Library Coq.Array.PArray

Require Import Int63.










Module Export PArrayNotations.

Delimit Scope array_scope with array.
Notation "t .[ i ]" := (get t i)
  (at level 2, left associativity, format "t .[ i ]").
Notation "t .[ i <- a ]" := (set t i a)
  (at level 2, left associativity, format "t .[ i <- a ]").

End PArrayNotations.

Local Open Scope int63_scope.
Local Open Scope array_scope.


Axioms
Axiom get_out_of_bounds : forall A (t:array A) i, (i <? length t) = false -> t.[i] = default t.

Axiom get_set_same : forall A t i (a:A), (i <? length t) = true -> t.[i<-a].[i] = a.
Axiom get_set_other : forall A t i j (a:A), i <> j -> t.[i<-a].[j] = t.[j].
Axiom default_set : forall A t i (a:A), default t.[i<-a] = default t.

Axiom get_make : forall A (a:A) size i, (make size a).[i] = a.

Axiom leb_length : forall A (t:array A), length t <=? max_length = true.

Axiom length_make : forall A size (a:A),
  length (make size a) = if size <=? max_length then size else max_length.
Axiom length_set : forall A t i (a:A),
  length t.[i<-a] = length t.

Axiom get_copy : forall A (t:array A) i, (copy t).[i] = t.[i].
Axiom length_copy : forall A (t:array A), length (copy t) = length t.

Axiom get_reroot : forall A (t:array A) i, (reroot t).[i] = t.[i].
Axiom length_reroot : forall A (t:array A), length (reroot t) = length t.

Axiom array_ext : forall A (t1 t2:array A),
  length t1 = length t2 ->
  (forall i, i <? length t1 = true -> t1.[i] = t2.[i]) ->
  default t1 = default t2 ->
  t1 = t2.


Lemma default_copy A (t:array A) : default (copy t) = default t.

Lemma default_make A (a : A) size : default (make size a) = a.

Lemma default_reroot A (t:array A) : default (reroot t) = default t.

Lemma get_set_same_default A (t : array A) (i : int) :
  t.[i <- default t].[i] = default t.

Lemma get_not_default_lt A (t:array A) x :
 t.[x] <> default t -> (x <? length t) = true.