create directory for saturated arithmetic in preparation for splitting into multiple files

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diff --git a/src/Arithmetic/Saturated/Core.v b/src/Arithmetic/Saturated/Core.v
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+++ b/src/Arithmetic/Saturated/Core.v
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+Require Import Coq.micromega.Lia.
+Require Import Coq.Init.Nat.
+Require Import Coq.ZArith.ZArith.
+Require Import Coq.Lists.List.
+Local Open Scope Z_scope.
+
+Require Import Crypto.Algebra.Nsatz.
+Require Import Crypto.Arithmetic.Core.
+Require Import Crypto.Util.LetIn Crypto.Util.CPSUtil.
+Require Import Crypto.Util.Tuple Crypto.Util.ListUtil.
+Require Import Crypto.Util.Tactics.BreakMatch.
+Require Import Crypto.Util.Decidable Crypto.Util.ZUtil.
+Require Import Crypto.Util.NatUtil.
+Require Import Crypto.Util.ZUtil.Definitions.
+Require Import Crypto.Util.ZUtil.AddGetCarry.
+Require Import Crypto.Util.ZUtil.Zselect.
+Require Import Crypto.Util.ZUtil.MulSplit.
+Require Import Crypto.Util.Tactics.SpecializeBy.
+Local Notation "A ^ n" := (tuple A n) : type_scope.
+
+(***
+
+Arithmetic on bignums that handles carry bits; this is useful for
+saturated limbs. Compatible with mixed-radix bases.
+
+Uses "columns" representation: a bignum has type [tuple (list Z) n].
+Associated with a weight function w, the bignum B represents:
+
+    \sum_{i=0}^{n}{w[i] * sum{B[i]}}
+
+Example: ([a21, a20],[],[a0]) with weight function (fun i => 10^i)
+represents
+
+    a0 + 10*0 + 100 * (a20 + a21)
+
+If you picture this representation with the weights on the bottom and
+the terms in each list stacked above the corresponding weight,
+
+                a20
+    a0          a21
+    ---------------
+     1    10    100
+
+it's easy to see how the lists can be called "columns".
+
+This is a particularly useful representation for adding partial
+products after multiplication, particularly when we want to do this
+using a carrying add. We want to add together the terms from each
+column, accumulating the carries together along the way. Then we want
+to add the carry accumulator to the next column, and repeat, producing
+a [tuple Z n] as output. This operation is called "compact".
+
+As an example, let's compact the product of 571 and 645 in base 10.
+At first, the partial products look like this:
+
+
+                   1*6
+            1*4    7*4     7*6
+     1*5    7*5    5*5     5*4      5*6
+    ------------------------------------
+       1     10    100    1000    10000
+
+                     6
+              4     28      42
+       5     35     25      20       30
+    ------------------------------------
+       1     10    100    1000    10000
+
+Now, we process the first column:
+
+     {carry_acc = 0; output =()}
+     STEP [5]
+     {carry_acc = 0; output=(5,)}
+
+Since we have only one term, there's no addition to do, and no carry
+bit. We add a 0 to the next column and continue.
+
+     STEP [0,4,35] (0 + 4 = 4)
+     {carry_acc = 0; output=(5,)}
+     STEP [4,35] (4 + 35 = 39)
+     {carry_acc = 3; output=(9,5)}
+
+This time, we have a carry. We add it to the third column and process
+that:
+
+     STEP [9,6,28,25] (9 + 6  = 15)
+     {carry_acc = 1; output=(9,5)}
+     STEP [5,28,25] (5 + 28 = 33)
+     {carry_acc = 4; output=(9,5)}
+     STEP [3,25] (3 + 25 = 28)
+     {carry_acc = 2; output=(8,9,5)}
+
+You're probably getting the idea, but here are the fourth and fifth
+columns:
+
+     STEP [2,42,20] (2 + 42 = 44)
+     {carry_acc = 4; output=(8,9,5)}
+     STEP [4,20] (4 + 20 = 24)
+     {carry_acc = 6; output=(4,8,9,5)}
+
+     STEP [6,30] (6 + 30 = 36)
+     {carry_acc = 3; output=(6,4,8,9,5)}
+
+The final result is the output plus the final carry, so we produce
+(6,4,8,9,5) and 3, representing the number 364895. A quick calculator
+check confirms our result.
+
+ ***)
+
+Module Associational.
+  Section Associational.
+    Context {mul_split : Z -> Z -> Z -> Z * Z} (* first argument is where to split output; [mul_split s x y] gives ((x * y) mod s, (x * y) / s) *)
+            {mul_split_mod : forall s x y,
+                fst (mul_split s x y)  = (x * y) mod s}
+            {mul_split_div : forall s x y,
+                snd (mul_split s x y)  = (x * y) / s}
+            .
+
+    Definition multerm_cps s (t t' : B.limb) {T} (f:list B.limb ->T) :=
+      dlet xy := mul_split s (snd t) (snd t') in
+      f ((fst t * fst t', fst xy) :: (fst t * fst t' * s, snd xy) :: nil).
+
+    Definition multerm s t t' := multerm_cps s t t' id.
+    Lemma multerm_id s t t' T f :
+      @multerm_cps s t t' T f = f (multerm s t t').
+    Proof. reflexivity. Qed.
+    Hint Opaque multerm : uncps.
+    Hint Rewrite multerm_id : uncps.
+
+    Definition mul_cps s (p q : list B.limb) {T} (f : list B.limb -> T) :=
+      flat_map_cps (fun t => @flat_map_cps _ _ (multerm_cps s t) q) p f.
+
+    Definition mul s p q := mul_cps s p q id.
+    Lemma mul_id s p q T f : @mul_cps s p q T f = f (mul s p q).
+    Proof. cbv [mul mul_cps]. autorewrite with uncps. reflexivity. Qed.
+    Hint Opaque mul : uncps.
+    Hint Rewrite mul_id : uncps.
+
+    Lemma eval_map_multerm s a q (s_nonzero:s<>0):
+      B.Associational.eval (flat_map (multerm s a) q) = fst a * snd a * B.Associational.eval q.
+    Proof.
+      cbv [multerm multerm_cps Let_In]; induction q;
+      repeat match goal with
+             | _ => progress (autorewrite with uncps push_id cancel_pair push_basesystem_eval in * )
+             | _ => progress simpl flat_map
+             | _ => progress rewrite ?IHq, ?mul_split_mod, ?mul_split_div
+             | _ => rewrite Z.mod_eq by assumption
+             | _ => ring_simplify; omega
+             end.
+    Qed.
+    Hint Rewrite eval_map_multerm using (omega || assumption)
+      : push_basesystem_eval.
+
+    Lemma eval_mul s p q (s_nonzero:s<>0):
+      B.Associational.eval (mul s p q) = B.Associational.eval p * B.Associational.eval q.
+    Proof.
+      cbv [mul mul_cps]; induction p; [reflexivity|].
+      repeat match goal with
+             | _ => progress (autorewrite with uncps push_id push_basesystem_eval in * )
+             | _ => progress simpl flat_map
+             | _ => rewrite IHp
+             | _ => progress change (fun x => multerm_cps s a x id)  with (multerm s a)
+             | _ => ring_simplify; omega
+             end.
+    Qed.
+    Hint Rewrite eval_mul : push_basesystem_eval.
+
+  End Associational.
+End Associational.
+Hint Opaque Associational.mul Associational.multerm : uncps.
+Hint Rewrite @Associational.mul_id @Associational.multerm_id : uncps.
+Hint Rewrite @Associational.eval_mul @Associational.eval_map_multerm using (omega || assumption) : push_basesystem_eval.
+
+
+Module Columns.
+  Section Columns.
+    Context (weight : nat->Z)
+            {weight_0 : weight 0%nat = 1}
+            {weight_nonzero : forall i, weight i <> 0}
+            {weight_positive : forall i, weight i > 0}
+            {weight_multiples : forall i, weight (S i) mod weight i = 0}
+            {weight_divides : forall i : nat, weight (S i) / weight i > 0}
+            (* add_get_carry takes in a number at which to split output *)
+            {add_get_carry: Z ->Z -> Z -> (Z * Z)}
+            {add_get_carry_mod : forall s x y,
+                fst (add_get_carry s x y)  = (x + y) mod s}
+            {add_get_carry_div : forall s x y,
+                snd (add_get_carry s x y)  = (x + y) / s}
+            {div modulo : Z -> Z -> Z}
+            {div_correct : forall a b, div a b = a / b}
+            {modulo_correct : forall a b, modulo a b = a mod b}
+    .
+    Hint Rewrite div_correct modulo_correct add_get_carry_mod add_get_carry_div : div_mod.
+
+    Definition eval {n} (x : (list Z)^n) : Z :=
+      B.Positional.eval weight (Tuple.map sum x).
+
+    Lemma eval_unit (x:unit) : eval (n:=0) x = 0.
