Merge pull request #64 from AeneasVerif/son/merge_back

Merge the forward/backward functions

2 files changed, 895 insertions, 0 deletions

diff --git a/backends/fstar/merge/Makefile b/backends/fstar/merge/Makefile
new file mode 100644
index 00000000..a16b0edb
--- /dev/null
+++ b/backends/fstar/merge/Makefile
@@ -0,0 +1,47 @@
+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_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/backends/fstar/merge/Primitives.fst b/backends/fstar/merge/Primitives.fst
new file mode 100644
index 00000000..fca80829
--- /dev/null
+++ b/backends/fstar/merge/Primitives.fst
@@ -0,0 +1,848 @@
+/// 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 core_mem_replace (a : Type0) (x : a) (y : a) : a & a = (x, x)
+
+// We don't really use raw pointers for now
+type mut_raw_ptr (t : Type0) = { v : t }
+type const_raw_ptr (t : Type0) = { v : t }
+
+(*** Scalars *)
+/// Rem.: most of the following code was partially generated
+
+assume val size_numbits : pos
+
+// TODO: we could use FStar.Int.int_t and FStar.UInt.int_t
+
+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 is_unsigned = function
+  | Isize | I8 | I16 | I32 | I64 | I128 -> false
+  | Usize | U8 | U16 | U32 | U64 | U128 -> true
+
+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)
+
+let scalar_xor (#ty : scalar_ty)
+    (x : scalar ty) (y : scalar ty) : scalar ty =
+  match ty with
+  | U8 -> FStar.UInt.logxor #8 x y
+  | U16 -> FStar.UInt.logxor #16 x y
+  | U32 -> FStar.UInt.logxor #32 x y
+  | U64 -> FStar.UInt.logxor #64 x y
+  | U128 -> FStar.UInt.logxor #128 x y
+  | Usize -> admit() // TODO
+  | I8 ->
+    // Encoding issues...
+    normalize_spec (FStar.Int.int_t 8);
+    normalize_spec (scalar I8);
+    FStar.Int.logxor #8 x y
+  | I16 ->
+    // Encoding issues...
+    normalize_spec (FStar.Int.int_t 16);
+    normalize_spec (scalar I16);
+    FStar.Int.logxor #16 x y
+  | I32 -> FStar.Int.logxor #32 x y
+  | I64 -> FStar.Int.logxor #64 x y
+  | I128 ->
+    // Encoding issues...
+    normalize_spec (FStar.Int.int_t 128);
+    normalize_spec (scalar I128);
+    FStar.Int.logxor #128 x y
+  | Isize -> admit() // TODO
+
+let scalar_or (#ty : scalar_ty)
+    (x : scalar ty) (y : scalar ty) : scalar ty =
+  match ty with
+  | U8 -> FStar.UInt.logor #8 x y
+  | U16 -> FStar.UInt.logor #16 x y
+  | U32 -> FStar.UInt.logor #32 x y
+  | U64 -> FStar.UInt.logor #64 x y
+  | U128 -> FStar.UInt.logor #128 x y
+  | Usize -> admit() // TODO
+  | I8 ->
+    // Encoding issues...
+    normalize_spec (FStar.Int.int_t 8);
+    normalize_spec (scalar I8);
+    FStar.Int.logor #8 x y
+  | I16 ->
+    // Encoding issues...
+    normalize_spec (FStar.Int.int_t 16);
+    normalize_spec (scalar I16);
+    FStar.Int.logor #16 x y
+  | I32 -> FStar.Int.logor #32 x y
+  | I64 -> FStar.Int.logor #64 x y
+  | I128 ->
+    // Encoding issues...
+    normalize_spec (FStar.Int.int_t 128);
+    normalize_spec (scalar I128);
+    FStar.Int.logor #128 x y
+  | Isize -> admit() // TODO
+
+let scalar_and (#ty : scalar_ty)
+    (x : scalar ty) (y : scalar ty) : scalar ty =
+  match ty with
+  | U8 -> FStar.UInt.logand #8 x y
+  | U16 -> FStar.UInt.logand #16 x y
+  | U32 -> FStar.UInt.logand #32 x y
+  | U64 -> FStar.UInt.logand #64 x y
+  | U128 -> FStar.UInt.logand #128 x y
+  | Usize -> admit() // TODO
+  | I8 ->
+    // Encoding issues...
+    normalize_spec (FStar.Int.int_t 8);
+    normalize_spec (scalar I8);
+    FStar.Int.logand #8 x y
+  | I16 ->
+    // Encoding issues...
