5 files changed, 134 insertions, 113 deletions

diff --git a/src/Algebra/Field.v b/src/Algebra/Field.v
index 76b2a9ed3..ebc92c0e5 100644
--- a/src/Algebra/Field.v
+++ b/src/Algebra/Field.v
@@ -231,17 +231,6 @@ Ltac goal_to_field_equality fld :=
          end
   end.
 
-Ltac _introduce_inverse fld d d_nz :=
-  let eq := match type of fld with Algebra.field(eq:=?eq) => eq end in
-  let mul := match type of fld with Algebra.field(mul:=?mul) => mul end in
-  let one := match type of fld with Algebra.field(one:=?one) => one end in
-  let inv := match type of fld with Algebra.field(inv:=?inv) => inv end in
-  match goal with [H: eq (mul d _) one |- _ ] => fail 1 | _ => idtac end;
-  let d_i := fresh "i" in
-  unique pose proof (right_multiplicative_inverse(H:=fld) _ d_nz);
-  set (inv d) as d_i in *;
-  clearbody d_i.
-
 Ltac inequalities_to_inverse_equations fld :=
   let eq := match type of fld with Algebra.field(eq:=?eq) => eq end in
   let zero := match type of fld with Algebra.field(zero:=?zero) => zero end in
@@ -250,52 +239,80 @@ Ltac inequalities_to_inverse_equations fld :=
   repeat match goal with
          | [H: not (eq _ _) |- _ ] =>
            lazymatch type of H with
-           | not (eq ?d zero) => _introduce_inverse fld d H
-           | not (eq zero ?d) => _introduce_inverse fld d (symmetry(R:=fun a b => not (eq a b)) H)
-           | not (eq ?x ?y) => _introduce_inverse fld (sub x y) (Ring.neq_sub_neq_zero _ _ H)
+           | not (eq ?d zero) =>
+             unique pose proof (right_multiplicative_inverse(H:=fld) _ H)
+           | not (eq zero ?d) =>
+             unique pose proof (right_multiplicative_inverse(H:=fld) _ (symmetry(R:=fun a b => not (eq a b)) H))
+           | not (eq ?x ?y) => 
+             unique pose proof (right_multiplicative_inverse(H:=fld) _ (Ring.neq_sub_neq_zero _ _ H))
            end
          end.
 
-Ltac _nonzero_tac fld :=
-  solve [trivial | IntegralDomain.solve_constant_nonzero | goal_to_field_equality fld; nsatz; IntegralDomain.solve_constant_nonzero].
+Ltac unique_pose_implication pf :=
+  let B := match type of pf with ?A -> ?B => B end in
+  match goal with
+             | [H:B|-_] => fail 1
+             | _ => unique pose proof pf
+  end.
 
-Ltac _inverse_to_equation_by fld d tac :=
+Ltac inverses_to_conditional_equations fld :=
   let eq := match type of fld with Algebra.field(eq:=?eq) => eq end in
-  let zero := match type of fld with Algebra.field(zero:=?zero) => zero end in
-  let one := match type of fld with Algebra.field(one:=?one) => one end in
-  let mul := match type of fld with Algebra.field(mul:=?mul) => mul end in
-  let div := match type of fld with Algebra.field(div:=?div) => div end in
   let inv := match type of fld with Algebra.field(inv:=?inv) => inv end in
-  let d_nz := fresh "nz" in
-  assert (not (eq d zero)) as d_nz by tac;
-  lazymatch goal with
-  | H: eq (mul ?di d) one |- _ => rewrite <-!(left_inv_unique(H:=fld) _ _ H) in *
-  | H: eq (mul d ?di) one |- _ => rewrite <-!(right_inv_unique(H:=fld) _ _ H) in *
-  | _ => _introduce_inverse fld d d_nz
-  end;
-  clear d_nz.
-
-Ltac inverses_to_equations_by fld tac :=
+  repeat match goal with
+         | |- context[inv ?d] =>
+           unique_pose_implication constr:(right_multiplicative_inverse(H:=fld) d)
+         | H: context[inv ?d] |- _ => 
+           unique_pose_implication constr:(right_multiplicative_inverse(H:=fld) d)
+         end.
+
+Ltac clear_hypotheses_with_nonzero_requirements fld :=
+  let eq := match type of fld with Algebra.field(eq:=?eq) => eq end in
+  let zero := match type of fld with Algebra.field(zero:=?zero) => zero end in
+  repeat match goal with
+           [H: not (eq _ zero) -> _ |- _ ] => clear H
+         end.
+
+Ltac forward_nonzero fld solver_tac :=
   let eq := match type of fld with Algebra.field(eq:=?eq) => eq end in
   let zero := match type of fld with Algebra.field(zero:=?zero) => zero end in
-  let inv := match type of fld with Algebra.field(inv:=?inv) => inv end in
   repeat match goal with
-         | |- context[inv ?d] => _inverse_to_equation_by fld d tac
-         | H: context[inv ?d] |- _ => _inverse_to_equation_by fld d tac
+         | [H: not (eq ?x zero) -> _ |- _ ]
+           => let H' := fresh in
+              assert (H' : not (eq x zero)) by (clear_hypotheses_with_nonzero_requirements; solver_tac); specialize (H H')
+         | [H: not (eq ?x zero) -> _ |- _ ]
+           => let H' := fresh in
+              assert (H' : not (eq x zero)) by (clear H; solver_tac); specialize (H H')
          end.
 
