include "basics/types.ma".
include "utilities/binary/positive.ma".
include "utilities/lists.ma".
include "utilities/extralib.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:identifierTag) : Type[0] ≝
  an_identifier : Pos → identifier tag.
*)

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

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

let rec fresh (tag:identifierTag) (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.

lemma eq_identifier_eq:
  ∀tag: identifierTag.
  ∀l.
  ∀r.
    eq_identifier tag l r = true → l = r.
  #tag #l #r cases l cases r
  #pos_l #pos_r
  cases pos_l cases pos_r
  [1:
    #_ %
  |2,3,4,7:
    #p1_l normalize in ⊢ (% → ?);
    #absurd destruct(absurd)
  |5,9:
    #p1_l #p1_r normalize in ⊢ (% → ?);
    #relevant lapply (eqb_true_to_eq … relevant) #EQ >EQ %
  |*:
    #p_l #p_r normalize in ⊢ (% → ?);
    #absurd destruct(absurd)
  ]
qed.

axiom neq_identifier_neq:
  ∀tag: identifierTag.
  ∀l, r: identifier tag.
    eq_identifier tag l r = false → (l = r → False).

include "basics/deqsets.ma".
definition Deq_identifier : identifierTag → DeqSet ≝ λtag.
  mk_DeqSet (identifier tag) (eq_identifier tag) ?.
#x#y @eq_identifier_elim /2 by conj/ * #H % [#ABS destruct(ABS) | #G elim (H G)]
qed.

unification hint 0 ≔ tag; D ≟ Deq_identifier tag
(*-----------------------------------------------------*)⊢
identifier tag ≡ carr D.

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: identifierTag.
  ∀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:identifierTag. ∀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:identifierTag. 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. *)

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:identifierTag) (A:Type[0]) : Type[0] ≝
  an_id_map : positive_map A → identifier_map tag A.
  
definition empty_map : ∀tag:identifierTag. ∀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].

definition member ≝
  λtag,A.λm:identifier_map tag A.λl:identifier tag.
  match lookup tag A m l with [ None ⇒ false | _ ⇒ true ].

interpretation "identifier map membership" 'mem a b = (member ?? b a).

definition lookup_safe : ∀tag,A.∀m : identifier_map tag A.∀i.i∈m → A ≝
λtag,A,m,i.
match lookup … m i return λx.match x in option return λ_.bool with [ _ ⇒ ?] → ? with
[ Some x ⇒ λ_.x
| None ⇒ λprf.⊥
]. @prf qed.

lemma lookup_eq_safe : ∀tag,A,m,i,prf.lookup tag A m i = Some ? (lookup_safe tag A m i prf).
#tag #A #m #i whd in match (i∈m);
whd in match lookup_safe; normalize nodelta
cases (lookup ????) normalize nodelta [*] // qed.

(* 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' ]) [ ].

(* Test a predicate on all of the entries in a map.  The predicate is given a
   proof that the entry appears in the map. *)

definition idmap_all : ∀tag,A. ∀m:identifier_map tag A. (∀id,a. lookup tag A m id = Some A a → bool) → bool ≝
λtag,A,m,f. pm_all A (match m with [ an_id_map m' ⇒ m' ])
                     (λp,a,H. f (an_identifier tag p) a ?).
cases m in H ⊢ %; #m' normalize //
qed.

inductive idmap_pred_graph : ∀tag,A,m,id,a,L. ∀f:(∀id,a. lookup tag A m id = Some A a → bool). bool → Prop ≝
| idmappg : ∀tag,A,m,id,a,L,f. idmap_pred_graph tag A m id a L f (f id a L).

lemma idmap_pred_irr : ∀tag,A,m,id,a,L,L'. ∀f:(∀id,a. lookup tag A m id = Some A a → bool).
  f id a L = f id a L'.
#tag #A #m #id #a #L #L' #f
cut (idmap_pred_graph tag A m id a L f (f id a L)) [ % ]
cases (f id a L) #H
cut (idmap_pred_graph tag A m id a L' f ?) [ 2,5: @H | 1,4: skip ] * //
qed.

lemma idmap_all_ok : ∀tag,A,m,f.
  bool_to_Prop (idmap_all tag A m f) ↔ (∀id,a,H. f id a H).
#tag #A * #m #f
whd in match (idmap_all ????); @(iff_trans … (pm_all_ok …)) %
[ #H * #id #a #PR lapply (H id a PR) #X @eq_true_to_b <X @idmap_pred_irr
| #H #p #a #PR @H
] qed.


(* 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')
  ].

