From 60db3c210aeaf66a4fe312544c6e5d4662681de7 Mon Sep 17 00:00:00 2001
From: Son Ho
Date: Fri, 4 Aug 2023 22:33:05 +0200
Subject: Generate the array test files for Coq and F*

---
 tests/fstar/array/Array.Clauses.Template.fst |  18 ++
 tests/fstar/array/Array.Clauses.fst          |  19 ++
 tests/fstar/array/Array.Funs.fst             | 226 ++++++++++++++++
 tests/fstar/array/Array.Types.fst            |   7 +
 tests/fstar/array/Makefile                   |  49 ++++
 tests/fstar/array/Primitives.fst             | 375 +++++++++++++++++++++++++++
 6 files changed, 694 insertions(+)
 create mode 100644 tests/fstar/array/Array.Clauses.Template.fst
 create mode 100644 tests/fstar/array/Array.Clauses.fst
 create mode 100644 tests/fstar/array/Array.Funs.fst
 create mode 100644 tests/fstar/array/Array.Types.fst
 create mode 100644 tests/fstar/array/Makefile
 create mode 100644 tests/fstar/array/Primitives.fst

(limited to 'tests/fstar/array')

diff --git a/tests/fstar/array/Array.Clauses.Template.fst b/tests/fstar/array/Array.Clauses.Template.fst
new file mode 100644
index 00000000..8a15e230
--- /dev/null
+++ b/tests/fstar/array/Array.Clauses.Template.fst
@@ -0,0 +1,18 @@
+(** THIS FILE WAS AUTOMATICALLY GENERATED BY AENEAS *)
+(** [array]: templates for the decreases clauses *)
+module Array.Clauses.Template
+open Primitives
+open Array.Types
+
+#set-options "--z3rlimit 50 --fuel 1 --ifuel 1"
+
+(** [array::sum]: decreases clause *)
+unfold
+let sum_loop_decreases (s : slice u32) (sum : u32) (i : usize) : nat = admit ()
+
+(** [array::sum2]: decreases clause *)
+unfold
+let sum2_loop_decreases (s : slice u32) (s2 : slice u32) (sum : u32)
+  (i : usize) : nat =
+  admit ()
+
diff --git a/tests/fstar/array/Array.Clauses.fst b/tests/fstar/array/Array.Clauses.fst
new file mode 100644
index 00000000..68cbf216
--- /dev/null
+++ b/tests/fstar/array/Array.Clauses.fst
@@ -0,0 +1,19 @@
+(** [array]: decreases clauses *)
+module Array.Clauses
+open Primitives
+open Array.Types
+open FStar.List.Tot
+
+#set-options "--z3rlimit 50 --fuel 1 --ifuel 1"
+
+(** [array::sum]: decreases clause *)
+unfold
+let sum_loop_decreases (s : slice u32) (sum : u32) (i : usize) : nat =
+  if i < length s then length s - i else 0
+
+(** [array::sum2]: decreases clause *)
+unfold
+let sum2_loop_decreases (s : slice u32) (s2 : slice u32) (sum : u32)
+  (i : usize) : nat =
+  if i < length s then length s - i else 0
+
diff --git a/tests/fstar/array/Array.Funs.fst b/tests/fstar/array/Array.Funs.fst
new file mode 100644
index 00000000..985633bc
--- /dev/null
+++ b/tests/fstar/array/Array.Funs.fst
@@ -0,0 +1,226 @@
+(** THIS FILE WAS AUTOMATICALLY GENERATED BY AENEAS *)
+(** [array]: function definitions *)
+module Array.Funs
+open Primitives
+include Array.Types
+include Array.Clauses
+
+#set-options "--z3rlimit 50 --fuel 1 --ifuel 1"
+
+(** [array::array_to_shared_slice_]: forward function *)
+let array_to_shared_slice__fwd
+  (t : Type0) (s : array t 32) : result (slice t) =
+  array_to_slice_shared t 32 s
+
