module Div

  use int.Int
  use mach.int.Int32
  use import mach.int.UInt64GMP as Limb
  use int.Power
  use ref.Ref
  use mach.c.C
  use array.Array
  use map.Map
  use types.Types
  use types.Int32Eq
  use types.UInt64Eq
  use lemmas.Lemmas
  use compare.Compare
  use util.UtilOld
  use add.AddOld
  use sub.SubOld
  use logical.LogicalUtil
  use logical.LogicalOld
  use int.EuclideanDivision

  (** Based on Niels Möller and Torbjörn Granlund, “Improved
      division by invariant integers” 2010 *)

  use int.MinMax as MM

  predicate reciprocal (v d:limb) =
    v = (div (radix*radix - 1) (d)) - radix

  let lemma fact_div (x y z:int)
    requires { y > 0 }
    ensures { div (x + y * z) y = (div x y) + z }
  =
    assert { div (x + y * z) y = (div x y) + z
             by x + y * z = y * (div (x + y * z) y) + mod (x + y * z) y
             so mod (x + y * z) y = mod (y * z + x) y = mod x y
             so x + y * z = y * (div (x + y * z) y) + mod x y
             so x = y * div x y + mod x y
             so x + y * z = y * div x y + mod x y + y * z
             so y * (div (x + y * z) y) + mod x y
                = y * div x y + mod x y + y * z
             so y * (div (x + y * z) y) = y * div x y + y * z
                                        = y * ((div x y) + z)
             so y <> 0
             so div (x + y * z) y = div x y + z }

  let invert_limb (d:limb) : limb
    requires { d >= div radix 2 }
    ensures { reciprocal result d }
  =
    let v = div2by1 Limb.uint64_max
                    (Limb.uint64_max - d)
                    d in
    fact_div (radix * radix - 1) (l2i d) (- radix);
    assert { v = (div (radix*radix - 1) (d)) - radix
             by
             radix - 1 + radix * (radix - 1 - d)
             = radix - 1 + radix * (radix - 1) - radix * d
             = radix - 1 + radix * radix - radix - radix * d
             = radix * radix - 1 - radix * d
             so
             radix - 1 + radix * (radix - 1 - d)
             = radix * radix - 1 - radix * d
             so
             v
             = div ((radix - 1) + radix * (radix - 1 - d)) (d)
             = div (radix * radix - 1 - radix * d) (d)
             = div (radix * radix - 1) (d) - radix
           };
   v

  (** Divide a two-word integer by a one-word integer given the
  reciprocal of the divisor. *)
  let div2by1_inv (uh ul d v:limb) : (limb,limb)
    requires { d >= div radix 2 }
    requires { uh < d }
    requires { reciprocal v d }
    returns { q, r -> l2i q * d + l2i r = ul + radix * uh }
    returns { _q, r -> 0 <= l2i r < d }
  =
    let ghost k = radix * radix - (radix + l2i v) * l2i d in
    let ghost u = l2i ul + radix * l2i uh in
    assert { 1 <= k <= d };
    let l,h = mul_double v uh in
    let sl,c = add_with_carry l ul 0 in
    let (sh,ghost c') = add_with_carry uh h c in  (* <c',sh,sl> = <uh, ul> + <h,l> *)
    assert { sl + radix * sh + radix * radix * c'
             = l + radix * h + ul + radix * uh };
    assert { c' = 0
             by
             uh < d
             so v * uh <= v * d
             so k = radix * radix - (radix + v) * d
                  = radix * radix - radix * d - v * d
             so v * d = radix * radix - radix * d - k
                              = radix * (radix - d) - k
             so k > 0
             so v * d < radix * (radix - d)
             so v * uh < radix * (radix - d)
             so l + radix * h = v * uh
             so l + radix * h < radix * (radix - d)
             so uh <= d - 1
             so radix * uh <= radix * (d - 1) = radix * d - radix
             so l + radix * h + radix * uh
                < radix * (radix - d) + radix * uh
                <= radix * (radix - d) + radix * d - radix
                <= radix * (radix - d + d) - radix = radix * radix - radix
             so ul < radix
             so l + radix * h + ul + radix * uh
                = l + radix * h + radix * uh + ul
                < radix * radix - radix + ul
                < radix * radix - radix + radix = radix * radix
             so sl + radix * sh + radix * radix * c'
                = l + radix * h + ul + radix * uh
                < radix * radix
             so radix * radix * c' <= sl + radix * sh + radix * radix * c'
             so radix * radix * c' < radix * radix
     };
    assert { sl + radix * sh = l + radix * h + ul + radix * uh
             = v * uh + ul + radix * uh
             = ul + (radix + v) * uh };
    let qh = ref (sh:limb) in
    let ql = ref sl in
    let ghost q0 = l2i !ql in
    let ghost cq = l2i sh + 1 in (*candidate quotient*)
    let ghost cr = l2i ul - cq * l2i d + radix * l2i uh in (*candidate remainder*)
    assert { cq * d + cr = u};
    qh := add_mod !qh 1;
    assert { !qh = mod cq radix };
    let p = mul_mod !qh d in
    let r = ref (sub_mod ul p) in
    let ghost r' = !r in
    assert { r' = mod cr radix
             by
             let a = (- div (!qh * d) radix) in
             r' = !r
             = mod (ul - p) radix
             = mod (ul - mod (!qh * d) radix) radix
             = mod (radix * a
                   + ul - mod (!qh * d) radix) radix
             = mod (ul - mod (!qh * d) radix
                           - radix * div (!qh * d) radix) radix
             = mod (ul - !qh * d) radix
             = mod (ul - mod cq radix * d) radix
             = mod (radix * (- (div cq radix)) * d + ul - mod cq radix * d) radix
             = mod (ul - (mod cq radix + radix * div cq radix) * d) radix
             = mod (ul - cq * d) radix
             = mod (radix * uh + ul - cq * d) radix
             = mod (ul - cq * d + radix * uh) radix
             = mod cr radix };
    assert { radix * cr = uh * k + ul * (radix - d) + q0 * d - radix * d };
    prod_compat_strict_r (l2i ul) radix (radix - l2i d);
    prod_compat_strict_r (l2i d) radix (radix - q0);
    assert { (* Theorem 2 of Möller&Granlund 2010 *)
             (MM.max (radix - d) (q0 + 1)) - radix <= cr < MM.max (radix - d) q0
             by radix * cr = uh * k + ul * (radix - d) + q0 * d - radix * d
             so (uh * k + ul * (radix - d) >= 0
                by uh >= 0 /\ k >= 0 /\ ul >= 0 /\ radix - d >= 0)
             so radix * cr >= q0 * d - radix * d
             so radix * cr >= - radix * d
             so cr >= - d
             so radix * cr >= q0 * d - radix * d = (q0 - radix) * d
             so radix > d
             so radix - q0 > 0
             so d * (radix-q0) < radix * (radix - q0)
             so (q0 - radix) * d > (q0 - radix) * radix
             so radix * cr > (q0 - radix) * radix
             so cr > q0 - radix
             so (let m = MM.max (radix - d) (q0 +1) in
                cr >= m - radix
                by (cr + radix >= - d + radix
                   /\ (cr + radix > q0 so cr + radix >= q0 + 1))
                   so cr + radix >= m)
             so 0 < k <= d so 0 <= uh < d
             so k * uh < k * d <= d * d
             so radix * cr < d * d + ul * (radix - d) + q0 * d - radix * d
             so ul * (radix - d) < radix * (radix - d)
             so radix * cr < d * d + radix * (radix - d) + q0 * d - radix * d
             so (radix * cr < (radix - d) * (radix - d) + q0 * d
                by
                d * d + radix * (radix - d) + q0 * d - radix * d
                        = radix * (radix - d) + d * d - radix * d + q0 * d
                        = radix * (radix - d) + (d - radix) * d +  q0 * d
                        = radix * (radix - d) - d * (radix - d) + q0 * d
                        = (radix - d) * (radix - d) + q0 * d)
             so let m = MM.max (radix - d) q0 in
                radix - d <= m
             so (radix - d) * (radix - d) <= m* (radix - d)
             so (q0 * d <= m * d by 0 <= q0 <= m /\ 0 < d)
             so radix * cr < (radix - d) * (radix - d) + q0 * d
                           <= m* (radix - d) + q0 * d
                           <= m* (radix - d) + m * d
                           = m * radix
             so cr < m
                           };
    assert { cr >= 0 -> r' = cr };
    assert { cr < 0 ->
           ( r' = cr + radix
             by cr >= MM.max (radix - d) (q0 + 1) - radix
             so cr >= - d
             so cr + radix >= radix - d >= 0
             so 0 <= cr + radix < radix
             so mod (cr + radix) radix = mod cr radix
             so r' = mod (cr + radix) radix ) };
    assert { cr < 0 ->
                ( !r > !ql
                by MM.max (radix - d) (q0 + 1) >= q0 + 1 > q0
                so cr >= (MM.max (radix - d) (q0 +1)) - radix > q0 - radix
                so r' = cr + radix > q0 - radix + radix = q0 ) };
    assert { 1 <= cq <= radix };
    assert { (!qh = cq \/ (!qh = 0 /\ cq = radix)
              by (1 <= cq < radix -> !qh = mod cq radix = cq)
              so (cq = radix -> !qh = 0) ) };
    assert { cq = radix ->
             (cr < 0
                by cq * d + cr = u
                so uh <= d - 1
                so 1 + uh <= d
                so ul < radix
                so u = ul + radix * uh
                     < radix + radix * uh
                     = radix * (1 + uh)
                     <= radix * d
                so u < radix * d
                so radix * d + cr = u
                so radix * d + cr < radix * d
                so cr < 0) };
    assert { 1 <= cq < radix -> !qh = cq /\ !qh * d + cr = u };
    if !r > !ql
    then
    begin
      qh := sub_mod !qh 1;
      r := add_mod !r d;
      assert { cr >= 0 ->
                  (!r = cr + d
                  by r' = cr
                  so r' < MM.max (radix - d) q0
                  so r' > q0
                  so 0 <= r' < radix - d
                  so d <= r' + d < radix
                  so !r = mod (r' + d) radix = r' + d) };
      assert { cr >= 0 ->
                  ( !r >= d
                  by r' = cr >= 0
                  so !r = r' + d >= d ) };
      assert { cr < 0 ->
                  ( !r = r' + d - radix
                  by r' = cr + radix < radix
                  so cr >= MM.max (radix - d) (q0 + 1) - radix
                        >= radix - d - radix = - d
                  so r' = cr + radix >= radix - d
                  so !r = mod (r' + d) radix
                  so radix + radix >= r' + d >= radix
                  so !r = mod (r' + d) radix = r' + d - radix ) };
      assert { cr < 0 ->
                  ( 0 <= !r < d
                  by r' = cr + radix < radix
                  so !r = mod (r' + d) radix = r' + d - radix
                  so !r >= 0
                  so !r = r' + d - radix < d ) };
      assert { cq = radix ->
                ( !qh * d + !r = u
                by cq * d + cr = u
                so cr < 0
                so r' = cr + radix
                so u = radix * d + cr
                     = (radix - 1) * d + d + cr
                     = (radix - 1) * d + d + r' - radix
                so r' = cr + radix >= MM.max (radix - d) (q0 + 1)
                   >= radix - d
                so radix + radix >= d + r' >= radix
                so !r = mod (d + r') radix = d + r' - radix
                so (radix - 1) * d + !r = u
                so !qh = mod ((mod cq radix) - 1) radix
                           = mod (-1) radix
                           = radix - 1
                so !qh * d + !r = u
               ) };
      assert { !r = cr + d by [@case_split] cr >= 0 \/ cr < 0 };
      assert { 1 <= cq <= radix ->
               ( !qh * d + !r = u
               by cq * d + cr = u
               so !qh = cq - 1
               so !qh * d + cr + d = u
               so !r = cr + d ) };
    end
    else
    begin
       assert { cr >= 0 };
       assert { 1 <= cq < radix };
    end;
    assert { !qh * d + !r = ul + radix * uh
            by [@case_split] cq = radix \/ 1 <= cq < radix };
    if !r >= d
    then begin
      assert { cr >= 0 };
      assert { !qh < radix - 1
               by
               !qh * d = ul + radix * uh  - !r
               so uh <= d - 1
               so ul + radix * uh - !r
                  <= ul + radix * (d - 1) - !r
                  = ul + radix * d - radix - !r
                  = (ul - radix)  + radix * d - !r
                  <  radix * d - !r
                  <= radix * d - d
                  = (radix - 1) * d
               so !qh * d < (radix - 1) * d
               so d > 0
               so !qh < radix - 1 };
      qh := !qh + 1;
      r := !r - d;
      assert { 0 <= !r < d };
      assert { !qh * d + !r = ul + radix * uh };
    end;
    assert { 0 <= !r < d };
    assert { !qh * d + !r = ul + radix * uh };
    (!qh,!r)

(** `wmpn_divrem_1 q x sz y` divides `(x, sz)` by `y`, writes the quotient
    in `(q, sz)` and returns the remainder. Corresponds to
    `mpn_divrem_1`. *)
(* TODO develop further decimal points (qxn) *)
let wmpn_divrem_1 (q x:t) (sz:int32) (y:limb) : limb
    requires { valid x sz }
    requires { valid q sz }
    requires { writable q }
    requires { 0 < sz }
    requires { 0 < y }
    ensures { value x sz
              = value q sz * y + result }
    ensures { result < y }
  =
    let msb = sz - 1 in
    let lx = ref 0 in
    let i = ref msb in
    let r = ref 0 in
    (*normalize divisor*)
    let clz = count_leading_zeros y in
    if (clz > 0)
    then begin
      let ghost mult = power 2 (p2i clz) in
      let ry = lsl y (Limb.of_int32 clz) in
      assert { ry = mult * y };
      let ghost tlum = power 2 (Limb.length - p2i clz) in
      assert { tlum * mult = radix };
      let v = invert_limb ry in
      while (!i >= 0) do
        variant   { !i }
        invariant { -1 <= !i <= msb }
        invariant { !r < ry }
        invariant { mult * value_sub (pelts x) (x.offset + !i + 1) (x.offset + sz)
                    = value_sub (pelts q) (q.offset + !i + 1)
                                          (q.offset + sz)
                      * ry
                      + !r }
        invariant { !r <= radix - mult }
        invariant { mod (!r) mult = 0 }
        assert { !i >= 0 };
        label StartLoop in
        lx := C.get_ofs x !i;
        (*TODO lshift in place would simplify things*)
        let l,h = lsld_ext !lx (Limb.of_int32 clz) in
        mod_mult mult (l2i y) 0;
        assert { !r + h < ry
                 by
                 let drm = div (!r) mult in
                 let dym = div (ry) mult in
                 mod (!r) mult = 0
                 so !r = mult * drm
                 so mod (ry) mult
                    = mod (mult * (y) + 0) mult
                    = mod 0 mult
                    = 0
                 so ry = mult * dym
                 so !r < ry
                 so 0 < ry - !r
                      = mult * dym - mult * drm
                      = mult * (dym - drm)
                 so mult > 0
                 so dym - drm > 0
                 so dym >= drm + 1
                 so h < mult
                 so !r + h = mult * drm + h
                    < mult * drm + mult
                    = mult * (drm + 1)
                    <= mult * dym = l2i ry };
        assert { !r + h < radix by
                 !r + h < ry < radix };
        let (qu,rem) = div2by1_inv (!r + h) l ry v in
        mod_mult mult (l2i y * l2i qu) (l2i rem);
        mod_mult mult (tlum * (l2i !r + l2i h)) (l2i l);
        assert { mod (rem) mult = 0
                 by
                 ry * qu + rem
                 = (radix * (!r + h) + l)
                 so
                 mult * y * qu + rem
                 = (mult * tlum * (!r + h) + l)
                 so mod (mult * y * qu + rem) mult
                    = mod (mult * tlum * (!r + h) + l) mult
                 so mult > 0
                 so mod (mult * (y * qu) + rem) mult
                    = mod (rem) mult
                 so mod (mult * tlum * (!r + h) + l) mult
                    = mod (l) mult
                    = 0
                 };
        let ghost mer = div (l2i rem) mult in
        assert { rem <= radix - mult
                 by
                 mod (rem) mult = 0
                 so mult * mer = l2i rem < radix = mult * tlum
                 so mult > 0
                 so 0 < mult * tlum - mult * mer = mult * (tlum - mer)
                 so tlum - mer > 0
                 so mer < tlum
                 so rem = mult * mer <= mult * (tlum - 1) = radix - mult
                 };
        r:=rem;
        assert { qu * ry + !r = l + radix * h + radix * (!r at StartLoop) };
               (* coerced div2by1 postcondition *)
        value_sub_update_no_change (pelts q) (q.offset + p2i !i)
                                   (q.offset + 1 + p2i !i)
                                   (q.offset + p2i sz) qu;
        C.set_ofs q !i qu;
        assert { mult * value_sub (pelts x) (x.offset + !i + 1) (x.offset + sz)
                    = value_sub (pelts q) (q.offset + !i + 1)
                                          (q.offset + sz)
                      * ry
                      + (!r at StartLoop) }; (* previous invariant is still true *)
        value_sub_head (pelts x) (x.offset + int32'int !i) (x.offset + p2i sz);
        value_sub_head (pelts q) (q.offset + int32'int !i) (q.offset + p2i sz);
        assert { l + radix * h = mult * !lx }; (*lsld_ext postcondition *)
        assert { mult * value_sub (pelts x) (x.offset + !i)
                                            (x.offset + sz)
                 = mult * !lx
                   + radix * (mult * value_sub (pelts x) (x.offset + !i + 1)
                                              (x.offset + sz))
                 by (pelts x)[x.offset + !i] = !lx
                 so value_sub (pelts x) (x.offset + !i) (x.offset + sz)
                    = !lx + radix * value_sub (pelts x) (x.offset + !i + 1)
                                              (x.offset + sz) }; (*nonlinear*)
        assert { value_sub (pelts q) (q.offset + !i) (q.offset + sz) * ry
                 = qu * ry
                   + radix
                      * (value_sub (pelts q) (q.offset + !i + 1) (q.offset + sz)
                        * ry)
                 by (pelts q)[q.offset + !i] = qu
                 so value_sub (pelts q) (q.offset + !i) (q.offset + sz)
                    = qu + radix * value_sub (pelts q) (q.offset + !i + 1)
                                              (q.offset + sz) }; (*nonlinear*)
        assert { mult * value_sub (pelts x) (x.offset + !i)
                                            (x.offset + sz)
                = value_sub (pelts q) (q.offset + !i) (q.offset + sz) * ry
                  + !r };
        i := !i - 1;
      done;
      let ghost res = lsr !r (Limb.of_int32 clz) in
      assert { value x sz = value q sz * y + res
               by !r = res * mult
               so mult * value x sz
                  = value q sz * ry + !r
                  = value q sz * y * mult + !r
                  = value q sz * y * mult + res * mult
                  = (value q sz * y + res) * mult };
      lsr !r (Limb.of_int32 clz) end
    else begin
      let v = invert_limb y in
      while (!i >= 0) do
        variant   { !i }
        invariant { -1 <= !i <= msb }
        invariant { !r < y }
        invariant { value_sub (pelts x) (x.offset + !i + 1) (x.offset + sz)
                    = value_sub (pelts q) (q.offset + !i + 1)
                                          (q.offset + sz)
                      * y
                      + !r }
        assert { !i >= 0 };
        label StartLoop in
        let ghost k = p2i !i in
        lx := C.get_ofs x !i;
        let (qu, rem) = div2by1_inv !r !lx y v in
        r := rem;
        value_sub_update_no_change (pelts q) (q.offset + p2i !i)
                                   (q.offset + 1 + p2i !i)
                                   (q.offset + p2i sz) qu;
        C.set_ofs q !i qu;
        i := !i - 1;
        value_sub_head (pelts x) (x.offset + k) (x.offset + p2i sz);
        value_sub_head (pelts q) (q.offset + k) (q.offset + p2i sz);
        assert { value_sub (pelts x) (x.offset + !i + 1) (x.offset + sz)
                    = value_sub (pelts q) (q.offset + !i + 1)
                                          (q.offset + sz)
                      * y
                      + !r
                 by (pelts q)[q.offset + k] = qu
                 so (pelts x)[x.offset + k] = !lx
                 so value_sub (pelts x) (x.offset + !i + 1) (x.offset + sz)
                    = value_sub (pelts x) (x.offset + k) (x.offset + sz)
                    = !lx + radix * value_sub (pelts x) (x.offset + k + 1)
                                                        (x.offset + sz)
                    = !lx + radix * (value_sub (pelts q) (q.offset + k + 1)
                                                         (q.offset + sz)
                                     * y + (!r at StartLoop))
                    = !lx + radix * (!r at StartLoop)
                          + radix * (value_sub (pelts q) (q.offset + k + 1)
                                                         (q.offset + sz)
                                     * y)
                    = qu * y + !r
                          + radix * (value_sub (pelts q) (q.offset + k + 1)
                                                         (q.offset + sz)
                                     * y)
                    = (qu + radix * value_sub (pelts q) (q.offset + k + 1)
                                                        (q.offset + sz))
                      * y
                      + !r
                    = value_sub (pelts q) (q.offset + !i + 1)
                                          (q.offset + sz)
                      * y
                      + !r };
      done;
      !r
    end


  predicate reciprocal_3by2 (v dh dl:limb) =
    v = div (radix*radix*radix -1) (dl + radix * dh) - radix