(* Remove an entry from a map (and leave it equivalent, otherwise) *)
definition remove : ∀tag,A. identifier_map tag A → identifier tag → identifier_map tag A ≝
λtag,A,m,id. an_id_map tag A (pm_set A (match id with [ an_identifier p ⇒ p ]) (None ?)
                                       (match m with [ an_id_map m ⇒ m ])).

lemma lookup_remove_hit : ∀tag,A,m,id.
  lookup tag A (remove tag A m id) id = None ?.
#tag #A * #m * #id @lookup_opt_pm_set_hit
qed.

lemma lookup_remove_miss : ∀tag,A,m,id,id'.
  id ≠ id' →
  lookup tag A (remove tag A m id') id = lookup tag A m id.
#tag #A * #m * #id * #id' #NE
@lookup_opt_pm_set_miss
/2 by not_to_not/
qed.

(* Fold over the entries in a map.  There are some lemmas to help reason about
   this near the bottom of the file (they require sets). *)

definition foldi:
  ∀A, B: Type[0].
  ∀tag: identifierTag.
  (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 ].

(* An informative, dependently-typed, fold. *)

definition fold_inf:
  ∀A, B: Type[0].
  ∀tag: identifierTag.
  ∀m:identifier_map tag A.
  (∀id:identifier tag. ∀a:A. lookup … m id = Some A a → B → B) → B → B ≝
λA,B,tag,m.
  match m return λm. (∀id:identifier tag. ∀a:A. lookup … m id = Some A a → B → B) → B → B with
  [ an_id_map m' ⇒ λf,b. pm_fold_inf A B m' (λbv,a,H. f (an_identifier ? bv) a H) b ].

(* Find one element of a map that satisfies a predicate *)
definition find : ∀tag,A. identifier_map tag A → (identifier tag → A → bool) →
  option (identifier tag × A) ≝
λtag,A,m,p.
  match m with [ an_id_map m' ⇒
    option_map … (λx. 〈an_identifier tag (\fst x), \snd x〉)
      (pm_find … m' (λid. p (an_identifier tag id))) ].

lemma find_lookup : ∀tag,A,m,p,id,a.
  find tag A m p = Some ? 〈id,a〉 →
  lookup … m id = Some ? a.
#tag #A * #m #p * #id #a #FIND
@(pm_find_lookup A (λid. p (an_identifier tag id)) id a m)
whd in FIND:(??%?); cases (pm_find ???) in FIND ⊢ %;
[ normalize #E destruct
| * #id' #a' normalize #E destruct %
] qed.

lemma find_none : ∀tag,A,m,p,id,a.
  find tag A m p = None ? →
  lookup … m id = Some ? a →
  ¬ p id a.
#tag #A * #m #p * #id #a #FIND
@(pm_find_none A m (λid. p (an_identifier tag id)))
whd in FIND:(??%?); cases (pm_find ???) in FIND ⊢ %;
[ normalize #E destruct %
| * #id' #a' normalize #E destruct
] qed.


lemma find_predicate : ∀tag,A,m,p,id,a.
  find tag A m p = Some ? 〈id,a〉 →
  p id a.
#tag #A * #m #p * #id #a #FIND whd in FIND:(??%?);
@(pm_find_predicate A m (λid. p (an_identifier tag id)) id a)
cases (pm_find ???) in FIND ⊢ %;
[ normalize #E destruct
| * #id' #a' normalize #E destruct %
] qed.

(* 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.

lemma present_member : ∀tag,A,m,id.
  present tag A m id → member tag A m id.
#tag #A #m #id whd in ⊢ (% → ?%); cases (lookup ????) // * #H cases (H (refl ??))
qed.

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.

lemma lookup_is_present : ∀tag,T,m,id,t.
  lookup tag T m id = Some T t →
  present ?? m id.
#tag #T #m #id #t #L normalize >L % #E destruct
qed.

lemma lookup_present_eq : ∀tag,T,m,id,t.
  lookup tag T m id = Some T t →
  ∀H. lookup_present tag T m id H = t.
#tag #T #m #id #t #L #H
lapply (lookup_lookup_present … H) >L #E destruct %
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.

lemma lookup_present_add_hit:
  ∀tag, A, map, k, v, k_pres.
    lookup_present tag A (add … map k v) k k_pres = v.
  #tag #a #map #k #v #k_pres
  lapply (lookup_lookup_present … (add … map k v) … k_pres)
  >lookup_add_hit #Some_assm destruct(Some_assm)
  <e0 %
qed.

lemma lookup_present_add_miss:
  ∀tag, A, map, k, k', v, k_pres', k_pres''.
    k' ≠ k →
      lookup_present tag A (add … map k v) k' k_pres' = lookup_present tag A map k' k_pres''.
  #tag #A #map #k #k' #v #k_pres' #k_pres'' #neq_assm
  lapply (lookup_lookup_present … (add … map k v) ? k_pres')
  >lookup_add_miss try assumption
  #Some_assm
  lapply (lookup_lookup_present … map k') >Some_assm #Some_assm'
  lapply (Some_assm' k_pres'') #Some_assm'' destruct assumption
qed.