+(** [array::array_to_mut_slice_]: forward function *)
+let array_to_mut_slice__fwd (t : Type0) (s : array t 32) : result (slice t) =
+  array_to_slice_mut_fwd t 32 s
+
+(** [array::array_to_mut_slice_]: backward function 0 *)
+let array_to_mut_slice__back
+  (t : Type0) (s : array t 32) (ret : slice t) : result (array t 32) =
+  array_to_slice_mut_back t 32 s ret
+
+(** [array::array_len]: forward function *)
+let array_len_fwd (t : Type0) (s : array t 32) : result usize =
+  let* s0 = array_to_slice_shared t 32 s in let i = slice_len t s0 in Return i
+
+(** [array::shared_array_len]: forward function *)
+let shared_array_len_fwd (t : Type0) (s : array t 32) : result usize =
+  let* s0 = array_to_slice_shared t 32 s in let i = slice_len t s0 in Return i
+
+(** [array::shared_slice_len]: forward function *)
+let shared_slice_len_fwd (t : Type0) (s : slice t) : result usize =
+  let i = slice_len t s in Return i
+
+(** [array::index_array_shared]: forward function *)
+let index_array_shared_fwd
+  (t : Type0) (s : array t 32) (i : usize) : result t =
+  array_index_shared t 32 s i
+
+(** [array::index_array_u32]: forward function *)
+let index_array_u32_fwd (s : array u32 32) (i : usize) : result u32 =
+  array_index_shared u32 32 s i
+
+(** [array::index_array_generic]: forward function *)
+let index_array_generic_fwd
+  (n : usize) (s : array u32 n) (i : usize) : result u32 =
+  array_index_shared u32 n s i
+
+(** [array::index_array_generic_call]: forward function *)
+let index_array_generic_call_fwd
+  (n : usize) (s : array u32 n) (i : usize) : result u32 =
+  index_array_generic_fwd n s i
+
+(** [array::index_array_copy]: forward function *)
+let index_array_copy_fwd (x : array u32 32) : result u32 =
+  array_index_shared u32 32 x 0
+
+(** [array::index_mut_array]: forward function *)
+let index_mut_array_fwd (t : Type0) (s : array t 32) (i : usize) : result t =
+  array_index_mut_fwd t 32 s i
+
+(** [array::index_mut_array]: backward function 0 *)
+let index_mut_array_back
+  (t : Type0) (s : array t 32) (i : usize) (ret : t) : result (array t 32) =
+  array_index_mut_back t 32 s i ret
+
+(** [array::index_slice]: forward function *)
+let index_slice_fwd (t : Type0) (s : slice t) (i : usize) : result t =
+  slice_index_shared t s i
+
+(** [array::index_mut_slice]: forward function *)
+let index_mut_slice_fwd (t : Type0) (s : slice t) (i : usize) : result t =
+  slice_index_mut_fwd t s i
+
+(** [array::index_mut_slice]: backward function 0 *)
+let index_mut_slice_back
+  (t : Type0) (s : slice t) (i : usize) (ret : t) : result (slice t) =
+  slice_index_mut_back t s i ret
+
+(** [array::slice_subslice_shared_]: forward function *)
+let slice_subslice_shared__fwd
+  (x : slice u32) (y : usize) (z : usize) : result (slice u32) =
+  slice_subslice_shared u32 x (Mkrange y z)
+
+(** [array::slice_subslice_mut_]: forward function *)
+let slice_subslice_mut__fwd
+  (x : slice u32) (y : usize) (z : usize) : result (slice u32) =
+  slice_subslice_mut_fwd u32 x (Mkrange y z)
+
+(** [array::slice_subslice_mut_]: backward function 0 *)