  let div3by2_inv (uh um ul dh dl v: limb) : (limb,limb,limb)
    requires { dh >= div radix 2 }
    requires { reciprocal_3by2 v dh dl }
    requires { um + radix * uh < dl + radix * dh }
    returns { q, rl, rh -> uint64'int q * dl + radix * q * dh
                           + uint64'int rl + radix * uint64'int rh
                  = ul + radix * um + radix * radix * uh }
    returns { _q, rl, rh -> 0 <= uint64'int rl + radix * uint64'int rh < dl + radix * dh }
  = [@vc:do_not_keep_trace] (* traceability info breaks the proof *)
    let ghost d = l2i dl + radix * l2i dh in
    let ghost u = l2i ul + radix * (l2i um + radix * l2i uh) in
    let q1 = ref 0 in
    let r0 = ref 0 in
    let r1 = ref 0 in
    let l,h = mul_double v uh in
    let sl, c = add_with_carry um l 0 in
    let sh, ghost c' = add_with_carry uh h c in
    assert { sl + radix * sh + radix * radix * c'
             = um + radix * uh + v * uh };
    assert { c' = 0
             by
             um + radix * uh < d
             so radix * uh < d
             so radix * radix * radix - 1
                = d * (div (radix * radix * radix - 1) d)
                  + mod (radix * radix * radix - 1) d
                >= d * (div (radix * radix * radix - 1) d)
             so radix * (um + radix * uh + v * uh)
                < radix * (d + v * uh)
                = radix * d + v * radix * uh
                <= radix * d + v * d
                = (div (radix * radix * radix - 1) d) * d
                <= radix * radix * radix - 1
                < radix * radix * radix
             so um + radix * uh + v * uh < radix * radix
             so sl + radix * sh + radix * radix * c' < radix * radix
             so radix * radix * c' < radix * radix
     };
    q1 := sh;
    let ghost q0 = l2i sl in
    let ghost cq = l2i !q1 + 1 in (*candidate quotient*)
    q1 := add_mod !q1 1;
    assert { !q1 = mod cq radix };
    let p = mul_mod dh sh in
    r1 := sub_mod um p;
    label CQuot in
    let ghost a = div (l2i um - l2i dh * l2i sh) radix in
    let tl, th = mul_double sh dl in
    let il, b = sub_with_borrow ul tl 0 in
    let (ih, ghost b') = sub_with_borrow !r1 th b in
    assert { il + radix * ih - radix * radix * b'
             = ul + radix * !r1 - sh * dl };
    let bl,b2 = sub_with_borrow il dl 0 in
    let bh, ghost b2' = sub_with_borrow ih dh b2 in
    assert { bl + radix * bh - radix * radix * b2'
             = il + radix * ih - dl - radix * dh };
    mod_mult (radix * radix) (uint64'int b')
             (uint64'int ul + radix * uint64'int !r1
                - uint64'int sh * uint64'int dl - uint64'int dl
                - radix * uint64'int dh);
    assert { bl + radix * bh
             = mod (ul + radix * !r1
                - sh * dl- dl
                - radix * dh) (radix * radix)
             by
             mod (il + radix * ih - dl - radix * dh) (radix * radix)
             = mod (radix * radix * (-b2') + bl + radix * bh) (radix * radix)
             = mod (bl + radix * bh) (radix * radix)
             = bl + radix * bh
             so
             bl + radix * bh
             = mod (il + radix * ih
                - dl - radix * dh) (radix * radix)
             so il + radix * ih
                = radix * radix * b' + ul + radix * !r1
                  - sh * dl
             so mod (il + radix * ih
                - dl - radix * dh) (radix * radix)
                = mod (radix * radix * b' + ul + radix * !r1
                      - sh * dl - dl - radix * dh)
                  (radix * radix)
                = mod (ul + radix * !r1
                - sh * dl - dl
                - radix * dh) (radix * radix) };
    r1 := bh;
    r0 := bl;
    let ghost r' = l2i !r0 + radix * l2i !r1 in
    let ghost cr = u - d * cq in
    assert {  r' = mod cr(radix * radix)
              by
              (!r1 at CQuot = mod (um - dh * sh) radix
                by let a' = div (dh * sh) radix in
                   dh * sh = p + radix * a'
                so !r1 at CQuot = mod (um - p) radix
                   = mod (radix * a' + um - dh * sh) radix
                   = mod (um - dh * sh) radix )
              so um - dh * sh = a * radix + !r1 at CQuot
              so !r0 + radix * !r1
                 = mod (ul + radix * (!r1 at CQuot)
                  - sh * dl - dl
                  - radix * dh) (radix * radix)
              so ul + radix * (!r1 at CQuot)
                   - sh * dl - dl - radix * dh
                 = ul + radix * (um - dh * sh - a * radix)
                   - sh * dl - dl - radix * dh
                 = ul + radix * um - radix * dh * sh
                   - radix * radix * a - sh * dl - dl
                   - radix * dh
                 = ul + radix * um - radix * dh * (sh + 1)
                   - radix * radix * a - sh * dl - dl
                 = ul + radix * um - radix * dh * (sh + 1)
                   - radix * radix * a - dl * (sh + 1)
                 = ul + radix * um
                   - (dl + radix * dh) * (sh + 1)
                   - radix * radix * a
                 = ul + radix * um - d * cq - radix * radix * a
                 = u - radix * radix * uh - d * cq - radix * radix * a
                 = cr + radix * radix * (- a - uh)
              so (*let y = - a - uh in*)
                 mod (ul + radix * (!r1 at CQuot)
                  - sh * dl - dl
                  - radix * dh) (radix * radix)
                 = mod (radix * radix * (-a - uh) + cr)
                       (radix * radix)
                 = mod cr (radix*radix)
               };
    let ghost m = MM.max (radix * radix - d) (q0 * radix) in
    assert { (* Theorem 3 of Moller&Granlund 2010 *)
             m - radix * radix <= cr < m
            by
            let k = radix * radix * radix - (radix + v) * d in
            reciprocal_3by2 v dh dl
            so let m3 = radix * radix * radix - 1 in
               (radix + v) * d = d * div m3 d = m3 - mod m3 d
            so (k = 1 + mod m3 d
                by k = radix * radix * radix - (radix + v) * d
                     = m3 + 1 - (radix + v) * d
                     = m3 + 1 - m3 + mod m3 d
                     = 1 + mod m3 d)
            so 1 <=  k <= d
            so q0 + radix * sh = (radix + v) * uh + um
            so cq = sh + 1
            so radix * cq = radix * sh + radix
               = (radix + v) * uh + um - q0 + radix
            so (radix * cr = k * uh + (radix * radix - d) * um
                 + radix * ul + d * q0 - d * radix
               by radix * cr = radix * (u - cq * d)
               = radix * u
                 - ((radix + v) * uh + um - q0 + radix) * d
               = radix * u - d * (radix + v) * uh
                 - d * um + d * q0 - d * radix
               = radix * u - (radix * radix * radix - k) * uh
                 - d * um + d * q0 - d * radix
               = (radix * radix * radix * uh + radix * radix * um
                 + radix * ul) - (radix * radix * radix - k) * uh
                 - d * um + d * q0 - d * radix
               = k * uh + radix * radix * um + radix * ul
                 - d * um + d * q0 - d * radix
               = k * uh + (radix * radix - d) * um + radix * ul
                 + d * q0 - d * radix )
            so (cr >= m - radix * radix
               by (
                  k >= 0 so radix * radix - d >= 0
               so uh >= 0 so um >= 0 so ul >= 0
               so  k * uh + (radix * radix - d) * um + radix * ul
                   >= 0
               so radix * cr >= d * q0 - d * radix
               so q0 >= 0 so d >= 0
               so d * q0 >= 0
               so radix * cr >= - d * radix
               so cr >= -d = radix * radix - d - radix * radix
               so radix * cr >= d * (q0 - radix)
               so (
                    (radix - q0) * d < (radix - q0) * radix * radix
                    by let rq = radix - q0 in let r2 = radix * radix in
                    rq > 0 /\ d < r2
                    so rq * d < rq * r2
                  )
               so d * (q0 - radix) > radix * radix * (q0 - radix)
               so radix * cr > radix * radix * (q0 - radix)
               so cr > radix * (q0 - radix) = radix * q0 - radix * radix
                  ))
            so cr < m
            by (
                let bbd = radix * radix - d in
                bbd > 0 /\ bbd <= m /\ q0 * radix <= m
                so (bbd * bbd <= bbd * m
                    by [@case_split]
                    (bbd = m \/ (bbd < m so bbd * bbd < bbd * m)))
                so (d*(radix * q0) <= d * m
                    by [@case_split]
                    (radix * q0 = m \/ (radix * q0 < m so d > 0 so d * (radix * q0) < d * m)))
                so if uh <= dh - 1
                then
                  let dm = dh - 1 in
                  uh <= dm
                  so
                  k * uh <= k * dm
                  so (k * dm <= d * dm
                     by k <= d /\ 0 <= dm
                     so [@case_split] (k = d \/ dm = 0 \/
                                     (k < d /\ dm > 0 so k * dm < d * dm)))
                  so k * uh <= d * dm
                  so
                  bbd * um <= bbd * (radix - 1)
                  so
                  radix * cr
                  = k * uh + (radix * radix - d) * um
                    + radix * ul + d * q0 - radix * d
                  <= d * dm + bbd * um
                    + radix * ul + d * q0 - radix * d
                  <= d * dm + bbd * (radix - 1)
                    + radix * ul + d * q0 - radix * d
                  < d * dm + bbd * (radix - 1)
                    + radix * radix + d * q0 - radix * d
                  so radix * radix * cr
                  < radix * (d * dm + bbd * (radix - 1)
                    + radix * radix + d * q0 - radix * d)
                  = d * radix * (dh - 1) + bbd * radix * (radix - 1)
                    + radix * radix * radix + radix * d * q0 - radix * radix * d
                  = d * radix * dh - d * radix + bbd * radix * (radix - 1)
                    + radix * radix * radix + radix * d * q0 - radix * radix * d
                  = d * (d - dl) - d * radix + bbd * radix * (radix - 1)
                    + radix * radix * radix + radix * d * q0 - radix * radix * d
                  = d * d - d * radix + bbd * radix * (radix - 1)
                    + radix * radix * radix + radix * d * q0 - radix * radix * d - d * dl
                  so (d * dl >= 0 by d >= 0 /\ dl >= 0)
                  so radix * radix * cr
                  < d * d - d * radix + bbd * radix * (radix - 1)
                    + radix * radix * radix + radix * d * q0 - radix * radix * d - d * dl
                  <= d * d - d * radix + bbd * radix * (radix - 1)
                    + radix * radix * radix + radix * d * q0 - radix * radix * d
                  = d * d - d * radix + bbd * (radix * radix - radix)
                    + radix * radix * radix + radix * d * q0 - radix * radix * d
                  = d * d - d * radix + bbd * radix * radix - (radix * radix - d) * radix
                    + radix * radix * radix + radix * d * q0 - radix * radix * d
                  = d * d - d * radix + bbd * radix * radix
                    + radix * d - radix * radix * radix
                    + radix * radix * radix + radix * d * q0 - radix * radix * d
                  = d * d + bbd * radix * radix - radix * radix * d + radix * d * q0
                  = bbd * radix * radix - d * (radix * radix - d) + radix * d * q0
                  = bbd * radix * radix - d * bbd + radix * d * q0
                  = bbd * bbd + d * (radix * q0)
                  <= bbd * m + d * (radix * q0)
                  <= bbd * m + d * m
                  = radix * radix * m
                  so cr < m
                 else
                  uh = dh
                  so
                  (um <= dl - 1
                  by um + radix * uh < dl + radix * dh)
                  so (radix * radix - d) * um <= (radix * radix - d) * (dl - 1)
                  so
                  ( radix * radix * cr
                    < radix * radix * m
                    - (radix - dl) * (radix * radix * radix - d * (1+ radix))
                 by radix * cr
                  = k * dh + (radix * radix - d) * um
                    + radix * ul + d * q0 - radix * d
                  <= d * dh + (radix * radix - d) * um
                    + radix * ul + d * q0 - radix * d
                  <= d * dh + (radix * radix - d) * (dl - 1)
                    + radix * ul + d * q0 - radix * d
                  < d * dh + (radix * radix - d) * (dl - 1)
                    + radix * radix + d * q0 - radix * d
                  so radix * radix * cr
                  < radix * (d * dh + (radix * radix - d) * (dl - 1)
                    + radix * radix + d * q0 - radix * d)
                  = d * radix * dh
                    + (radix * radix - d) * (dl - 1) * radix
                    + radix * radix * radix + d * q0 * radix - radix * radix * d
                  = d * (d - dl)
                    + (radix * radix - d) * (radix * dl - radix)
                    + radix * radix * radix + d * q0 * radix - radix * radix * d
                  = d * d - d * dl + radix * radix * radix * dl
                    - d * radix * dl + d * radix - radix * radix * radix
                    + radix * radix * radix + d * q0 * radix - radix * radix * d
                  = d * d - d * dl + radix * radix * radix * dl
                    - d * radix * dl + d * radix + d * q0 * radix
                    - radix * radix * d
                  = d * d - radix * radix * d + d * radix + d * q0 * radix
                    + dl * (radix * radix * radix - d - d * radix)
                  = d * (d - radix * radix) + d * radix + d * q0 * radix
                    + dl * (radix * radix * radix - d - d * radix)
                  = bbd * (-d) + d * radix + d * q0 * radix
                    + dl * (radix * radix * radix - d - d * radix)
                  = bbd * (bbd - radix * radix) + d * radix + d * q0 * radix
                    + dl * (radix * radix * radix - d - d * radix)
                  = bbd * bbd + d * q0 * radix
                    - bbd * radix * radix + d * radix
                    + dl * (radix * radix * radix - d * (1 + radix))
                  = bbd * bbd + d * q0 * radix
                    - (radix * radix - d) * radix * radix + d * radix
                    + dl * (radix * radix * radix - d * (1 + radix))
                  = bbd * bbd + d * q0 * radix
                    - radix * ((radix * radix - d) * radix - d)
                    + dl * (radix * radix * radix - d * (1 + radix))
                  = bbd * bbd + d * q0 * radix
                    - radix *  (radix * radix * radix - d * radix - d)
                    + dl * (radix * radix * radix - d * (1 + radix))
                  = bbd * bbd + d * q0 * radix
                    - radix * (radix * radix * radix - d * (1+ radix))
                    + dl * (radix * radix * radix - d * (1 + radix))
                  = bbd * bbd + d * q0 * radix
                    - (radix - dl) * (radix * radix * radix - d * (1+ radix))
                  <= bbd * m + d * q0 * radix
                    - (radix - dl) * (radix * radix * radix - d * (1+ radix))
                  <= bbd * m + d * m
                    - (radix - dl) * (radix * radix * radix - d * (1+ radix))
                  = (bbd + d) * m
                    - (radix - dl) * (radix * radix * radix - d * (1+ radix))
                  = radix * radix * m
                    - (radix - dl) * (radix * radix * radix - d * (1+ radix))
                  )
                  so
                    (cr < m by
                    if d <= radix * (radix - 1)
                    then (radix + 1) * d <= radix * (radix - 1) * (radix + 1)
                         = radix * (radix * radix - 1)
                         = radix * radix * radix - radix
                         < radix * radix * radix
                         so (radix * radix * radix - d * (1+ radix)) > 0
                         so radix - dl > 0
                         so (radix - dl) * (radix * radix * radix
                                               - d * (1+ radix))
                            > 0
                         so
                         radix * radix * cr
                         < radix * radix * m
                           - (radix - dl) * (radix * radix * radix
                           - d * (1+ radix))
                         < radix * radix * m
                         so radix * radix * cr < radix * radix * m
                    else
                        dl + radix * dh = d > radix * (radix - 1)
                        so dl < radix
                        so dl + radix * dh < radix * (1 + dh)
                        so radix - 1 < 1 + dh
                        so dh > radix - 2
                        so dh = radix - 1
                        so uh = dh
                        so d >= radix * (radix - 1) +1
                        so d * (radix + 1)
                           >= (radix * (radix - 1) + 1) * (radix +1)
                            = radix * (radix * radix - 1) + radix + 1
                            = radix * radix * radix - radix + radix + 1
                            = radix * radix * radix + 1
                        so
                        (d * div (radix * radix * radix - 1) d
                             <= d * (radix + 1)
                          by d * div (radix * radix * radix - 1) d
                             <= radix * radix * radix - 1
                             < radix * radix * radix + 1
                             <= d * (radix + 1))
                        so (let a = div (radix * radix * radix - 1) d in
                            a < radix + 1
                            by d > 0
                            so (forall x y z. x * z < y * z /\ z > 0 -> x < y)
                            so (forall x y. x * d < y * d -> x < y)
                            so let r = radix + 1 in
                               a * d < r * d
                               so a < r)
                        so v = div (radix * radix * radix - 1) d - radix
                                 < radix + 1 - radix = 1
                        so v = 0
                        so sh = uh = radix - 1
                        so cq = sh + 1 = radix
                        so cr = u - cq * d
                              = u - radix * d
                              = ul + radix * (um + radix * dh)
                                       - radix * (dl + radix * dh)
                              = ul + radix * (um - dl)
                        so um <= dl - 1
                        so 1 + um - dl <= 0
                        so ul < radix
                        so cr = ul + radix * (um - dl)
                              < radix + radix * (um - dl)
                              = radix * (1 + um - dl) <= 0
                        so cr < 0 <= m
                   )
                )
            };
    assert { cr >= 0 -> r' = cr };
    assert { cr < 0 -> r' = radix * radix + cr
             by
             m >= radix * radix - d
             so cr >= m - radix * radix >= -d
             so cr + radix * radix >= radix * radix - d >= 0
             so 0 <= cr + radix * radix < radix * radix
             so mod (radix * radix + cr) (radix*radix) = mod cr (radix*radix)
             so r' = mod (radix * radix + cr) (radix*radix) };
    assert { cr < 0 -> !r1 >= sl
             by m >= radix * q0
             so cr >= m - radix * radix >= radix * q0 - radix * radix
             so r' = radix * radix + cr >= radix * q0
             so r' = radix * !r1 + !r0 >= radix * q0
             so !r0 < radix
             so r' < radix * !r1 + radix = radix * (!r1 + 1)
             so radix * q0 < radix * (!r1 + 1)
             so sl = q0 < !r1 + 1 };
    assert { 1 <= cq <= radix };
    assert { 1 <= cq < radix -> !q1 = cq so !q1 * d + cr = u };
    assert { cq = radix ->
             (cr < 0
                by cq * d + cr = u
                so um + radix * uh <= d - 1
                so radix * d + cr = ul
                                    + radix * (um + radix * uh)
                                  <= ul + radix * (d - 1)
                                  = ul - radix + radix * d
                                  < radix * d
             )
           };
    label PreCorrections in
    if !r1 >= sl
    then begin
      q1 := sub_mod !q1 1;
      assert { !q1 = cq - 1
               by
               if cq = radix
               then
                 (!q1 at PreCorrections)
                 = mod cq radix = mod radix radix= 0
                 so !q1 = mod (0 - 1) radix = radix - 1 = cq - 1
               else
                 0 <= cq - 1 < radix - 1
                 so (!q1 at PreCorrections) = cq
                 so !q1 = mod (cq - 1) radix = cq - 1
                 };
      let rl, c = add_with_carry !r0 dl 0 in
      let rh, ghost c' = add_with_carry !r1 dh c in
      assert { rl + radix * rh = mod (r' + d) (radix * radix)
               by radix * radix * c' + rl + radix * rh
                  = r' + d
               so mod (r' + d) (radix * radix)
                  = mod (radix * radix * c' + rl + radix * rh)
                      (radix * radix)
                  = mod (rl + radix * rh) (radix * radix)  };
      assert { rl + radix * rh = cr + d
               by
               if cr >= 0
               then r' = cr
                    so rl + radix * rh = mod (cr + d) (radix * radix)
                    so cr < MM.max (radix * radix - d) (q0*radix)
                    so (cr >= q0 * radix
                       by
                          r' = radix * !r1 + !r0
                             >= radix * !r1
                             >= radix * q0)
                    so cr < radix * radix - d
                    so cr + d < radix * radix
                    so (cr + d >= 0  by cr + d >= cr)
                    so mod (cr + d) (radix * radix) = cr + d
               else
                    r' = cr + radix * radix
                    so cr >= m - radix * radix
                    so r' >= m >= radix * radix - d
                    so r' + d >= radix * radix
                    so r' < radix * radix
                    so d < radix * radix
                    so r' + d < radix * radix + radix * radix
                    so mod (r' + d) (radix * radix)
                       = r' + d - radix * radix
                       = cr + d
             };
      r1 := rh;
      r0 := rl;
      assert { !q1 * d + !r0 + radix * !r1 = u
               by
               cq * d + cr = u
               so !q1 = cq - 1
               so !r0 + radix * !r1 = cr + d
               so !q1 * d + !r0 + radix * !r1
                  = (cq - 1) * d + cr + d
                  = cq * d - d + cr + d
                  = cq * d + cr };
    end
    else assert { !q1 * d + r' = u
                  by cr >= 0
                  so r' = cr
                  so 1 <= cq < radix
                  so !q1 * d + cr = u };
    assert { !q1 * d + !r0 + radix * !r1 = u };
    label PreRemAdjust in
    if [@extraction:unlikely] (!r1 > dh) || (!r1 = dh && !r0 >= dl)
    then begin
      let bl, b = sub_with_borrow !r0 dl 0 in
      let bh, ghost b'= sub_with_borrow !r1 dh b in
      assert { b' = 0 };
      assert { bl + radix * bh = !r0 + radix * !r1 - d };
      assert { !q1 < radix - 1
               by !q1 * d + !r0 + radix * !r1 = u
               so !r0 + radix * !r1 >= d
               so um + radix * uh <= d - 1
               so u = ul + radix * (um + radix * uh)
                    <= ul + radix * (d - 1)
                    < radix + radix * (d-1)
                    = radix * d
               so (!q1 * d < (radix - 1) * d
                  by
                  !q1 * d = u - (!r0 + radix * !r1)
                              <= u - d
                              < radix * d - d
                              = (radix - 1) * d )
               };
      q1 := add_mod !q1 1;
      assert { !q1 = (!q1 at PreRemAdjust) + 1 };
      r1 := bh;
      r0 := bl;
      assert { !q1 * d + !r0 + radix * !r1 = u
               by
               !q1 * d + !r0 + radix * !r1
               = ((!q1 at PreRemAdjust) + 1) * d
                 + (!r0 + radix * !r1 at PreRemAdjust) - d
               = (!q1 * d + !r0 + radix * !r1 at PreRemAdjust)
                };
    end;
    assert { 0 <= !r0 + radix * !r1 < d };
    (!q1,!r0,!r1)

  let lemma bounds_imply_rec3by2 (v dh dl:limb)
    requires { radix * radix * radix - (dl + radix * dh)
               <= (radix + v) * (dl + radix * dh)
               < radix * radix * radix }
    ensures { reciprocal_3by2 v dh dl }
  = ()

  let reciprocal_word_3by2 (dh dl:limb) : limb
    requires { dh >= div radix 2 }
    ensures { reciprocal_3by2 result dh dl }
  =
    let ghost d = l2i dl + radix * l2i dh in
    let v = ref (invert_limb dh) in
    assert { radix * radix - dh
             <= (radix + !v) * dh
             < radix * radix
             by
             radix + !v = div (radix * radix - 1) (dh) };
    let p = ref (mul_mod dh !v) in
    assert { (radix + !v) * dh
             = radix * (radix-1)
               + !p
             by
             mod ((radix + !v) * dh) radix
             = mod (radix * dh + dh * !v) radix
             = mod (dh * !v) radix = l2i !p
             so
             div ((radix + !v) * dh) radix = radix - 1
             so
             (radix + !v) * dh
             = radix * div ((radix + !v) * dh) radix
               + mod (dh * !v) radix
             = radix * (radix - 1) + !p
             };
    label Estimate in
    p := add_mod !p dl;
    if !p < dl (* carry out *)
    then begin
      assert { (!p at Estimate) + dl >= radix };
      assert { (!p at Estimate) + dl = radix + !p };
      assert { !v >= 1
               by
               (!p at Estimate) + dl >= radix
               so (!p at Estimate) > 0
             };
      assert { (radix + !v) * dh + dl
               = radix * (radix - 1) + radix + !p };
      label Carry in
      if !p >= dh
      then begin
        v := !v - 1;
        p := !p - dh;
        assert { (radix + !v) * dh + dl
                 = radix * (radix - 1) + radix + !p
               };
      end;
      label Borrow in
      v := !v - 1;
      assert { !p < dh };
      p := sub_mod !p dh;
      assert { !p = radix + !p at Borrow - dh };
    end;
    assert { (radix + !v) * dh * radix + radix * dl
               = radix * radix * (radix - 1) + radix * !p
             by (radix + !v) * dh + dl
                 = radix * (radix - 1) + !p };
    assert { radix * radix - dh
             <= (radix + !v) * dh + dl
             < radix * radix };
    let tl, th = mul_double !v dl in
    label Adjust in
    p := add_mod !p th;
    if !p < th (* carry out *)
    then begin
      assert { (!p at Adjust) + th >= radix };
      assert { (!p at Adjust) + th = radix + !p
               by (!p at Adjust) + th < radix + radix
               so div ((!p at Adjust) + th) radix = 1
               so !p = mod ((!p at Adjust) + th) radix
               so (!p at Adjust) + th
                  = radix * div ((!p at Adjust) + th) radix
                    + mod ((!p at Adjust) + th) radix
                  = radix + !p
             };
      assert { !v >= 1
               by
               th <> 0
               so !v <> 0
             };
      if !p > dh || (!p = dh && tl >= dl)
      then begin
        assert { tl + radix * !p >= d };
        v := !v - 1;
        assert { (radix + !v) * dh * radix + radix * dl
                   + !v * dl
                 = radix * radix * radix
                   + radix * !p + tl - d
                 by
                 (radix + !v) * dh * radix + radix * dl
                   + !v * dl
                 = (radix + !v at Adjust - 1) * dh * radix
                   + radix * dl
                   + (!v at Adjust - 1) * dl
                 = (radix + !v at Adjust) * dh * radix
                   + radix *  dl
                   + (!v at Adjust) * dl - radix * dh
                   - dl
                 = radix * radix * (radix - 1) + radix * (!p at Adjust)
                   + (!v at Adjust) * dl - radix * dh
                   - dl
                 = radix * radix * (radix - 1) + radix * (!p at Adjust)
                   + radix * th + tl - d
                 = radix * radix * (radix - 1) + radix * (radix + !p)
                   + tl - d
                 = radix * radix * (radix - 1) + radix * radix + radix * !p
                   + tl - d
                 = radix * radix * radix + radix * !p + tl - d
            };
      end;
      assert { radix * radix * radix
               <= (radix + !v) * dh * radix + radix * dl
                   + !v * dl
               < radix * radix * radix + d };
      v := !v - 1;
    end;
    bounds_imply_rec3by2 !v dh dl;
    !v