+    Proof. reflexivity. Qed.
+    Hint Rewrite eval_unit : push_basesystem_eval.
+
+    Lemma eval_single (x:list Z) : eval (n:=1) x = sum x.
+    Proof.
+      cbv [eval]. simpl map. cbv - [Z.mul Z.add sum].
+      rewrite weight_0; ring.
+    Qed. Hint Rewrite eval_single : push_basesystem_eval.
+
+    Definition eval_from {n} (offset:nat) (x : (list Z)^n) : Z :=
+      B.Positional.eval (fun i => weight (i+offset)) (Tuple.map sum x).
+
+    Lemma eval_from_0 {n} x : @eval_from n 0 x = eval x.
+    Proof using Type. cbv [eval_from eval]. auto using B.Positional.eval_wt_equiv. Qed.
+
+    Lemma eval_from_S {n}: forall i (inp : (list Z)^(S n)),
+        eval_from i inp = eval_from (S i) (tl inp) + weight i * sum (hd inp).
+    Proof using Type.
+      intros i inp; cbv [eval_from].
+      replace inp with (append (hd inp) (tl inp))
+        by (simpl in *; destruct n; destruct inp; reflexivity).
+      rewrite map_append, B.Positional.eval_step, hd_append, tl_append.
+      autorewrite with natsimplify; ring_simplify; rewrite Group.cancel_left.
+      apply B.Positional.eval_wt_equiv; intros; f_equal; omega.
+    Qed.
+
+    (* Sums a list of integers using carry bits.
+     Output : carry, sum
+     *)
+    Fixpoint compact_digit_cps n (digit : list Z) {T} (f:Z * Z->T) :=
+      match digit with
+      | nil => f (0, 0)
+      | x :: nil => f (div x (weight (S n) / weight n), modulo x (weight (S n) / weight n))
+      | x :: y :: nil =>
+          dlet sum_carry := add_get_carry (weight (S n) / weight n) x y in
+          dlet carry := snd sum_carry in
+          f (carry, fst sum_carry)
+      | x :: tl =>
+        compact_digit_cps n tl
+          (fun rec =>
+            dlet sum_carry := add_get_carry (weight (S n) / weight n) x (snd rec) in
+            dlet carry' := (fst rec + snd sum_carry)%RT in
+            f (carry', fst sum_carry))
+      end.
+
+    Definition compact_digit n digit := compact_digit_cps n digit id.
+    Lemma compact_digit_id n digit: forall {T} f,
+        @compact_digit_cps n digit T f = f (compact_digit n digit).
+    Proof using Type.
+      induction digit; intros; cbv [compact_digit]; [reflexivity|];
+        simpl compact_digit_cps; break_match; rewrite ?IHdigit;
+          reflexivity.
+    Qed.
+    Hint Opaque compact_digit : uncps.
+    Hint Rewrite compact_digit_id : uncps.
+
+    Definition compact_step_cps (index:nat) (carry:Z) (digit: list Z)
+               {T} (f:Z * Z->T) :=
+      compact_digit_cps index (carry::digit) f.
+
+    Definition compact_step i c d := compact_step_cps i c d id.
+    Lemma compact_step_id i c d T f :
+      @compact_step_cps i c d T f = f (compact_step i c d).
+    Proof using Type. cbv [compact_step_cps compact_step]; autorewrite with uncps; reflexivity. Qed.
+    Hint Opaque compact_step : uncps.
+    Hint Rewrite compact_step_id : uncps.
+
+    Definition compact_cps {n} (xs : (list Z)^n) {T} (f:Z * Z^n->T) :=
+      Tuple.mapi_with_cps compact_step_cps 0 xs f.
+
+    Definition compact {n} xs := @compact_cps n xs _ id.
+    Lemma compact_id {n} xs {T} f : @compact_cps n xs T f = f (compact xs).
+    Proof using Type. cbv [compact_cps compact]; autorewrite with uncps; reflexivity. Qed.
+
+    Lemma compact_digit_mod i (xs : list Z) :
+      snd (compact_digit i xs)  = sum xs mod (weight (S i) / weight i).
+    Proof using add_get_carry_div add_get_carry_mod div_correct modulo_correct.
+      induction xs; cbv [compact_digit]; simpl compact_digit_cps;
+        cbv [Let_In];
+        repeat match goal with
+               | _ => progress autorewrite with div_mod
+               | _ => rewrite IHxs, <-Z.add_mod_r
+               | _ => progress (rewrite ?sum_cons, ?sum_nil in * )
+               | _ => progress (autorewrite with uncps push_id cancel_pair in * )
+               | _ => progress break_match; try discriminate
+               | _ => reflexivity
+               | _ => f_equal; ring
+               end.
+    Qed. Hint Rewrite compact_digit_mod : div_mod.
+
+    Lemma compact_digit_div i (xs : list Z) :
+      fst (compact_digit i xs)  = sum xs / (weight (S i) / weight i).
+    Proof using add_get_carry_div add_get_carry_mod div_correct modulo_correct weight_0 weight_divides.
+      induction xs; cbv [compact_digit]; simpl compact_digit_cps;
+        cbv [Let_In];
+        repeat match goal with
+               | _ => progress autorewrite with div_mod
+               | _ => rewrite IHxs
+               | _ => progress (rewrite ?sum_cons, ?sum_nil in * )
+               | _ => progress (autorewrite with uncps push_id cancel_pair in * )
+               | _ => progress break_match; try discriminate
+               | _ => reflexivity
+               | _ => f_equal; ring
+               end.
+      assert (weight (S i) / weight i <> 0) by auto using Z.positive_is_nonzero.
+      match goal with |- _ = (?a + ?X) / ?D =>
+                      transitivity  ((a + X mod D + D * (X / D)) / D);
+                        [| rewrite (Z.div_mod'' X D) at 3; f_equal; auto; ring]
+      end.
+      rewrite Z.div_add' by auto; nsatz.
+    Qed.
+
+    Lemma small_mod_eq a b n: a mod n = b mod n -> 0 <= a < n -> a = b mod n.
+    Proof. intros; rewrite <-(Z.mod_small a n); auto. Qed.
+
+    (* helper for some of the modular logic in compact *)
+    Lemma compact_mod_step a b c d: 0 < a -> 0 < b ->
+      a * ((c / a + d) mod b) + c mod a = (a * d + c) mod (a * b).
+    Proof.
+      intros Ha Hb. assert (a <= a * b) by (apply Z.le_mul_diag_r; omega).
+      pose proof (Z.mod_pos_bound c a Ha).
+      pose proof (Z.mod_pos_bound (c/a+d) b Hb).
+      apply small_mod_eq.
+      { rewrite <-(Z.mod_small (c mod a) (a * b)) by omega.
+        rewrite <-Z.mul_mod_distr_l with (c:=a) by omega.
+        rewrite Z.mul_add_distr_l, Z.mul_div_eq, <-Z.add_mod_full by omega.
+        f_equal; ring. }
+      { split; [zero_bounds|].
+        apply Z.lt_le_trans with (m:=a*(b-1)+a); [|ring_simplify; omega].
+        apply Z.add_le_lt_mono; try apply Z.mul_le_mono_nonneg_l; omega. }
+    Qed.
+
+    Lemma compact_div_step a b c d : 0 < a -> 0 < b ->
+      (c / a + d) / b = (a * d + c) / (a * b).
+    Proof.
+      intros Ha Hb.
+      rewrite <-Z.div_div by omega.
+      rewrite Z.div_add_l' by omega.
+      f_equal; ring.
+    Qed.
+
+    Lemma compact_div_mod {n} inp :
+      (B.Positional.eval weight (snd (compact inp))
+       = (eval inp) mod (weight n))
+        /\ (fst (compact inp) = eval (n:=n) inp / weight n).
+    Proof.
+      cbv [compact compact_cps compact_step compact_step_cps];
+        autorewrite with uncps push_id.
+      change (fun i s a => compact_digit_cps i (s :: a) id)
+        with (fun i s a => compact_digit i (s :: a)).
+
+      apply mapi_with'_linvariant; [|tauto].
+
+      clear n inp. intros n st x0 xs ys Hst Hys [Hmod Hdiv].
+      pose proof (weight_positive n). pose proof (weight_divides n).
+      autorewrite with push_basesystem_eval.
+      destruct n; cbv [mapi_with] in *; simpl tuple in *;
+        [destruct xs, ys; subst; simpl| cbv [eval] in *];
+        repeat match goal with
+               | _ => rewrite mapi_with'_left_step
+               | _ => rewrite compact_digit_div, sum_cons
+               | _ => rewrite compact_digit_mod, sum_cons
+               | _ => rewrite map_left_append
+               | _ => rewrite B.Positional.eval_left_append
+               | _ => rewrite weight_0, ?Z.div_1_r, ?Z.mod_1_r
+               | _ => rewrite Hdiv
+               | _ => rewrite Hmod
+               | _ => progress subst
+               | _ => progress autorewrite with natsimplify cancel_pair push_basesystem_eval
+               | _ => solve [split; ring_simplify; f_equal; ring]
+               end.
