source: src/common/Identifiers.ma @ 1878

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

let rec fresh (tag:String) (u:universe tag) on u : identifier tag × (universe tag) ≝
  let id ≝ next_identifier ? u in
  〈an_identifier tag id, mk_universe tag (succ id)〉.


let rec fresh_for_univ tag (id:identifier tag) (u:universe tag) on id : Prop ≝
  match id with [ an_identifier p ⇒ p < next_identifier … u ].


lemma fresh_is_fresh : ∀tag,id,u,u'.
  〈id,u〉 = fresh tag u' →
  fresh_for_univ tag id u.
#tag * #id * #u * #u' #E whd in E:(???%); destruct //
qed.

lemma fresh_remains_fresh : ∀tag,id,id',u,u'.
  fresh_for_univ tag id u →
  〈id',u'〉 = fresh tag u →
  fresh_for_univ tag id u'.
#tag * #id * #id' * #u * #u' normalize #H #E destruct /2 by le_S/
qed.

lemma fresh_distinct : ∀tag,id,id',u,u'.
  fresh_for_univ tag id u →
  〈id',u'〉 = fresh tag u →
  id ≠ id'.
#tag * #id * #id' * #u * #u' normalize #H #E destruct % #E' destruct /2 by absurd/
qed.


let rec env_fresh_for_univ tag A (env:list (identifier tag × A)) (u:universe tag) on u : Prop ≝
  All ? (λida. fresh_for_univ tag (\fst ida) u) env.

lemma fresh_env_extend : ∀tag,A,env,u,u',id,a.
  env_fresh_for_univ tag A env u →
  〈id,u'〉 = fresh tag u →
  env_fresh_for_univ tag A (〈id,a〉::env) u'.
#tag #A #env * #u * #u' #id #a
#H #E whd % [ @(fresh_is_fresh … E) | @(All_mp … H) * #id #a #H' /2 by fresh_remains_fresh/ ]
qed.

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 by / | * #H @F % #E destruct /2 by / ]
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 by not_to_not/
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 by absurd/ ]
qed.

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


(* States that all identifiers in an environment are distinct from one another. *)
let rec distinct_env tag (A:Type[0]) (l:list (identifier tag × A)) on l : Prop ≝
match l with
[ nil ⇒ True
| cons hd tl ⇒ All ? (λia. \fst hd ≠ \fst ia) tl ∧
               distinct_env tag A tl
].

lemma distinct_env_append_l : ∀tag,A,l,r. distinct_env tag A (l@r) → distinct_env tag A l.
#tag #A #l elim l
[ //
| * #id #a #tl #IH #r * #H1 #H2 % /2 by All_append_l/
] qed.

lemma distinct_env_append_r : ∀tag,A,l,r. distinct_env tag A (l@r) → distinct_env tag A r.
#tag #A #l elim l
[ //
| * #id #a #tl #IH #r * #H1 #H2 /2 by /
] qed.

(* check_distinct_env is quadratic - we could pay more attention when building maps to make sure that
   the original environment was distinct. *)

axiom DuplicateVariable : String.

let rec check_member_env tag (A:Type[0]) (id:identifier tag) (l:list (identifier tag × A)) on l : res (All ? (λia. id ≠ \fst ia) l) ≝
match l return λl.res (All ?? l) with
[ nil ⇒ OK ? I
| cons hd tl ⇒
    match identifier_eq tag id (\fst hd) with
    [ inl _ ⇒ Error ? [MSG DuplicateVariable; CTX ? id]
    | inr NE ⇒
        do Htl ← check_member_env tag A id tl;
        OK ? (conj ?? NE Htl)
    ]
].

let rec check_distinct_env tag (A:Type[0]) (l:list (identifier tag × A)) on l : res (distinct_env tag A l) ≝
match l return λl.res (distinct_env tag A l) with
[ nil ⇒ OK ? I
| cons hd tl ⇒
    do Hhd ← check_member_env tag A (\fst hd) tl;
    do Htl ← check_distinct_env tag A tl;
    OK ? (conj ?? Hhd Htl)
].




(* 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 by not_to_not/
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 by or_intror, sym_not_eq/
] 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.


let rec fresh_for_map tag A (id:identifier tag) (m:identifier_map tag A) on id : Prop ≝
  lookup … m id = None A.

lemma fresh_for_empty_map : ∀tag,A,id.
  fresh_for_map tag A id (empty_map tag A).
#tag #A * #id //
qed.

definition fresh_map_for_univ ≝
λtag,A. λm:identifier_map tag A. λu:universe tag.
  ∀id. present tag A m id → fresh_for_univ tag id u.

lemma fresh_fresh_for_map : ∀tag,A,m,id,u,u'.
  fresh_map_for_univ tag A m u →
  〈id,u'〉 = fresh tag u →
  fresh_for_map tag A id m.
#tag #A * #m * #id * #u * #u' whd in ⊢ (% → ???% → %);
#FMU #E destruct lapply (FMU (an_identifier tag u)) whd in ⊢ ((% → %) → ?);
generalize in ⊢ ((?(??%?) → ?) → ??%?); *
[ // | #a #H @False_ind lapply (H ?) /2 by absurd/ % #E destruct
qed.

lemma fresh_map_preserved : ∀tag,A,m,u,u',id.
  fresh_map_for_univ tag A m u →
  〈id,u'〉 = fresh tag u →
  fresh_map_for_univ tag A m u'.
#tag #A #m #u * #u' #id whd in ⊢ (% → ? → %); #H #E
#id' #PR @(fresh_remains_fresh … E) @H //
qed.

lemma fresh_map_add : ∀tag,A,m,u,id,a.
  fresh_map_for_univ tag A m u →
  fresh_for_univ tag id u →
  fresh_map_for_univ tag A (add tag A m id a) u.
#tag #A * #m #u #id #a #Hm #Hi
#id' #PR cases (identifier_eq tag id' id)
[ #E >E @Hi
| #NE @Hm whd in PR;
  change with (add tag A (an_id_map tag A m) id a) in PR:(?(??(???%?)?));
  >lookup_add_miss in PR; //
] qed.

lemma present_not_fresh : ∀tag,A,m,id,id'.
  present tag A m id →
  fresh_for_map tag A id' m →
  id ≠ id'.
#tag #A #m #id * #id' whd in ⊢ (% → % → ?);
* #NE #E % #E' destruct @(NE E)
qed.

lemma fresh_for_map_add : ∀tag,A,id,m,id',a.
  id ≠ id' →
  fresh_for_map tag A id m →
  fresh_for_map tag A id (add tag A m id' a).
#tag #A * #id #m #id' #a #NE #F
whd >lookup_add_miss //
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.