statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
comap_fun_is_locally_fraction (f : R →+* S)
(U : opens (prime_spectrum.Top R)) (V : opens (prime_spectrum.Top S))
(hUV : V.1 ⊆ (prime_spectrum.comap f) ⁻¹' U.1) (s : Π x : U, localizations R x)
(hs : (is_locally_fraction R).to_prelocal_predicate.pred s) :
(is_locally_fraction S).to_prelocal_predicate.pred (coma... | begin
rintro ⟨p, hpV⟩,
-- Since `s` is locally fraction, we can find a neighborhood `W` of `prime_spectrum.comap f p`
-- in `U`, such that `s = a / b` on `W`, for some ring elements `a, b : R`.
rcases hs ⟨prime_spectrum.comap f p, hUV hpV⟩ with ⟨W, m, iWU, a, b, h_frac⟩,
-- We claim that we can write our new ... | lemma | algebraic_geometry.structure_sheaf.comap_fun_is_locally_fraction | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"localization.local_ring_hom_to_map",
"prime_spectrum.comap",
"ring_hom.map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap (f : R →+* S) (U : opens (prime_spectrum.Top R))
(V : opens (prime_spectrum.Top S)) (hUV : V.1 ⊆ (prime_spectrum.comap f) ⁻¹' U.1) :
(structure_sheaf R).1.obj (op U) →+* (structure_sheaf S).1.obj (op V) | { to_fun := λ s, ⟨comap_fun f U V hUV s.1, comap_fun_is_locally_fraction f U V hUV s.1 s.2⟩,
map_one' := subtype.ext $ funext $ λ p, by
{ rw [subtype.coe_mk, subtype.val_eq_coe, comap_fun, (sections_subring R (op U)).coe_one,
pi.one_apply, ring_hom.map_one], refl },
map_zero' := subtype.ext $ funext $ λ p... | def | algebraic_geometry.structure_sheaf.comap | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"pi.mul_apply",
"pi.one_apply",
"prime_spectrum.comap",
"ring_hom.map_add",
"ring_hom.map_mul",
"ring_hom.map_one",
"ring_hom.map_zero",
"subtype.coe_mk",
"subtype.ext",
"subtype.val_eq_coe"
] | For a ring homomorphism `f : R →+* S` and open sets `U` and `V` of the prime spectra of `R` and
`S` such that `V ⊆ (comap f) ⁻¹ U`, the induced ring homomorphism from the structure sheaf of `R`
at `U` to the structure sheaf of `S` at `V`.
Explicitly, this map is given as follows: For a point `p : V`, if the section `s... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comap_apply (f : R →+* S) (U : opens (prime_spectrum.Top R))
(V : opens (prime_spectrum.Top S)) (hUV : V.1 ⊆ (prime_spectrum.comap f) ⁻¹' U.1)
(s : (structure_sheaf R).1.obj (op U)) (p : V) :
(comap f U V hUV s).1 p =
localization.local_ring_hom (prime_spectrum.comap f p.1).as_ideal _ f rfl
(s.1 ⟨(prime_spe... | rfl | lemma | algebraic_geometry.structure_sheaf.comap_apply | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"localization.local_ring_hom",
"prime_spectrum.comap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_const (f : R →+* S) (U : opens (prime_spectrum.Top R))
(V : opens (prime_spectrum.Top S)) (hUV : V.1 ⊆ (prime_spectrum.comap f) ⁻¹' U.1)
(a b : R) (hb : ∀ x : prime_spectrum R, x ∈ U → b ∈ x.as_ideal.prime_compl) :
comap f U V hUV (const R a b U hb) =
const S (f a) (f b) V (λ p hpV, hb (prime_spectrum.com... | subtype.eq $ funext $ λ p,
begin
rw [comap_apply, const_apply, const_apply],
erw localization.local_ring_hom_mk',
refl,
end | lemma | algebraic_geometry.structure_sheaf.comap_const | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"localization.local_ring_hom_mk'",
"prime_spectrum",
"prime_spectrum.comap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_id_eq_map (U V : opens (prime_spectrum.Top R)) (iVU : V ⟶ U) :
comap (ring_hom.id R) U V
(λ p hpV, le_of_hom iVU $ by rwa prime_spectrum.comap_id) =
(structure_sheaf R).1.map iVU.op | ring_hom.ext $ λ s, subtype.eq $ funext $ λ p,
begin
rw comap_apply,
-- Unfortunately, we cannot use `localization.local_ring_hom_id` here, because
-- `prime_spectrum.comap (ring_hom.id R) p` is not *definitionally* equal to `p`. Instead, we use
-- that we can write `s` as a fraction `a/b` in a small neighborho... | lemma | algebraic_geometry.structure_sheaf.comap_id_eq_map | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"localization.local_ring_hom_mk'",
"prime_spectrum.comap_id",
"ring_hom.ext",
"ring_hom.id"
] | For an inclusion `i : V ⟶ U` between open sets of the prime spectrum of `R`, the comap of the
identity from OO_X(U) to OO_X(V) equals as the restriction map of the structure sheaf.