 Ltac divisions_to_inverses fld :=
   rewrite ?(field_div_definition(field:=fld)) in *.
 
-Ltac fsatz :=
-  let fld := guess_field in
+Ltac fsatz_solve_on fld :=
   goal_to_field_equality fld;
-  inequalities_to_inverse_equations fld;
-  divisions_to_inverses fld;
-  inverses_to_equations_by fld ltac:(solve_debugfail ltac:(_nonzero_tac fld));
+  forward_nonzero fld ltac:(fsatz_solve_on fld);
   nsatz;
   solve_debugfail ltac:(IntegralDomain.solve_constant_nonzero).
 
+Ltac fsatz_solve :=
+  let fld := guess_field in
+  fsatz_solve_on fld.
+
+Ltac fsatz_prepare_hyps_on fld :=
+  divisions_to_inverses fld;
+  inequalities_to_inverse_equations fld;
+  inverses_to_conditional_equations fld;
+  forward_nonzero fld ltac:(fsatz_solve_on fld).
+
+Ltac fsatz_prepare_hyps :=
+  let fld := guess_field in
+  fsatz_prepare_hyps_on fld.
+
+Ltac fsatz :=
+  let fld := guess_field in
+  fsatz_prepare_hyps_on fld;
+  fsatz_solve_on fld.
+
+
 Section FieldSquareRoot.
   Context {T eq zero one opp add mul sub inv div} `{@field T eq zero one opp add sub mul inv div} {eq_dec:DecidableRel eq}.
   Local Infix "=" := eq : type_scope. Local Notation "a <> b" := (not (a = b)) : type_scope.
diff --git a/src/Algebra/Field_test.v b/src/Algebra/Field_test.v
index 13a0ffa95..2df673163 100644
--- a/src/Algebra/Field_test.v
+++ b/src/Algebra/Field_test.v
@@ -55,7 +55,16 @@ Module _fsatz_test.
     Lemma fractional_equation_no_solution x (A:x<>1) (B:x<>opp two) (C:x*x+x <> two) (X:nine/(x*x + x - two) = opp three/(x+two) + seven*inv(x-1)) : False.
     Proof. fsatz. Qed.
 
-    Local Notation "x ^ 2" := (x*x). Local Notation "x ^ 3" := (x^2*x).
+    Local Notation "x ^ 2" := (x*x).
+    Lemma recursive_nonzero_solving
+          (a sqrt_a d x y : F)
+          (Hpoly : a * x^2 + y^2 = one + d * x^2 * y^2)
+          (Hsqrt : sqrt_a^2 = a)
+          (Hfrac : (sqrt_a / y)^2 <> d)
+      : x^2 = (y^2 - one) / (d * y^2 - a).
+    Proof. fsatz. Qed.
+
+    Local Notation "x ^ 3" := (x^2*x).
     Lemma weierstrass_associativity_main a b x1 y1 x2 y2 x4 y4
           (A: y1^2=x1^3+a*x1+b)
           (B: y2^2=x2^3+a*x2+b)
@@ -77,6 +86,6 @@ Module _fsatz_test.
           x9 (Hx9: x9 = λ9^2-x1-x6)
           y9 (Hy9: y9 = λ9*(x1-x9)-y1)
       : x7 = x9 /\ y7 = y9.
-    Proof. split; fsatz. Qed.
+    Proof. fsatz_prepare_hyps; split; fsatz. Qed.
   End _test.
 End _fsatz_test.
\ No newline at end of file
diff --git a/src/CompleteEdwardsCurve/CompleteEdwardsCurveTheorems.v b/src/CompleteEdwardsCurve/CompleteEdwardsCurveTheorems.v
index a52bf38f6..996c5d672 100644
--- a/src/CompleteEdwardsCurve/CompleteEdwardsCurveTheorems.v
+++ b/src/CompleteEdwardsCurve/CompleteEdwardsCurveTheorems.v
@@ -95,14 +95,9 @@ Module E.
     Section PointCompression.
       Local Notation "x ^ 2" := (x*x).
 