lemma present_add_present:
  ∀tag, a, map, k, k', v.
    k' ≠ k →
      present tag a (add tag a map k v) k' → 
        present tag a map k'.
  #tag #a #map #k #k' #v #neq_hyp #present_hyp
  whd in match present; normalize nodelta
  whd in match present in present_hyp; normalize nodelta in present_hyp;
  cases (not_None_to_Some a … present_hyp) #v' #Some_eq_hyp
  lapply (lookup_add_cases tag ?????? Some_eq_hyp) *
  [1:
    * #k_eq_hyp @⊥ /2 by absurd/
  |2:
    #Some_eq_hyp' /2 by /
  ]
qed.

lemma present_add_hit:
  ∀tag, a, map, k, v.
    present tag a (add tag a map k v) k.
  #tag #a #map #k #v
  whd >lookup_add_hit
  % #absurd destruct
qed.

lemma present_add_miss:
  ∀tag, a, map, k, k', v.
    present tag a map k' → present tag a (add tag a map k v) k'.
  #tag #a #map #k #k' #v #present_assm
  whd @lookup_add_oblivious assumption
qed.

lemma present_add_cases: ∀tag,A,map,k,v,k'.
  present tag A (add tag A map k v) k' →
  k = k' ∨ (k ≠ k' ∧ present tag A map k').
#tag #A #map #k #v #k' normalize
cases (identifier_eq ? k k')
[ #E /2 by or_introl/
| #NE >lookup_add_miss /3 by or_intror, conj, absurd, nmk/
] 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.

(* Extending the domain of a map (without necessarily preserving contents). *)

definition extends_domain : ∀tag,A. identifier_map tag A → identifier_map tag A → Prop ≝
λtag,A,m1,m2. ∀l. present ?? m1 l → present ?? m2 l.

lemma extends_dom_trans : ∀tag,A,m1,m2,m3.
  extends_domain tag A m1 m2 → extends_domain tag A m2 m3 → extends_domain tag A m1 m3.
#tag #A #m1 #m2 #m3 #H1 #H2 #l #P1 @H2 @H1 @P1 qed.


(* Sets *)

definition identifier_set ≝ λtag.identifier_map tag unit.

definition empty_set : ∀tag.identifier_set tag ≝ λtag.empty_map ….


definition add_set : ∀tag.identifier_set tag → identifier tag → identifier_set tag ≝
  λtag,s,i.add … s i it.

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

let rec union_set (tag:identifierTag) A B (s:identifier_map tag A) (s':identifier_map tag B) on s : identifier_set tag ≝
  an_id_map tag unit (merge … (λo,o'.match o with [Some _ ⇒ Some ? it | None ⇒ !_ o'; return it])
    (match s with [ an_id_map s0 ⇒ s0 ])
    (match s' with [ an_id_map s1 ⇒ s1 ])).


(* set minus is generalised to maps *)
let rec minus_set (tag:identifierTag) A B (s:identifier_map tag A) (s':identifier_map tag B) on s : identifier_map tag A ≝
  an_id_map tag A (merge A B A (λo,o'.match o' with [None ⇒ o | Some _ ⇒ None ?])
    (match s with [ an_id_map s0 ⇒ s0 ])
    (match s' with [ an_id_map s1 ⇒ s1 ])).

notation "a ∖ b" left associative with precedence 55 for @{'setminus $a $b}.

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 map difference" 'setminus a b = (minus_set ??? a b).

definition IdentifierSet : identifierTag → Setoid ≝ λtag.
  mk_Setoid (identifier_set tag) (λs,s'.∀i.i ∈ s = (i ∈ s')) ???.
  // qed.

unification hint 0 ≔ tag;
S ≟ IdentifierSet tag 
(*-----------------------------*)⊢
identifier_set tag ≡ std_supp S.
unification hint 0 ≔ tag;
S ≟ IdentifierSet tag 
(*-----------------------------*)⊢
identifier_map tag unit ≡ std_supp S.

lemma mem_set_add : ∀tag,A.∀i,j : identifier tag.∀s,x.
  i ∈ add ? A s j x = (eq_identifier ? i j ∨ i ∈ s).
#tag #A *#i *#j *#s #x normalize
@(eqb_elim i j)
[#EQ destruct
  >(lookup_opt_insert_hit A x j)
|#NEQ >(lookup_opt_insert_miss … s NEQ)
] elim (lookup_opt  A j s) normalize // qed.

lemma mem_set_add_id : ∀tag,A,i,s,x.bool_to_Prop (i ∈ add tag A s i x).
#tag #A #i #s #x >mem_set_add
@eq_identifier_elim [#_ %| #ABS elim (absurd … (refl ? i) ABS)] qed.