  (* `(x, sz)` is normalized if its most significant bit is set. *)
  predicate normalized (x:t) (sz:int32) =
    valid x sz
    /\ (pelts x)[x.offset + sz - 1] >= div radix 2

  use mul.Mul

  let div_sb_qr (q x:t) (sx:int32) (y:t) (sy:int32) : limb
    requires { 3 <= sy <= sx }
    requires { valid x sx }
    requires { valid y sy }
    requires { valid q (sx - sy) }
    requires { writable q }
    requires { writable x }
    requires { normalized y sy }
    ensures { value (old x) sx =
              (value q (sx - sy)
              + power radix (sx - sy) * result)
                * value y sy
              + value x sy }
    ensures { value x sy < value y sy }
    ensures { 0 <= result <= 1 }
  = [@vc:do_not_keep_trace] (* traceability info breaks the proof *)
    let xp = ref (C.incr x (sx - 2)) in
    let qp = ref (C.incr q (sx - sy)) in
    let dh = C.get_ofs y (sy - 1) in
    assert { dh >= div radix 2 by normalized y sy };
    let dl = C.get_ofs y (sy - 2) in
    let v = reciprocal_word_3by2 dh dl in
    let i = ref (sx - sy) in
    let mdn = 2 - sy in
    let ql = ref 0 in
    let xd = C.incr !xp mdn in
    let ghost vy = value y (p2i sy) in
    let x1 = ref 0 in
    let x0 = ref 0 in
    let r = wmpn_cmp xd y sy in
    let qh = (*begin
               ensures { r >= 0 -> result = 1 }
               ensures { r < 0 -> result = 0 }*)
               if (r >= 0)
               then 1
               else 0
             (*end*) in
    label PreAdjust in
    begin
      ensures { value (old x) sx =
                  (value !qp (sx - sy - !i)
                   + qh * power radix (sx - sy - !i))
                  * vy * power radix !i
                  + value x (sy + !i - 1)
                  + power radix (sy + !i - 1) * !x1 }
      ensures { value_sub (pelts x) (!xp.offset + mdn)
                 (!xp.offset + mdn + sy - 1)
                 + power radix (sy - 1) * !x1
                 < vy }
      ensures { dl + radix * dh
                  >= (pelts x)[(!xp).offset] + radix * !x1 }
    let ghost ox = pelts x in
    begin [@vc:sp]
    if (not (qh = 0))
    then begin
         assert { qh = 1 };
         let (ghost b) = wmpn_sub_n_in_place xd y sy in
         begin
             ensures { value (x at PreAdjust) sx =
                  (value !qp (sx - sy - !i)
                   + qh * power radix (sx - sy - !i))
                  * vy * power radix !i
                  + value x sx }
             ensures { value_sub (pelts x) (!xp.offset + mdn)
                 (!xp.offset + mdn + sy)
                 < vy }
           value_sub_upper_bound (pelts x) xd.offset (xd.offset + p2i sy);
           assert { b = 0 };
           assert { value (xd at PreAdjust) sy
                    = value xd sy + vy };
           value_sub_concat (pelts x) x.offset xd.offset (xd.offset + p2i sy);
           value_sub_concat ox x.offset xd.offset (xd.offset + p2i sy);
           value_sub_frame (pelts x) ox x.offset xd.offset;
           assert { value (x at PreAdjust) sx
             = value x sx
             + power radix (sx - sy) * vy
             by
             value_sub (pelts (x at PreAdjust)) x.offset xd.offset
             = value_sub (pelts x) x.offset xd.offset
             so pelts (xd at PreAdjust) = pelts (x at PreAdjust)
             so value_sub (pelts (x at PreAdjust)) xd.offset (xd.offset + sy)
                = value (xd at PreAdjust) sy
             so value (x at PreAdjust) sx
             = value_sub (pelts (x at PreAdjust)) x.offset xd.offset
               + power radix (sx - sy)
               * value_sub (pelts (x at PreAdjust)) xd.offset (xd.offset + sy)
             = value_sub (pelts x) x.offset xd.offset
               + power radix (sx - sy)
               * value (xd at PreAdjust) sy
             = value_sub (pelts x) x.offset xd.offset
               + power radix (sx - sy)
               * (value xd sy + vy)
             = value_sub (pelts x) x.offset xd.offset
               + power radix (sx - sy)
               * (value_sub (pelts x) (xd.offset) (xd.offset + sy) + vy)
             = value_sub (pelts x) x.offset xd.offset
               + power radix (sx - sy)
               * value_sub (pelts x) (xd.offset) (xd.offset + sy)
               + power radix (sx - sy) * vy
             = value x sx
               + power radix (sx - sy) * vy
           };
           value_sub_tail (pelts x) x.offset (x.offset + p2i sy + p2i !i - 1);
           assert { value (x at PreAdjust) sx =
                  (value !qp (sx - sy - !i)
                   + qh * power radix (sx - sy - !i))
                  * vy * power radix !i
                  + value x sx
                 by
                   !i = sx - sy
                 so power radix (sx - sy - !i) = 1
                 so value !qp (sx - sy - !i) = 0 };
           value_sub_lower_bound_tight (pelts y) y.offset (y.offset + p2i sy);
           assert { value_sub (pelts x) (!xp.offset + mdn)
                 (!xp.offset + mdn + sy)
                 < vy
                 by
                 value_sub (pelts x) (!xp.offset + mdn)
                           (!xp.offset + mdn + sy)
                 = value xd sy
                 = value (xd at PreAdjust) sy - vy
                 so value (xd at PreAdjust) sy
                    < power radix sy
                 so vy >= dh * power radix (sy - 1)
                 so 2 * vy >= 2 * dh * power radix (sy - 1)
                 so 2 * dh >= radix
                 so 2 * dh * power radix (sy - 1) >= radix * power radix (sy - 1)
                 so 2 * vy >= radix * power radix (sy - 1) = power radix sy
                 so value (xd at PreAdjust) sy < 2 * vy
                 so value (xd at PreAdjust) sy - vy < vy };
           end
         end
    else begin
         assert { value_sub (pelts x) (!xp.offset + mdn)
                 (!xp.offset + mdn + sy)
                 < vy
                 by r < 0 };
         assert { value (x at PreAdjust) sx =
                  (value !qp (sx - sy - !i)
                   + qh * power radix (sx - sy - !i))
                   * vy * power radix !i
                   + value x sx
                 by qh = 0
                 so sx - sy - !i = 0
                 so (value !qp (sx - sy - !i)
                   + qh * power radix (sx - sy - !i)) = 0 };
         end
    end;
    let ghost gx1 = (C.get_ofs !xp 1) in
    value_sub_tail (pelts y) y.offset (y.offset + p2i sy - 1);
    value_sub_upper_bound_tight (pelts y) y.offset (y.offset + p2i sy - 1);
    value_sub_tail (pelts x) (!xp.offset) (!xp.offset + p2i sy - 1);
    value_sub_lower_bound_tight (pelts x) (!xp.offset) (!xp.offset + p2i sy - 1);
    assert { dl + radix * dh
             >= (pelts x)[(!xp).offset] + radix * gx1
             by value_sub (pelts x) (!xp.offset + mdn)
                                    (!xp.offset + mdn + sy)
                < vy
             so value y (sy - 1) < (dl + 1) * power radix (sy - 1 - 1)
             so vy = dh * power radix (sy - 1)
                     + value y (sy - 1)
                     < dh * power radix (sy - 1)
                     + (dl + 1) * power radix (sy - 1 - 1)
                   = power radix (sy - 2) * (dl+1 + radix * dh)
             so !xp.offset + mdn + sy - 1 = !xp.offset + 1
             so (pelts x)[!xp.offset + mdn + sy - 1]
                = (pelts x)[!xp.offset + 1] = gx1
             so value_sub (pelts x) (!xp.offset + mdn) (!xp.offset + mdn + sy)
                = gx1 * power radix (sy - 1)
                     + value_sub (pelts x) (!xp.offset + mdn)
                                           (!xp.offset + mdn + sy - 1)
                >= gx1 * power radix (sy - 1)
                   + (pelts x)[!xp.offset] * power radix (sy - 1 - 1)
                = power radix (sy - 2)
                  * ((pelts x) [!xp.offset] + radix * gx1)
             so power radix (sy - 2) * ((pelts x) [!xp.offset] + radix * gx1)
                < power radix (sy - 2) * (dl+1 + radix * dh)
             so (pelts x) [!xp.offset] + radix * gx1
                < dl + 1 + radix * dh
             };
    value_sub_tail (pelts x) (!xp.offset + p2i mdn)
                   (!xp.offset + p2i mdn + p2i sy - 1);
    value_sub_tail (pelts x) x.offset (x.offset + p2i sy + p2i !i - 1);
    x1 := (C.get_ofs !xp 1);
    assert { value_sub (pelts x) (!xp.offset + mdn)
                 (!xp.offset + mdn + sy - 1)
                 + power radix (sy - 1) * !x1
                 < vy
             by
             !xp.offset + mdn + sy - 1 = !xp.offset + 1
             so value_sub (pelts x) (!xp.offset + mdn)
                 (!xp.offset + mdn + sy - 1)
                 + power radix (sy - 1) * !x1
             = value_sub (pelts x) (!xp.offset + mdn)
                 (!xp.offset + mdn + sy - 1)
                 + power radix (sy - 1) * (pelts x)[!xp.offset + 1]
             = value_sub (pelts x) (!xp.offset + mdn)
                 (!xp.offset + mdn + sy - 1)
                 + power radix (sy - 1)
                   * (pelts x)[!xp.offset + mdn + sy - 1]
             = value_sub (pelts x) (!xp.offset + mdn)
                         (!xp.offset + mdn + sy)
             < vy };
    assert { value (x at PreAdjust) sx =
                  (value !qp (sx - sy - !i)
                   + qh * power radix (sx - sy - !i))
                  * vy * power radix !i
                  + value x (sy + !i - 1)
                  + power radix (sy + !i - 1) * !x1
                by value (x at PreAdjust) sx =
                   (value !qp (sx - sy - !i)
                   + qh * power radix (sx - sy - !i))
                   * vy * power radix !i
                   + value x sx
                so sx = sy + !i
                so x.offset + sy + !i - 1 = !xp.offset + 1
                so (pelts x)[x.offset + sy + !i - 1] =
                   (pelts x)[!xp.offset + 1]= !x1
                so value x sx
                   = value x (sx - 1)
                     + power radix (sx -1) * (pelts x)[x.offset + sx - 1]
                   = value x (sy + !i - 1)
                     + power radix (sy + !i - 1) * (pelts x)[x.offset + sy + !i - 1]
                so value x (sy + !i - 1)
                  + power radix (sy + !i - 1) * !x1
                  = value x sx
           };
    end;
    while (!i > 0) do
      variant { !i }
      invariant { 0 <= !i <= sx - sy }
      invariant { (!qp).offset = q.offset + !i }
      invariant { (!xp).offset = x.offset + sy + !i - 2 }
      invariant { plength !qp = plength q }
      invariant { !qp.min = q.min }
      invariant { !qp.max = q.max }
      invariant { plength !xp = plength x }
      invariant { !xp.min = x.min }
      invariant { !xp.max = x.max }
      invariant { pelts !qp = pelts q }
      invariant { pelts !xp = pelts x }
      invariant { writable !qp /\ writable !xp }
      invariant { value (old x) sx =
                  (value !qp (sx - sy - !i)
                   + qh * power radix (sx - sy - !i))
                  * vy * power radix !i
                  + value x (sy + !i - 1)
                  + power radix (sy + !i - 1) * !x1 }
      invariant { value_sub (pelts x) (!xp.offset + mdn)
                 (!xp.offset + mdn + sy - 1)
                 + power radix (sy - 1) * !x1
                 < vy }
      invariant { dl + radix * dh
                  >= (pelts x)[(!xp).offset] + radix * !x1 }
      label StartLoop in
      let ghost k = int32'int !i in
      i := !i - 1;
      let ghost s = int32'int sy + int32'int !i - 1 in
      xp.contents <- C.incr !xp (-1);
      let xd = C.incr !xp mdn in
      let nx0 = C.get_ofs !xp 1 in
      if [@extraction:unlikely] (!x1 = dh && nx0 = dl)
      then begin
        ql := Limb.uint64_max;
        value_sub_concat (pelts x) x.offset xd.offset (xd.offset + p2i sy);
        value_sub_tail (pelts xd) xd.offset (xd.offset + p2i sy - 1);
        let ghost vlx = value xd (p2i sy - 1) in
        assert { value xd sy
                 = vlx + power radix (sy - 1) * dl
                 by value xd sy
                    = vlx + power radix (sy - 1)
                            * (pelts xd)[xd.offset + sy - 1]
                 so xd.offset + sy - 1 = !xp.offset + mdn + sy - 1
                                           = !xp.offset + 1
                 so pelts xd = pelts !xp
                 so (pelts xd)[xd.offset + sy - 1]
                    = (pelts !xp)[!xp.offset + 1] = dl
               };
        value_sub_tail (pelts y) y.offset (y.offset + p2i sy - 1);
        value_sub_tail (pelts y) y.offset (y.offset + p2i sy - 2);
        let ghost vly = value y (p2i sy - 2) in
        assert { vy = vly + power radix (sy - 2) * dl
                          + power radix (sy - 1) * dh
                 by (pelts y)[y.offset + sy - 1] = dh
                 so (pelts y)[y.offset + sy - 2] = dl
                 so
                    vy = value y (sy - 1)
                          + power radix (sy - 1) * dh
                    = vly + power radix (sy - 2) * dl
                          + power radix (sy - 1) * dh };
        begin
          ensures { value_sub (pelts xd) (xd.offset+1) (xd.offset + p2i sy - 1)
                        + power radix (sy - 2) * dl
                        + power radix (sy - 1) * dh
                     < vy }
          value_sub_tail (pelts xd) (xd.offset + 1) (xd.offset + p2i sy - 2);
          assert { value_sub (pelts x) (!xp.offset at StartLoop + mdn)
                             (!xp.offset at StartLoop + mdn + sy - 1)
                   = value_sub (pelts xd) (xd.offset+1) (xd.offset + p2i sy - 1)
                        + power radix (sy - 2) * dl
                   by
                   pelts x = pelts xd
                   so !xp.offset at StartLoop + mdn = xd.offset + 1
                   so !xp.offset at StartLoop + mdn + sy - 1 = xd.offset + sy
                   so xd.offset + sy - 1 = !xp.offset + 1
                   so pelts xd = pelts !xp
                   so (pelts xd)[xd.offset + sy - 1] = (pelts !xp)[!xp.offset+1] = dl
                   so value_sub (pelts x) (!xp.offset at StartLoop + mdn)
                             (!xp.offset at StartLoop + mdn + sy - 1)
                    = value_sub (pelts xd) (xd.offset+1) (xd.offset + sy)
                    = value_sub (pelts xd) (xd.offset+1) (xd.offset + p2i sy - 1)
                      + power radix (sy - 2)
                        * (pelts xd)[xd.offset + p2i sy - 1]
                    = value_sub (pelts xd) (xd.offset+1) (xd.offset + p2i sy - 1)
                      + power radix (sy - 2) * dl
                 };
         assert { !x1 = dh };
        end;
        label SubMax in
        let ghost xc = Array.copy (x.data) in
        value_sub_frame (pelts x) xc.elts x.offset (x.offset + p2i !i);
        let (ghost b) = wmpn_submul_1 xd y sy !ql in
        begin
          ensures { value x !i
                 = value (x at SubMax) !i }
          assert { forall j. x.offset <= j < x.offset + !i
                          -> (pelts x)[j] = xc.elts[j]
                   by
                   (pelts x)[j] = (pelts x at SubMax)[j]
                   so
                   ((pelts x at SubMax)[j] = xc.elts[j]
                   by
                   0 <= j /\ j < xc.Array.length
                   ) };
          value_sub_frame (pelts x) xc.elts x.offset (x.offset + p2i !i);
        end;
        value_sub_upper_bound (pelts xd) xd.offset (xd.offset + p2i sy);
        value_sub_lower_bound (pelts xd) xd.offset (xd.offset + p2i sy);
        value_sub_head (pelts xd) xd.offset (xd.offset + p2i sy - 1);
        assert { vlx < radix * vly
                 by
                   vlx = value_sub (pelts xd at SubMax) xd.offset
                                   (xd.offset + sy - 1)
                       = (pelts xd at SubMax)[xd.offset]
                         + radix * value_sub (pelts xd at SubMax)
                                             (xd.offset + 1)
                                             (xd.offset + sy - 1)
                   so value_sub (pelts xd at SubMax) (xd.offset + 1)
                                           (xd.offset + sy - 1)
                      + power radix (sy - 2) * dl
                      + power radix (sy - 1) * dh
                      < vy
                      = vly + power radix (sy - 2) * dl
                            + power radix (sy - 1) * dh
                   so value_sub (pelts xd at SubMax) (xd.offset + 1)
                                           (xd.offset + sy - 1)
                      < vly
                   so value_sub (pelts xd at SubMax) (xd.offset + 1)
                                           (xd.offset + sy - 1)
                      <= vly - 1
                   so vlx = (pelts xd at SubMax)[xd.offset]
                            + radix * value_sub (pelts xd at SubMax)
                                                (xd.offset + 1)
                                                (xd.offset + sy - 1)
                          <= (pelts xd at SubMax)[xd.offset]
                            + radix * (vly - 1)
                          < radix + radix * (vly - 1)
                          = radix * vly
               };
        assert { b = dh
                 by
                   value xd sy
                 = value (xd at SubMax) sy
                   - (!ql) * vy
                   + power radix sy * b
                 so !ql = radix - 1
                 so 0 <= value xd sy < power radix sy
                 so radix * power radix (sy - 2) = power radix (sy - 1)
                 so radix * power radix (sy - 1) = power radix sy
                 so value xd sy
                    = power radix (sy - 1) * dl + vlx
                      - (radix - 1) * vy
                      + power radix sy * b
                    = power radix (sy - 1) * dl + vlx
                      - radix * (vly + power radix (sy - 2) * dl
                                + power radix (sy - 1) * dh)
                      + vy + power radix sy * b
                    = power radix (sy - 1) * dl + vlx
                      - radix * vly - radix * power radix (sy - 2) * dl
                      - radix * power radix (sy - 1) * dh
                      + vy + power radix sy * b
                    = power radix (sy - 1) * dl + vlx
                      - radix * vly - power radix (sy - 1) * dl
                      - power radix sy * dh
                      + vy + power radix sy * b
                    = power radix sy * (b - dh)
                      + vlx - radix * vly + vy
                 so vlx < radix * vly
                 so (0 <= vlx - radix * vly + vy < power radix sy
                      by
                      vy - radix * vly
                      = vly + power radix (sy - 2) * dl
                          + power radix (sy - 1) * dh
                          - radix * vly
                      = power radix (sy - 2) * (dl + radix * dh)
                        - vly * (radix - 1)
                      so let pr2 = power radix (sy - 2) in
                      0 <= vly < pr2
                      so 0 <= vly * (radix - 1) < pr2 * (radix - 1)
                      so vy - radix * vly
                         >= pr2 * (dl + radix * dh)
                           - pr2 * (radix - 1)
                         = pr2 * (dl + radix * dh - (radix - 1))
                      so dh + radix * dh - (radix - 1) >= 0
                      so pr2 >= 0
                      so vy - radix * vly
                         >= pr2 * (dl + radix * dh - (radix - 1)) >= 0
                      so vlx - radix * vly < 0
                      so vlx - radix * vly + vy < vy < power radix sy
                    )
                 so - (power radix sy)
                    < power radix sy * (b - dh)
                    < power radix sy
                 so - 1 < b - dh < 1
               };
        value_sub_concat (pelts x) x.offset xd.offset (x.offset + s);
        x1 := C.get_ofs !xp 1;
        qp.contents <- C.incr !qp (-1);
        value_sub_update_no_change (pelts q) (!qp).offset
                            ((!qp).offset + 1)
                            ((!qp).offset + p2i sx - p2i sy - p2i !i)
                            !ql;
        label QUp in
        C.set !qp !ql;
        assert { value_sub (pelts q) ((!qp).offset + 1)
                              ((!qp).offset + sx - sy - !i)
                    = value (!qp at StartLoop)
                              (sx - sy - k)
                by value_sub (pelts q) ((!qp).offset + 1)
                              ((!qp).offset + sx - sy - !i)
                   = value_sub (pelts q at QUp) (!qp.offset + 1)
                              ((!qp).offset + sx - sy - !i)
                   = value (!qp at StartLoop) (sx - sy - k) };
        value_sub_head (pelts q) (!qp).offset
          ((!qp).offset + p2i sx - p2i sy - p2i !i);
        value_sub_tail (pelts x) x.offset (x.offset + p2i sy + p2i !i - 1);
        assert { value xd (sy - 1)
                 + power radix (sy - 1) * !x1
                 = value (xd at SubMax) sy
                   + power radix sy * (!x1 at StartLoop)
                   - !ql * vy
                 by
                 value xd sy
                 = value (xd at SubMax) sy
                   - (!ql) * vy
                   + power radix sy * b
                 so b = dh = !x1 at StartLoop
                 so pelts !xp = pelts x = pelts xd
                 so ((pelts xd)[xd.offset + sy - 1] = !x1
                    by
                    xd.offset = x.offset + !i
                    so (!xp).offset = x.offset + !i + sy - 2
                    so (!xp).offset + 1 = xd.offset + sy - 1
                    so (pelts xd)[xd.offset + sy - 1]
                       = (pelts !xp)[(!xp).offset + 1]
                       = !x1
                    )
                 so value xd sy
                    = value xd (sy - 1)
                      + power radix (sy - 1) * (pelts xd)[xd.offset + sy - 1]
                    = value xd (sy - 1)
                      + power radix (sy - 1) * !x1
               };
        (* refl *)
        assert { value (old x) sx =
                  (value !qp (sx - sy - !i)
                   + qh * power radix (sx - sy - !i))
                  * vy * power radix !i
                  + value x (sy + !i - 1)
                  + power radix (sy + !i - 1) * !x1
                 by
                 pelts !xp = pelts x = pelts xd
                 so
                 value xd sy
                 = value (xd at SubMax) sy
                   - (!ql) * vy
                   + power radix sy * b
                 = value (xd at SubMax) sy
                   - (!ql) * vy
                   + power radix sy * dh
                 so (value x s
                    = value x !i
                      + power radix !i
                        * value xd (sy - 1)
                    by
                      xd.offset = x.offset + !i
                      so x.offset + s = xd.offset + sy - 1
                      so value_sub (pelts x) (x.offset + !i) (x.offset + s)
                         = value xd (sy - 1)
                      so value x s
                    = value x !i
                      + power radix !i
                        * value_sub (pelts x) (x.offset + !i)
                                              (x.offset + s)
                    = value x !i
                      + power radix !i
                        * value xd (sy - 1))
                 so (power radix s
                    = power radix !i * power radix (sy - 1)
                    by
                    let n = !i in
                    let m = sy - 1 in
                    let x = radix in
                    power x s = power x (n + m)
                    so (power x (n + m) = power x n * power x m
                       by 0 <= n
                       so 0 <= m
                       so forall x:int, n:int, m:int.
                                 0 <= n -> 0 <= m ->
                                 power x (n + m) = (power x n * power x m)))
                 so (value x s + power radix s * !x1
                    = value x !i
                      + power radix !i * (value xd sy)
                    by
                    value x s + power radix s * !x1
                    = value x !i
                      + power radix !i
                        * value xd (sy - 1)
                      + power radix (!i + sy - 1) * !x1
                    = value x !i
                      + power radix !i *
                        (value xd (sy - 1)
                         + power radix (sy - 1) * !x1)
                    = value x !i
                      + power radix !i * (value xd sy)
                    )
                 so (value (x at StartLoop) (sy + k - 1)
                    = value (x at SubMax) !i
                      + power radix !i
                        * value (xd at SubMax) sy
                    by
                    pelts xd at SubMax = pelts x at SubMax
                    so x.offset at SubMax + !i = xd.offset at SubMax
                    so
                    value (x at StartLoop) (sy + k - 1)
                    = value_sub (pelts x at SubMax) (x at SubMax).offset
                                (xd.offset at SubMax)
                      + power radix !i
                        * value_sub (pelts x at SubMax)
                           (xd.offset at SubMax)
                           (xd.offset at SubMax + sy)
                    so value_sub (pelts x at SubMax) (x at SubMax).offset
                                 (xd at SubMax).offset
                       = value (x at SubMax) !i
                    so value_sub (pelts x at SubMax) (xd.offset at SubMax)
                                                     (xd.offset at SubMax + sy)
                       = value (xd at SubMax) sy
                    )
                 so value x !i
                    = value (x at SubMax) !i
                 so value x s + power radix s * !x1
                    = value (x at StartLoop) (sy + k - 1)
                      + power radix !i
                        * (value xd sy
                           - value (xd at SubMax) sy)
                    = value (x at StartLoop) (sy + k - 1)
                      + power radix !i
                        * (- (!ql) * vy
                           + power radix sy * b)
                    = value (x at StartLoop) (sy + k - 1)
                      + power radix !i
                        * (- (!ql) * vy
                           + power radix sy * (!x1 at StartLoop))
                 so value !qp (sx - sy - !i)
                    = !ql + radix *
                          value_sub (pelts q) ((!qp).offset + 1)
                                    ((!qp).offset + sx - sy - !i)
                 so (value_sub (pelts q) ((!qp).offset + 1)
                              ((!qp).offset + sx - sy - !i)
                    = value (!qp at StartLoop)
                              (sx - sy - k)
                     by value (!qp at StartLoop) (sx - sy - k)
                      = value_sub (pelts q at StartLoop)
                                  (!qp.offset + 1) (!qp.offset + sx - sy - !i))
                 so value !qp (sx - sy - !i)
                    = !ql + radix * value (!qp at StartLoop)
                                    (sx - sy - k)
                 so power radix (sx - sy - !i)
                    = radix * power radix (sx - sy - k)
                 so radix * power radix !i = power radix k
                 so (power radix !i * power radix sy
                      = power radix (sy + k - 1)
                      by !i + sy = sy + k - 1
                      so power radix !i * power radix sy
                         = power radix (!i + sy))
                 so (value !qp (sx - sy - !i)
                      + qh * power radix (sx - sy - !i))
                      * vy * power radix !i
                      + value x (sy + !i - 1)
                      + power radix (sy + !i - 1) * !x1
                    = (!ql + radix * value (!qp at StartLoop)
                                    (sx - sy - k)
                      + qh * power radix (sx - sy - !i))
                      * vy * power radix !i
                      + value x (sy + !i - 1)
                      + power radix (sy + !i - 1) * !x1
                    = (!ql + radix * value (!qp at StartLoop)
                                    (sx - sy - k)
                      + radix * qh * power radix (sx - sy - k))
                      * vy * power radix !i
                      + value x (sy + !i - 1)
                      + power radix (sy + !i - 1) * !x1
                    = (!ql + radix * value (!qp at StartLoop)
                                    (sx - sy - k)
                      + radix * qh * power radix (sx - sy - k))
                      * vy * power radix !i
                      + value x s
