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Bitwalker


Authors: Claude Marché / Clément Fumex

Topics: Bitwise operations

Tools: Why3

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module Bitwalker
  use import int.Int
  use import int.EuclideanDivision
  use import array.Array
  use import ref.Ref
  (* use bv.BV32 *)
  (* use bv.BV8 *)
  (* use bv.BV64 *)
  use mach.bv.BVCheck8  as BV8
  use mach.bv.BVCheck32 as BV32
  use mach.bv.BVCheck64 as BV64
  use bv.BVConverter_32_64 as C32_64
  use bv.BVConverter_8_32 as C8_32

(* file missing in repository
  use import mach.bv.BV
*)

  function nth8_stream (stream : array BV8.t) (pos : int) : bool =
    BV8.nth stream[div pos 8] (7 - mod pos 8)

  lemma nth64:
    forall value:BV64.t, i:int. 0 <= i < 64 ->
      BV64.nth value i = BV64.nth_bv value (C32_64.toBig (BV32.of_int i))

  lemma nth8:
    forall value:BV8.t, i:int. 0 <= i < 8 ->
      BV8.nth value i = BV8.nth_bv value (C8_32.toSmall (BV32.of_int i))

  (*  *)
  function maxvalue (len : BV32.t) : BV64.t = BV64.lsl_bv (BV64.of_int 1) (C32_64.toBig len)

  let lemma nth_ultpre0 (x:BV64.t) (len:BV32.t)
    requires { BV32.to_uint len < 64}
    ensures { BV64.eq_sub x BV64.zeros (BV32.to_uint len) (64 - BV32.to_uint len)
          <-> BV64.to_uint x < BV64.to_uint (maxvalue len) }
  =
    assert { BV32.ult len (BV32.of_int 64) };
    assert { BV64.eq_sub_bv x BV64.zeros (C32_64.toBig len) (BV64.sub (BV64.of_int 64) (C32_64.toBig len))
         <-> BV64.ult x (maxvalue len) }

return value with the bit of index left from the left set to flag

  let poke_64bit_bv (value : BV64.t) (left : BV32.t) (flag : bool) : BV64.t
    requires { BV32.to_uint left < 64 }
    ensures  { forall i. 0 <= i < 64 /\ i <> 63 - BV32.to_uint left ->
                 BV64.nth result i = BV64.nth value i }
    ensures  { flag = BV64.nth result (63 - BV32.to_uint left) }
  =
    assert { BV32.ult left (BV32.of_int 64) };
    abstract
    ensures { forall i:BV32.t. i <> BV32.sub (BV32.of_int 63) left ->
               BV64.nth_bv result (C32_64.toBig i) =
                 BV64.nth_bv value (C32_64.toBig i) }
    ensures { flag = BV64.nth_bv result
                       (C32_64.toBig (BV32.sub (BV32.of_int 63) left)) }
      let mask =
        BV64.lsl_bv (BV64.int_check 1)
          (C32_64.toBig (BV32.sub_check (BV32.int_check 63) left))
      in
      match flag with
      | True -> BV64.bw_or value mask
      | False -> BV64.bw_and value (BV64.bw_not mask)
      end
    end

return value with the bit of index left from the left set to flag

(* version where [left] is an int and not a bitvector, which
  is closer to the result of the SPARK translation from signed
  integers *)
  let poke_64bit (value : BV64.t) (left : int) (flag : bool) : BV64.t
    requires { 0 <= left < 64 }
    ensures  { forall i. 0 <= i < 64 /\ i <> 63 - left ->
                 BV64.nth result i = BV64.nth value i }
    ensures  { flag = BV64.nth result (63 - left) }
  =
    let ghost left_bv = BV64.of_int left in
    assert { BV64.ult left_bv (BV64.of_int 64) };
    assert { (BV64.sub (BV64.of_int 63) left_bv) = BV64.of_int (63 - left) };
    abstract
    ensures { forall i:BV64.t. BV64.ule i (BV64.of_int 63) ->
               i <> BV64.sub (BV64.of_int 63) left_bv ->
               BV64.nth_bv result i = BV64.nth_bv value i }
    ensures { flag = BV64.nth_bv result (BV64.sub (BV64.of_int 63) left_bv) }
      let mask =
        BV64.lsl_bv (BV64.int_check 1) (BV64.of_int (63 - left))
      in
      match flag with
      | True -> BV64.bw_or value mask
      | False -> BV64.bw_and value (BV64.bw_not mask)
      end
    end