This is a generalization of the fact that, for fixed `U`, the comap of the identity from OO_X(U)
to OO_X(U) is the identity. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comap_id (U V : opens (prime_spectrum.Top R)) (hUV : U = V) :
comap (ring_hom.id R) U V (λ p hpV, by rwa [hUV, prime_spectrum.comap_id]) =
eq_to_hom (show (structure_sheaf R).1.obj (op U) = _, by rw hUV) | by erw [comap_id_eq_map U V (eq_to_hom hUV.symm), eq_to_hom_op, eq_to_hom_map] | lemma | algebraic_geometry.structure_sheaf.comap_id | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"prime_spectrum.comap_id",
"ring_hom.id"
] | The comap of the identity is the identity. In this variant of the lemma, two open subsets `U` and
`V` are given as arguments, together with a proof that `U = V`. This is be useful when `U` and `V`
are not definitionally equal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comap_id' (U : opens (prime_spectrum.Top R)) :
comap (ring_hom.id R) U U (λ p hpU, by rwa prime_spectrum.comap_id) =
ring_hom.id _ | by { rw comap_id U U rfl, refl } | lemma | algebraic_geometry.structure_sheaf.comap_id' | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"prime_spectrum.comap_id",
"ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_comp (f : R →+* S) (g : S →+* P) (U : opens (prime_spectrum.Top R))
(V : opens (prime_spectrum.Top S)) (W : opens (prime_spectrum.Top P))
(hUV : ∀ p ∈ V, prime_spectrum.comap f p ∈ U) (hVW : ∀ p ∈ W, prime_spectrum.comap g p ∈ V) :
comap (g.comp f) U W (λ p hpW, hUV (prime_spectrum.comap g p) (hVW p hpW)) =... | ring_hom.ext $ λ s, subtype.eq $ funext $ λ p,
begin
rw comap_apply,
erw localization.local_ring_hom_comp _ (prime_spectrum.comap g p.1).as_ideal,
-- refl works here, because `prime_spectrum.comap (g.comp f) p` is defeq to
-- `prime_spectrum.comap f (prime_spectrum.comap g p)`
refl,
end | lemma | algebraic_geometry.structure_sheaf.comap_comp | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"localization.local_ring_hom_comp",
"prime_spectrum.comap",
"ring_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_open_comp_comap (f : R →+* S)
(U : opens (prime_spectrum.Top R)) :
to_open R U ≫ comap f U (opens.comap (prime_spectrum.comap f) U) (λ _, id) =
CommRing.of_hom f ≫ to_open S _ | ring_hom.ext $ λ s, subtype.eq $ funext $ λ p,
begin
simp_rw [comp_apply, comap_apply, subtype.val_eq_coe],
erw localization.local_ring_hom_to_map,
refl,
end | lemma | algebraic_geometry.structure_sheaf.to_open_comp_comap | algebraic_geometry | src/algebraic_geometry/structure_sheaf.lean | [
"algebraic_geometry.prime_spectrum.basic",
"algebra.category.Ring.colimits",
"algebra.category.Ring.limits",
"topology.sheaves.local_predicate",
"ring_theory.localization.at_prime",
"ring_theory.subring.basic"
] | [
"CommRing.of_hom",
"localization.local_ring_hom_to_map",
"prime_spectrum.comap",
"ring_hom.ext",
"subtype.val_eq_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_simp : tactic unit | `[simp only [map_one, map_bit0, map_bit1, map_neg, map_add, map_sub, _root_.map_mul, map_pow,
map_div₀]] | def | map_simp | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [
"map_bit0",
"map_bit1",
"map_div₀",
"map_one",
"map_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eval_simp : tactic unit | `[simp only [eval_C, eval_X, eval_neg, eval_add, eval_sub, eval_mul, eval_pow]] | def | eval_simp | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
C_simp : tactic unit | `[simp only [C_1, C_bit0, C_bit1, C_neg, C_add, C_sub, C_mul, C_pow]] | def | C_simp | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
derivative_simp : tactic unit | `[simp only [derivative_C, derivative_X, derivative_X_pow, derivative_neg, derivative_add,
derivative_sub, derivative_mul, derivative_sq]] | def | derivative_simp | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_polynomial : R[X][Y] | -Y - C (C W.a₁ * X + C W.a₃) | def | weierstrass_curve.neg_polynomial | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [] | The polynomial $-Y - a_1X - a_3$ associated to negation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
neg_Y : R | -y₁ - W.a₁ * x₁ - W.a₃ | def | weierstrass_curve.neg_Y | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [] | The $Y$-coordinate of the negation of an affine point in `W`.
This depends on `W`, and has argument order: $x_1$, $y_1$. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
neg_Y_neg_Y : W.neg_Y x₁ (W.neg_Y x₁ y₁) = y₁ | by { simp only [neg_Y], ring1 } | lemma | weierstrass_curve.neg_Y_neg_Y | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
base_change_neg_Y :
(W.base_change A).neg_Y (algebra_map R A x₁) (algebra_map R A y₁)
= algebra_map R A (W.neg_Y x₁ y₁) | by { simp only [neg_Y], map_simp, refl } | lemma | weierstrass_curve.base_change_neg_Y | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [
"algebra_map",
"map_simp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
base_change_neg_Y_of_base_change (x₁ y₁ : A) :
(W.base_change B).neg_Y (algebra_map A B x₁) (algebra_map A B y₁)
= algebra_map A B ((W.base_change A).neg_Y x₁ y₁) | by rw [← base_change_neg_Y, base_change_base_change] | lemma | weierstrass_curve.base_change_neg_Y_of_base_change | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eval_neg_polynomial : (W.neg_polynomial.eval $ C y₁).eval x₁ = W.neg_Y x₁ y₁ | by { rw [neg_Y, sub_sub, neg_polynomial], eval_simp } | lemma | weierstrass_curve.eval_neg_polynomial | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [
"eval_simp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
line_polynomial : R[X] | C L * (X - C x₁) + C y₁ | def | weierstrass_curve.line_polynomial | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [] | The polynomial $L(X - x_1) + y_1$ associated to the line $Y = L(X - x_1) + y_1$,
with a slope of $L$ that passes through an affine point $(x_1, y_1)$.
This does not depend on `W`, and has argument order: $x_1$, $y_1$, $L$. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
XY_ideal_eq₁ : XY_ideal W x₁ (C y₁) = XY_ideal W x₁ (line_polynomial x₁ y₁ L) | begin
simp only [XY_ideal, X_class, Y_class, line_polynomial],
rw [← span_pair_add_mul_right $ adjoin_root.mk _ $ C $ C $ -L, ← _root_.map_mul, ← map_add],
apply congr_arg (_ ∘ _ ∘ _ ∘ _),
C_simp,
ring1
end | lemma | weierstrass_curve.XY_ideal_eq₁ | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [
"C_simp",
"adjoin_root.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_polynomial : R[X] | W.polynomial.eval $ line_polynomial x₁ y₁ L | def | weierstrass_curve.add_polynomial | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [] | The polynomial obtained by substituting the line $Y = L*(X - x_1) + y_1$, with a slope of $L$
that passes through an affine point $(x_1, y_1)$, into the polynomial $W(X, Y)$ associated to `W`.
