Projective coordinates for Weierstrass curves

This file defines the type of points on a Weierstrass curve as a tuple, consisting of an equivalence class of triples up to scaling by a unit, satisfying a Weierstrass equation with a nonsingular condition.

Mathematical background

Let W be a Weierstrass curve over a field F. A point on the projective plane is an equivalence class of triples $[x:y:z]$ with coordinates in F such that $(x,y,z)\sim (x^{\prime },y^{\prime },z^{\prime })$ precisely if there is some unit u of F such that $(x,y,z)=(u{x}^{\prime },u{y}^{\prime },u{z}^{\prime })$, with an extra condition that $(x,y,z)\ne (0,0,0)$. As described in Mathlib.AlgebraicGeometry.EllipticCurve.Affine, a rational point is a point on the projective plane satisfying a homogeneous Weierstrass equation, and being nonsingular means the partial derivatives $W_X(X,Y,Z)$, $W_Y(X,Y,Z)$, and $W_Z(X,Y,Z)$ do not vanish simultaneously. Note that the vanishing of the Weierstrass equation and its partial derivatives are independent of the representative for $[x:y:z]$, and the nonsingularity condition already implies that $(x,y,z)\ne (0,0,0)$, so a nonsingular rational point on W can simply be given by a tuple consisting of $[x:y:z]$ and the nonsingular condition on any representative.

As in Mathlib.AlgebraicGeometry.EllipticCurve.Affine, the set of nonsingular rational points forms an abelian group under the same secant-and-tangent process, but the polynomials involved are homogeneous, and any instances of division become multiplication in the $Z$-coordinate.

Main definitions

• WeierstrassCurve.Projective.PointClass: the equivalence class of a point representative.
• WeierstrassCurve.Projective.toAffine: the Weierstrass curve in affine coordinates.
• WeierstrassCurve.Projective.Nonsingular: the nonsingular condition on a point representative.
• WeierstrassCurve.Projective.NonsingularLift: the nonsingular condition on a point class.

Main statements

• WeierstrassCurve.Projective.polynomial_relation: Euler's homogeneous function theorem.

Implementation notes

A point representative is implemented as a term P of type Fin 3 → R, which allows for the vector notation ![x, y, z]. However, P is not definitionally equivalent to the expanded vector ![P x, P y, P z], so the lemmas fin3_def and fin3_def_ext can be used to convert between the two forms. The equivalence of two point representatives P and Q is implemented as an equivalence of orbits of the action of Rˣ, or equivalently that there is some unit u of R such that P = u • Q. However, u • Q is not definitionally equal to ![u * Q x, u * Q y, u * Q z], so the lemmas smul_fin3 and smul_fin3_ext can be used to convert between the two forms.

References

[J Silverman, The Arithmetic of Elliptic Curves][silverman2009]

Tags

elliptic curve, rational point, projective coordinates

Weierstrass curves

@[reducible, inline]

abbrev WeierstrassCurve.Projective (R : Type u) : Type u

An abbreviation for a Weierstrass curve in projective coordinates.

Equations

• WeierstrassCurve.Projective R = WeierstrassCurve R

@[reducible, inline]

abbrev WeierstrassCurve.toProjective {R : Type u} (W : WeierstrassCurve R) : WeierstrassCurve.Projective R

The coercion to a Weierstrass curve in projective coordinates.

Equations

• W.toProjective = W

Projective coordinates

theorem WeierstrassCurve.Projective.fin3_def {R : Type u} (P : Fin 3 → R) : ![P 0, P 1, P 2] = P

theorem WeierstrassCurve.Projective.fin3_def_ext {R : Type u} (X : R) (Y : R) (Z : R) : ![X, Y, Z] 0 = X ∧ ![X, Y, Z] 1 = Y ∧ ![X, Y, Z] 2 = Z

theorem WeierstrassCurve.Projective.comp_fin3 {R : Type u} {S : Type v} (f : R → S) (X : R) (Y : R) (Z : R) : f ∘ ![X, Y, Z] = ![f X, f Y, f Z]

theorem WeierstrassCurve.Projective.smul_fin3 {R : Type u} [CommRing R] (P : Fin 3 → R) (u : R) : u • P = ![u * P 0, u * P 1, u * P 2]

theorem WeierstrassCurve.Projective.smul_fin3_ext {R : Type u} [CommRing R] (P : Fin 3 → R) (u : R) : (u • P) 0 = u * P 0 ∧ (u • P) 1 = u * P 1 ∧ (u • P) 2 = u * P 2

def WeierstrassCurve.Projective.instSetoidPoint {R : Type u} [CommRing R] : Setoid (Fin 3 → R)

The equivalence setoid for a point representative.

