⇒
    match r with
    [ an_identifier r' ⇒
      eqb l' r'
    ]
  ].

lemma eq_identifier_elim : ∀P:bool → Type[0]. ∀t,x,y.
  (x = y → P true) → (x ≠ y → P false) →
  P (eq_identifier t x y).
#P #t * #x * #y #T #F
change with (P (eqb ??))
@(eqb_elim x y P) [ /2/ | * #H @F % #E destruct /2/ ]
qed.
    
definition word_of_identifier ≝
  λt.
  λl: identifier t.
  match l with    
  [ an_identifier l' ⇒ l'
  ].

lemma eq_identifier_refl : ∀tag,id. eq_identifier tag id id = true.
#tag * #id whd in ⊢ (??%?); >eqb_n_n @refl
qed.

axiom eq_identifier_sym:
  ∀tag: String.
  ∀l  : identifier tag.
  ∀r  : identifier tag.
    eq_identifier tag l r = eq_identifier tag r l.

lemma eq_identifier_false : ∀tag,x,y. x≠y → eq_identifier tag x y = false.
#tag * #x * #y #NE normalize @not_eq_to_eqb_false /2/
qed.

definition identifier_eq : ∀tag:String. ∀x,y:identifier tag. (x=y) + (x≠y).
#tag * #x * #y lapply (refl ? (eqb x y)) cases (eqb x y) in ⊢ (???% → %);
#E [ % | %2 ]
lapply E @eqb_elim
[ #H #_ >H @refl | 2,3: #_ #H destruct | #H #_ % #H' destruct /2/ ]
qed.

definition identifier_of_nat : ∀tag:String. nat → identifier tag ≝
  λtag,n. an_identifier tag (succ_pos_of_nat  n).


(* Maps from identifiers to arbitrary types. *)

include "common/PositiveMap.ma".

inductive identifier_map (tag:String) (A:Type[0]) : Type[0] ≝
  an_id_map : positive_map A → identifier_map tag A.
  
definition empty_map : ∀tag:String. ∀A. identifier_map tag A ≝
  λtag,A. an_id_map tag A (pm_leaf A).

let rec lookup tag A (m:identifier_map tag A) (l:identifier tag) on m : option A ≝
  lookup_opt A (match l with [ an_identifier l' ⇒ l' ])
               (match m with [ an_id_map m' ⇒ m' ]).

definition lookup_def ≝
λtag,A,m,l,d. match lookup tag A m l with [ None ⇒ d | Some x ⇒ x].

let rec member tag A (m:identifier_map tag A) (l:identifier tag) on m : bool ≝
  match lookup tag A m l with [ None ⇒ false | _ ⇒ true ].

(* Always adds the identifier to the map. *)
let rec add tag A (m:identifier_map tag A) (l:identifier tag) (a:A) on m : identifier_map tag A ≝
  an_id_map tag A (insert A (match l with [ an_identifier l' ⇒ l' ]) a
                            (match m with [ an_id_map m' ⇒ m' ])).

lemma lookup_add_hit : ∀tag,A,m,i,a.
  lookup tag A (add tag A m i a) i = Some ? a.
#tag #A * #m * #i #a
@lookup_opt_insert_hit
qed.

lemma lookup_def_add_hit : ∀tag,A,m,i,a,d.
  lookup_def tag A (add tag A m i a) i d = a.
#tag #A * #m * #i #a #d
@lookup_insert_hit
qed.

lemma lookup_add_miss : ∀tag,A,m,i,j,a.
  i ≠ j →
  lookup tag A (add tag A m j a) i = lookup tag A m i.
#tag #A * #m * #i * #j #a #H
@lookup_opt_insert_miss /2/
qed.

axiom lookup_def_add_miss : ∀tag,A,m,i,j,a,d.
  i ≠ j →
  lookup_def tag A (add tag A m j a) i d = lookup_def tag A m i d.

lemma lookup_add_oblivious : ∀tag,A,m,i,j,a.
  (lookup tag A m i ≠ None ?) →
  lookup tag A (add tag A m j a) i ≠ None ?.
#tag #A #m #i #j #a #H
cases (identifier_eq ? i j)
[ #E >E >lookup_add_hit % #N destruct
| #NE >lookup_add_miss //
] qed.

lemma lookup_add_cases : ∀tag,A,m,i,j,a,v.
  lookup tag A (add tag A m i a) j = Some ? v →
  (i=j ∧ v = a) ∨ lookup tag A m j = Some ? v.
#tag #A #m #i #j #a #v
cases (identifier_eq ? i j)
[ #E >E >lookup_add_hit #H %1 destruct % //
| #NE >lookup_add_miss /2/
] qed. 

(* Extract every identifier, value pair from the map. *)
definition elements : ∀tag,A. identifier_map tag A → list (identifier tag × A) ≝
λtag,A,m.
  fold ?? (λl,a,el. 〈an_identifier tag l, a〉::el)
          (match m with [ an_id_map m' ⇒ m' ]) [ ].

axiom MissingId : String.

