val lemma_little_endian_lt_2_128: b:bytes {Seq.length b <= 16} -> Lemma
  (requires True)
  (ensures  (little_endian b < pow2 128))
  [SMTPat (little_endian b)]
let lemma_little_endian_lt_2_128 b =
  lemma_little_endian_is_bounded b;
  if Seq.length b = 16 then ()
  else Math.Lemmas.pow2_lt_compat 128 (8 * Seq.length b)
val lemma_little_endian_lt_2_128: b:bytes {Seq.length b <= 16} -> Lemma
  (requires True)
  (ensures  (little_endian b < pow2 128))
  [SMTPat (little_endian b)]
let lemma_little_endian_lt_2_128 b =
lemma_little_endian_is_bounded b;
if Seq.length b = 16 then () else Math.Lemmas.pow2_lt_compat 128 (8 * Seq.length b)
FStar.Old.Endianness.fst.checked
prims.fst.checked FStar.UInt8.fsti.checked FStar.UInt64.fsti.checked FStar.UInt32.fsti.checked FStar.UInt128.fsti.checked FStar.Seq.fst.checked FStar.Pervasives.fsti.checked FStar.Mul.fst.checked FStar.Math.Lib.fst.checked FStar.Math.Lemmas.fst.checked FStar.Int.Cast.fst.checked FStar.HyperStack.ST.fsti.checked FStar.HyperStack.All.fst.checked FStar.Ghost.fsti.checked
FStar.Old.Endianness.fst
FStar.Old.Endianness.bytes Prims.b2t Prims.op_LessThanOrEqual FStar.Seq.Base.length FStar.UInt8.t Prims.op_Equality Prims.int Prims.bool FStar.Math.Lemmas.pow2_lt_compat FStar.Mul.op_Star Prims.unit FStar.Old.Endianness.lemma_little_endian_is_bounded
module FStar.Old.Endianness

module ST = FStar.HyperStack.ST

open FStar.HyperStack.All

open FStar.UInt32
open FStar.Ghost
open FStar.Mul
open FStar.Int.Cast

module U8 = FStar.UInt8

let u8  = UInt8.t
let u32 = UInt32.t
let u64 = UInt64.t

#set-options "--initial_fuel 0 --max_fuel 0"

type bytes  = Seq.seq U8.t // Pure sequence of bytes

type  lbytes  (l:nat) = b:bytes   {Seq.length b == l}

(* type lbuffer (l:nat) = b:buffer {Buffer.length b == l} *)

#reset-options

let uint128_to_uint8 (a:UInt128.t) : Tot (b:UInt8.t{UInt8.v b = UInt128.v a % pow2 8})
  = Math.Lemmas.pow2_modulo_modulo_lemma_2 (UInt128.v a) 64 8;
    uint64_to_uint8 (FStar.UInt128.uint128_to_uint64 a)

#reset-options "--z3rlimit 20 --initial_fuel 0 --max_fuel 0"


open FStar.Seq

(* Little endian integer value of a sequence of bytes *)
let rec little_endian (b:bytes) : Tot (n:nat) (decreases (Seq.length b)) =
  if Seq.length b = 0 then 0
  else
    UInt8.v (head b) + pow2 8 * little_endian (tail b)

(* Big endian integer value of a sequence of bytes *)
let rec big_endian (b:bytes) : Tot (n:nat) (decreases (Seq.length b)) = 
  if Seq.length b = 0 then 0 
  else
    UInt8.v (last b) + pow2 8 * big_endian (Seq.slice b 0 (Seq.length b - 1))

#reset-options "--initial_fuel 1 --max_fuel 1"

val little_endian_null: len:nat{len < 16} -> Lemma
  (little_endian (Seq.create len 0uy) == 0)
let rec little_endian_null len =
  if len = 0 then ()
  else
    begin
    Seq.lemma_eq_intro (Seq.slice (Seq.create len 0uy) 1 len)
		       (Seq.create (len - 1) 0uy);
    assert (little_endian (Seq.create len 0uy) ==
      0 + pow2 8 * little_endian (Seq.slice (Seq.create len 0uy) 1 len));
    little_endian_null (len - 1)
    end

val little_endian_singleton: n:UInt8.t -> Lemma
  (little_endian (Seq.create 1 n) == UInt8.v n)
let little_endian_singleton n =
  assert (little_endian (Seq.create 1 n) ==
    UInt8.v (Seq.index (Seq.create 1 n) 0) + pow2 8 *
    little_endian (Seq.slice (Seq.create 1 n) 1 1))


