module Toom

  use array.Array
  use map.Map
  use mach.c.C
  use ref.Ref
  use mach.int.Int32
  use import mach.int.UInt64GMP as Limb
  use int.Int
  use int.Power
  use valuation.Valuation
  use int.ComputerDivision
  use types.Types
  use lemmas.Lemmas
  use compare.Compare
  use util.Util
  use util.UtilOld
  use add_1.Add_1
  use add.AddOld
  use sub_1.Sub_1
  use sub.SubOld
  use mul.Mul
  use mul.Mul_basecase
  use logical.Logical

let constant toom22_threshold : int32 = 29

let lemma no_borrow (x y r b m:int)
  requires { 0 <= y <= x }
  requires { 0 <= r < m }
  requires { r - m * b = x - y }
  requires { 0 <= b }
  ensures  { b = 0 }
= assert { b <= 0 by m * b < m * 1 }

let lemma no_borrow_ptr (x y r: ptr limb) (nx ny:int) (b:limb)
  requires { 0 < ny <= nx }
  requires { value y ny <= value x nx }
  requires { value r nx - (power radix nx) * b = value x nx - value y ny }
  requires { 0 <= b }
  ensures  { b = 0 }
= no_borrow (value x nx) (value y ny) (value r nx) (l2i b) (power radix nx)

let rec wmpn_toom22_mul (r x:ptr limb) (sx:int32) (y:ptr limb) (sy:int32)
                        (scratch: ptr limb) (ghost k: int) : unit
  requires { valid x sx }
  requires { valid y sy }
  requires { valid r (sx + sy) }
  requires { toom22_threshold < sy }
  requires { 0 < k <= 64 }
  requires { sx <= toom22_threshold * power 2 k }
  requires { valid scratch (2 * (sx + k)) } (*TODO faire en fonction de sy *)
  requires { writable r /\ writable scratch }
  requires { 8 * sx < max_int32 }
  requires { 2 < sy <= sx < sy + sy - 1 }
  requires { 4 * sx < 5 * sy }
  ensures  { min r = old min r }
  ensures  { max r = old max r }
  ensures  { plength r = old plength r }
  ensures  { min scratch = old min scratch }
  ensures  { max scratch = old max scratch }
  ensures  { plength scratch = old plength scratch }
  ensures  { value r (sx + sy) = value x sx * value y sy }
  ensures  { forall j. min r <= j < offset r \/ offset r + sx + sy <= j < max r
                       -> (pelts r)[j] = old (pelts r)[j] }
  ensures  { forall j. min scratch <= j < offset scratch
                       -> (pelts scratch)[j] = old (pelts scratch)[j] }
  variant { k + k }
=
  let s = sx / 2 in (* TODO sx >> 1 *)
  let n = sx - s in
  let t = sy - n in
  assert { 0 < s };
  assert { n-1 <= s <= n };
  assert { 0 < t <= s };
  let x0 = x in
  let x1 = C.incr x n in
  let y0 = y in
  let y1 = C.incr y n in
  let ghost a0 = value x0 (int32'int n) in
  let ghost a1 = value x1 (int32'int s) in
  let ghost b0 = value y0 (int32'int n) in
  let ghost b1 = value y1 (int32'int t) in
  let ghost m = power radix (int32'int n) in
  value_concat x n sx;
  value_concat y n sy;
  assert { value x sx = a0 + m * a1 };
  assert { value y sy = b0 + m * b1 };
  let r' = decr_split r 0 in
  let ro = C.incr_split r (sx + sy) in
  let scratch' = decr_split scratch 0 in
  assert { min r = offset r /\ max r = offset r + sx + sy };
  let s_out = C.incr_split scratch (n + n) in
  let vinf = C.incr_split r (n + n) in
  label ASM1 in
  let xsm1 = r in
  let ysm1 = C.incr_split r n in
  let vm1_neg = ref false in
  begin ensures { (!vm1_neg /\ value xsm1 n = a1 - a0)
                  \/ (not !vm1_neg /\ value xsm1 n = a0 - a1) }
        ensures { min scratch = old min scratch }
        ensures { max scratch = old max scratch }
        ensures { plength scratch = old plength scratch }
    if (s = n)
    then
      if begin ensures { result <->  value x0 n < value x1 n }
           (wmpn_cmp x0 x1 n) < 0
         end
      then begin
        let ghost b = wmpn_sub_n xsm1 x1 x0 n in
        no_borrow_ptr x1 x0 xsm1 (p2i n) (p2i n) b;
        vm1_neg := true end
      else let ghost b = wmpn_sub_n xsm1 x0 x1 n in
           no_borrow_ptr x0 x1 xsm1 (p2i n) (p2i n) b
    else
      (* n-s=1*)
      if ((get_ofs x0 s) = 0) &&
         ((wmpn_cmp x0 x1 s) < 0)
      then begin
        assert { value x0 s < value x1 s };
        value_tail x0 s;
        assert { value x0 n = value x0 s };
        let ghost b = wmpn_sub_n xsm1 x1 x0 s in
        no_borrow_ptr x1 x0 xsm1 (p2i s) (p2i s) b;
        value_sub_shift_no_change (pelts xsm1) (xsm1.offset) (p2i s) (p2i s) 0;
        set_ofs xsm1 s 0;
        vm1_neg := true;
        value_tail xsm1 s
        end
      else begin
        let b = wmpn_sub_n xsm1 x0 x1 s in
        label Borrow in
        value_sub_shift_no_change (pelts xsm1) (xsm1.offset) (p2i s) (p2i s) b;
        let lx = get_ofs x0 s in
        set_ofs xsm1 s (lx - b);
        assert { value xsm1 s = value (xsm1 at Borrow) s };
        value_tail x0 s;
        value_tail xsm1 s;
        end
  end;
  label BSM1 in
  begin ensures { (!vm1_neg = (!vm1_neg at BSM1) /\ value ysm1 n = b0 - b1)
                  \/ (!vm1_neg = not (!vm1_neg at BSM1) /\ value ysm1 n = b1 - b0) }
        ensures { value xsm1 n = (value xsm1 n at BSM1) }
        ensures { min scratch = old min scratch }
        ensures { max scratch = old max scratch }
        ensures { plength scratch = old plength scratch }
    if (t = n)
    then
      if begin ensures { result <-> value y0 n < value y1 n }
           (wmpn_cmp y0 y1 n) < 0
         end
      then begin
        let ghost b = wmpn_sub_n ysm1 y1 y0 n in
        no_borrow_ptr y1 y0 ysm1 (p2i n) (p2i n) b;
        vm1_neg := not !vm1_neg end
      else
        let ghost b = wmpn_sub_n ysm1 y0 y1 n in
        no_borrow_ptr y0 y1 ysm1 (p2i n) (p2i n) b;
    else
      let y0t = C.incr y0 t in
      let c0 = ((wmpn_zero_p y0t (n - t)) = 1) in
      let c1 = ((wmpn_cmp y0 y1 t) < 0) in
      if c0 && c1
      then begin
        assert { value y0 t < value y1 t };
        value_concat y0 t n;
        assert { value y0 n = value y0 t };
        let ghost b = wmpn_sub_n ysm1 y1 y0 t in
        no_borrow_ptr y1 y0 ysm1 (p2i t) (p2i t) b;
        assert { forall j. (j < offset r \/ offset r + sx + sy <= j)
                             -> (pelts r)[j] = (pelts r)[j] at BSM1 };
        label Zero in
        let ghost ysm1z = { ysm1 } in
        let ysm1t = C.incr ysm1 t in
        wmpn_zero ysm1t (n - t);
        value_sub_frame_shift (pelts ysm1) (pelts ysm1z)
                              (offset ysm1) (offset ysm1z) (p2i t);
        assert { value ysm1 t = value ysm1 t at Zero };
        assert { value xsm1 n = value xsm1 n at Zero };
        value_concat ysm1 t n;
        assert { value ysm1 n = value ysm1 t };
        vm1_neg := not !vm1_neg end
      else begin
        value_concat y0 t n;
        assert { value y0 n = value y0 t + power radix t * value y0t (n-t) };
        assert { value y1 t <= value y0 n
                 by (not c0
                     so 1 <= value y0t (n - t)
                     so power radix t * 1 <= power radix t * value y0t (n-t)
                     so power radix t <= value y0 n
                     so value y1 t < power radix t)
                  \/ (not c1
                      so value y1 t <= value y0 t
                      so value y0 t <= value y0 n) };
        let ghost b = wmpn_sub ysm1 y0 n y1 t in
        no_borrow_ptr y0 y1 ysm1 (p2i n) (p2i t) b;
        end
  end;
  let ghost asm1_abs = value xsm1 (int32'int n) in
  let ghost bsm1_abs = value ysm1 (int32'int n) in
  label RecM1 in
  wmpn_toom22_mul_n_rec scratch xsm1 ysm1 s_out n (k-1);
  assert { value scratch (n+n) = asm1_abs * bsm1_abs };
  join r ysm1;
  assert { min scratch = offset scratch };
  assert { max scratch = max scratch at ASM1 };
  assert { plength scratch = old plength scratch };
  assert { min r = offset r };
  assert { max r = max r at ASM1 };
  assert { plength r = old plength r };
  assert { forall j. min scratch <= j < offset scratch -> (pelts scratch)[j]
            = old (pelts scratch)[j] };
  let v0 = r in
  label Rec in
  begin ensures { value scratch (n+n) = asm1_abs * bsm1_abs }
        ensures { value v0 (n+n) = a0 * b0 }
        ensures { value vinf (s+t) = a1 * b1 }
        ensures { min scratch = old min scratch }
        ensures { max scratch = old max scratch }
        ensures { plength scratch = old plength scratch }
        ensures { min s_out = old min s_out }
        ensures { max s_out = old max s_out }
        ensures { min vinf = old min vinf }
        ensures { max vinf = old max vinf }
        ensures { plength vinf = old plength vinf }
        ensures { min r = old min r }
        ensures { max r = old max r }
        ensures { plength r = old plength r }
    let lemma valid_monotonous (s n:int32)
      requires { valid s_out (2*(n+(k-1))) }
      requires { 0 <= s <= n }
      ensures  { valid s_out (2*(s+(k-1))) }
    = assert { 0 <= 2*(s+(k-1)) <= 2*(n+(k-1)) };
      assert { forall p: ptr limb, n1 n2. 0 <= n1 <= n2 -> valid p n2 -> valid p n1 } in
    valid_monotonous s n;
    valid_monotonous t n;
    (if s > t
     then wmpn_toom22_mul_rec vinf x1 s y1 t s_out (k-1)
     else wmpn_toom22_mul_n_rec vinf x1 y1 s_out s (k-1));
    wmpn_toom22_mul_n_rec v0 x0 y0 s_out n (k-1);
  end;
  label Adds in
  value_concat v0 n (n + n);
  value_concat vinf n (s + t);
  let v0n = incr_split v0 n in
  let vinfn = incr_split vinf n in
  let ghost lv0 = value v0 (int32'int n) in
  let ghost hv0 = value v0n (int32'int n) in
  let ghost lvinf = value vinf (int32'int n) in
  let ghost hvinf = value vinfn (int32'int s + int32'int t- int32'int n) in
  assert { lv0 + m * hv0 = a0 * b0 };
  assert { lvinf + m * hvinf = a1 * b1 };
  let cy = ref (wmpn_add_n_in_place vinf v0n n) in
  assert { value vinf n = lvinf + hv0 - m * !cy };
  let c = wmpn_add_n v0n vinf v0 n in
  let cy2 = c + !cy in
  assert { value v0n n = lvinf + hv0 + lv0 - m * cy2
           by value v0n n = lv0 + value vinf n - m * c
                          = lvinf + hv0 + lv0 - m * !cy - m * c
                          = lvinf + hv0 + lv0 - m * cy2 };
  label Add3 in
  let c' = wmpn_add_in_place vinf n vinfn ((s+t) - n) in
  cy := !cy + c';
  assert { value vinf n = hvinf + lvinf + hv0 - m * !cy
           by m * (!cy at Add3) + m * c' = m * !cy
           so value vinf n = value vinf n at Add3 + hvinf - m * c'
              = lvinf + hv0 - m * (!cy at Add3) + hvinf - m * c'
              = hvinf + lvinf + hv0 - m * !cy };
  assert { !cy <= 2 };
  label JoinMid in
  let ghost v0nj = { v0n } in
  let ghost vinfj = { vinf } in
  join v0n vinf;
  value_sub_frame (pelts v0n) (pelts v0nj) (offset v0n) (offset v0n + p2i n);
  assert { value v0n n = value v0nj n };
  value_sub_frame (pelts v0n) (pelts vinfj) (offset vinf) (offset vinf + p2i n);
  assert { value_sub (pelts v0n) (offset v0n + n) (offset v0n + n + n)
           = value vinfj n };
  value_concat v0n n (n + n);
  assert { value v0n (n+n) = a1 * b1 + a0 * b0 + hv0 + m * lvinf
             - m * cy2 - m * m * !cy
           by value v0n (n+n)
           = value v0n n at JoinMid + m * value vinf n at JoinMid
           = lvinf + hv0 + lv0 - m * cy2
             + m * (hvinf + lvinf + hv0 - m * !cy)
           = lvinf + hv0 + lv0 - m * cy2 + m * hvinf + m * lvinf
             + m * hv0 - m * m * !cy
           = a1 * b1 + a0 * b0 + hv0 + m * lvinf
             - m * cy2 - m * m * !cy };
  label AddSub in
  begin ensures { !cy <= 3 (*2?*)
                   /\ value v0n (n+n) = a1 * b1 + a0 * b0 - (a0 - a1)*(b0 - b1)
                                      + hv0 + m * lvinf - m * cy2 - m * m * !cy
                  \/ !cy = radix - 1
                      /\ value v0n (n+n) = a1 * b1 + a0 * b0  - (a0 - a1)*(b0 - b1)
                         + hv0 + m * lvinf - m * cy2 + m * m
                      /\ !cy at AddSub = 0 }
    assert { !vm1_neg /\ value scratch (n+n) = - (a0-a1)*(b0-b1)
             \/ not !vm1_neg /\ value scratch (n+n) = (a0-a1)*(b0-b1)
             by value scratch (n+n) = asm1_abs * bsm1_abs
             so [@case_split]
                 (!vm1_neg at BSM1 /\ !vm1_neg
               \/ !vm1_neg at BSM1 /\ not !vm1_neg
               \/ not !vm1_neg at BSM1 /\ !vm1_neg