lemma in_map_domain : ∀tag,A.∀m : identifier_map tag A.∀i.
  if i ∈ m then (∃s.lookup … m i = Some ? s) else (lookup … m i = None ?).
#tag #A * #m * #i normalize
elim (lookup_opt A i m) normalize
[ % | #x %{x} % ]
qed.
(*
lemma union_empty_l : ∀tag.∀s:identifier_set tag. ∅ ∪ s = s.
#tag * normalize #m >map_opt_id_eq_ext // * %
qed.

lemma union_empty_r : ∀tag.∀s:identifier_set tag. s ∪ ∅ = s.
#tag * * [//] *[2: *] #l#r normalize
>map_opt_id_eq_ext [1,3: >map_opt_id_eq_ext [2,4: *] |*: *] //
qed.

lemma minus_empty_l : ∀tag,A.∀s:identifier_map tag A. ∅ ∖ s ≅ ∅.
#tag #A * * [//] *[2:#x]#l#r * * normalize [1,4://]
#p >lookup_opt_map elim (lookup_opt ???) normalize //
qed.

lemma minus_empty_r : ∀tag,A.∀s:identifier_map tag A. s ∖ ∅ = s.
#tag #A * * [//] *[2:#x]#l#r normalize
>map_opt_id >map_opt_id //
qed.
*)
lemma mem_set_union : ∀tag.∀i : identifier tag.∀s,s' : identifier_set tag.
  i ∈ (s ∪ s') = (i ∈ s ∨ i ∈ s').
#tag * #i * #s * #s' normalize
>lookup_opt_merge [2: @refl]
elim (lookup_opt ???)
elim (lookup_opt ???)
normalize // qed.

lemma mem_set_minus : ∀tag,A,B.∀i : identifier tag.∀s : identifier_map tag A.
  ∀s' : identifier_map tag B.
  i ∈ (s ∖ s') = (i ∈ s ∧ ¬ i ∈ s').
#tag #A #B * #i * #s * #s' normalize
>lookup_opt_merge [2: @refl]
elim (lookup_opt ???)
elim (lookup_opt ???)
normalize // qed.

lemma set_eq_ext_node : ∀tag.∀o,o',l,l',r,r'.
  an_id_map tag ? (pm_node ? o l r) ≅ an_id_map … (pm_node ? o' l' r') →
    o = o' ∧ an_id_map tag ? l ≅ an_id_map … l' ∧ an_id_map tag ? r ≅ an_id_map … r'.
#tag#o#o'#l#l'#r#r'#H
%[
%[ lapply (H (an_identifier ? one))
   elim o [2: *] elim o' [2,4: *] normalize // #EQ destruct
 | *#p lapply (H (an_identifier ? (p0 p))) normalize //
]| *#p lapply (H (an_identifier ? (p1 p))) normalize //
]
qed.

lemma set_eq_ext_leaf : ∀tag,A.∀o,l,r.
  (∀i.i∈an_id_map tag A (pm_node ? o l r) = false) →
    o = None ? ∧ (∀i.i∈an_id_map tag ? l = false) ∧ (∀i.i∈an_id_map tag ? r = false).
#tag#A#o#l#r#H
%[
%[ lapply (H (an_identifier ? one))
   elim o [2: #a] normalize // #EQ destruct
 | *#p lapply (H (an_identifier ? (p0 p))) normalize //
]| *#p lapply (H (an_identifier ? (p1 p))) normalize //
]
qed.


definition id_map_size : ∀tag : identifierTag.∀A. identifier_map tag A → ℕ ≝
  λtag,A,s.match s with [an_id_map p ⇒ |p|].

interpretation "identifier map domain size" 'card s = (id_map_size ?? s).

lemma set_eq_ext_empty_to_card : ∀tag,A.∀s : identifier_map tag A. (∀i.i∈s = false) → |s| = 0.
#tag#A * #s elim s [//]
#o#l#r normalize in ⊢((?→%)→(?→%)→?); #Hil #Hir #H
elim (set_eq_ext_leaf … H) * #EQ destruct #Hl #Hr normalize
>(Hil Hl) >(Hir Hr) // qed.

lemma set_eq_ext_to_card : ∀tag.∀s,s' : identifier_set tag. s ≅ s' → |s| = |s'|.
#tag *#s elim s
[** [//] #o#l#r #H
  >(set_eq_ext_empty_to_card … (std_symm … H)) //
| #o#l#r normalize in ⊢((?→?→??%?)→(?→?→??%?)→?);
  #Hil #Hir **
  [#H @(set_eq_ext_empty_to_card … H)]
  #o'#l'#r' #H elim (set_eq_ext_node … H) * #EQ destruct(EQ) #Hl #Hr
  normalize >(Hil ? Hl) >(Hir ? Hr) //
] qed.

lemma add_size: ∀tag,A,s,i,x.
  |add tag A s i x| = (if i ∈ s then 0 else 1) + |s|.
#tag #A *#s *#i #x
lapply (insert_size ? i x s)
lapply (refl ? (lookup_opt ? i s))
generalize in ⊢ (???%→?); * [2: #x']
normalize #EQ >EQ normalize //
qed.

lemma mem_set_O_lt_card : ∀tag,A.∀i.∀s : identifier_map tag A. i ∈ s → |s| > 0.
#tag #A * #i * #s normalize #H
@(lookup_opt_O_lt_size … i)
% #EQ >EQ in H; normalize *
qed.

