20 files changed, 231 insertions, 24 deletions

diff --git a/tests/coq/betree/BetreeMain_Funs.v b/tests/coq/betree/BetreeMain_Funs.v
index 86a9d5f2..85aecfc8 100644
--- a/tests/coq/betree/BetreeMain_Funs.v
+++ b/tests/coq/betree/BetreeMain_Funs.v
@@ -3,6 +3,8 @@
 Require Import Primitives.
 Import Primitives.
 Require Import Coq.ZArith.ZArith.
+Require Import List.
+Import ListNotations.
 Local Open Scope Primitives_scope.
 Require Export BetreeMain_Types.
 Import BetreeMain_Types.
@@ -1190,4 +1192,7 @@ Definition betree_be_tree_lookup_back
 Definition main_fwd : result unit :=
   Return tt.
 
+(** Unit test for [betree_main::main] *)
+Check (main_fwd )%return.
+
 End BetreeMain_Funs .
diff --git a/tests/coq/betree/BetreeMain_Opaque.v b/tests/coq/betree/BetreeMain_Opaque.v
index bd49500b..ecd81b9d 100644
--- a/tests/coq/betree/BetreeMain_Opaque.v
+++ b/tests/coq/betree/BetreeMain_Opaque.v
@@ -3,6 +3,8 @@
 Require Import Primitives.
 Import Primitives.
 Require Import Coq.ZArith.ZArith.
+Require Import List.
+Import ListNotations.
 Local Open Scope Primitives_scope.
 Require Export BetreeMain_Types.
 Import BetreeMain_Types.
diff --git a/tests/coq/betree/BetreeMain_Types.v b/tests/coq/betree/BetreeMain_Types.v
index 25f280dd..c8af54cd 100644
--- a/tests/coq/betree/BetreeMain_Types.v
+++ b/tests/coq/betree/BetreeMain_Types.v
@@ -3,6 +3,8 @@
 Require Import Primitives.
 Import Primitives.
 Require Import Coq.ZArith.ZArith.
+Require Import List.
+Import ListNotations.
 Local Open Scope Primitives_scope.
 Module BetreeMain_Types.
 
diff --git a/tests/coq/betree/Primitives.v b/tests/coq/betree/Primitives.v
index ae961ac2..71a2d9c3 100644
--- a/tests/coq/betree/Primitives.v
+++ b/tests/coq/betree/Primitives.v
@@ -394,13 +394,15 @@ Notation "x s< y" := (scalar_ltb x y)  (at level 80) : Primitives_scope.
 Notation "x s>= y" := (scalar_geb x y)  (at level 80) : Primitives_scope.
 Notation "x s> y" := (scalar_gtb x y)  (at level 80) : Primitives_scope.
 
-(*** Vectors *)
-
-Definition vec T := { l: list T | Z.of_nat (length l) <= usize_max }.
+(*** Range *)
+Record range (T : Type) := mk_range {
+  start: T;
+  end_: T;
+}.
+Arguments mk_range {_}.
 
-Definition vec_to_list {T: Type} (v: vec T) : list T := proj1_sig v.
-
-Definition vec_length {T: Type} (v: vec T) : Z := Z.of_nat (length (vec_to_list v)).
+(*** Arrays *)
+Definition array T (n : usize) := { l: list T | Z.of_nat (length l) = to_Z n}.
 
 Lemma le_0_usize_max : 0 <= usize_max.
 Proof.
@@ -409,6 +411,47 @@ Proof.
   lia.
 Qed.
 
