From bd34d676010ebf6dd4aa5076537f89572803dd3d Mon Sep 17 00:00:00 2001
From: jadep <jade.philipoom@gmail.com>
Date: Wed, 3 Apr 2019 14:45:32 -0400
Subject: partition -> Partition.partition to prevent confusion with
 List.partition

---
 src/Arithmetic/BarrettReduction.v              | 52 +++++++++++++-------------
 src/Arithmetic/BaseConversion.v                | 16 ++++----
 src/Arithmetic/FancyMontgomeryReduction.v      |  4 +-
 src/Arithmetic/Freeze.v                        | 18 ++++-----
 src/Arithmetic/Partition.v                     | 16 +++++---
 src/Arithmetic/Saturated.v                     | 26 ++++++-------
 src/Arithmetic/UniformWeight.v                 | 16 ++++----
 src/Arithmetic/WordByWordMontgomery.v          | 22 +++++------
 src/PushButtonSynthesis/BarrettReduction.v     |  4 +-
 src/PushButtonSynthesis/Primitives.v           |  2 +-
 src/PushButtonSynthesis/WordByWordMontgomery.v |  4 +-
 11 files changed, 92 insertions(+), 88 deletions(-)

diff --git a/src/Arithmetic/BarrettReduction.v b/src/Arithmetic/BarrettReduction.v
index 2f1f272ac..54740308b 100644
--- a/src/Arithmetic/BarrettReduction.v
+++ b/src/Arithmetic/BarrettReduction.v
@@ -35,7 +35,7 @@ Section Generic.
           (width_pos : 0 < width)
           (strong_bound : b ^ 1 * (b ^ (2 * k) mod M) <= b ^ (k + 1) - mu).
   Local Notation weight := (uweight width).
-  Local Notation partition := (partition weight).
+  Local Notation partition := (Partition.partition weight).
   Context (q1 : list Z -> list Z)
           (q1_correct :
              forall x,
@@ -116,7 +116,7 @@ Module Fancy.
             (k_eq : k = width * Z.of_nat sz).
     (* sz = 1, width = k = 256 *)
     Local Notation w := (uweight width). Local Notation eval := (Positional.eval w).
-    Context (mut Mt : list Z) (mut_correct : mut = partition w (sz+1) mu) (Mt_correct : Mt = partition w sz M).
+    Context (mut Mt : list Z) (mut_correct : mut = Partition.partition w (sz+1) mu) (Mt_correct : Mt = Partition.partition w sz M).
     Context (mu_eq : mu = 2 ^ (2 * k) / M) (muHigh_one : mu / w sz = 1) (M_range : 2^(k-1) < M < 2^k).
 
     Local Lemma wprops : @weight_properties w. Proof. apply uwprops; auto with lia. Qed.
@@ -201,7 +201,7 @@ Module Fancy.
       Lemma shiftr'_correct m n :
         forall t tn,
           (m <= tn)%nat -> 0 <= t < w tn -> 0 <= n < width ->
-          shiftr' m (partition w tn t) n = partition w m (t / 2 ^ n).
+          shiftr' m (Partition.partition w tn t) n = Partition.partition w m (t / 2 ^ n).
       Proof.
         cbv [shiftr']. induction m; intros; [ reflexivity | ].
         rewrite !partition_step, seq_snoc.
@@ -232,7 +232,7 @@ Module Fancy.
         forall t tn,
           (Z.to_nat (n / width) <= tn)%nat ->
           (m <= tn - Z.to_nat (n / width))%nat -> 0 <= t < w tn -> 0 <= n ->
-          shiftr m (partition w tn t) n = partition w m (t / 2 ^ n).
+          shiftr m (Partition.partition w tn t) n = Partition.partition w m (t / 2 ^ n).
       Proof.
         cbv [shiftr]; intros.
         break_innermost_match; [ | solve [auto using shiftr'_correct with zarith] ].
@@ -256,7 +256,7 @@ Module Fancy.
       (* 2 ^ (k + 1) bits fit in sz + 1 limbs because we know 2^k bits fit in sz and 1 <= width *)
       Lemma q1_correct x :
         0 <= x < w (sz * 2) ->
-        q1 (partition w (sz*2)%nat x) = partition w (sz+1)%nat (x / 2 ^ (k - 1)).
+        q1 (Partition.partition w (sz*2)%nat x) = Partition.partition w (sz+1)%nat (x / 2 ^ (k - 1)).
       Proof.
         cbv [q1]; intros. assert (1 <= Z.of_nat sz) by (destruct sz; lia).
         assert (Z.to_nat ((k-1) / width) < sz)%nat. {
@@ -266,13 +266,13 @@ Module Fancy.
         autorewrite with pull_partition. reflexivity.
       Qed.
 
-      Lemma low_correct n a : (sz <= n)%nat -> low (partition w n a) = partition w sz a.
+      Lemma low_correct n a : (sz <= n)%nat -> low (Partition.partition w n a) = Partition.partition w sz a.
       Proof. cbv [low]; auto using uweight_firstn_partition with lia. Qed.
-      Lemma high_correct a : high (partition w (sz*2) a) = partition w sz (a / w sz).
+      Lemma high_correct a : high (Partition.partition w (sz*2) a) = Partition.partition w sz (a / w sz).
       Proof. cbv [high]. rewrite uweight_skipn_partition by lia. f_equal; lia. Qed.
       Lemma fill_correct n m a :
         (n <= m)%nat ->
-        fill m (partition w n a) = partition w m (a mod w n).
+        fill m (Partition.partition w n a) = Partition.partition w m (a mod w n).
       Proof.
         cbv [fill]; intros. distr_length.
         rewrite <-partition_0 with (weight:=w).
@@ -282,21 +282,21 @@ Module Fancy.
       Hint Rewrite low_correct high_correct fill_correct using lia : pull_partition.
 