+    normalize_spec (FStar.Int.int_t 16);
+    normalize_spec (scalar I16);
+    FStar.Int.logand #16 x y
+  | I32 -> FStar.Int.logand #32 x y
+  | I64 -> FStar.Int.logand #64 x y
+  | I128 ->
+    // Encoding issues...
+    normalize_spec (FStar.Int.int_t 128);
+    normalize_spec (scalar I128);
+    FStar.Int.logand #128 x y
+  | Isize -> admit() // TODO
+
+// Shift left
+let scalar_shl (#ty0 #ty1 : scalar_ty)
+    (x : scalar ty0) (y : scalar ty1) : result (scalar ty0) =
+  admit()
+
+// Shift right
+let scalar_shr (#ty0 #ty1 : scalar_ty)
+    (x : scalar ty0) (y : scalar ty1) : result (scalar ty0) =
+  admit()
+
+(** 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
+
+// This can't fail, but for now we make all casts faillible (easier for the translation)
+let scalar_cast_bool (tgt_ty : scalar_ty) (x : bool) : result (scalar tgt_ty) =
+  mk_scalar tgt_ty (if x then 1 else 0)
+
+/// 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
+
+
+let core_isize_min : isize = isize_min
+let core_isize_max : isize = isize_max
+let core_i8_min    : i8 = i8_min
+let core_i8_max    : i8 = i8_max
+let core_i16_min   : i16 = i16_min
+let core_i16_max   : i16 = i16_max
+let core_i32_min   : i32 = i32_min
+let core_i32_max   : i32 = i32_max
+let core_i64_min   : i64 = i64_min
+let core_i64_max   : i64 = i64_max
+let core_i128_min  : i128 = i128_min
+let core_i128_max  : i128 = i128_max
+
+let core_usize_min : usize = usize_min
+let core_usize_max : usize = usize_max
+let core_u8_min    : u8 = u8_min
+let core_u8_max    : u8 = u8_max
+let core_u16_min   : u16 = u16_min
+let core_u16_max   : u16 = u16_max
+let core_u32_min   : u32 = u32_min
+let core_u32_max   : u32 = u32_max
+let core_u64_min   : u64 = u64_min
+let core_u64_max   : u64 = u64_max
+let core_u128_min  : u128 = u128_min
+let core_u128_max  : u128 = u128_max
+
+/// 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
+
+/// Subtraction
+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
+
+/// Xor
+let u8_xor = scalar_xor #U8
+let u16_xor = scalar_xor #U16
+let u32_xor = scalar_xor #U32
+let u64_xor = scalar_xor #U64
+let u128_xor = scalar_xor #U128
+let usize_xor = scalar_xor #Usize
+let i8_xor = scalar_xor #I8
+let i16_xor = scalar_xor #I16
+let i32_xor = scalar_xor #I32
+let i64_xor = scalar_xor #I64
+let i128_xor = scalar_xor #I128
+let isize_xor = scalar_xor #Isize
+
+/// Or
+let u8_or = scalar_or #U8
+let u16_or = scalar_or #U16
+let u32_or = scalar_or #U32
+let u64_or = scalar_or #U64
+let u128_or = scalar_or #U128
+let usize_or = scalar_or #Usize
+let i8_or = scalar_or #I8
+let i16_or = scalar_or #I16
+let i32_or = scalar_or #I32
+let i64_or = scalar_or #I64
+let i128_or = scalar_or #I128
+let isize_or = scalar_or #Isize
+
+/// And
+let u8_and = scalar_and #U8
+let u16_and = scalar_and #U16
+let u32_and = scalar_and #U32
+let u64_and = scalar_and #U64
+let u128_and = scalar_and #U128
+let usize_and = scalar_and #Usize
+let i8_and = scalar_and #I8
+let i16_and = scalar_and #I16
+let i32_and = scalar_and #I32
+let i64_and = scalar_and #I64
+let i128_and = scalar_and #I128
+let isize_and = scalar_and #Isize
+
+/// Shift left
+let u8_shl #ty = scalar_shl #U8 #ty
+let u16_shl #ty = scalar_shl #U16 #ty
+let u32_shl #ty = scalar_shl #U32 #ty
+let u64_shl #ty = scalar_shl #U64 #ty
+let u128_shl #ty = scalar_shl #U128 #ty
+let usize_shl #ty = scalar_shl #Usize #ty
+let i8_shl #ty = scalar_shl #I8 #ty
+let i16_shl #ty = scalar_shl #I16 #ty
+let i32_shl #ty = scalar_shl #I32 #ty
+let i64_shl #ty = scalar_shl #I64 #ty
+let i128_shl #ty = scalar_shl #I128 #ty
+let isize_shl #ty = scalar_shl #Isize #ty