+let slice_subslice_mut__back
+  (x : slice u32) (y : usize) (z : usize) (ret : slice u32) :
+  result (slice u32)
+  =
+  slice_subslice_mut_back u32 x (Mkrange y z) ret
+
+(** [array::array_to_slice_shared_]: forward function *)
+let array_to_slice_shared__fwd (x : array u32 32) : result (slice u32) =
+  array_to_slice_shared u32 32 x
+
+(** [array::array_to_slice_mut_]: forward function *)
+let array_to_slice_mut__fwd (x : array u32 32) : result (slice u32) =
+  array_to_slice_mut_fwd u32 32 x
+
+(** [array::array_to_slice_mut_]: backward function 0 *)
+let array_to_slice_mut__back
+  (x : array u32 32) (ret : slice u32) : result (array u32 32) =
+  array_to_slice_mut_back u32 32 x ret
+
+(** [array::array_subslice_shared_]: forward function *)
+let array_subslice_shared__fwd
+  (x : array u32 32) (y : usize) (z : usize) : result (slice u32) =
+  array_subslice_shared u32 32 x (Mkrange y z)
+
+(** [array::array_subslice_mut_]: forward function *)
+let array_subslice_mut__fwd
+  (x : array u32 32) (y : usize) (z : usize) : result (slice u32) =
+  array_subslice_mut_fwd u32 32 x (Mkrange y z)
+
+(** [array::array_subslice_mut_]: backward function 0 *)
+let array_subslice_mut__back
+  (x : array u32 32) (y : usize) (z : usize) (ret : slice u32) :
+  result (array u32 32)
+  =
+  array_subslice_mut_back u32 32 x (Mkrange y z) ret
+
+(** [array::index_slice_0]: forward function *)
+let index_slice_0_fwd (t : Type0) (s : slice t) : result t =
+  slice_index_shared t s 0
+
+(** [array::index_array_0]: forward function *)
+let index_array_0_fwd (t : Type0) (s : array t 32) : result t =
+  array_index_shared t 32 s 0
+
+(** [array::index_index_array]: forward function *)
+let index_index_array_fwd
+  (s : array (array u32 32) 32) (i : usize) (j : usize) : result u32 =
+  let* a = array_index_shared (array u32 32) 32 s i in
+  array_index_shared u32 32 a j
+
+(** [array::update_update_array]: forward function *)
+let update_update_array_fwd
+  (s : array (array u32 32) 32) (i : usize) (j : usize) : result unit =
+  let* a = array_index_mut_fwd (array u32 32) 32 s i in
+  let* a0 = array_index_mut_back u32 32 a j 0 in
+  let* _ = array_index_mut_back (array u32 32) 32 s i a0 in
+  Return ()
+
+(** [array::array_local_deep_copy]: forward function *)
+let array_local_deep_copy_fwd (x : array u32 32) : result unit =
+  Return ()
+
+(** [array::f0]: forward function *)
+let f0_fwd : result unit =
+  let* s = array_to_slice_mut_fwd u32 2 (mk_array u32 2 [ 1; 2 ]) in
+  let* s0 = slice_index_mut_back u32 s 0 1 in
+  let* _ = array_to_slice_mut_back u32 2 (mk_array u32 2 [ 1; 2 ]) s0 in
+  Return ()
+
+(** [array::f1]: forward function *)
+let f1_fwd : result unit =
+  let* _ = array_index_mut_back u32 2 (mk_array u32 2 [ 1; 2 ]) 0 1 in
+  Return ()
+
+(** [array::sum]: loop 0: forward function *)
+let rec sum_loop_fwd
+  (s : slice u32) (sum : u32) (i : usize) :
+  Tot (result u32) (decreases (sum_loop_decreases s sum i))
+  =
+  let i0 = slice_len u32 s in
+  if i < i0
+  then