+Lemma eqb_imp_eq (x y : Z) : Z.eqb x y = true -> x = y.
+Proof.
+  lia.
+Qed.
+
+(* TODO: finish the definitions *)
+Axiom mk_array : forall (T : Type) (n : usize) (l : list T), array T n.
+
+Axiom array_index_shared : forall (T : Type) (n : usize) (x : array T n) (i : usize), result T.
+Axiom array_index_mut_fwd : forall (T : Type) (n : usize) (x : array T n) (i : usize), result T.
+Axiom array_index_mut_back : forall (T : Type) (n : usize) (x : array T n) (i : usize) (nx : T), result (array T n).
+
+(*** Slice *)
+Definition slice T := { l: list T | Z.of_nat (length l) <= usize_max}.
+
+Axiom slice_len : forall (T : Type) (s : slice T), usize.
+Axiom slice_index_shared : forall (T : Type) (x : slice T) (i : usize), result T.
+Axiom slice_index_mut_fwd : forall (T : Type) (x : slice T) (i : usize), result T.
+Axiom slice_index_mut_back : forall (T : Type) (x : slice T) (i : usize) (nx : T), result (slice T).
+
+(*** Subslices *)
+
+Axiom array_to_slice_shared : forall (T : Type) (n : usize) (x : array T n), result (slice T).
+Axiom array_to_slice_mut_fwd : forall (T : Type) (n : usize) (x : array T n), result (slice T).
+Axiom array_to_slice_mut_back : forall (T : Type) (n : usize) (x : array T n) (s : slice T), result (array T n).
+
+Axiom array_subslice_shared: forall (T : Type) (n : usize) (x : array T n) (r : range usize), result (slice T).
+Axiom array_subslice_mut_fwd: forall (T : Type) (n : usize) (x : array T n) (r : range usize), result (slice T).
+Axiom array_subslice_mut_back: forall (T : Type) (n : usize) (x : array T n) (r : range usize) (ns : slice T), result (array T n).
+Axiom slice_subslice_shared: forall (T : Type) (x : slice T) (r : range usize), result (slice T).
+Axiom slice_subslice_mut_fwd: forall (T : Type) (x : slice T) (r : range usize), result (slice T).
+Axiom slice_subslice_mut_back: forall (T : Type) (x : slice T) (r : range usize) (ns : slice T), result (slice T).
+
+(*** Vectors *)
+
+Definition vec T := { l: list T | Z.of_nat (length l) <= usize_max }.
+
+Definition vec_to_list {T: Type} (v: vec T) : list T := proj1_sig v.
+
+Definition vec_length {T: Type} (v: vec T) : Z := Z.of_nat (length (vec_to_list v)).
+
 Definition vec_new (T: Type) : vec T := (exist _ [] le_0_usize_max).
 
 Lemma vec_len_in_usize {T} (v: vec T) : usize_min <= vec_length v <= usize_max.
diff --git a/tests/coq/hashmap/Hashmap_Funs.v b/tests/coq/hashmap/Hashmap_Funs.v
index c8630eb6..c412abcd 100644
--- a/tests/coq/hashmap/Hashmap_Funs.v
+++ b/tests/coq/hashmap/Hashmap_Funs.v
@@ -3,6 +3,8 @@
 Require Import Primitives.
 Import Primitives.
 Require Import Coq.ZArith.ZArith.
+Require Import List.
+Import ListNotations.
 Local Open Scope Primitives_scope.
 Require Export Hashmap_Types.
 Import Hashmap_Types.
diff --git a/tests/coq/hashmap/Hashmap_Types.v b/tests/coq/hashmap/Hashmap_Types.v
index ce6e7dab..dbde6be9 100644
--- a/tests/coq/hashmap/Hashmap_Types.v
+++ b/tests/coq/hashmap/Hashmap_Types.v
@@ -3,6 +3,8 @@
 Require Import Primitives.
 Import Primitives.
 Require Import Coq.ZArith.ZArith.
+Require Import List.
+Import ListNotations.
 Local Open Scope Primitives_scope.
 Module Hashmap_Types.
 
diff --git a/tests/coq/hashmap/Primitives.v b/tests/coq/hashmap/Primitives.v
index ae961ac2..71a2d9c3 100644
--- a/tests/coq/hashmap/Primitives.v
+++ b/tests/coq/hashmap/Primitives.v
@@ -394,13 +394,15 @@ Notation "x s< y" := (scalar_ltb x y)  (at level 80) : Primitives_scope.
 Notation "x s>= y" := (scalar_geb x y)  (at level 80) : Primitives_scope.
 Notation "x s> y" := (scalar_gtb x y)  (at level 80) : Primitives_scope.
 
-(*** Vectors *)
-
-Definition vec T := { l: list T | Z.of_nat (length l) <= usize_max }.
+(*** Range *)
+Record range (T : Type) := mk_range {
+  start: T;
+  end_: T;
+}.
+Arguments mk_range {_}.
 
-Definition vec_to_list {T: Type} (v: vec T) : list T := proj1_sig v.
-
-Definition vec_length {T: Type} (v: vec T) : Z := Z.of_nat (length (vec_to_list v)).
+(*** Arrays *)
+Definition array T (n : usize) := { l: list T | Z.of_nat (length l) = to_Z n}.
 
