The Coq Proof Assistant

# Library Coq.Array.PArray

Require Import Uint63.
Primitive array := #array_type.
Primitive make : forall A, int -> A -> array A := #array_make.
Arguments make {_} _ _.
Primitive get : forall A, array A -> int -> A := #array_get.
Arguments get {_} _ _.
Primitive default : forall A, array A -> A:= #array_default.
Arguments default {_} _.
Primitive set : forall A, array A -> int -> A -> array A := #array_set.
Arguments set {_} _ _ _.
Primitive length : forall A, array A -> int := #array_length.
Arguments length {_} _.
Primitive copy : forall A, array A -> array A := #array_copy.
Arguments copy {_} _.
Module Export PArrayNotations.
Declare Scope array_scope.
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 uint63_scope.
Local Open Scope array_scope.
Primitive max_length := #array_max_length.
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 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 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.