  (* return the bit of [byte] at position [left] starting from the
  left *)

  let peek_8bit_bv (byte : BV8.t) (left : BV32.t) : bool
    requires { 0 <= BV32.to_uint left < 8 }
    ensures  { result = BV8.nth byte (7 - BV32.to_uint left) }
  =
    assert {BV32.ult left (BV32.of_int 8)};
    abstract
      ensures {
        result = BV8.nth_bv
                   byte (BV8.sub (BV8.of_int 7) (C8_32.toSmall left))}
    let mask =
      BV32.lsl_bv (BV32.int_check 1) (BV32.sub_check (BV32.int_check 7) left)
    in
    BV32.bw_and (C8_32.toBig byte) mask <> BV32.zeros
    end

  (* return the bit of the [left]/8 element of [addr] at position mod [left] 8 starting from the left *)
  let peek_8bit_array (addr : array BV8.t) (left : BV32.t) : bool
    requires { 8 * (length addr) < BV32.two_power_size }
    requires { BV32.to_uint left < 8 * length addr }
    ensures  { result = nth8_stream addr (BV32.to_uint left) }
  =
    peek_8bit_bv (addr[ BV32.to_uint (BV32.udiv_check left (BV32.int_check 8)) ]) (BV32.urem_check left (BV32.int_check 8))

  (* return a bitvector of 64 bits with its [len] bits of the right
    set to the bits between [start] and [start] + [len] of [addr] *)
  let peek (start : BV32.t) (len : BV32.t) (addr : array BV8.t) : BV64.t
    requires { BV32.to_uint len <= 64 }
    requires { BV32.to_uint start + BV32.to_uint len < BV32.two_power_size }
    requires { 8 * length addr < BV32.two_power_size }
    ensures  { BV32.to_uint start + BV32.to_uint len > (8 * length addr) ->
      result = BV64.zeros }
    ensures  { BV32.to_uint start + BV32.to_uint len <= (8 * length addr) ->
      (forall i:int. 0 <= i < BV32.to_uint len ->
         BV64.nth result i
         = nth8_stream addr (BV32.to_uint start + BV32.to_uint len - i - 1))
      /\
      (forall i:int. BV32.to_uint len <= i < 64 -> BV64.nth result i = False) }
  =
    if (BV32.to_uint (BV32.add_check start len) > ( 8 *length addr ))
    then
      BV64.zeros
    else

    let retval = ref BV64.zeros in
    let i = ref BV32.zeros in
    let lstart = BV32.sub_check (BV32.of_int 64) len in

    while BV32.ult !i len do variant{ !i with BV32.ugt }
    invariant {0 <= BV32.to_uint !i <= BV32.to_uint len}
    invariant {forall j:int. BV32.to_uint len - BV32.to_uint !i <= j < BV32.to_uint len ->
                 BV64.nth !retval j
               = nth8_stream addr (BV32.to_uint start + BV32.to_uint len - j - 1)}
    invariant {forall j:int. 0 <= j < BV32.to_uint len - BV32.to_uint !i ->
                 BV64.nth !retval j
               = False}
    invariant {forall j:int. BV32.to_uint len <= j < 64 ->
                 BV64.nth !retval j
               = False}

      let flag = peek_8bit_array addr (BV32.add_check start !i) in
      retval := poke_64bit_bv !retval (BV32.add_check lstart !i) flag;

      i := BV32.add_check !i (BV32.int_check 1);

    done;

    !retval

  let peek_64bit (value : BV64.t) (left : BV32.t) : bool
    requires {BV32.to_uint left < 64}
    ensures {result = BV64.nth value (63 - BV32.to_uint left)}
  =
     assert {BV32.ult left (BV32.of_int 64)};
     abstract
     ensures {result = BV64.nth_bv value
                       (BV64.sub (BV64.of_int 63) (C32_64.toBig left))}
       let mask = BV64.lsl_bv (BV64.int_check 1)
                  (C32_64.toBig (BV32.sub_check (BV32.int_check 63) left)) in
       BV64.bw_and value mask <> BV64.zeros
     end

(*
  static inline uint8_t PokeBit8(uint8_t byte, uint32_t left, int flag)
  {
    uint8_t mask = ((uint8_t) 1) << (7u - left);

    return (flag == 0) ? (byte & ~mask) : (byte | mask);
  }
*)