+
+/// Shift right
+let u8_shr #ty = scalar_shr #U8 #ty
+let u16_shr #ty = scalar_shr #U16 #ty
+let u32_shr #ty = scalar_shr #U32 #ty
+let u64_shr #ty = scalar_shr #U64 #ty
+let u128_shr #ty = scalar_shr #U128 #ty
+let usize_shr #ty = scalar_shr #Usize #ty
+let i8_shr #ty = scalar_shr #I8 #ty
+let i16_shr #ty = scalar_shr #I16 #ty
+let i32_shr #ty = scalar_shr #I32 #ty
+let i64_shr #ty = scalar_shr #I64 #ty
+let i128_shr #ty = scalar_shr #I128 #ty
+let isize_shr #ty = scalar_shr #Isize #ty
+
+(*** core::ops *)
+
+// Trait declaration: [core::ops::index::Index]
+noeq type core_ops_index_Index (self idx : Type0) = {
+  output : Type0;
+  index : self → idx → result output
+}
+
+// Trait declaration: [core::ops::index::IndexMut]
+noeq type core_ops_index_IndexMut (self idx : Type0) = {
+  indexInst : core_ops_index_Index self idx;
+  index_mut : self → idx → result (indexInst.output & (indexInst.output → result self));
+}
+
+// Trait declaration [core::ops::deref::Deref]
+noeq type core_ops_deref_Deref (self : Type0) = {
+  target : Type0;
+  deref : self → result target;
+}
+
+// Trait declaration [core::ops::deref::DerefMut]
+noeq type core_ops_deref_DerefMut (self : Type0) = {
+  derefInst : core_ops_deref_Deref self;
+  deref_mut : self → result (derefInst.target & (derefInst.target → result self));
+}
+
+type core_ops_range_Range (a : Type0) = {
+  start : a;
+  end_ : a;
+}
+
+(*** [alloc] *)
+
+let alloc_boxed_Box_deref (t : Type0) (x : t) : result t = Return x
+let alloc_boxed_Box_deref_mut (t : Type0) (x : t) : result (t & (t -> result t)) =
+  Return (x, (fun x -> Return x))
+
+// Trait instance
+let alloc_boxed_Box_coreopsDerefInst (self : Type0) : core_ops_deref_Deref self = {
+  target = self;
+  deref = alloc_boxed_Box_deref self;
+}
+
+// Trait instance
+let alloc_boxed_Box_coreopsDerefMutInst (self : Type0) : core_ops_deref_DerefMut self = {
+  derefInst = alloc_boxed_Box_coreopsDerefInst self;
+  deref_mut = alloc_boxed_Box_deref_mut self;
+}
+
+(*** 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_usize (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_update_usize (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
+
+let array_index_mut_usize (a : Type0) (n : usize) (x : array a n) (i : usize) :
+  result (a & (a -> result (array a n))) =
+  match array_index_usize a n x i with
+  | Fail e -> Fail e
+  | Return v ->
+    Return (v, array_update_usize a n x i)
+
+(*** 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_usize (a : Type0) (x : slice a) (i : usize) : result a =
+  if i < length x then Return (index x i)
+  else Fail Failure
+
+let slice_update_usize (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
+
+let slice_index_mut_usize (a : Type0) (s : slice a) (i : usize) :
+  result (a & (a -> result (slice a))) =
+  match slice_index_usize a s i with
+  | Fail e -> Fail e
+  | Return x ->
+    Return (x, slice_update_usize a s i)
+
+(*** Subslices *)
+
+let array_to_slice (a : Type0) (n : usize) (x : array a n) : result (slice a) = Return x
+let array_from_slice (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
+
+let array_to_slice_mut (a : Type0) (n : usize) (x : array a n) :
+  result (slice a & (slice a -> result (array a n))) =
+  Return (x, array_from_slice a n x)
+
+// TODO: finish the definitions below (there lacks [List.drop] and [List.take] in the standard library *)
+let array_subslice (a : Type0) (n : usize) (x : array a n) (r : core_ops_range_Range usize) : result (slice a) =
+  admit()
+
+let array_update_subslice (a : Type0) (n : usize) (x : array a n) (r : core_ops_range_Range usize) (ns : slice a) : result (array a n) =