       Lemma wideadd_correct a b :
-        wideadd (partition w (sz*2) a) (partition w (sz*2) b) = partition w (sz*2) (a + b).
+        wideadd (Partition.partition w (sz*2) a) (Partition.partition w (sz*2) b) = Partition.partition w (sz*2) (a + b).
       Proof.
         cbv [wideadd]. rewrite Rows.add_partitions by (distr_length; auto).
         autorewrite with push_eval.
         apply partition_eq_mod; auto with zarith.
       Qed.
       Lemma widesub_correct a b :
-        widesub (partition w (sz*2) a) (partition w (sz*2) b) = partition w (sz*2) (a - b).
+        widesub (Partition.partition w (sz*2) a) (Partition.partition w (sz*2) b) = Partition.partition w (sz*2) (a - b).
       Proof.
         cbv [widesub]. rewrite Rows.sub_partitions by (distr_length; auto).
         autorewrite with push_eval.
         apply partition_eq_mod; auto with zarith.
       Qed.
       Lemma widemul_correct a b :
-        widemul (partition w sz a) (partition w sz b) = partition w (sz*2) ((a mod w sz) * (b mod w sz)).
+        widemul (Partition.partition w sz a) (Partition.partition w sz b) = Partition.partition w (sz*2) ((a mod w sz) * (b mod w sz)).
       Proof.
         cbv [widemul]. rewrite BaseConversion.widemul_inlined_correct; (distr_length; auto).
         autorewrite with push_eval. reflexivity.
@@ -327,8 +327,8 @@ Module Fancy.
       Lemma mul_high_correct a b
             (Ha : a / w sz = 1)
             a0b1 (Ha0b1 : a0b1 = a mod w sz * (b / w sz)) :
-        mul_high (partition w (sz*2) a) (partition w (sz*2) b) (partition w (sz*2) a0b1) =
-        partition w (sz*2) (a * b / w sz).
+        mul_high (Partition.partition w (sz*2) a) (Partition.partition w (sz*2) b) (Partition.partition w (sz*2) a0b1) =
+        Partition.partition w (sz*2) (a * b / w sz).
       Proof.
         cbv [mul_high Let_In].
         erewrite mul_high_idea by auto using Z.div_mod with zarith.
@@ -359,7 +359,7 @@ Module Fancy.
 
       Lemma muSelect_correct x :
         0 <= x < w (sz * 2) ->
-        muSelect (partition w (sz*2) x) = partition w sz (mu mod (w sz) * (x / 2 ^ (k - 1) / (w sz))).
+        muSelect (Partition.partition w (sz*2) x) = Partition.partition w sz (mu mod (w sz) * (x / 2 ^ (k - 1) / (w sz))).
       Proof.
         cbv [muSelect]; intros;
           repeat match goal with
@@ -391,7 +391,7 @@ Module Fancy.
       Qed.
 
       Lemma q3_correct x (Hx : 0 <= x < w (sz * 2)) q1 (Hq1 : q1 = x / 2 ^ (k - 1)) :
-        q3 (partition w (sz*2) x) (partition w (sz+1) q1) = partition w (sz+1) ((mu*q1) / 2 ^ (k + 1)).
+        q3 (Partition.partition w (sz*2) x) (Partition.partition w (sz+1) q1) = Partition.partition w (sz+1) ((mu*q1) / 2 ^ (k + 1)).
       Proof.
         cbv [q3 Let_In]. intros. pose proof mu_q1_range x ltac:(lia).
         pose proof mu_range'. pose proof q1_range x ltac:(lia).
@@ -408,8 +408,8 @@ Module Fancy.
       Qed.
 