+        remember (weight (S (S n)) / weight (S n)) as bound.
+        replace (weight (S (S n))) with (weight (S n) * bound)
+          by (subst bound; rewrite Z.mul_div_eq by omega;
+              rewrite weight_multiples; ring).
+        split; [apply compact_mod_step | apply compact_div_step]; omega.
+    Qed.
+
+    Lemma compact_mod {n} inp :
+      (B.Positional.eval weight (snd (compact inp))
+       = (eval (n:=n) inp) mod (weight n)).
+    Proof. apply (proj1 (compact_div_mod inp)). Qed.
+    Hint Rewrite @compact_mod : push_basesystem_eval.
+
+    Lemma compact_div {n} inp :
+      fst (compact inp) = eval (n:=n) inp / weight n.
+    Proof. apply (proj2 (compact_div_mod inp)). Qed.
+    Hint Rewrite @compact_div : push_basesystem_eval.
+
+    (* TODO : move to tuple *)
+    Lemma hd_to_list {A n} a (t : A^(S n)) : List.hd a (to_list (S n) t) = hd t.
+    Proof.
+      rewrite (subst_append t), to_list_append, hd_append. reflexivity.
+    Qed.
+
+    Definition cons_to_nth_cps {n} i (x:Z) (t:(list Z)^n)
+               {T} (f:(list Z)^n->T) :=
+      @on_tuple_cps _ _ nil (update_nth_cps i (cons x)) n n t _ f.
+
+    Definition cons_to_nth {n} i x t := @cons_to_nth_cps n i x t _ id.
+    Lemma cons_to_nth_id {n} i x t T f :
+      @cons_to_nth_cps n i x t T f = f (cons_to_nth i x t).
+    Proof using Type.
+      cbv [cons_to_nth_cps cons_to_nth].
+      assert (forall xs : list (list Z), length xs = n ->
+                 length (update_nth_cps i (cons x) xs id) = n) as Hlen.
+      { intros. autorewrite with uncps push_id distr_length. assumption. }
+      rewrite !on_tuple_cps_correct with (H:=Hlen)
+        by (intros; autorewrite with uncps push_id; reflexivity). reflexivity.
+    Qed.
+    Hint Opaque cons_to_nth : uncps.
+    Hint Rewrite @cons_to_nth_id : uncps.
+
+    Lemma map_sum_update_nth l : forall i x,
+      List.map sum (update_nth i (cons x) l) =
+      update_nth i (Z.add x) (List.map sum l).
+    Proof using Type.
+      induction l as [|a l IHl]; intros i x; destruct i; simpl; rewrite ?IHl; reflexivity.
+    Qed.
+
+    Lemma cons_to_nth_add_to_nth n i x t :
+      map sum (@cons_to_nth n i x t) = B.Positional.add_to_nth i x (map sum t).
+    Proof using weight.
+      cbv [B.Positional.add_to_nth B.Positional.add_to_nth_cps cons_to_nth cons_to_nth_cps on_tuple_cps].
+      induction n; [simpl; rewrite !update_nth_cps_correct; reflexivity|].
+      specialize (IHn (tl t)). autorewrite with uncps push_id in *.
+      apply to_list_ext. rewrite <-!map_to_list.
+      erewrite !from_list_default_eq, !to_list_from_list.
+      rewrite map_sum_update_nth. reflexivity.
+      Unshelve.
+      distr_length.
+      distr_length.
+    Qed.
+
+    Lemma eval_cons_to_nth n i x t : (i < n)%nat ->
+      eval (@cons_to_nth n i x t) = weight i * x + eval t.
+    Proof using Type.
+      cbv [eval]; intros. rewrite cons_to_nth_add_to_nth.
+      auto using B.Positional.eval_add_to_nth.
+    Qed.
+    Hint Rewrite eval_cons_to_nth using omega : push_basesystem_eval.
+
+    Definition nils n : (list Z)^n := Tuple.repeat nil n.
+
+    Lemma map_sum_nils n : map sum (nils n) = B.Positional.zeros n.
+    Proof using Type.
+      cbv [nils B.Positional.zeros]; induction n as [|n]; [reflexivity|].
+      change (repeat nil (S n)) with (@nil Z :: repeat nil n).
+      rewrite map_repeat, sum_nil. reflexivity.
+    Qed.
+
+    Lemma eval_nils n : eval (nils n) = 0.
+    Proof using Type. cbv [eval]. rewrite map_sum_nils, B.Positional.eval_zeros. reflexivity. Qed. Hint Rewrite eval_nils : push_basesystem_eval.
+
+    Definition from_associational_cps n (p:list B.limb)
+               {T} (f:(list Z)^n -> T) :=
+      fold_right_cps
+        (fun t st =>
+           B.Positional.place_cps weight t (pred n)
+             (fun p=> cons_to_nth_cps (fst p) (snd p) st id))
+        (nils n) p f.
+
+    Definition from_associational n p := from_associational_cps n p id.
+    Lemma from_associational_id n p T f :
+      @from_associational_cps n p T f = f (from_associational n p).
+    Proof using Type.
+      cbv [from_associational_cps from_associational].
+      autorewrite with uncps push_id; reflexivity.
+    Qed.
+    Hint Opaque from_associational : uncps.
+    Hint Rewrite from_associational_id : uncps.
+
+    Lemma eval_from_associational n p (n_nonzero:n<>0%nat):
+      eval (from_associational n p) = B.Associational.eval p.
+    Proof using weight_0 weight_nonzero.
+      cbv [from_associational_cps from_associational]; induction p;
+        autorewrite with uncps push_id push_basesystem_eval; [reflexivity|].
+        pose proof (B.Positional.weight_place_cps weight weight_0 weight_nonzero a (pred n)).
+        pose proof (B.Positional.place_cps_in_range weight a (pred n)).
+        rewrite Nat.succ_pred in * by assumption. simpl.
+        autorewrite with uncps push_id push_basesystem_eval in *.
+        rewrite eval_cons_to_nth by omega. nsatz.
+    Qed.
+  End Columns.
+  Hint Rewrite
+      @Columns.compact_id
+      @Columns.from_associational_id
+  : uncps.
+  Hint Rewrite
+      @Columns.compact_mod
+      @Columns.compact_div
+      @Columns.eval_from_associational
+  using (assumption || omega): push_basesystem_eval.
+
+  Section Wrappers.
+    Context (weight : nat->Z).
+
+    Definition add_cps {n1 n2 n3} (p : Z^n1) (q : Z^n2)
+               {T} (f : (Z*Z^n3)->T) :=
+      B.Positional.to_associational_cps weight p
+        (fun P => B.Positional.to_associational_cps weight q
+        (fun Q => from_associational_cps weight n3 (P++Q)
+        (fun R => compact_cps (div:=div) (modulo:=modulo) (add_get_carry:=Z.add_get_carry_full) weight R f))).
+
+    Definition unbalanced_sub_cps {n1 n2 n3} (p : Z^n1) (q:Z^n2)
+               {T} (f : (Z*Z^n3)->T) :=
+      B.Positional.to_associational_cps weight p
+        (fun P => B.Positional.negate_snd_cps weight q
+        (fun nq => B.Positional.to_associational_cps weight nq
+        (fun Q => from_associational_cps weight n3 (P++Q)
+        (fun R => compact_cps (div:=div) (modulo:=modulo) (add_get_carry:=Z.add_get_carry_full) weight R f)))).
+
+    Definition mul_cps {n1 n2 n3} s (p : Z^n1) (q : Z^n2)
+               {T} (f : (Z*Z^n3)->T) :=
+      B.Positional.to_associational_cps weight p
+        (fun P => B.Positional.to_associational_cps weight q
+        (fun Q => Associational.mul_cps (mul_split := Z.mul_split) s P Q
+        (fun PQ => from_associational_cps weight n3 PQ
+        (fun R => compact_cps (div:=div) (modulo:=modulo) (add_get_carry:=Z.add_get_carry_full) weight R f)))).
+
+    Definition conditional_add_cps {n1 n2 n3} mask cond (p:Z^n1) (q:Z^n2)
+               {T} (f:_->T) :=
+      B.Positional.select_cps mask cond q
+                 (fun qq => add_cps (n3:=n3) p qq f).
+
+  End Wrappers.
+  Hint Unfold add_cps unbalanced_sub_cps mul_cps conditional_add_cps.
+
+End Columns.
+Hint Unfold
+     Columns.conditional_add_cps
+     Columns.add_cps
+     Columns.unbalanced_sub_cps
+     Columns.mul_cps.