               \/ not !vm1_neg at BSM1 /\ not !vm1_neg) };
    assert { power radix (n+n) = m * m };
    if !vm1_neg
    then
      let c'' = wmpn_add_n_in_place v0n scratch (n+n) in
      assert { value v0n (n+n)
               = value v0n (n+n) at AddSub + value scratch (n+n)
                 - power radix (n+n) * c'' };
      cy := !cy + c'';
      assert { value v0n (n+n) = a1 * b1 + a0 * b0 - (a0 - a1)*(b0 - b1)
                 + hv0 + m * lvinf - m * cy2 - m * m * !cy
               by - m * m * c'' - m * m * !cy at AddSub = - m * m * !cy
               so value scratch (n+n) = -(a0-a1)*(b0-b1)
               so value v0n (n+n)
                  = value v0n (n+n) at AddSub + value scratch (n+n)
                    - power radix (n+n) * c''
                  = value v0n (n+n) at AddSub - (a0 - a1)*(b0-b1) - m * m * c''
                  = a1 * b1 + a0 * b0 - (a0 - a1)*(b0 - b1)
                    + hv0 + m * lvinf - m * cy2 - m * m * !cy }
    else
      let b = wmpn_sub_n_in_place v0n scratch (n+n) in
      assert { value v0n (n+n)
               = value v0n (n+n) at AddSub - value scratch (n+n)
                 + power radix (n+n) * b };
      cy := sub_mod !cy b;
      assert { !cy <= 2 /\ !cy = !cy at AddSub - b
               \/ !cy = radix - 1 /\ !cy at AddSub = 0 /\ b = 1
               by [@case_split]
                  ((!cy at AddSub = 0 /\ b = 1
                   so !cy = EuclideanDivision.mod (-1) radix = radix - 1)
                   \/
                   (1 <= !cy at AddSub \/ b = 0
                   so 0 <= !cy at AddSub - b < radix
                   so !cy = !cy at AddSub - b)) };
      assert { !cy <= 2 ->
               (value v0n (n+n) = a1 * b1 + a0 * b0 - (a0 - a1)*(b0 - b1)
                                  + hv0 + m * lvinf - m * cy2 - m * m * !cy
               by !cy = !cy at AddSub - b
               so m * m * b - m * m * !cy at AddSub = - m * m * !cy
               so value scratch (n+n) = (a0-a1)*(b0-b1)
               so value v0n (n+n)
                  = value v0n (n+n) at AddSub - value scratch (n+n)
                    + power radix (n+n) * b
                  = value v0n (n+n) at AddSub - (a0 - a1)*(b0-b1) + m * m * b
                  = a1 * b1 + a0 * b0 - (a0 - a1)*(b0 - b1)
                    + hv0 + m * lvinf - m * cy2 - m * m * !cy) };
      assert { !cy = radix - 1 ->
               (value v0n (n+n) = a1 * b1 + a0 * b0 - (a0 - a1)*(b0 - b1)
                                  - m * cy2 + hv0 + m * lvinf + m * m
                by b = 1 so power radix (n+n) * b = m * m
                so m * m * !cy at AddSub = 0
                so value scratch (n+n) = (a0-a1)*(b0-b1)
                so value v0n (n+n)
                   = value v0n (n+n) at AddSub - value scratch (n+n)
                     + power radix (n+n) * b
                   = value v0n (n+n) at AddSub - (a0 - a1)*(b0-b1) + m*m
                   = a1 * b1 + a0 * b0 + hv0 + m * lvinf
                     - m * cy2 - m * m * (!cy at AddSub) - (a0 - a1)*(b0-b1) + m*m
                   = a1 * b1 + a0 * b0 + hv0 - (a0 - a1)*(b0 - b1)
                     + m * lvinf - m * cy2 + m * m) }
  end;
  label Join in
  let ghost rj = { r } in
  let ghost v0nj = { v0n } in
  let ghost vinfnj = { vinfn } in
  join r v0n;
  value_sub_frame (pelts r) (pelts rj) (offset r) (offset r + p2i n);
  assert { value r n = value rj n = lv0 };
  value_concat r n (3 * n);
  value_sub_frame (pelts r) (pelts v0nj) (offset r + p2i n)
                                         (offset r + 3 * p2i n);
  assert { value r (3*n) = value r n + m * value (v0n at Join) (n+n)
           by offset v0nj = offset r + n
           so offset v0nj + (n + n) = offset r + 3 * n
           so value_sub (pelts r) (offset r + n) (offset r + 3*n)
              = value v0nj (n+n) };
  label JoinH in
  let ghost rh = { r } in
  join r vinfn;
  value_sub_frame (pelts r) (pelts rh) (offset r) (offset r + 3 * p2i n);
  assert { value r (3*n) = value r (3*n) at JoinH };
  value_sub_frame (pelts r) (pelts rh) (offset r) (offset r + p2i n);
  assert { value r n = value r n at JoinH };
  value_concat r (3*n) ((3 * n) + ((s + t) - n));
  assert { forall i. offset r + 3 * n <= i < offset r + 3 * n + s + t - n ->
           min vinfnj <= i < max vinfnj
           by max vinfnj >= offset r + sx + sy
              = offset r + n + s + n + t
              = offset r + 3 * n + s + t - n };
  value_sub_frame (pelts r) (pelts vinfnj) (offset r + 3 * p2i n)
                            (offset r + 3 * p2i n + p2i s + p2i t - p2i n);
  assert { value r (sx + sy)
           = value r (3*n + s + t - n)
           = value r (3*n) + m*m*m* value (vinfn at Join) (s+t-n)
           by offset vinfnj = offset r + 3*n
           so offset vinfnj + s + t - n = offset r + 3*n + s + t - n
           so m * m * m = power radix n * power radix n * power radix n
              = power radix (n+n+n) = power radix (3 * n)
           so value_sub (pelts r) (offset r + 3*n) (offset r + 3*n + s + t - n)
              = value vinfnj (s+t-n) };
  join scratch s_out;
  assert { a1 * b1 + a0 * b0 - (a0 - a1)*(b0 - b1) = a0 * b1 + a1 * b0 };
  assert { value x sx * value y sy
           = (a0 +  m * a1)*(b0 + m * b1)
           = a0 * b0 + m * (a0 * b1 + a1 * b0) + m * m * (a1 * b1) };
  assert { !cy <= 3 /\ value r (sx + sy) = value x sx * value y sy
                                            - m * m * cy2 - m * m * m * !cy
           \/ !cy = radix - 1 /\ value r (sx + sy) = value x sx * value y sy
                                  - m * m * cy2 + m * m * m
            by value r n = lv0
               so value vinfnj (s+t-n) = hvinf
               so value r (sx + sy)
               = value r (3 * n) + m * m * m * value vinfnj (s+t-n)
               = value r n + m * value v0nj (n+n)
                 + m*m*m * value vinfnj (s+t-n)
               = lv0 + m * value v0nj (n+n) + m * m * m * hvinf
            so (lv0 + m * (a1*b1 + a0*b0 - (a0 - a1)*(b0 - b1) + hv0 + m*lvinf)
                + m * m * m * hvinf = value x sx * value y sy
               by lv0 + m * (a0 * b1 + a1 * b0 + hv0 + m * lvinf)
                      + m * m * m * hvinf
                  = lv0 + m * hv0 + m * (a0 * b1 + a1 * b0)
                     + m * m * lvinf + m * m * m * hvinf
                  = lv0 + m * hv0 + m * (a0 * b1 + a1 * b0)
                    + m * m * (lvinf + m * hvinf)
                  = a0 * b0 + m * (a0 * b1 + a1 * b0)
                    + m * m * a1 * b1
                  = value x sx * value y sy)
            so [@case_split] (* TODO *)
               ((!cy <= 3
                so value v0nj (n+n) = a1 * b1 + a0 * b0 - (a0 - a1)*(b0 - b1)
                                     + hv0 + m * lvinf - m * cy2 - m * m * !cy
                   = a0 * b1 + a1 * b0 + hv0 + m * lvinf - m * cy2 - m * m * !cy
                so value r (sx + sy)
                   = lv0 + m * value v0nj (n+n) + m * m * m * hvinf
                   = lv0 + m * (a0 * b1 + a1 * b0 + hv0 + m * lvinf
                               - m * cy2 - m * m * !cy)
                               + m * m * m * hvinf
                   = lv0 + m * (a0 * b1 + a1 * b0 + hv0 + m * lvinf)
                      + m * m * m * hvinf
                      - m * m * cy2 - m * m * m * !cy
                   = value x sx * value y sy
                     - m * m * cy2 - m * m * m * !cy)
                 \/
                 (!cy = radix - 1
                  so value v0nj (n+n) = a1 * b1 + a0 * b0 - (a0 - a1)*(b0 - b1)
                                     + hv0 + m * lvinf - m * cy2 + m * m
                   = a0 * b1 + a1 * b0 + hv0 + m * lvinf - m * cy2 + m * m
                  so value r (sx + sy)
                   = lv0 + m * value v0nj (n+n) + m * m * m * hvinf
                   = lv0 + m * (a0 * b1 + a1 * b0 + hv0 + m * lvinf
                               - m * cy2 + m * m)
                               + m * m * m * hvinf
                   = lv0 + m * (a0 * b1 + a1 * b0 + hv0 + m * lvinf)
                      + m * m * m * hvinf
                      - m * m * cy2 + m * m * m
                   = value x sx * value y sy
                     - m * m * cy2 + m * m * m)) };
  let vinf0 = C.incr r (n+n) in
  value_sub_upper_bound (pelts x1) (offset x1) (offset x1 + int32'int s);
  value_sub_upper_bound (pelts y1) (offset y1) (offset y1 + int32'int t);
  assert { s + t - n > 0
           by s >= n-1
           so 4 * sx < 5 * sy
           so 4 * n + 4 * s < 5 * n + 5 * t
           so 5 * t > 4 * s - n >= 4 * n - 4 - n = 3 * n - 4
           so 5 * t > 3 * n - 4
           so n > 3
           so t > 1 };
  let ghost m' = power radix (p2i s + p2i t - p2i n) in
  begin ensures { value r (sx + sy) + m * m * cy2  < m * m * m * m' }
        ensures { value x sx * value y sy < m * m * m * m' }
  assert { power radix (s+t) = m * m' };
  value_sub_upper_bound (pelts r) (offset r) (offset r + int32'int n);
  value_sub_upper_bound (pelts v0nj) (offset v0nj)
                        (offset v0nj + int32'int n + int32'int n);
  value_sub_upper_bound (pelts x) (offset x) (offset x + int32'int sx);
  value_sub_upper_bound (pelts y) (offset y) (offset y + int32'int sy);
  assert { !cy = radix - 1 -> hvinf <= m' - 2
           by value v0nj (n+n) < power radix (n+n) = m * m
           so value v0nj (n+n) = a1 * b1 + a0 * b0 - (a0 - a1)*(b0 - b1)
                                     + hv0 + m * lvinf - m * cy2 + m * m
                   = a0 * b1 + a1 * b0 + hv0 + m * lvinf - m * cy2 + m * m
                              so (0 <= a0 * b1 by 0 <= a0 /\ 0 <= b1)
           so (0 <= a1 * b0 by 0 <= a1 /\ 0 <= b0)
           so hv0 >= 0
           so value v0nj (n+n) >= m * lvinf - m * cy2 + m * m
           so m * lvinf - m * cy2 + m * m < m * m
           so m * (lvinf - cy2) = m * lvinf - m * cy2 < 0
           so lvinf - cy2 < 0
           so (cy2 <= 1 by !cy at AddSub = 0 so !cy at Add3 = 0)
           so (lvinf = 0 by 0 <= lvinf)
           so a1 * b1 = m * hvinf
           so if a1*b1 <= 0
           then (hvinf <= m' - 2
                 by m*hvinf=0 so 0 <> m so hvinf = 0
                 so 1 <= s+t-n
                 so radix = power radix 1 <= power radix (s+t-n)
                 so m' > 2
                 so hvinf <= m' - 2)
           else hvinf <= m'-2 by
                (forall s. 0 <= s -> power radix s = power 2 (64*s)
                by radix = power 2 64)
           so (m = power 2 (64 * n)
               by m = power radix n)
           so m >= 1
           so (hvinf >= 1 by m*hvinf > 0)
           so valuation m 2 = 64*n
           so 64*n <= valuation (a1*b1) 2 = valuation a1 2 + valuation b1 2
           so if 64*n < valuation (a1*b1) 2
           then hvinf <= m'-2
                 by valuation (m*hvinf) 2 > 64*n
                 so valuation m 2 = 64*n
                 so if valuation hvinf 2 = 0
                    then false by valuation (m*hvinf) 2 = valuation m 2
                    else hvinf <= m'-2
                    by even hvinf
                    so (odd (m'-1)
                       by m' = power radix (s+t-n)
                             = power 2 (64*(s+t-n))
                             = 2 * power 2 (64*(s+t-n) - 1)
                          so even m')
                    so hvinf <> m'-1
                    so (hvinf < m'
                        by hvinf = value vinfnj (s+t-n)
                                 < power radix ((offset vinfnj +(s+t-n))
                                                 - offset vinfnj)
                                 = power radix (s+t-n))
                    so hvinf < m'-1
           else hvinf <= m'-2
           by power radix s = power 2 (64*s)
           so power radix t = power 2 (64*t)
           so m' = power radix (s+t-n) = power 2 (64*(s+t-n))
           so let k = valuation a1 2 in
              let l = valuation b1 2 in
              let a1' = div a1 (power 2 k) in