+    let* i1 = slice_index_shared u32 s i in
+    let* sum0 = u32_add sum i1 in
+    let* i2 = usize_add i 1 in
+    sum_loop_fwd s sum0 i2
+  else Return sum
+
+(** [array::sum]: forward function *)
+let sum_fwd (s : slice u32) : result u32 =
+  sum_loop_fwd s 0 0
+
+(** [array::sum2]: loop 0: forward function *)
+let rec sum2_loop_fwd
+  (s : slice u32) (s2 : slice u32) (sum : u32) (i : usize) :
+  Tot (result u32) (decreases (sum2_loop_decreases s s2 sum i))
+  =
+  let i0 = slice_len u32 s in
+  if i < i0
+  then
+    let* i1 = slice_index_shared u32 s i in
+    let* i2 = slice_index_shared u32 s2 i in
+    let* i3 = u32_add i1 i2 in
+    let* sum0 = u32_add sum i3 in
+    let* i4 = usize_add i 1 in
+    sum2_loop_fwd s s2 sum0 i4
+  else Return sum
+
+(** [array::sum2]: forward function *)
+let sum2_fwd (s : slice u32) (s2 : slice u32) : result u32 =
+  let i = slice_len u32 s in
+  let i0 = slice_len u32 s2 in
+  if not (i = i0) then Fail Failure else sum2_loop_fwd s s2 0 0
+
+(** [array::f2]: forward function *)
+let f2_fwd (i : u32) : result unit =
+  Return ()
+
+(** [array::f4]: forward function *)
+let f4_fwd (x : array u32 32) (y : usize) (z : usize) : result (slice u32) =
+  array_subslice_shared u32 32 x (Mkrange y z)
+
+(** [array::f3]: forward function *)
+let f3_fwd : result u32 =
+  let* i = array_index_shared u32 2 (mk_array u32 2 [ 1; 2 ]) 0 in
+  let* _ = f2_fwd i in
+  let* s = array_to_slice_shared u32 2 (mk_array u32 2 [ 1; 2 ]) in
+  let* s0 =
+    f4_fwd
+      (mk_array u32 32 [
+        0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0;
+        0; 0; 0; 0; 0; 0; 0; 0
+        ]) 16 18 in
+  sum2_fwd s s0
+
diff --git a/tests/fstar/array/Array.Types.fst b/tests/fstar/array/Array.Types.fst
new file mode 100644
index 00000000..d20f1718
--- /dev/null
+++ b/tests/fstar/array/Array.Types.fst
@@ -0,0 +1,7 @@
+(** THIS FILE WAS AUTOMATICALLY GENERATED BY AENEAS *)
+(** [array]: type definitions *)
+module Array.Types
+open Primitives
+
+#set-options "--z3rlimit 50 --fuel 1 --ifuel 1"
+
diff --git a/tests/fstar/array/Makefile b/tests/fstar/array/Makefile
new file mode 100644
index 00000000..fa7d1f36
--- /dev/null
+++ b/tests/fstar/array/Makefile
@@ -0,0 +1,49 @@
+# This file was automatically generated - modify ../Makefile.template instead
+INCLUDE_DIRS = .
+
+FSTAR_INCLUDES = $(addprefix --include ,$(INCLUDE_DIRS))
+
+FSTAR_HINTS ?= --use_hints --use_hint_hashes --record_hints
+
+FSTAR_OPTIONS = $(FSTAR_HINTS) \
+  --cache_checked_modules $(FSTAR_INCLUDES) --cmi \
+  --warn_error '+241@247+285-274' \
+
+FSTAR_EXE ?= fstar.exe
+FSTAR_NO_FLAGS = $(FSTAR_EXE) --already_cached 'Prims FStar LowStar Steel' --odir obj --cache_dir obj
+
+FSTAR = $(FSTAR_NO_FLAGS) $(FSTAR_OPTIONS)
+
+# The F* roots are used to compute the dependency graph, and generate the .depend file
+FSTAR_ROOTS ?= $(wildcard *.fst *.fsti)
+
+# Build all the files
+all: $(addprefix obj/,$(addsuffix .checked,$(FSTAR_ROOTS)))
+
+# This is the right way to ensure the .depend file always gets re-built.