(* NB: no control on values if applied to maps *)
definition set_subset ≝ λtag,A,B.λs : identifier_map tag A.
  λs' : identifier_map tag B. ∀i.i ∈ s → (bool_to_Prop (i ∈ s')).

interpretation "identifier set subset" 'subseteq s s' = (set_subset ??? s s').

lemma add_subset :
  ∀tag,A,B.∀i : identifier tag.∀x.∀s : identifier_map ? A.∀s' : identifier_map ? B.
    i ∈ s' → s ⊆ s' → add … s i x ⊆ s'.
#tag#A#B#i#x#s#s' #H #G #j
>mem_set_add
@eq_identifier_elim #H' [* >H' @H | #js @(G ? js)]
qed.

definition set_forall : ∀tag,A.(identifier tag → Prop) →
  identifier_map tag A → Prop ≝ λtag,A,P,m.∀i. i ∈ m → P i.
  
lemma set_forall_add : ∀tag,P,m,i.set_forall tag ? P m → P i →
  set_forall tag ? P (add_set ? m i).
#tag#P#m#i#Pm#Pi#j
>mem_set_add
@eq_identifier_elim
[#EQ destruct(EQ) #_ @Pi
|#_ @Pm
]
qed.

include "utilities/proper.ma".

lemma minus_subset : ∀tag,A,B.minus_set tag A B ⊨ set_subset … ++> set_subset … -+> set_subset ….
#tag#A#B#s#s' #H #s'' #s''' #G #i
>mem_set_minus >mem_set_minus
#H' elim (andb_Prop_true … H') -H' #is #nis''
>(H … is) 
elim (true_or_false_Prop (i∈s'''))
[ #is''' >(G … is''') in nis''; *
| #nis''' >nis''' %
]
qed.

lemma subset_node : ∀tag,A,B.∀o,o',l,l',r,r'.
  an_id_map tag A (pm_node ? o l r) ⊆ an_id_map tag B (pm_node ? o' l' r') →
    opt_All ? (λ_.o' ≠ None ?) o ∧ an_id_map tag ? l ⊆ an_id_map tag  ? l' ∧
      an_id_map tag ? r ⊆ an_id_map tag ? r'.
#tag#A#B#o#o'#l#l'#r#r'#H
%[%
  [ lapply (H (an_identifier ? (one))) elim o [2: #a] elim o' [2:#b]
    normalize // [#_ % #ABS destruct(ABS) | #G lapply (G I) *]
  | *#p lapply (H (an_identifier ? (p0 p)))
  ]
 | *#p lapply (H (an_identifier ? (p1 p)))
] #H @H
qed.

lemma subset_leaf : ∀tag,A.∀o,l,r.
  an_id_map tag A (pm_node ? o l r) ⊆ ∅ →
    o = None ? ∧ (∀i.i∈an_id_map tag ? l = false) ∧ (∀i.i∈an_id_map tag ? r = false).
#tag#A#o#l#r#H
%[
%[ lapply (H (an_identifier ? one))
   elim o [2: #a] normalize // #EQ lapply(EQ I) *
 | *#p lapply (H (an_identifier ? (p0 p)))
 ]
|  *#p lapply (H (an_identifier ? (p1 p)))
] normalize elim (lookup_opt ? p ?) normalize
// #a #H lapply (H I) *
qed.

lemma subset_card : ∀tag,A,B.∀s : identifier_map tag A.∀s' : identifier_map tag B.
  s ⊆ s' → |s| ≤ |s'|.
#tag #A #B *#s elim s
[ //
| #o#l#r #Hil #Hir **
  [ #H elim (subset_leaf … H) * #EQ >EQ #Hl #Hr
    lapply (set_eq_ext_empty_to_card … Hl)
    lapply (set_eq_ext_empty_to_card … Hr)
    normalize //
  | #o' #l' #r' #H elim (subset_node … H) *
    elim o [2: #a] elim o' [2,4: #a']
    [3: #G normalize in G; elim(absurd ? (refl ??) G)
    |*: #_ #Hl #Hr lapply (Hil ? Hl) lapply (Hir ? Hr)
      normalize #H1 #H2
      [@le_S_S | @(transitive_le … (|l'|+|r'|)) [2: / by /]]
      @le_plus assumption
    ]
  ]
]
qed.