              let b1' = div b1 (power 2 l) in
              a1 = (power 2 k) * a1'
           so b1 = (power 2 l) * b1'
           so 64*n = k + l
           so 1 <= a1 /\ 1 <= b1
           so 0 <= k /\ 0 <= l
           so 1 <= a1' /\ 1 <= b1'
           so (k <= 64*s by power 2 k <= a1 < power 2 (64*s)
                         so power 2 k < power 2 (64*s)
                         so not 64*s < k)
           so (l <= 64*t by power 2 l <= b1 < power 2 (64*t)
                         so power 2 l < power 2 (64*t)
                         so not 64*t < l)
           so (forall a b c m:int. 0 <= a -> 0 <= b -> 0 < c -> 0 < m
              -> a * c < m -> b * c >= m -> a < b)
           so a1' < power 2 (64*s - k)
              (*(by a1' * (power 2 k) = a1 < power radix s = power 2 (64*s)
              so a1' * power 2 k < power 2 (64*s)
              so power 2 (64*s-k) * power 2 k = power 2 (64*s))*)
           so a1' <= power 2 (64*s - k) - 1
           so b1' < power 2 (64*t - l)
           so b1' <= power 2 (64*t - l) - 1
           so a1' * b1'
              <= (power 2 (64*s - k) - 1) * b1'
              <= (power 2 (64*s - k) - 1) * (power 2 (64*t - l) - 1)
              = (power 2 (64*s-k))*(power 2 (64*t -l))
                - power 2 (64*s-k)
                - power 2 (64*t-l)
                + 1
              <= (power 2 (64*s-k))*(power 2 (64*t -l)) - 2
              = power 2 (64*(s+t) - (k+l)) - 2
              = power 2 (64*(s+t) - 64*n) - 2
              = power 2 (64*(s+t-n)) - 2
              = power radix (s+t-n) - 2
              = m' - 2
           so a1 * b1 = (power 2 k) * a1' * (power 2 l) * b1'
              = power 2 k * power 2 l * a1' * b1'
              = (power 2 (k+l)) * a1' * b1'
              = power 2 (64*n) * a1' * b1'
              = power radix n * a1' * b1'
              = m * (a1' * b1')
           so a1 * b1 = m * hvinf = m * (a1' * b1')
           so hvinf = a1' * b1' <= m' - 2 };
  assert { value x sx * value y sy < m * m * m * m'
           by (m * m * m * m' = power radix (n+n+s+t)
              by m * m * m * m'
                 = power radix n * power radix n * power radix n
                   * power radix (s+t-n)
                 = power radix n * power radix n * power radix (s+t)
                 = power radix n * power radix (n+s+t)
                 = power radix (n+n+s+t))
           so value x sx < power radix sx = power radix (n+s)
           so value y sy < power radix sy = power radix (n+t)
           so (forall a b c d. 0 <= a < b -> 0 <= c < d -> a * c < b * d
               by [@case_split]
                  (((a=0\/c=0) so a*c=0 so b*d>0 so a*c < b*d)
                  \/
                  ((0<a /\ 0<c) so a * c < a * d = d*a < d * b = b * d)))
           so value x sx * value y sy < power radix (n+s) * power radix (n+t)
              = power radix (n+n+s+t) };
  assert { value r (sx + sy) + m * m * cy2  < m * m * m * m'
           by [@case_split]
           ((!cy <= 3
           so 0 <= m * m * m * !cy
           so value r (sx + sy) + m * m * cy2
           = value x sx * value y sy - m * m * cy2 - m * m * m * !cy
               + m * m * cy2
           = value x sx * value y sy - m * m * m * !cy
           <= value x sx * value y sy)
           \/
           (!cy = radix - 1
           so hvinf <= m' - 2
           so value r (sx + sy) = value r n + m * (value v0nj (n+n))
                                   + m*m*m * value vinfnj (s+t-n)
              = value r n + m * (value v0nj (n+n)) + m*m*m * hvinf
           so value r n < power radix n
           so value r n < m
           so value v0nj (n+n) < power radix (n+n) = m * m
           so value v0nj (n+n) <= m * m - 1
           so m * value v0nj (n+n) <= m * (m * m - 1)
           so value r n + m * (value v0nj (n+n))
              < m + m * (m * m -1) = m * m * m
           so m * m * m * hvinf <= m * m * m * (m'-2)
           so value r (sx + sy) < m * m * m + m * m * m * (m'-2)
              = m * m * m * m' - m * m * m
           so m * m * cy2 < m * m * m)) };
  end;
  value_concat r (n + n) (sx + sy);
  assert { value_sub (pelts r) (offset r + n + n) (offset r + sx + sy)
           = value vinf0 (s+t) };
  assert { value r (sx + sy) = value r (n+n) + m * m * value vinf0 (s+t) };
  assert { value vinf0 (s+t) + cy2 < m * m'
           by value r (n+n) >= 0
           so m * m * m * m' > value r (sx + sy) + m * m * cy2
              = value r (n+n) + m * m * value vinf0 (s+t) + m * m * cy2
              >= m * m * value vinf0 (s+t) + m * m * cy2
              = m * m * (value vinf0 (s+t) + cy2)
              so (m * m) * (value vinf0 (s+t) + cy2) < (m * m) * (m * m') };
  let ghost ri = { r } in
  label IncrM in
  wmpn_incr vinf0 (s + t) cy2;
  value_concat r (n + n) (sx + sy);
  assert { value_sub (pelts r) (offset r + n + n) (offset r + sx + sy)
           = value vinf0 (s+t) };
  assert { value r (sx + sy) = value r (n+n) + m * m * value vinf0 (s+t) };
  assert { forall j. offset r <= j < offset r + (n+n)
           -> (pelts r)[j] = (pelts r)[j] at IncrM };
  value_sub_frame (pelts r) (pelts ri) (offset r)
                  (offset r + (int32'int n + int32'int n));
  assert { value r (sx + sy) = value r (sx+sy) at IncrM + m * m * cy2
           by value vinf0 (s+t) = value vinf0 (s+t) at IncrM + cy2
           so value r (n+n) = value r (n+n) at IncrM
           so value r (sx + sy) = value r (n+n) + m * m * value vinf0 (s+t)
              = (value r (n+n) at IncrM)
                + m * m * (value vinf0 (s+t) at IncrM + cy2)
              = (value r (n+n) + m * m * value vinf0 (s+t) at IncrM)
                + m * m * cy2
              = value r (sx+sy) at IncrM + m * m * cy2 };
  assert { forall j. min r <= j < offset r \/ offset r + sx + sy <= j < max r
             -> (pelts r)[j] = (pelts r)[j] at IncrM
                by (pelts r)[j] = (pelts vinf0)[j]
                   = (pelts vinf0)[j] at IncrM };
  assert { !cy <= 3 /\ value r (sx + sy)
                       = value x sx * value y sy - m * m * m * !cy
           \/ value r (sx + sy) = value x sx * value y sy + m * m * m };
  let rh = { r } in
  let vinfn = C.incr r (3 * n) in
  label IncrH in
  assert { valid vinfn (s+t-n) };
  value_concat r (3 * n) (sx + sy);
  assert { value_sub (pelts r) (offset r + 3*n) (offset r + sx + sy)
           = value vinfn (s+t-n)
           by pelts r = pelts vinfn
           so offset r + 3*n = offset vinfn
           so offset r + sx + sy = offset vinfn + s + t - n };
  assert { value r (sx + sy) = value r (3*n)
                               + power radix (3*n) * value vinfn (s+t-n)};
  assert { power radix (3*n) = m * m * m
           by power radix (3*n) = power radix (n+n+n)
              = power radix (n+n) * power radix n
              = power radix n * power radix n * power radix n };
  if ([@likely] !cy <= 3)
  then begin
    assert { value r (sx+sy) = value x sx * value y sy
                               - power radix (3*n) * !cy
              by value r (sx+sy) = value x sx * value y sy - m * m * m * !cy };
    assert { value r (sx+sy) + power radix (3*n)* !cy < power radix (3*n) * m'
             by value r (sx+sy) + power radix (3*n) * !cy
                = value x sx * value y sy
                < m*m*m*m' = power radix (3*n) * m' };
    assert { value vinfn (s+t-n) + !cy < m'
             by power radix (3*n) * m'
                > value r (sx+sy) + power radix (3*n) * !cy
                = value r (3*n) + power radix (3*n) * value vinfn (s+t-n)
                                + power radix (3*n) * !cy
                = value r (3*n) + power radix (3*n) * (value vinfn (s+t-n) + !cy)
                >= power radix (3*n) * (value vinfn (s+t-n) + !cy)
                so power radix (3*n) * (value vinfn (s+t-n) + !cy)
                   < power radix (3*n) * m'};
    wmpn_incr vinfn ((s + t) - n) !cy;
    value_concat r (3 * n) (sx + sy);
    assert { value_sub (pelts r) (offset r + 3*n) (offset r + sx + sy)
             = value vinfn (s+t-n)
             by pelts r = pelts vinfn
             so offset r + 3*n = offset vinfn
             so offset r + sx + sy = offset vinfn + s + t - n };
    assert { value r (sx + sy) = value r (3*n)
                                + power radix (3*n) * value vinfn (s+t-n)};
    assert { forall j. offset r <= j < offset r + (3*n)
             -> (pelts r)[j] = (pelts r)[j] at IncrH };
    value_sub_frame (pelts r) (pelts rh) (offset r)
                    (offset r + (3 * int32'int n));
    assert { value r (sx + sy) = value x sx * value y sy
             by value vinfn (s+t-n) = (value vinfn (s+t-n) at IncrH) + !cy
             so value r (3*n) = value r (3*n) at IncrH
             so value r (sx + sy)
                = value r (3*n) + power radix (3*n) * value vinfn (s+t-n)
                = value r (3*n) at IncrH
                  + power radix (3*n) * (value vinfn (s+t-n) at IncrH + !cy)
                = (value r (3*n) + power radix (3*n) * value vinfn (s+t-n)
                         at IncrH)
                  + power radix (3*n) * !cy
                = value r (sx+sy) at IncrH + power radix (3*n) * !cy
                = value x sx * value y sy - power radix (3*n) * !cy
                                          + power radix (3*n) * !cy };
  end
  else begin
    assert { !cy = radix - 1 };
    assert { value r (sx+sy) = value x sx * value y sy
                               + power radix (3*n)
              by value r (sx+sy) = value x sx * value y sy + m * m * m };
    value_sub_upper_bound (pelts r) (offset r) (offset r + 3 * int32'int n);
    assert { 0 <= value vinfn (s+t-n) - 1
             by 0 <= value x sx so 0 <= value y sy
             so 0 <= value x sx * value y sy
             so value r (sx + sy) >= power radix (3*n)
             so value r (sx + sy)
                = value r (3*n) + power radix (3*n) * value vinfn (s+t-n)
             so value r (3*n) < power radix (3*n)
             so power radix (3*n) * value vinfn (s+t-n) > 0
             so value vinfn (s+t-n) > 0 };
    wmpn_decr_1 vinfn ((s + t) - n);
    value_concat r (3 * n) (sx + sy);
    assert { value_sub (pelts r) (offset r + 3*n) (offset r + sx + sy)
             = value vinfn (s+t-n)
             by pelts r = pelts vinfn
             so offset r + 3*n = offset vinfn
             so offset r + sx + sy = offset vinfn + s + t - n };
    assert { value r (sx + sy) = value r (3*n)
                                + power radix (3*n) * value vinfn (s+t-n)};
    assert { forall j. offset r <= j < offset r + (3*n)
             -> (pelts r)[j] = (pelts r)[j] at IncrH };
    value_sub_frame (pelts r) (pelts rh) (offset r)
                    (offset r + (3 * int32'int n));
    assert { value r (sx + sy) = value x sx * value y sy
             by value vinfn (s+t-n) = (value vinfn (s+t-n) at IncrH) - 1
             so value r (3*n) = value r (3*n) at IncrH
             so value r (sx + sy)
                = value r (3*n) + power radix (3*n) * value vinfn (s+t-n)
                = value r (3*n) at IncrH
                  + power radix (3*n) * (value vinfn (s+t-n) at IncrH - 1)
                = (value r (3*n) + power radix (3*n) * value vinfn (s+t-n)
                         at IncrH)
                  - power radix (3*n)
                = value r (sx+sy) at IncrH - power radix (3*n)
                = value x sx * value y sy };
  end;
  label JoinRight in
  let rf = { r } in
  C.join r ro;
  label JoinLeft in
  C.join_r r' r;
  assert { forall j. offset r <= j < offset r + sx + sy ->
           (pelts r)[j] = (pelts rf)[j]
           by (pelts r)[j] = (pelts r)[j] at JoinLeft
              = (pelts rf)[j] };
  value_sub_frame (pelts r) (pelts rf) (offset r) (offset r + p2i sx + p2i sy);
  assert { value r (sx + sy) = value r (sx + sy) at JoinRight };
  C.join_r scratch' scratch


  (* Choose which multiplication algorithm is called recursively.