+Hint Rewrite
+     @Columns.compact_digit_id
+     @Columns.compact_step_id
+     @Columns.compact_id
+     @Columns.cons_to_nth_id
+     @Columns.from_associational_id
+  : uncps.
+Hint Rewrite
+     @Columns.compact_mod
+     @Columns.compact_div
+     @Columns.eval_cons_to_nth
+     @Columns.eval_from_associational
+     @Columns.eval_nils
+  using (assumption || omega): push_basesystem_eval.
+
+Section Freeze.
+    Context (weight : nat->Z)
+            {weight_0 : weight 0%nat = 1}
+            {weight_nonzero : forall i, weight i <> 0}
+            {weight_positive : forall i, weight i > 0}
+            {weight_multiples : forall i, weight (S i) mod weight i = 0}
+            {weight_divides : forall i : nat, weight (S i) / weight i > 0}
+    .
+
+
+  (*
+    The input to [freeze] should be less than 2*m (this can probably
+    be accomplished by a single carry_reduce step, for most moduli).
+
+    [freeze] has the following steps:
+    (1) subtract modulus in a carrying loop (in our framework, this
+    consists of two steps; [Columns.unbalanced_sub_cps] combines the
+    input p and the modulus m such that the ith limb in the output is
+    the list [p[i];-m[i]]. We can then call [Columns.compact].)
+    (2) look at the final carry, which should be either 0 or -1. If
+    it's -1, then we add the modulus back in. Otherwise we add 0 for
+    constant-timeness.
+    (3) discard the carry after this last addition; it should be 1 if
+    the carry in step 3 was -1, so they cancel out.
+   *)
+  Definition freeze_cps {n} mask (m:Z^n) (p:Z^n) {T} (f : Z^n->T) :=
+    Columns.unbalanced_sub_cps (n3:=n) weight p m
+      (fun carry_p => Columns.conditional_add_cps (n3:=n) weight mask (fst carry_p) (snd carry_p) m
+      (fun carry_r => f (snd carry_r)))
+  .
+
+  Definition freeze {n} mask m p :=
+    @freeze_cps n mask m p _ id.
+  Lemma freeze_id {n} mask m p T f:
+    @freeze_cps n mask m p T f = f (freeze mask m p).
+  Proof.
+    cbv [freeze_cps freeze]; repeat progress autounfold;
+      autorewrite with uncps push_id; reflexivity.
+  Qed.
+  Hint Opaque freeze : uncps.
+  Hint Rewrite @freeze_id : uncps.
+
+  Lemma freezeZ m s c y y0 z z0 c0 a :
+    m = s - c ->
+    0 < c < s ->
+    s <> 0 ->
+    0 <= y < 2*m ->
+    y0 = y - m ->
+    z = y0 mod s ->
+    c0 = y0 / s ->
+    z0 = z + (if (dec (c0 = 0)) then 0 else m) ->
+    a = z0 mod s ->
+    a mod m = y0 mod m.
+  Proof.
+    clear. intros. subst. break_match.
+    { rewrite Z.add_0_r, Z.mod_mod by omega.
+      assert (-(s-c) <= y - (s-c) < s-c) by omega.
+      match goal with H : s <> 0 |- _ =>
+                      rewrite (proj2 (Z.mod_small_iff _ s H))
+                              by (apply Z.div_small_iff; assumption)
+      end.
+      reflexivity. }
+    { rewrite <-Z.add_mod_l, Z.sub_mod_full.
+      rewrite Z.mod_same, Z.sub_0_r, Z.mod_mod by omega.
+      rewrite Z.mod_small with (b := s)
+      by (pose proof (Z.div_small (y - (s-c)) s); omega).
+      f_equal. ring. }
+  Qed.
+
+  Lemma eval_freeze {n} c mask m p
+        (n_nonzero:n<>0%nat)
+        (Hc : 0 < B.Associational.eval c < weight n)
+        (Hmask : Tuple.map (Z.land mask) m = m)
+        modulus (Hm : B.Positional.eval weight m = Z.pos modulus)
+        (Hp : 0 <= B.Positional.eval weight p < 2*(Z.pos modulus))
+        (Hsc : Z.pos modulus = weight n - B.Associational.eval c)
+    :
+      mod_eq modulus
+             (B.Positional.eval weight (@freeze n mask m p))
+             (B.Positional.eval weight p).
+  Proof.
+    cbv [freeze_cps freeze].
+    repeat progress autounfold.
+    pose proof Z.add_get_carry_full_mod.
+    pose proof Z.add_get_carry_full_div.
+    pose proof div_correct. pose proof modulo_correct.
+    autorewrite with uncps push_id push_basesystem_eval.
+
+    pose proof (weight_nonzero n).
+
+    remember (B.Positional.eval weight p) as y.
+    remember (y + -B.Positional.eval weight m) as y0.
+    rewrite Hm in *.
+
+    transitivity y0; cbv [mod_eq].
+    { eapply (freezeZ (Z.pos modulus) (weight n) (B.Associational.eval c) y y0);
+        try assumption; reflexivity. }
+    { subst y0.
+      assert (Z.pos modulus <> 0) by auto using Z.positive_is_nonzero, Zgt_pos_0.
+      rewrite Z.add_mod by assumption.
+      rewrite Z.mod_opp_l_z by auto using Z.mod_same.
+      rewrite Z.add_0_r, Z.mod_mod by assumption.
+      reflexivity. }
+  Qed.
+End Freeze.
+
+Section UniformWeight.
+  Context (bound : Z) {bound_pos : bound > 0}.
+
+  Definition uweight : nat -> Z := fun i => bound ^ Z.of_nat i.
+  Lemma uweight_0 : uweight 0%nat = 1. Proof. reflexivity. Qed.
+  Lemma uweight_positive i : uweight i > 0.
+  Proof. apply Z.lt_gt, Z.pow_pos_nonneg; omega. Qed.
+  Lemma uweight_nonzero i : uweight i <> 0.
+  Proof. auto using Z.positive_is_nonzero, uweight_positive. Qed.
+  Lemma uweight_multiples i : uweight (S i) mod uweight i = 0.
+  Proof. apply Z.mod_same_pow; rewrite Nat2Z.inj_succ; omega. Qed.
+  Lemma uweight_divides i : uweight (S i) / uweight i > 0.
+  Proof.
+    cbv [uweight]. rewrite <-Z.pow_sub_r by (rewrite ?Nat2Z.inj_succ; omega).
+    apply Z.lt_gt, Z.pow_pos_nonneg; rewrite ?Nat2Z.inj_succ; omega.
+  Qed.
+
+  (* TODO : move to Positional *)
+  Lemma eval_from_eq {n} (p:Z^n) wt offset :
+    (forall i, wt i = uweight (i + offset)) ->
+    B.Positional.eval wt p = B.Positional.eval_from uweight offset p.
+  Proof. cbv [B.Positional.eval_from]. auto using B.Positional.eval_wt_equiv. Qed.
+
+  Lemma uweight_eval_from {n} (p:Z^n): forall offset,
+    B.Positional.eval_from uweight offset p = uweight offset * B.Positional.eval uweight p.
+  Proof.
+    induction n; intros; cbv [B.Positional.eval_from];
+      [|rewrite (subst_append p)];
+    repeat match goal with
+           | _ => destruct p
+           | _ => rewrite B.Positional.eval_unit; [ ]
+           | _ => rewrite B.Positional.eval_step; [ ]
+           | _ => rewrite IHn; [ ]
+           | _ => rewrite eval_from_eq with (offset0:=S offset)
+               by (intros; f_equal; omega)
+           | _ => rewrite eval_from_eq with
+                  (wt:=fun i => uweight (S i)) (offset0:=1%nat)
+               by (intros; f_equal; omega)
+           | _ => ring
+           end.
+    repeat match goal with
+           | _ => cbv [uweight]; progress autorewrite with natsimplify
+           | _ => progress (rewrite ?Nat2Z.inj_succ, ?Nat2Z.inj_0, ?Z.pow_0_r)
+           | _ => rewrite !Z.pow_succ_r by (try apply Nat2Z.is_nonneg; omega)
+           | _ => ring
+           end.
+  Qed.
+
+  Lemma uweight_eval_step {n} (p:Z^S n):
+    B.Positional.eval uweight p = hd p + bound * B.Positional.eval uweight (tl p).
+  Proof.
+    rewrite (subst_append p) at 1; rewrite B.Positional.eval_step.
+    rewrite eval_from_eq with (offset := 1%nat) by (intros; f_equal; omega).
+    rewrite uweight_eval_from. cbv [uweight]; rewrite Z.pow_0_r, Z.pow_1_r.
+    ring.
+  Qed.
+
+  Definition small {n} (p : Z^n) : Prop :=
+    forall x, In x (to_list _ p) -> 0 <= x < bound.
+
+End UniformWeight.
+
+Module Positional.
+  Section Positional.
+    Context {s:Z}. (* s is bitwidth *)
+    Let small {n} := @small s n.