lemma mem_set_empty : ∀tag,A.∀i: identifier tag. i∈empty_map tag A = false.
#tag #A * #i normalize %
qed.

lemma mem_set_singl_to_eq : ∀tag.∀i,j : identifier tag.i∈{(j)} → i = j.
#tag
#i #j >mem_set_add >mem_set_empty
#H elim (orb_true_l … H) -H
[@eq_identifier_elim [//] #_] #EQ destruct
qed.

lemma subset_add_set : ∀tag,i,s.s ⊆ add_set tag s i.
#tag#i#s#j #H >mem_set_add >H
>commutative_orb %
qed.

lemma add_set_monotonic : ∀tag,i,s,s'.s ⊆ s' → add_set tag s i ⊆ add_set tag s' i.
#tag#i#s#s' #H #j >mem_set_add >mem_set_add
@orb_elim elim (eq_identifier ???)
whd lapply (H j) /2 by /
qed.

lemma transitive_subset : ∀tag,A.transitive ? (set_subset tag A A).
#tag#A#s#s'#s''#H#G#i #is
@(G … (H … is))
qed.

definition set_from_list : ∀tag.list (identifier tag) → identifier_map tag unit ≝
  λtag.foldl … (add_set ?) ∅.

coercion id_set_from_list : ∀tag.∀l : list (identifier tag).identifier_map tag unit ≝
  set_from_list on _l : list (identifier ?) to identifier_map ? unit.

lemma mem_map_domain : ∀tag,A.∀m : identifier_map tag A.∀i.
i∈m → lookup … m i ≠ None ?.
#tag#A * #m #i
whd in match (i∈?);
elim (lookup ????) normalize [2: #x]
* % #EQ destruct(EQ)
qed.



lemma mem_list_as_set : ∀tag.∀l : list (identifier tag).
  ∀i.i ∈ l → In ? l i.
#tag #l @(list_elim_left … l)
[ #i *
| #t #h #Hi  #i
  whd in ⊢ (?(???%?)→?);
  >foldl_append
  whd in ⊢ (?(???%?)→?);
  >mem_set_add
  @eq_identifier_elim
  [ #EQi destruct(EQi)
    #_ @Exists_append_r % %
  | #_ #H @Exists_append_l @Hi assumption
  ]
]
qed.

lemma list_as_set_mem : ∀tag.∀l : list (identifier tag).
  ∀i.In ? l i → i ∈ l.
#tag #l @(list_elim_left … l)
[ #i *
| #t #h #Hi #i #H
  whd in ⊢ (?(???%?));
  >foldl_append
  whd in ⊢ (?(???%?));
  elim (Exists_append … H) -H
  [ #H >mem_set_add
    @eq_identifier_elim [//] #_ normalize
    @Hi @H
  | * [2: *] #EQi destruct(EQi) >mem_set_add_id %
  ]
]
qed.

lemma list_as_set_All : ∀tag,P.∀ l : list (identifier tag).
  (∀i.i ∈ l → P i) → All ? P l.
#tag #P #l @(list_elim_left … l)
[ #_ %
| #x #l' #Hi
  whd in match (set_from_list … (l'@[x]));
  >foldl_append
  #H @All_append
  [ @Hi #i #G @H
    whd in ⊢ (?(???%?));
    >mem_set_add @orb_Prop_r @G
  | % [2: %]
    @H
    whd in ⊢ (?(???%?));
    @mem_set_add_id
  ]
]
qed.

lemma All_list_as_set : ∀tag,P.∀ l : list (identifier tag).
  All ? P l → ∀i.i ∈ l → P i.
#tag #P #l @(list_elim_left … l)
[ * #i *
| #x #l' #Hi #H
  lapply (All_append_l … H)
  lapply (All_append_r … H)
  * #Px * #Pl' #i
  whd in match (set_from_list … (l'@[x]));
  >foldl_append
  >mem_set_add
  @eq_identifier_elim
  [ #EQx >EQx #_ @Px
  | #_ whd in match (?∨?); @Hi @Pl'
  ]
]
qed.  

lemma map_mem_prop : 
  ∀tag,A.∀m : identifier_map tag A.∀i.
  lookup ?? m i ≠ None ? → i ∈ m.
#p #globals #m #i
lapply (in_map_domain … m i)
cases (i∈m)
[ * #x #_ #_ %
| #EQ >EQ * #ABS @ABS %
] qed.