 Lemma le_0_usize_max : 0 <= usize_max.
 Proof.
@@ -409,6 +411,47 @@ Proof.
   lia.
 Qed.
 
+Lemma eqb_imp_eq (x y : Z) : Z.eqb x y = true -> x = y.
+Proof.
+  lia.
+Qed.
+
+(* TODO: finish the definitions *)
+Axiom mk_array : forall (T : Type) (n : usize) (l : list T), array T n.
+
+Axiom array_index_shared : forall (T : Type) (n : usize) (x : array T n) (i : usize), result T.
+Axiom array_index_mut_fwd : forall (T : Type) (n : usize) (x : array T n) (i : usize), result T.
+Axiom array_index_mut_back : forall (T : Type) (n : usize) (x : array T n) (i : usize) (nx : T), result (array T n).
+
+(*** Slice *)
+Definition slice T := { l: list T | Z.of_nat (length l) <= usize_max}.
+
+Axiom slice_len : forall (T : Type) (s : slice T), usize.
+Axiom slice_index_shared : forall (T : Type) (x : slice T) (i : usize), result T.
+Axiom slice_index_mut_fwd : forall (T : Type) (x : slice T) (i : usize), result T.
+Axiom slice_index_mut_back : forall (T : Type) (x : slice T) (i : usize) (nx : T), result (slice T).
+
+(*** Subslices *)
+
+Axiom array_to_slice_shared : forall (T : Type) (n : usize) (x : array T n), result (slice T).
+Axiom array_to_slice_mut_fwd : forall (T : Type) (n : usize) (x : array T n), result (slice T).
+Axiom array_to_slice_mut_back : forall (T : Type) (n : usize) (x : array T n) (s : slice T), result (array T n).
+
+Axiom array_subslice_shared: forall (T : Type) (n : usize) (x : array T n) (r : range usize), result (slice T).
+Axiom array_subslice_mut_fwd: forall (T : Type) (n : usize) (x : array T n) (r : range usize), result (slice T).
+Axiom array_subslice_mut_back: forall (T : Type) (n : usize) (x : array T n) (r : range usize) (ns : slice T), result (array T n).
+Axiom slice_subslice_shared: forall (T : Type) (x : slice T) (r : range usize), result (slice T).
+Axiom slice_subslice_mut_fwd: forall (T : Type) (x : slice T) (r : range usize), result (slice T).
+Axiom slice_subslice_mut_back: forall (T : Type) (x : slice T) (r : range usize) (ns : slice T), result (slice T).
+
+(*** Vectors *)
+
+Definition vec T := { l: list T | Z.of_nat (length l) <= usize_max }.
+
+Definition vec_to_list {T: Type} (v: vec T) : list T := proj1_sig v.
+
+Definition vec_length {T: Type} (v: vec T) : Z := Z.of_nat (length (vec_to_list v)).
+
 Definition vec_new (T: Type) : vec T := (exist _ [] le_0_usize_max).
 
 Lemma vec_len_in_usize {T} (v: vec T) : usize_min <= vec_length v <= usize_max.
diff --git a/tests/coq/hashmap_on_disk/HashmapMain_Funs.v b/tests/coq/hashmap_on_disk/HashmapMain_Funs.v
index 1b7304cc..e6095fe1 100644
--- a/tests/coq/hashmap_on_disk/HashmapMain_Funs.v
+++ b/tests/coq/hashmap_on_disk/HashmapMain_Funs.v
@@ -3,6 +3,8 @@
 Require Import Primitives.
 Import Primitives.
 Require Import Coq.ZArith.ZArith.
+Require Import List.
+Import ListNotations.
 Local Open Scope Primitives_scope.
 Require Export HashmapMain_Types.
 Import HashmapMain_Types.
diff --git a/tests/coq/hashmap_on_disk/HashmapMain_Opaque.v b/tests/coq/hashmap_on_disk/HashmapMain_Opaque.v
index 1ad9c697..2d17cc29 100644
--- a/tests/coq/hashmap_on_disk/HashmapMain_Opaque.v
+++ b/tests/coq/hashmap_on_disk/HashmapMain_Opaque.v
@@ -3,6 +3,8 @@
 Require Import Primitives.
 Import Primitives.
 Require Import Coq.ZArith.ZArith.
+Require Import List.
+Import ListNotations.
 Local Open Scope Primitives_scope.
 Require Export HashmapMain_Types.
 Import HashmapMain_Types.
diff --git a/tests/coq/hashmap_on_disk/HashmapMain_Types.v b/tests/coq/hashmap_on_disk/HashmapMain_Types.v
index b92cbf3a..36aaaf25 100644
--- a/tests/coq/hashmap_on_disk/HashmapMain_Types.v
+++ b/tests/coq/hashmap_on_disk/HashmapMain_Types.v
@@ -3,6 +3,8 @@
 Require Import Primitives.
 Import Primitives.
 Require Import Coq.ZArith.ZArith.
+Require Import List.
+Import ListNotations.
 Local Open Scope Primitives_scope.
 Module HashmapMain_Types.
 