     Equivalent to the macros WMPN_TOOM22_MUL_REC and WMPN_TOOM22_MUL_N_REC
     respectively, in wmpn_toom22_mul.c *)
with wmpn_toom22_mul_rec (r x: ptr limb) (sx:int32) (y:ptr limb) (sy: int32)
                         (scratch:ptr limb) (ghost k: int)
     : unit
  requires { valid x sx }
  requires { valid y sy }
  requires { valid r (sx + sy) }
  requires { writable r /\ writable scratch }
  requires { 0 < sy <= sx <= sy + sy }
  requires { 8 * sx < max_int32 }
  requires { 0 <= k <= 64 }
  requires { sx <= toom22_threshold * power 2 k }
  requires { valid scratch (2 * (sx + k)) }
  ensures  { value r (sx + sy) = value x sx * value y sy }
  ensures  { forall j. min r <= j < offset r \/ offset r + sx + sy <= j < max r
                       -> (pelts r)[j] = old (pelts r)[j] }
  ensures  { forall j. min scratch <= j < offset scratch
             -> (pelts scratch)[j] = old (pelts scratch)[j] }
  ensures  { min r = old min r }
  ensures  { max r = old max r }
  ensures  { plength r = old plength r }
  ensures  { min scratch = old min scratch }
  ensures  { max scratch = old max scratch }
  ensures  { plength scratch = old plength scratch }
  variant  { k + k + 1 }
=
  if sy <= toom22_threshold
  then wmpn_mul_basecase r x sx y sy
  else
    if (4 * sx < 5 * sy) (* ? *)
    then wmpn_toom22_mul r x sx y sy scratch k
    else wmpn_toom32_mul r x sx y sy scratch k

with wmpn_toom22_mul_n_rec (r x y scratch: ptr limb) (sz:int32) (ghost k: int) : unit
  requires { valid x sz }
  requires { valid y sz }
  requires { valid r (sz + sz) }
  requires { writable r /\ writable scratch }
  requires { 0 < sz }
  requires { 8 * sz < max_int32 }
  requires { 0 <= k <= 64 }
  requires { sz <= toom22_threshold * power 2 k }
  requires { valid scratch (2 * (sz + k)) }
  ensures  { value r (sz + sz) = value x sz * value y sz }
  ensures  { forall j. min r <= j < offset r \/ offset r + sz + sz <= j < max r
                       -> (pelts r)[j] = old (pelts r)[j] }
  ensures  { forall j. min scratch <= j < offset scratch
             -> (pelts scratch)[j] = old (pelts scratch)[j] }
  ensures  { min r = old min r }
  ensures  { max r = old max r }
  ensures  { plength r = old plength r }
  ensures  { min scratch = old min scratch }
  ensures  { max scratch = old max scratch }
  ensures  { plength scratch = old plength scratch }
  variant  { k + k + 1 }
=
  if sz <= toom22_threshold
  then wmpn_mul_basecase r x sz y sz
  else wmpn_toom22_mul r x sz y sz scratch k

with wmpn_toom32_mul (r x:ptr limb) (sx:int32) (y:ptr limb) (sy:int32)
                     (scratch: ptr limb) (ghost k: int) : unit
  requires { valid x sx }
  requires { valid y sy }
  requires { valid r (sx + sy) }
  requires { writable r /\ writable scratch }
  requires { toom22_threshold < sy }
  requires { 0 < k <= 64 }
  requires { sx <= toom22_threshold * power 2 k }
  requires { valid scratch (2 * (sx + k)) }
  requires { 8 * sx < max_int32 }
  requires { 4 < sy + 2 <= sx }
  requires { sx + 6 <= 3 * sy }
  ensures  { min r = old min r }
  ensures  { max r = old max r }
  ensures  { plength r = old plength r }
  ensures  { min scratch = old min scratch }
  ensures  { max scratch = old max scratch }
  ensures  { plength scratch = old plength scratch }
  ensures  { value r (sx + sy) = value x sx * value y sy }
  ensures  { forall j. min r <= j < offset r \/ offset r + sx + sy <= j < max r
                       -> (pelts r)[j] = old (pelts r)[j] }
  ensures  { forall j. min scratch <= j < offset scratch
                       -> (pelts scratch)[j] = old (pelts scratch)[j] }
  variant { k + k }
=
  let n = 1 + (if 2 * sx >= 3 * sy
               then (sx - 1) / 3
               else (sy - 1) / 2) in
  let s = sx - 2 * n in
  let t = sy - n in
  assert { 0 < s <= n };
  assert { 0 < t <= n };
  assert { s + t >= n };
  let x0 = x in
  let x1 = C.incr x n in
  let x2 = C.incr x1 n in
  let y0 = y in
  let y1 = C.incr y n in
  let ghost a0 = value x0 (int32'int n) in
  let ghost a1 = value x1 (int32'int n) in
  let ghost a2 = value x2 (int32'int s) in
  let ghost b0 = value y0 (int32'int n) in
  let ghost b1 = value y1 (int32'int t) in
  let ghost m  = power radix (int32'int n) in
  value_concat x1 n (sx - n);
  value_concat x n sx;
  value_concat y n sy;
  assert { value y sy = b0 + m * b1 };
  assert { value x sx = a0 + m * a1 + m * m * a2 };
  let rol = decr_split r 0 in
  let ror = incr_split r (sx + sy) in
  let sol = decr_split scratch 0 in
  let sor = incr_split scratch ((n + n) + 1) in
  (* xp1 | yp1 | xm1 | ym1 *)
  let xp1 = r in                  (* x(1) = a0 + a1 + a2  *)
  let yp1 = incr_split r n in     (* y(1) = b0 + b1       *)
  let xm1 = incr_split yp1 n in   (* x(-1) = a0 - a1 + a2 *)
  let ym1 = incr_split xm1 n in   (* y(-1) = b0 - b1      *)
  let v1  = scratch in            (* x(1)*y(1)            *)
  let vm1 = r in                  (* x(-1)*y(-1)          *)
  let xp1_hi = ref 0 in           (* high limb of xp1     *)
  let yp1_hi = ref 0 in           (* high bit of yp1      *)
  let hi = ref 0 in               (* high bit of xm1      *)
  let vm1_neg = ref false in      (* negate vm1 ?         *)
  begin ensures { value xp1 n + m * !xp1_hi = a0 + a1 + a2 }
        ensures { (!vm1_neg /\ value xm1 n + m * !hi = a1 - (a0 + a2))
                  \/ (not !vm1_neg /\ value xm1 n + m * !hi = a0 - a1 + a2) }
        ensures { 0 <= !xp1_hi <= 2 }
        ensures { 0 <= !hi <= 1 }
    xp1_hi := wmpn_add xp1 x0 n x2 s;
    begin
      let cmp = wmpn_cmp xp1 x1 n in
      if (*begin ensures { result <-> a0 + a2 < a1 }*)
           !xp1_hi = 0
           && (cmp < 0)
         (*end*)
      then begin
        assert { value xp1 n < value x1 n };
        assert { value xp1 n = a0 + a2 };
        let ghost b = wmpn_sub_n xm1 x1 xp1 n in
        assert { b = 0 };
        hi := 0;
        vm1_neg := true end
      else begin
        let b = wmpn_sub_n xm1 xp1 x1 n in
        assert { !xp1_hi = 0 -> b = 0 };
        hi := !xp1_hi - b;
        assert { value xm1 n + m * !hi = a0 - a1 + a2
                 by value xm1 n + m * !hi
                  = value xm1 n - m * b + m * !xp1_hi
                  = value xp1 n - a1 + m * !xp1_hi
                  = a0 - a1 + a2 };
        end
    end;
    label Addx1 in
    assert { value xp1 n + m * !xp1_hi = a0 + a2 };
    let c = wmpn_add_n_in_place xp1 x1 n in
    xp1_hi := !xp1_hi + c;
    assert { value xp1 n + m * !xp1_hi = a0 + a2 + a1
             by value xp1 n + m * !xp1_hi
                = value xp1 n + m * c + m * (!xp1_hi at Addx1)
                = (value xp1 n at Addx1) + value x1 n + m * (!xp1_hi at Addx1)
                = a0 + a2 + value x1 n = a0 + a2 + a1 };
  end;
  label B1 in
  begin ensures { value yp1 n + m * !yp1_hi = b0 + b1 }
        ensures { (!vm1_neg = (!vm1_neg at B1) /\ value ym1 n = b0 - b1)
                  \/ (!vm1_neg = not (!vm1_neg at B1) /\ value ym1 n = b1 - b0) }
        ensures { 0 <= !yp1_hi <= 1 }
    if (t = n)
    then begin
      yp1_hi := wmpn_add_n yp1 y0 y1 n;
      let cmp = wmpn_cmp y0 y1 n in
      if (cmp < 0)
      then begin
        let ghost b = wmpn_sub_n ym1 y1 y0 n in
        assert { b = 0 };
        vm1_neg := not !vm1_neg end
      else begin
        let ghost b = wmpn_sub_n ym1 y0 y1 n in
        assert { b = 0 }
        end
      end
    else begin
      yp1_hi := wmpn_add yp1 y0 n y1 t;
      let y0t = C.incr y0 t in
      let c0 = ((wmpn_zero_p y0t (n - t)) = 1) in
      let cmp = wmpn_cmp y0 y1 t in
      let c1 = (cmp < 0) in
      if c0 && c1
      then begin
        value_concat y0 t n;
        assert { value y0 t = value y0 n };
        assert { value y0 t < value y1 t };
        let ghost b = wmpn_sub_n ym1 y1 y0 t in
        assert { b = 0 };
        let ghost ym1z = { ym1 } in
        let ym1t = C.incr ym1 t in
        label Zero in
        wmpn_zero ym1t (n - t);
        assert { forall i. 0 <= i < t ->
                 (pelts ym1)[offset ym1 + i] = (pelts ym1z)[offset ym1z + i]
                 by offset ym1 + i < offset ym1t
                 so (pelts ym1)[offset ym1 + i]
                     = (pelts ym1t)[offset ym1 + i]
                     = (pelts ym1t at Zero)[offset ym1 + i]
                     = (pelts ym1 at Zero)[offset ym1 + i] };
        value_sub_frame_shift (pelts ym1) (pelts ym1z)
                              (offset ym1) (offset ym1z) (p2i t);
        assert { value ym1 t = value ym1 t at Zero };
        value_concat ym1 t n;
        assert { value ym1 n = value ym1 t
                 by value_sub (pelts ym1) (offset ym1 + t) (offset ym1 + n)
                    = value ym1t (n-t) = 0 };
        vm1_neg := not !vm1_neg end
      else begin
        value_concat y0 t n;
        assert { value y0 n = value y0 t + power radix t * value y0t (n-t) };
        assert { value y1 t <= value y0 n
                 by (not c0
                     so 1 <= value y0t (n - t)
                     so power radix t * 1 <= power radix t * value y0t (n-t)
                     so power radix t <= value y0 n
                     so value y1 t < power radix t)
                  \/ (not c1
                      so value y1 t <= value y0 t
                      so value y0 t <= value y0 n) };
        let ghost b = wmpn_sub ym1 y0 n y1 t in
        assert { b = 0 }
        end;
      end
  end;
  let ghost am1_abs = value xm1 (int32'int n) + m * (uint64'int !hi) in
  let ghost bm1_abs = value ym1 (int32'int n) in
  label RecP1 in
  wmpn_toom22_mul_n_rec v1 xp1 yp1 sor n (k-1);
  let cy = ref 0 in
  begin ensures { value scratch (2 * n) + power radix (n + n) * !cy
                  = (a0 + a1 + a2) * (b0 + b1) }
    assert { value scratch (n+n) = value xp1 n * value yp1 n };
    begin ensures { value scratch (n + n) + power radix (n + n) * !cy
                    = (a0 + a1 + a2) * (value yp1 n) }
          ensures { 0 <= !cy <= 3 }
          (* actually 2, but this is enough to prove there is no overflow *)
    if (!xp1_hi = 1)
    then begin
      let sa = { scratch } in
      let sn = C.incr scratch n in
      label Adjust1 in
      value_concat scratch n (n+n);
      assert { pelts scratch = pelts sn };
      let c = wmpn_add_n_in_place sn yp1 n in
      assert { pelts scratch = pelts sn };
      value_concat scratch n (n+n);
      value_sub_frame_shift (pelts scratch) (pelts sa)
                            (offset scratch) (offset sa) (p2i n);
      assert { value scratch n = value (scratch at Adjust1) n };
      assert { m * m = power radix (n+n) };
      assert { value scratch (n+n) + m * m * c = (a0 + a1 + a2) * value yp1 n
               by value sn n = value_sub (pelts scratch) (offset scratch + n)
                                         (offset scratch + (n+n))
               so value scratch (n+n) = value scratch n + m * value sn n
               so value (sn at Adjust1) n
                   = value_sub (pelts sa) (offset scratch + n)
                                          (offset scratch + (n+n))
               so value (scratch at Adjust1) (n+n)
                   = value scratch n + m * value (sn at Adjust1) n
               so value sn n + m * c = value (sn at Adjust1) n + value yp1 n
               so value scratch (n+n) + m * m * c
                  = value scratch n + m * (value sn n + m * c)
                  = value scratch n + m * value (sn at Adjust1) n + m * value yp1 n
                  = value (scratch at Adjust1) (n+n) + m * value yp1 n
                  = value xp1 n * value yp1 n + m * value yp1 n
                  = (value xp1 n + m * !xp1_hi) * value yp1 n
                  = (a0 + a1 + a2) * value yp1 n };
      cy := c end
    else begin
      if (!xp1_hi = 2)
      then begin
      let sa = { scratch } in
      let sn = C.incr scratch n in
      label Adjust2 in
      value_concat scratch n (n+n);
      assert { pelts scratch = pelts sn };
      let c = wmpn_addmul_1 sn yp1 n 2 in
      assert { pelts scratch = pelts sn };
      value_concat scratch n (n+n);
      value_sub_frame_shift (pelts scratch) (pelts sa)
                            (offset scratch) (offset sa) (p2i n);
      assert { value scratch n = value (scratch at Adjust2) n };
      assert { m * m = power radix (n+n) };
      assert { value scratch (n+n) + m * m * c = (a0 + a1 + a2) * value yp1 n
               by value sn n = value_sub (pelts scratch) (offset scratch + n)
                                         (offset scratch + (n+n))
               so value scratch (n+n) = value scratch n + m * value sn n
               so value (sn at Adjust2) n
                   = value_sub (pelts sa) (offset scratch + n)
                                          (offset scratch + (n+n))
