Make Karatsuba depend on Arithmetic/Core to make calling it less of a pain

2 files changed, 71 insertions, 186 deletions

diff --git a/src/Arithmetic/Karatsuba.v b/src/Arithmetic/Karatsuba.v
index 7b2004a2d..f17623da7 100644
--- a/src/Arithmetic/Karatsuba.v
+++ b/src/Arithmetic/Karatsuba.v
@@ -1,48 +1,19 @@
 Require Import Coq.ZArith.ZArith.
 Require Import Crypto.Algebra.Nsatz.
-Require Import Crypto.Util.ZUtil Crypto.Util.LetIn Crypto.Util.CPSUtil.
+Require Import Crypto.Util.ZUtil Crypto.Util.LetIn Crypto.Util.CPSUtil Crypto.Util.Tactics.
+Require Import Crypto.Arithmetic.Core. Import B. Import Positional.
+Require Import Crypto.Util.Tuple.
 Local Open Scope Z_scope.
 
 Section Karatsuba.
-  (* T is the "half-length" type, T2 is the "full-length" type *)
-  Context {T T2 : Type} (eval : T -> Z) (eval2 : T2 -> Z).
-
-  (* multiplication takes half-length inputs to full-length output *)
-  Context {mul_cps : T -> T -> forall {R}, (T2->R)->R}
-          {mul : T -> T -> T2}
-          {mul_id : forall x y {R} f, @mul_cps x y R f = f (mul x y)}
-          {eval_mul : forall x y, eval2 (mul x y) = eval x * eval y}.
-
-  (* splitting takes full-length input to half-length outputs *)
-  Context {split_cps : Z -> T2 -> forall {R}, ((T * T)->R)->R}
-          {split : Z -> T2 -> T * T}
-          {split_id : forall s x R f, @split_cps s x R f = f (split s x)}
-          {eval_split : forall s x, s <> 0 -> eval (fst (split s x)) + s * (eval (snd (split s x))) = eval2 x}.
-
-  (* half-length add *)
-  Context {add_cps : T -> T -> forall {R}, (T->R)->R}
-          {add : T -> T -> T}
-          {add_id : forall x y {R} f, @add_cps x y R f = f (add x y)}
-          {eval_add : forall x y, eval (add x y) = eval x + eval y}.
-
-  (* full-length operations: sub, add, scmul *)
-  Context {sub2_cps : T2 -> T2 -> forall {R}, (T2->R)->R}
-          {sub2 : T2 -> T2 -> T2}
-          {sub2_id : forall x y {R} f, @sub2_cps x y R f = f (sub2 x y)} 
-          {eval_sub2 : forall x y, eval2 (sub2 x y) = eval2 x - eval2 y}
-          {add2_cps : T2 -> T2 -> forall {R}, (T2->R)->R}
-          {add2 : T2 -> T2 -> T2}
-          {add2_id : forall x y {R} f, @add2_cps x y R f = f (add2 x y)} 
-          {eval_add2 : forall x y, eval2 (add2 x y) = eval2 x + eval2 y}
-          {scmul2_cps : Z -> T2 -> forall {R}, (T2->R)->R}
-          {scmul2 : Z -> T2 -> T2}
-          {scmul2_id : forall z x {R} f, @scmul2_cps z x R f = f (scmul2 z x)}
-          {eval_scmul2 : forall c x, eval2 (scmul2 c x) = c * eval2 x}.
-
-  Local Ltac rewrite_id :=
-    repeat progress rewrite ?mul_id, ?split_id, ?add_id, ?sub2_id, ?add2_id, ?scmul2_id.
-  Local Ltac rewrite_eval :=
-    repeat progress rewrite ?eval_mul, ?eval_split, ?eval_add, ?eval_sub2, ?eval_add2, ?eval_scmul2.
+Context (weight : nat -> Z)
+        (weight_0 : weight 0%nat = 1%Z)
+        (weight_nonzero : forall i, weight i <> 0).
+  (* [tuple Z n] is the "half-length" type,
+     [tuple Z n2] is the "full-length" type *)
+  Context {n n2 : nat} (n_nonzero : n <> 0%nat) (n2_nonzero : n2 <> 0%nat).
+  Let T := tuple Z n.
+  Let T2 := tuple Z n2.
 