  (* return [byte] with the bit at index [left] from the left set to [flag] *)
  let poke_8bit (byte : BV8.t) (left : BV32.t) (flag : bool) : BV8.t
    requires { BV32.to_uint left < 8 }
    ensures  { forall i:int. 0 <= i < 8 -> i <> 7 - BV32.to_uint left ->
               BV8.nth result i = BV8.nth byte i }
    ensures  { BV8.nth result (7 - BV32.to_uint left) = flag }
  =
    assert { BV32.ult left (BV32.of_int 8) };
    abstract
    ensures { forall i:BV32.t. BV32.ult i (BV32.of_int 8) ->
               i <> BV32.sub (BV32.of_int 7) left ->
               BV8.nth_bv result (C8_32.toSmall i)
             = BV8.nth_bv byte (C8_32.toSmall i) }
    ensures { BV8.nth_bv result (BV8.sub (BV8.of_int 7) (C8_32.toSmall left))
            = flag }
      let mask = BV8.lsl_bv (BV8.int_check 1) (BV8.sub_check (BV8.int_check 7) (C8_32.toSmall left)) in
      match flag with
      | True -> BV8.bw_or byte mask
      | False -> BV8.bw_and byte (BV8.bw_not mask)
      end
    end

  let poke_8bit_array (addr : array BV8.t) (left : BV32.t) (flag : bool)
    writes {addr}
    requires { 8 * (length addr) < BV32.two_power_size }
    requires { BV32.to_uint left < 8 * length addr }
    ensures  { forall i:int. 0 <= i < 8 * length addr -> i <> BV32.to_uint left ->
               nth8_stream addr i = nth8_stream (old addr) i}
    ensures  { nth8_stream addr (BV32.to_uint left) = flag }
  =
    let i = BV32.udiv_check left (BV32.int_check 8) in
    let k = BV32.urem_check left (BV32.int_check 8) in
    addr[BV32.to_uint i] <- poke_8bit addr[BV32.to_uint i] k flag

  let poke (start : BV32.t) (len : BV32.t) (addr : array BV8.t) (value : BV64.t)
    writes  { addr }
    requires{ BV32.to_uint len < 64 } (* could be lower or equal if maxvalue and the condition to return -2 is corrected *)
    requires{ BV32.to_uint start + BV32.to_uint len < BV32.two_power_size }
    requires{ 8 * length addr < BV32.two_power_size }
    ensures { -2 <= result <= 0 }
    ensures { result = -1 <-> BV32.to_uint start + BV32.to_uint len > 8 * length addr }
    ensures { result = -2 <-> BV64.to_uint (maxvalue len) <= BV64.to_uint value /\ BV32.to_uint start + BV32.to_uint len <= 8 * length addr }
    ensures { result =  0 <-> BV64.to_uint (maxvalue len) > BV64.to_uint value /\ BV32.to_uint start + BV32.to_uint len <= 8 * length addr }
    ensures { result =  0 ->
              (forall i:int. 0 <= i < BV32.to_uint start ->
                nth8_stream (old addr) i
              = nth8_stream addr i)
           /\
              (forall i:int. BV32.to_uint start <= i < BV32.to_uint start + BV32.to_uint len ->
                nth8_stream addr i
              = BV64.nth value (BV32.to_uint len - i - 1 + BV32.to_uint start))
           /\
              (forall i:int. BV32.to_uint start + BV32.to_uint len <= i < 8 * length addr ->
                nth8_stream addr i
              = nth8_stream (old addr) i) }
  =
    if BV32.to_uint (BV32.add_check start len) > 8 * length addr
    then
      -1                        (*error: invalid_bit_sequence*)
    else

    if BV64.uge value (maxvalue len) (* should be len <> 64 && _..._ *)
    then
      -2                        (*error: value_too_big*)
    else

    let lstart = BV32.sub_check (BV32.of_int 64) len in
    let i = ref BV32.zeros in

    'Init:
    while BV32.ult !i len do variant { !i with BV32.ugt }
    invariant {0 <= BV32.to_uint !i <= BV32.to_uint len}
    invariant {forall j:int. 0 <= j < BV32.to_uint start ->
                 nth8_stream (at addr 'Init) j
               = nth8_stream addr j}
    invariant {forall j:int. BV32.to_uint start <= j < BV32.to_uint start + BV32.to_uint !i ->
                 nth8_stream addr j
               = BV64.nth value (BV32.to_uint len - j - 1 + BV32.to_uint start) }
    invariant {forall j:int. BV32.to_uint start + BV32.to_uint !i <= j < 8 * length addr ->
                 nth8_stream addr j
               = nth8_stream (at addr 'Init) j }

      let flag = peek_64bit value (BV32.add_check lstart !i) in

      poke_8bit_array addr (BV32.add_check start !i) flag;

      assert {nth8_stream addr (BV32.to_uint start + BV32.to_uint !i)
            = BV64.nth value ((BV32.to_uint len - BV32.to_uint !i - 1))};
      assert { forall k. BV32.to_uint start <= k < BV32.to_uint start + BV32.to_uint !i ->
               k <> BV32.to_uint start + BV32.to_uint !i &&
               0 <= k < 8 * length addr &&
                   nth8_stream addr k
                 = BV64.nth value (BV32.to_uint start + BV32.to_uint len - k - 1)};

      i := BV32.add_check !i (BV32.int_check 1);
    done;

