diff --git a/crypto-specs b/crypto-specs
index a82b8d51..3710c5ff 160000
--- a/crypto-specs
+++ b/crypto-specs
@@ -1 +1 @@
-Subproject commit a82b8d510b34d0939543ab340b9b0e3907c14f51
+Subproject commit 3710c5ff92a84a3a150c70a89d8d2fd3486bd735
diff --git a/proof/correctness/MLKEMFCLib.ec b/proof/correctness/MLKEMFCLib.ec
index 772028e6..7d5a1b9a 100644
--- a/proof/correctness/MLKEMFCLib.ec
+++ b/proof/correctness/MLKEMFCLib.ec
@@ -221,7 +221,7 @@ lemma initEq16 (f g: int -> 'a) :
 
 (*-----------------------------------------------------------------------------*)
 
-lemma nosmt set_neqiE (t : coeff Array256.t) x y a :
+lemma set_neqiE (t : coeff Array256.t) x y a :
   y <> x => t.[x <- a].[y] = t.[y].
 proof. by rewrite get_set_if => /neqF ->. qed.
 
diff --git a/proof/correctness/MLKEM_Poly.ec b/proof/correctness/MLKEM_Poly.ec
index 084bf15f..b6942a76 100644
--- a/proof/correctness/MLKEM_Poly.ec
+++ b/proof/correctness/MLKEM_Poly.ec
@@ -1733,7 +1733,8 @@ have  /= [#] redbl6 redbh6 redv6 :=
 have  /= [#] redbl7 redbh7 redv7 :=
     (SREDCp_corr (to_sint r6 * to_sint (- jzetas.[64 + to_uint i{hr} %/ 4])) hq _).
 + rewrite /R /=; move : (zeta_bound (64 + to_uint i{hr} %/ 4)); rewrite /minimum_residues /bpos16 => zb.
-  rewrite to_sintN /=; do split; smt().  
+  case/(_ _): zb => *; ~-1:smt().
+  by rewrite to_sintN /=; do split; smt().
 
 have  /= [#] redbl8 redbh8 redv8 :=
     (SREDCp_corr (to_sint ap{hr}.[to_uint i{hr}+2] * to_sint bp{hr}.[to_uint i{hr}+2]) hq _).
@@ -1801,7 +1802,7 @@ case (k < to_uint i{hr} %/4).
 + move => kbb; move: (vprev k _); 1:smt(); rewrite !mapiE /=; 1..12:smt().
   rewrite /doublemul /cmplx_mul_169 /=.
   move => /> vprev0 vprev1 vprev2 vprev3.
-  by rewrite !set_neqiE /#. 
+  by rewrite !set_neqiE //#.
 
 move => *; have kval : (k = to_uint i{hr} %/ 4) by smt().
 have -> : 4 * k = to_uint i{hr} by smt().
diff --git a/proof/correctness/Montgomery.ec b/proof/correctness/Montgomery.ec
index a8e40220..ffdf369b 100644
--- a/proof/correctness/Montgomery.ec
+++ b/proof/correctness/Montgomery.ec
@@ -164,7 +164,7 @@ op BREDC(a bits : int) =
 
 require import Barrett_mlkem_general.
 
-lemma nosmt BREDCp_corr a bits:
+lemma BREDCp_corr a bits:
    0 < 2 * q < R %/2 =>
    R < 2^bits =>
    2 ^ bits %/ q * q < 2 ^ bits =>
@@ -222,7 +222,7 @@ op SREDC (a: int) : int =
   let t = smod (a - u %/ R * q) (R^2)in
       smod (t %/ R %% (R^2)) R.
 
-lemma nosmt SREDCp_corr a:
+lemma SREDCp_corr a:
    0 < q < R %/2 =>
    -R %/ 2 * q <= a < R %/2 * q =>
    -q   <= SREDC a < q /\
@@ -353,7 +353,7 @@ op REDC' (T: int) : int =
  let m = ((T %% R)*_N') %% R
  in (T + m*_N) %/ R.
 
-lemma nosmt aux_divR T:
+lemma aux_divR T:
  let m = ((T %% R)*_N') %% R
  in (T + m*_N) %% R = 0.
 proof.
@@ -364,7 +364,7 @@ smt().
 qed.
 
 
-lemma nosmt REDC'_congr T:
+lemma REDC'_congr T:
  REDC' T %% _N = T * Rinv %% _N.
 proof.
 pose m := ((T %% R)*_N') %% R.
@@ -378,7 +378,7 @@ have t_modN: t %% _N = T*Rinv %% _N.
 by rewrite /REDC'.
 qed.
 