(* Attempt to choose an entry in the map/set, and if successful return the entry
   and the map/set without it. *)

definition choose : ∀tag,A. identifier_map tag A → option (identifier tag × A × (identifier_map tag A)) ≝
λtag,A,m.
  match pm_choose A (match m with [ an_id_map m' ⇒ m' ]) with
  [ None ⇒ None ?
  | Some x ⇒ Some ? 〈〈an_identifier tag (\fst (\fst x)), \snd (\fst x)〉, an_id_map tag A (\snd x)〉
  ].

lemma choose_empty : ∀tag,A,m.
  choose tag A m = None ? ↔ ∀id. lookup tag A m id = None ?.
#tag #A * #m lapply (pm_choose_empty A m) * #H1 #H2 % 
[ normalize #C * @H1 cases (pm_choose A m) in C ⊢ %; [ // | normalize #x #E destruct ]
| normalize #L lapply (pm_choose_empty A m) cases (pm_choose A m)
  [ * #H1 #H2 normalize // | #x * #_ #H lapply (H ?) [ #p @(L (an_identifier ? p)) | #E destruct ] ]
] qed.

lemma choose_some : ∀tag,A,m,id,a,m'.
  choose tag A m = Some ? 〈〈id,a〉,m'〉 →
  lookup tag A m id = Some A a ∧
  lookup tag A m' id = None A ∧
  (∀id'. id = id' ∨ lookup tag A m id' = lookup tag A m' id').
#tag #A * #m * #id #a * #m' #C
lapply (pm_choose_some A m id a m' ?)
[ whd in C:(??%?); cases (pm_choose A m) in C ⊢ %; normalize [ #E destruct | * * #x #y #z #E destruct % ] ]
* * * #L1 #L2 #L3 #_
% [ % [ @L1 | @L2 ] | * #id' cases (L3 id') [ /2 by or_introl/ | #L4 %2 @L4 ] ]
qed.

lemma choose_some_subset : ∀tag,A,m,id,a,m'.
  choose tag A m = Some ? 〈〈id,a〉,m'〉 →
  m' ⊆ m.
#tag #A #m #id #a #m' #C
cases (choose_some … m' C) * #L1 #L2 #L3
#id' whd in ⊢ (?% → ?%);
cases (L3 id')
[ #E destruct >L2 *
| #L4 >L4 //
] qed.

lemma choose_some_card : ∀tag,A,m,id,a,m'.
  choose tag A m = Some ? 〈〈id,a〉,m'〉 →
  |m| = S (|m'|).
#tag #A * #m * #id #a * #m' #C
lapply (pm_choose_some A m id a m' ?)
[ whd in C:(??%?); cases (pm_choose A m) in C ⊢ %; normalize [ #E destruct | * * #x #y #z #E destruct % ] ]
* #_ #H @H
qed.

(* Remove an element from a map/set, returning the element and a new map/set. *)

definition try_remove : ∀tag,A. identifier_map tag A → identifier tag → option (A × (identifier_map tag A)) ≝
λtag,A,m,id.
  match pm_try_remove A (match id with [ an_identifier id' ⇒ id']) (match m with [ an_id_map m' ⇒ m' ]) with
  [ None ⇒ None ?
  | Some x ⇒ Some ? 〈\fst x, an_id_map tag A (\snd x)〉
  ].

lemma try_remove_empty : ∀tag,A,m,id.
  try_remove tag A m id = None ? ↔ lookup tag A m id = None ?.
#tag #A * #m * #id lapply (pm_try_remove_none A id m) * #H1 #H2 % 
[ normalize #C @H1 cases (pm_try_remove A id m) in C ⊢ %; [ // | normalize #x #E destruct ]
| normalize #L >H2 //
] qed.

lemma try_remove_some : ∀tag,A,m,id,a,m'.
  try_remove tag A m id = Some ? 〈a,m'〉 →
  lookup tag A m id = Some A a ∧
  lookup tag A m' id = None A ∧
  (∀id'. id = id' ∨ lookup tag A m id' = lookup tag A m' id').
#tag #A * #m * #id #a * #m' #C
lapply (pm_try_remove_some A id m a m' ?)
[ whd in C:(??%?); cases (pm_try_remove A id m) in C ⊢ %; normalize [ #E destruct | * #x #y #E destruct % ] ]
* * * #L1 #L2 #L3 #_
% [ % [ @L1 | @L2 ] | * #id' cases (L3 id') [ /2 by or_introl/ | #L4 %2 @L4 ] ]
qed.

lemma try_remove_some_card : ∀tag,A,m,id,a,m'.
  try_remove tag A m id = Some ? 〈a,m'〉 →
  |m| = S (|m'|).
#tag #A * #m * #id #a * #m' #C
lapply (pm_try_remove_some A id m a m' ?)
[ whd in C:(??%?); cases (pm_try_remove A id m) in C ⊢ %; normalize [ #E destruct | * #x #y #E destruct % ] ]
* #_ #H @H
qed.

lemma try_remove_this : ∀tag,A,m,id,a.
  lookup tag A m id = Some A a →
  ∃m'. try_remove tag A m id = Some ? 〈a,m'〉.
#tag #A * #m * #id #a #L
cases (pm_try_remove_some' A id m a L)
#m' #R %{(an_id_map tag A m')} whd in ⊢ (??%?); >R %
qed.
  