               so value (scratch at Adjust2) (n+n)
                   = value scratch n + m * value (sn at Adjust2) n
               so value sn n + m * c = value (sn at Adjust2) n + !xp1_hi * value yp1 n
               so value scratch (n+n) + m * m * c
                  = value scratch n + m * (value sn n + m * c)
                  = value scratch n + m * value (sn at Adjust2) n
                                    + m * !xp1_hi * value yp1 n
                  = value (scratch at Adjust2) (n+n) + m * !xp1_hi * value yp1 n
                  = value xp1 n * value yp1 n + m * !xp1_hi * value yp1 n
                  = (value xp1 n + m * !xp1_hi) * value yp1 n
                  = (a0 + a1 + a2) * value yp1 n };
      assert { c <= 3
               by value sn n + m * c = value (sn at Adjust2) n + (value yp1 n) * 2
               so 0 <= value sn n
               so value (sn at Adjust2) n < m
               so value yp1 n < m
               so m * c < m * 3
               so c <= 3 };
      cy := c end
    else assert { !xp1_hi = 0 }  end
    end;
    begin ensures { value scratch (n + n) + power radix (n + n) * !cy
                    = (a0 + a1 + a2) * (b0 + b1) }
    if not (!yp1_hi = 0)
    then begin
      let sa = { scratch } in
      let sn = C.incr scratch n in
      label Adjust3 in
      value_concat scratch n (n+n);
      assert { pelts scratch = pelts sn };
      let c = wmpn_add_n_in_place sn xp1 n in
      value_concat scratch n (n+n);
      assert { pelts scratch = pelts sn };
      value_sub_frame_shift (pelts scratch) (pelts sa)
                            (offset scratch) (offset sa) (p2i n);
      assert { value scratch n = value (scratch at Adjust3) n };
      cy := !xp1_hi * !yp1_hi + c + !cy;
      assert { m * m = power radix (n+n) };
      assert { value scratch (n + n) + power radix (n + n) * !cy
                    = (a0 + a1 + a2) * (b0 + b1)
               by value sn n = value_sub (pelts scratch) (offset scratch + n)
                                         (offset scratch + (n+n))
               so value scratch (n+n) = value scratch n + m * value sn n
               so value (sn at Adjust3) n
                   = value_sub (pelts sa) (offset scratch + n)
                                          (offset scratch + (n+n))
               so value (scratch at Adjust3) (n+n)
                   = value scratch n + m * value (sn at Adjust3) n
               so !yp1_hi = 1
               so value sn n + m * c
                  = value (sn at Adjust3) n + !yp1_hi * value xp1 n
               so !yp1_hi * value xp1 n + m * !xp1_hi * !yp1_hi
                  = (a0 + a1 + a2) * !yp1_hi
               so value scratch (n+n) + m * m * !cy
                  = value scratch (n+n) + m * m * c
                       + m * m * !xp1_hi * !yp1_hi + m * m * (!cy at Adjust3)
                  = value scratch n + m * (value sn n + m * c)
                       + m * m * !xp1_hi * !yp1_hi + m * m * (!cy at Adjust3)
                  = value scratch n
                       + m * (value (sn at Adjust3) n + !yp1_hi * value xp1 n)
                       + m * m * !xp1_hi * !yp1_hi + m * m * (!cy at Adjust3)
                  = value scratch n + m * value (sn at Adjust3) n
                       + m * !yp1_hi * value xp1 n
                       + m * m * !xp1_hi * !yp1_hi + m * m * (!cy at Adjust3)
                  = value (scratch at Adjust3) (n+n)
                       + m * !yp1_hi * value xp1 n
                       + m * m * !xp1_hi * !yp1_hi + m * m * (!cy at Adjust3)
                  = (a0 + a1 + a2) * value yp1 n
                       + m * !yp1_hi * value xp1 n
                       + m * m * !xp1_hi * !yp1_hi
                  = (a0 + a1 + a2) * value yp1 n + m * (a0 + a1 + a2) * !yp1_hi
                  = (a0 + a1 + a2) * (value yp1 n + m * !yp1_hi)
                  = (a0 + a1 + a2) * (b0 + b1) } end
      else assert { b0 + b1 = value yp1 n }
    end;
  end;
  label RecM1 in
  join vm1 yp1;
  wmpn_toom22_mul_n_rec vm1 xm1 ym1 sor n (k-1);
  begin ensures { value vm1 (2*n) + m * m * !hi = am1_abs * bm1_abs }
        ensures { min r = old min r }
        ensures { max r = old max r }
        ensures { plength r = old plength r }
        ensures { 0 <= !hi <= 1 }
  if (not (!hi = 0))
  then begin
    assert { !hi = 1 };
    value_concat vm1 n (2*n);
    label HiSplit in
    let vm1n = incr_split vm1 n in
    let vm1l = { vm1 } in
    assert { value vm1 (2*n) at HiSplit = value vm1l n + m * value vm1n n
             by value vm1n n = value_sub (pelts vm1 at HiSplit)
                               (offset vm1 + n) (offset vm1 + 2*n) };
    label HiAdd in
    let c = wmpn_add_n_in_place vm1n ym1 n in
    label HiJoin in
    let vm1nj = { vm1n } in
    join vm1 vm1n;
    value_concat vm1 n (2*n);
    value_sub_frame (pelts vm1) (pelts vm1l) (offset vm1) (offset vm1 + p2i n);
    value_sub_frame (pelts vm1) (pelts vm1nj) (offset vm1 + p2i n)
                                              (offset vm1 + 2 * p2i n);
    assert { value vm1 n = value vm1l n };
    assert { value vm1 (2*n) = value vm1l n + m * value vm1nj n
             by value vm1nj n = value_sub (pelts vm1) (offset vm1 + p2i n)
                                                  (offset vm1 + 2 * p2i n) };
    assert { value vm1 (2*n) + m * m * c = am1_abs * bm1_abs
             by value vm1 (2 * n) + m * m * c
                = value vm1l n + m * (value (vm1n at HiJoin) n + m * c)
                = value vm1l n + m * (value (vm1n at HiAdd) n + value ym1 n)
                = value vm1 (2*n) at HiSplit + m * value ym1 n
                = value xm1 n * value ym1 n + m * value ym1 n
                = (value xm1 n + m * !hi) * value ym1 n
                = am1_abs * bm1_abs };
    hi := c;
    end
  end;
  let ghost vx0 = a0 * b0 in
  let ghost vx1 = a1 * b0 + a0 * b1 in
  let ghost vx2 = a2 * b0 + a1 * b1 in
  let ghost vx3 = a2 * b1 in
  assert { 0 <= vx0 /\ 0 <= vx1 /\ 0 <= vx2 /\ 0 <= vx3
           by 0 <= a0 /\ 0 <= a1 /\ 0 <= a2 /\ 0 <= b0 /\ 0 <= b1 };
  assert { (a0 + m * a1 + m * m * a2) * (b0 + m * b1)
           = vx0 + m * vx1 + m * m * vx2 + m * m * m * vx3 };
  assert { (a0 + a1 + a2) * (b0 + b1) = vx0 + vx1 + vx2 + vx3 };
  assert { vx0 + vx2 < 3 * m * m
            by (0 <= a0 < m /\ 0 <= a1 < m /\ 0 <= a2 < m
                /\ 0 <= b0 < m /\ 0 <= b1 < m
                by a2 < power radix s <= m
                so b1 < power radix t <= m)
            so (vx0 < m * m
                by vx0 = a0 * b0 <= a0 * m = m * a0 < m * m)
            so (vx2 < 2 * m * m
                by vx2 = a2 * b0 + a1 * b1
                so a2 * b0 <= a2 * m = m * a2 < m * m
                so a1 * b1 <= a1 * m = m * a1 < m * m) };
  begin ensures { value scratch (2*n + 1) = vx0 + vx2 }
        ensures { if !vm1_neg
                  then am1_abs*bm1_abs = (a0 - a1 + a2) * (b1 - b0)
                  else am1_abs*bm1_abs = (a0 - a1 + a2) * (b0 - b1) }
    begin ensures { value scratch (2*n + 1) = 2 * (vx0 + vx2) }
          ensures { if !vm1_neg
                    then am1_abs*bm1_abs = (a0 - a1 + a2) * (b1 - b0)
                    else am1_abs*bm1_abs = (a0 - a1 + a2) * (b0 - b1) }
    if !vm1_neg
    then begin
      assert { am1_abs * bm1_abs = (a0 - a1 + a2) * (b1 - b0)
               by if !vm1_neg at B1
               then am1_abs * bm1_abs = (a0 - a1 + a2) * (b1 - b0)
                    by am1_abs = a1 - (a0 + a2) /\ bm1_abs = b0 - b1
               else am1_abs * bm1_abs = (a0 - a1 + a2) * (b1 - b0)
                    by am1_abs = a0 - a1 + a2 /\ bm1_abs = b1 - b0 };
      assert { (a0 - a1 + a2) * (b1 - b0) = vx1 -vx0 + vx3 - vx2 };
      label Sub in
      let b = wmpn_sub_n_in_place scratch vm1 (2*n) in
      let r, ghost b' = sub_with_borrow !cy !hi b in
      assert { (b' = 0 /\ value scratch (2*n) + m * m * r = 2 * (vx0 + vx2))
               by r - radix * b' = !cy - !hi - b
               so m * m * r - m * m * radix * b'
                  = m * m * !cy - m * m * !hi - m * m * b
               so (m * m * b
                    = value scratch (2*n) - value (scratch at Sub) (2*n)
                      + value vm1 (2*n)
                   by value scratch (2*n) - (m * m) * b
                      = value (scratch at Sub) (2*n) - value vm1 (2*n))
               so m * m * r - m * m * radix * b' + value scratch (2 * n)
                  = m * m * !cy - m * m * !hi + value (scratch at Sub) (2*n)
                    - value vm1 (2*n)
                  = (value (scratch at Sub) (2*n) + m * m * !cy)
                    - (value vm1 (2*n) + m * m * !hi)
                  = (vx0 + vx1 + vx2 + vx3) - (vx1 - vx0 + vx3 - vx2)
                  = 2 * (vx0 + vx2)
                  >= 0
               so m * m * r - m * m * radix * b' + value scratch (2 * n) >= 0
               so value scratch (2 * n) < m * m
               so (m * m) * r <= (m * m) * (radix - 1)
               so value scratch (2 * n) + m * m * r < m * m * radix
               so m * m * radix * (1 - b') > 0
               so b' < 1
               so b' = 0
               };
      label Set in
      value_sub_shift_no_change (pelts scratch) (offset scratch)
                                (2 * p2i n)
                                (2 * p2i n) !cy;
      set_ofs scratch (2*n) r;
      assert { value scratch (2 * n) = value (scratch at Set) (2 * n) };
      value_tail scratch (2*n);
      assert { value scratch (2*n + 1) = 2 * (vx0 + vx2)
               by value scratch (2*n + 1) = value scratch (2*n) + (m*m) * r };
      end
    else begin
      assert { am1_abs * bm1_abs = (a0 - a1 + a2) * (b0 - b1)
               by if !vm1_neg at B1
               then am1_abs * bm1_abs = (a0 - a1 + a2) * (b0 - b1)
                    by am1_abs = a1 - (a0 + a2) /\ bm1_abs = b1 - b0
               else am1_abs * bm1_abs = (a0 - a1 + a2) * (b0 - b1)
                    by am1_abs = a0 - a1 + a2 /\ bm1_abs = b0 - b1 };
      assert { (a0 - a1 + a2) * (b0 - b1) = vx0 - vx1 + vx2 - vx3 };
      label Add in
      let c = wmpn_add_n_in_place scratch vm1 (2*n) in
      let r, ghost c' = add_with_carry !cy !hi c in
      assert { (c' = 0 /\ value scratch (2*n) + (m*m) * r = 2 * (vx0 + vx2))
                by r + radix * c' = !cy + !hi + c
                so m * m * r + m * m * radix * c'
                   = m * m * !cy + m * m * !hi + m * m * c
                so (m * m * c
                    = value (scratch at Add) (2*n) - value scratch (2*n)
                      + value vm1 (2*n)
                    by value scratch (2*n) + (m*m)*c
                       = value (scratch at Add) (2*n) + value vm1 (2*n))
                so m * m * r + m * m * radix * c' + value scratch (2*n)
                   = m * m * !cy + m * m * !hi + value (scratch at Add) (2*n)
                     + value vm1 (2*n)
                   = (value (scratch at Add) (2*n) + m * m * !cy)
                     + (value vm1 (2*n) + m * m * !hi)
                   = (vx0 + vx1 + vx2 + vx3) + (am1_abs * bm1_abs)
                   = (vx0 + vx1 + vx2 + vx3) + (vx0 - vx1 + vx2 - vx3)
                   = 2 * (vx0 + vx2)
                so 2 * (vx0 + vx2) < 6 * m * m < m * m * radix
                so (m * m * radix * c' < m * m * radix
                    by 0 <= r so 0 <= m * m * r
                    so 0 <= value scratch (2*n)
                    so m * m * radix * c'
                       <= m * m * r + m * m * radix * c' + value scratch (2*n)
                       < m * m * radix)
                so c' < 1
                so c' = 0
      };
      label Set in
      value_sub_shift_no_change (pelts scratch) (offset scratch)
                                (2 * p2i n)
                                (2 * p2i n) !cy;
      set_ofs scratch (2*n) r;
      assert { value scratch (2 * n) = value (scratch at Set) (2 * n) };
      value_tail scratch (2*n);
      assert { value scratch (2*n + 1) = 2 * (vx0 + vx2)
               by value scratch (2*n + 1) = value scratch (2*n) + (m*m) * r };
    end
    end;
    label Shift in
    let s = (2 * n) + 1 in
    let ghost low = wmpn_rshift scratch scratch s 1 in
    assert { low = 0 /\ value scratch s = vx0 + vx2
             by (low + radix * value scratch s)
                = value (scratch at Shift) s * power 2 (Limb.length - 1)
                = 2 * (vx0 + vx2) * power 2 (Limb.length - 1)
                = (vx0 + vx2) * power 2 Limb.length
                = (vx0 + vx2) * radix
             so divides radix ((vx0 + vx2) * radix)
             so divides radix (low + radix * value scratch s)
             so divides radix low }
  end;
  let ghost vy = vx1 + vx3 + (vx0 + vx2) * m in