-lemma nosmt REDC'_bnds T n:
+lemma REDC'_bnds T n:
  0 <= n =>
  0 <= T < _N + _N * R^(n+1) =>
  0 <= REDC' T < _N + _N*R^n.
diff --git a/proof/correctness/Montgomery16.ec b/proof/correctness/Montgomery16.ec
index 55c58f4d..f1ca7ec9 100644
--- a/proof/correctness/Montgomery16.ec
+++ b/proof/correctness/Montgomery16.ec
@@ -6,7 +6,7 @@ require import MLKEMFCLib.
 
 (* @JBA: MOVE THIS *)
 
-lemma nosmt modzB_eq0 (x y m:int):
+lemma modzB_eq0 (x y m:int):
  0 < m => (x-y) %% m = 0 =>
  x%%m = y%%m.
 proof.
@@ -53,7 +53,7 @@ case: ( 2 ^ (16 - 1) <= x) => ?//.
 by rewrite -modzDmr -modzNm modzz.
 qed.
 
-lemma nosmt to_sint_mod x:
+lemma to_sint_mod x:
  W16.to_sint x %% W16.modulus = to_uint x.
 proof.
 rewrite /to_sint /smod.
@@ -69,7 +69,7 @@ lemma to_sintK (w : W16.t) :
   W16.of_int (W16.to_sint w) = w.
 proof. by rewrite -of_int_mod to_sint_mod to_uintK //. qed.
 
-lemma nosmt to_sint_eq (w1 w2: W16.t):
+lemma to_sint_eq (w1 w2: W16.t):
  to_sint w1 = to_sint w2 <=> w1=w2.
 proof.
 rewrite !to_sintE /smod /=.
@@ -82,7 +82,7 @@ case: (32768 <= to_uint w2) => CB /=.
 by rewrite to_uint_eq.
 qed.
 
-lemma nosmt smod_small (x: int):
+lemma smod_small (x: int):
  -2^(16-1) <= x < 2^(16-1) => 
  W16.smod (x %% W16.modulus) = x.
 proof.
@@ -92,7 +92,7 @@ rewrite /smod; case: (x < 0) => C.
 by have ->/#: ! 2 ^ (16 - 1) <= x %% W16.modulus by smt().
 qed.
 
-lemma nosmt modzM_sint (x y: W16.t):
+lemma modzM_sint (x y: W16.t):
  (to_sint x * to_sint y) %% W16.modulus
  = (to_uint x * to_uint y) %% W16.modulus.
 proof.
@@ -110,14 +110,14 @@ by rewrite modzMm.
 done.
 qed.
 
-lemma nosmt to_sintM (x y: W16.t):
+lemma to_sintM (x y: W16.t):
  to_sint (x*y)
  = W16.smod ((to_sint x * to_sint y) %% W16.modulus).
 proof.
 by rewrite {1}/W16.to_sint to_uintM modzM_sint.
 qed.
 
-lemma nosmt modzD_sint (x y: W16.t):
+lemma modzD_sint (x y: W16.t):
  (to_sint x + to_sint y) %% W16.modulus
  = (to_uint x + to_uint y) %% W16.modulus.
 proof.
@@ -126,14 +126,14 @@ case: (2 ^ (16 - 1) <= to_uint x);
  case: (2 ^ (16 - 1) <= to_uint y); smt().
 qed.
 
-lemma nosmt to_sintD (x y: W16.t):
+lemma to_sintD (x y: W16.t):
  to_sint (x+y)
  = W16.smod ((to_sint x + to_sint y)%%W16.modulus). 
 proof.
 by rewrite {1}/W16.to_sint to_uintD modzD_sint.
 qed.
 
-lemma nosmt modzN_sint (x: W16.t):
+lemma modzN_sint (x: W16.t):
  (- to_sint x) %% W16.modulus
  = (- to_uint x) %% W16.modulus.
 proof.
@@ -141,7 +141,7 @@ rewrite /to_sint /smod.
 case: (2 ^ (16 - 1) <= to_uint x); smt().
 qed.
 
-lemma nosmt to_sintN (x: W16.t):
+lemma to_sintN (x: W16.t):
  to_sint (-x)
  = W16.smod ((-to_sint x) %% W16.modulus).
 proof.
@@ -164,7 +164,7 @@ have X: 32768 <= (- to_uint x) %% 65536 by smt().
 smt().
 qed.
 
-lemma nosmt modzB_sint (x y: W16.t):
+lemma modzB_sint (x y: W16.t):
  (to_sint x - to_sint y) %% W16.modulus
  = (to_uint x - to_uint y) %% W16.modulus.
 proof.
@@ -173,14 +173,14 @@ case: (2 ^ (16 - 1) <= to_uint x);
  case: (2 ^ (16 - 1) <= to_uint y); smt().
 qed.
 