                      + power radix s * !x1
                    = !ql * vy * power radix !i
                      + radix * (value (!qp at StartLoop)
                                    (sx - sy - k)
                                 + qh * power radix (sx - sy - k))
                        * vy * power radix !i
                      + value x s
                      + power radix s * !x1
                    = !ql * vy * power radix !i
                      + (value (!qp at StartLoop)
                                    (sx - sy - k)
                                 + qh * power radix (sx - sy - k))
                        * vy * radix * power radix !i
                      + value x s
                      + power radix s * !x1
                    = !ql * vy * power radix !i
                      + (value (!qp at StartLoop)
                                    (sx - sy - k)
                                 + qh * power radix (sx - sy - k))
                        * vy * power radix k
                      + value x s
                      + power radix s * !x1
                    = !ql * vy * power radix !i
                      + (value (!qp at StartLoop)
                                    (sx - sy - k)
                                 + qh * power radix (sx - sy - k))
                        * vy * power radix k
                      + value (x at StartLoop) (sy + k - 1)
                      + power radix !i
                        * (- (!ql) * vy
                           + power radix sy * (!x1 at StartLoop))
                    = (value (!qp at StartLoop)
                                    (sx - sy - k)
                                 + qh * power radix (sx - sy - k))
                        * vy * power radix k
                      + value (x at StartLoop) (sy + k - 1)
                      + power radix !i * power radix sy
                        * (!x1 at StartLoop)
                    = (value (!qp at StartLoop)
                                    (sx - sy - k)
                                 + qh * power radix (sx - sy - k))
                        * vy * power radix k
                      + value (x at StartLoop) (sy + k - 1)
                      + power radix (sy + k - 1) * (!x1 at StartLoop)
                    = value (old x) sx
                  };
        assert { value_sub (pelts x) (!xp.offset + mdn)
                   (!xp.offset + mdn + sy - 1)
                   + power radix (sy - 1) * !x1
                   < vy
                   by
                     pelts x = pelts xd
                   so xd.offset = !xp.offset + mdn
                   so !xp.offset + mdn + sy - 1 = xd.offset + sy - 1
                   so
                     value xd (sy - 1)
                     = value_sub (pelts xd) xd.offset (xd.offset + sy - 1)
                     = value_sub (pelts x) (!xp.offset + mdn)
                       (!xp.offset + mdn + sy - 1)
                   so value_sub (pelts x) (!xp.offset + mdn)
                                (!xp.offset + mdn + sy - 1)
                      + power radix (sy - 1) * !x1
                      = value (xd at SubMax) sy
                        + power radix sy * (!x1 at StartLoop)
                        - !ql * vy
                   so value (xd at SubMax) sy =
                      vlx + power radix (sy - 1) * dl
                   so vlx < radix * vly
                   so (value (xd at SubMax) sy
                        + power radix sy * (!x1 at StartLoop)
                      < radix * vy
                      by
                      !x1 at StartLoop = dh
                      so power radix sy = radix * power radix (sy - 1)
                      so power radix (sy - 1) = radix * power radix (sy - 2)
                      so value (xd at SubMax) sy
                          + power radix sy * (!x1 at StartLoop)
                         = vlx + power radix (sy - 1) * dl
                               + power radix sy * dh
                         < radix * vly + power radix (sy - 1) * dl
                                       + power radix sy * dh
                         = radix * vly + radix * power radix (sy - 2) * dl
                                       + radix * power radix (sy - 1) * dh
                         = radix * (vly + power radix (sy - 2) * dl
                                        + power radix (sy - 1) * dh)
                         = radix * vy
                      )
                   so !ql = radix - 1
                   so value (xd at SubMax) sy
                        + power radix sy * (!x1 at StartLoop)
                        - !ql * vy
                      < radix * vy - (radix - 1) * vy
                      = vy
               };
          value_sub_tail (pelts x) (!xp.offset + p2i mdn) (!xp.offset);
          value_sub_upper_bound (pelts y) (y.offset) (y.offset + p2i sy - 2);
          value_sub_lower_bound (pelts x) (!xp.offset + p2i mdn) (!xp.offset);
          assert { dl + radix * dh
                   >= (pelts x)[(!xp).offset] + radix * !x1
                   by
                   vy = vly + power radix (sy - 2)
                              * (dl + radix * dh)
                   so value_sub (pelts x) (!xp.offset + mdn)
                   (!xp.offset + mdn + sy - 1)
                   + power radix (sy - 1) * !x1
                   < vy
                   so !xp.offset + mdn + sy - 1 = !xp.offset + 1
                   so power radix (sy - 1) = power radix (sy - 2) * radix
                   so - mdn = sy - 2
                   so vy
                      > value_sub (pelts x) (!xp.offset + mdn)
                                            (!xp.offset + mdn + sy - 1)
                        + power radix (sy - 1) * !x1
                      = value_sub (pelts x) (!xp.offset + mdn)
                                            (!xp.offset + 1)
                        + power radix (sy - 1) * !x1
                      = value_sub (pelts x) (!xp.offset + mdn) (!xp.offset)
                        + power radix (- mdn) * (pelts x)[(!xp).offset]
                        + power radix (sy - 1) * !x1
                      = value_sub (pelts x) (!xp.offset + mdn) (!xp.offset)
                        + power radix (sy - 2) * (pelts x)[(!xp).offset]
                        + power radix (sy - 1) * !x1
                      = value_sub (pelts x) (!xp.offset + mdn) (!xp.offset)
                        + power radix (sy - 2) * (pelts x)[(!xp).offset]
                        + power radix (sy - 2) * radix * !x1
                      = value_sub (pelts x) (!xp.offset + mdn) (!xp.offset)
                        + power radix (sy - 2)
                          * ((pelts x)[(!xp).offset] + radix * !x1)
                      >= power radix (sy - 2)
                          * ((pelts x)[(!xp).offset] + radix * !x1)
                   so vly < power radix (sy - 2)
                   so vy < power radix (sy - 2)
                           + power radix (sy - 2)
                              * (dl + radix * dh)
                           = power radix (sy - 2)
                             * (1 + dl + radix * dh)
                   so power radix (sy - 2)
                      * ((pelts x)[(!xp).offset] + radix * !x1)
                      < power radix (sy - 2) * (1 + dl + radix * dh)
                   so power radix (sy - 2)
                      * ((pelts x)[(!xp).offset] + radix * !x1
                        - (1 + dl + radix * dh))
                      < 0
                   so (pelts x)[(!xp).offset] + radix * !x1
                        - (1 + dl + radix * dh)
                      < 0
                 };
      end
      else begin
        assert { dl + radix * dh
                  > (pelts x)[(!xp).offset + 1] + radix * !x1
                  by
                  dl + radix * dh
                  >= (pelts x)[(!xp).offset + 1] + radix * !x1
                  so dh >= !x1
                  so [@case_split] dh <> !x1
                   \/ (dh = !x1
                      /\ dl <> (pelts x)[(!xp).offset + 1])
                  so
                   [@case_split] dh > !x1 \/
                   (dh = !x1 /\ dl > (pelts x)[(!xp).offset + 1])
        };
        label SmallDiv in
        let ghost vlx = value xd (p2i sy - 2) in
        let xp0 = C.get !xp in
        let xp1 = C.get_ofs !xp 1 in
        begin
          ensures { value xd sy =
                    vlx
                    + power radix (sy - 2) * (xp0 + radix * xp1) }
        value_sub_tail (pelts xd) xd.offset (xd.offset + p2i sy - 1);
        value_sub_tail (pelts xd) xd.offset (xd.offset + p2i sy - 2);
        value_sub_upper_bound (pelts xd) xd.offset (xd.offset + p2i sy - 2);
        assert { value xd sy
                 = vlx + power radix (sy - 2)
                         * (xp0 + radix * xp1)
                 by xd.offset + sy - 2 = !xp.offset
                 so (pelts xd)[xd.offset + sy - 1] = xp1
                 so (pelts xd)[xd.offset + sy - 2] = xp0
                 so pelts xd = pelts !xp
                 so value xd sy
                    = value xd (sy - 1)
                      + power radix (sy - 1)
                            * (pelts xd)[xd.offset + sy - 1]
                    = value xd (sy - 2)
                      + power radix (sy - 2)
                        * (pelts xd)[xd.offset + sy - 2]
                      + power radix (sy - 1)
                        * (pelts xd)[xd.offset + sy - 1]
                    = vlx
                      + power radix (sy - 2) * xp0
                      + power radix (sy - 1) * xp1
                    = value xd (sy - 2)
                      + power radix (sy - 2) * xp0
                      + power radix (sy - 2) * radix * xp1
                    = vlx + power radix (sy - 2)
                         * (xp0 + radix * xp1)
               };
        end;
        let qu, rl, rh =
            div3by2_inv !x1 xp1 xp0 dh dl v in
        ql := qu;
        x1 := rh;
        x0 := rl;
        label SubProd in
        value_sub_concat (pelts x) x.offset xd.offset
                           (x.offset + p2i sy + k - 1);
        let ghost xc = Array.copy (x.data) in
        value_sub_frame (pelts x) xc.elts x.offset (x.offset + p2i !i);
        let cy = wmpn_submul_1 xd y (sy - 2) !ql in
        label PostSub in
        begin
          ensures { value x !i
                 = value (x at SubProd) !i }
          assert { forall j. x.offset <= j < x.offset + !i
                          -> (pelts x)[j] = xc.elts[j]
                   by
                   (pelts x)[j] = (pelts x at SubProd)[j]
                   so
                   ((pelts x at SubProd)[j] = xc.elts[j]
                   by
                   0 <= j /\ j < xc.Array.length
                   ) };
          value_sub_frame (pelts x) xc.elts x.offset (x.offset + p2i !i);
        end;
        let (cy1:limb) = [@vc:sp] if (!x0 < cy) then 1 else 0 in
        x0 := sub_mod !x0 cy;
        let (cy2:limb) = [@vc:sp] if (!x1 < cy1) then 1 else 0 in
        x1 := sub_mod !x1 cy1;
        assert { 0 <= cy2 <= 1 };
        (* assert { cy2 = 1 -> rh = 0 }; (* and cy > rl *)*)
        value_sub_update (pelts x) (!xp).offset xd.offset
                                   (xd.offset + p2i sy - 1) !x0;
        value_sub_update_no_change (pelts x) (!xp).offset
                                   x.offset (x.offset + p2i !i) !x0;
        value_sub_update_no_change (pelts x) (!xp).offset
                                   xd.offset (xd.offset + p2i sy - 2) !x0;
        C.set !xp !x0;
        assert { value x !i
                 = value (x at SubProd) !i
                 by
                 value x !i
                 = value (x at PostSub) !i
                 = value (x at SubProd) !i };
        value_sub_tail (pelts x) xd.offset (xd.offset + p2i sy - 1);
        begin
          ensures { value xd (sy - 1)
                   + power radix (sy - 1) * !x1
                   - power radix sy * cy2
                 = value (xd at SubProd) sy
                   + power radix sy * (!x1 at StartLoop)
                   - !ql * vy }
          assert { value xd (sy - 2)
                   = value (xd at PostSub) (sy - 2) };
          value_sub_tail (pelts y) y.offset (y.offset + p2i sy - 1);
          value_sub_tail (pelts y) y.offset (y.offset + p2i sy - 2);
          let ghost vly = value y (p2i sy - 2) in
          assert { vy = vly + power radix (sy - 2)
                              * (dl + radix * dh)
                   by (pelts y)[y.offset + sy - 1] = dh
                   so (pelts y)[y.offset + sy - 2] = dl
                   so
                      vy = value y (sy - 1)
                            + power radix (sy - 1) * dh
                         = vly + power radix (sy - 2) * dl
                            + power radix (sy - 1) * dh
                   so power radix (sy - 1)
                      = power radix (sy - 2) * radix };
          assert { value xd (sy - 2)
                   - power radix (sy - 2) * cy
                   = vlx - !ql * vly
                   by
                   value xd (sy - 2)
                     - power radix (sy - 2) * cy
                   = value (xd at PostSub) (sy - 2)
                     - power radix (sy - 2) * cy
                   = vlx - !ql * vly
                 };
          assert { power radix sy
                   = power radix (sy - 2) * radix * radix };
          assert { xp0 + radix * xp1
                   + radix * radix * !x1 at StartLoop
                   - !ql * (dl + radix * dh)
                  = rl + radix * rh };
          begin ensures { value (xd at SubProd) sy
                     + power radix sy * (!x1 at StartLoop)
                     - !ql * vy
                   = value xd (sy - 2)
                     - power radix (sy - 2) * cy
                     + power radix (sy - 2) *
                       (rl + radix * rh) }
            assert { value (xd at SubProd) sy
                     = vlx + power radix (sy - 2) * xp0
                           + power radix (sy - 1) * xp1 }; (*nonlinear*)
            assert { !ql * vy = !ql * vly
                                + power radix (sy - 2)
                                  * (!ql * (dl + radix * dh)) }; (*nonlinear*)
           end;
           begin ensures { value xd (sy - 2)
                     - power radix (sy - 2) * cy
                     + power radix (sy - 2) * (rl + radix * rh)
                    =  value xd (sy - 1)
                    + power radix (sy - 1) * !x1
                    - power radix sy * cy2 }
           value_sub_tail (pelts xd) xd.offset (xd.offset + p2i sy - 2);
           assert { value xd (sy - 1)
                    = value xd (sy - 2)
                      + power radix (sy - 2) * !x0
                    by (pelts xd)[xd.offset + sy - 2] = !x0
                    so value xd (sy - 1)
                       = value_sub (pelts xd) xd.offset (xd.offset + sy - 1)
                       = value_sub (pelts xd) xd.offset (xd.offset + sy - 2)
                         + power radix (sy - 2) * !x0
                       = value xd (sy - 2)
                         + power radix (sy - 2) * !x0 };
           assert { rl + radix * rh - cy
                     = !x0 + radix * !x1 - power radix 2 * cy2
                     by
                        (!x0 - radix * cy1 = rl - cy
                         by
                         !x0 = mod (rl - cy) radix
                         so - radix < rl - cy < radix
                         so (if rl < cy
                             then cy1 = 1
                                  /\ (- radix < rl - cy < 0
                                     so
                                     div (rl - cy) radix = - 1
                                     so rl - cy
                                     = radix * div (rl - cy) radix
                                       + mod (rl - cy) radix
                                     = !x0 - radix
                                     = !x0 - radix * cy1)
                             else cy1 = 0 /\ rl - cy = l2i !x0)) }
                    (* nonlinear *)
           end;
        end;
        if [@extraction:unlikely] (not (cy2 = 0))
        then begin
          label Adjust in
          assert { cy2 = 1 };
          begin ensures { !ql > 0 }
            value_sub_lower_bound (pelts y) y.offset (y.offset + p2i sy - 1);
            value_sub_tail (pelts y) y.offset (y.offset + p2i sy - 1);
            value_sub_upper_bound (pelts xd) xd.offset (xd.offset + p2i sy - 1);
            assert { !ql > 0
                      by
                        (value xd (sy - 1)
                         + power radix (sy - 1) * !x1
                         - power radix sy * cy2
                         < 0
                         by
                         value xd (sy - 1) < power radix (sy - 1)
                         so !x1 <= radix - 1
                         so  value xd (sy - 1)
                             + power radix (sy - 1) * !x1
                           < power radix (sy - 1)
                             + power radix (sy - 1) * !x1
                           = power radix (sy - 1) * (1 + !x1)
                           <= power radix (sy - 1) * radix
                           = power radix sy
                         so  value xd (sy - 1)
                             + power radix (sy - 1) * !x1
                             - power radix sy * cy2
                           < power radix sy - power radix sy * cy2
                           = 0
                       )
                      so value (xd at SubProd) sy
                         + power radix sy * (!x1 at StartLoop)
                         - !ql * vy
                         < 0
                      so (value (xd at SubProd) sy
                         + power radix sy * (!x1 at StartLoop) >= 0
                         by value (xd at SubProd) sy >= 0
                         so !x1 at StartLoop >= 0
                         so power radix sy * (!x1 at StartLoop) >= 0
                         )
                      so !ql * vy > 0
                      so vy = value_sub (pelts y)
                                y.offset (y.offset + sy - 1)
                              + power radix (sy - 1) * dh
                      so dh > 0
                      so vy > 0
                   };
          end;
          value_sub_tail (pelts y) y.offset (y.offset + p2i sy - 1);
          value_sub_tail (pelts y) y.offset (y.offset + p2i sy - 2);
          let ghost vly = value y (p2i sy - 2) in
          assert { vy = vly + power radix (sy - 2) * dl
                            + power radix (sy - 1) * dh
                   by (pelts y)[y.offset + sy - 1] = dh
                   so (pelts y)[y.offset + sy - 2] = dl
                   so
                      vy = value y (sy - 1)
                            + power radix (sy - 1) * dh
                         = vly + power radix (sy - 2) * dl
                            + power radix (sy - 1) * dh };
          begin
            ensures { value xd (sy - 1)
                      + power radix (sy - 1) * !x1
                      >= power radix sy - vy }
            assert { value xd (sy - 1)
                      + power radix (sy - 1) * !x1
                     = power radix sy + value (xd at SubProd) sy
                       + power radix sy * (!x1 at StartLoop)
                       - !ql * vy };
            assert {  value (xd at SubProd) sy
                       + power radix sy * (!x1 at StartLoop)
                       - !ql * vy
                      >= - vy
                      by
                         value (xd at SubProd) sy
                         = vlx + power radix (sy - 2) * (xp0 + radix * xp1)
                      so xp0 + radix * xp1 + radix * radix * (!x1 at StartLoop)
                         = !ql * (dl + radix * dh) + rl + radix * rh
                      so power radix (sy - 1) = power radix (sy - 2) * radix
                      so vy = vly + power radix (sy - 2) * (dl + radix * dh)
                      so (!ql * vly < vy
                           by
                          vly <= power radix (sy - 2)
                          so !ql < radix
                          so !ql * vly <= !ql * power radix (sy - 2)
                                       < radix * power radix (sy - 2)
                                       = power radix (sy - 1)
                          so vy = vly + power radix (sy - 2) * (dl + radix * dh)
                          so dh >= div radix 2 > 1
                          so vly >= 0
                          so dl >= 0
                          so vy >= power radix (sy - 2) * radix * dh
                                > power radix (sy - 2) * radix * 1
                                = power radix (sy - 1)
                          )
                      so - !ql * vly > - vy
                      so vlx >= 0
                      so power radix sy = power radix (sy - 2) * radix * radix
                      so value (xd at SubProd) sy
                       + power radix sy * (!x1 at StartLoop)
                       - !ql * vy
                         = vlx + power radix (sy - 2) * (xp0 + radix * xp1)
                           + power radix sy * (!x1 at StartLoop)
                           - !ql * vy
                         = vlx + power radix (sy - 2) * (xp0 + radix * xp1)
                           + power radix (sy - 2)
                             * radix * radix * (!x1 at StartLoop)
                           - !ql * vy
                         = vlx + power radix (sy - 2)
                                 * (xp0 + radix * xp1
                                     + radix * radix * (!x1 at StartLoop))
                           - !ql * vy
                         = vlx + power radix (sy - 2) *
                               (!ql * (dl + radix * dh) + rl + radix * rh)
                           - !ql * vy
                         = vlx + power radix (sy - 2) *
                               (!ql * (dl + radix * dh) + rl + radix * rh)
                           - !ql * (vly
                                    + power radix (sy - 2) * (dl + radix * dh))
                         = vlx + power radix (sy - 2) * (rl + radix * rh)
                           - !ql * vly
                         >= power radix (sy - 2) * (rl + radix * rh)
                            - !ql * vly
                         >= - !ql * vly > - vy
                   };
          end;
          let ghost xc = Array.copy (x.data) in
          assert { forall j. x.offset <= j < x.offset + !i
                          -> (pelts x)[j] = xc.elts[j]
                   by
                   0 <= x.offset <= j /\ j < x.offset + !i <= xc.Array.length
                   so 0 <= j < xc.Array.length
                 } ;
          value_sub_frame (pelts x) xc.elts x.offset (x.offset + p2i !i);
          let c = wmpn_add_n_in_place xd y (sy - 1) in
          begin
          ensures { value x !i
                 = value (x at Adjust) !i }
          assert { forall j. x.offset <= j < x.offset + !i
                          -> (pelts x)[j] = xc.elts[j]
                   by
                   pelts (xd at Adjust) = pelts (x at Adjust)
                   so pelts x = pelts xd
                   so (pelts x)[j] = (pelts x at Adjust)[j]
                   so
                   ((pelts x at Adjust)[j] = xc.elts[j]
                   by
                   0 <= j /\ j < xc.Array.length
                   ) } ;
          value_sub_frame (pelts x) xc.elts x.offset (x.offset + p2i !i);
          end;
          label MidAdd in
          begin
            ensures { value xd (sy - 1) + power radix (sy - 1) * !x1
                      = value (xd at Adjust) (sy - 1)
                      + power radix (sy - 1) * (!x1 at Adjust)
                      + vy
                      - power radix sy }
            assert { 0 <= c <= 1
                     by
                     value xd (sy - 1) + c * power radix (sy - 1)
                     = value (xd at Adjust) (sy - 1)
                       + value y (sy - 1)
                     so
                        value (xd at Adjust) (sy - 1)
                        < power radix (sy - 1)
                     so value y (sy - 1) < power radix (sy - 1)
                     so value xd (sy - 1) >= 0
                     so c * power radix (sy - 1) < 2 * power radix (sy - 1)
                     so let p = power radix (sy - 1) in
                        (c < 2 by c * p < 2 * p so p > 0)
                   };
            let ghost c' = div (l2i !x1 + l2i dh + l2i c) radix in
            x1 := add_mod !x1 (add_mod dh c);
            assert { !x1 + c' * radix = !x1 at Adjust + dh + c
                     by
                        (!x1 = mod (!x1 at Adjust + dh + c) radix
                        by
                        !x1 = mod (!x1 at Adjust + (mod (dh + c) radix)) radix
                        so mod (div (dh + c) radix * radix + !x1 at Adjust
                                + mod (dh + c) radix) radix
                           = mod (!x1 at Adjust + (mod (dh + c) radix)) radix
                        so !x1 = mod (div (dh + c) radix * radix + !x1 at Adjust
                                + mod (dh + c) radix) radix
                               = mod (!x1 at Adjust + dh + c) radix
                        )
                     so (!x1 at Adjust) + dh + c
                        = div (!x1 at Adjust + dh + c) radix * radix
                          + mod (!x1 at Adjust + dh + c) radix
                        = c' * radix + !x1
                     };
            assert { 0 <= c' <= 1 };
            value_sub_tail (pelts y) y.offset (y.offset + p2i sy - 1);
            assert { value xd (sy - 1) + power radix (sy - 1) * !x1
                     = value (xd at Adjust) (sy - 1)
                       + power radix (sy - 1) * (!x1 at Adjust)
                       + vy
                       - power radix sy
                     by
                     value xd (sy - 1) + power radix (sy - 1) * c
                     = value (xd at Adjust) (sy - 1)
                       + value y (sy - 1)
                     so vy = value y (sy - 1)
                             + power radix (sy - 1) * dh
                     so value xd (sy - 1) + power radix (sy - 1) * c
                         + power radix (sy - 1) * (!x1 at Adjust)
                         + power radix (sy - 1) * dh
                        = value (xd at Adjust) (sy - 1)
                           + value y (sy - 1)
                           + power radix (sy - 1) * (!x1 at Adjust)
                           + power radix (sy - 1) * dh
                        = value (xd at Adjust) (sy - 1)
                           + power radix (sy - 1) * (!x1 at Adjust)
                           + vy
                     so value xd (sy - 1) + power radix (sy - 1) * c
                         + power radix (sy - 1) * (!x1 at Adjust)
                         + power radix (sy - 1) * dh
                        = value xd (sy - 1)
                          + power radix (sy - 1) * (c + dh + !x1 at Adjust)
                        = value xd (sy - 1)
                          + power radix (sy - 1) * (!x1 + radix * c')
                        = value xd (sy - 1)
                          + power radix (sy - 1) * !x1
                          + power radix sy * c'
                      so  value xd (sy - 1)
                            + power radix (sy - 1) * !x1
                            + power radix sy * c'
                          = value (xd at Adjust) (sy - 1)
                            + power radix (sy - 1) * (!x1 at Adjust)
                            + vy
                      so value (xd at Adjust) (sy - 1)
                            + power radix (sy - 1) * (!x1 at Adjust)
                         >= power radix sy - vy
                      so value xd (sy - 1) < power radix (sy - 1)
                      so !x1 <= radix - 1
                      so power radix (sy - 1) * !x1
                         <= power radix (sy - 1) * (radix - 1)
                      so value xd (sy - 1)
                            + power radix (sy - 1) * !x1
                         <= value xd (sy - 1)
                            + power radix (sy - 1) * (radix - 1)
                         < power radix (sy - 1)
                           + power radix (sy - 1) * (radix - 1)
                         = power radix sy
                      so c' <> 0
                      so c' = 1
                      };
          end;
          ql := !ql - 1;
          (* todo refl *)
          assert { value xd (sy - 1) + power radix (sy - 1) * !x1
                   =  value (xd at SubProd) sy
                   + power radix sy * (!x1 at StartLoop)
                   - !ql * vy
                   by