+  admit()
+
+let array_repeat (a : Type0) (n : usize) (x : a) : array a n =
+  admit()
+
+let slice_subslice (a : Type0) (x : slice a) (r : core_ops_range_Range usize) : result (slice a) =
+  admit()
+
+let slice_update_subslice (a : Type0) (x : slice a) (r : core_ops_range_Range usize) (ns : slice a) : result (slice a) =
+  admit()
+
+(*** Vector *)
+type alloc_vec_Vec (a : Type0) = v:list a{length v <= usize_max}
+
+let alloc_vec_Vec_new (a  : Type0) : alloc_vec_Vec a = assert_norm(length #a [] == 0); []
+let alloc_vec_Vec_len (a : Type0) (v : alloc_vec_Vec a) : usize = length v
+
+// Helper
+let alloc_vec_Vec_index_usize (#a : Type0) (v : alloc_vec_Vec a) (i : usize) : result a =
+  if i < length v then Return (index v i) else Fail Failure
+// Helper
+let alloc_vec_Vec_update_usize (#a : Type0) (v : alloc_vec_Vec a) (i : usize) (x : a) : result (alloc_vec_Vec a) =
+  if i < length v then Return (list_update v i x) else Fail Failure
+
+let alloc_vec_Vec_index_mut_usize (#a : Type0) (v: alloc_vec_Vec a) (i: usize) :
+  result (a & (a → result (alloc_vec_Vec a))) =
+  match alloc_vec_Vec_index_usize v i with
+  | Return x ->
+    Return (x, alloc_vec_Vec_update_usize v i)
+  | Fail e -> Fail e
+
+let alloc_vec_Vec_push (a  : Type0) (v : alloc_vec_Vec a) (x : a) :
+  Pure (result (alloc_vec_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
+
+let alloc_vec_Vec_insert (a : Type0) (v : alloc_vec_Vec a) (i : usize) (x : a) : result (alloc_vec_Vec a) =
+  if i < length v then Return (list_update v i x) else Fail Failure
+
+// Trait declaration: [core::slice::index::private_slice_index::Sealed]
+type core_slice_index_private_slice_index_Sealed (self : Type0) = unit
+
+// Trait declaration: [core::slice::index::SliceIndex]
+noeq type core_slice_index_SliceIndex (self t : Type0) = {
+  sealedInst : core_slice_index_private_slice_index_Sealed self;
+  output : Type0;
+  get : self → t → result (option output);
+  get_mut : self → t → result (option output & (option output -> result t));
+  get_unchecked : self → const_raw_ptr t → result (const_raw_ptr output);
+  get_unchecked_mut : self → mut_raw_ptr t → result (mut_raw_ptr output);
+  index : self → t → result output;
+  index_mut : self → t → result (output & (output -> result t));
+}
+
+// [core::slice::index::[T]::index]: forward function
+let core_slice_index_Slice_index
+  (t idx : Type0) (inst : core_slice_index_SliceIndex idx (slice t))
+  (s : slice t) (i : idx) : result inst.output =
+  let* x = inst.get i s in
+  match x with
+  | None -> Fail Failure
+  | Some x -> Return x
+
+// [core::slice::index::Range:::get]: forward function
+let core_slice_index_RangeUsize_get (t : Type0) (i : core_ops_range_Range usize) (s : slice t) :
+  result (option (slice t)) =
+  admit () // TODO
+
+// [core::slice::index::Range::get_mut]: forward function
+let core_slice_index_RangeUsize_get_mut (t : Type0) :
+  core_ops_range_Range usize → slice t → result (option (slice t) & (option (slice t) -> result (slice t))) =
+  admit () // TODO
+
+// [core::slice::index::Range::get_unchecked]: forward function
+let core_slice_index_RangeUsize_get_unchecked
+  (t : Type0) :
+  core_ops_range_Range usize → const_raw_ptr (slice t) → result (const_raw_ptr (slice t)) =
+  // Don't know what the model should be - for now we always fail to make
+  // sure code which uses it fails
+  fun _ _ -> Fail Failure
+
+// [core::slice::index::Range::get_unchecked_mut]: forward function
+let core_slice_index_RangeUsize_get_unchecked_mut
+  (t : Type0) :