+    Section GenericOp.
+      Context {op : Z -> Z -> Z}
+              {op_get_carry : Z -> Z -> Z * Z} (* no carry in, carry out *)
+              {op_with_carry : Z -> Z -> Z -> Z * Z}. (* carry in, carry out  *)
+
+      Fixpoint chain_op'_cps {n}:
+        option Z->Z^n->Z^n->forall T, (Z*Z^n->T)->T :=
+        match n with
+        | O => fun c p _ _ f =>
+                 let carry := match c with | None => 0 | Some x => x end in
+                 f (carry,p)
+        | S n' =>
+          fun c p q _ f =>
+            (* for the first call, use op_get_carry, then op_with_carry *)
+            let op' := match c with
+                       | None => op_get_carry
+                       | Some x => op_with_carry x end in
+            dlet carry_result := op' (hd p) (hd q) in
+              chain_op'_cps (Some (snd carry_result)) (tl p) (tl q) _
+                            (fun carry_pq =>
+                               f (fst carry_pq,
+                                  append (fst carry_result) (snd carry_pq)))
+        end.
+      Definition chain_op' {n} c p q := @chain_op'_cps n c p q _ id.
+      Definition chain_op_cps {n} p q {T} f := @chain_op'_cps n None p q T f.
+      Definition chain_op {n} p q : Z * Z^n := chain_op_cps p q id.
+
+      Lemma chain_op'_id {n} : forall c p q T f,
+        @chain_op'_cps n c p q T f = f (chain_op' c p q).
+      Proof.
+        cbv [chain_op']; induction n; intros; destruct c;
+          simpl chain_op'_cps; cbv [Let_In]; try reflexivity.
+        { etransitivity; rewrite IHn; reflexivity. }
+        { etransitivity; rewrite IHn; reflexivity. }
+      Qed.
+
+      Lemma chain_op_id {n} p q T f :
+        @chain_op_cps n p q T f = f (chain_op p q).
+      Proof. apply chain_op'_id. Qed.
+    End GenericOp.
+
+    Section AddSub.
+      Let eval {n} := B.Positional.eval (n:=n) (uweight s).
+
+      Definition sat_add_cps {n} p q T (f:Z*Z^n->T) :=
+        chain_op_cps (op_get_carry := Z.add_get_carry_full s)
+                     (op_with_carry := Z.add_with_get_carry_full s)
+                     p q f.
+      Definition sat_add {n} p q := @sat_add_cps n p q _ id.
+
+      Lemma sat_add_id n p q T f :
+        @sat_add_cps n p q T f = f (sat_add p q).
+      Proof. cbv [sat_add sat_add_cps]. rewrite !chain_op_id. reflexivity. Qed.
+
+      Lemma sat_add_mod n p q :
+        eval (snd (@sat_add n p q)) = (eval p + eval q) mod (uweight s n).
+      Admitted.
+
+      Lemma sat_add_div n p q :
+        fst (@sat_add n p q) = (eval p + eval q) / (uweight s n).
+      Admitted.
+
+      Lemma small_sat_add n p q : small (snd (@sat_add n p q)).
+      Admitted.
+
+      Definition sat_sub_cps {n} p q T (f:Z*Z^n->T) :=
+        chain_op_cps (op_get_carry := Z.sub_get_borrow_full s)
+                     (op_with_carry := Z.sub_with_get_borrow_full s)
+                     p q f.
+      Definition sat_sub {n} p q := @sat_sub_cps n p q _ id.
+
+      Lemma sat_sub_id n p q T f :
+        @sat_sub_cps n p q T f = f (sat_sub p q).
+      Proof. cbv [sat_sub sat_sub_cps]. rewrite !chain_op_id. reflexivity. Qed.
+
+      Lemma sat_sub_mod n p q :
+        eval (snd (@sat_sub n p q)) = (eval p - eval q) mod (uweight s n).
+      Admitted.
+
+      Lemma sat_sub_div n p q :
+        fst (@sat_sub n p q) = - ((eval p - eval q) / uweight s n).
+      Admitted.
+
+      Lemma small_sat_sub n p q : small (snd (@sat_sub n p q)).
+      Admitted.
+
+    End AddSub.
+  End Positional.
+End Positional.
+Hint Opaque Positional.sat_sub Positional.sat_add Positional.chain_op Positional.chain_op' : uncps.
+Hint Rewrite @Positional.sat_sub_id @Positional.sat_add_id @Positional.chain_op_id @Positional.chain_op' : uncps.
+Hint Rewrite @Positional.sat_sub_mod @Positional.sat_sub_div @Positional.sat_add_mod @Positional.sat_add_div using (omega || assumption) : push_basesystem_eval.
+
+Section API.
+  Context (bound : Z) {bound_pos : bound > 0}.
+  Definition T : nat -> Type := tuple Z.
+
+  (* lowest limb is less than its bound; this is required for [divmod]
+  to simply separate the lowest limb from the rest and be equivalent
+  to normal div/mod with [bound]. *)
+  Local Notation small := (@small bound).
+
+  Definition zero {n:nat} : T n := B.Positional.zeros n.
+
+  (** Returns 0 iff all limbs are 0 *)
+  Definition nonzero_cps {n} (p : T n) {cpsT} (f : Z -> cpsT) : cpsT
+    := CPSUtil.to_list_cps _ p (fun p => CPSUtil.fold_right_cps runtime_lor 0%Z p f).
+  Definition nonzero {n} (p : T n) : Z
+    := nonzero_cps p id.
+
+  Definition join0_cps {n:nat} (p : T n) {R} (f:T (S n) -> R)
+    := Tuple.left_append_cps 0 p f.
+  Definition join0 {n} p : T (S n) := @join0_cps n p _ id.
+
+  Definition divmod_cps {n} (p : T (S n)) {R} (f:T n * Z->R) : R
+    := Tuple.tl_cps p (fun d => Tuple.hd_cps p (fun m =>  f (d, m))).
+  Definition divmod {n} p : T n * Z := @divmod_cps n p _ id.
+
+  Definition drop_high_cps {n : nat} (p : T (S n)) {R} (f:T n->R)
+    := Tuple.left_tl_cps p f.
+  Definition drop_high {n} p : T n := @drop_high_cps n p _ id.
+
+  Definition scmul_cps {n} (c : Z) (p : T n) {R} (f:T (S n)->R) :=
+    Columns.mul_cps (n1:=1) (n3:=S n) (uweight bound) bound c p
+      (* The carry that comes out of Columns.mul_cps will be 0, since
+      (S n) limbs is enough to hold the result of the
+      multiplication, so we can safely discard it. *)
+      (fun carry_result =>f (snd carry_result)).
+  Definition scmul {n} c p : T (S n) := @scmul_cps n c p _ id.
+
+  Definition add_cps {n} (p q: T n) {R} (f:T (S n)->R) :=
+    Positional.sat_add_cps (s:=bound) p q _
+      (* join the last carry *)
+      (fun carry_result => Tuple.left_append_cps (fst carry_result) (snd carry_result) f).
+  Definition add {n} p q : T (S n) := @add_cps n p q _ id.
+
+  (* Wrappers for additions with slightly uneven limb counts *)
+  Definition add_S1_cps {n} (p: T (S n)) (q: T n) {R} (f:T (S (S n))->R) :=
+    join0_cps q (fun Q => add_cps p Q f).
+  Definition add_S1 {n} p q := @add_S1_cps n p q _ id.
+  Definition add_S2_cps {n} (p: T n) (q: T (S n)) {R} (f:T (S (S n))->R) :=
+    join0_cps p (fun P => add_cps P q f).
+  Definition add_S2 {n} p q := @add_S2_cps n p q _ id.
+>>>>>>> addsubchains
+
+  Definition sub_then_maybe_add_cps {n} mask (p q r : T n)
+             {R} (f:T n -> R) :=
+    Positional.sat_sub_cps (s:=bound) p q _
+      (* the carry will be 0 unless we underflow--we do the addition only
+         in the underflow case *)
+      (fun carry_result =>
+         B.Positional.select_cps mask (fst carry_result) r
+      (fun selected => join0_cps selected
+      (fun selected' =>
+         Positional.sat_sub_cps (s:=bound) (left_append (fst carry_result) (snd carry_result)) selected' _
+      (* We can now safely discard the carry and the highest digit.
+         This relies on the precondition that p - q + r < bound^n. *)
+      (fun carry_result' => drop_high_cps (snd carry_result') f)))).
+  Definition sub_then_maybe_add {n} mask (p q r : T n) :=
+    sub_then_maybe_add_cps mask p q r id.