-      Lemma a_d_y2_nonzero y : d * y^2 - a <> 0.
-      Proof.
-        destruct square_a as [sqrt_a], (dec (y=0));
-          pose proof nonzero_a; pose proof (nonsquare_d (sqrt_a/y)); fsatz.
-      Qed.
-
-      Lemma solve_correct : forall x y, onCurve x y <-> (x^2 = (y^2-1) / (d*y^2-a)).
-      Proof. pose proof a_d_y2_nonzero; t. Qed.
+      Lemma solve_correct x y : onCurve x y <-> (x^2 = (y^2-1) / (d*y^2-a)).
+      Proof. destruct square_a as [sqrt_a]; pose proof (nonsquare_d (sqrt_a/y));
+               split; intros; fsatz. Qed.
 
       (* TODO: move *)
       Definition exist_option {A} (P : A -> Prop) (x : option A)
diff --git a/src/Spec/MontgomeryCurve.v b/src/Spec/MontgomeryCurve.v
index 2717f6bbc..cff35104c 100644
--- a/src/Spec/MontgomeryCurve.v
+++ b/src/Spec/MontgomeryCurve.v
@@ -60,6 +60,8 @@ Module M.
             end.
     Next Obligation. Proof. t. Qed.
 
+    Program Definition zero : point := ∞.
+
     Program Definition opp (P:point) : point :=
       match P return F*F+∞ with
       | (x, y) => (x, -y)
@@ -73,23 +75,55 @@ Module M.
     Local Notation "27" := (3*9).
     Context {char_ge_28:@Ring.char_ge F Feq Fzero Fone Fopp Fadd Fsub Fmul 28}.
 
-    Let WeierstrassA := ((3-a^2)/(3*b^2)).
-    Let WeierstrassB := ((2*a^3-9*a)/(27*b^3)).
+    Local Notation WeierstrassA := ((3-a^2)/(3*b^2)).
+    Local Notation WeierstrassB := ((2*a^3-9*a)/(27*b^3)).
     Local Notation Wpoint := (@W.point F Feq Fadd Fmul WeierstrassA WeierstrassB).
     Local Notation Wadd := (@W.add F Feq Fzero Fone Fopp Fadd Fsub Fmul Finv Fdiv field Feq_dec char_ge_3 WeierstrassA WeierstrassB).
 
+    Program Definition to_Weierstrass (P:point) : Wpoint :=
+      match coordinates P return F*F+∞ with
+      | (x, y) => ((x + a/3)/b, y/b)
+      | _ => ∞
+      end.
+    Next Obligation.
+    Proof. clear char_ge_3; destruct P; t. Qed.
+
     Program Definition of_Weierstrass (P:Wpoint) : point :=
       match W.coordinates P return F*F+∞ with
       | (x,y) => (b*x-a/3, b*y)
       | _ => ∞
       end.
     Next Obligation.
-    Proof. clear char_ge_3; subst WeierstrassA; subst WeierstrassB; destruct P; t. Qed.
+    Proof. clear char_ge_3; destruct P; t. Qed.
 