   (* 
      If x = x0 + sx1 and y = y0 + sy1, then xy = s^2 * z2 + s * z1 + s * z0, 
@@ -60,34 +31,45 @@ Section Karatsuba.
      z1 = mul_sumxy - sum_z
   *)
   Definition karatsuba_mul_cps s (x y : T2) {R} (f:T2->R) :=
-    split_cps s x _
-      (fun x0_x1 => split_cps s y _
-      (fun y0_y1 => mul_cps (fst x0_x1) (fst y0_y1) _
-      (fun z0 => mul_cps (snd x0_x1) (snd y0_y1) _
-      (fun z2 => add2_cps z0 z2 _
-      (fun sum_z => add_cps (fst x0_x1) (snd x0_x1) _
-      (fun sum_x => add_cps (fst y0_y1) (snd y0_y1) _
-      (fun sum_y => mul_cps sum_x sum_y _
-      (fun mul_sumxy => sub2_cps mul_sumxy sum_z _
-      (fun z1 => scmul2_cps s z1 _
-      (fun sz1 => scmul2_cps (s^2) z2 _
-      (fun s2z2 => add2_cps s2z2 sz1 _
-      (fun add_s2z2_sz1 => add2_cps add_s2z2_sz1 z0 _ f)))))))))))).
+    split_cps (n:=n2) (m1:=n) (m2:=n) weight s x
+      (fun x0_x1 => split_cps weight s y
+      (fun y0_y1 => mul_cps weight (fst x0_x1) (fst y0_y1)
+      (fun z0 => mul_cps weight(snd x0_x1) (snd y0_y1)
+      (fun z2 => add_cps weight z0 z2
+      (fun sum_z => add_cps weight (fst x0_x1) (snd x0_x1)
+      (fun sum_x => add_cps weight (fst y0_y1) (snd y0_y1)
+      (fun sum_y => mul_cps weight sum_x sum_y
+      (fun mul_sumxy => unbalanced_sub_cps weight mul_sumxy sum_z
+      (fun z1 => scmul_cps weight s z1
+      (fun sz1 => scmul_cps weight (s^2) z2
+      (fun s2z2 => add_cps weight s2z2 sz1
+      (fun add_s2z2_sz1 => add_cps weight add_s2z2_sz1 z0 f)))))))))))).
 
   Definition karatsuba_mul s x y := @karatsuba_mul_cps s x y _ id.
   Lemma karatsuba_mul_id s x y R f :
     @karatsuba_mul_cps s x y R f = f (karatsuba_mul s x y).
   Proof.
-    cbv [karatsuba_mul karatsuba_mul_cps]. rewrite_id.
+    cbv [karatsuba_mul karatsuba_mul_cps].
+    repeat autounfold.
+    autorewrite with cancel_pair push_id uncps.
     reflexivity.
   Qed.
 
   Lemma eval_karatsuba_mul s x y (s_nonzero:s <> 0) :
-    eval2 (karatsuba_mul s x y) = eval2 x * eval2 y.
+    eval weight (karatsuba_mul s x y) = eval weight x * eval weight y.
   Proof.
-    cbv [karatsuba_mul karatsuba_mul_cps]. rewrite_id.
-    repeat rewrite push_id. rewrite_eval.
-    rewrite <-(eval_split s x), <-(eval_split s y) by assumption; ring.
+    cbv [karatsuba_mul karatsuba_mul_cps]; repeat autounfold.
+    autorewrite with cancel_pair push_id uncps push_basesystem_eval.
+    repeat match goal with
+           | _ => rewrite <-eval_to_associational
+           | |- context [(to_associational ?w ?x)] =>
+             rewrite <-(Associational.eval_split
+                          s (to_associational w x)) by assumption
+           | _ => rewrite <-Associational.eval_split by assumption
+           | _ => setoid_rewrite Associational.eval_nil
+           end.
+    ring_simplify.
+    nsatz.
   Qed.
 