+ifeq (,$(filter %-in,$(MAKECMDGOALS)))
+ifndef NODEPEND
+ifndef MAKE_RESTARTS
+.depend: .FORCE
+	$(FSTAR_NO_FLAGS) --dep full $(notdir $(FSTAR_ROOTS)) > $@
+
+.PHONY: .FORCE
+.FORCE:
+endif
+endif
+
+include .depend
+endif
+
+# For the interactive mode
+%.fst-in %.fsti-in:
+	@echo $(FSTAR_OPTIONS)
+
+# Generete the .checked files in batch mode
+%.checked:
+	$(FSTAR) $(FSTAR_OPTIONS) $< && \
+	touch -c $@
+
+.PHONY: clean
+clean:
+	rm -f obj/*
diff --git a/tests/fstar/array/Primitives.fst b/tests/fstar/array/Primitives.fst
new file mode 100644
index 00000000..9db82069
--- /dev/null
+++ b/tests/fstar/array/Primitives.fst
@@ -0,0 +1,375 @@
+/// This file lists primitive and assumed functions and types
+module Primitives
+open FStar.Mul
+open FStar.List.Tot
+
+#set-options "--z3rlimit 15 --fuel 0 --ifuel 1"
+
+(*** Utilities *)
+val list_update (#a : Type0) (ls : list a) (i : nat{i < length ls}) (x : a) :
+  ls':list a{
+    length ls' = length ls /\
+    index ls' i == x
+  }
+#push-options "--fuel 1"
+let rec list_update #a ls i x =
+  match ls with
+  | x' :: ls -> if i = 0 then x :: ls else x' :: list_update ls (i-1) x
+#pop-options
+
+(*** Result *)
+type error : Type0 =
+| Failure
+| OutOfFuel
+
+type result (a : Type0) : Type0 =
+| Return : v:a -> result a
+| Fail : e:error -> result a
+
+// Monadic return operator
+unfold let return (#a : Type0) (x : a) : result a = Return x
+
+// Monadic bind operator.
+// Allows to use the notation:
+// ```
+// let* x = y in
+// ...
+// ```
+unfold let (let*) (#a #b : Type0) (m: result a)
+  (f: (x:a) -> Pure (result b) (requires (m == Return x)) (ensures fun _ -> True)) :
+  result b =
+  match m with
+  | Return x -> f x
+  | Fail e   -> Fail e
+
+// Monadic assert(...)
+let massert (b:bool) : result unit = if b then Return () else Fail Failure
+
+// Normalize and unwrap a successful result (used for globals).
+let eval_global (#a : Type0) (x : result a{Return? (normalize_term x)}) : a = Return?.v x
+
+(*** Misc *)
+type char = FStar.Char.char
+type string = string
+
+let is_zero (n: nat) : bool = n = 0
+let decrease (n: nat{n > 0}) : nat = n - 1
+
+let mem_replace_fwd (a : Type0) (x : a) (y : a) : a = x
+let mem_replace_back (a : Type0) (x : a) (y : a) : a = y
+
+(*** Scalars *)
+/// Rem.: most of the following code was partially generated
+
+let isize_min : int = -9223372036854775808 // TODO: should be opaque
+let isize_max : int = 9223372036854775807 // TODO: should be opaque
+let i8_min : int = -128
+let i8_max : int = 127
+let i16_min : int = -32768
+let i16_max : int = 32767
+let i32_min : int = -2147483648
+let i32_max : int = 2147483647
+let i64_min : int = -9223372036854775808
+let i64_max : int = 9223372036854775807
+let i128_min : int = -170141183460469231731687303715884105728