If such a line intersects `W` at another point $(x_2, y_2)$, then the roots of this polynomial are
precisely $x_1$, $x_2$, and ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
C_add_polynomial :
C (W.add_polynomial x₁ y₁ L)
= (Y - C (line_polynomial x₁ y₁ L)) * (W.neg_polynomial - C (line_polynomial x₁ y₁ L))
+ W.polynomial | by { rw [add_polynomial, line_polynomial, weierstrass_curve.polynomial, neg_polynomial], eval_simp,
C_simp, ring1 } | lemma | weierstrass_curve.C_add_polynomial | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [
"C_simp",
"eval_simp",
"weierstrass_curve.polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coordinate_ring.C_add_polynomial :
adjoin_root.mk W.polynomial (C (W.add_polynomial x₁ y₁ L))
= adjoin_root.mk W.polynomial
((Y - C (line_polynomial x₁ y₁ L)) * (W.neg_polynomial - C (line_polynomial x₁ y₁ L))) | adjoin_root.mk_eq_mk.mpr ⟨1, by rw [C_add_polynomial, add_sub_cancel', mul_one]⟩ | lemma | weierstrass_curve.coordinate_ring.C_add_polynomial | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [
"adjoin_root.mk",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_polynomial_eq : W.add_polynomial x₁ y₁ L = -cubic.to_poly
⟨1, -L ^ 2 - W.a₁ * L + W.a₂,
2 * x₁ * L ^ 2 + (W.a₁ * x₁ - 2 * y₁ - W.a₃) * L + (-W.a₁ * y₁ + W.a₄),
-x₁ ^ 2 * L ^ 2 + (2 * x₁ * y₁ + W.a₃ * x₁) * L - (y₁ ^ 2 + W.a₃ * y₁ - W.a₆)⟩ | by { rw [add_polynomial, line_polynomial, weierstrass_curve.polynomial, cubic.to_poly], eval_simp,
C_simp, ring1 } | lemma | weierstrass_curve.add_polynomial_eq | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [
"C_simp",
"cubic.to_poly",
"eval_simp",
"weierstrass_curve.polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_X : R | L ^ 2 + W.a₁ * L - W.a₂ - x₁ - x₂ | def | weierstrass_curve.add_X | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [] | The $X$-coordinate of the addition of two affine points $(x_1, y_1)$ and $(x_2, y_2)$ in `W`,
where the line through them is not vertical and has a slope of $L$.
This depends on `W`, and has argument order: $x_1$, $x_2$, $L$. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
base_change_add_X :
(W.base_change A).add_X (algebra_map R A x₁) (algebra_map R A x₂) (algebra_map R A L)
= algebra_map R A (W.add_X x₁ x₂ L) | by { simp only [add_X], map_simp, refl } | lemma | weierstrass_curve.base_change_add_X | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [
"algebra_map",
"map_simp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
base_change_add_X_of_base_change (x₁ x₂ L : A) :
(W.base_change B).add_X (algebra_map A B x₁) (algebra_map A B x₂) (algebra_map A B L)
= algebra_map A B ((W.base_change A).add_X x₁ x₂ L) | by rw [← base_change_add_X, base_change_base_change] | lemma | weierstrass_curve.base_change_add_X_of_base_change | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_Y' : R | L * (W.add_X x₁ x₂ L - x₁) + y₁ | def | weierstrass_curve.add_Y' | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [] | The $Y$-coordinate, before applying the final negation, of the addition of two affine points
$(x_1, y_1)$ and $(x_2, y_2)$, where the line through them is not vertical and has a slope of $L$.
This depends on `W`, and has argument order: $x_1$, $x_2$, $y_1$, $L$. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
base_change_add_Y' :
(W.base_change A).add_Y' (algebra_map R A x₁) (algebra_map R A x₂) (algebra_map R A y₁)
(algebra_map R A L) = algebra_map R A (W.add_Y' x₁ x₂ y₁ L) | by { simp only [add_Y', base_change_add_X], map_simp } | lemma | weierstrass_curve.base_change_add_Y' | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [
"algebra_map",
"map_simp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
base_change_add_Y'_of_base_change (x₁ x₂ y₁ L : A) :
(W.base_change B).add_Y' (algebra_map A B x₁) (algebra_map A B x₂) (algebra_map A B y₁)
(algebra_map A B L) = algebra_map A B ((W.base_change A).add_Y' x₁ x₂ y₁ L) | by rw [← base_change_add_Y', base_change_base_change] | lemma | weierstrass_curve.base_change_add_Y'_of_base_change | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_Y : R | W.neg_Y (W.add_X x₁ x₂ L) (W.add_Y' x₁ x₂ y₁ L) | def | weierstrass_curve.add_Y | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [] | The $Y$-coordinate of the addition of two affine points $(x_1, y_1)$ and $(x_2, y_2)$ in `W`,
where the line through them is not vertical and has a slope of $L$.
This depends on `W`, and has argument order: $x_1$, $x_2$, $y_1$, $L$. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
base_change_add_Y :
(W.base_change A).add_Y (algebra_map R A x₁) (algebra_map R A x₂) (algebra_map R A y₁)
(algebra_map R A L) = algebra_map R A (W.add_Y x₁ x₂ y₁ L) | by simp only [add_Y, base_change_add_Y', base_change_add_X, base_change_neg_Y] | lemma | weierstrass_curve.base_change_add_Y | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
base_change_add_Y_of_base_change (x₁ x₂ y₁ L : A) :
(W.base_change B).add_Y (algebra_map A B x₁) (algebra_map A B x₂) (algebra_map A B y₁)
(algebra_map A B L) = algebra_map A B ((W.base_change A).add_Y x₁ x₂ y₁ L) | by rw [← base_change_add_Y, base_change_base_change] | lemma | weierstrass_curve.base_change_add_Y_of_base_change | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
XY_ideal_add_eq :
XY_ideal W (W.add_X x₁ x₂ L) (C (W.add_Y x₁ x₂ y₁ L))