  assert { vy = (vx0 + vx2) * m + vx0 + vx2 - (vx0 - vx1 + vx2 - vx3) };
  join xm1 ym1;
  (* (    r    |    xm1  ) *)
  let ghost ss = (2 * n) + 1 in
  assert { value scratch ss = vx0 + vx2 };
  let vy0 = scratch in
  let ghost l02 = value scratch (int32'int n) in
  let vy1 = xm1 in
  let vy2 = incr_split scratch n in
  let ghost h02 = value vy2 (int32'int n) in
  let t02 = get_ofs vy2 n in
  begin ensures { value vy0 n + m * value vy1 n + m * m * value vy2 (n+1) = vy }
        ensures { value vy2 (n+1) < (power radix n) * 6 }
  label Vy in
  let os = { vy0 } in
  value_tail vy2 n;
  value_concat scratch n ss;
  assert { l02 + m * h02 + m * m * t02 = vx0 + vx2
           by vx0 + vx2 = value scratch ss
              = value scratch n + m * value_sub (pelts scratch)
                                      (offset scratch + n) (offset scratch + ss)
              = value scratch n + m * value vy2 (n+1)
              = l02 + m * (h02 + m * t02) };
  assert { t02 < 3
           by l02 + m * h02 + m * m * t02 < m * m * 3
           so 0 <= l02 /\ 0 <= h02 /\ 0 <= m
           so 0 <= l02 + m * h02
           so m * m * t02 < m * m * 3 };
  let c = wmpn_add_n vy1 vy0 vy2 n in
  assert { value vy2 (n+1) < (power radix n) * 4
           by value vy2 (n+1) = value vy2 n + (power radix n) * t02
              < power radix n + (power radix n) * t02
              <= (power radix n) * 4 };
  assert { value vy2 (n+1) + (c + t02) < (power radix n) * 5
           by c + t02 <= power radix n
           so value vy2 (n+1) + (c+t02) <= (power radix n) * 5 };
  wmpn_incr vy2 (n+1) (c + t02);
  assert { value vy2 (n+1) < (power radix n) * 5 };
  value_sub_frame (pelts vy0) (pelts os) (offset scratch) (offset scratch + int32'int n);
  assert { value vy0 n + m * value vy1 n + m * m * value vy2 (n+1)
           = l02 + m * (l02 + h02) + m * m * (h02 + t02 + m * t02)
           = (vx0 + vx2) * m + vx0 + vx2
           by value vy0 n = l02
           so value vy1 n + m * c = l02 + h02
           so value vy2 (n+1) = value vy2 (n+1) at Vy + c + t02
              = h02 + t02 + m * t02 + c
           so value vy0 n + m * value vy1 n + m * m * value vy2 (n+1) + m * c
              = l02 + m * (l02 + h02) + m * m * (h02 + t02 + m * t02) + m * c };
  let vm1n = incr vm1 n in
  value_concat vm1 n (2*n);
  assert { value vm1 n + m * value vm1n n + m * m * !hi = am1_abs * bm1_abs
           by value vm1 n + m * value vm1n n = value vm1 (2*n) };
  begin ensures { value vy0 n + m * value vy1 n + m * m * value vy2 (n+1)
                  = old (value vy0 n + m * value vy1 n + m * m * value vy2 (n+1))
                    - (vx0 - vx1 + vx2 - vx3) }
        ensures { value vy2 (n+1) < (power radix n) * 6 }
  label Vm1 in
  let ovy2 = { vy2 } in
  if !vm1_neg
  then begin
    assert { am1_abs*bm1_abs = (a0 - a1 + a2) * (b1 - b0)
             = - (vx0 - vx1 + vx2 - vx3) };
    let c1 = wmpn_add_n_in_place scratch vm1 n in
    assert { value scratch n = value scratch n at Vm1 + value vm1 n - m * c1 };
    let c2 = wmpn_add_n_in_place vy1 vm1n n in
    assert { value vy1 n = value vy1 n at Vm1 + value vm1n n - m * c2};
    hi := !hi + c2;
    let c3 = wmpn_add_1_in_place vy1 n c1 in
    assert { value vy1 n
             = value vy1 n at Vm1 + value vm1n n + c1 - m * (c2 + c3) };
    hi := !hi + c3;
    wmpn_incr vy2 (n+1) !hi;
    assert { value vy2 (n+1) = value ovy2 (n+1) + c2 + c3 + !hi at Vm1 };
    assert { value vy0 n + m * value vy1 n + m * m * value vy2 (n+1)
             = (value vy0 n + m * value vy1 n + m * m * value vy2 (n+1)) at Vm1
               - (vx0 - vx1 + vx2 - vx3)
             by value vy0 n + m * value vy1 n + m * m * value vy2 (n+1)
             = value vy0 n at Vm1 + value vm1 n - m * c1
               + m * (value vy1 n at Vm1 + value vm1n n + c1 -  m * (c2 + c3))
               + m * m * (value ovy2 (n+1) + c2 + c3 + !hi at Vm1)
             = (value vy0 n + m * value vy1 n + m * m * value vy2 (n+1)) at Vm1
               + (value vm1 n + m * value vm1n n + m * m * !hi at Vm1)
             = (value vy0 n + m * value vy1 n + m * m * value vy2 (n+1)) at Vm1
               + am1_abs * bm1_abs
             = (value vy0 n + m * value vy1 n + m * m * value vy2 (n+1)) at Vm1
                - (vx0 - vx1 + vx2 - vx3) };
    end
  else begin
    assert { am1_abs*bm1_abs = (a0 - a1 + a2) * (b0 - b1)
             = vx0 - vx1 + vx2 - vx3 };
    let b1 = wmpn_sub_n_in_place scratch vm1 n in
    assert { value scratch n = value scratch n at Vm1 - value vm1 n + m * b1 };
    let b2 = wmpn_sub_n_in_place vy1 vm1n n in
    assert { value vy1 n = value vy1 n at Vm1 - value vm1n n + m * b2 };
    hi := !hi + b2;
    let b3 = wmpn_sub_1_in_place vy1 n b1 in
    assert { value vy1 n
             = value vy1 n at Vm1 - value vm1n n - b1 + m * (b2 + b3) };
    hi := !hi + b3;
    assert { value vy0 n + m * value vy1 n + m * m * (value vy2 (n+1) - !hi)
             = (value vy0 n + m * value vy1 n + m * m * value vy2 (n+1)) at Vm1
               - (vx0 - vx1 + vx2 - vx3)
             by value vy0 n + m * value vy1 n + m * m * (value vy2 (n+1) - !hi)
             = value vy0 n + m * value vy1 n
               + m * m * (value ovy2 (n+1) - b2 - b3 - !hi at Vm1)
             = value vy0 n at Vm1 - value vm1 n + m * b1
               + m * (value vy1 n at Vm1 - value vm1n n - b1 +  m * (b2 + b3))
               + m * m * (value ovy2 (n+1) - b2 - b3 - !hi at Vm1)
             = (value vy0 n + m * value vy1 n + m * m * value vy2 (n+1)) at Vm1
               - (value vm1 n + m * value vm1n n + m * m * !hi at Vm1)
             = (value vy0 n + m * value vy1 n + m * m * value vy2 (n+1)) at Vm1
               - am1_abs * bm1_abs
             = (value vy0 n + m * value vy1 n + m * m * value vy2 (n+1)) at Vm1
               - (vx0 - vx1 + vx2 - vx3) };
    assert { value vy2 (n+1) - !hi >= 0
             by value vy0 n + m * value vy1 n + m * m * (value vy2 (n+1) - !hi)
                = vy
             so (vy >= 0 by 0 <= vx1 /\ 0 <= vx1 /\ 0 <= vx2 /\ 0 <= vx3)
             so value vy0 n < m
             so value vy1 n <= m - 1
             so m * value vy1 n <= m * (m-1)
             so value vy0 n + m * value vy1 n <= value vy0 n + m * (m-1)
                < m + m * (m-1) = m * m
             so 0 <= value vy0 n + m * value vy1 n + m * m * (value vy2 (n+1) - !hi)
                < m * m * (1 + value vy2 (n+1) - !hi)
             so 0 < (m * m) * (1 + value vy2 (n+1) - !hi)
             so 0 < (1 + value vy2 (n+1) - !hi) };
    wmpn_decr vy2 (n+1) !hi;
    assert { value vy2 (n+1) = value ovy2 (n+1) - b2 - b3 - !hi at Vm1 };
    end;
  end
  end;
  label Split3 in
  let ghost ovy1 = { vy1 } in
  wmpn_toom22_mul_n_rec r x y sor n (k-1);
  let r3n = incr_split xm1 n in
  value_sub_frame (pelts vy1) (pelts ovy1) (offset vy1)
                              (offset vy1 + int32'int n);
  assert { value vy1 n = value ovy1 n };
  assert { value r (n+n) = vx0
           by value x n = a0 so value y n = b0 };
  begin
    ensures { value r3n (s+t) = vx3 }
    ensures { min r3n = old min r3n /\ max r3n = old max r3n }
    ensures { plength r3n = old plength r3n }
    if (s > t)
    then let ghost _ = wmpn_mul r3n x2 s y1 t (k-1) in ()
    else let ghost _ = wmpn_mul r3n y1 t x2 s (k-1) in ()
  end;
  assert { (a0 + m * a1 + m * m * a2) * (b0 + m * b1)
           = m * vy + vx0 + m * m * m * vx3 - m * m * vx0 - m * vx3 };
  assert { 0 <= vx0 < m * m /\ 0 <= vx3 < m * m
           by a0 * b0 <= a0 * m = m * a0 < m * m
           so a2 < power radix s <= m
           so b1 < power radix t <= m
           so a2 * b1 <= a2 * m = m * a2 < m * m };
  let ghost or = { r } in
  let r1n = incr_split r n in
  value_sub_frame (pelts r) (pelts or) (offset r) (offset r + int32'int n);
  value_sub_frame (pelts r1n) (pelts or) (offset r1n) (offset r1n + int32'int n);
  let ghost lvx0 = value r (int32'int n) in
  let ghost hvx0 = value r1n (int32'int n) in
  value_concat or n (n + n);
  assert { vx0 = lvx0 + m * hvx0 };
  let ghost or3n = { r3n } in
  let r4n = incr_split r3n n in
  let r2n = xm1 in
  value_sub_frame (pelts r3n) (pelts or3n) (offset r3n)
                                           (offset r3n + int32'int n);
  value_sub_frame (pelts r4n) (pelts or3n) (offset r4n)
                  (offset r4n + int32'int s + int32'int t - int32'int n);
  let ghost lvx3 = value r3n (int32'int n) in
  let ghost hvx3 = value r4n (int32'int s + int32'int t- int32'int n) in
  value_concat or3n n (s + t);
  assert { vx3 = lvx3 + m * hvx3 };
  let ghost vvy0 = value vy0 (int32'int n) in
  let ghost vvy1 = value vy1 (int32'int n) in
  let ghost vvy2 = value vy2 (int32'int n+1) in
  assert { vy = vvy0 + m * vvy1 + m * m * vvy2 };
  assert { m * vy + vx0 + m * m * m * vx3 - m * m * vx0 - m * vx3
           = lvx0 + m * (vvy0 + hvx0 - lvx3) + m * m * (vvy1 - lvx0 - hvx3)
             + m * m * m * (vvy2 - (hvx0 - lvx3)) + m * m * m * m * hvx3 };
  label R1 in
  let bo1 = wmpn_sub_n_in_place r1n r3n n in
  let bo = ref bo1 in
  assert { value r1n n - m * bo1 = hvx0 - lvx3 };
  let ly2 = get_ofs vy2 n in
  value_tail vy2 n;
  assert { 0 <= ly2 < 6
           by 0 <= value vy2 n
           so value vy2 n + (power radix n) * ly2 = value vy2 (n+1)
           so (power radix n) * ly2 <= value vy2 (n+1) < (power radix n) * 6 };
  let h = ref (Limb.to_int64 (ly2 + bo1)) in
  label R2 in
  let bo2 = wmpn_sub_n_in_place r2n r n in
  let bo2' = wmpn_sub_1_in_place r2n n !bo in
  bo := bo2 + bo2';
  assert { value r2n n - m * !bo = vvy1 - lvx0 - (!bo at R2) };
  assert { value r1n n + m * value r2n n - m * m * !bo
           = hvx0 - lvx3 + m * (vvy1 - lvx0) };
  label R3 in
  let bo3 = wmpn_sub_n r3n vy2 r1n n in
  let bo3' = wmpn_sub_1_in_place r3n n !bo in
  bo := bo3 + bo3';
  assert { value r3n n - m * !bo = value vy2 n - value r1n n - (!bo at R3) };
  assert { value r1n n + m * value r2n n + m * m * value r3n n - m * m *m * !bo
           = hvx0 - lvx3 + m * (vvy1 - lvx0) + m * m * value vy2 n
             - m * m * (hvx0 - lvx3 + m * bo1) };
  h := Int64.(-) !h (Limb.to_int64 !bo);
  label Join3 in
  let ghost or1n = { r1n } in
  let ghost or2n = { r2n } in
  let ghost or3n = { r3n } in
  join r2n r3n;
  value_concat r2n n (n + n);
  join r1n r2n;
  value_sub_frame (pelts r1n) (pelts or1n)
                  (offset r1n) (offset r1n + int32'int n);
  value_sub_frame (pelts r1n) (pelts or2n)
                  (offset r1n + int32'int n) (offset r1n + 2 * int32'int n);
  value_sub_frame (pelts r1n) (pelts or3n)
                  (offset r1n + 2 * int32'int n) (offset r1n + 3 * int32'int n);
  value_concat r1n n (3 * n);
  value_sub_concat (pelts r1n) (offset r1n + int32'int n)
                   (offset r1n + int32'int (2 * n))
                   (offset r1n + int32'int (3 * n));
  assert { value r1n (3*n)
           = value r1n n + m * value r2n n + m * m * value r3n n at Join3
           by offset r2n at Join3 = offset r1n + n
           so offset r3n at Join3 = offset r1n + 2 * n
           so value r1n n = value r1n n at Join3
           so value_sub (pelts r1n) (offset r1n + n) (offset r1n + 2*n)
              = value r2n n at Join3
           so value_sub (pelts r1n) (offset r1n + 2*n) (offset r1n + 3*n)
              = value_sub (pelts or3n) (offset r1n + 2*n) (offset r1n + 3*n)
              = value r3n n at Join3
           so value r1n (3*n)
              = value r1n n
                + m * value_sub (pelts r1n) (offset r1n + n) (offset r1n + 3*n)
              = value r1n n
                + m *
                (value_sub (pelts r1n) (offset r1n + n) (offset r1n + 2*n)
                 + m * value_sub (pelts r1n) (offset r1n + 2*n)
                                             (offset r1n + 3*n))
              = value r1n n
                + m * value_sub (pelts r1n) (offset r1n + n) (offset r1n + 2*n)
                + m * m * value_sub (pelts r1n) (offset r1n + 2*n)
                                                (offset r1n + 3*n) };
  assert { value r1n (3*n) + m * m * m * !h
           = hvx0 - lvx3 + m * (vvy1 - lvx0) + m * m * vvy2
             - m * m * (hvx0 - lvx3)
           by value r1n (3*n) - m * m * m * !bo
              = hvx0 - lvx3 + m * (vvy1 - lvx0) + m * m * value vy2 n
               - m * m * (hvx0 - lvx3 + m * bo1)
           so !h = ly2 + bo1 - !bo
           so value r1n (3*n) + m * m * m * !h
              = value r1n (3*n) - m * m * m * !bo
                + m * m * m * ly2 + m * m * m * bo1
           so m * m * value vy2 n + m * m * m * ly2 = m * m * vvy2 };
  label Addy0 in
  let c = wmpn_add_in_place r1n (3 * n) scratch n in
  h := Int64.(+) !h (Limb.to_int64 c);