   (*
@@ -109,23 +91,25 @@ Section Karatsuba.
    
   *)
   Definition goldilocks_mul_cps s (xs ys : T2) {R} (f:T2->R) :=
-    split_cps s xs _
-      (fun x0_x1 => split_cps s ys _
-      (fun y0_y1 => mul_cps (fst x0_x1) (fst y0_y1) _
-      (fun z0 => mul_cps (snd x0_x1) (snd y0_y1) _
-      (fun z2 => add2_cps z0 z2 _
-      (fun sum_z => add_cps (fst x0_x1) (snd x0_x1) _
-      (fun sum_x => add_cps (fst y0_y1) (snd y0_y1) _
-      (fun sum_y => mul_cps sum_x sum_y _
-      (fun mul_sumxy => sub2_cps mul_sumxy z0 _
-      (fun z1 => scmul2_cps s z1 _
-      (fun sz1 => add2_cps sum_z sz1 _ f)))))))))).
+    split_cps (m1:=n) (m2:=n) weight s xs
+      (fun x0_x1 => split_cps weight s ys
+      (fun y0_y1 => mul_cps weight (fst x0_x1) (fst y0_y1)
+      (fun z0 => mul_cps weight (snd x0_x1) (snd y0_y1)
+      (fun z2 => add_cps weight z0 z2
+      (fun sum_z => add_cps weight (fst x0_x1) (snd x0_x1)
+      (fun sum_x => add_cps weight (fst y0_y1) (snd y0_y1)
+      (fun sum_y => mul_cps weight sum_x sum_y
+      (fun mul_sumxy => unbalanced_sub_cps weight mul_sumxy z0
+      (fun z1 => scmul_cps weight s z1
+      (fun sz1 => add_cps weight sum_z sz1 f)))))))))).
 
   Definition goldilocks_mul s xs ys := @goldilocks_mul_cps s xs ys _ id.
   Lemma goldilocks_mul_id s xs ys {R} f :
     @goldilocks_mul_cps s xs ys R f = f (goldilocks_mul s xs ys).
   Proof.
-    cbv [goldilocks_mul goldilocks_mul_cps]. rewrite_id.
+    cbv [goldilocks_mul goldilocks_mul_cps].
+    repeat autounfold.
+    autorewrite with cancel_pair push_id uncps.
     reflexivity.
   Qed.
     
@@ -135,11 +119,20 @@ Section Karatsuba.
         Z.modulo_equiv_modulo_Proper.
 
   Lemma goldilocks_mul_correct (p : Z) (p_nonzero : p <> 0) s (s_nonzero : s <> 0) (s2_modp : (s^2) mod p = (s+1) mod p) xs ys :
-    (eval2 (goldilocks_mul s xs ys)) mod p = (eval2 xs * eval2 ys) mod p.
+    (eval weight (goldilocks_mul s xs ys)) mod p = (eval weight xs * eval weight ys) mod p.
   Proof.
     cbv [goldilocks_mul_cps goldilocks_mul]; Zmod_to_equiv_modulo.
-    rewrite_id. rewrite push_id. rewrite_eval.
-    repeat progress rewrite <-?(eval_split s xs), <-?(eval_split s ys) by assumption; ring_simplify.
+    repeat autounfold; autorewrite with push_id cancel_pair uncps push_basesystem_eval.
+    repeat match goal with
+           | _ => rewrite <-eval_to_associational
+           | |- context [(to_associational ?w ?x)] =>
+             rewrite <-(Associational.eval_split
+                          s (to_associational w x)) by assumption
+           | _ => rewrite <-Associational.eval_split by assumption
+           | _ => setoid_rewrite Associational.eval_nil
+           end.
+
+    ring_simplify.
     setoid_rewrite s2_modp.
     apply f_equal2; nsatz.
   Qed.
diff --git a/src/Specific/Karatsuba.v b/src/Specific/Karatsuba.v
index 0f205f253..70834e9d7 100644
--- a/src/Specific/Karatsuba.v
+++ b/src/Specific/Karatsuba.v
@@ -149,119 +149,8 @@ Section Ops51.
     solve_op_F wt x. reflexivity.
   Defined.
 