= span {adjoin_root.mk W.polynomial $ W.neg_polynomial - C (line_polynomial x₁ y₁ L)}
⊔ X_ideal W (W.add_X x₁ x₂ L) | begin
simp only [XY_ideal, X_ideal, X_class, Y_class, add_Y, add_Y', neg_Y, neg_polynomial,
line_polynomial],
conv_rhs { rw [sub_sub, ← neg_add', map_neg, span_singleton_neg, sup_comm, ← span_insert] },
rw [← span_pair_add_mul_right $ adjoin_root.mk _ $ C $ C $ W.a₁ + L, ← _root_.map_mul, ← map_add],... | lemma | weierstrass_curve.XY_ideal_add_eq | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [
"C_simp",
"adjoin_root.mk",
"sup_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equation_add_iff :
W.equation (W.add_X x₁ x₂ L) (W.add_Y' x₁ x₂ y₁ L)
↔ (W.add_polynomial x₁ y₁ L).eval (W.add_X x₁ x₂ L) = 0 | by { rw [equation, add_Y', add_polynomial, line_polynomial, weierstrass_curve.polynomial],
eval_simp } | lemma | weierstrass_curve.equation_add_iff | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [
"eval_simp",
"weierstrass_curve.polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonsingular_add_of_eval_derivative_ne_zero
(hx' : W.equation (W.add_X x₁ x₂ L) (W.add_Y' x₁ x₂ y₁ L))
(hx : (derivative $ W.add_polynomial x₁ y₁ L).eval (W.add_X x₁ x₂ L) ≠ 0) :
W.nonsingular (W.add_X x₁ x₂ L) (W.add_Y' x₁ x₂ y₁ L) | begin
rw [nonsingular, and_iff_right hx', add_Y', polynomial_X, polynomial_Y],
eval_simp,
contrapose! hx,
rw [add_polynomial, line_polynomial, weierstrass_curve.polynomial],
eval_simp,
derivative_simp,
simp only [zero_add, add_zero, sub_zero, zero_mul, mul_one],
eval_simp,
linear_combination hx.left +... | lemma | weierstrass_curve.nonsingular_add_of_eval_derivative_ne_zero | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [
"derivative_simp",
"eval_simp",
"mul_one",
"weierstrass_curve.polynomial",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
point
| zero
| some {x y : R} (h : W.nonsingular x y) | inductive | weierstrass_curve.point | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [] | A nonsingular rational point on a Weierstrass curve `W` over `R`. This is either the point at
infinity `weierstrass_curve.point.zero` or an affine point `weierstrass_curve.point.some` $(x, y)$
satisfying the equation $y^2 + a_1xy + a_3y = x^3 + a_2x^2 + a_4x + a_6$ of `W`. For an algebraic
extension `S` of `R`, the typ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_def : (zero : W.point) = 0 | rfl | lemma | weierstrass_curve.point.zero_def | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equation_neg_iff : W.equation x₁ (W.neg_Y x₁ y₁) ↔ W.equation x₁ y₁ | by { rw [equation_iff, equation_iff, neg_Y], congr' 2, ring1 } | lemma | weierstrass_curve.equation_neg_iff | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equation_neg_of (h : W.equation x₁ $ W.neg_Y x₁ y₁) : W.equation x₁ y₁ | equation_neg_iff.mp h | lemma | weierstrass_curve.equation_neg_of | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equation_neg (h : W.equation x₁ y₁) : W.equation x₁ $ W.neg_Y x₁ y₁ | equation_neg_iff.mpr h | lemma | weierstrass_curve.equation_neg | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [] | The negation of an affine point in `W` lies in `W`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonsingular_neg_iff : W.nonsingular x₁ (W.neg_Y x₁ y₁) ↔ W.nonsingular x₁ y₁ | begin
rw [nonsingular_iff, equation_neg_iff, ← neg_Y, neg_Y_neg_Y, ← @ne_comm _ y₁, nonsingular_iff],
exact and_congr_right' ((iff_congr not_and_distrib.symm not_and_distrib.symm).mpr $
not_iff_not_of_iff $ and_congr_left $ λ h, by rw [← h])
end | lemma | weierstrass_curve.nonsingular_neg_iff | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [
"and_congr_left",
"and_congr_right'",
"ne_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonsingular_neg_of (h : W.nonsingular x₁ $ W.neg_Y x₁ y₁) : W.nonsingular x₁ y₁ | nonsingular_neg_iff.mp h | lemma | weierstrass_curve.nonsingular_neg_of | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonsingular_neg (h : W.nonsingular x₁ y₁) : W.nonsingular x₁ $ W.neg_Y x₁ y₁ | nonsingular_neg_iff.mpr h | lemma | weierstrass_curve.nonsingular_neg | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [] | The negation of a nonsingular affine point in `W` is nonsingular. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
neg : W.point → W.point | | 0 := 0
| (some h) := some $ nonsingular_neg h | def | weierstrass_curve.point.neg | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [] | The negation of a nonsingular rational point.
Given a nonsingular rational point `P`, use `-P` instead of `neg P`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
neg_def (P : W.point) : P.neg = -P | rfl | lemma | weierstrass_curve.point.neg_def | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_zero : (-0 : W.point) = 0 | rfl | lemma | weierstrass_curve.point.neg_zero | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_some (h : W.nonsingular x₁ y₁) : -some h = some (nonsingular_neg h) | rfl | lemma | weierstrass_curve.point.neg_some | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
slope : F | if hx : x₁ = x₂ then if hy : y₁ = W.neg_Y x₂ y₂ then 0
else (3 * x₁ ^ 2 + 2 * W.a₂ * x₁ + W.a₄ - W.a₁ * y₁) / (y₁ - W.neg_Y x₁ y₁)
else (y₁ - y₂) / (x₁ - x₂) | def | weierstrass_curve.slope | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [
"slope"
] | The slope of the line through two affine points $(x_1, y_1)$ and $(x_2, y_2)$ in `W`.