diff --git a/tests/coq/hashmap_on_disk/Primitives.v b/tests/coq/hashmap_on_disk/Primitives.v
index ae961ac2..71a2d9c3 100644
--- a/tests/coq/hashmap_on_disk/Primitives.v
+++ b/tests/coq/hashmap_on_disk/Primitives.v
@@ -394,13 +394,15 @@ Notation "x s< y" := (scalar_ltb x y)  (at level 80) : Primitives_scope.
 Notation "x s>= y" := (scalar_geb x y)  (at level 80) : Primitives_scope.
 Notation "x s> y" := (scalar_gtb x y)  (at level 80) : Primitives_scope.
 
-(*** Vectors *)
-
-Definition vec T := { l: list T | Z.of_nat (length l) <= usize_max }.
+(*** Range *)
+Record range (T : Type) := mk_range {
+  start: T;
+  end_: T;
+}.
+Arguments mk_range {_}.
 
-Definition vec_to_list {T: Type} (v: vec T) : list T := proj1_sig v.
-
-Definition vec_length {T: Type} (v: vec T) : Z := Z.of_nat (length (vec_to_list v)).
+(*** Arrays *)
+Definition array T (n : usize) := { l: list T | Z.of_nat (length l) = to_Z n}.
 
 Lemma le_0_usize_max : 0 <= usize_max.
 Proof.
@@ -409,6 +411,47 @@ Proof.
   lia.
 Qed.
 
+Lemma eqb_imp_eq (x y : Z) : Z.eqb x y = true -> x = y.
+Proof.
+  lia.
+Qed.
+
+(* TODO: finish the definitions *)
+Axiom mk_array : forall (T : Type) (n : usize) (l : list T), array T n.
+
+Axiom array_index_shared : forall (T : Type) (n : usize) (x : array T n) (i : usize), result T.
+Axiom array_index_mut_fwd : forall (T : Type) (n : usize) (x : array T n) (i : usize), result T.
+Axiom array_index_mut_back : forall (T : Type) (n : usize) (x : array T n) (i : usize) (nx : T), result (array T n).
+
+(*** Slice *)
+Definition slice T := { l: list T | Z.of_nat (length l) <= usize_max}.
+
+Axiom slice_len : forall (T : Type) (s : slice T), usize.
+Axiom slice_index_shared : forall (T : Type) (x : slice T) (i : usize), result T.
+Axiom slice_index_mut_fwd : forall (T : Type) (x : slice T) (i : usize), result T.
+Axiom slice_index_mut_back : forall (T : Type) (x : slice T) (i : usize) (nx : T), result (slice T).
+
+(*** Subslices *)
+
+Axiom array_to_slice_shared : forall (T : Type) (n : usize) (x : array T n), result (slice T).
+Axiom array_to_slice_mut_fwd : forall (T : Type) (n : usize) (x : array T n), result (slice T).
+Axiom array_to_slice_mut_back : forall (T : Type) (n : usize) (x : array T n) (s : slice T), result (array T n).
+
+Axiom array_subslice_shared: forall (T : Type) (n : usize) (x : array T n) (r : range usize), result (slice T).
+Axiom array_subslice_mut_fwd: forall (T : Type) (n : usize) (x : array T n) (r : range usize), result (slice T).
+Axiom array_subslice_mut_back: forall (T : Type) (n : usize) (x : array T n) (r : range usize) (ns : slice T), result (array T n).
+Axiom slice_subslice_shared: forall (T : Type) (x : slice T) (r : range usize), result (slice T).
+Axiom slice_subslice_mut_fwd: forall (T : Type) (x : slice T) (r : range usize), result (slice T).
+Axiom slice_subslice_mut_back: forall (T : Type) (x : slice T) (r : range usize) (ns : slice T), result (slice T).
+
+(*** Vectors *)
+
+Definition vec T := { l: list T | Z.of_nat (length l) <= usize_max }.
+
+Definition vec_to_list {T: Type} (v: vec T) : list T := proj1_sig v.
+
+Definition vec_length {T: Type} (v: vec T) : Z := Z.of_nat (length (vec_to_list v)).
+
 Definition vec_new (T: Type) : vec T := (exist _ [] le_0_usize_max).
 