+let i128_max : int = 170141183460469231731687303715884105727
+let usize_min : int = 0
+let usize_max : int = 4294967295 // TODO: should be opaque
+let u8_min : int = 0
+let u8_max : int = 255
+let u16_min : int = 0
+let u16_max : int = 65535
+let u32_min : int = 0
+let u32_max : int = 4294967295
+let u64_min : int = 0
+let u64_max : int = 18446744073709551615
+let u128_min : int = 0
+let u128_max : int = 340282366920938463463374607431768211455
+
+type scalar_ty =
+| Isize
+| I8
+| I16
+| I32
+| I64
+| I128
+| Usize
+| U8
+| U16
+| U32
+| U64
+| U128
+
+let scalar_min (ty : scalar_ty) : int =
+  match ty with
+  | Isize -> isize_min
+  | I8 -> i8_min
+  | I16 -> i16_min
+  | I32 -> i32_min
+  | I64 -> i64_min
+  | I128 -> i128_min
+  | Usize -> usize_min
+  | U8 -> u8_min
+  | U16 -> u16_min
+  | U32 -> u32_min
+  | U64 -> u64_min
+  | U128 -> u128_min
+
+let scalar_max (ty : scalar_ty) : int =
+  match ty with
+  | Isize -> isize_max
+  | I8 -> i8_max
+  | I16 -> i16_max
+  | I32 -> i32_max
+  | I64 -> i64_max
+  | I128 -> i128_max
+  | Usize -> usize_max
+  | U8 -> u8_max
+  | U16 -> u16_max
+  | U32 -> u32_max
+  | U64 -> u64_max
+  | U128 -> u128_max
+
+type scalar (ty : scalar_ty) : eqtype = x:int{scalar_min ty <= x && x <= scalar_max ty}
+
+let mk_scalar (ty : scalar_ty) (x : int) : result (scalar ty) =
+  if scalar_min ty <= x && scalar_max ty >= x then Return x else Fail Failure
+
+let scalar_neg (#ty : scalar_ty) (x : scalar ty) : result (scalar ty) = mk_scalar ty (-x)
+
+let scalar_div (#ty : scalar_ty) (x : scalar ty) (y : scalar ty) : result (scalar ty) =
+  if y <> 0 then mk_scalar ty (x / y) else Fail Failure
+
+/// The remainder operation
+let int_rem (x : int) (y : int{y <> 0}) : int =
+  if x >= 0 then (x % y) else -(x % y)
+
+(* Checking consistency with Rust *)
+let _ = assert_norm(int_rem 1 2 = 1)
+let _ = assert_norm(int_rem (-1) 2 = -1)
+let _ = assert_norm(int_rem 1 (-2) = 1)
+let _ = assert_norm(int_rem (-1) (-2) = -1)
+
+let scalar_rem (#ty : scalar_ty) (x : scalar ty) (y : scalar ty) : result (scalar ty) =
+  if y <> 0 then mk_scalar ty (int_rem x y) else Fail Failure
+
+let scalar_add (#ty : scalar_ty) (x : scalar ty) (y : scalar ty) : result (scalar ty) =
+  mk_scalar ty (x + y)
+
+let scalar_sub (#ty : scalar_ty) (x : scalar ty) (y : scalar ty) : result (scalar ty) =
+  mk_scalar ty (x - y)
+
+let scalar_mul (#ty : scalar_ty) (x : scalar ty) (y : scalar ty) : result (scalar ty) =
+  mk_scalar ty (x * y)
+
+(** Cast an integer from a [src_ty] to a [tgt_ty] *)
+// TODO: check the semantics of casts in Rust
+let scalar_cast (src_ty : scalar_ty) (tgt_ty : scalar_ty) (x : scalar src_ty) : result (scalar tgt_ty) =
+  mk_scalar tgt_ty x
+
+/// The scalar types
+type isize : eqtype = scalar Isize
+type i8 : eqtype = scalar I8
+type i16 : eqtype = scalar I16