-  Check goldilocks_mul_cps.
   Definition half_sz : nat := Eval compute in (sz / 2).
 
-  (* TODO: move *)
-  Definition Positional_split_cps {n m1 m2} (s:Z) (p : tuple Z n)
-             {T} (f:(tuple Z m1 * tuple Z m2) -> T) :=
-        Positional.to_associational_cps wt p
-            (fun P => Associational.split_cps s P
-            (fun split_P =>
-               f (Positional.from_associational wt m1 (fst split_P),
-                  (Positional.from_associational wt m2 (snd split_P))))).
-  Definition Positional_scmul_cps {n} (x : Z) (p: tuple Z n)
-             {T} (f:tuple Z n->T) :=
-    Positional.to_associational_cps wt p
-        (fun P => Associational.mul_cps P [(1, x)]
-        (fun R => Positional.from_associational_cps wt n R f)).
-  Definition Positional_sub_cps {n} (p q: tuple Z n)
-             {T} (f:tuple Z n->T) :=
-    Positional.to_associational_cps wt p
-        (fun P => Positional.to_associational_cps wt q
-        (fun Q => Associational.negate_snd_cps Q
-        (fun negQ => Positional.from_associational_cps wt n (P ++ negQ) f))).
-  Definition goldilocks448_cps :=
-    (goldilocks_mul_cps
-           (T := tuple Z half_sz) (T2 := tuple Z sz)
-           (mul_cps := Positional.mul_cps (n:=half_sz) wt)
-           (add_cps := Positional.add_cps (n:=half_sz) wt)
-           (add2_cps := Positional.add_cps (n:=sz) wt)
-           (split_cps := Positional_split_cps (n:=sz) (m1:=half_sz) (m2 := half_sz))
-           (scmul2_cps := Positional_scmul_cps (n:=sz))
-           (sub2_cps := Positional_sub_cps (n:=sz))
-    ).
-  Hint Unfold goldilocks448_cps.
-  Check goldilocks_mul_id.
-  Definition goldilocks448_id
-             mul_id add_id add2_id split_id scmul2_id sub2_id
-    :=
-    (goldilocks_mul_id
-       (T := tuple Z half_sz) (T2 := tuple Z sz)
-       (mul_cps := Positional.mul_cps (n:=half_sz) wt)
-       (add_cps := Positional.add_cps (n:=half_sz) wt)
-       (add2_cps := Positional.add_cps (n:=sz) wt)
-       (split_cps := Positional_split_cps (n:=sz) (m1:=half_sz) (m2 := half_sz))
-       (scmul2_cps := Positional_scmul_cps (n:=sz))
-       (sub2_cps := Positional_sub_cps (n:=sz))
-       (mul := fun a b => Positional.mul_cps (n:= half_sz) wt a b id)
-       (add := fun a b => Positional.add_cps (n:=half_sz) wt a b id)
-       (add2 := fun a b => Positional.add_cps (n:=sz) wt a b id)
-       (split := fun s a => Positional_split_cps (n:=sz) (m1:=half_sz) (m2 := half_sz) s a id)
-       (scmul2 := fun x a => Positional_scmul_cps (n:=sz) x a id)
-       (sub2 := fun a b => Positional_sub_cps (n:=sz) a b id)
-       (mul_id := mul_id)
-       (add_id := add_id)
-       (add2_id := add2_id)
-       (split_id := split_id)
-       (scmul2_id := scmul2_id)
-       (sub2_id := sub2_id)
-    ).
-  Definition goldilocks448_correct'
-             mul_id add_id add2_id split_id scmul2_id sub2_id
-             eval_mul eval_add eval_add2 eval_split eval_scmul2 eval_sub2
-    :=
-    (goldilocks_mul_correct
-       (T := tuple Z half_sz) (T2 := tuple Z sz)