+
+   (* Subtract q if and only if p >= q. We rely on the preconditions
+    that 0 <= p < 2*q and q < bound^n (this ensures the output is less
+    than bound^n). *)
+  Definition conditional_sub_cps {n} (p:Z^S n) (q:Z^n) R (f:Z^n->R) :=
+    join0_cps q
+    (fun qq => Positional.sat_sub_cps (s:=bound) p qq _
+      (* if carry is zero, we select the result of the subtraction,
+      otherwise the first input *)
+      (fun carry_result =>
+        Tuple.map2_cps (Z.zselect (fst carry_result)) (snd carry_result) p
+      (* in either case, since our result must be < q and therefore <
+      bound^n, we can drop the high digit *)
+      (fun r => drop_high_cps r f))).
+  Definition conditional_sub {n} p q := @conditional_sub_cps n p q _ id.
+
+  Hint Opaque join0 divmod drop_high scmul add sub_then_maybe_add conditional_sub : uncps.
+
+  Section CPSProofs.
+
+    Local Ltac prove_id :=
+      repeat autounfold; autorewrite with uncps; reflexivity.
+
+    Lemma nonzero_id n p {cpsT} f : @nonzero_cps n p cpsT f = f (@nonzero n p).
+    Proof. cbv [nonzero nonzero_cps]. prove_id. Qed.
+
+    Lemma join0_id n p R f :
+      @join0_cps n p R f = f (join0 p).
+    Proof. cbv [join0_cps join0]. prove_id. Qed.
+
+    Lemma divmod_id n p R f :
+      @divmod_cps n p R f = f (divmod p).
+    Proof. cbv [divmod_cps divmod]; prove_id. Qed.
+
+    Lemma drop_high_id n p R f :
+      @drop_high_cps n p R f = f (drop_high p).
+    Proof. cbv [drop_high_cps drop_high]; prove_id. Qed.
+    Hint Rewrite drop_high_id : uncps.
+
+    Lemma scmul_id n c p R f :
+      @scmul_cps n c p R f = f (scmul c p).
+    Proof. cbv [scmul_cps scmul]. prove_id. Qed.
+
+    Lemma add_id n p q R f :
+      @add_cps n p q R f = f (add p q).
+    Proof. cbv [add_cps add Let_In]. prove_id. Qed.
+    Hint Rewrite add_id : uncps.
+
+    Lemma add_S1_id n p q R f :
+      @add_S1_cps n p q R f = f (add_S1 p q).
+    Proof. cbv [add_S1_cps add_S1 join0_cps]. prove_id. Qed.
+
+    Lemma add_S2_id n p q R f :
+      @add_S2_cps n p q R f = f (add_S2 p q).
+    Proof. cbv [add_S2_cps add_S2 join0_cps]. prove_id. Qed.
+
+    Lemma sub_then_maybe_add_id n mask p q r R f :
+      @sub_then_maybe_add_cps n mask p q r R f = f (sub_then_maybe_add mask p q r).
+    Proof. cbv [sub_then_maybe_add_cps sub_then_maybe_add join0_cps Let_In]. prove_id. Qed.
+
+    Lemma conditional_sub_id n p q R f :
+      @conditional_sub_cps n p q R f = f (conditional_sub p q).
+    Proof. cbv [conditional_sub_cps conditional_sub join0_cps Let_In]. prove_id. Qed.
+
+  End CPSProofs.
+  Hint Rewrite nonzero_id join0_id divmod_id drop_high_id scmul_id add_id sub_then_maybe_add_id conditional_sub_id : uncps.
+
+  Section Proofs.
+
+    Definition eval {n} (p : T n) : Z :=
+      B.Positional.eval (uweight bound) p.
+
+    Lemma eval_small n (p : T n) (Hsmall : small p) :
+      0 <= eval p < uweight bound n.
+    Proof.
+      cbv [small eval] in *; intros.
+      induction n; cbv [T uweight] in *; [destruct p|rewrite (subst_left_append p)];
+      repeat match goal with
+             | _ => progress autorewrite with push_basesystem_eval
+             | _ => rewrite Z.pow_0_r
+             | _ => specialize (IHn (left_tl p))
+             | _ =>
+               let H := fresh "H" in
+               match type of IHn with
+                 ?P -> _ => assert P as H by auto using Tuple.In_to_list_left_tl;
+                              specialize (IHn H)
+               end
+             | |- context [?b ^ Z.of_nat (S ?n)] =>
+               replace (b ^ Z.of_nat (S n)) with (b ^ Z.of_nat n * b) by
+                   (rewrite Nat2Z.inj_succ, <-Z.add_1_r, Z.pow_add_r,
+                    Z.pow_1_r by (omega || auto using Nat2Z.is_nonneg);
+                    reflexivity)
+             | _ => omega
+             end.
+
+        specialize (Hsmall _ (Tuple.In_left_hd _ p)).
+        split; [Z.zero_bounds; omega |].
+        apply Z.lt_le_trans with (m:=bound^Z.of_nat n * (left_hd p+1)).
+        { rewrite Z.mul_add_distr_l.
+          apply Z.add_le_lt_mono; omega. }
+        { apply Z.mul_le_mono_nonneg; omega. }
+    Qed.
+
+    Lemma eval_zero n : eval (@zero n) = 0.
+    Proof.
+      cbv [eval zero].
+      autorewrite with push_basesystem_eval.
+      reflexivity.
+    Qed.
+
+    Lemma small_zero n : small (@zero n).
+    Proof.
+      cbv [zero small B.Positional.zeros]. destruct n; [simpl;tauto|].
+      rewrite to_list_repeat.
+      intros x H; apply repeat_spec in H; subst x; omega.
+    Qed.
+
+    Lemma eval_pair n (p : T (S (S n))) : small p -> (snd p = 0 /\ eval (n:=S n) (fst p) = 0) <-> eval p = 0.
+    Admitted.
+
+    Lemma eval_nonzero n p : small p -> @nonzero n p = 0 <-> eval p = 0.
+    Proof.
+      destruct n as [|n].
+      { compute; split; trivial. }
+      induction n as [|n IHn].
+      { simpl; rewrite Z.lor_0_r; unfold eval, id.
+        cbv -[Z.add iff].
+        rewrite Z.add_0_r.
+        destruct p; omega. }
+      { destruct p as [ps p]; specialize (IHn ps).
+        unfold nonzero, nonzero_cps in *.
+        autorewrite with uncps in *.
+        unfold id in *.
+        setoid_rewrite to_list_S.
+        set (k := S n) in *; simpl in *.
+        intro Hsmall.
+        rewrite Z.lor_eq_0_iff, IHn
+          by (hnf in Hsmall |- *; simpl in *; eauto);
+          clear IHn.
+        exact (eval_pair n (ps, p) Hsmall). }
+    Qed.
+
+    Lemma eval_join0 n p
+      : eval (@join0 n p) = eval p.
+    Proof.
+    Admitted.
+
+    Local Ltac pose_uweight bound :=
+      match goal with H : bound > 0 |- _ =>
+                      pose proof (uweight_0 bound);
+                      pose proof (@uweight_positive bound H);
+                      pose proof (@uweight_nonzero bound H);
+                      pose proof (@uweight_multiples bound);
+                      pose proof (@uweight_divides bound H)
+      end.
+
+    Local Ltac pose_all :=
+      pose_uweight bound;
+      pose proof Z.add_get_carry_full_div;
+      pose proof Z.add_get_carry_full_mod;
+      pose proof Z.mul_split_div; pose proof Z.mul_split_mod;
+      pose proof div_correct; pose proof modulo_correct.
+
+    Lemma eval_add_nz n p q :
+      n <> 0%nat ->
+      eval (@add n p q) = eval p + eval q.
+    Proof.
+      intros. pose_all.
+      repeat match goal with
+             | _ => progress (cbv [add_cps add eval Let_In] in *; repeat autounfold)
+             | _ => progress autorewrite with uncps push_id cancel_pair push_basesystem_eval
+             | _ => rewrite B.Positional.eval_left_append
+
+             | _ => progress
+                      (rewrite <-!from_list_default_eq with (d:=0);
+                       erewrite !length_to_list, !from_list_default_eq,
+                       from_list_to_list)
+             | _ => apply Z.mod_small; omega
+             end.
+    Admitted.
+
+    Lemma eval_add_z n p q :
+      n = 0%nat ->
+      eval (@add n p q) = eval p + eval q.
+    Proof. intros; subst; reflexivity. Qed.
+
+    Lemma eval_add n p q
+      :  eval (@add n p q) = eval p + eval q.
+    Proof.
+      destruct (Nat.eq_dec n 0%nat); intuition auto using eval_add_z, eval_add_nz.
+    Qed.
+    Lemma eval_add_same n p q
+      :  eval (@add n p q) = eval p + eval q.
+    Proof. apply eval_add; omega. Qed.
+    Lemma eval_add_S1 n p q
+      :  eval (@add_S1 n p q) = eval p + eval q.
+    Proof.
+      cbv [add_S1 add_S1_cps]. autorewrite with uncps push_id.
+      (*rewrite eval_add; rewrite eval_join0; [reflexivity|assumption].*)
+    Admitted.