                   value xd (sy - 1) + power radix (sy - 1) * !x1
                   = value (xd at Adjust) (sy - 1)
                       + power radix (sy - 1) * (!x1 at Adjust)
                       + vy
                       - power radix sy
                   = value (xd at SubProd) sy
                     + power radix sy * (!x1 at StartLoop)
                     - (!ql at Adjust) * vy
                     + vy
                   = value (xd at SubProd) sy
                     + power radix sy * (!x1 at StartLoop)
                     - (!ql + 1) * vy
                     + vy
                   = value (xd at SubProd) sy
                     + power radix sy * (!x1 at StartLoop)
                     - !ql * vy };
          qp.contents <- C.incr !qp (-1);
          value_sub_update_no_change (pelts q) (!qp).offset
                            ((!qp).offset + 1)
                            ((!qp).offset + p2i sx - p2i sy - p2i !i)
                            !ql;
          C.set !qp !ql;
          value_sub_head (pelts q) (!qp).offset
            ((!qp).offset + p2i sx - p2i sy - p2i !i);
          value_sub_tail (pelts x) x.offset (x.offset + p2i sy + p2i !i - 1);
          value_sub_concat (pelts x) x.offset xd.offset (x.offset + s);
          (* todo refl *)
          assert { value (old x) sx =
                  (value !qp (sx - sy - !i)
                   + qh * power radix (sx - sy - !i))
                  * vy * power radix !i
                  + value x (sy + !i - 1)
                  + power radix (sy + !i - 1) * !x1
                 by
                    value !qp (sx - sy - !i)
                    = !ql + radix *
                          value_sub (pelts q) ((!qp).offset + 1)
                                    ((!qp).offset + sx - sy - !i)
                 so (value_sub (pelts q) ((!qp).offset + 1)
                              ((!qp).offset + sx - sy - !i)
                    = value (!qp at StartLoop)
                              (sx - sy - k)
                     by
                     (!qp at StartLoop).offset = (!qp).offset + 1
                     so ((!qp).offset + sx - sy - !i)
                         - ((!qp).offset + 1)
                        = sx - sy - k
                     )
                 so value !qp (sx - sy - !i)
                    = !ql + radix * value (!qp at StartLoop)
                                    (sx - sy - k)
                 so (value x s
                    = value x !i
                      + power radix !i
                        * value xd (sy - 1)
                     by
                        xd.offset = x.offset + !i
                     so x.offset + s = xd.offset + sy - 1
                     so pelts x = pelts xd
                     so x.offset + s - xd.offset = sy - 1
                     so value_sub (pelts x) xd.offset (x.offset + s)
                        = value xd (sy - 1)
                     so value x s
                        = value_sub (pelts x) x.offset xd.offset
                          + power radix !i * value_sub (pelts x) xd.offset (x.offset + s)
                        = value x !i
                          + power radix !i * value xd (sy - 1)
                        )
                 so (power radix s
                    = power radix !i * power radix (sy - 1)
                    by
                    let n = !i in
                    let m = sy - 1 in
                    let x = radix in
                    power x s = power x (n + m)
                    so (power x (n + m) = power x n * power x m
                       by 0 <= n
                       so 0 <= m
                       so forall x:int, n:int, m:int.
                                 0 <= n -> 0 <= m -> power x (n + m) = (power x n * power x m)))
                 so (value x s + power radix s * !x1
                    = value x !i
                      + power radix !i *
                        (value (xd at SubProd) sy
                        + power radix sy * (!x1 at StartLoop)
                        - !ql * vy)
                    by  value xd (sy - 1)
                         + power radix (sy - 1) * !x1
                       = value (xd at SubProd) sy
                         + power radix sy * (!x1 at StartLoop)
                         - !ql * vy
                    so value x s + power radix s * !x1
                    = value x !i
                      + power radix !i
                        * value xd (sy - 1)
                      + power radix (!i + sy - 1) * !x1
                    = value x !i
                      + power radix !i
                        * value xd (sy - 1)
                      + power radix !i
                        * power radix (sy - 1) * !x1
                    = value x !i
                      + power radix !i *
                        (value xd (sy - 1)
                         + power radix (sy - 1) * !x1)
                    = value x !i
                      + power radix !i *
                        (value (xd at SubProd) sy
                        + power radix sy * (!x1 at StartLoop)
                        - !ql * vy)
                    )
                 so (value (x at StartLoop) (sy + k - 1)
                    = value (x at SubProd) !i
                      + power radix !i
                        * value (xd at SubProd) sy
                    by
                     value (x at StartLoop) (sy + k - 1)
                     = value_sub (pelts x at SubProd) (x at SubProd).offset
                                 ((x at SubProd).offset + sy + k - 1)
                     = value_sub (pelts x at SubProd) (x at SubProd).offset xd.offset
                       + power radix (xd.offset - (x at SubProd).offset)
                         * value_sub (pelts x at SubProd) xd.offset
                           ((x at SubProd).offset + sy + k - 1)
                     so (x at SubProd).offset = x.offset
                     so xd.offset = x.offset + !i
                     so value_sub (pelts x at SubProd) (x at SubProd).offset xd.offset
                        = value (x at SubProd) !i
                     so power radix (xd.offset - x.offset) = power radix !i
                     so x.offset + sy + k - 1 - xd.offset = p2i sy
                     so value_sub (pelts x at SubProd) xd.offset
                                  (x.offset + sy + k - 1)
                        = value (xd at SubProd) sy
                    )
                 so (value x !i
                    = value (x at SubProd) !i
                    by
                    value x !i
                    = value (x at Adjust) !i
                    = value (x at SubProd) !i
                    )
                 so power radix !i * power radix sy = power radix (!i + sy)
                 so value x s + power radix s * !x1
                    - value (x at StartLoop) (sy + k - 1)
                    = value x !i
                      + power radix !i *
                        (value (xd at SubProd) sy
                        + power radix sy * (!x1 at StartLoop)
                        - !ql * vy)
                      - (value (x at SubProd) !i
                         + power radix !i
                         * value (xd at SubProd) sy)
                    = value x !i
                      + power radix !i *
                        (value (xd at SubProd) sy
                        + power radix sy * (!x1 at StartLoop)
                        - !ql * vy)
                      - (value x !i
                         + power radix !i
                         * value (xd at SubProd) sy)
                    =  power radix !i
                       * (power radix sy * (!x1 at StartLoop)
                          - !ql * vy)
                    = power radix !i * power radix sy * (!x1 at StartLoop)
                      - power radix !i * !ql * vy
                    = power radix (!i + sy) * (!x1 at StartLoop)
                      - power radix !i * !ql * vy
                    = power radix (sy + k - 1) * (!x1 at StartLoop)
                      - power radix !i * !ql * vy
                 so value x s + power radix s * !x1
                    = value (x at StartLoop) (sy + k - 1)
                      + power radix (sy + k - 1) * (!x1 at StartLoop)
                      - power radix !i * !ql * vy
                 so power radix (sx - sy - !i)
                    = radix * power radix (sx - sy - k)
                 so radix * power radix !i = power radix k
                 so  (value !qp (sx - sy - !i)
                       + qh * power radix (sx - sy - !i))
                       * vy * power radix !i
                       + value x (sy + !i - 1)
                       + power radix (sy + !i - 1) * !x1
                    = (!ql + radix * value (!qp at StartLoop)
                                                      (sx - sy - k)
                       + qh * power radix (sx - sy - !i))
                       * vy * power radix !i
                       + value x (sy + !i - 1)
                       + power radix (sy + !i - 1) * !x1
                    = (!ql + radix * value (!qp at StartLoop)
                                                      (sx - sy - k)
                       + qh * radix * power radix (sx - sy - k))
                       * vy * power radix !i
                       + value x (sy + !i - 1)
                       + power radix (sy + !i - 1) * !x1
                    = !ql * vy * power radix !i
                      + (value (!qp at StartLoop)
                                                 (sx - sy - k)
                                 + qh * power radix (sx - sy - k))
                        * vy * radix * power radix !i
                      + value x (sy + !i - 1)
                      + power radix (sy + !i - 1) * !x1
                    = !ql * vy * power radix !i
                      + (value (!qp at StartLoop)
                                                 (sx - sy - k)
                                 + qh * power radix (sx - sy - k))
                        * vy * power radix k
                      + value x (sy + !i - 1)
                      + power radix (sy + !i - 1) * !x1
                    = !ql * vy * power radix !i
                      + (value (!qp at StartLoop)
                                                 (sx - sy - k)
                                 + qh * power radix (sx - sy - k))
                        * vy * power radix k
                      + value x s
                      + power radix s * !x1
                    = !ql * vy * power radix !i
                      + (value (!qp at StartLoop)
                                                 (sx - sy - k)
                                 + qh * power radix (sx - sy - k))
                        * vy * power radix k
                      +  value (x at StartLoop) (sy + k - 1)
                      + power radix (sy + k - 1) * (!x1 at StartLoop)
                      - power radix !i * !ql * vy
                    = (value (!qp at StartLoop)
                                                 (sx - sy - k)
                                 + qh * power radix (sx - sy - k))
                        * vy * power radix k
                      +  value (x at StartLoop) (sy + k - 1)
                      + power radix (sy + k - 1) * (!x1 at StartLoop)
                    = value (old x) sx
                };
        assert { value_sub (pelts x) (!xp.offset + mdn)
                 (!xp.offset + mdn + sy - 1)
                 + power radix (sy - 1) * !x1
                 < vy
                 by
                 (value xd (sy - 1) + power radix (sy - 1) * !x1 < vy
                   by
                   value xd (sy - 1) + power radix (sy - 1) * !x1
                   = value (xd at Adjust) (sy - 1)
                     + power radix (sy - 1) * (!x1 at Adjust)
                     + vy
                     - power radix sy
                   so value (xd at Adjust) (sy - 1)
                      < power radix (sy - 1)
                   so 1 + (!x1 at Adjust) <= radix
                   so value (xd at Adjust) (sy - 1)
                      + power radix (sy - 1) * (!x1 at Adjust)
                      + vy
                      - power radix sy
                    < power radix (sy - 1)
                      + power radix (sy - 1) * (!x1 at Adjust)
                      + vy
                      - power radix sy
                    = power radix (sy - 1) * (1 + !x1 at Adjust)
                      + vy
                      - power radix sy
                    <= power radix (sy - 1) * radix
                      + vy
                      - power radix sy
                    = vy
                 )
                 so pelts x = pelts xd
                 so xd.offset = !xp.offset + mdn
                 so value xd (sy - 1)
                    = value_sub (pelts x) (!xp.offset + mdn)
                                          (!xp.offset + mdn + sy - 1)
               };
        assert { dl + radix * dh
                 >= (pelts x)[(!xp).offset] + radix * !x1
                 by
                    vy = vly + power radix (sy - 2)
                             * (dl + radix * dh)
                 so value_sub (pelts x) (!xp.offset + mdn)
                              (!xp.offset + mdn + sy - 1)
                    + power radix (sy - 1) * !x1
                    < vy
                 so !xp.offset + mdn + sy - 1 = !xp.offset + 1
                 so power radix (sy - 1) = power radix (sy - 2) * radix
                 so - mdn = sy - 2
                 so vy
                    > value_sub (pelts x) (!xp.offset + mdn)
                                          (!xp.offset + mdn + sy - 1)
                      + power radix (sy - 1) * !x1
                    = value_sub (pelts x) (!xp.offset + mdn)
                                          (!xp.offset + 1)
                      + power radix (sy - 1) * !x1
                    = value_sub (pelts x) (!xp.offset + mdn) (!xp.offset)
                      + power radix (- mdn) * (pelts x)[(!xp).offset]
                      + power radix (sy - 1) * !x1
                    = value_sub (pelts x) (!xp.offset + mdn) (!xp.offset)
                      + power radix (sy - 2) * (pelts x)[(!xp).offset]
                      + power radix (sy - 1) * !x1
                    = value_sub (pelts x) (!xp.offset + mdn) (!xp.offset)
                      + power radix (sy - 2) * (pelts x)[(!xp).offset]
                      + power radix (sy - 2) * radix * !x1
                    = value_sub (pelts x) (!xp.offset + mdn) (!xp.offset)
                      + power radix (sy - 2)
                        * ((pelts x)[(!xp).offset] + radix * !x1)
                    >= power radix (sy - 2)
                        * ((pelts x)[(!xp).offset] + radix * !x1)
                 so vly < power radix (sy - 2)
                 so vy < power radix (sy - 2)
                         + power radix (sy - 2)
                           * (dl + radix * dh)
                         = power radix (sy - 2)
                           * (1 + dl + radix * dh)
                 so power radix (sy - 2)
                    * ((pelts x)[(!xp).offset] + radix * !x1)
                    < power radix (sy - 2) * (1 + dl + radix * dh)
                 so power radix (sy - 2)
                    * ((pelts x)[(!xp).offset] + radix * !x1
                      - (1 + dl + radix * dh))
                    < 0
                 so (pelts x)[(!xp).offset] + radix * !x1
                      - (1 + dl + radix * dh)
                    < 0
               };
        end
        else begin
          qp.contents <- C.incr !qp (-1);
          value_sub_update_no_change (pelts q) (!qp).offset
                            ((!qp).offset + 1)
                            ((!qp).offset + p2i sx - p2i sy - p2i !i)
                            !ql;
          C.set !qp !ql;
          value_sub_head (pelts q) (!qp).offset
            ((!qp).offset + p2i sx - p2i sy - p2i !i);
          assert { value !qp (sx - sy - !i) * vy
                    = !ql * vy + radix *
                          (value_sub (pelts q) ((!qp).offset + 1)
                                    ((!qp).offset + sx - sy - !i) * vy) }; (*nonlinear*)
          assert { value_sub (pelts q) ((!qp).offset + 1)
                                    ((!qp).offset + sx - sy - !i) * vy
                   =  (value !qp (sx - sy - !i) * vy at StartLoop) }; (*nonlinear*)
          value_tail x (sy + !i - 1);
          value_sub_concat (pelts x) x.offset xd.offset (x.offset + s);
          (* todo refl *)
          assert { cy2 = 0 };
          assert { value x !i = value (x at SubProd) !i };
          assert { value x s = value x !i + power radix !i * value xd (sy-1)
                     by xd.offset = x.offset + !i
                     so x.offset + s = xd.offset + sy - 1
                     so pelts x = pelts xd
                     so x.offset + s - xd.offset = sy - 1
                     so value_sub (pelts x) xd.offset (x.offset + s)
                        = value xd (sy - 1)
                     so value x s
                        = value_sub (pelts x) x.offset xd.offset
                          + power radix !i * value_sub (pelts x) xd.offset (x.offset + s)
                        = value x !i
                          + power radix !i * value xd (sy - 1)}; (*lifted from assertion*)
          assert { (value !qp (sx - sy - !i) + qh * power radix (sx - sy - !i))
                   * vy
                   = value !qp (sx - sy - !i) * vy
                     + qh * vy * power radix (sx - sy - !i)  }; (*nonlinear*)
          assert { ((value !qp (sx - sy - !i) + qh * power radix (sx - sy - !i))
                   * vy at StartLoop)
                   = (value !qp (sx - sy - !i) * vy
                     + qh * vy * power radix (sx - sy - !i) at StartLoop)  }; (*nonlinear*)
          assert { value x s = value x (sy + !i - 1) };
          assert { value (xd at SmallDiv) sy =
                    vlx + power radix (sy - 2) * xp0
                    + power radix (sy - 1) * xp1 };  (*nonlinear*)
          assert { value (x at SubProd) (sy + (!i at StartLoop) - 1)
                   = value (x at SubProd) !i + power radix !i * value (xd at SubProd) sy };
          assert { value (old x) sx =
                  (value !qp (sx - sy - !i)
                   + qh * power radix (sx - sy - !i))
                  * vy * power radix !i
                  + value x (sy + !i - 1)
                  + power radix (sy + !i - 1) * !x1 };
          value_sub_tail (pelts y) y.offset (y.offset + p2i sy - 1);
          value_sub_tail (pelts y) y.offset (y.offset + p2i sy - 2);
          let ghost vly = value y (p2i sy - 2) in
          assert { vy = vly + power radix (sy - 2) * dl
                            + power radix (sy - 1) * dh
                   by (pelts y)[y.offset + sy - 1] = dh
                   so (pelts y)[y.offset + sy - 2] = dl
                   so
                      vy = value y (sy - 1)
                            + power radix (sy - 1) * dh
                         = vly + power radix (sy - 2) * dl
                            + power radix (sy - 1) * dh };
          assert { value_sub (pelts x) (!xp.offset + mdn)
                   (!xp.offset + mdn + sy - 1)
                   + power radix (sy - 1) * !x1
                   < vy
                   by
                     pelts x = pelts xd
                   so xd.offset = !xp.offset + mdn
                   so !xp.offset + mdn + sy - 1 = xd.offset + sy - 1
                   so
                     value xd (sy - 1)
                     = value_sub (pelts xd) xd.offset (xd.offset + sy - 1)
                     = value_sub (pelts x) (!xp.offset + mdn)
                       (!xp.offset + mdn + sy - 1)
                   so value xd (sy - 1)
                      + power radix (sy - 1) * !x1
                      - power radix sy * cy2
                      = value (xd at SubProd) sy
                      + power radix sy * (!x1 at StartLoop)
                      - !ql * vy
                   so cy2 = 0
                   so value xd (sy - 1)
                      + power radix (sy - 1) * !x1
                      = value (xd at SubProd) sy
                      + power radix sy * (!x1 at StartLoop)
                      - !ql * vy
                   so !ql * (dl + radix * dh)
                      + (rl + radix * rh)
                      = xp0
                        + radix * xp1
                        + radix * radix * (!x1 at StartLoop)
                   so vy = vly + power radix (sy - 2)
                                 * (dl + radix * dh)
                   so !ql * vy
                      = power radix (sy - 2) *
                          (xp0
                          + radix * xp1
                          + radix * radix * (!x1 at StartLoop))
                        - power radix (sy - 2) * (rl + radix * rh)
                        + !ql * vly
                   so value (xd at SubProd) sy
                       = vlx
                        + power radix (sy - 2) * (xp0 + radix * xp1)
                   so power radix sy
                      = power radix (sy - 2) * radix * radix
                   so (value (xd at SubProd) sy
                      + power radix sy * (!x1 at StartLoop)
                      - !ql * vy
                      < vy
                      by
                      (!ql * vly >= 0
                       by !ql >= 0 so vly >= 0)
                      so (power radix (sy - 2) * (rl + radix * rh)
                         <= power radix (sy - 2)
                            * (dl + radix * dh)
                            - power radix (sy - 2)
                         by
                         rl + radix * rh <= dl + radix * dh - 1
                         so power radix (sy - 2) >= 0
                         so power radix (sy - 2) * (rl + radix * rh)
                            <= power radix (sy - 2)
                               * (dl + radix * dh - 1)
                            = power radix (sy - 2)
                              * (dl + radix * dh)
                              - power radix (sy - 2)
                         )
                      so vlx < power radix (sy - 2)
                      so value (xd at SubProd) sy
                        + power radix sy * (!x1 at StartLoop)
                        - !ql * vy
                      = vlx
                        + power radix (sy - 2) * (xp0 + radix * xp1)
                        + power radix sy * (!x1 at StartLoop)
                        - !ql * vy
                      = vlx
                        + power radix (sy - 2) *
                                (xp0 + radix * xp1
                                  + radix * radix * (!x1 at StartLoop))
                        - !ql * vy
                      = vlx
                        + power radix (sy - 2) *
                                (xp0 + radix * xp1
                                  + radix * radix * (!x1 at StartLoop))
                        -  (power radix (sy - 2) *
                           (xp0
                            + radix * xp1
                            + radix * radix * (!x1 at StartLoop))
                          - power radix (sy - 2) * (rl + radix * rh)
                          + !ql * vly)
                      = vlx
                        + power radix (sy - 2) * (rl + radix * rh)
                        - !ql * vly
                      <= vlx
                        + power radix (sy - 2) * (rl + radix * rh)
                      <= vlx
                         +  power radix (sy - 2)
                            * (dl + radix * dh)
                         - power radix (sy - 2)
                      <  power radix (sy - 2)
                         +  power radix (sy - 2)
                            * (dl + radix * dh)
                         - power radix (sy - 2)
                      = power radix (sy - 2) * (dl + radix * dh)
                      = vy - vly <= vy
                      )
                    so value_sub (pelts x) (!xp.offset + mdn)
                       (!xp.offset + mdn + sy - 1)
                       + power radix (sy - 1) * !x1
                       = value xd (sy - 1)
                         + power radix (sy - 1) * !x1
                       = value (xd at SubProd) sy
                        + power radix sy * (!x1 at StartLoop)
                        - !ql * vy
                       < vy
                 };
          value_sub_tail (pelts x) (!xp.offset + p2i mdn) (!xp.offset);
          value_sub_upper_bound (pelts y) (y.offset) (y.offset + p2i sy - 2);
          value_sub_lower_bound (pelts x) (!xp.offset + p2i mdn) (!xp.offset);
          assert { dl + radix * dh
                   >= (pelts x)[(!xp).offset] + radix * !x1
                   by
                   vy = vly + power radix (sy - 2)
                              * (dl + radix * dh)
                   so value_sub (pelts x) (!xp.offset + mdn)
                   (!xp.offset + mdn + sy - 1)
                   + power radix (sy - 1) * !x1
                   < vy
                   so !xp.offset + mdn + sy - 1 = !xp.offset + 1
                   so power radix (sy - 1) = power radix (sy - 2) * radix
                   so - mdn = sy - 2
                   so vy
                      > value_sub (pelts x) (!xp.offset + mdn)
                                            (!xp.offset + mdn + sy - 1)
                        + power radix (sy - 1) * !x1
                      = value_sub (pelts x) (!xp.offset + mdn)
                                            (!xp.offset + 1)
                        + power radix (sy - 1) * !x1
                      = value_sub (pelts x) (!xp.offset + mdn) (!xp.offset)
                        + power radix (- mdn) * (pelts x)[(!xp).offset]
                        + power radix (sy - 1) * !x1
                      = value_sub (pelts x) (!xp.offset + mdn) (!xp.offset)
                        + power radix (sy - 2) * (pelts x)[(!xp).offset]
                        + power radix (sy - 1) * !x1
                      = value_sub (pelts x) (!xp.offset + mdn) (!xp.offset)
                        + power radix (sy - 2) * (pelts x)[(!xp).offset]
                        + power radix (sy - 2) * radix * !x1
                      = value_sub (pelts x) (!xp.offset + mdn) (!xp.offset)
                        + power radix (sy - 2)
                          * ((pelts x)[(!xp).offset] + radix * !x1)
                      >= power radix (sy - 2)
                          * ((pelts x)[(!xp).offset] + radix * !x1)
                   so vly < power radix (sy - 2)
                   so vy < power radix (sy - 2)
                           + power radix (sy - 2)
                              * (dl + radix * dh)
                           = power radix (sy - 2)
                             * (1 + dl + radix * dh)
                   so power radix (sy - 2)
                      * ((pelts x)[(!xp).offset] + radix * !x1)
                      < power radix (sy - 2) * (1 + dl + radix * dh)
                   so power radix (sy - 2)
                      * ((pelts x)[(!xp).offset] + radix * !x1
                        - (1 + dl + radix * dh))
                      < 0
                   so (pelts x)[(!xp).offset] + radix * !x1
                        - (1 + dl + radix * dh)
                      < 0
                 };
        end;
      end;
    done;
    label EndLoop in
    assert { !i = 0 };
    assert { !xp.offset = x.offset + sy - 2 };
    value_sub_update_no_change (pelts x) (!xp.offset + 1)
                               x.offset (!xp.offset) !x1;
    C.set_ofs !xp 1 !x1;
    assert { value x (sy - 1) =
             value (x at EndLoop) (sy - 1)
             by pelts x = Map.set (pelts x at EndLoop) (x.offset + sy - 1) !x1 };
    value_sub_tail (pelts x) x.offset (!xp.offset+1);
    (* todo refl *)
    assert { value (old x) sx =
              (value q (sx - sy)
              + power radix (sx - sy) * qh)
                * value y sy
              + value x sy
             by
               value x sy
                = value x (sy - 1)
                  + power radix (sy - 1) * !x1
             so vy = value y sy
             so value (old x) sx
                = (value !qp (sx - sy - !i)
                  + qh * power radix (sx - sy - !i))
                  * vy * power radix !i
                  + value x (sy + !i - 1)
                  + power radix (sy + !i - 1) * !x1
                = (value !qp (sx - sy)
                  + qh * power radix (sx - sy))
                  * vy * 1
                  + value x (sy - 1)
                  + power radix (sy - 1) * !x1
                = (value !qp (sx - sy)
                  + qh * power radix (sx - sy))
                  * value y sy
                  + value x sy };
    qh