       Lemma cond_sub_correct a b :
-        cond_sub (partition w (sz*2) a) (partition w sz b)
-        = partition w sz (if dec ((a / w sz) mod 2 = 0)
+        cond_sub (Partition.partition w (sz*2) a) (Partition.partition w sz b)
+        = Partition.partition w sz (if dec ((a / w sz) mod 2 = 0)
                           then a
                           else a - b).
       Proof.
@@ -425,8 +425,8 @@ Module Fancy.
       Qed.
       Hint Rewrite cond_sub_correct : pull_partition.
       Lemma cond_subM_correct a :
-        cond_subM (partition w sz a)
-        = partition w sz (if dec (a mod w sz < M)
+        cond_subM (Partition.partition w sz a)
+        = Partition.partition w sz (if dec (a mod w sz < M)
                           then a
                           else a - M).
       Proof.
@@ -479,7 +479,7 @@ Module Fancy.
         0 <= x < M * 2 ^ k ->
         0 <= q3 ->
         (exists b : bool, q3 = x / M + (if b then -1 else 0)) ->
-        r (partition w (sz*2) x) (partition w (sz+1) q3) = partition w sz (x mod M).
+        r (Partition.partition w (sz*2) x) (Partition.partition w (sz+1) q3) = Partition.partition w sz (x mod M).
       Proof.
         intros; cbv [r Let_In]. pose proof M_range'. assert (0 < 2^(k-1)) by Z.zero_bounds.
         autorewrite with pull_partition. Z.rewrite_mod_small.
@@ -515,7 +515,7 @@ Module Fancy.
       Lemma fancy_reduce_muSelect_first_correct x :
         0 <= x < M * 2^k ->
         2 * (2 ^ (2 * k) mod M) <= 2 ^ (k + 1) - mu ->
-        fancy_reduce_muSelect_first (partition w (sz*2) x) = partition w sz (x mod M).
+        fancy_reduce_muSelect_first (Partition.partition w (sz*2) x) = Partition.partition w sz (x mod M).
       Proof.
         intros. pose proof w_eq_22k.
         erewrite <-reduce_correct with (b:=2) (k:=k) (mu:=mu) by
@@ -528,7 +528,7 @@ Module Fancy.
                forall x,
                  0 <= x < M * 2^k ->
                  2 * (2 ^ (2 * k) mod M) <= 2 ^ (k + 1) - mu ->
-                 fancy_reduce' (partition w (sz*2) x) = partition w sz (x mod M))
+                 fancy_reduce' (Partition.partition w (sz*2) x) = Partition.partition w sz (x mod M))
              As fancy_reduce'_correct.
       Proof.
         intros. assert (k = width) as width_eq_k by nia.
@@ -546,9 +546,9 @@ Module Fancy.
       Lemma partition_2 xLow xHigh :
         0 <= xLow < 2 ^ k ->
         0 <= xHigh < M ->
-        partition w 2 (xLow + 2^k * xHigh) = [xLow;xHigh].
+        Partition.partition w 2 (xLow + 2^k * xHigh) = [xLow;xHigh].
       Proof.
-        replace k with width in M_range |- * by nia; intros. cbv [partition map seq].
+        replace k with width in M_range |- * by nia; intros. cbv [Partition.partition map seq].
         rewrite !uweight_S, !weight_0 by auto with zarith lia.
         autorewrite with zsimplify.
         rewrite <-Z.mod_pull_div by Z.zero_bounds.
@@ -567,10 +567,10 @@ Module Fancy.
         intros. cbv [fancy_reduce]. rewrite <-partition_2 by lia.
         replace 2%nat with (sz*2)%nat by lia.
         rewrite fancy_reduce'_correct by nia.
-        rewrite sz_eq_1; cbv [partition map seq hd].
+        rewrite sz_eq_1; cbv [Partition.partition map seq hd].
         rewrite !uweight_S, !weight_0 by auto with zarith lia.
         autorewrite with zsimplify. reflexivity.
       Qed.
     End Def.
   End Fancy.
-End Fancy.
\ No newline at end of file
+End Fancy.
diff --git a/src/Arithmetic/BaseConversion.v b/src/Arithmetic/BaseConversion.v
index dde04ae2a..ca5890705 100644
--- a/src/Arithmetic/BaseConversion.v
+++ b/src/Arithmetic/BaseConversion.v
@@ -41,13 +41,13 @@ Module BaseConversion.
     Lemma convert_bases_partitions sn dn p
           (dw_unique : forall i j : nat, (i <= dn)%nat -> (j <= dn)%nat -> dw i = dw j -> i = j)
           (p_bounded : 0 <= eval sw sn p < dw dn)
-      : convert_bases sn dn p = partition dw dn (eval sw sn p).
+      : convert_bases sn dn p = Partition.partition dw dn (eval sw sn p).
     Proof using dwprops.
       apply list_elementwise_eq; intro i.
       destruct (lt_dec i dn); [ | now rewrite !nth_error_length_error by distr_length ].
       erewrite !(@nth_error_Some_nth_default _ _ 0) by (break_match; distr_length).
       apply f_equal.
-      cbv [convert_bases partition].
+      cbv [convert_bases Partition.partition].
       unshelve erewrite map_nth_default, nth_default_chained_carries_no_reduce_pred;
         repeat first [ progress autorewrite with distr_length push_eval
                      | rewrite eval_from_associational, eval_to_associational
@@ -117,7 +117,7 @@ Module BaseConversion.
     Hint Rewrite eval_from_associational using solve [push_eval; distr_length] : push_eval.
 
     Lemma from_associational_partitions n idxs p  (_:n<>0%nat):
-      from_associational idxs n p = partition sw n (Associational.eval p).
+      from_associational idxs n p = Partition.partition sw n (Associational.eval p).
     Proof using dwprops swprops.
       intros. cbv [from_associational].
       rewrite Rows.flatten_correct with (n:=n) by eauto using Rows.length_from_associational.
@@ -181,7 +181,7 @@ Module BaseConversion.
 
       Lemma mul_converted_partitions n1 n2 m1 m2 n3 idxs p1 p2  (_:n3<>0%nat) (_:m1<>0%nat) (_:m2<>0%nat):
         length p1 = n1 -> length p2 = n2 ->
-        mul_converted n1 n2 m1 m2 n3 idxs p1 p2 = partition sw n3 (Positional.eval sw n1 p1 * Positional.eval sw n2 p2).
+        mul_converted n1 n2 m1 m2 n3 idxs p1 p2 = Partition.partition sw n3 (Positional.eval sw n1 p1 * Positional.eval sw n2 p2).
       Proof using dwprops swprops.
         intros; cbv [mul_converted].
         rewrite from_associational_partitions by auto. push_eval.
@@ -215,14 +215,14 @@ Module BaseConversion.
     Lemma widemul_correct a b :
       length a = m ->
       length b = m ->
-      widemul a b = partition sw nout (seval m a * seval m b). 
+      widemul a b = Partition.partition sw nout (seval m a * seval m b). 
     Proof. apply mul_converted_partitions; auto with zarith. Qed.
 