(* Link a map with the set consisting of its domain. *)

definition id_set_of_map : ∀tag,A. identifier_map tag A → identifier_set tag ≝
λtag,A,m. an_id_map tag unit (map … (λ_. it) (match m with [ an_id_map m' ⇒ m'])).

lemma id_set_of_map_subset : ∀tag,A,m.
  id_set_of_map tag A m ⊆ m.
#tag #A * #m * #id normalize
>lookup_opt_map normalize cases (lookup_opt ???) //
qed.

lemma id_set_of_map_present : ∀tag,A,m,id.
  present tag A m id ↔ present tag unit (id_set_of_map … m) id.
#tag #A * #m * #id %
normalize @not_to_not
>lookup_opt_map cases (lookup_opt ???) normalize //
#a #E destruct
qed.

lemma id_set_of_map_card : ∀tag,A,m.
  |m| = |id_set_of_map tag A m|.
#tag #A * #m whd in ⊢ (??%%); >map_size //
qed.


(* Transforming a list into a set. *)

definition set_of_list : ∀tag. list (identifier tag) → identifier_set tag ≝
λtag,l. foldl ?? (λs,id. add_set tag s id) ∅ l.

lemma fold_add_set_monotone : ∀tag,l,s,id.
  present tag unit s id →
  present tag unit (foldl ?? (λs,id. add_set tag s id) s l) id.
#tag #l elim l
[ //
| #h #t #IH #s #id #PR
  whd in ⊢ (???%?); @IH
  @lookup_add_oblivious
  @PR
] qed. 

lemma in_set_of_list : ∀tag,l,id.
  Exists ? (λid'. id' = id) l →
  present ?? (set_of_list tag l) id.
#tag #l #id whd in match (set_of_list ??); generalize in match ∅; elim l
[ #s *
| #id' #tl #IH #s *
  [ #E whd in ⊢ (???%?); destruct
    @fold_add_set_monotone //
  | @IH
  ]
] qed.

lemma in_set_of_list' : ∀tag,l,id.
  present ?? (set_of_list tag l) id →
  Exists ? (λid'. id = id') l.
#tag #l #id whd in match (set_of_list ??);
cut (¬present ?? ∅ id) [ /3 by refl, absurd, nmk/ ]
generalize in match ∅;
elim l
[ #s #F #T @⊥ @(absurd … T F)
| #id' #tl #IH #s #F #PR whd in PR:(???%?);
  cases (identifier_eq … id id')
  [ #E destruct /2 by or_introl/
  | #NE %2 @(IH … PR)
    @(not_to_not … F) /2 by present_add_present/
  ]
] qed. 


(* Returns the domain of a map as the canonical set (one made only from the
   empty set and addition. *)
definition domain_of_map : ∀tag,A. identifier_map tag A → identifier_set tag ≝
λtag,A,m. an_id_map tag unit (domain_of_pm A (match m with [ an_id_map m ⇒ m ])).
lemma domain_of_map_present : ∀tag,A,m,id.
  present tag A m id ↔ present tag unit (domain_of_map … m) id.
#tag #A * #m * #p @domain_of_pm_present
qed.

(* Some lemmas for reasoning about folds. *)

lemma foldi_ind : ∀A,B,tag,f,m,b. ∀P:B → identifier_set tag → Prop.
  P b (empty_set …) →
  (∀k,ks,a,b. ¬present ?? ks k → lookup ?? m k = Some ? a → P b ks → P (f k a b) (add_set tag ks k)) →
  P (foldi A B tag f m b) (domain_of_map … m).
#A #B #tag #f * #m #b #P #P0 #STEP
whd in match (foldi ??????); change with (an_id_map ?? (domain_of_pm A m)) in match (domain_of_map ???);
@pm_fold_ind
[ @P0
| #p #ps #a #b0 #FR #L #pre @(STEP (an_identifier tag p) (an_id_map tag unit ps))
  [ normalize >FR /3 by absurd, nmk/
  | @L
  | @pre
  ]
] qed.

lemma foldi_ind' : ∀A,B,tag,f,m,b,b'. ∀P:B → identifier_set tag → Prop.
  P b (empty_set …) →
  (∀k,ks,a,b. ¬present ?? ks k → lookup ?? m k = Some ? a → P b ks → P (f k a b) (add_set tag ks k)) →
  foldi A B tag f m b = b' →
  P b' (domain_of_map … m).
#H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9 #H10 #H11  destruct @foldi_ind /2 by /
qed.