If $x_1 \ne x_2$, then this line is the secant of `W` through $(x_1, y_1)$ and $(x_2, y_2)$,
and has slope $(y_1 - y_2) / (x_1 - x_2)$. Otherwise, if $y_1 \ne -y_1 - a_1x_1 - a_3$,
then this line is the tangent of `W` at $(x_1, y_1) =... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
slope_of_Y_eq (hx : x₁ = x₂) (hy : y₁ = W.neg_Y x₂ y₂) :
W.slope x₁ x₂ y₁ y₂ = 0 | by rw [slope, dif_pos hx, dif_pos hy] | lemma | weierstrass_curve.slope_of_Y_eq | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [
"slope"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
slope_of_Y_ne (hx : x₁ = x₂) (hy : y₁ ≠ W.neg_Y x₂ y₂) :
W.slope x₁ x₂ y₁ y₂ = (3 * x₁ ^ 2 + 2 * W.a₂ * x₁ + W.a₄ - W.a₁ * y₁) / (y₁ - W.neg_Y x₁ y₁) | by rw [slope, dif_pos hx, dif_neg hy] | lemma | weierstrass_curve.slope_of_Y_ne | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [
"slope"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
slope_of_X_ne (hx : x₁ ≠ x₂) : W.slope x₁ x₂ y₁ y₂ = (y₁ - y₂) / (x₁ - x₂) | by rw [slope, dif_neg hx] | lemma | weierstrass_curve.slope_of_X_ne | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [
"slope"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
slope_of_Y_ne_eq_eval (hx : x₁ = x₂) (hy : y₁ ≠ W.neg_Y x₂ y₂) :
W.slope x₁ x₂ y₁ y₂
= -(W.polynomial_X.eval $ C y₁).eval x₁ / (W.polynomial_Y.eval $ C y₁).eval x₁ | by { rw [slope_of_Y_ne hx hy, eval_polynomial_X, neg_sub], congr' 1, rw [neg_Y, eval_polynomial_Y],
ring1 } | lemma | weierstrass_curve.slope_of_Y_ne_eq_eval | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
base_change_slope :
(W.base_change K).slope (algebra_map F K x₁) (algebra_map F K x₂) (algebra_map F K y₁)
(algebra_map F K y₂) = algebra_map F K (W.slope x₁ x₂ y₁ y₂) | begin
by_cases hx : x₁ = x₂,
{ by_cases hy : y₁ = W.neg_Y x₂ y₂,
{ rw [slope_of_Y_eq hx hy, slope_of_Y_eq $ congr_arg _ hx, map_zero],
{ rw [hy, base_change_neg_Y] } },
{ rw [slope_of_Y_ne hx hy, slope_of_Y_ne $ congr_arg _ hx],
{ map_simp,
simpa only [base_change_neg_Y] },
{ rw [b... | lemma | weierstrass_curve.base_change_slope | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [
"algebra_map",
"map_simp",
"no_zero_smul_divisors.algebra_map_injective",
"slope"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
base_change_slope_of_base_change {R : Type u} [comm_ring R] (W : weierstrass_curve R)
(F : Type v) [field F] [algebra R F] (K : Type w) [field K] [algebra R K] [algebra F K]
[is_scalar_tower R F K] (x₁ x₂ y₁ y₂ : F) :
(W.base_change K).slope (algebra_map F K x₁) (algebra_map F K x₂) (algebra_map F K y₁)
(alge... | by rw [← base_change_slope, base_change_base_change] | lemma | weierstrass_curve.base_change_slope_of_base_change | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [
"algebra",
"algebra_map",
"comm_ring",
"field",
"is_scalar_tower",
"slope",
"weierstrass_curve"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Y_eq_of_X_eq (hx : x₁ = x₂) : y₁ = y₂ ∨ y₁ = W.neg_Y x₂ y₂ | begin
rw [equation_iff] at h₁' h₂',
rw [← sub_eq_zero, ← @sub_eq_zero _ _ y₁, ← mul_eq_zero, neg_Y],
linear_combination h₁' - h₂' with { normalization_tactic := `[rw [hx], ring1] }
end | lemma | weierstrass_curve.Y_eq_of_X_eq | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [
"mul_eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Y_eq_of_Y_ne (hx : x₁ = x₂) (hy : y₁ ≠ W.neg_Y x₂ y₂) : y₁ = y₂ | or.resolve_right (Y_eq_of_X_eq h₁' h₂' hx) hy | lemma | weierstrass_curve.Y_eq_of_Y_ne | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
XY_ideal_eq₂ (hxy : x₁ = x₂ → y₁ ≠ W.neg_Y x₂ y₂) :
XY_ideal W x₂ (C y₂) = XY_ideal W x₂ (line_polynomial x₁ y₁ $ W.slope x₁ x₂ y₁ y₂) | begin
have hy₂ : y₂ = (line_polynomial x₁ y₁ $ W.slope x₁ x₂ y₁ y₂).eval x₂ :=
begin
by_cases hx : x₁ = x₂,
{ rcases ⟨hx, Y_eq_of_Y_ne h₁' h₂' hx $ hxy hx⟩ with ⟨rfl, rfl⟩,
field_simp [line_polynomial, sub_ne_zero_of_ne (hxy rfl)] },
{ field_simp [line_polynomial, slope_of_X_ne hx, sub_ne_zero_of_... | lemma | weierstrass_curve.XY_ideal_eq₂ | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [
"C_simp",
"adjoin_root.mk",
"eval_simp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_polynomial_slope (hxy : x₁ = x₂ → y₁ ≠ W.neg_Y x₂ y₂) :
W.add_polynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂)
= -((X - C x₁) * (X - C x₂) * (X - C (W.add_X x₁ x₂ $ W.slope x₁ x₂ y₁ y₂))) | begin
rw [add_polynomial_eq, neg_inj, cubic.prod_X_sub_C_eq, cubic.to_poly_injective],
by_cases hx : x₁ = x₂,
{ rcases ⟨hx, Y_eq_of_Y_ne h₁' h₂' hx (hxy hx)⟩ with ⟨rfl, rfl⟩,
rw [equation_iff] at h₁' h₂',
rw [slope_of_Y_ne rfl $ hxy rfl],
rw [neg_Y, ← sub_ne_zero] at hxy,
ext,
{ refl },
{ ... | lemma | weierstrass_curve.add_polynomial_slope | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [
"cubic.prod_X_sub_C_eq",
"cubic.to_poly_injective",
"mul_right_injective₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coordinate_ring.C_add_polynomial_slope (hxy : x₁ = x₂ → y₁ ≠ W.neg_Y x₂ y₂) :
adjoin_root.mk W.polynomial (C $ W.add_polynomial x₁ y₁ $ W.slope x₁ x₂ y₁ y₂)
= -(X_class W x₁ * X_class W x₂ * X_class W (W.add_X x₁ x₂ $ W.slope x₁ x₂ y₁ y₂)) | by simpa only [add_polynomial_slope h₁' h₂' hxy, map_neg, neg_inj, _root_.map_mul] | lemma | weierstrass_curve.coordinate_ring.C_add_polynomial_slope | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [
"adjoin_root.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