     Derive widemul_inlined
            SuchThat (forall a b,
                         length a = m ->
                         length b = m ->
-                        widemul_inlined a b = partition sw nout (seval m a * seval m b))
+                        widemul_inlined a b = Partition.partition sw nout (seval m a * seval m b))
            As widemul_inlined_correct.
     Proof.
       intros.
@@ -238,7 +238,7 @@ Module BaseConversion.
            SuchThat (forall a b,
                         length a = m ->
                         length b = m ->
-                        widemul_inlined_reverse a b = partition sw nout (seval m a * seval m b))
+                        widemul_inlined_reverse a b = Partition.partition sw nout (seval m a * seval m b))
            As widemul_inlined_reverse_correct.
     Proof.
       intros.
@@ -258,4 +258,4 @@ Module BaseConversion.
         reflexivity. }
     Qed.
   End widemul.
-End BaseConversion.
\ No newline at end of file
+End BaseConversion.
diff --git a/src/Arithmetic/FancyMontgomeryReduction.v b/src/Arithmetic/FancyMontgomeryReduction.v
index 51b578bed..f69f7c1d4 100644
--- a/src/Arithmetic/FancyMontgomeryReduction.v
+++ b/src/Arithmetic/FancyMontgomeryReduction.v
@@ -91,7 +91,7 @@ Module MontgomeryReduction.
       autorewrite with widemul.
       rewrite Rows.add_partitions, Rows.add_div by (distr_length; apply wprops; omega).
       (* rewrite R_two_pow. *)
-      cbv [partition seq].
+      cbv [Partition.partition seq].
       repeat match goal with
                | _ => progress rewrite ?eval1, ?eval2
                | _ => progress rewrite ?Z.zselect_correct, ?Z.add_modulo_correct
@@ -127,4 +127,4 @@ Module MontgomeryReduction.
       apply reduce_via_partial_in_range; omega.
     Qed.
   End MontRed'.
-End MontgomeryReduction.
\ No newline at end of file
+End MontgomeryReduction.
diff --git a/src/Arithmetic/Freeze.v b/src/Arithmetic/Freeze.v
index e766e7aea..bda62617a 100644
--- a/src/Arithmetic/Freeze.v
+++ b/src/Arithmetic/Freeze.v
@@ -116,7 +116,7 @@ Module Freeze.
           (Hp : 0 <= Positional.eval weight n p < 2*modulus)
           (Hplen : length p = n)
           (Hmlen : length m = n)
-      : @freeze n mask m p = partition weight n (Positional.eval weight n p mod modulus).
+      : @freeze n mask m p = Partition.partition weight n (Positional.eval weight n p mod modulus).
     Proof using wprops.
       pose proof (@weight_positive weight wprops n).
       pose proof (fun v => Z.mod_pos_bound v (weight n) ltac:(lia)).
@@ -252,7 +252,7 @@ Section freeze_mod_ops.
     : forall (f : list Z)
              (Hf : length f = n)
              (Hf_small : 0 <= eval weight n f < weight n),
-      to_bytes f = partition bytes_weight bytes_n (Positional.eval weight n f).
+      to_bytes f = Partition.partition bytes_weight bytes_n (Positional.eval weight n f).
   Proof using Hn_nz limbwidth_good.
     clear -Hn_nz limbwidth_good.
     intros; cbv [to_bytes].
@@ -265,7 +265,7 @@ Section freeze_mod_ops.
              (Hf : length f = n)
              (Hf_small : 0 <= eval weight n f < weight n),
       eval bytes_weight bytes_n (to_bytesmod f) = eval weight n f
-      /\ to_bytesmod f = partition bytes_weight bytes_n (Positional.eval weight n f).
+      /\ to_bytesmod f = Partition.partition bytes_weight bytes_n (Positional.eval weight n f).
   Proof using Hn_nz limbwidth_good.
     split; apply eval_to_bytes || apply to_bytes_partitions; assumption.
   Qed.
@@ -275,7 +275,7 @@ Section freeze_mod_ops.
         (Hf : length f = n)
         (Hf_bounded : 0 <= eval weight n f < 2 * m),
       (eval bytes_weight bytes_n (freeze_to_bytesmod f)) = (eval weight n f) mod m
-      /\ freeze_to_bytesmod f = partition bytes_weight bytes_n (Positional.eval weight n f mod m).
+      /\ freeze_to_bytesmod f = Partition.partition bytes_weight bytes_n (Positional.eval weight n f mod m).
   Proof using m_enc_correct Hs limbwidth_good Hn_nz c_small Hm_enc_len m_enc_bounded.
     clear -m_enc_correct Hs limbwidth_good Hn_nz c_small Hm_enc_len m_enc_bounded.
     intros; subst m s.
@@ -299,7 +299,7 @@ Section freeze_mod_ops.
     : forall (f : list Z)
         (Hf : length f = n)
         (Hf_bounded : 0 <= eval weight n f < 2 * m),
-      freeze_to_bytesmod f = partition bytes_weight bytes_n (Positional.eval weight n f mod m).
+      freeze_to_bytesmod f = Partition.partition bytes_weight bytes_n (Positional.eval weight n f mod m).
   Proof using m_enc_correct Hs limbwidth_good Hn_nz c_small Hm_enc_len m_enc_bounded.
     intros; now apply eval_freeze_to_bytesmod_and_partitions.
   Qed.
@@ -320,7 +320,7 @@ Section freeze_mod_ops.
   Lemma from_bytes_partitions
     : forall (f : list Z)
              (Hf_small : 0 <= eval bytes_weight bytes_n f < weight n),
-      from_bytes f = partition weight n (Positional.eval bytes_weight bytes_n f).
+      from_bytes f = Partition.partition weight n (Positional.eval bytes_weight bytes_n f).
   Proof using limbwidth_good.
     clear -limbwidth_good.
     intros; cbv [from_bytes].
@@ -337,7 +337,7 @@ Section freeze_mod_ops.
   Lemma from_bytesmod_partitions
     : forall (f : list Z)
              (Hf_small : 0 <= eval bytes_weight bytes_n f < weight n),
-      from_bytesmod f = partition weight n (Positional.eval bytes_weight bytes_n f).
+      from_bytesmod f = Partition.partition weight n (Positional.eval bytes_weight bytes_n f).
   Proof using limbwidth_good. apply from_bytes_partitions. Qed.
 