#reset-options "--initial_fuel 1 --max_fuel 1 --z3rlimit 50"

val little_endian_append: w1:bytes -> w2:bytes -> Lemma
  (requires True)
  (ensures
    little_endian (Seq.append w1 w2) ==
    little_endian w1 + pow2 (8 * Seq.length w1) * little_endian w2)
  (decreases (Seq.length w1))
let rec little_endian_append w1 w2 =
  let open FStar.Seq in
  if length w1 = 0 then
    begin
    assert_norm (pow2 (8 * 0) == 1);
    Seq.lemma_eq_intro (append w1 w2) w2
    end
  else
    begin
    let w1' = slice w1 1 (length w1) in
    assert (length w1' == length w1 - 1);
    little_endian_append w1' w2;
    assert (index (append w1 w2) 0 == index w1 0);
    Seq.lemma_eq_intro
      (append w1' w2)
      (Seq.slice (append w1 w2) 1 (length (append w1 w2)));
    assert (little_endian (append w1 w2) ==
      UInt8.v (index w1 0) + pow2 8 * little_endian (append w1' w2));
    assert (little_endian (append w1' w2) ==
      little_endian w1' + pow2 (8 * length w1') * little_endian w2);
    assert (UInt8.v (index w1 0) + pow2 8 * little_endian (append w1' w2) ==
      UInt8.v (index w1 0) +
      pow2 8 * (little_endian w1' + pow2 (8 * length w1') * little_endian w2));
    Math.Lemmas.pow2_plus 8 (8 * (length w1 - 1));
    assert (pow2 8 * pow2 (8 * length w1') == pow2 (8 * length w1));
    assert (UInt8.v (index w1 0) + pow2 8 * little_endian (append w1' w2) ==
      UInt8.v (index w1 0) +
      pow2 8 * little_endian w1' + pow2 (8 * length w1) * little_endian w2);
    assert (UInt8.v (index w1 0) + pow2 8 * little_endian w1' == little_endian w1)
    end


#reset-options "--initial_fuel 1 --max_fuel 1 --z3rlimit 50"

val lemma_euclidean_division: r:nat -> b:nat -> q:pos -> Lemma
  (requires (r < q))
  (ensures  (r + q * b < q * (b+1)))
let lemma_euclidean_division r b q = ()

private val lemma_factorise: a:nat -> b:nat -> Lemma (a + a * b == a * (b + 1))
let lemma_factorise a b = ()

val lemma_little_endian_is_bounded: b:bytes -> Lemma
  (requires True)
  (ensures  (little_endian b < pow2 (8 * Seq.length b)))
  (decreases (Seq.length b))
let rec lemma_little_endian_is_bounded b =
  if Seq.length b = 0 then ()
  else
    begin
    let s = Seq.slice b 1 (Seq.length b) in
    assert(Seq.length s = Seq.length b - 1);
    lemma_little_endian_is_bounded s;
    assert(UInt8.v (Seq.index b 0) < pow2 8);
    assert(little_endian s < pow2 (8 * Seq.length s));
    assert(little_endian b < pow2 8 + pow2 8 * pow2 (8 * (Seq.length b - 1)));
    lemma_euclidean_division (UInt8.v (Seq.index b 0)) (little_endian s) (pow2 8);
    assert(little_endian b <= pow2 8 * (little_endian s + 1));
    assert(little_endian b <= pow2 8 * pow2 (8 * (Seq.length b - 1)));
    Math.Lemmas.pow2_plus 8 (8 * (Seq.length b - 1));
    lemma_factorise 8 (Seq.length b - 1)
    end

val lemma_big_endian_is_bounded: b:bytes -> Lemma
  (requires True)
  (ensures  (big_endian b < pow2 (8 * Seq.length b)))
  (decreases (Seq.length b))
  [SMTPat (big_endian b)]
let rec lemma_big_endian_is_bounded b =
  if Seq.length b = 0 then ()
  else
    begin
    let s = Seq.slice b 0 (Seq.length b - 1) in
    assert(Seq.length s = Seq.length b - 1);
    lemma_big_endian_is_bounded s;
    assert(UInt8.v (Seq.last b) < pow2 8);
    assert(big_endian s < pow2 (8 * Seq.length s));
    assert(big_endian b < pow2 8 + pow2 8 * pow2 (8 * (Seq.length b - 1)));
    lemma_euclidean_division (UInt8.v (Seq.last b)) (big_endian s) (pow2 8);
    assert(big_endian b <= pow2 8 * (big_endian s + 1));
    assert(big_endian b <= pow2 8 * pow2 (8 * (Seq.length b - 1)));
    Math.Lemmas.pow2_plus 8 (8 * (Seq.length b - 1));
    lemma_factorise 8 (Seq.length b - 1)
    end


#reset-options "--initial_fuel 0 --max_fuel 0"

val lemma_little_endian_lt_2_128: b:bytes {Seq.length b <= 16} -> Lemma
  (requires True)
  (ensures  (little_endian b < pow2 128))
  [SMTPat (little_endian b)]
val lemma_little_endian_lt_2_128: b:bytes {Seq.length b <= 16} -> Lemma
  (requires True)
  (ensures  (little_endian b < pow2 128))
  [SMTPat (little_endian b)]
b: FStar.Old.Endianness.bytes{FStar.Seq.Base.length b <= 16}
  -> FStar.Pervasives.Lemma (ensures FStar.Old.Endianness.little_endian b < Prims.pow2 128)
      [SMTPat (FStar.Old.Endianness.little_endian b)]