 Lemma vec_len_in_usize {T} (v: vec T) : usize_min <= vec_length v <= usize_max.
diff --git a/tests/coq/misc/Constants.v b/tests/coq/misc/Constants.v
index 14c05c61..710ae1d9 100644
--- a/tests/coq/misc/Constants.v
+++ b/tests/coq/misc/Constants.v
@@ -3,6 +3,8 @@
 Require Import Primitives.
 Import Primitives.
 Require Import Coq.ZArith.ZArith.
+Require Import List.
+Import ListNotations.
 Local Open Scope Primitives_scope.
 Module Constants.
 
diff --git a/tests/coq/misc/External_Funs.v b/tests/coq/misc/External_Funs.v
index f18bbd1f..28370b2b 100644
--- a/tests/coq/misc/External_Funs.v
+++ b/tests/coq/misc/External_Funs.v
@@ -3,6 +3,8 @@
 Require Import Primitives.
 Import Primitives.
 Require Import Coq.ZArith.ZArith.
+Require Import List.
+Import ListNotations.
 Local Open Scope Primitives_scope.
 Require Export External_Types.
 Import External_Types.
diff --git a/tests/coq/misc/External_Opaque.v b/tests/coq/misc/External_Opaque.v
index 1224f426..d2ee42d4 100644
--- a/tests/coq/misc/External_Opaque.v
+++ b/tests/coq/misc/External_Opaque.v
@@ -3,6 +3,8 @@
 Require Import Primitives.
 Import Primitives.
 Require Import Coq.ZArith.ZArith.
+Require Import List.
+Import ListNotations.
 Local Open Scope Primitives_scope.
 Require Export External_Types.
 Import External_Types.
diff --git a/tests/coq/misc/External_Types.v b/tests/coq/misc/External_Types.v
index cec5b88e..1883fa6c 100644
--- a/tests/coq/misc/External_Types.v
+++ b/tests/coq/misc/External_Types.v
@@ -3,6 +3,8 @@
 Require Import Primitives.
 Import Primitives.
 Require Import Coq.ZArith.ZArith.
+Require Import List.
+Import ListNotations.
 Local Open Scope Primitives_scope.
 Module External_Types.
 
diff --git a/tests/coq/misc/Loops.v b/tests/coq/misc/Loops.v
index f17eb986..82e57576 100644
--- a/tests/coq/misc/Loops.v
+++ b/tests/coq/misc/Loops.v
@@ -3,6 +3,8 @@
 Require Import Primitives.
 Import Primitives.
 Require Import Coq.ZArith.ZArith.
+Require Import List.
+Import ListNotations.
 Local Open Scope Primitives_scope.
 Module Loops.
 
diff --git a/tests/coq/misc/NoNestedBorrows.v b/tests/coq/misc/NoNestedBorrows.v
index 470a2cde..f93254e1 100644
--- a/tests/coq/misc/NoNestedBorrows.v
+++ b/tests/coq/misc/NoNestedBorrows.v
@@ -3,6 +3,8 @@
 Require Import Primitives.
 Import Primitives.
 Require Import Coq.ZArith.ZArith.
+Require Import List.
+Import ListNotations.
 Local Open Scope Primitives_scope.
 Module NoNestedBorrows.
 
diff --git a/tests/coq/misc/Paper.v b/tests/coq/misc/Paper.v
index 0f854f31..175a523d 100644
--- a/tests/coq/misc/Paper.v
+++ b/tests/coq/misc/Paper.v
@@ -3,6 +3,8 @@
 Require Import Primitives.
 Import Primitives.
 Require Import Coq.ZArith.ZArith.
+Require Import List.
+Import ListNotations.
 Local Open Scope Primitives_scope.
 Module Paper.
 
diff --git a/tests/coq/misc/PoloniusList.v b/tests/coq/misc/PoloniusList.v
index e94b6dcb..54021bdf 100644
--- a/tests/coq/misc/PoloniusList.v
+++ b/tests/coq/misc/PoloniusList.v
@@ -3,6 +3,8 @@
 Require Import Primitives.
 Import Primitives.
 Require Import Coq.ZArith.ZArith.
+Require Import List.
+Import ListNotations.
 Local Open Scope Primitives_scope.
 Module PoloniusList.
 
