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

1 files changed, 0 insertions, 1421 deletions

diff --git a/src/Arithmetic/Saturated.v b/src/Arithmetic/Saturated.v
deleted file mode 100644
index 0c059b93d..000000000
--- a/src/Arithmetic/Saturated.v
+++ /dev/null
@@ -1,1421 +0,0 @@
-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