+  core_ops_range_Range usize → mut_raw_ptr (slice t) → result (mut_raw_ptr (slice t)) =
+  // Don't know what the model should be - for now we always fail to make
+  // sure code which uses it fails
+  fun _ _ -> Fail Failure
+
+// [core::slice::index::Range::index]: forward function
+let core_slice_index_RangeUsize_index
+  (t : Type0) : core_ops_range_Range usize → slice t → result (slice t) =
+  admit () // TODO
+
+// [core::slice::index::Range::index_mut]: forward function
+let core_slice_index_RangeUsize_index_mut (t : Type0) :
+  core_ops_range_Range usize → slice t → result (slice t & (slice t -> result (slice t))) =
+  admit () // TODO
+
+// [core::slice::index::[T]::index_mut]: forward function
+let core_slice_index_Slice_index_mut
+  (t idx : Type0) (inst : core_slice_index_SliceIndex idx (slice t)) :
+  slice t → idx → result (inst.output & (inst.output -> result (slice t))) =
+  admit () // 
+
+// [core::array::[T; N]::index]: forward function
+let core_array_Array_index
+  (t idx : Type0) (n : usize) (inst : core_ops_index_Index (slice t) idx)
+  (a : array t n) (i : idx) : result inst.output =
+  admit () // TODO
+
+// [core::array::[T; N]::index_mut]: forward function
+let core_array_Array_index_mut
+  (t idx : Type0) (n : usize) (inst : core_ops_index_IndexMut (slice t) idx)
+  (a : array t n) (i : idx) :
+  result (inst.indexInst.output & (inst.indexInst.output -> result (array t n))) =
+  admit () // TODO
+
+// Trait implementation: [core::slice::index::private_slice_index::Range]
+let core_slice_index_private_slice_index_SealedRangeUsizeInst
+  : core_slice_index_private_slice_index_Sealed (core_ops_range_Range usize) = ()
+
+// Trait implementation: [core::slice::index::Range]
+let core_slice_index_SliceIndexRangeUsizeSliceTInst (t : Type0) :
+  core_slice_index_SliceIndex (core_ops_range_Range usize) (slice t) = {
+  sealedInst = core_slice_index_private_slice_index_SealedRangeUsizeInst;
+  output = slice t;
+  get = core_slice_index_RangeUsize_get t;
+  get_mut = core_slice_index_RangeUsize_get_mut t;
+  get_unchecked = core_slice_index_RangeUsize_get_unchecked t;
+  get_unchecked_mut = core_slice_index_RangeUsize_get_unchecked_mut t;
+  index = core_slice_index_RangeUsize_index t;
+  index_mut = core_slice_index_RangeUsize_index_mut t;
+}
+
+// Trait implementation: [core::slice::index::[T]]
+let core_ops_index_IndexSliceTIInst (t idx : Type0)
+  (inst : core_slice_index_SliceIndex idx (slice t)) :
+  core_ops_index_Index (slice t) idx = {
+  output = inst.output;
+  index = core_slice_index_Slice_index t idx inst;
+}
+
+// Trait implementation: [core::slice::index::[T]]
+let core_ops_index_IndexMutSliceTIInst (t idx : Type0)
+  (inst : core_slice_index_SliceIndex idx (slice t)) :
+  core_ops_index_IndexMut (slice t) idx = {
+  indexInst = core_ops_index_IndexSliceTIInst t idx inst;
+  index_mut = core_slice_index_Slice_index_mut t idx inst;
+}
+
+// Trait implementation: [core::array::[T; N]]
+let core_ops_index_IndexArrayInst (t idx : Type0) (n : usize)
+  (inst : core_ops_index_Index (slice t) idx) :
+  core_ops_index_Index (array t n) idx = {
+  output = inst.output;
+  index = core_array_Array_index t idx n inst;
+}
+
+// Trait implementation: [core::array::[T; N]]
+let core_ops_index_IndexMutArrayIInst (t idx : Type0) (n : usize)
+  (inst : core_ops_index_IndexMut (slice t) idx) :
+  core_ops_index_IndexMut (array t n) idx = {
+  indexInst = core_ops_index_IndexArrayInst t idx n inst.indexInst;
+  index_mut = core_array_Array_index_mut t idx n inst;
+}
+
+// [core::slice::index::usize::get]: forward function
+let core_slice_index_usize_get