    0

  let peekthenpoke (start len : BV32.t) (addr : array BV8.t)
    requires {8 * length addr < BV32.two_power_size}
    requires {0 <= BV32.to_uint len < 64}
    requires {BV32.to_uint start + BV32.to_uint len <= 8 * length addr}
    ensures {result = 0}
    ensures {forall i. 0 <= i < 8 * length addr ->
               nth8_stream addr i = nth8_stream (old addr) i}
  =
    'Init:
    let value = peek start len addr in
    let res = poke start len addr value in

    assert {res = 0};

    assert {forall i. BV32.to_uint start <= i < BV32.to_uint start + BV32.to_uint len ->
         nth8_stream addr i
       = nth8_stream (at addr 'Init) i};

    res

  let pokethenpeek (start len : BV32.t) (addr : array BV8.t) (value : BV64.t)
    writes   {addr}
    requires {8 * length addr < BV32.two_power_size}
    requires {0 <= BV32.to_uint len < 64}
    requires {BV32.to_uint start + BV32.to_uint len <= 8 * length addr}
    requires {BV64.to_uint value < BV64.to_uint (maxvalue len)}
    ensures  {result = value}
  =
    assert { forall i:int. BV32.to_uint len <= i < 64 -> BV64.nth value i = False };
    let poke_result = poke start len addr value in
    assert {poke_result = 0};
    let peek_result = peek start len addr in

    assert {BV64.eq peek_result value};

    peek_result

end

download ZIP archive

Why3 Proof Results for Project "bitwalker"