derivative_add_polynomial_slope (hxy : x₁ = x₂ → y₁ ≠ W.neg_Y x₂ y₂) :
derivative (W.add_polynomial x₁ y₁ $ W.slope x₁ x₂ y₁ y₂)
= -((X - C x₁) * (X - C x₂) + (X - C x₁) * (X - C (W.add_X x₁ x₂ $ W.slope x₁ x₂ y₁ y₂))
+ (X - C x₂) * (X - C (W.add_X x₁ x₂ $ W.slope x₁ x₂ y₁ y₂))) | by { rw [add_polynomial_slope h₁' h₂' hxy], derivative_simp, ring1 } | lemma | weierstrass_curve.derivative_add_polynomial_slope | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [
"derivative_simp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equation_add' (hxy : x₁ = x₂ → y₁ ≠ W.neg_Y x₂ y₂) :
W.equation (W.add_X x₁ x₂ $ W.slope x₁ x₂ y₁ y₂) (W.add_Y' x₁ x₂ y₁ $ W.slope x₁ x₂ y₁ y₂) | by { rw [equation_add_iff, add_polynomial_slope h₁' h₂' hxy], eval_simp,
rw [neg_eq_zero, sub_self, mul_zero] } | lemma | weierstrass_curve.equation_add' | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [
"eval_simp",
"mul_zero"
] | The addition of two affine points in `W` on a sloped line,
before applying the final negation that maps $Y$ to $-Y - a_1X - a_3$, lies in `W`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equation_add (hxy : x₁ = x₂ → y₁ ≠ W.neg_Y x₂ y₂) :
W.equation (W.add_X x₁ x₂ $ W.slope x₁ x₂ y₁ y₂) (W.add_Y x₁ x₂ y₁ $ W.slope x₁ x₂ y₁ y₂) | equation_neg $ equation_add' h₁' h₂' hxy | lemma | weierstrass_curve.equation_add | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [] | The addition of two affine points in `W` on a sloped line lies in `W`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonsingular_add' (hxy : x₁ = x₂ → y₁ ≠ W.neg_Y x₂ y₂) :
W.nonsingular (W.add_X x₁ x₂ $ W.slope x₁ x₂ y₁ y₂) (W.add_Y' x₁ x₂ y₁ $ W.slope x₁ x₂ y₁ y₂) | begin
by_cases hx₁ : W.add_X x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₁,
{ rwa [add_Y', hx₁, sub_self, mul_zero, zero_add] },
{ by_cases hx₂ : W.add_X x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₂,
{ by_cases hx : x₁ = x₂,
{ subst hx,
contradiction },
{ rwa [add_Y', ← neg_sub, mul_neg, hx₂, slope_of_X_ne hx,
... | lemma | weierstrass_curve.nonsingular_add' | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [
"div_mul_cancel",
"eval_simp",
"mul_ne_zero",
"mul_neg",
"mul_zero"
] | The addition of two nonsingular affine points in `W` on a sloped line,
before applying the final negation that maps $Y$ to $-Y - a_1X - a_3$, is nonsingular. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonsingular_add (hxy : x₁ = x₂ → y₁ ≠ W.neg_Y x₂ y₂) :
W.nonsingular (W.add_X x₁ x₂ $ W.slope x₁ x₂ y₁ y₂) (W.add_Y x₁ x₂ y₁ $ W.slope x₁ x₂ y₁ y₂) | nonsingular_neg $ nonsingular_add' h₁ h₂ hxy | lemma | weierstrass_curve.nonsingular_add | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [] | The addition of two nonsingular affine points in `W` on a sloped line is nonsingular. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add : W.point → W.point → W.point | | 0 P := P
| P 0 := P
| (@some _ _ _ x₁ y₁ h₁) (@some _ _ _ x₂ y₂ h₂) :=
if hx : x₁ = x₂ then if hy : y₁ = W.neg_Y x₂ y₂ then 0
else some $ nonsingular_add h₁ h₂ $ λ _, hy
else some $ nonsingular_add h₁ h₂ $ λ h, (hx h).elim | def | weierstrass_curve.point.add | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [] | The addition of two nonsingular rational points.
Given two nonsingular rational points `P` and `Q`, use `P + Q` instead of `add P Q`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_def (P Q : W.point) : P.add Q = P + Q | rfl | lemma | weierstrass_curve.point.add_def | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
some_add_some_of_Y_eq (hx : x₁ = x₂) (hy : y₁ = W.neg_Y x₂ y₂) :
some h₁ + some h₂ = 0 | by rw [← add_def, add, dif_pos hx, dif_pos hy] | lemma | weierstrass_curve.point.some_add_some_of_Y_eq | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
some_add_self_of_Y_eq (hy : y₁ = W.neg_Y x₁ y₁) : some h₁ + some h₁ = 0 | some_add_some_of_Y_eq rfl hy | lemma | weierstrass_curve.point.some_add_self_of_Y_eq | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
some_add_some_of_Y_ne (hx : x₁ = x₂) (hy : y₁ ≠ W.neg_Y x₂ y₂) :
some h₁ + some h₂ = some (nonsingular_add h₁ h₂ $ λ _, hy) | by rw [← add_def, add, dif_pos hx, dif_neg hy] | lemma | weierstrass_curve.point.some_add_some_of_Y_ne | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
some_add_some_of_Y_ne' (hx : x₁ = x₂) (hy : y₁ ≠ W.neg_Y x₂ y₂) :
some h₁ + some h₂ = -some (nonsingular_add' h₁ h₂ $ λ _, hy) | some_add_some_of_Y_ne hx hy | lemma | weierstrass_curve.point.some_add_some_of_Y_ne' | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
some_add_self_of_Y_ne (hy : y₁ ≠ W.neg_Y x₁ y₁) :
some h₁ + some h₁ = some (nonsingular_add h₁ h₁ $ λ _, hy) | some_add_some_of_Y_ne rfl hy | lemma | weierstrass_curve.point.some_add_self_of_Y_ne | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
some_add_self_of_Y_ne' (hy : y₁ ≠ W.neg_Y x₁ y₁) :
some h₁ + some h₁ = -some (nonsingular_add' h₁ h₁ $ λ _, hy) | some_add_some_of_Y_ne rfl hy | lemma | weierstrass_curve.point.some_add_self_of_Y_ne' | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
some_add_some_of_X_ne (hx : x₁ ≠ x₂) :
some h₁ + some h₂ = some (nonsingular_add h₁ h₂ $ λ h, (hx h).elim) | by rw [← add_def, add, dif_neg hx] | lemma | weierstrass_curve.point.some_add_some_of_X_ne | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
some_add_some_of_X_ne' (hx : x₁ ≠ x₂) :