  assert { power radix (3*n) = m * m * m };
  assert { value r1n (3*n) + m * m * m * !h
           = vy + hvx0 - lvx3 - m * lvx0 - m * m * (hvx0 - lvx3)
           = vy + hvx0 - lvx3 - m * vx0 + m * m * lvx3
           by value r1n (3*n) + m * m * m * !h
              = value r1n (3*n) + (m * m * m * !h at Addy0) + m * m * m * c
              = (value r1n (3*n) + m * m * m * !h at Addy0)
                 + vvy0
              = vvy0 + hvx0 - lvx3 + m * (vvy1 - lvx0) + m * m * vvy2
                - m * m * (hvx0 - lvx3) };
  label Join1 in
  let ghost or = { r } in
  let ghost or1n = { r1n } in
  join r r1n;
  value_sub_frame (pelts r) (pelts or) (offset r) (offset r + int32'int n);
  value_sub_frame (pelts r) (pelts or1n)
                  (offset r + int32'int n)
                  (offset r + 4 * int32'int n);
  value_concat r n (4*n);
  assert { value r (4*n)
           = lvx0 + m * value r1n (3*n) at Join1
           by offset or1n = offset r + n
           so value r n = lvx0
           so value r1n (3*n) at Join1
              = value_sub (pelts r) (offset r + n) (offset r + 4*n) };
  assert { m * m * m * m = power radix (4*n)
           by m * m = power radix (2*n) };
  assert { value r (4*n) + m * m * m * m * !h
           = vx0 + m * vy - m * lvx3 - m * m * vx0 + m * m * m * lvx3 };
  let rs = s + t - n in
  begin ensures { value r (4*n) + m * m * m * m * value r4n rs
                  = (value x sx) * (value y sy) }
  if [@extraction:likely] (s + t > n)
  then begin
    let r2n = incr r (2 * n) in
    label Sub2 in
    value_concat r (2 * n) (4 * n);
    assert { value r (4*n) = value r (2*n) + m*m * value r2n (2*n)
             by m * m = power radix (2*n)
             so value_sub (pelts r) (offset r + 2*n) (offset r + 4*n)
                = value r2n (2*n) };
    assert { value r4n rs = hvx3 };
    let ghost or = { r } in
    let b = wmpn_sub_in_place r2n (2 * n) r4n rs in
    value_sub_frame (pelts r) (pelts or) (offset r) (offset r + 2 * int32'int n);
    assert { value r (2*n) = value or (2*n) };
    value_concat r (2 * n) (4 * n);
    assert { value r (4*n) = value r (2*n) + m*m * value r2n (2*n)
             by m * m = power radix (2*n)
             so pelts r2n = pelts r
             so offset r2n = offset r + 2*n
             so offset r2n + 2*n = offset r + 4*n
             so value_sub (pelts r) (offset r + 2*n) (offset r + 4*n)
                = value r2n (2*n) };
    h := Int64.(-) !h (Limb.to_int64 b);
    assert { value r2n (2*n) - m * m * b = (value r2n (2*n) at Sub2) - hvx3
             by m * m = power radix (2*n) };
    assert { value r (4*n) - m * m * m * m * b
             = value r (4*n) at Sub2 - m * m * hvx3 };
    assert { value r (4*n) + m * m * m * m * !h + m * m * m * m * value r4n rs
             = value x sx * value y sy
             by value r (4*n) + m * m * m * m * !h
             = value r (4*n) + m * m * m * m * (!h at Sub2) - m * m * m * m * b
             = value r (4*n) at Sub2 + m * m * m * m * !h at Sub2 - m * m * hvx3
             = vx0 + m * vy - m * lvx3 - m * m * vx0 + m * m * m * lvx3
               - m * m * hvx3
             = vx0 + m * vy - m * vx3 - m * m * vx0 + m * m * m * lvx3
             so value r (4*n) + m * m * m * m * !h
                + m * m * m * m * value r4n rs
             = vx0 + m * vy - m * vx3 - m * m * vx0 + m * m * m * lvx3
                + m * m * m * m * value r4n rs
             = vx0 + m * vy - m * vx3 - m * m * vx0 + m * m * m * vx3
             = (a0 + m * a1 + m * m * a2) * (b0 + m * b1)
             = value x sx * value y sy };
    (if Int64.(<) !h 0
    then begin
      assert { 0 <= value r4n rs + !h
               by 0 <= value x sx so 0 <= value y sy
               so 0 <= value x sx * value y sy
               so value r (4*n) < power radix (4*n) = m * m * m * m
               so 0 <= value r (4*n) + m * m * m * m * (value r4n rs + !h)
                    < (m * m * m * m) * (1 + value r4n rs + !h)
               so 0 < (m * m * m * m) * (1 + value r4n rs + !h)
               so 0 < (m * m * m * m)
               so 0 < 1 + value r4n rs + !h };
      wmpn_decr r4n rs (Limb.of_int64 (Int64.(-_) !h))
      end
    else begin
      assert { value r4n rs + !h < power radix rs
               by value x sx < power radix sx
               so value y sy < power radix sy
               so value x sx * value y sy
                  <= value x sx * power radix sy
                  < power radix sx * power radix sy
                  = power radix (sx + sy)
                  = power radix (3*n+s+t)
               so 0 <= value r (4*n)
               so power radix (4*n) * (value r4n rs + !h)
                  = m * m * m * m * (value r4n rs + !h)
                  <= value r (4*n) + m * m * m * m * (value r4n rs + !h)
                  < power radix (3*n + s + t) = power radix (4*n + rs)
                  = power radix (4*n) * power radix rs
               so power radix (4*n) * (value r4n rs + !h)
                  < power radix (4*n) * power radix rs
               so 0 < power radix (4*n) };
      wmpn_incr r4n rs (Limb.of_int64 !h)
      end)
  end
  else begin
    assert { !h = 0
             by rs = 0
             so sx + sy = 4 * n
             so hvx3 = 0
             so vx3 = lvx3
             so value r (4*n) + power radix (4*n) * !h = value x sx * value y sy
             so 0 <= value x sx < power radix sx
             so 0 <= value y sy < power radix sy
             so 0 <= value x sx * value y sy
             so value x sx * value y sy
                <= value x sx * power radix sy
                < power radix sx * power radix sy
                = power radix (sx + sy)
                = power radix (4*n)
             so 0 <= value r (4*n) < power radix (4*n)
             so 0 <= value x sx * value y sy
                < power radix (4*n) + power radix (4*n) * !h
                = power radix (4*n) * (1 + !h)
             so 0 < power radix (4*n) * (1 + !h)
             so 0 < 1 + !h
             so power radix (4*n) * 1 > value x sx * value y sy
                >= power radix (4*n) * !h
             so power radix (4*n) * !h < power radix (4*n) * 1
             so !h < 1 };
  end
  end;
  label Join4 in
  let ghost or = { r } in
  let ghost or4n = { r4n } in
  join r r4n;
  value_sub_frame (pelts r) (pelts or) (offset r) (offset r + 4 * int32'int n);
  value_sub_frame (pelts r) (pelts or4n) (offset r + 4 * int32'int n)
                  (offset r + 3 * int32'int n + int32'int s + int32'int t);
  value_concat r (4*n) (3*n + s + t);
  assert { value r (3*n + s + t) = value x sx * value y sy
           by offset or4n = offset r + 4*n
           so offset r4n + rs = offset r + 3*n + s + t
           so value_sub (pelts r) (offset r + 4*n) (offset r + 3*n + s + t)
              = value_sub (pelts or4n) (offset r + 4*n) (offset r + 3*n + s + t)
              = value or4n rs
           so value r (3*n + s + t)
              = value or (4*n) + m * m * m * m * value or4n rs };
  join scratch vy2;
  join scratch sor;
  join_r sol scratch;
  label JoinR in
  let ghost or = { r } in
  join r ror;
  value_sub_frame (pelts r) (pelts or) (offset r)
                  (offset r + int32'int sx + int32'int sy);
  label JoinL in
  join_r rol r;
  value_sub_frame (pelts r) (pelts or) (offset r)
                  (offset r + int32'int sx + int32'int sy);
  assert { value r (sx + sy) = value x sx * value y sy
           by value r (sx + sy) = value r (sx + sy) at JoinL
              = value r (sx + sy) at JoinR }

  with wmpn_mul (r x:ptr limb) (sx:int32) (y:ptr limb) (sy:int32) (ghost k: int) : limb
    requires { valid x sx }
    requires { valid y sy }
    requires { valid r (sx + sy) }
    requires { writable r }
    requires { 0 < sy <= sx }
    requires { 8 * sx < max_int32 }
    requires { sx <= toom22_threshold * power 2 k }
    requires { 0 <= k <= 64 }
    ensures  { min r = old min r }
    ensures  { max r = old max r }
    ensures  { plength r = old plength r }
    ensures  { value r (sx + sy) = value x sx * value y sy }
    ensures  { result = (pelts r)[offset r + sx + sy - 1] }
    ensures  { forall j. min r <= j < offset r \/ offset r + sx + sy <= j < max r
                         -> (pelts r)[j] = old (pelts r)[j] }
    variant  { k+k+1 }
  =
    if sy <= toom22_threshold
    then
    (* TODO block product if sx large, for memory locality according to GMP *)
     wmpn_mul_basecase r x sx y sy
    else begin
      (* this would be faster with salloc *)
      (*let ghost k = 64 in*) (* is always enough *)
      let scratch = salloc (UInt32.of_int32 ((5 * sy) + 128)) in
      (* c_assert (is_not_null scratch); *)
      let rol = decr_split r 0 in
      let ror = incr_split r (sx + sy) in
      if ((2 * sx) >= (5 * sy))
      then begin
        let un = ref sx in
        let su = (3 * sy) / 2 in
        assert { 0 < su };
        let ghost sr = su + sy in
        let ws = salloc (UInt32.of_int32 (4 * sy)) in
        (* c_assert (is_not_null ws); *)
        wmpn_toom32_mul r x su y sy scratch k;
        un := !un - su;
        let up = ref (C.incr x su) in
        let rp = ref (C.incr r su) in
        let ghost ou = ref su in
        let ghost or = ref sr in
        while (!un >= (2 * sy)) do (* 5/2?*)
          invariant { min_int32 <= 2 * !un <= max_int32 }
          invariant { !ou + !un = sx }
          invariant { !or = !ou + sy }
          invariant { su <= !ou < sx }
          invariant { !un < sx }
          invariant { 2 * sy - su <= !un }
          invariant { value r !or = value x !ou * value y sy }
          invariant { offset !rp = offset r + !ou }
          invariant { offset !up = offset x + !ou }
          invariant { min !up = min x /\ max !up = max x }
          invariant { plength !up = plength x }
          invariant { min !rp = min r /\ max !rp = max r }
          invariant { plength !rp = plength r }
          invariant { writable !rp }
          invariant { min ws = 0 /\ max ws = plength ws = 4 * sy }
          invariant { min scratch = 0 /\ max scratch = plength scratch }
          invariant { plength scratch = 5 * sy + 128 }
          invariant { pelts !rp = pelts r }
          invariant { pelts !up = pelts x }
          variant { p2i !un }
          (*wmpn_copyi ws !rp sy;
          let rr = rp.contents in
          wmpn_toom32_mul rr !up y scratch su sy k;
          let cy = wmpn_add_in_place rr ws sy sy in
          let rpn = C.incr rr sy in
          wmpn_incr rpn cy su;
          un := !un - su;
          up.contents <- C.incr !up su;
          ou := !ou + su;
          or := !or + su;
          rp.contents <- C.incr !rp su;*)
          label StartLoop in
          let ghost o_r = { r } in
          let ghost rrp = !rp in
          let ghost o_rp = { rrp } in (* TODO why not { !rp } ? *)
          value_concat r !ou !or;
          assert { value r !or = value r !ou + power radix !ou * value !rp sy };
          wmpn_toom32_mul ws !up su y sy scratch k;
          let cy = wmpn_add_n_in_place !rp ws sy in
          value_sub_frame (pelts r) (pelts o_r) (offset r) (offset r + p2i !ou);
          assert { value r !ou = value o_r !ou };
          assert { value !rp sy + (power radix sy) * cy
                   = value o_rp sy + value ws sy };
          let rpn = C.incr !rp sy in
          let wsy = C.incr ws sy in
          let orp = { rpn } in
          label Copy in
          wmpn_copyi rpn wsy su;
          value_sub_frame_shift (pelts rpn) (pelts wsy)
                                (offset rpn) (offset wsy) (int32'int su);
          value_sub_frame (pelts r) (pelts orp) (offset r) (offset r + p2i !ou);
          value_sub_frame (pelts rpn) (pelts orp)
                          (offset !rp) (offset !rp + p2i sy);
          assert { value rpn su = value wsy su };
          assert { value !rp sy = value (!rp at Copy) sy };
          assert { value r !ou = value o_r !ou };
          value_concat ws sy sr;
          assert { value ws sr
                   = value ws (sy + su)
                   = value ws sy + power radix sy * value wsy su };
          value_concat r !ou (!ou + sr);
          assert { value r (!or + su) = value r (!ou + sr)
                   = value r !ou + power radix !ou * value !rp sr };
          value_concat !rp sy sr;
          assert { value !rp sr = value !rp sy + power radix sy * value rpn su };
          value_concat x !ou (!ou + su);
          assert { value x (!ou + su)
                   = value x !ou + power radix !ou * value !up su };
          assert { value r (!ou + sr) + (power radix !or) * cy
                   = value x (!ou + su) * value y sy