Equations

• WeierstrassCurve.Projective.instSetoidPoint = MulAction.orbitRel Rˣ (Fin 3 → R)

@[reducible, inline]

abbrev WeierstrassCurve.Projective.PointClass (R : Type u) [CommRing R] : Type u

The equivalence class of a point representative.

Equations

• WeierstrassCurve.Projective.PointClass R = MulAction.orbitRel.Quotient Rˣ (Fin 3 → R)

theorem WeierstrassCurve.Projective.smul_equiv {R : Type u} [CommRing R] (P : Fin 3 → R) {u : R} (hu : IsUnit u) : u • P ≈ P

@[simp]

theorem WeierstrassCurve.Projective.smul_eq {R : Type u} [CommRing R] (P : Fin 3 → R) {u : R} (hu : IsUnit u) : ⟦u • P⟧ = ⟦P⟧

@[reducible, inline]

abbrev WeierstrassCurve.Projective.toAffine {R : Type u} (W' : WeierstrassCurve.Projective R) : WeierstrassCurve.Affine R

The coercion to a Weierstrass curve in affine coordinates.

Equations

• W'.toAffine = W'

theorem WeierstrassCurve.Projective.equiv_iff_eq_of_Z_eq' {R : Type u} [CommRing R] {P : Fin 3 → R} {Q : Fin 3 → R} (hz : P 2 = Q 2) (mem : Q 2 ∈ nonZeroDivisors R) : P ≈ Q ↔ P = Q

theorem WeierstrassCurve.Projective.equiv_iff_eq_of_Z_eq {R : Type u} [CommRing R] [NoZeroDivisors R] {P : Fin 3 → R} {Q : Fin 3 → R} (hz : P 2 = Q 2) (hQz : Q 2 ≠ 0) : P ≈ Q ↔ P = Q

theorem WeierstrassCurve.Projective.Z_eq_zero_of_equiv {R : Type u} [CommRing R] {P : Fin 3 → R} {Q : Fin 3 → R} (h : P ≈ Q) : P 2 = 0 ↔ Q 2 = 0

theorem WeierstrassCurve.Projective.X_eq_of_equiv {R : Type u} [CommRing R] {P : Fin 3 → R} {Q : Fin 3 → R} (h : P ≈ Q) : P 0 * Q 2 = Q 0 * P 2

theorem WeierstrassCurve.Projective.Y_eq_of_equiv {R : Type u} [CommRing R] {P : Fin 3 → R} {Q : Fin 3 → R} (h : P ≈ Q) : P 1 * Q 2 = Q 1 * P 2

theorem WeierstrassCurve.Projective.not_equiv_of_Z_eq_zero_left {R : Type u} [CommRing R] {P : Fin 3 → R} {Q : Fin 3 → R} (hPz : P 2 = 0) (hQz : Q 2 ≠ 0) : ¬P ≈ Q

theorem WeierstrassCurve.Projective.not_equiv_of_Z_eq_zero_right {R : Type u} [CommRing R] {P : Fin 3 → R} {Q : Fin 3 → R} (hPz : P 2 ≠ 0) (hQz : Q 2 = 0) : ¬P ≈ Q

theorem WeierstrassCurve.Projective.not_equiv_of_X_ne {R : Type u} [CommRing R] {P : Fin 3 → R} {Q : Fin 3 → R} (hx : P 0 * Q 2 ≠ Q 0 * P 2) : ¬P ≈ Q

theorem WeierstrassCurve.Projective.not_equiv_of_Y_ne {R : Type u} [CommRing R] {P : Fin 3 → R} {Q : Fin 3 → R} (hy : P 1 * Q 2 ≠ Q 1 * P 2) : ¬P ≈ Q

theorem WeierstrassCurve.Projective.equiv_of_X_eq_of_Y_eq {F : Type v} [Field F] {P : Fin 3 → F} {Q : Fin 3 → F} (hPz : P 2 ≠ 0) (hQz : Q 2 ≠ 0) (hx : P 0 * Q 2 = Q 0 * P 2) (hy : P 1 * Q 2 = Q 1 * P 2) : P ≈ Q

theorem WeierstrassCurve.Projective.equiv_some_of_Z_ne_zero {F : Type v} [Field F] {P : Fin 3 → F} (hPz : P 2 ≠ 0) : P ≈ ![P 0 / P 2, P 1 / P 2, 1]

theorem WeierstrassCurve.Projective.X_eq_iff {F : Type v} [Field F] {P : Fin 3 → F} {Q : Fin 3 → F} (hPz : P 2 ≠ 0) (hQz : Q 2 ≠ 0) : P 0 * Q 2 = Q 0 * P 2 ↔ P 0 / P 2 = Q 0 / Q 2

theorem WeierstrassCurve.Projective.Y_eq_iff {F : Type v} [Field F] {P : Fin 3 → F} {Q : Fin 3 → F} (hPz : P 2 ≠ 0) (hQz : Q 2 ≠ 0) : P 1 * Q 2 = Q 1 * P 2 ↔ P 1 / P 2 = Q 1 / Q 2

