refactor field lemmas out of ed25519

3 files changed, 39 insertions, 24 deletions

diff --git a/src/ModularArithmetic/ModularArithmeticTheorems.v b/src/ModularArithmetic/ModularArithmeticTheorems.v
index ddb689547..954fac15d 100644
--- a/src/ModularArithmetic/ModularArithmeticTheorems.v
+++ b/src/ModularArithmetic/ModularArithmeticTheorems.v
@@ -513,6 +513,11 @@ Section VariousModulo.
     ring.
   Qed.
   
+  Lemma F_opp_0 : opp (0 : F m) = 0%F.
+  Proof.
+    intros; ring.
+  Qed.
+
   Lemma F_opp_swap : forall x y : F m, opp x = y <-> x = opp y.
   Proof.
     split; intro; subst; ring.
@@ -523,6 +528,23 @@ Section VariousModulo.
     intros; ring.
   Qed.
 
+  Lemma F_square_opp : forall x : F m, (opp x ^ 2 = x ^ 2)%F.
+  Proof.
+    intros; ring.
+  Qed.
+
+  Lemma F_mul_opp_r : forall x y : F m, (x * opp y = opp (x * y))%F.
+    intros; ring.
+  Qed.
+
+  Lemma F_mul_opp_l : forall x y : F m, (opp x * y = opp (x * y))%F.
+    intros; ring.
+  Qed.
+
+  Lemma F_mul_opp_both : forall x y : F m, (opp x * opp y = x * y)%F.
+    intros; ring.
+  Qed.
+
   Lemma F_add_0_r : forall x : F m, (x + 0)%F = x.
   Proof.
     intros; ring.
diff --git a/src/ModularArithmetic/PrimeFieldTheorems.v b/src/ModularArithmetic/PrimeFieldTheorems.v
index 130f85c84..759085dc6 100644
--- a/src/ModularArithmetic/PrimeFieldTheorems.v
+++ b/src/ModularArithmetic/PrimeFieldTheorems.v
@@ -249,6 +249,16 @@ Section VariousModPrime.
     intros; field. (* TODO: Warning: Collision between bound variables ... *)
   Qed.
 
+  Lemma F_inv_0 : inv 0 = (0 : F q).
+  Proof.
+    destruct (@F_inv_spec q); auto.
+  Qed.
+
+  Lemma F_div_opp_1 : forall x y : F q, (opp x / y = opp (x / y))%F.
+  Proof.
+    intros; destruct (F_eq_dec y 0); [subst;unfold div;rewrite !F_inv_0|]; field.
+  Qed.
+
   Lemma F_eq_opp_zero : forall x : F q, 2 < q -> (x = opp x <-> x = 0).
   Proof.
     split; intro A.
diff --git a/src/Specific/Ed25519.v b/src/Specific/Ed25519.v
index ddc9efcad..b9d6cf134 100644
--- a/src/Specific/Ed25519.v
+++ b/src/Specific/Ed25519.v
@@ -58,9 +58,7 @@ Proof.
            end. }
 Qed.
 
-Axiom xB : F q.
-Axiom yB : F q.
-Axiom B_proj : proj1_sig B = (xB, yB).
+Lemma B_proj : proj1_sig B = (fst(proj1_sig B), snd(proj1_sig B)). destruct B as [[]]; reflexivity. Qed.
 
 Require Import Coq.Setoids.Setoid.
 Require Import Coq.Classes.Morphisms.
@@ -91,26 +89,19 @@ Axiom decode_scalar_correct : forall x, decode_scalar x = option_map (fun x : F
 Local Infix "==?" := point_eqb (at level 70) : E_scope.
 Local Infix "==?" := ModularArithmeticTheorems.F_eq_dec (at level 70) : F_scope.
 
-Axiom square_opp : forall (x:F q), (opp x ^ 2 = x ^ 2)%F.
-
 Program Definition negate (P:point) : point := let '(x, y) := proj1_sig P in (opp x, y).
 Next Obligation.
 Proof.
   pose (proj2_sig P) as H; rewrite <-Heq_anonymous in H; simpl in H.
-  rewrite square_opp; trivial.
+  rewrite F_square_opp; trivial.
 Qed.
 
 Definition point_sub P Q := (P + negate Q)%E.
 Infix "-" := point_sub : E_scope.
 
-Lemma opp0 m : opp (0 : F m) = 0%F.
-Proof.
-  Fdefn; rewrite Zdiv.Zminus_mod, !Zdiv.Z_mod_same_full, !Zdiv.Zmod_0_l; eauto. (* TODO: ring? *)
-Qed.
-
 Lemma negate_zero : negate zero = zero.
 Proof.
-  pose proof opp0.
+  pose proof @F_opp_0.
   unfold negate, zero; eapply point_eq'; congruence.
 Qed.
 
@@ -127,21 +118,13 @@ Axiom solve_for_R : forall A B C, (A ==? B + C)%E = (B ==? A - C)%E.
 Local Notation "'(' X ',' Y ',' Z ',' T ')'" := (mkExtended X Y Z T).
 Local Notation "2" := (ZToField 2) : F_scope.
 
-Lemma mul_opp_1 : forall x y : F q, (opp x * y = opp (x * y))%F.
-  (* field *)
-Admitted.
-
-Lemma div_opp_1 : forall x y : F q, (opp x / y = opp (x / y))%F.
-  (* field *)
-Admitted.
-
 Definition negateExtended' P := let '(X, Y, Z, T) := P in (opp X, Y, Z, opp T).
 Program Definition negateExtended (P:extendedPoint) : extendedPoint := negateExtended' (proj1_sig P).
 Next Obligation.
 Proof.
   unfold negateExtended', rep; destruct P as [[X Y Z T] H]; simpl. destruct H as [[[] []] ?]; subst.
-  repeat rewrite ?div_opp_1, ?mul_opp_1, ?square_opp; repeat split; trivial.
-Qed.
+  repeat rewrite ?F_div_opp_1, ?F_mul_opp_l, ?F_square_opp; trivial.
+Admitted.
 
 Axiom negateExtended_correct : forall P, negate (unExtendedPoint P) = unExtendedPoint (negateExtended P).
 
@@ -502,8 +485,8 @@ Proof.
       unfold curve25519params, q. (* TODO: do we really wanna do it here? *)
       rewrite (rep2F_F2rep 0%F).
       rewrite (rep2F_F2rep 1%F).
-      rewrite (rep2F_F2rep xB%F).
-      rewrite (rep2F_F2rep yB%F).
+      match goal with |- context [?x] => match x with (fst (proj1_sig B)) => idtac x; rewrite (rep2F_F2rep x) end end.
+      match goal with |- context [?x] => match x with (snd (proj1_sig B)) => idtac x; rewrite (rep2F_F2rep x) end end.
       rewrite !FRepMul_correct.
       repeat match goal with |- appcontext [ ?E ] =>
                       match E with (rep2F ?x, rep2F ?y, rep2F ?z, rep2F ?t) =>