                   by value r (!ou + sr) + (power radix !or) * cy
                      = value r !ou + power radix !ou * value !rp sr
                        + power radix !ou * (power radix sy) * cy
                      = value r !ou + power radix !ou * value !rp sy
                        + power radix !ou * power radix sy * value wsy su
                        + power radix !ou * power radix sy * cy
                      = value r !ou
                        + power radix !ou * (value !rp sy + power radix sy * cy)
                        + power radix !ou * power radix sy * value wsy su
                      = value r !ou
                        + power radix !ou * (value o_rp sy + value ws sy)
                        + power radix !ou * power radix sy * value wsy su
                      = value r !ou + power radix !ou * value o_rp sy
                        + power radix !ou
                          * (value ws sy + power radix sy * value wsy su)
                      = value o_r !or
                        + power radix !ou * value ws sr
                      = value x !ou * value y sy
                        + power radix !ou * value !up su * value y sy
                      = (value x !ou + power radix !ou * value !up su)
                        * value y sy
                      = value x (!ou + su) * value y sy };
          value_concat r !or (!ou + sr);
          assert { value r (!ou + sr)
                      = value r !or + power radix !or * value rpn su 
                   by value rpn su = value_sub (pelts r) (offset r + !or) (offset r + !ou + sr) };
          assert { value rpn su + cy < power radix su
                   by value x (!ou + su) < power radix (!ou + su)
                   so value y sy < power radix sy
                   so value x (!ou + su) * value y sy
                      < power radix (!ou + su) * power radix sy
                      = power radix (!ou + su + sy)
                      = power radix (!or + su)
                   so value r (!ou + sr) + power radix !or * cy
                      = value r !or + power radix !or * (value rpn su)
                        + power radix !or * cy
                      = value r !or + power radix !or * (value rpn su + cy)
                   so value r !or + power radix !or * (value rpn su + cy)
                      < power radix (!or + su)
                   so 0 <= value r !or
                   so power radix !or * (value rpn su + cy)
                      < power radix (!or + su)
                      = power radix !or * power radix su };
          label Incr in
          let orp = { rpn } in
          wmpn_incr rpn su cy;
          value_sub_frame (pelts r) (pelts orp) (offset r) (offset r + p2i !or);
          assert { value r !or = value r !or at Incr };
          value_concat r !or (!ou + sr);
          assert { value r (!ou + sr) = value x (!ou + su) * value y sy
                   by value r (!ou + sr)
                   = value r !or + power radix !or * value rpn su
                   = value r !or at Incr + power radix !or * value rpn su
                   = value r !or at Incr
                     + power radix !or * (value rpn su at Incr + cy)
                   = value r !or at Incr
                     + power radix !or * value rpn su at Incr
                     + power radix !or * cy
                   = value r (!ou + sr) at Incr + power radix !or * cy };
          un := !un - su;
          up.contents <- C.incr !up su;
          ou := !ou + su;
          or := !or + su;
          rp.contents <- C.incr !rp su;
        done;
 (*       wmpn_copyi ws !rp sy;*)
        (* sy <= !un <= 2.5 * sy *)
        value_concat r !ou !or;
        assert { value r !or = value r !ou + power radix !ou * value !rp sy };
        let ghost o_r = { r } in
        let ghost rrp = !rp in
        let ghost o_rp = { rrp } in (* TODO why not { !rp } ? *)
        begin
          ensures { value ws (!un + sy) = value !up !un * value y sy }
          ensures { min ws = old min ws /\ max ws = old max ws
                    /\ plength ws = old plength ws }
          ensures { min scratch = old min scratch
                    /\ max scratch = old max scratch
                    /\ plength scratch = old plength scratch }
          if sy <= !un
          then begin
            if ((4 * !un) < (5 * sy))
            then wmpn_toom22_mul ws !up !un y sy scratch k
            else wmpn_toom32_mul ws !up !un y sy scratch k
          end
          else let _ = wmpn_mul ws y sy !up !un (k-1) in ()
        end;
        let cy = wmpn_add_n_in_place !rp ws sy in
        value_sub_frame (pelts r) (pelts o_r) (offset r) (offset r + p2i !ou);
        assert { value r !ou = value o_r !ou };
        assert { value !rp sy + (power radix sy) * cy
                 = value o_rp sy + value ws sy };
        assert { valid !rp sy };
        let rpn = C.incr !rp sy in
        let wsy = C.incr ws sy in
        let orp = { rpn } in
        label Copy in
        wmpn_copyi rpn wsy !un;
        value_sub_frame_shift (pelts rpn) (pelts wsy)
                              (offset rpn) (offset wsy) (int32'int !un);
        value_sub_frame (pelts r) (pelts orp) (offset r) (offset r + p2i !ou);
        value_sub_frame (pelts rpn) (pelts orp)
                        (offset !rp) (offset !rp + p2i sy);
        assert { value rpn !un = value wsy !un };
        assert { value !rp sy = value (!rp at Copy) sy };
        assert { value r !ou = value o_r !ou };
        let ghost sr = sy + !un in
        value_concat ws sy sr;
        assert { value ws sr
                 = value ws sy + power radix sy * value wsy !un };
        value_concat r !ou (!ou + sr);
        assert { value r (!or + !un) = value r (!ou + sr)
                 = value r !ou + power radix !ou * value !rp sr };
        value_concat !rp sy sr;
        assert { value !rp sr = value !rp sy + power radix sy * value rpn !un };
        value_concat x !ou (!ou + !un);
        assert { value x (!ou + !un)
                 = value x !ou + power radix !ou * value !up !un };
        assert { value r (!ou + sr) + (power radix !or) * cy
                 = value x (!ou + !un) * value y sy
                 by value r (!ou + sr) + (power radix !or) * cy
                    = value r !ou + power radix !ou * value !rp sr
                      + power radix !ou * (power radix sy) * cy
                    = value r !ou + power radix !ou * value !rp sy
                      + power radix !ou * power radix sy * value wsy !un
                      + power radix !ou * power radix sy * cy
                    = value r !ou
                      + power radix !ou * (value !rp sy + power radix sy * cy)
                      + power radix !ou * power radix sy * value wsy !un
                    = value r !ou
                      + power radix !ou * (value o_rp sy + value ws sy)
                      + power radix !ou * power radix sy * value wsy !un
                    = value r !ou + power radix !ou * value o_rp sy
                      + power radix !ou
                        * (value ws sy + power radix sy * value wsy !un)
                    = value o_r !or
                      + power radix !ou * value ws sr
                    = value x !ou * value y sy
                      + power radix !ou * value !up !un * value y sy
                    = (value x !ou + power radix !ou * value !up !un)
                      * value y sy
                    = value x (!ou + !un) * value y sy };
        value_concat r !or (!ou + sr);
        assert { value r (!ou + sr)
                    = value r !or + power radix !or * value rpn !un };
        assert { value rpn !un + cy < power radix !un
                 by value x (!ou + !un) < power radix (!ou + !un)
                 so value y sy < power radix sy
                 so value x (!ou + !un) * value y sy
                    < power radix (!ou + !un) * power radix sy
                    = power radix (!ou + !un + sy)
                    = power radix (!or + !un)
                 so value r (!ou + sr) + power radix !or * cy
                    = value r !or + power radix !or * (value rpn !un)
                      + power radix !or * cy
                    = value r !or + power radix !or * (value rpn !un + cy)
                 so value r !or + power radix !or * (value rpn !un + cy)
                    < power radix (!or + !un)
                 so 0 <= value r !or
                 so power radix !or * (value rpn !un + cy)
                    < power radix (!or + !un)
                    = power radix !or * power radix !un };
        let orp = { rpn } in
        label Incr in
        wmpn_incr rpn !un cy;
        value_sub_frame (pelts r) (pelts orp) (offset r) (offset r + p2i !or);
        assert { value r !or = value r !or at Incr };
        value_concat r !or (!ou + sr);
        assert { value r (!ou + sr) = value x (!ou + !un) * value y sy
                 by value r (!ou + sr)
                 = value r !or + power radix !or * value rpn !un
                 = value r !or at Incr + power radix !or * value rpn !un
                 = value r !or at Incr
                   + power radix !or * (value rpn !un at Incr + cy)
                 = value r !or at Incr
                   + power radix !or * value rpn !un at Incr
                   + power radix !or * cy
                 = value r (!ou + sr) at Incr + power radix !or * cy };
        assert { value r (sx + sy) = value x sx * value y sy
                 by !ou + sr = sx + sy /\ !ou + !un = sx };
        sfree ws;
      end
      else begin
        if ((4 * sx) < (5 * sy))
        then wmpn_toom22_mul r x sx y sy scratch k
        else wmpn_toom32_mul r x sx y sy scratch k
      end;
      sfree scratch;
      label JoinR in
      let ghost or = { r } in
      join r ror;
      value_sub_frame (pelts r) (pelts or) (offset r)
                      (offset r + int32'int sx + int32'int sy);
      label JoinL in
      join_r rol r;
      value_sub_frame (pelts r) (pelts or) (offset r)
                      (offset r + int32'int sx + int32'int sy);
      assert { value r (sx + sy) = value x sx * value y sy
               by value r (sx + sy) = value r (sx + sy) at JoinL
                = value r (sx + sy) at JoinR }
    end;
  C.get_ofs r (sx + sy - 1)
        (* sy <= !un <= 2.5 * sy *)
       (* if sy <= !un
        then begin
           if ((4 * !un) < (5 * sy))
           then wmpn_toom22_mul ws !up y scratch !un sy k
           else wmpn_toom32_mul ws !up y scratch !un sy k
        end
        else wmpn_mul ws y !up sy !un (k-1);
        let cy = wmpn_add_in_place !rp ws sy sy in
        let rpn = C.incr !rp sy in
        wmpn_copyi rpn (C.incr ws sy) !un;
        wmpn_incr rpn cy !un;
        sfree ws;
      end
      else begin
        if ((4 * sx) < (5 * sy))
        then wmpn_toom22_mul r x y scratch sx sy k
        else wmpn_toom32_mul r x y scratch sx sy k
      end;
      sfree scratch
    end*)

  (** `wmpn_mul_n r x y sz` multiplies `(x, sz)` and `(y, sz)` and
  writes the result to `(r, 2*sz)`. `r` must not overlap with either
  `x` or `y`. Corresponds to `mpn_mul_n`.  *)
  let wmpn_mul_n (r x y:t) (sz:int32) (ghost k: int) : unit
    requires { sz > 0 }
    requires { valid x sz }
    requires { valid y sz }
    requires { valid r (sz + sz) }
    requires { writable r }
    requires { 8 * sz < max_int32 }
    requires { sz <= toom22_threshold * power 2 k }
    requires { 0 <= k <= 64 }
    ensures { value r (sz + sz) = value x sz * value y sz }
    ensures  { min r = old min r }
    ensures  { max r = old max r }
    ensures  { plength r = old plength r }
    ensures  { forall j. min r <= j < offset r \/ offset r + sz + sz <= j < max r
                         -> (pelts r)[j] = old (pelts r)[j] }
  =
    if sz <= toom22_threshold
    then wmpn_mul_basecase r x sz y sz
    else
      let ws = salloc (UInt32.of_int32 (2 * (sz + 64))) in
      wmpn_toom22_mul r x sz y sz ws 64

  meta remove_prop axiom no_borrow
  meta remove_prop axiom no_borrow_ptr
  meta remove_prop lemma power_ge_1
  meta remove_prop axiom valuation'def
  meta remove_prop axiom valuation'spec
  meta remove_prop lemma valuation_mul
  meta remove_prop axiom valuation_times_pow
  meta remove_prop axiom valuation_lower_bound
  meta remove_prop axiom valuation_monotonous
  meta remove_prop lemma valuation_nondiv
  meta remove_prop lemma valuation_zero_prod
  meta remove_prop axiom valuation_times_nondiv
  meta remove_prop lemma valuation_split
  meta remove_prop lemma valuation_prod
  meta remove_prop lemma valuation_mod
  meta remove_prop lemma valuation_decomp
  meta remove_prop lemma valuation_pow

end