+    Lemma eval_add_S2 n p q
+      :  eval (@add_S2 n p q) = eval p + eval q.
+    Proof.
+      cbv [add_S2 add_S2_cps]. autorewrite with uncps push_id.
+      (*rewrite eval_add; rewrite eval_join0; [reflexivity|assumption].*)
+    Admitted.
+>>>>>>> addsubchains
+    Hint Rewrite eval_add_same eval_add_S1 eval_add_S2 using (omega || assumption): push_basesystem_eval.
+
+    Lemma uweight_le_mono n m : (n <= m)%nat ->
+      uweight bound n <= uweight bound m.
+    Proof.
+      unfold uweight; intro; Z.peel_le; omega.
+    Qed.
+
+    Lemma uweight_lt_mono (bound_gt_1 : bound > 1) n m : (n < m)%nat ->
+      uweight bound n < uweight bound m.
+    Proof.
+      clear bound_pos.
+      unfold uweight; intro; apply Z.pow_lt_mono_r; omega.
+    Qed.
+
+    Lemma uweight_succ n : uweight bound (S n) = bound * uweight bound n.
+    Proof.
+      unfold uweight.
+      rewrite Nat2Z.inj_succ, Z.pow_succ_r by auto using Nat2Z.is_nonneg; reflexivity.
+    Qed.
+
+    Local Definition compact {n} := Columns.compact (n:=n) (add_get_carry:=Z.add_get_carry_full) (div:=div) (modulo:=modulo) (uweight bound).
+    Local Definition compact_digit := Columns.compact_digit (add_get_carry:=Z.add_get_carry_full) (div:=div) (modulo:=modulo) (uweight bound).
+    Lemma small_compact {n} (p:(list Z)^n) : small (snd (compact p)).
+    Proof.
+      pose_all.
+      match goal with
+        |- ?G => assert (G /\ fst (compact p) = fst (compact p)); [|tauto]
+      end. (* assert a dummy second statement so that fst (compact x) is in context *)
+      cbv [compact Columns.compact Columns.compact_cps small
+                   Columns.compact_step Columns.compact_step_cps];
+        autorewrite with uncps push_id.
+      change (fun i s a => Columns.compact_digit_cps (uweight bound) i (s :: a) id)
+      with (fun i s a => compact_digit i (s :: a)).
+      remember (fun i s a => compact_digit i (s :: a)) as f.
+
+      apply @mapi_with'_linvariant with (n:=n) (f:=f) (inp:=p);
+        intros; [|simpl; tauto]. split; [|reflexivity].
+      let P := fresh "H" in
+      match goal with H : _ /\ _ |- _ => destruct H end.
+      destruct n0; subst f.
+      { cbv [compact_digit uweight to_list to_list' In].
+        rewrite Columns.compact_digit_mod by assumption.
+        rewrite Z.pow_0_r, Z.pow_1_r, Z.div_1_r. intros x ?.
+        match goal with
+          H : _ \/ False |- _ => destruct H; [|exfalso; assumption] end.
+        subst x. apply Z.mod_pos_bound, Z.gt_lt, bound_pos. }
+      { rewrite Tuple.to_list_left_append.
+        let H := fresh "H" in
+        intros x H; apply in_app_or in H; destruct H;
+          [solve[auto]| cbv [In] in H; destruct H;
+                        [|exfalso; assumption] ].
+        subst x. cbv [compact_digit].
+        rewrite Columns.compact_digit_mod by assumption.
+        rewrite !uweight_succ, Z.div_mul by
+            (apply Z.neq_mul_0; split; auto; omega).
+        apply Z.mod_pos_bound, Z.gt_lt, bound_pos. }
+    Qed.
+
+    Lemma small_add n a b :
+      (2 <= bound) ->
+      small a -> small b -> small (@add n a b).
+    Proof.
+      intros. pose_all.
+      cbv [add_cps add Let_In].
+      autorewrite with uncps push_id.
+      apply Positional.small_sat_add.
+      (*apply Positional.small_sat_add.*)
+    Admitted.
+
+    Lemma small_add_S1 n a b :
+      (2 <= bound) ->
+      small a -> small b -> small (@add_S1 n a b).
+    Proof.
+      intros. pose_all.
+      cbv [add_cps add add_S1 Let_In].
+      autorewrite with uncps push_id.
+      (*apply Positional.small_sat_add.*)
+    Admitted.
+
+    Lemma small_add_S2 n a b :
+      (2 <= bound) ->
+      small a -> small b -> small (@add_S2 n a b).
+    Proof.
+      intros. pose_all.
+      cbv [add_cps add add_S2 Let_In].
+      autorewrite with uncps push_id.
+      (*apply Positional.small_sat_add.*)
+>>>>>>> addsubchains
+    Admitted.
+
+    Lemma small_left_tl n (v:T (S n)) : small v -> small (left_tl v).
+    Proof. cbv [small]. auto using Tuple.In_to_list_left_tl. Qed.
+
+    Lemma small_divmod n (p: T (S n)) (Hsmall : small p) :
+      left_hd p = eval p / uweight bound n /\ eval (left_tl p) = eval p mod (uweight bound n).
+    Admitted.
+
+    Lemma eval_drop_high n v :
+      small v -> eval (@drop_high n v) = eval v mod (uweight bound n).
+    Proof.
+      cbv [drop_high drop_high_cps eval].
+      rewrite Tuple.left_tl_cps_correct, push_id. (* TODO : for some reason autorewrite with uncps doesn't work here *)
+      intro H. apply small_left_tl in H.
+      rewrite (subst_left_append v) at 2.
+      autorewrite with push_basesystem_eval.
+      apply eval_small in H.
+      rewrite Z.mod_add_l' by (pose_uweight bound; auto).
+      rewrite Z.mod_small; auto.
+    Qed.
+
+    Lemma small_drop_high n v : small v -> small (@drop_high n v).
+    Proof.
+      cbv [drop_high drop_high_cps].
+      rewrite Tuple.left_tl_cps_correct, push_id.
+      apply small_left_tl.
+    Qed.
+
+    Lemma div_nonzero_neg_iff x y : x < y -> 0 < y -> x / y <> 0 <-> x < 0.
+    Proof.
+      repeat match goal with
+             | _ => progress intros
+             | _ => rewrite Z.div_small_iff by omega
+             | _ => split
+             | _ => omega
+             end.
+   Qed.
+
+    Lemma eval_sub_then_maybe_add_nz n mask p q r:
+      small p -> small q -> small r -> (n<>0)%nat ->
+      (map (Z.land mask) r = r) ->
+      (0 <= eval p < eval r) -> (0 <= eval q < eval r) ->
+      eval (@sub_then_maybe_add n mask p q r) = eval p - eval q + (if eval p - eval q <? 0 then eval r else 0).
+    Proof.
+      pose_all.
+      repeat match goal with
+             | _ => progress (cbv [sub_then_maybe_add sub_then_maybe_add_cps eval] in *; intros)
+             | _ => progress autounfold
+             | _ => progress autorewrite with uncps push_id push_basesystem_eval
+             | _ => rewrite eval_drop_high
+             | _ => rewrite eval_join0
+             | H : small _ |- _ => apply eval_small in H
+             | _ => progress break_match
+             | _ => (rewrite Z.add_opp_r in * )
+             | H : _ |- _ => rewrite Z.ltb_lt in H;
+                               rewrite <-div_nonzero_neg_iff with
+                               (y:=uweight bound n) in H by (auto; omega)
+             | H : _ |- _ => rewrite Z.ltb_ge in H
+             | _ => rewrite Z.mod_small by omega
+             | _ => omega
+             | _ => progress autorewrite with zsimplify; [ ]
+             end.
+    Admitted.
+
+    Lemma eval_sub_then_maybe_add n mask p q r :
+      small p -> small q -> small r ->
+      (map (Z.land mask) r = r) ->
+      (0 <= eval p < eval r) -> (0 <= eval q < eval r) ->
+      eval (@sub_then_maybe_add n mask p q r) = eval p - eval q + (if eval p - eval q <? 0 then eval r else 0).
+    Proof.
+      destruct n; [|solve[auto using eval_sub_then_maybe_add_nz]].
+      destruct p, q, r; reflexivity.
+    Qed.
+
+   Lemma small_sub_then_maybe_add n mask (p q r : T n) :
+      small (sub_then_maybe_add mask p q r).
+   Proof.
+     cbv [sub_then_maybe_add_cps sub_then_maybe_add]; intros.
+     repeat progress autounfold. autorewrite with uncps push_id.
+     apply small_drop_high, Positional.small_sat_sub.
+   Qed.
+
+   (* TODO : remove if unneeded when all admits are proven
+    Lemma small_highest_zero_iff {n} (p: T (S n)) (Hsmall : small p) :
+      (left_hd p = 0 <-> eval p < uweight bound n).