+  (t : Type0) : usize → slice t → result (option t) =
+  admit () // TODO
+
+// [core::slice::index::usize::get_mut]: forward function
+let core_slice_index_usize_get_mut (t : Type0) :
+  usize → slice t → result (option t & (option t -> result (slice t))) =
+  admit () // TODO
+
+// [core::slice::index::usize::get_unchecked]: forward function
+let core_slice_index_usize_get_unchecked
+  (t : Type0) : usize → const_raw_ptr (slice t) → result (const_raw_ptr t) =
+  admit () // TODO
+
+// [core::slice::index::usize::get_unchecked_mut]: forward function
+let core_slice_index_usize_get_unchecked_mut
+  (t : Type0) : usize → mut_raw_ptr (slice t) → result (mut_raw_ptr t) =
+  admit () // TODO
+
+// [core::slice::index::usize::index]: forward function
+let core_slice_index_usize_index (t : Type0) : usize → slice t → result t =
+  admit () // TODO
+
+// [core::slice::index::usize::index_mut]: forward function
+let core_slice_index_usize_index_mut (t : Type0) :
+  usize → slice t → result (t & (t -> result (slice t))) =
+  admit () // TODO
+
+// Trait implementation: [core::slice::index::private_slice_index::usize]
+let core_slice_index_private_slice_index_SealedUsizeInst
+  : core_slice_index_private_slice_index_Sealed usize = ()
+
+// Trait implementation: [core::slice::index::usize]
+let core_slice_index_SliceIndexUsizeSliceTInst (t : Type0) :
+  core_slice_index_SliceIndex usize (slice t) = {
+  sealedInst = core_slice_index_private_slice_index_SealedUsizeInst;
+  output = t;
+  get = core_slice_index_usize_get t;
+  get_mut = core_slice_index_usize_get_mut t;
+  get_unchecked = core_slice_index_usize_get_unchecked t;
+  get_unchecked_mut = core_slice_index_usize_get_unchecked_mut t;
+  index = core_slice_index_usize_index t;
+  index_mut = core_slice_index_usize_index_mut t;
+}
+
+// [alloc::vec::Vec::index]: forward function
+let alloc_vec_Vec_index (t idx : Type0) (inst : core_slice_index_SliceIndex idx (slice t))
+  (self : alloc_vec_Vec t) (i : idx) : result inst.output =
+  admit () // TODO
+
+// [alloc::vec::Vec::index_mut]: forward function
+let alloc_vec_Vec_index_mut (t idx : Type0) (inst : core_slice_index_SliceIndex idx (slice t))
+  (self : alloc_vec_Vec t) (i : idx) :
+  result (inst.output & (inst.output -> result (alloc_vec_Vec t))) =
+  admit () // TODO
+
+// Trait implementation: [alloc::vec::Vec]
+let alloc_vec_Vec_coreopsindexIndexInst (t idx : Type0)
+  (inst : core_slice_index_SliceIndex idx (slice t)) :
+  core_ops_index_Index (alloc_vec_Vec t) idx = {
+  output = inst.output;
+  index = alloc_vec_Vec_index t idx inst;
+}
+
+// Trait implementation: [alloc::vec::Vec]
+let alloc_vec_Vec_coreopsindexIndexMutInst (t idx : Type0)
+  (inst : core_slice_index_SliceIndex idx (slice t)) :
+  core_ops_index_IndexMut (alloc_vec_Vec t) idx = {
+  indexInst = alloc_vec_Vec_coreopsindexIndexInst t idx inst;
+  index_mut = alloc_vec_Vec_index_mut t idx inst;
+}
+
+(*** Theorems *)
+
+let alloc_vec_Vec_index_eq (#a : Type0) (v : alloc_vec_Vec a) (i : usize) :
+  Lemma (
+    alloc_vec_Vec_index a usize (core_slice_index_SliceIndexUsizeSliceTInst a) v i ==
+      alloc_vec_Vec_index_usize v i)
+  [SMTPat (alloc_vec_Vec_index a usize (core_slice_index_SliceIndexUsizeSliceTInst a) v i)]
+  =
+  admit()
+
+let alloc_vec_Vec_index_mut_eq (#a : Type0) (v : alloc_vec_Vec a) (i : usize) :
+  Lemma (
+    alloc_vec_Vec_index_mut a usize (core_slice_index_SliceIndexUsizeSliceTInst a) v i ==
+      alloc_vec_Vec_index_mut_usize v i)
+  [SMTPat (alloc_vec_Vec_index_mut a usize (core_slice_index_SliceIndexUsizeSliceTInst a) v i)]
+  =
+  admit()