(* Only updates an existing entry; fails with an error otherwise. *)
definition update : ∀tag,A. identifier_map tag A → identifier tag → A → res (identifier_map tag A) ≝
λtag,A,m,l,a.
  match update A (match l with [ an_identifier l' ⇒ l' ]) a
                 (match m with [ an_id_map m' ⇒ m' ]) with
  [ None ⇒ Error ? ([MSG MissingId; CTX tag l]) (* missing identifier *)
  | Some m' ⇒ OK ? (an_id_map tag A m')
  ].

definition foldi:
  ∀A, B: Type[0].
  ∀tag: String.
  (identifier tag -> A -> B -> B) -> identifier_map tag A -> B -> B ≝
λA,B,tag,f,m,b.
  match m with
  [ an_id_map m' ⇒ fold A B (λbv. f (an_identifier ? bv)) m' b ].

(* A predicate that an identifier is in a map, and a failure-avoiding lookup
   and update using it. *)

definition present : ∀tag,A. identifier_map tag A → identifier tag → Prop ≝
λtag,A,m,i. lookup … m i ≠ None ?.

lemma member_present : ∀tag,A,m,id.
  member tag A m id = true → present tag A m id.
#tag #A * #m #id normalize cases (lookup_opt A ??) normalize
[ #E destruct
| #x #E % #E' destruct
] qed.

include "ASM/Util.ma".

definition lookup_present : ∀tag,A. ∀m:identifier_map tag A. ∀id. present ?? m id → A ≝
λtag,A,m,id. match lookup ?? m id return λx. x ≠ None ? → ? with [ Some a ⇒ λ_. a | None ⇒ λH.⊥ ].
cases H #H'  cases (H' (refl ??)) qed.

lemma lookup_lookup_present : ∀tag,A,m,id,p.
  lookup tag A m id = Some ? (lookup_present tag A m id p).
#tag #A #m #id #p
whd in p ⊢ (???(??%));
cases (lookup tag A m id) in p ⊢ %;
[ * #H @⊥ @H @refl
| #a #H @refl
] qed.

definition update_present : ∀tag,A. ∀m:identifier_map tag A. ∀id. present ?? m id → A → identifier_map tag A ≝
λtag,A,m,l,p,a.
  let l' ≝ match l with [ an_identifier l' ⇒ l' ] in
  let m' ≝ match m with [ an_id_map m' ⇒ m' ] in
  let u' ≝ update A l' a m' in
  match u' return λx. update ???? = x → ? with
  [ None ⇒ λE.⊥
  | Some m' ⇒ λ_. an_id_map tag A m'
  ] (refl ? u').
cases l in p E; cases m; -l' -m' #m' #l'
whd in ⊢ (% → ?);
 whd in ⊢ (?(??(???%%)?) → ??(??%?%)? → ?);
#NL #U cases NL #H @H @(update_fail … U)
qed.

lemma update_still_present : ∀tag,A,m,id,a,id'.
  ∀H:present tag A m id.
  ∀H':present tag A m id'.
  present tag A (update_present tag A m id' H' a) id.
#tag #A * #m * #id #a * #id' #H #H'
whd whd in ⊢ (?(??(???(%??????)?)?)); normalize nodelta
cases (identifier_eq ? (an_identifier tag id) (an_identifier tag id'))
[ #E >E @refute_none_by_refl #m' #U whd in ⊢ (?(??%?)); >(update_lookup_opt_same ????? U)
  % #E' destruct
| #NE @refute_none_by_refl #m' #U whd in ⊢ (?(??%?)); whd in ⊢ (?(??(??%%)?));
  <(update_lookup_opt_other ????? U id) [ @H | % #E cases NE >E #H @H @refl ]
] qed.

(* Sets *)

inductive identifier_set (tag:String) : Type[0] ≝
  an_id_set : positive_map unit → identifier_set tag.

definition empty_set : ∀tag:String. identifier_set tag ≝
λtag. an_id_set tag (pm_leaf unit).

let rec add_set (tag:String) (s:identifier_set tag) (i:identifier tag) on s : identifier_set tag ≝
  an_id_set tag (insert unit (match i with [ an_identifier i' ⇒ i' ])
                          it (match s with [ an_id_set s' ⇒ s' ])).

definition singleton_set : ∀tag:String. identifier tag → identifier_set tag ≝
λtag,i. add_set tag (empty_set tag) i.

let rec mem_set (tag:String) (s:identifier_set tag) (i:identifier tag) on s : bool ≝
  match lookup_opt ? (match i with [ an_identifier i' ⇒ i' ])
                     (match s with [ an_id_set s' ⇒ s' ]) with
  [ None ⇒ false
  | Some _ ⇒ true
  ].

let rec union_set (tag:String) (s:identifier_set tag) (s':identifier_set tag) on s : identifier_set tag ≝
  an_id_set tag (merge unit (match s with [ an_id_set s0 ⇒ s0 ])
                            (match s' with [ an_id_set s1 ⇒ s1 ])).

interpretation "identifier set union" 'union a b = (union_set ? a b).
notation "∅" non associative with precedence 90 for @{ 'empty }.
interpretation "empty identifier set" 'empty = (empty_set ?).
interpretation "singleton identifier set" 'singl a = (add_set ? (empty_set ?) a).
interpretation "identifier set membership" 'mem a b = (mem_set ? b a).

lemma union_empty_l : ∀tag.∀s:identifier_set tag. ∅ ∪ s = s.
#tag * //
qed.

lemma union_empty_r : ∀tag.∀s:identifier_set tag. s ∪ ∅ = s.
#tag * * // qed.