diff --git a/tests/coq/misc/Primitives.v b/tests/coq/misc/Primitives.v
index ae961ac2..71a2d9c3 100644
--- a/tests/coq/misc/Primitives.v
+++ b/tests/coq/misc/Primitives.v
@@ -394,13 +394,15 @@ Notation "x s< y" := (scalar_ltb x y)  (at level 80) : Primitives_scope.
 Notation "x s>= y" := (scalar_geb x y)  (at level 80) : Primitives_scope.
 Notation "x s> y" := (scalar_gtb x y)  (at level 80) : Primitives_scope.
 
-(*** Vectors *)
-
-Definition vec T := { l: list T | Z.of_nat (length l) <= usize_max }.
+(*** Range *)
+Record range (T : Type) := mk_range {
+  start: T;
+  end_: T;
+}.
+Arguments mk_range {_}.
 
-Definition vec_to_list {T: Type} (v: vec T) : list T := proj1_sig v.
-
-Definition vec_length {T: Type} (v: vec T) : Z := Z.of_nat (length (vec_to_list v)).
+(*** Arrays *)
+Definition array T (n : usize) := { l: list T | Z.of_nat (length l) = to_Z n}.
 
 Lemma le_0_usize_max : 0 <= usize_max.
 Proof.
@@ -409,6 +411,47 @@ Proof.
   lia.
 Qed.
 
+Lemma eqb_imp_eq (x y : Z) : Z.eqb x y = true -> x = y.
+Proof.
+  lia.
+Qed.
+
+(* TODO: finish the definitions *)
+Axiom mk_array : forall (T : Type) (n : usize) (l : list T), array T n.
+
+Axiom array_index_shared : forall (T : Type) (n : usize) (x : array T n) (i : usize), result T.
+Axiom array_index_mut_fwd : forall (T : Type) (n : usize) (x : array T n) (i : usize), result T.
+Axiom array_index_mut_back : forall (T : Type) (n : usize) (x : array T n) (i : usize) (nx : T), result (array T n).
+
+(*** Slice *)
+Definition slice T := { l: list T | Z.of_nat (length l) <= usize_max}.
+
+Axiom slice_len : forall (T : Type) (s : slice T), usize.
+Axiom slice_index_shared : forall (T : Type) (x : slice T) (i : usize), result T.
+Axiom slice_index_mut_fwd : forall (T : Type) (x : slice T) (i : usize), result T.
+Axiom slice_index_mut_back : forall (T : Type) (x : slice T) (i : usize) (nx : T), result (slice T).
+
+(*** Subslices *)
+
+Axiom array_to_slice_shared : forall (T : Type) (n : usize) (x : array T n), result (slice T).
+Axiom array_to_slice_mut_fwd : forall (T : Type) (n : usize) (x : array T n), result (slice T).
+Axiom array_to_slice_mut_back : forall (T : Type) (n : usize) (x : array T n) (s : slice T), result (array T n).
+
+Axiom array_subslice_shared: forall (T : Type) (n : usize) (x : array T n) (r : range usize), result (slice T).
+Axiom array_subslice_mut_fwd: forall (T : Type) (n : usize) (x : array T n) (r : range usize), result (slice T).
+Axiom array_subslice_mut_back: forall (T : Type) (n : usize) (x : array T n) (r : range usize) (ns : slice T), result (array T n).
+Axiom slice_subslice_shared: forall (T : Type) (x : slice T) (r : range usize), result (slice T).
+Axiom slice_subslice_mut_fwd: forall (T : Type) (x : slice T) (r : range usize), result (slice T).
+Axiom slice_subslice_mut_back: forall (T : Type) (x : slice T) (r : range usize) (ns : slice T), result (slice T).
+
+(*** Vectors *)
+
+Definition vec T := { l: list T | Z.of_nat (length l) <= usize_max }.
+
+Definition vec_to_list {T: Type} (v: vec T) : list T := proj1_sig v.
+
+Definition vec_length {T: Type} (v: vec T) : Z := Z.of_nat (length (vec_to_list v)).
+
 Definition vec_new (T: Type) : vec T := (exist _ [] le_0_usize_max).
 
 Lemma vec_len_in_usize {T} (v: vec T) : usize_min <= vec_length v <= usize_max.