-    Lemma of_Weierstrass_add P1 P2 :
-      eq (of_Weierstrass (W.add P1 P2))
-         (add (of_Weierstrass P1) (of_Weierstrass P2)).
-    Proof. cbv [WeierstrassA WeierstrassB eq of_Weierstrass W.add add coordinates W.coordinates proj1_sig] in *; clear char_ge_3; t. Qed.
+    (* TODO: move *)
+    Program Definition Wopp (P:Wpoint) : Wpoint :=
+      match P return F*F+∞ with
+      | (x, y) => (x, -y)
+      | ∞ => ∞
+      end.
+    Next Obligation. destruct P; t. Qed.
+
+    Axiom Wgroup : @Algebra.group Wpoint (@W.eq F Feq Fadd Fmul WeierstrassA WeierstrassB)
+         Wadd (@W.zero F Feq Fadd Fmul WeierstrassA WeierstrassB) Wopp.
+    Program Definition _MW : _ /\ _ /\ _ :=
+      @Group.group_from_redundant_representation
+        Wpoint W.eq Wadd W.zero Wopp
+        Wgroup
+        point eq add zero opp
+        of_Weierstrass
+        to_Weierstrass
+        _ _ _ _ _
+        .
+    Next Obligation. cbv [W.eq eq to_Weierstrass of_Weierstrass W.add add coordinates W.coordinates proj1_sig] in *; t. Qed.
+    Next Obligation. cbv [W.eq eq to_Weierstrass of_Weierstrass W.add add coordinates W.coordinates proj1_sig] in *. clear char_ge_3. t. 2:intuition idtac. 2:intuition idtac. 2:intuition idtac.
+                     { repeat split; destruct_head' and; t. } Qed.
+    Next Obligation.
+      (* addition case, same issue as in Weierstrass associativity *) 
+      cbv [W.eq eq to_Weierstrass of_Weierstrass W.add add coordinates W.coordinates proj1_sig] in *.
+      clear char_ge_3. t. Qed.
+    Next Obligation. cbv [W.eq eq to_Weierstrass of_Weierstrass W.add add Wopp opp coordinates W.coordinates proj1_sig] in *. clear char_ge_3. t. Qed.
+    Next Obligation. cbv [W.eq eq to_Weierstrass of_Weierstrass W.add add Wopp opp coordinates W.coordinates proj1_sig] in *. clear char_ge_3. t. Qed.
     
     Section AddX.
       Lemma homogeneous_x_differential_addition_releations P1 P2 :
diff --git a/src/WeierstrassCurve/WeierstrassCurveTheorems.v b/src/WeierstrassCurve/WeierstrassCurveTheorems.v
index aa444c9ee..b9f7458b0 100644
--- a/src/WeierstrassCurve/WeierstrassCurveTheorems.v
+++ b/src/WeierstrassCurve/WeierstrassCurveTheorems.v
@@ -10,74 +10,40 @@ Module W.
     Context {F Feq Fzero Fone Fopp Fadd Fsub Fmul Finv Fdiv} {a b:F}
             {field:@Algebra.field F Feq Fzero Fone Fopp Fadd Fsub Fmul Finv Fdiv}
             {char_ge_3:@Ring.char_ge F Feq Fzero Fone Fopp Fadd Fsub Fmul (BinNat.N.succ_pos (BinNat.N.two))}
-            {char_ge_12:@Ring.char_ge F Feq Fzero Fone Fopp Fadd Fsub Fmul 12%positive} (* FIXME: we shouldn't need this *)
+            {char_ge_12:@Ring.char_ge F Feq Fzero Fone Fopp Fadd Fsub Fmul 12%positive} (* FIXME: shouldn't need we need 4, not 12? *)
             {Feq_dec:DecidableRel Feq}.
     Local Infix "=" := Feq : type_scope. Local Notation "a <> b" := (not (a = b)) : type_scope.
     Local Notation "0" := Fzero.  Local Notation "1" := Fone.
-    Local Infix "+" := Fadd. Local Infix "*" := Fmul.
-    Local Infix "-" := Fsub. Local Infix "/" := Fdiv.
-    Local Notation eq := (@W.eq F Feq Fadd Fmul a b).
-    Local Notation point := (@W.point F Feq Fadd Fmul a b).
-    Local Notation "0" := Fzero.  Local Notation "1" := Fone.
-    Local Notation "2" := (1+1). Local Notation "3" := (1+2). Local Notation "4" := (1+3).
-    Local Notation "8" := (1+(1+(1+(1+4)))). Local Notation "12" := (1+(1+(1+(1+8)))).
-    Local Notation "16" := (1+(1+(1+(1+12)))). Local Notation "20" := (1+(1+(1+(1+16)))).
-    Local Notation "24" := (1+(1+(1+(1+20)))). Local Notation "27" := (1+(1+(1+24))).
-    Local Notation "x ^ 2" := (x*x) (at level 30). Local Notation "x ^ 3" := (x*x^2) (at level 30).
-    Context {discriminant_nonzero:4*a^3 + 27*b^2 <> 0}.
+    Local Infix "+" := Fadd. Local Infix "-" := Fsub. Local Infix "*" := Fmul.
+    Local Notation "4" := (1+1+1+1). Local Notation "27" := (4*4 + 4+4 +1+1+1).
+    Context {discriminant_nonzero:4*a*a*a + 27*b*b <> 0}.
 