-       (Positional.eval (n:=half_sz) wt)
-       (Positional.eval (n:=sz) wt)
-       (mul_cps := Positional.mul_cps (n:=half_sz) wt)
-       (add_cps := Positional.add_cps (n:=half_sz) wt)
-       (add2_cps := Positional.add_cps (n:=sz) wt)
-       (split_cps := Positional_split_cps (n:=sz) (m1:=half_sz) (m2 := half_sz))
-       (scmul2_cps := Positional_scmul_cps (n:=sz))
-       (sub2_cps := Positional_sub_cps (n:=sz))
-       (mul := fun a b => Positional.mul_cps (n:= half_sz) wt a b id)
-       (add := fun a b => Positional.add_cps (n:=half_sz) wt a b id)
-       (add2 := fun a b => Positional.add_cps (n:=sz) wt a b id)
-       (split := fun s a => Positional_split_cps (n:=sz) (m1:=half_sz) (m2 := half_sz) s a id)
-       (scmul2 := fun x a => Positional_scmul_cps (n:=sz) x a id)
-       (sub2 := fun a b => Positional_sub_cps (n:=sz) a b id)
-       (mul_id := mul_id)
-       (add_id := add_id)
-       (add2_id := add2_id)
-       (split_id := split_id)
-       (scmul2_id := scmul2_id)
-       (sub2_id := sub2_id)
-       (eval_mul := eval_mul)
-       (eval_add := eval_add)
-       (eval_add2 := eval_add2)
-       (eval_split := eval_split)
-       (eval_scmul2 := eval_scmul2)
-       (eval_sub2 := eval_sub2)
-    ).
-  Check goldilocks448_correct'.
-  Hint Unfold Positional_split_cps Positional_scmul_cps Positional_sub_cps.
-  Lemma goldilocks448_correct : 
-       forall p : positive,
-       forall s : Z,
-       s <> 0 ->
-       s ^ 2 mod p = (s + 1) mod p ->
-       forall xs ys : Z ^ sz,
-         mod_eq (Z.to_pos p)
-                (Positional.eval wt (goldilocks448_cps s xs ys _ id))
-                (Positional.eval wt xs * Positional.eval wt ys).
-  Proof.
-    pose proof wt_nonzero.
-    intros; autounfold. cbv [mod_eq].
-    rewrite goldilocks448_id by (intros; autounfold; autorewrite with uncps push_id; reflexivity). autorewrite with push_id.
-    apply goldilocks448_correct'; try assumption; intros; autounfold;
-      autorewrite with uncps push_id cancel_pair push_basesystem_eval;
-      try reflexivity.
-    { setoid_rewrite Associational.eval_nil. ring. }
-    { rewrite Pos2Z.id; congruence. }
-  Qed.
-
   Definition mul_sig :
     {mul : (Z^sz -> Z^sz -> Z^sz)%type |
                forall a b : Z^sz,
@@ -270,16 +159,19 @@ Section Ops51.
   Proof.
     eexists; cbv beta zeta; intros.
     pose proof wt_nonzero.
+    Print goldilocks_mul_cps.
     let x := constr:(
-               goldilocks448_cps (2^224) a b _ id) in
+               goldilocks_mul_cps (n:=half_sz) (n2:=sz) wt (2^224) a b id) in
     F_mod_eq;
       transitivity (Positional.eval wt x); repeat autounfold;
 
    [ 
    | autorewrite with uncps push_id push_basesystem_eval;
-     apply goldilocks448_correct; cbv; congruence ].
+     apply goldilocks_mul_correct; try assumption; cbv; congruence ].
    cbv[mod_eq]; apply f_equal2;
      [  | reflexivity ]; apply f_equal.
+   cbv [goldilocks_mul_cps].
+   repeat autounfold.
    basesystem_partial_evaluation_RHS.
    do_replace_match_with_destructuring_match_in_goal.
    reflexivity.