diff options
Diffstat (limited to 'src/Algebra')
-rw-r--r-- | src/Algebra/Field.v | 95 | ||||
-rw-r--r-- | src/Algebra/Field_test.v | 13 |
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.
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