   Lemma eval_from_bytesmod_and_partitions
@@ -345,9 +345,9 @@ Section freeze_mod_ops.
              (Hf : length f = bytes_n)
              (Hf_small : 0 <= eval bytes_weight bytes_n f < weight n),
       eval weight n (from_bytesmod f) = eval bytes_weight bytes_n f
-      /\ from_bytesmod f = partition weight n (Positional.eval bytes_weight bytes_n f).
+      /\ from_bytesmod f = Partition.partition weight n (Positional.eval bytes_weight bytes_n f).
   Proof using limbwidth_good Hn_nz.
     now (split; [ apply eval_from_bytesmod | apply from_bytes_partitions ]).
   Qed.
 End freeze_mod_ops.
-Hint Rewrite eval_freeze_to_bytesmod eval_to_bytes eval_to_bytesmod eval_from_bytes eval_from_bytesmod : push_eval.
\ No newline at end of file
+Hint Rewrite eval_freeze_to_bytesmod eval_to_bytes eval_to_bytesmod eval_from_bytes eval_from_bytesmod : push_eval.
diff --git a/src/Arithmetic/Partition.v b/src/Arithmetic/Partition.v
index 2d2fb87fa..ed2c45f9e 100644
--- a/src/Arithmetic/Partition.v
+++ b/src/Arithmetic/Partition.v
@@ -9,11 +9,15 @@ Require Import Crypto.Util.ZUtil.Modulo Crypto.Util.ZUtil.Div.
 Require Import Crypto.Util.Notations.
 Import ListNotations Weight. Local Open Scope Z_scope.
 
-Section Partition.
-  Context weight {wprops : @weight_properties weight}.
-
-  Definition partition n x :=
+(* extra name wrapper so partition won't be confused with List.partition *)
+Module Partition.
+  Definition partition (weight : nat -> Z) n x :=
     map (fun i => (x mod weight (S i)) / weight i) (seq 0 n).
+End Partition.
+
+Section PartitionProofs.
+  Context weight {wprops : @weight_properties weight}.
+  Local Notation partition := (Partition.partition weight).
 
   Lemma partition_step n x :
     partition (S n) x = partition n x ++ [(x mod weight (S n)) / weight n].
@@ -124,6 +128,6 @@ Section Partition.
     autorewrite with zsimplify; reflexivity.
   Qed.
 
-End Partition.
+End PartitionProofs.
 Hint Rewrite length_partition length_recursive_partition : distr_length.
-Hint Rewrite eval_partition using (solve [auto; distr_length]) : push_eval.
\ No newline at end of file
+Hint Rewrite eval_partition using (solve [auto; distr_length]) : push_eval.
diff --git a/src/Arithmetic/Saturated.v b/src/Arithmetic/Saturated.v
index c82f6afa9..dc258cfd9 100644
--- a/src/Arithmetic/Saturated.v
+++ b/src/Arithmetic/Saturated.v
@@ -219,7 +219,7 @@ Module Columns.
       Lemma flatten_correct inp:
         forall n,
           length inp = n ->
-          flatten inp = (partition weight n (eval n inp),
+          flatten inp = (Partition.partition weight n (eval n inp),
                          eval n inp / (weight n)).
       Proof using wprops.
         induction inp using rev_ind; intros;
@@ -654,10 +654,10 @@ Module Rows.
             nm = (n + m)%nat ->
             let eval := Positional.eval weight in
             snd (fst start_state) = (eval m row1' + eval m row2') / weight m ->
-            (fst (fst start_state) = partition weight m (eval m row1' + eval m row2')) ->
+            (fst (fst start_state) = Partition.partition weight m (eval m row1' + eval m row2')) ->
             let sum := eval nm (row1' ++ row1) + eval nm (row2' ++ row2) in
             sum_rows' start_state row1 row2
-            = (partition weight nm sum, sum / weight nm, nm) .
+            = (Partition.partition weight nm sum, sum / weight nm, nm) .
         Proof using wprops.
           destruct start_state as [ [acc rem] m].
           cbn [fst snd]. revert acc rem m.
@@ -687,7 +687,7 @@ Module Rows.
         Lemma sum_rows_correct row1: forall row2 n,
             length row1 = n -> length row2 = n ->
             let sum := Positional.eval weight n row1 + Positional.eval weight n row2 in
-            sum_rows row1 row2 = (partition weight n sum, sum / weight n).
+            sum_rows row1 row2 = (Partition.partition weight n sum, sum / weight n).
         Proof using wprops.
           cbv [sum_rows]; intros.
           erewrite sum_rows'_correct with (nm:=n) (row1':=nil) (row2':=nil)by (cbn; distr_length; reflexivity).
@@ -755,7 +755,7 @@ Module Rows.
         (forall row, In row inp -> length row = n) ->
         inp <> nil ->
         let sum := Positional.eval weight n (fst start_state) + eval n inp + weight n * snd start_state in
-        flatten' start_state inp = (partition weight n sum, sum / weight n).
+        flatten' start_state inp = (Partition.partition weight n sum, sum / weight n).
       Proof using wprops.
         induction inp using rev_ind; push. subst sum.
         destruct (dec (inp = nil)); [ subst inp; cbn | ];
@@ -776,7 +776,7 @@ Module Rows.
 