Weierstrass equations

noncomputable def WeierstrassCurve.Projective.polynomial {R : Type u} (W' : WeierstrassCurve.Projective R) [CommRing R] : MvPolynomial (Fin 3) R

The polynomial $W(X,Y,Z):=Y^{2}Z+a_{1}XYZ+a_{3}Y{Z}^2-(X^3+a_2{X}^{2}Z+a_{4}X{Z}^2+a_6{Z}^3)$ associated to a Weierstrass curve W' over R. This is represented as a term of type MvPolynomial (Fin 3) R, where X 0, X 1, and X 2 represent $X$, $Y$, and $Z$ respectively.

Equations

• One or more equations did not get rendered due to their size.

theorem WeierstrassCurve.Projective.eval_polynomial {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] (P : Fin 3 → R) : (MvPolynomial.eval P) W'.polynomial = P 1 ^ 2 * P 2 + W'.a₁ * P 0 * P 1 * P 2 + W'.a₃ * P 1 * P 2 ^ 2 - (P 0 ^ 3 + W'.a₂ * P 0 ^ 2 * P 2 + W'.a₄ * P 0 * P 2 ^ 2 + W'.a₆ * P 2 ^ 3)

theorem WeierstrassCurve.Projective.eval_polynomial_of_Z_ne_zero {F : Type v} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} (hPz : P 2 ≠ 0) : (MvPolynomial.eval P) W.polynomial / P 2 ^ 3 = Polynomial.evalEval (P 0 / P 2) (P 1 / P 2) W.toAffine.polynomial

def WeierstrassCurve.Projective.Equation {R : Type u} (W' : WeierstrassCurve.Projective R) [CommRing R] (P : Fin 3 → R) : Prop

The proposition that a point representative $(x,y,z)$ lies in W'. In other words, $W(x,y,z)=0$.

Equations

• W'.Equation P = ((MvPolynomial.eval P) W'.polynomial = 0)

theorem WeierstrassCurve.Projective.equation_iff {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] (P : Fin 3 → R) : W'.Equation P ↔ P 1 ^ 2 * P 2 + W'.a₁ * P 0 * P 1 * P 2 + W'.a₃ * P 1 * P 2 ^ 2 - (P 0 ^ 3 + W'.a₂ * P 0 ^ 2 * P 2 + W'.a₄ * P 0 * P 2 ^ 2 + W'.a₆ * P 2 ^ 3) = 0

theorem WeierstrassCurve.Projective.equation_smul {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] (P : Fin 3 → R) {u : R} (hu : IsUnit u) : W'.Equation (u • P) ↔ W'.Equation P

theorem WeierstrassCurve.Projective.equation_of_equiv {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] {P : Fin 3 → R} {Q : Fin 3 → R} (h : P ≈ Q) : W'.Equation P ↔ W'.Equation Q

theorem WeierstrassCurve.Projective.equation_of_Z_eq_zero {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] {P : Fin 3 → R} (hPz : P 2 = 0) : W'.Equation P ↔ P 0 ^ 3 = 0

theorem WeierstrassCurve.Projective.equation_zero {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] : W'.Equation ![0, 1, 0]

theorem WeierstrassCurve.Projective.equation_some {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] (X : R) (Y : R) : W'.Equation ![X, Y, 1] ↔ W'.toAffine.Equation X Y

theorem WeierstrassCurve.Projective.equation_of_Z_ne_zero {F : Type v} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} (hPz : P 2 ≠ 0) : W.Equation P ↔ W.toAffine.Equation (P 0 / P 2) (P 1 / P 2)

theorem WeierstrassCurve.Projective.X_eq_zero_of_Z_eq_zero {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] [NoZeroDivisors R] {P : Fin 3 → R} (hP : W'.Equation P) (hPz : P 2 = 0) : P 0 = 0

Nonsingular Weierstrass equations

noncomputable def WeierstrassCurve.Projective.polynomialX {R : Type u} (W' : WeierstrassCurve.Projective R) [CommRing R] : MvPolynomial (Fin 3) R

The partial derivative $W_X(X,Y,Z)$ of $W(X,Y,Z)$ with respect to $X$.