  let wmpn_divrem_2 (q x y:t) (sx:int32) : limb
    requires { 2 <= sx }
    requires { valid x sx }
    requires { valid y 2 }
    requires { valid q (sx - 2) }
    requires { (pelts y)[y.offset + 1] >= div radix 2 }
    requires { writable q /\ writable x }
    ensures { value (old x) sx =
              (value q (sx - 2)
               + power radix (sx - 2) * result)
              * value y 2
              + value x 2 }
    ensures { value x 2 < value y 2 }
    ensures { 0 <= result <= 1 }
  =
    let xp = ref (C.incr x (sx - 2)) in
    let dh = C.get_ofs y 1 in
    let dl = C.get y in
    let rh = ref (C.get_ofs !xp 1) in
    let rl = ref (C.get !xp) in
    let qh = ref 0 in
    let lx = ref 0 in
    assert { value y 2 = dl + radix * dh };
    let i = ref (sx - 2) in
    let dinv = reciprocal_word_3by2 dh dl in
    ([@vc:sp] if (!rh >= dh && ([@vc:sp] !rh > dh || !rl >= dl))
    then
      label Adjust in
      begin
        ensures { value x sx
                  = (value_sub (pelts q) (q.offset + !i) (q.offset + sx - 2)
                      + !qh * power radix (sx - 2 - !i))
                    * value y 2 * power radix !i
                    + value x !i
                    + power radix !i * (!rl + radix * !rh) }
        ensures { !rl + radix * !rh < dl + radix * dh }
        ensures { !qh = 1 }
        let (r0, b) = sub_with_borrow !rl dl 0 in
        let (r1, ghost b') = sub_with_borrow !rh dh b in
        assert { b' = 0 };
        assert { r0 + radix * r1 = !rl + radix * !rh - (dl + radix * dh) };
        value_sub_tail (pelts x) x.offset (x.offset + p2i sx - 1);
        value_sub_tail (pelts x) x.offset (x.offset + p2i sx - 2);
        rh := r1;
        rl := r0;
        qh := 1;
        assert { value x sx
                  = (value_sub (pelts q) (q.offset + !i) (q.offset + sx - 2)
                      + !qh * power radix (sx - 2 - !i))
                    * value y 2 * power radix !i
                    + value x !i
                    + power radix !i * (!rl + radix * !rh)
                 by
                 value_sub (pelts q) (q.offset + !i) (q.offset + sx - 2) = 0
                 so (value_sub (pelts q) (q.offset + !i) (q.offset + sx - 2)
                      + !qh * power radix (sx - 2 - !i))
                    * value y 2 * power radix !i
                    + value x !i
                    + power radix !i * (!rl + radix * !rh)
                  = value y 2 * power radix !i
                    + value x !i
                    + power radix !i * (!rl + radix * !rh)
                  = value x !i
                    + power radix !i * (dl + radix * dh + !rl + radix * !rh)
                  = value x !i
                    + power radix !i * (!rl at Adjust + radix * !rh at Adjust)
                  = value x !i
                    + power radix !i * !rl at Adjust
                    + power radix (!i+1) * !rh at Adjust
                  = value x sx
               };
      end
    else
      begin
        ensures { value x sx
                  = (value_sub (pelts q) (q.offset + !i) (q.offset + sx - 2)
                      + !qh * power radix (sx - 2 - !i))
                    * value y 2 * power radix !i
                    + value x !i
                    + power radix !i * (!rl + radix * !rh) }
        ensures { !rl + radix * !rh < dl + radix * dh }
        ensures { !qh = 0 }
        value_sub_tail (pelts x) x.offset (x.offset + p2i sx - 1);
        value_sub_tail (pelts x) x.offset (x.offset + p2i sx - 2);
      end);
    while (!i > 0) do
      variant { !i }
      invariant { 0 <= !i <= sx - 2 }
      invariant { !xp.offset = x.offset + !i }
      invariant { plength !xp = plength x }
      invariant { !xp.min = x.min }
      invariant { !xp.max = x.max }
      invariant { pelts !xp = pelts x }
      invariant { writable !xp }
      invariant { value x sx
                  = (value_sub (pelts q) (q.offset + !i) (q.offset + sx - 2)
                      + !qh * power radix (sx - 2 - !i))
                    * value y 2 * power radix !i
                    + value x !i
                    + power radix !i * (!rl + radix * !rh) }
      invariant { !rl + radix * !rh < dl + radix * dh }
      label StartLoop in
      let ghost k = p2i !i in
      xp.contents <- C.incr !xp (-1);
      lx := C.get !xp;
      label Got in
      let (qu, r0, r1) = div3by2_inv !rh !rl !lx dh dl dinv in
      rh := r1;
      rl := r0;
      i := !i - 1;
      C.set_ofs q !i qu;
      assert { qu * (dl + radix * dh) + r0 + radix * r1
               = !lx + radix * (!rl at StartLoop)
                     + radix * radix * (!rh at StartLoop)
               by
                 radix * ((!rl at StartLoop) + radix * (!rh at StartLoop))
                 = radix * (!rl at StartLoop) + radix * radix * (!rh at StartLoop)
               so
                 qu * (dl + radix * dh) + r0 + radix * r1
               = !lx + radix * ((!rl at StartLoop) + radix * (!rh at StartLoop))
               = !lx + radix * (!rl at StartLoop)
                     + radix * radix * (!rh at StartLoop)
             };
      value_sub_head (pelts q) (q.offset + p2i !i) (q.offset + p2i sx - 2);
      value_sub_tail (pelts x) x.offset (x.offset + p2i !i);
      assert { value x sx
                  = (value_sub (pelts q) (q.offset + !i) (q.offset + sx - 2)
                      + !qh * power radix (sx - 2 - !i))
                    * value y 2 * power radix !i
                    + value x !i
                    + power radix !i * (!rl + radix * !rh)
               by
                  value x k = value x !i + power radix !i * !lx
               so value_sub (pelts q) (q.offset + !i) (q.offset + sx - 2)
                 = qu + radix
                        * value_sub (pelts q) (q.offset + k) (q.offset + sx - 2)
               so power radix (sx - 2 - !i) = radix * power radix (sx - 2 - k)
               so
                 (value_sub (pelts q) (q.offset + !i) (q.offset + sx - 2)
                        + !qh * power radix (sx - 2 - !i))
                 = qu + radix
                        * (value_sub (pelts q) (q.offset + k) (q.offset + sx - 2)
                           + !qh * power radix (sx - 2 - k))
               so power radix !i * radix  = power radix k
               so ((value_sub (pelts q) (q.offset + !i) (q.offset + sx - 2)
                      + !qh * power radix (sx - 2 - !i))
                    * value y 2 * power radix !i
                  = power radix !i * qu * (dl + radix * dh)
                    + (value_sub (pelts q) (q.offset + k)
                                                   (q.offset + sx - 2)
                               + !qh * power radix (sx - 2 - k))
                      * value y 2 * power radix k
                  by
                  (value_sub (pelts q) (q.offset + !i) (q.offset + sx - 2)
                      + !qh * power radix (sx - 2 - !i))
                    * value y 2 * power radix !i
                  = (qu + radix
                         * (value_sub (pelts q) (q.offset + k)
                                                (q.offset + sx - 2)
                             + !qh * power radix (sx - 2 - k)))
                    * value y 2 * power radix !i
                  = power radix !i * qu * (dl + radix * dh)
                    + radix * (value_sub (pelts q) (q.offset + k)
                                                   (q.offset + sx - 2)
                               + !qh * power radix (sx - 2 - k))
                      * value y 2 * power radix !i
                  = power radix !i * qu * (dl + radix * dh)
                    + (value_sub (pelts q) (q.offset + k)
                                                   (q.offset + sx - 2)
                               + !qh * power radix (sx - 2 - k))
                      * value y 2 * power radix k)
               so (value_sub (pelts q) (q.offset + !i) (q.offset + sx - 2)
                      + !qh * power radix (sx - 2 - !i))
                    * value y 2 * power radix !i
                    + value x !i
                    + power radix !i * (!rl + radix * !rh)
                  = power radix !i * qu * (dl + radix * dh)
                    + (value_sub (pelts q) (q.offset + k)
                                           (q.offset + sx - 2)
                               + !qh * power radix (sx - 2 - k))
                      * value y 2 * power radix k
                    + value x !i
                    + power radix !i * (!rl + radix * !rh)
                  = (value_sub (pelts q) (q.offset + k)
                                           (q.offset + sx - 2)
                               + !qh * power radix (sx - 2 - k))
                      * value y 2 * power radix k
                    + value x !i
                    + power radix !i * (qu * (dl + radix * dh)
                                        + !rl + radix * !rh)
                  = (value_sub (pelts q) (q.offset + k)
                                           (q.offset + sx - 2)
                               + !qh * power radix (sx - 2 - k))
                      * value y 2 * power radix k
                    + value x !i
                    + power radix !i
                      * (!lx + radix * (!rl at StartLoop)
                         + radix * radix * (!rh at StartLoop))
                  = (value_sub (pelts q) (q.offset + k)
                                           (q.offset + sx - 2)
                               + !qh * power radix (sx - 2 - k))
                      * value y 2 * power radix k
                    + value x !i
                    + power radix !i * !lx
                    + power radix !i * (radix * (!rl at StartLoop
                                                 + radix * !rh at StartLoop))
                  = (value_sub (pelts q) (q.offset + k)
                                           (q.offset + sx - 2)
                               + !qh * power radix (sx - 2 - k))
                      * value y 2 * power radix k
                    + value x k
                    + power radix !i * (radix * (!rl at StartLoop
                                                 + radix * !rh at StartLoop))
                  = (value_sub (pelts q) (q.offset + k)
                                           (q.offset + sx - 2)
                               + !qh * power radix (sx - 2 - k))
                      * value y 2 * power radix k
                    + value x k
                    + power radix k * (!rl at StartLoop
                                       + radix * !rh at StartLoop)
                  = value x sx
          };
    done;
    assert { !i = 0 };
    assert { value x sx
             = (value_sub (pelts q) q.offset (q.offset + sx - 2)
                      + !qh * power radix (sx - 2))
                    * value y 2
                    + !rl + radix * !rh
             by power radix !i = 1 };
    C.set_ofs x 1 !rh;
    C.set x !rl;
    assert { value x 2 = !rl + radix * !rh
             by (pelts x)[x.offset] = !rl
                /\ (pelts x)[x.offset + 1] = !rh};
    !qh