       Lemma flatten_correct inp n :
         (forall row, In row inp -> length row = n) ->
-        flatten n inp = (partition weight n (eval n inp), eval n inp / weight n).
+        flatten n inp = (Partition.partition weight n (eval n inp), eval n inp / weight n).
       Proof using wprops.
         intros; cbv [flatten].
         destruct inp; [|destruct inp]; cbn [hd tl];
@@ -877,7 +877,7 @@ Module Rows.
 
       Lemma add_partitions n p q :
         length p = n -> length q = n ->
-        fst (add n p q) = partition weight n (Positional.eval weight n p + Positional.eval weight n q).
+        fst (add n p q) = Partition.partition weight n (Positional.eval weight n p + Positional.eval weight n q).
       Proof using wprops. solver. Qed.
 
       Lemma add_div n p q :
@@ -888,7 +888,7 @@ Module Rows.
       Lemma conditional_add_partitions n mask cond p q :
         length p = n -> length q = n -> map (Z.land mask) q = q ->
         fst (conditional_add n mask cond p q)
-        = partition weight n (Positional.eval weight n p + if dec (cond = 0) then 0 else Positional.eval weight n q).
+        = Partition.partition weight n (Positional.eval weight n p + if dec (cond = 0) then 0 else Positional.eval weight n q).
       Proof using wprops.
         cbv [conditional_add]; intros; rewrite add_partitions by (distr_length; auto).
         autorewrite with push_eval; reflexivity.
@@ -917,7 +917,7 @@ Module Rows.
 
       Lemma sub_partitions n p q :
         length p = n -> length q = n ->
-        fst (sub n p q) = partition weight n (Positional.eval weight n p - Positional.eval weight n q).
+        fst (sub n p q) = Partition.partition weight n (Positional.eval weight n p - Positional.eval weight n q).
       Proof using wprops. solver. Qed.
 
       Lemma sub_div n p q :
@@ -926,10 +926,10 @@ Module Rows.
       Proof using wprops. solver. Qed.
 
       Lemma conditional_sub_partitions n p q
-        (Hp : p = partition weight n (Positional.eval weight n p)) :
+        (Hp : p = Partition.partition weight n (Positional.eval weight n p)) :
         length q = n ->
         0 <= Positional.eval weight n q < weight n ->
-        conditional_sub n p q = partition weight n (if Positional.eval weight n q <=? Positional.eval weight n p then Positional.eval weight n p - Positional.eval weight n q else Positional.eval weight n p).
+        conditional_sub n p q = Partition.partition weight n (if Positional.eval weight n q <=? Positional.eval weight n p then Positional.eval weight n p - Positional.eval weight n q else Positional.eval weight n p).
       Proof using wprops.
         cbv [conditional_sub]; intros.
         rewrite (surjective_pairing (sub _ _ _)).
@@ -952,7 +952,7 @@ Module Rows.
         map (Z.land mask) r = r ->
         0 <= Positional.eval weight n p < weight n ->
         0 <= Positional.eval weight n q < weight n ->
-        fst (sub_then_maybe_add n mask p q r) = partition weight n (sub_then_maybe_add_Z (Positional.eval weight n p) (Positional.eval weight n q) (Positional.eval weight n r)).
+        fst (sub_then_maybe_add n mask p q r) = Partition.partition weight n (sub_then_maybe_add_Z (Positional.eval weight n p) (Positional.eval weight n q) (Positional.eval weight n r)).
       Proof using wprops.
         cbv [sub_then_maybe_add]. subst sub_then_maybe_add_Z.
         intros.
@@ -969,7 +969,7 @@ Module Rows.
 
       Lemma mul_partitions base n m p q :
         base <> 0 -> m <> 0%nat -> length p = n -> length q = n ->
-        fst (mul base n m p q) = partition weight m (Positional.eval weight n p * Positional.eval weight n q).
+        fst (mul base n m p q) = Partition.partition weight m (Positional.eval weight n p * Positional.eval weight n q).
       Proof using wprops. solver. Qed.
 
       Lemma mul_div base n m p q :
diff --git a/src/Arithmetic/UniformWeight.v b/src/Arithmetic/UniformWeight.v
index 9af083994..25d6fb92d 100644
--- a/src/Arithmetic/UniformWeight.v
+++ b/src/Arithmetic/UniformWeight.v
@@ -80,7 +80,7 @@ Proof using Type.
   erewrite IHn. reflexivity.
 Qed.
 Lemma uweight_recursive_partition_equiv lgr (Hr : 0 < lgr) n i x:
-  partition (uweight lgr) n x =
+  Partition.partition (uweight lgr) n x =
   recursive_partition (uweight lgr) n i x.
 Proof using Type.
   rewrite recursive_partition_equiv by auto using uwprops.
@@ -88,9 +88,9 @@ Proof using Type.
 Qed.
 