some h₁ + some h₂ = -some (nonsingular_add' h₁ h₂ $ λ h, (hx h).elim) | some_add_some_of_X_ne hx | lemma | weierstrass_curve.point.some_add_some_of_X_ne' | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
XY_ideal_neg_mul : XY_ideal W x₁ (C $ W.neg_Y x₁ y₁) * XY_ideal W x₁ (C y₁) = X_ideal W x₁ | begin
have Y_rw :
(Y - C (C y₁)) * (Y - C (C (W.neg_Y x₁ y₁))) - C (X - C x₁)
* (C (X ^ 2 + C (x₁ + W.a₂) * X + C (x₁ ^ 2 + W.a₂ * x₁ + W.a₄)) - C (C W.a₁) * Y)
= W.polynomial * 1 :=
by linear_combination congr_arg C (congr_arg C ((W.equation_iff _ _).mp h₁.left).symm)
with { normalization_tacti... | lemma | weierstrass_curve.XY_ideal_neg_mul | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [
"C_simp",
"adjoin_root.mk",
"mul_assoc",
"mul_comm",
"mul_inv_cancel",
"mul_right_inj'",
"mul_sup",
"set.image_insert_eq",
"set.image_singleton",
"weierstrass_curve.polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
XY_ideal'_mul_inv :
(XY_ideal W x₁ (C y₁) : fractional_ideal W.coordinate_ring⁰ W.function_field)
* (XY_ideal W x₁ (C $ W.neg_Y x₁ y₁) * (X_ideal W x₁)⁻¹) = 1 | by rw [← mul_assoc, ← fractional_ideal.coe_ideal_mul, mul_comm $ XY_ideal W _ _,
XY_ideal_neg_mul h₁, X_ideal,
fractional_ideal.coe_ideal_span_singleton_mul_inv W.function_field $ X_class_ne_zero W x₁] | lemma | weierstrass_curve.XY_ideal'_mul_inv | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [
"fractional_ideal",
"fractional_ideal.coe_ideal_mul",
"fractional_ideal.coe_ideal_span_singleton_mul_inv",
"mul_assoc",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
XY_ideal_mul_XY_ideal (hxy : x₁ = x₂ → y₁ ≠ W.neg_Y x₂ y₂) :
X_ideal W (W.add_X x₁ x₂ $ W.slope x₁ x₂ y₁ y₂) * (XY_ideal W x₁ (C y₁) * XY_ideal W x₂ (C y₂))
= Y_ideal W (line_polynomial x₁ y₁ $ W.slope x₁ x₂ y₁ y₂)
* XY_ideal W (W.add_X x₁ x₂ $ W.slope x₁ x₂ y₁ y₂)
(C $ W.add_Y x₁ x₂ y₁ $ W.slope x₁... | begin
have sup_rw : ∀ a b c d : ideal W.coordinate_ring, a ⊔ (b ⊔ (c ⊔ d)) = a ⊔ d ⊔ b ⊔ c :=
λ _ _ c _, by rw [← sup_assoc, @sup_comm _ _ c, sup_sup_sup_comm, ← sup_assoc],
rw [XY_ideal_add_eq, X_ideal, mul_comm, W.XY_ideal_eq₁ x₁ y₁ $ W.slope x₁ x₂ y₁ y₂, XY_ideal,
XY_ideal_eq₂ h₁' h₂' hxy, XY_ideal, span... | lemma | weierstrass_curve.XY_ideal_mul_XY_ideal | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [
"C_simp",
"adjoin_root.mk",
"adjoin_root.mk_self",
"adjoin_root.mk_surjective",
"ideal",
"ideal.map_sup",
"map_one",
"mul_assoc",
"mul_comm",
"mul_inv_cancel",
"mul_neg",
"mul_right_inj'",
"mul_sup",
"mul_zero",
"pow_ne_zero",
"set.image_singleton",
"sup_assoc",
"sup_comm",
"sup_... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
XY_ideal' : (fractional_ideal W.coordinate_ring⁰ W.function_field)ˣ | units.mk_of_mul_eq_one _ _ $ XY_ideal'_mul_inv h₁ | def | weierstrass_curve.XY_ideal' | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [
"fractional_ideal",
"units.mk_of_mul_eq_one"
] | The non-zero fractional ideal $\langle X - x, Y - y \rangle$ of $F(W)$ for some $x, y \in F$. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
XY_ideal'_eq :
(XY_ideal' h₁ : fractional_ideal W.coordinate_ring⁰ W.function_field) = XY_ideal W x₁ (C y₁) | rfl | lemma | weierstrass_curve.XY_ideal'_eq | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [
"fractional_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_XY_ideal'_mul_mk_XY_ideal'_of_Y_eq :
class_group.mk (XY_ideal' $ nonsingular_neg h₁) * class_group.mk (XY_ideal' h₁) = 1 | begin
rw [← _root_.map_mul],
exact (class_group.mk_eq_one_of_coe_ideal $
by exact (fractional_ideal.coe_ideal_mul _ _).symm.trans
(fractional_ideal.coe_ideal_inj.mpr $ XY_ideal_neg_mul h₁)).mpr
⟨_, X_class_ne_zero W _, rfl⟩
end | lemma | weierstrass_curve.mk_XY_ideal'_mul_mk_XY_ideal'_of_Y_eq | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [
"class_group.mk",
"class_group.mk_eq_one_of_coe_ideal",
"fractional_ideal.coe_ideal_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_XY_ideal'_mul_mk_XY_ideal' (hxy : x₁ = x₂ → y₁ ≠ W.neg_Y x₂ y₂) :
class_group.mk (XY_ideal' h₁) * class_group.mk (XY_ideal' h₂)
= class_group.mk (XY_ideal' $ nonsingular_add h₁ h₂ hxy) | begin
rw [← _root_.map_mul],
exact (class_group.mk_eq_mk_of_coe_ideal (by exact (fractional_ideal.coe_ideal_mul _ _).symm) $
XY_ideal'_eq _).mpr ⟨_, _, X_class_ne_zero W _, Y_class_ne_zero W _,
XY_ideal_mul_XY_ideal h₁.left h₂.left hxy⟩
end | lemma | weierstrass_curve.mk_XY_ideal'_mul_mk_XY_ideal' | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [
"class_group.mk",
"class_group.mk_eq_mk_of_coe_ideal",
"fractional_ideal.coe_ideal_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_class_fun : W.point → additive (class_group W.coordinate_ring) | | 0 := 0
| (some h) := additive.of_mul $ class_group.mk $ XY_ideal' h | def | weierstrass_curve.point.to_class_fun | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [
"additive",
"additive.of_mul",
"class_group",
"class_group.mk"
] | The set function mapping an affine point $(x, y)$ of `W` to the class of the non-zero fractional
ideal $\langle X - x, Y - y \rangle$ of $F(W)$ in the class group of $F[W]$. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_class : W.point →+ additive (class_group W.coordinate_ring) | { to_fun := to_class_fun,
map_zero' := rfl,
map_add' :=
begin
rintro (_ | @⟨x₁, y₁, h₁⟩) (_ | @⟨x₂, y₂, h₂⟩),
any_goals { simp only [zero_def, to_class_fun, _root_.zero_add, _root_.add_zero] },
by_cases hx : x₁ = x₂,
{ by_cases hy : y₁ = W.neg_Y x₂ y₂,
{ substs hx hy,