+type i32 : eqtype = scalar I32
+type i64 : eqtype = scalar I64
+type i128 : eqtype = scalar I128
+type usize : eqtype = scalar Usize
+type u8 : eqtype = scalar U8
+type u16 : eqtype = scalar U16
+type u32 : eqtype = scalar U32
+type u64 : eqtype = scalar U64
+type u128 : eqtype = scalar U128
+
+/// Negation
+let isize_neg = scalar_neg #Isize
+let i8_neg = scalar_neg #I8
+let i16_neg = scalar_neg #I16
+let i32_neg = scalar_neg #I32
+let i64_neg = scalar_neg #I64
+let i128_neg = scalar_neg #I128
+
+/// Division
+let isize_div = scalar_div #Isize
+let i8_div = scalar_div #I8
+let i16_div = scalar_div #I16
+let i32_div = scalar_div #I32
+let i64_div = scalar_div #I64
+let i128_div = scalar_div #I128
+let usize_div = scalar_div #Usize
+let u8_div = scalar_div #U8
+let u16_div = scalar_div #U16
+let u32_div = scalar_div #U32
+let u64_div = scalar_div #U64
+let u128_div = scalar_div #U128
+
+/// Remainder
+let isize_rem = scalar_rem #Isize
+let i8_rem = scalar_rem #I8
+let i16_rem = scalar_rem #I16
+let i32_rem = scalar_rem #I32
+let i64_rem = scalar_rem #I64
+let i128_rem = scalar_rem #I128
+let usize_rem = scalar_rem #Usize
+let u8_rem = scalar_rem #U8
+let u16_rem = scalar_rem #U16
+let u32_rem = scalar_rem #U32
+let u64_rem = scalar_rem #U64
+let u128_rem = scalar_rem #U128
+
+/// Addition
+let isize_add = scalar_add #Isize
+let i8_add = scalar_add #I8
+let i16_add = scalar_add #I16
+let i32_add = scalar_add #I32
+let i64_add = scalar_add #I64
+let i128_add = scalar_add #I128
+let usize_add = scalar_add #Usize
+let u8_add = scalar_add #U8
+let u16_add = scalar_add #U16
+let u32_add = scalar_add #U32
+let u64_add = scalar_add #U64
+let u128_add = scalar_add #U128
+
+/// Substraction
+let isize_sub = scalar_sub #Isize
+let i8_sub = scalar_sub #I8
+let i16_sub = scalar_sub #I16
+let i32_sub = scalar_sub #I32
+let i64_sub = scalar_sub #I64
+let i128_sub = scalar_sub #I128
+let usize_sub = scalar_sub #Usize
+let u8_sub = scalar_sub #U8
+let u16_sub = scalar_sub #U16
+let u32_sub = scalar_sub #U32
+let u64_sub = scalar_sub #U64
+let u128_sub = scalar_sub #U128
+
+/// Multiplication
+let isize_mul = scalar_mul #Isize
+let i8_mul = scalar_mul #I8
+let i16_mul = scalar_mul #I16
+let i32_mul = scalar_mul #I32
+let i64_mul = scalar_mul #I64
+let i128_mul = scalar_mul #I128
+let usize_mul = scalar_mul #Usize
+let u8_mul = scalar_mul #U8
+let u16_mul = scalar_mul #U16
+let u32_mul = scalar_mul #U32
+let u64_mul = scalar_mul #U64
+let u128_mul = scalar_mul #U128
+
+(*** Range *)
+type range (a : Type0) = {
+  start : a;
+  end_ : a;
+}
+
+(*** Array *)
+type array (a : Type0) (n : usize) = s:list a{length s = n}
+
+// We tried putting the normalize_term condition as a refinement on the list
+// but it didn't work. It works with the requires clause.