 Lemma uweight_firstn_partition lgr (Hr : 0 < lgr) n x m (Hm : (m <= n)%nat) :
-  firstn m (partition (uweight lgr) n x) = partition (uweight lgr) m x.
+  firstn m (Partition.partition (uweight lgr) n x) = Partition.partition (uweight lgr) m x.
 Proof.
-  cbv [partition];
+  cbv [Partition.partition];
     repeat match goal with
            | _ => progress intros
            | _ => progress autorewrite with push_firstn natsimplify zsimplify_fast
@@ -101,9 +101,9 @@ Proof.
 Qed.
 
 Lemma uweight_skipn_partition lgr (Hr : 0 < lgr) n x m :
-  skipn m (partition (uweight lgr) n x) = partition (uweight lgr) (n - m) (x / uweight lgr m).
+  skipn m (Partition.partition (uweight lgr) n x) = Partition.partition (uweight lgr) (n - m) (x / uweight lgr m).
 Proof.
-  cbv [partition];
+  cbv [Partition.partition];
     repeat match goal with
            | _ => progress intros
            | _ => progress autorewrite with push_skipn natsimplify zsimplify_fast
@@ -116,7 +116,7 @@ Qed.
 
 Lemma uweight_partition_unique lgr (Hr : 0 < lgr) n ls :
   length ls = n -> (forall x, List.In x ls -> 0 <= x <= 2^lgr - 1) ->
-  ls = partition (uweight lgr) n (Positional.eval (uweight lgr) n ls).
+  ls = Partition.partition (uweight lgr) n (Positional.eval (uweight lgr) n ls).
 Proof using Type.
   intro; subst n.
   rewrite uweight_recursive_partition_equiv with (i:=0%nat) by assumption.
@@ -169,8 +169,8 @@ Lemma uweight_eval_app lgr (Hr : 0 <= lgr) n m x y :
 Proof using Type. intros. subst m. apply uweight_eval_app'; lia. Qed.
 
 Lemma uweight_partition_app lgr (Hr : 0 < lgr) n m a b :
-  partition (uweight lgr) n a ++ partition (uweight lgr) m b
-  = partition (uweight lgr) (n+m) (a mod uweight lgr n + b * uweight lgr n).
+ Partition.partition (uweight lgr) n a ++ Partition.partition (uweight lgr) m b
+  = Partition.partition (uweight lgr) (n+m) (a mod uweight lgr n + b * uweight lgr n).
 Proof.
   assert (0 < uweight lgr n) by auto using uwprops.
   match goal with |- _ = ?rhs => rewrite <-(firstn_skipn n rhs) end.
diff --git a/src/Arithmetic/WordByWordMontgomery.v b/src/Arithmetic/WordByWordMontgomery.v
index 3fb3437c1..5b1a8a256 100644
--- a/src/Arithmetic/WordByWordMontgomery.v
+++ b/src/Arithmetic/WordByWordMontgomery.v
@@ -126,7 +126,7 @@ Module WordByWordMontgomery.
     Let R := (r^Z.of_nat R_numlimbs).
     Transparent T.
     Definition small {n} (v : T n) : Prop
-      := v = partition weight n (eval v).
+      := v = Partition.partition weight n (eval v).
     Context (small_N : small N)
             (N_lt_R : eval N < R)
             (N_nz : 0 < eval N)
@@ -161,9 +161,9 @@ Module WordByWordMontgomery.
     Qed.
 
     Lemma mask_r_sub1 n x :
-      map (Z.land (r - 1)) (partition weight n x) = partition weight n x.
+      map (Z.land (r - 1)) (Partition.partition weight n x) = Partition.partition weight n x.
     Proof using lgr_big.
-      clear - lgr_big. cbv [partition].
+      clear - lgr_big. cbv [Partition.partition].
       rewrite map_map. apply map_ext; intros.
       rewrite uweight_S by omega.
       rewrite <-Z.mod_pull_div by auto with zarith.
@@ -248,7 +248,7 @@ Module WordByWordMontgomery.
     Qed.
     Local Lemma small_zero : forall n, small (@zero n).
     Proof using Type.
-      etransitivity; [ eapply Positional.zeros_ext_map | rewrite eval_zero ]; cbv [partition]; cbn; try reflexivity; autorewrite with distr_length; reflexivity.
+      etransitivity; [ eapply Positional.zeros_ext_map | rewrite eval_zero ]; cbv [Partition.partition]; cbn; try reflexivity; autorewrite with distr_length; reflexivity.
     Qed.
     Local Hint Immediate small_zero.
 
@@ -288,7 +288,7 @@ Module WordByWordMontgomery.
     Qed.
 
     Definition canon_rep {n} x (v : T n) : Prop :=
-      (v = partition weight n x) /\ (0 <= x < weight n).
+      (v = Partition.partition weight n x) /\ (0 <= x < weight n).
     Lemma eval_canon_rep n x v : @canon_rep n x v -> eval v = x.
     Proof using lgr_big.
       clear - lgr_big.
@@ -974,7 +974,7 @@ Module WordByWordMontgomery.
     Let r := 2^bitwidth.
     Local Notation weight := (uweight bitwidth).
     Local Notation eval := (@eval bitwidth n).
-    Let m_enc := partition weight n m.
+    Let m_enc := Partition.partition weight n m.
     Local Coercion Z.of_nat : nat >-> Z.
     Context (r' : Z)
             (m' : Z)
@@ -1059,7 +1059,7 @@ Module WordByWordMontgomery.
         t_fin.
     Qed.
 
-    Definition onemod : list Z := partition weight n 1.
+    Definition onemod : list Z := Partition.partition weight n 1.
 