simpa only ... | def | weierstrass_curve.point.to_class | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [
"additive",
"class_group"
] | The group homomorphism mapping an affine point $(x, y)$ of `W` to the class of the non-zero
fractional ideal $\langle X - x, Y - y \rangle$ of $F(W)$ in the class group of $F[W]$. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_class_zero : to_class (0 : W.point) = 0 | rfl | lemma | weierstrass_curve.point.to_class_zero | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_class_some : to_class (some h₁) = class_group.mk (XY_ideal' h₁) | rfl | lemma | weierstrass_curve.point.to_class_some | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [
"class_group.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_eq_zero (P Q : W.point) : P + Q = 0 ↔ P = -Q | begin
rcases ⟨P, Q⟩ with ⟨_ | @⟨x₁, y₁, _⟩, _ | @⟨x₂, y₂, _⟩⟩,
any_goals { refl },
{ rw [zero_def, zero_add, ← neg_eq_iff_eq_neg, neg_zero, eq_comm] },
{ simp only [neg_some],
split,
{ intro h,
by_cases hx : x₁ = x₂,
{ by_cases hy : y₁ = W.neg_Y x₂ y₂,
{ exact ⟨hx, hy⟩ },
{ r... | lemma | weierstrass_curve.point.add_eq_zero | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_left_neg (P : W.point) : -P + P = 0 | by rw [add_eq_zero] | lemma | weierstrass_curve.point.add_left_neg | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_add_eq_zero (P Q : W.point) : -P + Q = 0 ↔ P = Q | by rw [add_eq_zero, neg_inj] | lemma | weierstrass_curve.point.neg_add_eq_zero | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_class_eq_zero (P : W.point) : to_class P = 0 ↔ P = 0 | ⟨begin
intro hP,
rcases P with (_ | @⟨_, _, ⟨h, _⟩⟩),
{ refl },
{ rcases (class_group.mk_eq_one_of_coe_ideal $ by refl).mp hP with ⟨p, h0, hp⟩,
apply (p.nat_degree_norm_ne_one _).elim,
rw [← finrank_quotient_span_eq_nat_degree_norm W^.coordinate_ring.basis h0,
← (quotient_equiv_alg_of_eq F hp).t... | lemma | weierstrass_curve.point.to_class_eq_zero | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [
"class_group.mk_eq_one_of_coe_ideal",
"finite_dimensional.finrank_self",
"finrank_quotient_span_eq_nat_degree_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_class_injective : function.injective $ @to_class _ _ W | begin
rintro (_ | h) _ hP,
all_goals { rw [← neg_add_eq_zero, ← to_class_eq_zero, map_add, ← hP] },
{ exact zero_add 0 },
{ exact mk_XY_ideal'_mul_mk_XY_ideal'_of_Y_eq h }
end | lemma | weierstrass_curve.point.to_class_injective | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_comm (P Q : W.point) : P + Q = Q + P | to_class_injective $ by simp only [map_add, add_comm] | lemma | weierstrass_curve.point.add_comm | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_assoc (P Q R : W.point) : P + Q + R = P + (Q + R) | to_class_injective $ by simp only [map_add, add_assoc] | lemma | weierstrass_curve.point.add_assoc | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_base_change_fun : W⟮F⟯ → W⟮K⟯ | | 0 := 0
| (some h) := some $ (nonsingular_iff_base_change_of_base_change W F K _ _).mp h | def | weierstrass_curve.point.of_base_change_fun | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [] | The function from `W⟮F⟯` to `W⟮K⟯` induced by a base change from `F` to `K`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_base_change : W⟮F⟯ →+ W⟮K⟯ | { to_fun := of_base_change_fun W F K,
map_zero' := rfl,
map_add' :=
begin
rintro (_ | @⟨x₁, y₁, _⟩) (_ | @⟨x₂, y₂, _⟩),
any_goals { refl },
by_cases hx : x₁ = x₂,
{ by_cases hy : y₁ = (W.base_change F).neg_Y x₂ y₂,
{ simp only [some_add_some_of_Y_eq hx hy, of_base_change_fun],
rw... | def | weierstrass_curve.point.of_base_change | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [
"no_zero_smul_divisors.algebra_map_injective"
] | The group homomorphism from `W⟮F⟯` to `W⟮K⟯` induced by a base change from `F` to `K`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_base_change_injective : function.injective $ of_base_change W F K | begin
rintro (_ | _) (_ | _) h,
{ refl },
any_goals { contradiction },
simp only,
exact ⟨no_zero_smul_divisors.algebra_map_injective F K (some.inj h).left,
no_zero_smul_divisors.algebra_map_injective F K (some.inj h).right⟩
end | lemma | weierstrass_curve.point.of_base_change_injective | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [
"no_zero_smul_divisors.algebra_map_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk {x y : R} (h : E.equation x y) : E.point | weierstrass_curve.point.some $ E.nonsingular h | def | elliptic_curve.point.mk | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/point.lean | [
"algebraic_geometry.elliptic_curve.weierstrass",
"linear_algebra.free_module.norm",
"ring_theory.class_group"
] | [] | An affine point on an elliptic curve `E` over `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_simp : tactic unit | `[simp only [map_one, map_bit0, map_bit1, map_neg, map_add, map_sub, map_mul, map_pow]] | def | map_simp | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [
"map_bit0",
"map_bit1",
"map_mul",
"map_one",
"map_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eval_simp : tactic unit | `[simp only [eval_C, eval_X, eval_add, eval_sub, eval_mul, eval_pow]] | def | eval_simp | algebraic_geometry.elliptic_curve | src/algebraic_geometry/elliptic_curve/weierstrass.lean | [
"algebra.cubic_discriminant",
"ring_theory.norm",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.