+let mk_array (a : Type0) (n : usize)
+  (l : list a) :
+  Pure (array a n)
+  (requires (normalize_term(FStar.List.Tot.length l) = n))
+  (ensures (fun _ -> True)) =
+  normalize_term_spec (FStar.List.Tot.length l);
+  l
+
+let array_index_shared (a : Type0) (n : usize) (x : array a n) (i : usize) : result a =
+  if i < length x then Return (index x i)
+  else Fail Failure
+
+let array_index_mut_fwd (a : Type0) (n : usize) (x : array a n) (i : usize) : result a =
+  if i < length x then Return (index x i)
+  else Fail Failure
+
+let array_index_mut_back (a : Type0) (n : usize) (x : array a n) (i : usize) (nx : a) : result (array a n) =
+  if i < length x then Return (list_update x i nx)
+  else Fail Failure
+
+(*** Slice *)
+type slice (a : Type0) = s:list a{length s <= usize_max}
+
+let slice_len (a : Type0) (s : slice a) : usize = length s
+
+let slice_index_shared (a : Type0) (x : slice a) (i : usize) : result a =
+  if i < length x then Return (index x i)
+  else Fail Failure
+
+let slice_index_mut_fwd (a : Type0) (x : slice a) (i : usize) : result a =
+  if i < length x then Return (index x i)
+  else Fail Failure
+
+let slice_index_mut_back (a : Type0) (x : slice a) (i : usize) (nx : a) : result (slice a) =
+  if i < length x then Return (list_update x i nx)
+  else Fail Failure
+
+(*** Subslices *)
+
+let array_to_slice_shared (a : Type0) (n : usize) (x : array a n) : result (slice a) = Return x
+let array_to_slice_mut_fwd (a : Type0) (n : usize) (x : array a n) : result (slice a) = Return x
+let array_to_slice_mut_back (a : Type0) (n : usize) (x : array a n) (s : slice a) : result (array a n) =
+  if length s = n then Return s
+  else Fail Failure
+
+// TODO: finish the definitions below (there lacks [List.drop] and [List.take] in the standard library *)
+let array_subslice_shared (a : Type0) (n : usize) (x : array a n) (r : range usize) : result (slice a) =
+  admit()
+
+let array_subslice_mut_fwd (a : Type0) (n : usize) (x : array a n) (r : range usize) : result (slice a) =
+  admit()
+
+let array_subslice_mut_back (a : Type0) (n : usize) (x : array a n) (r : range usize) (ns : slice a) : result (array a n) =
+  admit()
+
+let slice_subslice_shared (a : Type0) (x : slice a) (r : range usize) : result (slice a) =
+  admit()
+
+let slice_subslice_mut_fwd (a : Type0) (x : slice a) (r : range usize) : result (slice a) =
+  admit()
+
+let slice_subslice_mut_back (a : Type0) (x : slice a) (r : range usize) (ns : slice a) : result (slice a) =
+  admit()
+
+(*** Vector *)
+type vec (a : Type0) = v:list a{length v <= usize_max}
+
+let vec_new (a  : Type0) : vec a = assert_norm(length #a [] == 0); []
+let vec_len (a : Type0) (v : vec a) : usize = length v
+
+// The **forward** function shouldn't be used
+let vec_push_fwd (a  : Type0) (v : vec a) (x : a) : unit = ()
+let vec_push_back (a  : Type0) (v : vec a) (x : a) :
+  Pure (result (vec a))
+  (requires True)
+  (ensures (fun res ->
+    match res with
+    | Fail e -> e == Failure
+    | Return v' -> length v' = length v + 1)) =
+  if length v < usize_max then begin
+    (**) assert_norm(length [x] == 1);
+    (**) append_length v [x];
+    (**) assert(length (append v [x]) = length v + 1);
+    Return (append v [x])
+    end
+  else Fail Failure
+
+// The **forward** function shouldn't be used
+let vec_insert_fwd (a : Type0) (v : vec a) (i : usize) (x : a) : result unit =
+  if i < length v then Return () else Fail Failure
+let vec_insert_back (a : Type0) (v : vec a) (i : usize) (x : a) : result (vec a) =
+  if i < length v then Return (list_update v i x) else Fail Failure
+
+// The **backward** function shouldn't be used
+let vec_index_fwd (a : Type0) (v : vec a) (i : usize) : result a =
+  if i < length v then Return (index v i) else Fail Failure
+let vec_index_back (a : Type0) (v : vec a) (i : usize) (x : a) : result unit =
+  if i < length v then Return () else Fail Failure
+
+let vec_index_mut_fwd (a : Type0) (v : vec a) (i : usize) : result a =
+  if i < length v then Return (index v i) else Fail Failure
+let vec_index_mut_back (a : Type0) (v : vec a) (i : usize) (nx : a) : result (vec a) =
+  if i < length v then Return (list_update v i nx) else Fail Failure
-- 
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