Add fieldwise_eq_edwards_extended_add_coordinates_carry_nocarry

2 files changed, 66 insertions, 16 deletions

diff --git a/src/Specific/GF25519.v b/src/Specific/GF25519.v
index 790fa4571..68e4ff6bb 100644
--- a/src/Specific/GF25519.v
+++ b/src/Specific/GF25519.v
@@ -13,6 +13,7 @@ Require Import Crypto.Util.Tactics.
 Require Import Crypto.Util.LetIn.
 Require Import Crypto.Util.Notations.
 Require Import Crypto.Util.Decidable.
+Require Import Crypto.Util.Relations.
 Require Import Crypto.Algebra.
 Require Crypto.Spec.Ed25519.
 Import ListNotations.
@@ -750,23 +751,69 @@ Eval cbv iota beta delta [
          add sub mul
      ] in
     (@ExtendedCoordinates.Extended.add_coordinates _ add sub mul td P Q).
+Definition edwards_extended_carry_add_coordinates td P Q :=
+Eval cbv iota beta delta [
+       Extended.add_coordinates
+         carry_add carry_sub mul
+     ] in
+    (@ExtendedCoordinates.Extended.add_coordinates _ carry_add carry_sub mul td P Q).
 Print edwards_extended_add_coordinates.
 
 Local Existing Instance field25519.
 Create HintDb edwards_extended_add_coordinates_correct discriminated.
-Hint Rewrite
-     (Ring.homomorphism_mul(is_homomorphism:=homomorphism_F25519_decode))
-     (Ring.homomorphism_add(H1             :=homomorphism_F25519_decode))
-     (Ring.homomorphism_sub(H1             :=homomorphism_F25519_decode))
-  : edwards_extended_add_coordinates_correct .
-Lemma edwards_extended_add_coordinates_correct td P Q :
-  Tuple.map (n:=4) decode (edwards_extended_add_coordinates td P Q)
-  = (@ExtendedCoordinates.Extended.add_coordinates _ F.add F.sub F.mul (decode td) (Tuple.map (n:=4) decode P) (Tuple.map (n:=4) decode Q)).
-Proof.
-  change (edwards_extended_add_coordinates td P Q)
-  with (@ExtendedCoordinates.Extended.add_coordinates _ add sub mul td P Q).
-  destruct_head' prod.
-  simpl.
-  rewrite_strat topdown hints edwards_extended_add_coordinates_correct.
-  reflexivity.
+Section noncarry.
+  Hint Rewrite
+        (Ring.homomorphism_mul(is_homomorphism:=homomorphism_F25519_decode))
+        (Ring.homomorphism_add(H1             :=homomorphism_F25519_decode))
+        (Ring.homomorphism_sub(H1             :=homomorphism_F25519_decode))
+  : edwards_extended_add_coordinates_correct.
+  Lemma edwards_extended_add_coordinates_correct td P Q :
+    Tuple.map (n:=4) decode (edwards_extended_add_coordinates td P Q)
+    = (@ExtendedCoordinates.Extended.add_coordinates _ F.add F.sub F.mul (decode td) (Tuple.map (n:=4) decode P) (Tuple.map (n:=4) decode Q)).
+  Proof.
+    change (edwards_extended_add_coordinates td P Q)
+    with (@ExtendedCoordinates.Extended.add_coordinates _ add sub mul td P Q).
+    destruct_head' prod.
+    simpl.
+    rewrite_strat topdown hints edwards_extended_add_coordinates_correct.
+    reflexivity.
+  Qed.
+End noncarry.
+Section carry.
+  Local Existing Instance carry_field25519.
+  Hint Rewrite
+        (Ring.homomorphism_mul(is_homomorphism:=homomorphism_carry_F25519_decode))
+        (Ring.homomorphism_add(H1             :=homomorphism_carry_F25519_decode))
+        (Ring.homomorphism_sub(H1             :=homomorphism_carry_F25519_decode))
+  : edwards_extended_add_coordinates_correct.
+  Lemma edwards_extended_carry_add_coordinates_correct td P Q :
+    Tuple.map (n:=4) decode (edwards_extended_carry_add_coordinates td P Q)
+    = (@ExtendedCoordinates.Extended.add_coordinates _ F.add F.sub F.mul (decode td) (Tuple.map (n:=4) decode P) (Tuple.map (n:=4) decode Q)).
+  Proof.
+    change (edwards_extended_carry_add_coordinates td P Q)
+    with (@ExtendedCoordinates.Extended.add_coordinates _ carry_add carry_sub mul td P Q).
+    destruct_head' prod.
+    simpl.
+    rewrite_strat topdown hints edwards_extended_add_coordinates_correct.
+    reflexivity.
+  Qed.
+End carry.
+
+Lemma fieldwise_eq_edwards_extended_add_coordinates_carry_nocarry td P Q :
+  Tuple.fieldwise
+    (n:=4) eq
+    (edwards_extended_carry_add_coordinates td P Q)
+    (edwards_extended_add_coordinates td P Q).
+Proof.
+  pose proof (edwards_extended_carry_add_coordinates_correct td P Q) as H0.
+  pose proof (edwards_extended_add_coordinates_correct td P Q) as H1.
+  rewrite <- H0 in H1; clear H0.
+  assert (fieldwise
+            (fun x y => x = y)
+            (Tuple.map (n:=4) decode (edwards_extended_carry_add_coordinates td P Q))
+            (Tuple.map (n:=4) decode (edwards_extended_add_coordinates td P Q)))
+    by (rewrite H1; reflexivity).
+  clear H1.
+  destruct (edwards_extended_carry_add_coordinates td P Q), (edwards_extended_add_coordinates td P Q).
+  destruct_head' prod; simpl; unfold eq; trivial.
 Qed.
diff --git a/src/Util/Relations.v b/src/Util/Relations.v
index 4926bd74c..8ab045cfe 100644
--- a/src/Util/Relations.v
+++ b/src/Util/Relations.v
@@ -47,4 +47,7 @@ Proof.
            | _ => progress (cbv [fst snd] in * )
            | _ => solve[eauto]
          end.
-Qed.
\ No newline at end of file
+Qed.
+
+Global Instance eq_eta_Reflexive {T} : Reflexive (fun x y : T => x = y) | 1
+  := eq_Reflexive.