     Definition onemod_correct : eval onemod = 1 /\ valid onemod.
     Proof using n_nz m_big bitwidth_big.
@@ -1070,7 +1070,7 @@ Module WordByWordMontgomery.
     Lemma eval_onemod : eval onemod = 1.
     Proof. apply onemod_correct. Qed.
 
-    Definition R2mod : list Z := partition weight n ((r^n * r^n) mod m).
+    Definition R2mod : list Z := Partition.partition weight n ((r^n * r^n) mod m).
 
     Definition R2mod_correct : eval R2mod mod m = (r^n*r^n) mod m /\ valid R2mod.
     Proof using n_nz m_small m_big m'_correct bitwidth_big.
@@ -1137,7 +1137,7 @@ Module WordByWordMontgomery.
     Proof. apply squaremod_correct. Qed.
 
     Definition encodemod (v : Z) : list Z
-      := mulmod (partition weight n v) R2mod.
+      := mulmod (Partition.partition weight n v) R2mod.
 
     Local Ltac t_valid v :=
       cbv [valid]; repeat apply conj;
@@ -1260,7 +1260,7 @@ Module WordByWordMontgomery.
     Lemma to_bytesmod_correct
       : (forall a (_ : valid a), Positional.eval (uweight 8) (bytes_n bitwidth 1 n) (to_bytesmod a)
                                  = eval a mod m)
-        /\ (forall a (_ : valid a), to_bytesmod a = partition (uweight 8) (bytes_n bitwidth 1 n) (eval a mod m)).
+        /\ (forall a (_ : valid a), to_bytesmod a = Partition.partition (uweight 8) (bytes_n bitwidth 1 n) (eval a mod m)).
     Proof using n_nz m_small bitwidth_big.
       clear -n_nz m_small bitwidth_big.
       generalize (@length_small bitwidth n);
@@ -1279,4 +1279,4 @@ Module WordByWordMontgomery.
             = eval a mod m).
     Proof. apply to_bytesmod_correct. Qed.
   End modops.
-End WordByWordMontgomery.
\ No newline at end of file
+End WordByWordMontgomery.
diff --git a/src/PushButtonSynthesis/BarrettReduction.v b/src/PushButtonSynthesis/BarrettReduction.v
index 42be4b7d4..990a0de03 100644
--- a/src/PushButtonSynthesis/BarrettReduction.v
+++ b/src/PushButtonSynthesis/BarrettReduction.v
@@ -75,7 +75,7 @@ Section rbarrett_red.
 
   Lemma mut_correct :
     0 < machine_wordsize ->
-    partition (uweight machine_wordsize) (1 + 1) (muLow + 2 ^ machine_wordsize) = [muLow; 1].
+    Partition.partition (uweight machine_wordsize) (1 + 1) (muLow + 2 ^ machine_wordsize) = [muLow; 1].
   Proof.
     intros; cbn. subst muLow.
     assert (0 < 2^machine_wordsize) by ZeroBounds.Z.zero_bounds.
@@ -90,7 +90,7 @@ Section rbarrett_red.
   Lemma Mt_correct :
     0 < machine_wordsize ->
     2^(machine_wordsize - 1) < M < 2^machine_wordsize ->
-    partition (uweight machine_wordsize) 1 M = [M].
+    Partition.partition (uweight machine_wordsize) 1 M = [M].
   Proof.
     intros; cbn. assert (0 < 2^(machine_wordsize-1)) by ZeroBounds.Z.zero_bounds.
     rewrite !uweight_S, weight_0; auto using uwprops with lia.
diff --git a/src/PushButtonSynthesis/Primitives.v b/src/PushButtonSynthesis/Primitives.v
index 93c2e4c69..a3df39c25 100644
--- a/src/PushButtonSynthesis/Primitives.v
+++ b/src/PushButtonSynthesis/Primitives.v
@@ -574,7 +574,7 @@ Module CorrectnessStringification.
   Ltac stringify ctx correctness fname arg_var_data out_var_data :=
     let G := match goal with |- ?G => G end in
     let correctness := (eval hnf in correctness) in
-    let correctness := (eval cbv [partition WordByWordMontgomery.valid WordByWordMontgomery.small] in correctness) in
+    let correctness := (eval cbv [Partition.partition WordByWordMontgomery.valid WordByWordMontgomery.small] in correctness) in
     let correctness := strip_bounds_info correctness in
     let arg_var_names := constr:(type.map_for_each_lhs_of_arrow (@ToString.C.OfPHOAS.names_of_var_data) arg_var_data) in
     let out_var_names := constr:(ToString.C.OfPHOAS.names_of_base_var_data out_var_data) in
diff --git a/src/PushButtonSynthesis/WordByWordMontgomery.v b/src/PushButtonSynthesis/WordByWordMontgomery.v
index ebf250664..ea67f8b9c 100644
--- a/src/PushButtonSynthesis/WordByWordMontgomery.v
+++ b/src/PushButtonSynthesis/WordByWordMontgomery.v
@@ -103,9 +103,9 @@ Section __.
             end.
   Let n_bytes := bytes_n machine_wordsize 1 n.
   Let prime_upperbound_list : list Z
-    := partition (uweight machine_wordsize) n (s-1).
+    := Partition.partition (uweight machine_wordsize) n (s-1).
   Let prime_bytes_upperbound_list : list Z
-    := partition (weight 8 1) n_bytes (s-1).
+    := Partition.partition (weight 8 1) n_bytes (s-1).
   Let upperbounds : list Z := prime_upperbound_list.
   Definition prime_bound : ZRange.type.option.interp (base.type.Z)
     := Some r[0~>m-1]%zrange.
-- 
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