-lemma nosmt to_sintB (x y: W16.t):
+lemma to_sintB (x y: W16.t):
  to_sint (x-y)
  = W16.smod ((to_sint x - to_sint y)%%W16.modulus). 
 proof.
 rewrite {1}/W16.to_sint to_uintD to_uintN modzB_sint /#.
 qed.
 
-lemma nosmt wmulsE (x y: W16.t):
+lemma wmulsE (x y: W16.t):
  to_sint x * to_sint y
  = to_sint (wmulhs x y) * W16.modulus
    + to_uint (x * y).
@@ -193,7 +193,7 @@ have /=?:= to_sint_cmp y.
 smt().
 qed.
 
-lemma nosmt to_sint_wmulhs x y:
+lemma to_sint_wmulhs x y:
  to_sint (W16.wmulhs x y) = to_sint x * to_sint y %/ W16.modulus.
 proof.
 rewrite wmulsE divzDl 1:/# mulzK 1:/#; ring.
@@ -221,7 +221,7 @@ lemma sint_bndW (x: W16.t) (xL1 xH1 xL2 xH2: int):
  sint_bnd xL2 xH2 x
 by smt().
 
-lemma nosmt to_sintD_small (x y: W16.t):
+lemma to_sintD_small (x y: W16.t):
  W16.min_sint <= to_sint x + to_sint y <= W16.max_sint =>
  to_sint (x+y) = to_sint x + to_sint y.
 proof.
@@ -229,7 +229,7 @@ move=> /=?; rewrite to_sintD smod_small /= /#.
 qed.
 
 (* a version of [to_sintD_small] with bounds *)
-lemma nosmt to_sintD_small' (xL xH yL yH: int) (x y: W16.t):
+lemma to_sintD_small' (xL xH yL yH: int) (x y: W16.t):
  sint_bnd xL xH x =>
  sint_bnd yL yH y =>
  W16.min_sint <= xL+yL =>
@@ -239,7 +239,7 @@ proof.
 by move=> /= *; rewrite to_sintD_small /#.
 qed.
 
-lemma nosmt to_sintN_small (x: W16.t):
+lemma to_sintN_small (x: W16.t):
  W16.min_sint < to_sint x =>
  to_sint (-x) = - to_sint x.
 proof.
@@ -384,7 +384,7 @@ qed.
 abbrev sint32_bnd xL xH x =
  xL <= W32.to_sint x <= xH.
 
-lemma nosmt modzD_sint32 (x y: W32.t):
+lemma modzD_sint32 (x y: W32.t):
  (to_sint x + to_sint y) %% W32.modulus
  = (to_uint x + to_uint y) %% W32.modulus.
 proof.
@@ -393,14 +393,14 @@ case: (2 ^ (32 - 1) <= to_uint x);
  case: (2 ^ (32 - 1) <= to_uint y); smt().
 qed.
 
-lemma nosmt to_sint32D (x y: W32.t):
+lemma to_sint32D (x y: W32.t):
  to_sint (x+y)
  = W32.smod ((to_sint x + to_sint y)%%W32.modulus). 
 proof.
 by rewrite {1}/W32.to_sint to_uintD modzD_sint32.
 qed.
 
-lemma nosmt smod32_small (x: int):
+lemma smod32_small (x: int):
  -2^(32-1) <= x < 2^(32-1) => 
  W32.smod (x %% W32.modulus) = x.
 proof.
@@ -410,7 +410,7 @@ rewrite /smod; case: (x < 0) => C.
 by have ->/#: ! 2 ^ (32 - 1) <= x %% W32.modulus by smt().
 qed.
 
-lemma nosmt to_sint32D_small (x y: W32.t):
+lemma to_sint32D_small (x y: W32.t):
  W32.min_sint <= to_sint x + to_sint y <= W32.max_sint =>
  to_sint (x+y) = to_sint x + to_sint y.
 proof.
@@ -427,7 +427,7 @@ proof.
 by move=> /> *; rewrite to_sint32D_small /#. 
 qed.
 (*
-lemma nosmt modzN_sint32 (x: W32.t):
+lemma modzN_sint32 (x: W32.t):
  (- to_sint x) %% W32.modulus
  = (- to_uint x) %% W32.modulus.
 proof.
@@ -435,14 +435,14 @@ rewrite /to_sint /smod.
 case: (2 ^ (32 - 1) <= to_uint x); smt().
 qed.
 
-lemma nosmt to_sint32N (x: W32.t):
+lemma to_sint32N (x: W32.t):
  to_sint (-x)
  = W32.smod ((-to_sint x) %% W32.modulus).
 proof.
 by rewrite {1}/W32.to_sint to_uintN modzN_sint32.
 qed.
 *)
-lemma nosmt modzB_sint32 (x y: W32.t):
+lemma modzB_sint32 (x y: W32.t):
  (to_sint x - to_sint y) %% W32.modulus
  = (to_uint x - to_uint y) %% W32.modulus.
 proof.
@@ -451,7 +451,7 @@ case: (2 ^ (32 - 1) <= to_uint x);
  case: (2 ^ (32 - 1) <= to_uint y); smt().
 qed.
 
