2 files changed, 67 insertions, 41 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