Theory "bitwalker.Bitwalker": fully verified in 39.99 s

ObligationsAlt-Ergo (0.99.1)Alt-Ergo (1.00.prv)Alt-Ergo (1.10.prv)CVC3 (2.4.1)CVC4 (1.4)CVC4 (1.4 noBV)Z3 (4.4.0)
nth640.10------------------
nth80.30------------------
VC for nth_ultpre0---------------------
split_goal_wp
  1. assertion0.03------------------
2. assertion------------0.16------
3. postcondition0.36------------0.12---
VC for poke_64bit_bv---------------------
split_goal_wp
  1. assertion0.04------------------
2. precondition0.02---------0.01------
3. precondition0.02---------0.02------
4. precondition0.03---------0.01------
5. VC for poke_64bit_bv------------0.08------
6. VC for poke_64bit_bv------------0.08------
7. VC for poke_64bit_bv------------0.08------
8. VC for poke_64bit_bv------------0.06------
9. postcondition0.34------------0.09---
10. postcondition0.29------------0.12---
VC for poke_64bit---------------------
split_goal_wp
  1. assertion0.04------------------
2. assertion0.27------------------
3. precondition0.04------------------
4. VC for poke_64bit------------0.31------
5. VC for poke_64bit------------0.18------
6. VC for poke_64bit------------0.32------
7. VC for poke_64bit------------0.07------
8. postcondition0.48------------------
9. postcondition0.08------------------
VC for peek_8bit_bv---------------------
split_goal_wp
  1. assertion0.03------------------
2. precondition0.05---------0.02------
3. precondition0.06---------0.02------
4. precondition0.06---------0.01------
5. VC for peek_8bit_bv------------0.09------
6. postcondition1.22------------------
VC for peek_8bit_array---------------------
split_goal_wp
  1. precondition0.02------0.060.04------
2. precondition0.06------0.040.04------
3. precondition0.03------0.040.04------
4. precondition0.05------0.040.04------
5. index in array bounds0.26------0.05---------
6. precondition0.35------0.530.08------
7. postcondition0.14------------------
VC for peek---------------------
split_goal_wp
  1. precondition0.06------0.03---------
2. postcondition0.03------0.070.04------
3. postcondition0.03------0.090.10------
4. precondition0.04------0.04---------
5. loop invariant init0.05------0.09---------
6. loop invariant init4.64------0.04---------
7. loop invariant init------0.030.05---0.060.01
8. loop invariant init0.03------0.04---------
9. precondition0.05------0.06---------
10. precondition0.06------0.090.18------
11. precondition0.06------0.12---------
12. precondition---------0.05---------
13. precondition---------0.72---------
14. precondition0.10------0.120.06------
15. precondition0.11------0.130.10------
16. loop invariant preservation0.19------0.12---------
17. loop invariant preservation---1.65---------------
18. loop invariant preservation------------------0.01
19. loop invariant preservation---------------0.110.02
20. loop variant decrease0.14------0.14---------
21. postcondition0.05------0.080.14------
22. postcondition---------0.11---------
VC for peek_64bit---------------------
split_goal_wp
  1. assertion0.03------0.04---------
2. precondition0.02------0.080.06------
3. precondition0.05------0.040.12------
4. precondition0.02------0.070.06------
5. VC for peek_64bit------------0.22------
6. postcondition0.07------------------
VC for poke_8bit---------------------
split_goal_wp
  1. assertion0.03------0.04---------
2. precondition0.02------0.080.06------
3. precondition0.10------0.120.10------
4. precondition0.02------0.080.06------
5. VC for poke_8bit------------0.08------
6. VC for poke_8bit------------0.09------
7. VC for poke_8bit------------0.08------
8. VC for poke_8bit------------0.08------
9. postcondition---------0.83---------
10. postcondition---------0.09---------
VC for poke_8bit_array---------------------
split_goal_wp
  1. precondition0.03------0.080.03---0.00
2. precondition0.05------0.040.04---0.00
3. precondition0.02------0.060.04---0.00
4. precondition0.04------0.040.03---0.00
5. index in array bounds0.27------0.05---------
6. precondition0.34------1.260.08---0.01
7. index in array bounds0.03------0.090.09---0.01
8. postcondition------------------0.01
9. postcondition0.08---------------0.01
VC for poke---------------------
split_goal_wp
  1. precondition0.06------0.04---------
2. postcondition0.03------0.070.05------
3. postcondition0.05------0.100.07------
4. postcondition0.06------0.090.05------
5. postcondition0.04------0.090.10------
6. postcondition0.03------0.100.06------
7. postcondition0.03------0.090.06------
8. postcondition0.05------0.070.10------
9. postcondition0.04------0.05---------
10. postcondition0.04------0.09---------
11. postcondition0.03------0.090.02------
12. precondition0.04------0.04---------
13. loop invariant init0.06------0.09---------
14. loop invariant init---------0.050.10------
15. precondition0.05------0.05---------
16. precondition---------1.03---------
17. precondition---------0.07---------
18. type invariant0.06------0.100.16------
19. precondition0.06------0.090.16------
20. precondition0.04------0.11---------
21. assertion---0.44---0.06---------
22. assertion---------------------
split_goal_wp
  1. assertion0.05------0.130.06------
2. assertion---------0.06---------
3. assertion0.13------0.120.10------
4. assertion---------------0.100.02
23. precondition0.04------0.050.02------
24. precondition0.05------0.150.22------
25. loop invariant preservation0.22------0.06---------
26. loop invariant preservation0.29------------------
27. loop invariant preservation1.16------------------
28. loop invariant preservation0.49------------------
29. loop variant decrease0.20------0.86---------
30. type invariant0.05------0.080.08------
31. postcondition0.03------0.080.04------
32. postcondition0.05------0.080.12------
33. postcondition0.04------0.08---------
34. postcondition0.03------0.08---------
35. postcondition---------------------
split_goal_wp
  1. VC for poke0.08------0.07---------
2. VC for poke0.19------------------
3. VC for poke0.16------------------
VC for peekthenpoke---------------------
split_goal_wp
  1. precondition0.03------0.080.11------
2. precondition0.02------0.090.11------
3. precondition0.02------0.080.11------
4. precondition0.02------0.090.13------
5. precondition0.02------0.090.13------
6. precondition0.02------0.100.13------
7. assertion------------------0.00
8. assertion---------0.76---------
9. postcondition0.02------0.040.07------
10. postcondition0.08------0.08---------
VC for pokethenpeek---------------------
split_goal_wp
  1. assertion------------------0.01
2. precondition0.02------0.080.15------
3. precondition0.02------0.050.11------
4. precondition0.02------0.050.11------
5. assertion0.02------0.070.14------
6. precondition0.02------0.060.14------
7. precondition0.03------0.080.13------
8. precondition0.03------0.080.13------
9. assertion1.52------------------
10. postcondition0.18------0.05---------