  (** `div_qr q r x y sx sy` divides `(x, sx)` by `(y, sy)`, writes the quotient
    in `(q, (sx-sy))` and the remainder in `(r, sy)`. Corresponds to
    `mpn_tdiv_qr`. *)
  let div_qr (q r x y nx ny:t) (sx sy:int32) : unit
    requires { 1 <= sy <= sx <= (Int32.max_int32 - 1) }
    requires { valid x sx }
    requires { valid y sy }
    requires { valid q (sx - sy + 1) }
    requires { valid r sy }
    requires { valid nx (sx + 1) }
    requires { valid ny sy }
    requires { writable nx /\ writable ny }
    requires { (pelts y)[y.offset + sy - 1] > 0 }
    requires { writable q /\ writable r }
    ensures { value x sx
              = value q (sx - sy + 1) * value y sy
                + value r sy }
    ensures { value r sy < value y sy }
  =
    label Start in
    value_sub_tail (pelts y) y.offset (y.offset + p2i sy - 1);
    value_sub_lower_bound (pelts y) y.offset (y.offset + p2i sy - 1);
    assert { value y sy >= power radix (sy - 1) };
    if (sy = 1)
    then
      let lr = wmpn_divrem_1 q x sx (C.get y) in
      C.set r lr
    else
    if (sy = 2)
    then
      let clz = clz_ext (C.get_ofs y (sy - 1)) in
      let ghost p = power 2 (p2i clz) in
      if clz = 0
      then begin
        wmpn_copyi nx x sx;
        value_sub_shift_no_change (pelts x) x.offset (p2i sx) (p2i sx) 0;
        C.set_ofs nx sx 0;
        value_sub_frame_shift (pelts x) (pelts nx) x.offset nx.offset (p2i sx);
        label Div2_ns in
        let (ghost _qh) = wmpn_divrem_2 q nx y (sx + 1) in
        wmpn_copyi r nx sy;
        assert { value x sx
                 = value q (sx - sy + 1) * value y sy
                 + value r sy
               by value r sy = value nx sy
               so value (nx at Div2_ns) (sx + 1) < power radix sx
               so value (nx at Div2_ns) (sx + 1)
                  = value (nx at Div2_ns) sx
               so (_qh = 0
                   by
                   power radix sx
                   > value (nx at Div2_ns) (sx + 1)
                   = (value q (sx - 1) + power radix (sx - 1) * _qh)
                    * value y 2
                    + value nx 2
                   so value nx 2 >= 0
                   so value y 2 >= radix
                   so value q (sx - 1) >= 0
                   so _qh >= 0
                   so (value q (sx - 1)
                           + power radix (sx - 1) * _qh) >= 0
                   so (value q (sx - 1) + power radix (sx - 1) * _qh)
                      * value y 2
                      + value nx 2
                      >= (value q (sx - 1)
                           + power radix (sx - 1) * _qh)
                         * value y 2
                      >= (value q (sx - 1)
                           + power radix (sx - 1) * _qh)
                         * radix
                      >= power radix (sx - 1) * _qh * radix
                      = power radix sx * _qh
                   so power radix sx > power radix sx * _qh
                  )
               so value x sx = value (nx at Div2_ns) sx
               };
        ()
      end
      else begin
        let (ghost _c) = wmpn_lshift ny y sy (Limb.of_int32 clz) in
        begin
          ensures { normalized ny sy }
          ensures { value ny sy = power 2 clz * value y sy }
          let ghost dh = (pelts y)[y.offset + p2i sy - 1] in
          assert { value y sy
                   = value y (sy - 1) + power radix (sy - 1) * dh };
          value_sub_upper_bound (pelts y) y.offset (y.offset + p2i sy - 1);
          value_sub_tail (pelts ny) ny.offset (ny.offset + p2i sy - 1);
          value_sub_upper_bound (pelts ny) ny.offset (ny.offset + p2i sy - 1);
          let ghost ndh = (pelts ny)[ny.offset + p2i sy - 1] in
          assert { normalized ny sy
                   /\ value ny sy = power 2 clz * value y sy
                   by
                   value y sy < (dh + 1) * power radix (sy - 1)
                   so value ny sy + (power radix sy) * _c
                      = power 2 clz * value y sy
                      = power 2 clz
                        * (value y (sy - 1)
                           + dh * power radix (sy - 1))
                   so power 2 clz * dh <= radix - power 2 clz
                   so value ny sy + (power radix sy) * _c
                      = power 2 clz * value y (sy - 1)
                        + power 2 clz * dh * power radix (sy - 1)
                      < power 2 clz * power radix (sy - 1)
                        + power 2 clz * dh * power radix (sy - 1)
                      <= power 2 clz * power radix (sy - 1)
                        + (radix - power 2 clz) * power radix (sy - 1)
                       = radix * power radix (sy - 1)
                       = power radix sy
                   so _c = 0
                   so value ny sy
                      = power 2 clz * value y sy
                   so value y sy >= dh * power radix (sy - 1)
                   so value ny sy
                      >= power 2  clz * dh * power radix (sy - 1)
                   so value ny sy =
                      value ny (sy - 1) + power radix (sy - 1) * ndh
                      < power radix (sy - 1) + power radix (sy - 1) * ndh
                      = power radix (sy - 1) * (ndh + 1)
                   so power radix (sy - 1) * (ndh + 1)
                      > power radix (sy - 1) * (power 2 clz * dh)
                   so ndh + 1 > power 2 clz * dh
                   so ndh >= power 2 clz * dh
                   so 2 * power 2 clz * dh >= radix
                   so 2 * ndh >= radix
                   so ndh >= div radix 2
                 };
        end;
        let h = wmpn_lshift nx x sx (Limb.of_int32 clz) in
        C.set_ofs nx sx h;
        begin
          ensures { value nx (sx + 1)
                    = p * value x sx  }
          value_sub_tail (pelts nx) nx.offset (nx.offset + p2i sx);
          assert { value nx (sx + 1)
                    = p * value x sx
                   by
                   value nx sx + power radix sx * h
                    = p * value x sx
                   so value nx (sx + 1)
                      = value nx sx + power radix sx * h
                 }
        end;
        label Div2_s in
        (* TODO don't add 1 when not needed, cf "adjust" in GMP algo *)
        let (ghost _qh) = wmpn_divrem_2 q nx ny (sx + 1) in
        let (ghost _l) = wmpn_rshift r nx sy (Limb.of_int32 clz) in
        begin ensures { value nx 2 = p * value r 2 }
          assert { _l = 0
                 by
                   (mod (value nx sy) p = 0
                    by
                      value (nx at Div2_s) (sx + 1)
                   = (value q (sx - 1) + power radix (sx - 1) * _qh)
                      * value ny sy
                      + value nx sy
                   so value (nx at Div2_s) (sx + 1)
                    = p * value x sx
                   so value ny sy = p * value y sy
                   so value nx sy
                      = value (nx at Div2_s) (sx + 1)
                        - (value q (sx - 1)
                          + power radix (sx - 1) * _qh)
                          * value ny sy
                      = p * value x sx
                        - p * (value q (sx - 1)
                               + power radix (sx - 1) * _qh)
                            * value y sy
                      = p * (value x sx
                             - (value q (sx - 1)
                               + power radix (sx - 1) * _qh)
                               * value y sy)
                   so let n = (value x sx
                                - (value q (sx - 1)
                                   + power radix (sx - 1) * _qh)
                                   * value y sy)
                      in
                      value nx sy = p * n
                      so value nx sy >= 0
                      so p > 0
                      so n >= 0
                      so mod (value nx sy) p
                         = mod (p * n) p
                         = mod ((p*n)+0) p
                         = mod 0 p
                         = 0
                 )
                 so _l + radix * value r sy
                    = power 2 (Limb.length - clz) * (value nx sy)
                 so let a = div (value nx sy) p in
                    value nx sy = p * a
                 so power 2 (Limb.length - clz) * p = radix
                 so power 2 (Limb.length - clz) * (value nx sy)
                    = power 2 (Limb.length - clz) * (p * a)
                    = (power 2 (Limb.length - clz) * p) * a
                    = radix * a
                 so mod (radix * value r sy + _l) radix
                    = mod _l radix
                 so mod (radix * value r sy + _l) radix
                    = mod (radix * a) radix = 0
                 so mod _l radix = 0
                 so 0 <= _l < radix
               };
          assert { value nx 2 = p * value r 2
                   by
                   radix * value r 2
                   = power 2 (Limb.length - clz) * value nx 2
                   so p * power 2 (Limb.length - clz)
                      = radix
                   so p * radix * value r 2
                      = p * power 2 (Limb.length - clz) * value nx 2
                      = radix * value nx 2
                   so p * value r 2 = value nx 2
                 }
        end;
        assert { value x sx
                 = value q (sx - sy + 1) * value y sy
                 + value r sy
                by
                 value (nx at Div2_s) (sx + 1)
                 = (value q (sx - 1) + power radix (sx - 1) * _qh)
                   * value ny 2
                   + value nx 2
                so value (nx at Div2_s) (sx + 1)
                    = p * value x sx
                so value ny 2 = p * value y 2
                so (_qh = 0
                    by
                    value x sx < power radix sx
                    so value y 2 >= radix
                    so value ny 2 >= p * radix
                    so value q (sx - 1) >= 0
                    so value nx 2 >= 0
                    so (value q (sx - 1) + power radix (sx - 1) * _qh)
                        >= 0
                    so (value q (sx - 1) + power radix (sx - 1) * _qh)
                         * value ny 2
                       + value nx 2
                       >= (value q (sx - 1)
                           + power radix (sx - 1) * _qh)
                          * value ny 2
                       >= (value q (sx - 1)
                           + power radix (sx - 1) * _qh)
                          * (p * radix)
                       >= power radix (sx - 1) * _qh * p * radix
                       = power radix sx * p * _qh
                    so power radix sx * p
                       > value (nx at Div2_s) (sx + 1)
                       >= power radix sx * p * _qh
                    )
                so value nx 2 = p * value r 2
                so p * value x sx
                   = value q (sx - 1) * p * value y 2
                     + p * value r 2
                   = p * (value q (sx - 1) * value y 2
                          + value r 2)
               };
        ()
      end
    else
     (* let qn = ref (Int32.(-) (Int32.(+) sx 1) sy) in
      if (Int32.(>=)  (Int32.(+) !qn !qn) sx)
      then*) begin
        let adjust =
          if (get_ofs x (sx - 1)) >= (get_ofs y (sy - 1))
          then 1
          else 0
        in
        let clz = clz_ext (C.get_ofs y (sy - 1)) in
        let ghost p = power 2 (p2i clz) in
        if clz = 0
        then begin
          wmpn_copyi nx x sx;
          value_sub_shift_no_change (pelts x) x.offset
                                    (p2i sx) (p2i sx) 0;
          C.set_ofs nx sx 0;
          value_sub_frame_shift (pelts x) (pelts nx) x.offset nx.offset (p2i sx);
          assert { value y sy * (power radix (sx - sy + adjust))
                   > value nx (sx + adjust)
                   by
                   let dh = (pelts y)[y.offset + sy - 1] in
                   value y sy >= dh * power radix (sy - 1)
                   so value nx (sx + adjust) = value nx sx = value x sx
                   so [@case_split]
                      ((adjust = 1
                           so value x sx < power radix sx
                           so value y sy * power radix (sx - sy + adjust)
                              >= dh * power radix (sy - 1)
                                    * power radix (sx - sy + adjust)
                              = dh * power radix ((sy - 1) + (sx - sy + adjust))
                              = dh * power radix sx
                           so dh >= div radix 2 > 1
                           so dh * power radix sx > power radix sx )
                      \/
                      (adjust = 0
                           so let ah = (pelts x)[x.offset + sx - 1] in
                              value x sx < (ah + 1) * power radix (sx - 1)
                              so ah + 1 <= dh
                              so value x sx < dh * power radix (sx - 1)
                              so value y sy * power radix (sx - sy + adjust)
                                 = value y sy * power radix (sx - sy)
                                 >= dh * power radix (sy - 1)
                                       * power radix (sx - sy)
                                 = dh * power radix (sy - 1 + sx - sy)
                                 = dh * power radix (sx - 1))) };
          label Div_ns in
          let (ghost _qh) = div_sb_qr q nx (sx + adjust) y sy in
          wmpn_copyi r nx sy;
          assert { value x sx
                   = value q (sx - sy + adjust) * value y sy
                   + value r sy
                 by value r sy = value nx sy
                 so value (nx at Div_ns) (sx + adjust) = value x sx < power radix sx
                 so value (nx at Div_ns) (sx + adjust)
                    = value (nx at Div_ns) sx
                 so (_qh = 0
                     by
                     value (nx at Div_ns) (sx + adjust)
                     = (value q (sx - sy + adjust)
                        + power radix (sx - sy + adjust) * _qh)
                      * value y sy
                      + value nx sy
                     so value nx sy >= 0
                     so value q (sx - sy + adjust) >= 0
                     so _qh >= 0
                     so (value q (sx - sy + adjust)
                             + power radix (sx - sy + adjust) * _qh) >= 0
                     so (value q (sx - sy + adjust)
                         + power radix (sx - sy + adjust) * _qh)
                        * value y sy
                        + value nx sy
                        >= (value q (sx - sy + adjust)
                             + power radix (sx - sy + adjust) * _qh)
                           * value y sy
                        >= power radix (sx - sy + adjust) * _qh * value y sy
                     so _qh <> 1)
                 so value x sx = value (nx at Div_ns) sx
                 };
           label Ret_ns in
           begin
             ensures { value q (sx - sy + 1)
                       = value (q at Ret_ns) (sx - sy + adjust) }
             if (adjust = 0)
             then begin
               value_sub_shift_no_change (pelts x) x.offset
                                         (p2i sx) (p2i sx) 0;
               set_ofs q (sx - sy) 0;
               value_sub_tail (pelts q) q.offset (q.offset + p2i sx - p2i sy);
               ()
             end
           end
        end
        else begin
          let (ghost _c) = wmpn_lshift ny y sy (Limb.of_int32 clz) in
          begin
            ensures { normalized ny sy }
            ensures { value ny sy
                      = power 2 clz * value y sy }
            let ghost dh = (pelts y)[y.offset + p2i sy - 1] in
            assert { value y sy
                     = value y (sy - 1) + power radix (sy - 1) * dh };
            value_sub_upper_bound (pelts y) y.offset (y.offset + p2i sy - 1);
            value_sub_tail (pelts ny) ny.offset (ny.offset + p2i sy - 1);
            value_sub_upper_bound (pelts ny) ny.offset (ny.offset + p2i sy - 1);
            let ghost ndh = (pelts ny)[ny.offset + p2i sy - 1] in
            assert { normalized ny sy
                     /\ value ny sy
                        = power 2 clz * value y sy
                     by
                     value y sy < (dh + 1) * power radix (sy - 1)
                     so value ny sy + (power radix sy) * _c
                        = power 2 clz * value y sy
                        = power 2 clz
                          * (value y (sy - 1)
                             + dh * power radix (sy - 1))
                     so power 2 clz * dh <= radix - power 2 clz
                     so (_c = 0
                        by
                          value ny sy + (power radix sy) * _c
                          = power 2 clz * value y (sy - 1)
                            + power 2 clz * dh * power radix (sy - 1)
                          < power 2 clz * power radix (sy - 1)
                            + power 2 clz * dh * power radix (sy - 1)
                          <= power 2 clz * power radix (sy - 1)
                            + (radix - power 2 clz) * power radix (sy - 1)
                           = radix * power radix (sy - 1)
                           = power radix sy
                        so value ny sy >= 0
                        so power radix sy * _c < power radix sy
                        so power radix sy > 0
                        so _c >= 0
                        )
                     so value ny sy
                        = power 2 clz * value y sy
                     so value y sy >= dh * power radix (sy - 1)
                     so value ny sy
                        >= power 2  clz * dh * power radix (sy - 1)
                     so value ny sy =
                        value ny (sy - 1) + power radix (sy - 1) * ndh
                        < power radix (sy - 1) + power radix (sy - 1) * ndh
                        = power radix (sy - 1) * (ndh + 1)
                     so power radix (sy - 1) * (ndh + 1)
                        > power radix (sy - 1) * (power 2 clz * dh)
                     so ndh + 1 > power 2 clz * dh
                     so ndh >= power 2 clz * dh
                     so 2 * power 2 clz * dh >= radix
                     so 2 * ndh >= radix
                     so ndh >= div radix 2
                   };
          end;
          let h = wmpn_lshift nx x sx (Limb.of_int32 clz) in
          label Shifted in
          C.set_ofs nx sx h;
          begin
            ensures { value nx (sx + adjust)
                      = p * value x sx  }
            if (adjust = 1)
            then begin
              value_sub_tail (pelts nx) nx.offset (nx.offset + p2i sx);
              assert { value nx (sx + 1)
                        = p * value x sx
                       by
                       value nx sx + power radix sx * h
                        = p * value x sx
                       so value nx (sx + 1)
                          = value nx sx + power radix sx * h
                     } end
            else begin
              assert { adjust = 0 };
              assert { h = 0
                       by
                       let dh = (pelts y)[y.offset + sy - 1] in
                       let ah = (pelts x)[x.offset + sx - 1] in
                       p * dh < radix
                       so 0 <= ah <= dh
                       so p * ah < radix
                       so (p * ah <= radix - p
                           by
                           let q = power 2 (Limb.length - clz) in
                           radix = p * q
                           so p * ah < p * q
                           so ah < q
                           so ah <= q - 1
                           so p * ah <= p * (q - 1) = radix - p
                           )
                       so p * (ah + 1) <= radix
                       so let s = power radix (sx - 1) in
                          value x sx < (ah + 1) * s
                       so p * value x sx < p * (ah + 1) * s
                       so (p * (ah + 1) * s
                           <= radix * s
                           by
                           [@case_split]
                           (p * (ah + 1) = radix
                           \/ (p * (ah + 1) < radix
                               so s > 0
                               so p * (ah + 1) * s
                                 < radix * s)))
                       so radix * power radix (sx - 1) = power radix sx
                       so value (nx at Shifted) sx + power radix sx * h
                          < power radix sx
                       so power radix sx * h < power radix sx * 1
                       so (h < 1 by power radix sx > 0)
                      }
            end
          end;
          label Div_s in
          assert { value ny sy * (power radix (sx - sy + adjust))
                   > value nx (sx + adjust)
                   by
                   let dh = (pelts y)[y.offset + sy - 1] in
                   value ny sy >= p * dh * power radix (sy - 1)
                   so value nx (sx + adjust) = p * value x sx
                   so p > 0
                   so [@case_split]
                      ((adjust = 1
                           so value x sx < power radix sx
                           so p * value x sx < p * power radix sx
                           so value ny sy * power radix (sx - sy + adjust)
                              >= p * dh * power radix (sy - 1)
                                    * power radix (sx - sy + adjust)
                              = p * dh * power radix ((sy - 1) + (sx - sy + adjust))
                              = p * dh * power radix sx
                           so dh >= 1
                           so p * dh * power radix sx >= p * power radix sx )
                      \/
                      (adjust = 0
                           so let ah = (pelts x)[x.offset + sx - 1] in
                              value x sx < (ah + 1) * power radix (sx - 1)
                              so ah + 1 <= dh
                              so value x sx < dh * power radix (sx - 1)
                              so p * value x sx < p * dh * power radix (sx - 1)
                              so value ny sy * power radix (sx - sy + adjust)
                                 = value ny sy * power radix (sx - sy)
                                 >= p * dh * power radix (sy - 1)
                                       * power radix (sx - sy)
                                 = p * dh * power radix (sy - 1 + sx - sy)
                                 = p * dh * power radix (sx - 1))) };
          let (ghost _qh) = div_sb_qr q nx (sx + adjust) ny sy in
          let (ghost _l) = wmpn_rshift r nx sy (Limb.of_int32 clz) in
          begin ensures { value nx sy = p * value r sy }
            assert { _l = 0
                   by
                     (mod (value nx sy) p = 0
                      by
                        value (nx at Div_s) (sx + adjust)
                     = (value q (sx - sy + adjust)
                         + power radix (sx - sy + adjust) * _qh)
                        * value ny sy
                        + value nx sy
                     so value (nx at Div_s) (sx + adjust)
                      = p * value x sx
                     so value ny sy = p * value y sy
                     so value nx sy
                        = value (nx at Div_s) (sx + adjust)
                          - (value q (sx - sy + adjust)
                            + power radix (sx - sy + adjust) * _qh)
                            * value ny sy
                        = p * value x sx
                          - p * (value q (sx - sy + adjust)
                                 + power radix (sx - sy + adjust) * _qh)
                              * value y sy
                        = p * (value x sx
                               - (value q (sx - sy + adjust)
                                 + power radix (sx - sy + adjust) * _qh)
                                 * value y sy)
                     so let n = (value x sx
                                  - (value q (sx - sy + adjust)
                                     + power radix (sx - sy + adjust) * _qh)
                                     * value y sy)
                        in
                        value nx sy = p * n
                        so value nx sy >= 0
                        so p > 0
                        so n >= 0
                        so mod (value nx sy) p
                           = mod (p * n) p
                           = mod ((p*n)+0) p
                           = mod 0 p
                           = 0
                   )
                   so _l + radix * value r sy
                      = power 2 (Limb.length - clz) * (value nx sy)
                   so let a = div (value nx sy) p in
                      value nx sy = p * a
                   so power 2 (Limb.length - clz) * p = radix
                   so power 2 (Limb.length - clz) * (value nx sy)
                      = power 2 (Limb.length - clz) * (p * a)
                      = (power 2 (Limb.length - clz) * p) * a
                      = radix * a
                   so mod (radix * value r sy + _l) radix
                      = mod _l radix
                   so mod (radix * value r sy + _l) radix
                      = mod (radix * a) radix = 0
                   so mod _l radix = 0
                   so 0 <= _l < radix
                 };
            assert { value nx sy = p * value r sy
                     by
                     radix * value r sy
                     = power 2 (Limb.length - clz) * value nx sy
                     so p * power 2 (Limb.length - clz)
                        = radix
                     so p * radix * value r sy
                        = p * power 2 (Limb.length - clz) * value nx sy
                        = radix * value nx sy
                     so p * value r sy = value nx sy
                   }
          end;
          assert { value x sx
                   = value q (sx - sy + adjust) * value y sy
                   + value r sy
                  by
                   value (nx at Div_s) (sx + adjust)
                   = (value q (sx - sy + adjust)
                       + power radix (sx - sy + adjust) * _qh)
                     * value ny sy
                     + value nx sy
                  so value (nx at Div_s) (sx + adjust)
                      = p * value x sx
                  so power radix (sx - sy + 1) * power radix (sy - 1)
                     = power radix sx
                  so value ny sy = p * value y sy
                  so (_qh = 0
                     by
                     value (nx at Div_s) (sx + adjust)
                     = (value q (sx - sy + adjust)
                        + power radix (sx - sy + adjust) * _qh)
                      * value ny sy
                      + value nx sy
                     so value nx sy >= 0
                     so value q (sx - sy + adjust) >= 0
                     so _qh >= 0
                     so (value q (sx - sy + adjust)
                             + power radix (sx - sy + adjust) * _qh) >= 0
                     so (value q (sx - sy + adjust)
                         + power radix (sx - sy + adjust) * _qh)
                        * value ny sy
                        + value nx sy
                        >= (value q (sx - sy + adjust)
                             + power radix (sx - sy + adjust) * _qh)
                           * value ny sy
                        >= power radix (sx - sy + adjust) * _qh * value ny sy
                     so _qh <> 1)
                  so value nx sy = p * value r sy
                  so p * value x sx
                     = value q (sx - sy + adjust) * p * value y sy
                       + p * value r sy
                     = p * (value q (sx - sy + adjust)
                       * value y sy
                            + value r sy)
                 };
          label Ret_s in
          begin
            ensures { value q (sx - sy + 1)
                      = value (q at Ret_s) (sx - sy + adjust) }
            if (adjust = 0)
            then begin
              value_sub_shift_no_change (pelts x) x.offset
                                        (p2i sx) (p2i sx) 0;
              set_ofs q (sx - sy) 0;
              value_sub_tail (pelts q) q.offset (q.offset + p2i sx - p2i sy);
              assert { value q (sx - sy + 1) = value (q at Ret_s) (sx - sy)
                       by value q (sx - sy + 1)
                          = value (q at Ret_s) (sx - sy) + power radix (sx - sy) * 0
                          = value (q at Ret_s) (sx - sy) }
            end
          end;
          ()
        end
        end