+    Proof.
+      destruct (small_divmod _ p Hsmall) as [Hdiv Hmod].
+      pose proof Hsmall as Hsmalltl. apply eval_small in Hsmall.
+      apply small_left_tl, eval_small in Hsmalltl. rewrite Hdiv.
+      rewrite (Z.div_small_iff (eval p) (uweight bound n))
+        by auto using uweight_nonzero.
+      split; [|intros; left; omega].
+      let H := fresh "H" in intro H; destruct H; [|omega].
+      omega.
+    Qed.
+    *)
+
+    Lemma map2_zselect n cond x y :
+      Tuple.map2 (n:=n) (Z.zselect cond) x y = if dec (cond = 0) then x else y.
+    Proof.
+      unfold Z.zselect.
+      break_innermost_match; Z.ltb_to_lt; subst; try omega;
+        [ rewrite Tuple.map2_fst, Tuple.map_id
+        | rewrite Tuple.map2_snd, Tuple.map_id ];
+        reflexivity.
+    Qed.
+
+    Lemma eval_conditional_sub_nz n (p:T (S n)) (q:T n)
+          (n_nonzero: (n <> 0)%nat) (psmall : small p) (qsmall : small q):
+      0 <= eval p < eval q + uweight bound n ->
+      eval (conditional_sub p q) = eval p + (if eval q <=? eval p then - eval q else 0).
+    Proof.
+      cbv [conditional_sub conditional_sub_cps]. intros. pose_all.
+      repeat autounfold. apply eval_small in qsmall.
+      pose proof psmall; apply eval_small in psmall.
+      cbv [eval] in *. autorewrite with uncps push_id push_basesystem_eval.
+      rewrite map2_zselect.
+      let H := fresh "H" in let X := fresh "P" in
+      match goal with |- context [?x / ?y] =>
+                      pose proof (div_nonzero_neg_iff x y) end;
+        repeat match type of H with ?P -> _ =>
+                             assert P as X by omega; specialize (H X);
+                               clear X end.
+
+      break_match;
+        repeat match goal with
+               | _ => progress cbv [eval]
+               | H : (_ <=? _) = true |- _ => apply Z.leb_le in H
+               | H : (_ <=? _) = false |- _ => apply Z.leb_gt in H
+               | _ => rewrite eval_drop_high by auto using Positional.small_sat_sub
+               | _ => (rewrite eval_join0 in * )
+               | _ => progress autorewrite with uncps push_id push_basesystem_eval
+               | _ => repeat rewrite Z.mod_small; omega
+               | _ => omega
+               end.
+    Admitted.
+
+    Lemma eval_conditional_sub n (p:T (S n)) (q:T n)
+           (psmall : small p) (qsmall : small q) :
+      0 <= eval p < eval q + uweight bound n ->
+      eval (conditional_sub p q) = eval p + (if eval q <=? eval p then - eval q else 0).
+    Proof.
+      destruct n; [|solve[auto using eval_conditional_sub_nz]].
+      repeat match goal with
+             | _ => progress (intros; cbv [T tuple tuple'] in p, q)
+             | q : unit |- _ => destruct q
+             | _ => progress (cbv [conditional_sub conditional_sub_cps eval] in * )
+             | _ => progress autounfold
+             | _ => progress (autorewrite with uncps push_id push_basesystem_eval in * )
+             | _ => (rewrite uweight_0 in * )
+             | _ => assert (p = 0) by omega; subst p; break_match; ring
+             end.
+    Qed.
+
+    Lemma small_conditional_sub n (p:T (S n)) (q:T n)
+           (psmall : small p) (qsmall : small q) :
+      0 <= eval p < eval q + uweight bound n ->
+      small (conditional_sub p q).
+    Admitted.
+
+    Lemma eval_scmul n a v : small v -> 0 <= a < bound ->
+      eval (@scmul n a v) = a * eval v.
+    Proof.
+      intro Hsmall. pose_all. apply eval_small in Hsmall.
+      intros. cbv [scmul scmul_cps eval] in *. repeat autounfold.
+      autorewrite with uncps push_id push_basesystem_eval.
+      rewrite uweight_0, Z.mul_1_l. apply Z.mod_small.
+      split; [solve[Z.zero_bounds]|]. cbv [uweight] in *.
+      rewrite !Nat2Z.inj_succ, Z.pow_succ_r by auto using Nat2Z.is_nonneg.
+      apply Z.mul_lt_mono_nonneg; omega.
+    Qed.
+
+    Lemma small_scmul n a v : small (@scmul n a v).
+    Proof.
+      cbv [scmul scmul_cps eval] in *. repeat autounfold.
+      autorewrite with uncps push_id push_basesystem_eval.
+      apply small_compact.
+    Qed.
+
+    (* TODO : move to tuple *)
+    Lemma from_list_tl {A n} (ls : list A) H H':
+      from_list n (List.tl ls) H = tl (from_list (S n) ls H').
+    Proof.
+      induction ls; distr_length. simpl List.tl.
+      rewrite from_list_cons, tl_append, <-!(from_list_default_eq a ls).
+      reflexivity.
+    Qed.
+
+    Lemma small_hd n p : @small (S n) p -> 0 <= hd p < bound.
+    Proof.
+      cbv [small]. let H := fresh "H" in intro H; apply H.
+      rewrite (subst_append p). rewrite to_list_append, hd_append.
+      apply in_eq.
+    Qed.
+
+
+    Lemma eval_div n p : small p -> eval (fst (@divmod n p)) = eval p / bound.
+    Proof.
+      cbv [divmod divmod_cps eval]. intros.
+      autorewrite with uncps push_id cancel_pair.
+      rewrite (subst_append p) at 2.
+      rewrite uweight_eval_step. rewrite hd_append, tl_append.
+      rewrite Z.div_add' by omega. rewrite Z.div_small by auto using small_hd.
+      ring.
+    Qed.
+
+    Lemma eval_mod n p : small p -> snd (@divmod n p) = eval p mod bound.
+    Proof.
+      cbv [divmod divmod_cps eval]. intros.
+      autorewrite with uncps push_id cancel_pair.
+      rewrite (subst_append p) at 2.
+      rewrite uweight_eval_step, Z.mod_add'_full, hd_append.
+      rewrite Z.mod_small by auto using small_hd. reflexivity.
+    Qed.
+
+    Lemma small_div n v : small v -> small (fst (@divmod n v)).
+    Admitted.
+
+  End Proofs.
+End API.
+Hint Rewrite nonzero_id join0_id divmod_id drop_high_id scmul_id add_id add_S1_id add_S2_id sub_then_maybe_add_id conditional_sub_id : uncps.
+
+(*
+(* Just some pretty-printing *)
+Local Notation "fst~ a" := (let (x,_) := a in x) (at level 40, only printing).
+Local Notation "snd~ a" := (let (_,y) := a in y) (at level 40, only printing).
+
+(* Simple example : base 10, multiply two bignums and compact them *)
+Definition base10 i := Eval compute in 10^(Z.of_nat i).
+Eval cbv -[runtime_add runtime_mul Let_In] in
+    (fun adc a0 a1 a2 b0 b1 b2 =>
+       Columns.mul_cps (weight := base10) (n:=3) (a2,a1,a0) (b2,b1,b0) (fun ab => Columns.compact (n:=5) (add_get_carry:=adc) (weight:=base10) ab)).
+
+(* More complex example : base 2^56, 8 limbs *)
+Definition base2pow56 i := Eval compute in 2^(56*Z.of_nat i).
+Time Eval cbv -[runtime_add runtime_mul Let_In] in
+    (fun adc a0 a1 a2 a3 a4 a5 a6 a7 b0 b1 b2 b3 b4 b5 b6 b7 =>
+       Columns.mul_cps (weight := base2pow56) (n:=8) (a7,a6,a5,a4,a3,a2,a1,a0) (b7,b6,b5,b4,b3,b2,b1,b0) (fun ab => Columns.compact (n:=15) (add_get_carry:=adc) (weight:=base2pow56) ab)). (* Finished transaction in 151.392 secs *)
+
+(* Mixed-radix example : base 2^25.5, 10 limbs *)
+Definition base2pow25p5 i := Eval compute in 2^(25*Z.of_nat i + ((Z.of_nat i + 1) / 2)).
+Time Eval cbv -[runtime_add runtime_mul Let_In] in
+    (fun adc a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 b0 b1 b2 b3 b4 b5 b6 b7 b8 b9 =>
+       Columns.mul_cps (weight := base2pow25p5) (n:=10) (a9,a8,a7,a6,a5,a4,a3,a2,a1,a0) (b9,b8,b7,b6,b5,b4,b3,b2,b1,b0) (fun ab => Columns.compact (n:=19) (add_get_carry:=adc) (weight:=base2pow25p5) ab)). (* Finished transaction in 97.341 secs *)
+*)
\ No newline at end of file