-    Program Definition inv (P:point) : point
+    Program Definition inv (P:@W.point F Feq Fadd Fmul a b) : @W.point F Feq Fadd Fmul a b
       := match W.coordinates P return F*F+_ with
          | inl (x1, y1) => inl (x1, Fopp y1)
          | _ => P
          end.
     Next Obligation. destruct P as [[[??]|[]]?]; cbv; trivial; fsatz. Qed.
 
-    Lemma same_x_same_y
-          (xA yA : F)
-          (A : yA ^ 2 = xA ^ 3 + a * xA + b)
-          (xB yB : F)
-          (B : yB ^ 2 = xB ^ 3 + a * xB + b)
-          (Hx: xA = xB)
-          (Hy:yB <> Fopp yA)
-      : yB = yA.
-    Proof. fsatz. Qed.
-
-    Let is_redundant {T} (x:T) := x.
-    Ltac clear_marked_redundant :=
+    Global Instance commutative_group : abelian_group(eq:=W.eq)(op:=W.add)(id:=W.zero)(inv:=inv).
+    Proof.
       repeat match goal with
-               [H:?P, Hr:is_redundant ?P |- _] => clear H Hr
+             | _ => solve [ contradiction | trivial | exact _ ]
+             | _ => intro
+             | |- Equivalence _ => split
+             | |- abelian_group => split | |- group => split | |- monoid => split
+             | |- is_associative => split | |- is_commutative => split 
+             | |- is_left_inverse => split | |- is_right_inverse => split
+             | |- is_left_identity => split | |- is_right_identity => split
+             | _ => progress destruct_head' @W.point
+             | _ => progress destruct_head' sum
+             | _ => progress destruct_head' prod
+             | _ => progress destruct_head' unit
+             | _ => progress destruct_head' and
+             | _ => progress cbv [inv W.eq W.zero W.add W.coordinates proj1_sig]in*
+             | _ => progress break_match
              end.
-    Ltac t_step :=
-      match goal with
-      | _ => solve [ contradiction | trivial | exact _ ]
-      | _ => intro
-      | [ A : ?yA ^ 2 = ?xA ^ 3 + a * ?xA + b,
-              B : ?yB ^ 2 = ?xB ^ 3 + a * ?xB + b,
-                  Hx: ?xA = ?xB,
-                      Hy: ?yB <> Fopp ?yA
-          |- _] => unique pose proof (same_x_same_y _ _ A _ _ B Hx Hy)
-      | |- Equivalence _ => split
-      | |- abelian_group => split | |- group => split | |- monoid => split
-      | |- is_associative => split | |- is_commutative => split 
-      | |- is_left_inverse => split | |- is_right_inverse => split
-      | |- is_left_identity => split | |- is_right_identity => split
-      | p:point |- _ => destruct p
-      | _ => progress destruct_head' sum
-      | _ => progress destruct_head' prod
-      | _ => progress destruct_head' unit
-      | _ => progress destruct_head' and
-      | |- context[?P] =>
-        unique pose proof (proj2_sig P);
-        unique pose proof (proj2_sig P:(is_redundant _))
-      | _ => progress cbv [inv W.eq W.zero W.add W.coordinates proj1_sig] in *
-      | _ => progress break_match
-      | |- _ /\ _ => split | |- _ <-> _ => split
-      end.
-    Ltac t := repeat t_step; clear_marked_redundant.
-
-    Global Instance commutative_group : abelian_group(eq:=W.eq)(op:=W.add)(id:=W.zero)(inv:=inv).
-    Proof. t. all:try (abstract fsatz). Qed.
-    
+      all: try abstract(fsatz_prepare_hyps; repeat split; fsatz_solve).
+    Qed.
   End W.
 End W.