  let partial wmpn_tdiv_qr (q r:t) (qxn:int32) (x:t) (sx:int32) (y:t) (sy:int32) : unit
    requires { 1 <= sy <= sx <= (Int32.max_int32 - 1) }
    requires { valid x sx }
    requires { valid y sy }
    requires { valid q (sx - sy + 1) }
    requires { valid r sy }
    requires { writable q /\ writable r }
    requires { qxn = 0 }
    requires { (pelts y)[y.offset + sy - 1] > 0 }
    ensures { value x sx
              = value q (sx - sy + 1) * value y sy
                + value r sy }
    ensures { value r sy < value y sy }
  =
    let nx = malloc (UInt32.(+) (UInt32.of_int32 sx) 1) in
    c_assert (is_not_null nx);
    let ny = malloc (UInt32.of_int32 sy) in
    c_assert (is_not_null ny);
    div_qr q r x y nx ny sx sy;
    free nx;
    free ny

  let div_qr_in_place (q x y nx ny:t) (sx sy:int32) : unit
    requires { 1 <= sy <= sx <= (Int32.max_int32 - 1) }
    requires { valid x sx }
    requires { valid y sy }
    requires { valid q (sx - sy + 1) }
    requires { valid nx (sx + 1) }
    requires { valid ny sy }
    requires { writable q /\ writable x }
    requires { writable nx /\ writable ny }
    requires { (pelts y)[y.offset + sy - 1] > 0 }
    ensures { value (old x) sx
              = value q (sx - sy + 1) * value y sy
                + value x sy }
    ensures { value x sy < value y sy }
  =
    label Start in
    value_sub_tail (pelts y) y.offset (y.offset + p2i sy - 1);
    value_sub_lower_bound (pelts y) y.offset (y.offset + p2i sy - 1);
    assert { value y sy >= power radix (sy - 1) };
    let ghost ox = pure { x } in
    if (sy = 1)
    then
      let lr = wmpn_divrem_1 q x sx (C.get y) in
      C.set x lr
    else
    if (sy = 2)
    then
      let clz = clz_ext (C.get_ofs y (sy - 1)) in
      let ghost p = power 2 (p2i clz) in
      if clz = 0
      then begin
        wmpn_copyi nx x sx;
        value_sub_shift_no_change (pelts x) x.offset (p2i sx) (p2i sx) 0;
        C.set_ofs nx sx 0;
        value_sub_frame_shift (pelts x) (pelts nx) x.offset nx.offset (p2i sx);
        label Div2_ns in
        let (ghost _qh) = wmpn_divrem_2 q nx y (sx + 1) in
        wmpn_copyi x nx sy;
        assert { value ox sx
                 = value q (sx - sy + 1) * value y sy
                 + value x sy
               by value x sy = value nx sy
               so value (nx at Div2_ns) (sx + 1) < power radix sx
               so value (nx at Div2_ns) (sx + 1)
                  = value (nx at Div2_ns) sx
               so (_qh = 0
                   by
                   power radix sx
                   > value (nx at Div2_ns) (sx + 1)
                   = (value q (sx - 1) + power radix (sx - 1) * _qh)
                    * value y 2
                    + value nx 2
                   so value nx 2 >= 0
                   so value y 2 >= radix
                   so value q (sx - 1) >= 0
                   so _qh >= 0
                   so (value q (sx - 1)
                           + power radix (sx - 1) * _qh) >= 0
                   so (value q (sx - 1) + power radix (sx - 1) * _qh)
                      * value y 2
                      + value nx 2
                      >= (value q (sx - 1)
                           + power radix (sx - 1) * _qh)
                         * value y 2
                      >= (value q (sx - 1)
                           + power radix (sx - 1) * _qh)
                         * radix
                      >= power radix (sx - 1) * _qh * radix
                      = power radix sx * _qh
                   so power radix sx > power radix sx * _qh
                  )
               so value ox sx = value (nx at Div2_ns) sx
               };
        ()
      end
      else begin
        let (ghost _c) = wmpn_lshift ny y sy (Limb.of_int32 clz) in
        begin
          ensures { normalized ny sy }
          ensures { value ny sy = power 2 clz * value y sy }
          let ghost dh = (pelts y)[y.offset + p2i sy - 1] in
          assert { value y sy
                   = value y (sy - 1) + power radix (sy - 1) * dh };
          value_sub_upper_bound (pelts y) y.offset (y.offset + p2i sy - 1);
          value_sub_tail (pelts ny) ny.offset (ny.offset + p2i sy - 1);
          value_sub_upper_bound (pelts ny) ny.offset (ny.offset + p2i sy - 1);
          let ghost ndh = (pelts ny)[ny.offset + p2i sy - 1] in
          assert { normalized ny sy
                   /\ value ny sy = power 2 clz * value y sy
                   by
                   value y sy < (dh + 1) * power radix (sy - 1)
                   so value ny sy + (power radix sy) * _c
                      = power 2 clz * value y sy
                      = power 2 clz
                        * (value y (sy - 1)
                           + dh * power radix (sy - 1))
                   so power 2 clz * dh <= radix - power 2 clz
                   so value ny sy + (power radix sy) * _c
                      = power 2 clz * value y (sy - 1)
                        + power 2 clz * dh * power radix (sy - 1)
                      < power 2 clz * power radix (sy - 1)
                        + power 2 clz * dh * power radix (sy - 1)
                      <= power 2 clz * power radix (sy - 1)
                        + (radix - power 2 clz) * power radix (sy - 1)
                       = radix * power radix (sy - 1)
                       = power radix sy
                   so _c = 0
                   so value ny sy
                      = power 2 clz * value y sy
                   so value y sy >= dh * power radix (sy - 1)
                   so value ny sy
                      >= power 2  clz * dh * power radix (sy - 1)
                   so value ny sy =
                      value ny (sy - 1) + power radix (sy - 1) * ndh
                      < power radix (sy - 1) + power radix (sy - 1) * ndh
                      = power radix (sy - 1) * (ndh + 1)
                   so power radix (sy - 1) * (ndh + 1)
                      > power radix (sy - 1) * (power 2 clz * dh)
                   so ndh + 1 > power 2 clz * dh
                   so ndh >= power 2 clz * dh
                   so 2 * power 2 clz * dh >= radix
                   so 2 * ndh >= radix
                   so ndh >= div radix 2
                 };
        end;
        let h = wmpn_lshift nx x sx (Limb.of_int32 clz) in
        C.set_ofs nx sx h;
        begin
          ensures { value nx (sx + 1)
                    = p * value ox sx  }
          value_sub_tail (pelts nx) nx.offset (nx.offset + p2i sx);
          assert { value nx (sx + 1)
                    = p * value ox sx
                   by
                   value nx sx + power radix sx * h
                    = p * value ox sx
                   so value nx (sx + 1)
                      = value nx sx + power radix sx * h
                 }
        end;
        label Div2_s in
        (* TODO don't add 1 when not needed, cf "adjust" in GMP algo *)
        let (ghost _qh) = wmpn_divrem_2 q nx ny (sx + 1) in
        let (ghost _l) = wmpn_rshift x nx sy (Limb.of_int32 clz) in
        begin ensures { value nx 2 = p * value x 2 }
          assert { _l = 0
                 by
                   (mod (value nx sy) p = 0
                    by
                      value (nx at Div2_s) (sx + 1)
                   = (value q (sx - 1) + power radix (sx - 1) * _qh)
                      * value ny sy
                      + value nx sy
                   so value (nx at Div2_s) (sx + 1)
                    = p * value ox sx
                   so value ny sy = p * value y sy
                   so value nx sy
                      = value (nx at Div2_s) (sx + 1)
                        - (value q (sx - 1)
                          + power radix (sx - 1) * _qh)
                          * value ny sy
                      = p * value ox sx
                        - p * (value q (sx - 1)
                               + power radix (sx - 1) * _qh)
                            * value y sy
                      = p * (value ox sx
                             - (value q (sx - 1)
                               + power radix (sx - 1) * _qh)
                               * value y sy)
                   so let n = (value ox sx
                                - (value q (sx - 1)
                                   + power radix (sx - 1) * _qh)
                                   * value y sy)
                      in
                      value nx sy = p * n
                      so value nx sy >= 0
                      so p > 0
                      so n >= 0
                      so mod (value nx sy) p
                         = mod (p * n) p
                         = mod ((p*n)+0) p
                         = mod 0 p
                         = 0
                 )
                 so _l + radix * value x sy
                    = power 2 (Limb.length - clz) * (value nx sy)
                 so let a = div (value nx sy) p in
                    value nx sy = p * a
                 so power 2 (Limb.length - clz) * p = radix
                 so power 2 (Limb.length - clz) * (value nx sy)
                    = power 2 (Limb.length - clz) * (p * a)
                    = (power 2 (Limb.length - clz) * p) * a
                    = radix * a
                 so mod (radix * value x sy + _l) radix
                    = mod _l radix
                 so mod (radix * value x sy + _l) radix
                    = mod (radix * a) radix = 0
                 so mod _l radix = 0
                 so 0 <= _l < radix
               };
          assert { value nx 2 = p * value x 2
                   by
                   radix * value x 2
                   = power 2 (Limb.length - clz) * value nx 2
                   so p * power 2 (Limb.length - clz)
                      = radix
                   so p * radix * value x 2
                      = p * power 2 (Limb.length - clz) * value nx 2
                      = radix * value nx 2
                   so p * value x 2 = value nx 2
                 }
        end;
        assert { value ox sx
                 = value q (sx - sy + 1) * value y sy
                 + value x sy
                by
                 value (nx at Div2_s) (sx + 1)
                 = (value q (sx - 1) + power radix (sx - 1) * _qh)
                   * value ny 2
                   + value nx 2
                so value (nx at Div2_s) (sx + 1)
                    = p * value ox sx
                so value ny 2 = p * value y 2
                so (_qh = 0
                    by
                    value ox sx < power radix sx
                    so value y 2 >= radix
                    so value ny 2 >= p * radix
                    so value q (sx - 1) >= 0
                    so value nx 2 >= 0
                    so (value q (sx - 1) + power radix (sx - 1) * _qh)
                        >= 0
                    so (value q (sx - 1) + power radix (sx - 1) * _qh)
                         * value ny 2
                       + value nx 2
                       >= (value q (sx - 1)
                           + power radix (sx - 1) * _qh)
                          * value ny 2
                       >= (value q (sx - 1)
                           + power radix (sx - 1) * _qh)
                          * (p * radix)
                       >= power radix (sx - 1) * _qh * p * radix
                       = power radix sx * p * _qh
                    so power radix sx * p
                       > value (nx at Div2_s) (sx + 1)
                       >= power radix sx * p * _qh
                    )
                so value nx 2 = p * value x 2
                so p * value ox sx
                   = value q (sx - 1) * p * value y 2
                     + p * value x 2
                   = p * (value q (sx - 1) * value y 2
                          + value x 2)
               };
        ()
      end
    else
     (* let qn = ref (Int32.(-) (Int32.(+) sx 1) sy) in
      if (Int32.(>=)  (Int32.(+) !qn !qn) sx)
      then*) begin
        let adjust =
          if (get_ofs x (sx - 1)) >= (get_ofs y (sy - 1))
          then 1
          else 0
        in
        let clz = clz_ext (C.get_ofs y (sy - 1)) in
        let ghost p = power 2 (p2i clz) in
        if clz = 0
        then begin
          wmpn_copyi nx x sx;
          value_sub_shift_no_change (pelts x) x.offset
                                    (p2i sx) (p2i sx) 0;
          C.set_ofs nx sx 0;
          value_sub_frame_shift (pelts x) (pelts nx) x.offset nx.offset (p2i sx);
          assert { value y sy * (power radix (sx - sy + adjust))
                   > value nx (sx + adjust)
                   by
                   let dh = (pelts y)[y.offset + sy - 1] in
                   value y sy >= dh * power radix (sy - 1)
                   so value nx (sx + adjust) = value nx sx = value ox sx
                   so [@case_split]
                      ((adjust = 1
                           so value ox sx < power radix sx
                           so value y sy * power radix (sx - sy + adjust)
                              >= dh * power radix (sy - 1)
                                    * power radix (sx - sy + adjust)
                              = dh * power radix ((sy - 1) + (sx - sy + adjust))
                              = dh * power radix sx
                           so dh >= div radix 2 > 1
                           so dh * power radix sx > power radix sx )
                      \/
                      (adjust = 0
                           so let ah = (pelts x)[x.offset + sx - 1] in
                              value ox sx < (ah + 1) * power radix (sx - 1)
                              so ah + 1 <= dh
                              so value ox sx < dh * power radix (sx - 1)
                              so value y sy * power radix (sx - sy + adjust)
                                 = value y sy * power radix (sx - sy)
                                 >= dh * power radix (sy - 1)
                                       * power radix (sx - sy)
                                 = dh * power radix (sy - 1 + sx - sy)
                                 = dh * power radix (sx - 1))) };
          label Div_ns in
          let (ghost _qh) = div_sb_qr q nx (sx + adjust) y sy in
          wmpn_copyi x nx sy;
          assert { value ox sx
                   = value q (sx - sy + adjust) * value y sy
                   + value x sy
                 by value x sy = value nx sy
                 so value (nx at Div_ns) (sx + adjust) = value ox sx < power radix sx
                 so value (nx at Div_ns) (sx + adjust)
                    = value (nx at Div_ns) sx
                 so (_qh = 0
                     by
                     value (nx at Div_ns) (sx + adjust)
                     = (value q (sx - sy + adjust)
                        + power radix (sx - sy + adjust) * _qh)
                      * value y sy
                      + value nx sy
                     so value nx sy >= 0
                     so value q (sx - sy + adjust) >= 0
                     so _qh >= 0
                     so (value q (sx - sy + adjust)
                             + power radix (sx - sy + adjust) * _qh) >= 0
                     so (value q (sx - sy + adjust)
                         + power radix (sx - sy + adjust) * _qh)
                        * value y sy
                        + value nx sy
                        >= (value q (sx - sy + adjust)
                             + power radix (sx - sy + adjust) * _qh)
                           * value y sy
                        >= power radix (sx - sy + adjust) * _qh * value y sy
                     so _qh <> 1)
                 so value ox sx = value (nx at Div_ns) sx
                 };
           label Ret_ns in
           begin
             ensures { value q (sx - sy + 1)
                       = value (q at Ret_ns) (sx - sy + adjust) }
             if (adjust = 0)
             then begin
               value_sub_shift_no_change (pelts x) x.offset
                                         (p2i sx) (p2i sx) 0;
               set_ofs q (sx - sy) 0;
               value_sub_tail (pelts q) q.offset (q.offset + p2i sx - p2i sy);
               ()
             end
           end
        end
        else begin
          let (ghost _c) = wmpn_lshift ny y sy (Limb.of_int32 clz) in
          begin
            ensures { normalized ny sy }
            ensures { value ny sy
                      = power 2 clz * value y sy }
            let ghost dh = (pelts y)[y.offset + p2i sy - 1] in
            assert { value y sy
                     = value y (sy - 1) + power radix (sy - 1) * dh };
            value_sub_upper_bound (pelts y) y.offset (y.offset + p2i sy - 1);
            value_sub_tail (pelts ny) ny.offset (ny.offset + p2i sy - 1);
            value_sub_upper_bound (pelts ny) ny.offset (ny.offset + p2i sy - 1);
            let ghost ndh = (pelts ny)[ny.offset + p2i sy - 1] in
            assert { normalized ny sy
                     /\ value ny sy
                        = power 2 clz * value y sy
                     by
                     value y sy < (dh + 1) * power radix (sy - 1)
                     so value ny sy + (power radix sy) * _c
                        = power 2 clz * value y sy
                        = power 2 clz
                          * (value y (sy - 1)
                             + dh * power radix (sy - 1))
                     so power 2 clz * dh <= radix - power 2 clz
                     so (_c = 0
                        by
                          value ny sy + (power radix sy) * _c
                          = power 2 clz * value y (sy - 1)
                            + power 2 clz * dh * power radix (sy - 1)
                          < power 2 clz * power radix (sy - 1)
                            + power 2 clz * dh * power radix (sy - 1)
                          <= power 2 clz * power radix (sy - 1)
                            + (radix - power 2 clz) * power radix (sy - 1)
                           = radix * power radix (sy - 1)
                           = power radix sy
                        so value ny sy >= 0
                        so power radix sy * _c < power radix sy
                        so power radix sy > 0
                        so _c >= 0
                        )
                     so value ny sy
                        = power 2 clz * value y sy
                     so value y sy >= dh * power radix (sy - 1)
                     so value ny sy
                        >= power 2  clz * dh * power radix (sy - 1)
                     so value ny sy =
                        value ny (sy - 1) + power radix (sy - 1) * ndh
                        < power radix (sy - 1) + power radix (sy - 1) * ndh
                        = power radix (sy - 1) * (ndh + 1)
                     so power radix (sy - 1) * (ndh + 1)
                        > power radix (sy - 1) * (power 2 clz * dh)
                     so ndh + 1 > power 2 clz * dh
                     so ndh >= power 2 clz * dh
                     so 2 * power 2 clz * dh >= radix
                     so 2 * ndh >= radix
                     so ndh >= div radix 2
                   };
          end;
          let h = wmpn_lshift nx x sx (Limb.of_int32 clz) in
          label Shifted in
          C.set_ofs nx sx h;
          begin
            ensures { value nx (sx + adjust)
                      = p * value ox sx  }
            if (adjust = 1)
            then begin
              value_sub_tail (pelts nx) nx.offset (nx.offset + p2i sx);
              assert { value nx (sx + 1)
                        = p * value ox sx
                       by
                       value nx sx + power radix sx * h
                        = p * value ox sx
                       so value nx (sx + 1)
                          = value nx sx + power radix sx * h
                     } end
            else begin
              assert { adjust = 0 };
              assert { h = 0
                       by
                       let dh = (pelts y)[y.offset + sy - 1] in
                       let ah = (pelts x)[x.offset + sx - 1] in
                       p * dh < radix
                       so 0 <= ah <= dh
                       so p * ah < radix
                       so (p * ah <= radix - p
                           by
                           let q = power 2 (Limb.length - clz) in
                           radix = p * q
                           so p * ah < p * q
                           so ah < q
                           so ah <= q - 1
                           so p * ah <= p * (q - 1) = radix - p
                           )
                       so p * (ah + 1) <= radix
                       so let s = power radix (sx - 1) in
                          value ox sx < (ah + 1) * s
                       so p * value ox sx < p * (ah + 1) * s
                       so (p * (ah + 1) * s
                           <= radix * s
                           by
                           [@case_split]
                           (p * (ah + 1) = radix
                           \/ (p * (ah + 1) < radix
                               so s > 0
                               so p * (ah + 1) * s
                                 < radix * s)))
                       so radix * power radix (sx - 1) = power radix sx
                       so value (nx at Shifted) sx + power radix sx * h
                          < power radix sx
                       so power radix sx * h < power radix sx * 1
                       so (h < 1 by power radix sx > 0)
                      }
            end
          end;
          label Div_s in
          assert { value ny sy * (power radix (sx - sy + adjust))
                   > value nx (sx + adjust)
                   by
                   let dh = (pelts y)[y.offset + sy - 1] in
                   value ny sy >= p * dh * power radix (sy - 1)
                   so value nx (sx + adjust) = p * value ox sx
                   so p > 0
                   so [@case_split]
                      ((adjust = 1
                           so value ox sx < power radix sx
                           so p * value ox sx < p * power radix sx
                           so value ny sy * power radix (sx - sy + adjust)
                              >= p * dh * power radix (sy - 1)
                                    * power radix (sx - sy + adjust)
                              = p * dh * power radix ((sy - 1) + (sx - sy + adjust))
                              = p * dh * power radix sx
                           so dh >= 1
                           so p * dh * power radix sx >= p * power radix sx )
                      \/
                      (adjust = 0
                           so let ah = (pelts x)[x.offset + sx - 1] in
                              value ox sx < (ah + 1) * power radix (sx - 1)
                              so ah + 1 <= dh
                              so value ox sx < dh * power radix (sx - 1)
                              so p * value ox sx < p * dh * power radix (sx - 1)
                              so value ny sy * power radix (sx - sy + adjust)
                                 = value ny sy * power radix (sx - sy)
                                 >= p * dh * power radix (sy - 1)
                                       * power radix (sx - sy)
                                 = p * dh * power radix (sy - 1 + sx - sy)
                                 = p * dh * power radix (sx - 1))) };
          let (ghost _qh) = div_sb_qr q nx (sx + adjust) ny sy in
          let (ghost _l) = wmpn_rshift x nx sy (Limb.of_int32 clz) in
          begin ensures { value nx sy = p * value x sy }
            assert { _l = 0
                   by
                     (mod (value nx sy) p = 0
                      by
                        value (nx at Div_s) (sx + adjust)
                     = (value q (sx - sy + adjust)
                         + power radix (sx - sy + adjust) * _qh)
                        * value ny sy
                        + value nx sy
                     so value (nx at Div_s) (sx + adjust)
                      = p * value ox sx
                     so value ny sy = p * value y sy
                     so value nx sy
                        = value (nx at Div_s) (sx + adjust)
                          - (value q (sx - sy + adjust)
                            + power radix (sx - sy + adjust) * _qh)
                            * value ny sy
                        = p * value ox sx
                          - p * (value q (sx - sy + adjust)
                                 + power radix (sx - sy + adjust) * _qh)
                              * value y sy
                        = p * (value ox sx
                               - (value q (sx - sy + adjust)
                                 + power radix (sx - sy + adjust) * _qh)
                                 * value y sy)
                     so let n = (value ox sx
                                  - (value q (sx - sy + adjust)
                                     + power radix (sx - sy + adjust) * _qh)
                                     * value y sy)
                        in
                        value nx sy = p * n
                        so value nx sy >= 0
                        so p > 0
                        so n >= 0
                        so mod (value nx sy) p
                           = mod (p * n) p
                           = mod ((p*n)+0) p
                           = mod 0 p
                           = 0
                   )
                   so _l + radix * value x sy
                      = power 2 (Limb.length - clz) * (value nx sy)
                   so let a = div (value nx sy) p in
                      value nx sy = p * a
                   so power 2 (Limb.length - clz) * p = radix
                   so power 2 (Limb.length - clz) * (value nx sy)
                      = power 2 (Limb.length - clz) * (p * a)
                      = (power 2 (Limb.length - clz) * p) * a
                      = radix * a
                   so mod (radix * value x sy + _l) radix
                      = mod _l radix
                   so mod (radix * value x sy + _l) radix
                      = mod (radix * a) radix = 0
                   so mod _l radix = 0
                   so 0 <= _l < radix
                 };
            assert { value nx sy = p * value x sy
                     by
                     radix * value x sy
                     = power 2 (Limb.length - clz) * value nx sy
                     so p * power 2 (Limb.length - clz)
                        = radix
                     so p * radix * value x sy
                        = p * power 2 (Limb.length - clz) * value nx sy
                        = radix * value nx sy
                     so p * value x sy = value nx sy
                   }
          end;
          assert { value ox sx
                   = value q (sx - sy + adjust) * value y sy
                   + value x sy
                  by
                   value (nx at Div_s) (sx + adjust)
                   = (value q (sx - sy + adjust)
                       + power radix (sx - sy + adjust) * _qh)
                     * value ny sy
                     + value nx sy
                  so value (nx at Div_s) (sx + adjust)
                      = p * value ox sx
                  so power radix (sx - sy + 1) * power radix (sy - 1)
                     = power radix sx
                  so value ny sy = p * value y sy
                  so (_qh = 0
                     by
                     value (nx at Div_s) (sx + adjust)
                     = (value q (sx - sy + adjust)
                        + power radix (sx - sy + adjust) * _qh)
                      * value ny sy
                      + value nx sy
                     so value nx sy >= 0
                     so value q (sx - sy + adjust) >= 0
                     so _qh >= 0
                     so (value q (sx - sy + adjust)
                             + power radix (sx - sy + adjust) * _qh) >= 0
                     so (value q (sx - sy + adjust)
                         + power radix (sx - sy + adjust) * _qh)
                        * value ny sy
                        + value nx sy
                        >= (value q (sx - sy + adjust)
                             + power radix (sx - sy + adjust) * _qh)
                           * value ny sy
                        >= power radix (sx - sy + adjust) * _qh * value ny sy
                     so _qh <> 1)
                  so value nx sy = p * value x sy
                  so p * value ox sx
                     = value q (sx - sy + adjust) * p * value y sy
                       + p * value x sy
                     = p * (value q (sx - sy + adjust)
                       * value y sy
                            + value x sy)
                 };
          label Ret_s in
          begin
            ensures { value q (sx - sy + 1)
                      = value (q at Ret_s) (sx - sy + adjust) }
            if (adjust = 0)
            then begin
              value_sub_shift_no_change (pelts x) x.offset
                                        (p2i sx) (p2i sx) 0;
              set_ofs q (sx - sy) 0;
              value_sub_tail (pelts q) q.offset (q.offset + p2i sx - p2i sy);
              assert { value q (sx - sy + 1) = value (q at Ret_s) (sx - sy)
                       by value q (sx - sy + 1)
                          = value (q at Ret_s) (sx - sy) + power radix (sx - sy) * 0
                          = value (q at Ret_s) (sx - sy) }
            end
          end;
          ()
        end
        end

  let partial wmpn_tdiv_qr_in_place (q:t) (qxn:int32) (x:t) (sx:int32) (y:t) (sy:int32)
      : unit
    requires { 1 <= sy <= sx <= (Int32.max_int32 - 1) }
    requires { valid x sx }
    requires { valid y sy }
    requires { valid q (sx - sy + 1) }
    requires { writable x /\ writable q }
    requires { qxn = 0 }
    requires { (pelts y)[y.offset + sy - 1] > 0 }
    ensures { value (old x) sx
              = value q (sx - sy + 1) * value y sy
                + value x sy }
    ensures { value x sy < value y sy }
  =
    let nx = malloc (UInt32.(+) (UInt32.of_int32 sx) 1) in
    c_assert (is_not_null nx);
    let ny = malloc (UInt32.of_int32 sy) in
    c_assert (is_not_null ny);
    div_qr_in_place q x y nx ny sx sy;
    free nx;
    free ny


end