-lemma nosmt to_sint32B (x y: W32.t):
+lemma to_sint32B (x y: W32.t):
  to_sint (x-y)
  = W32.smod ((to_sint x - to_sint y)%%W32.modulus). 
 proof.
@@ -512,7 +512,7 @@ rewrite -{1}unpack16K /unpack16 /=; congr.
 by rewrite init_of_list -JUtils.iotaredE /=.
 qed.
 
-lemma nosmt modz_sint32 (x: W32.t):
+lemma modz_sint32 (x: W32.t):
  (to_sint x) %% W16.modulus
  = (to_uint x) %% W16.modulus.
 proof.
@@ -567,7 +567,7 @@ op REDC16 (xyL xyH: W16.t): W16.t =
  in xyH - (wmulhs m (W16.of_int q)).
 
 (* general bounds... *)
-lemma nosmt REDC16_correct bL bR (xyL xyH: W16.t):
+lemma REDC16_correct bL bR (xyL xyH: W16.t):
  W16.min_sint + 1664 <= bL <= 0 =>
  0 <= bR <= W16.max_sint - 1665 =>
  sint_bnd bL bR xyH =>
@@ -608,7 +608,7 @@ by rewrite eq_sym -modzDmr -Domain.mulNr -modzMm modzz mod0z.
 qed.
 
 (* useful specific case *)
-lemma nosmt REDC16_correct_q (xyL xyH: W16.t):
+lemma REDC16_correct_q (xyL xyH: W16.t):
  sint_bnd (-q%/2) (q%/2 - 1) xyH =>
  to_sint (REDC16 xyL xyH) %% q
  = (to_sint xyH * R + to_uint xyL) * Rinv %% q
@@ -625,7 +625,7 @@ abbrev REDCmul16 (x y: W16.t): W16.t = REDC16 (x*y) (wmulhs x y).
 
 (* correctness result for multiplication, for
  the specific case of a reduced argument *)
-lemma nosmt REDCmul16_correct (x y: W16.t):
+lemma REDCmul16_correct (x y: W16.t):
  sint_bnd 0 (q-1) y =>
  to_sint (REDCmul16 x y) %% q
  = to_sint x * to_sint y * Rinv %% q
diff --git a/proof/correctness/avx2/MLKEM_InnerPKE_avx2.ec b/proof/correctness/avx2/MLKEM_InnerPKE_avx2.ec
index 9eaad408..91e3775a 100644
--- a/proof/correctness/avx2/MLKEM_InnerPKE_avx2.ec
+++ b/proof/correctness/avx2/MLKEM_InnerPKE_avx2.ec
@@ -841,7 +841,7 @@ move: (noise_coef_avx2_aux bytes j) => /=.
 by rewrite /noise_coef C' to_sintE /smod to_uint_shr //= => <- /#.
 qed.
 
-lemma nosmt to_sint8_mod x:
+lemma to_sint8_mod x:
  W8.to_sint x %% W8.modulus = to_uint x.
 proof.
 rewrite /to_sint /smod.
diff --git a/proof/correctness/avx2/NTT_AVX_j.ec b/proof/correctness/avx2/NTT_AVX_j.ec
index d7528c9f..6ff926f5 100644
--- a/proof/correctness/avx2/NTT_AVX_j.ec
+++ b/proof/correctness/avx2/NTT_AVX_j.ec
@@ -1068,7 +1068,11 @@ lemma wmuls16P n x y _x _y:
  Iu16_sb n x _x =>
  Iu16_sb n y _y =>
  sint32_bnd (-n*n*q*q) (n*n*q*q) (wmuls16 x y).
-proof. by move => [??] [??]; rewrite to_sint_wmuls16 /#. qed.
+proof.
+move=> [? [??]] [? [??]]; rewrite to_sint_wmuls16.
+have ->: (n * n * q * q) = (n * q) * (n * q) by ring.
+by rewrite &(ler_norml) normrM ler_pmul // 1,2:normr_ge0 /#.
+qed.
 
 phoare wmul_16u16_ph n _x _y:
   [Jkem_avx2.M(Jkem_avx2.Syscall).__wmul_16u16:
@@ -1506,7 +1510,7 @@ qed.
 
 (** Butterfly *)
 
-lemma nosmt REDCmul16coeff (x y: W16.t):
+lemma REDCmul16coeff (x y: W16.t):
  sint_bnd 0 (q-1) y =>
  incoeffW16 (Montgomery16.REDCmul16 x y)
  = incoeffW16 x * incoeffW16 y * incoeff Montgomery16.Rinv