Equations

• W'.polynomialX = (MvPolynomial.pderiv 0) W'.polynomial

theorem WeierstrassCurve.Projective.polynomialX_eq {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] : W'.polynomialX = MvPolynomial.C W'.a₁ * MvPolynomial.X 1 * MvPolynomial.X 2 - (MvPolynomial.C 3 * MvPolynomial.X 0 ^ 2 + MvPolynomial.C (2 * W'.a₂) * MvPolynomial.X 0 * MvPolynomial.X 2 + MvPolynomial.C W'.a₄ * MvPolynomial.X 2 ^ 2)

theorem WeierstrassCurve.Projective.eval_polynomialX {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] (P : Fin 3 → R) : (MvPolynomial.eval P) W'.polynomialX = W'.a₁ * P 1 * P 2 - (3 * P 0 ^ 2 + 2 * W'.a₂ * P 0 * P 2 + W'.a₄ * P 2 ^ 2)

theorem WeierstrassCurve.Projective.eval_polynomialX_of_Z_ne_zero {F : Type v} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} (hPz : P 2 ≠ 0) : (MvPolynomial.eval P) W.polynomialX / P 2 ^ 2 = Polynomial.evalEval (P 0 / P 2) (P 1 / P 2) W.toAffine.polynomialX

noncomputable def WeierstrassCurve.Projective.polynomialY {R : Type u} (W' : WeierstrassCurve.Projective R) [CommRing R] : MvPolynomial (Fin 3) R

The partial derivative $W_Y(X,Y,Z)$ of $W(X,Y,Z)$ with respect to $Y$.

Equations

• W'.polynomialY = (MvPolynomial.pderiv 1) W'.polynomial

theorem WeierstrassCurve.Projective.polynomialY_eq {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] : W'.polynomialY = MvPolynomial.C 2 * MvPolynomial.X 1 * MvPolynomial.X 2 + MvPolynomial.C W'.a₁ * MvPolynomial.X 0 * MvPolynomial.X 2 + MvPolynomial.C W'.a₃ * MvPolynomial.X 2 ^ 2

theorem WeierstrassCurve.Projective.eval_polynomialY {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] (P : Fin 3 → R) : (MvPolynomial.eval P) W'.polynomialY = 2 * P 1 * P 2 + W'.a₁ * P 0 * P 2 + W'.a₃ * P 2 ^ 2

theorem WeierstrassCurve.Projective.eval_polynomialY_of_Z_ne_zero {F : Type v} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} (hPz : P 2 ≠ 0) : (MvPolynomial.eval P) W.polynomialY / P 2 ^ 2 = Polynomial.evalEval (P 0 / P 2) (P 1 / P 2) W.toAffine.polynomialY

noncomputable def WeierstrassCurve.Projective.polynomialZ {R : Type u} (W' : WeierstrassCurve.Projective R) [CommRing R] : MvPolynomial (Fin 3) R

The partial derivative $W_Z(X,Y,Z)$ of $W(X,Y,Z)$ with respect to $Z$.

Equations

• W'.polynomialZ = (MvPolynomial.pderiv 2) W'.polynomial

theorem WeierstrassCurve.Projective.polynomialZ_eq {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] : W'.polynomialZ = MvPolynomial.X 1 ^ 2 + MvPolynomial.C W'.a₁ * MvPolynomial.X 0 * MvPolynomial.X 1 + MvPolynomial.C (2 * W'.a₃) * MvPolynomial.X 1 * MvPolynomial.X 2 - (MvPolynomial.C W'.a₂ * MvPolynomial.X 0 ^ 2 + MvPolynomial.C (2 * W'.a₄) * MvPolynomial.X 0 * MvPolynomial.X 2 + MvPolynomial.C (3 * W'.a₆) * MvPolynomial.X 2 ^ 2)

theorem WeierstrassCurve.Projective.eval_polynomialZ {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] (P : Fin 3 → R) : (MvPolynomial.eval P) W'.polynomialZ = P 1 ^ 2 + W'.a₁ * P 0 * P 1 + 2 * W'.a₃ * P 1 * P 2 - (W'.a₂ * P 0 ^ 2 + 2 * W'.a₄ * P 0 * P 2 + 3 * W'.a₆ * P 2 ^ 2)

theorem WeierstrassCurve.Projective.polynomial_relation {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] (P : Fin 3 → R) : 3 * (MvPolynomial.eval P) W'.polynomial = P 0 * (MvPolynomial.eval P) W'.polynomialX + P 1 * (MvPolynomial.eval P) W'.polynomialY + P 2 * (MvPolynomial.eval P) W'.polynomialZ

Euler's homogeneous function theorem.

def WeierstrassCurve.Projective.Nonsingular {R : Type u} (W' : WeierstrassCurve.Projective R) [CommRing R] (P : Fin 3 → R) : Prop

The proposition that a point representative $(x,y,z)$ in W' is nonsingular. In other words, either $W_X(x,y,z)\ne 0$, $W_Y(x,y,z)\ne 0$, or $W_Z(x,y,z)\ne 0$.

Note that this definition is only mathematically accurate for fields.

Equations

• W'.Nonsingular P = (W'.Equation P ∧ ((MvPolynomial.eval P) W'.polynomialX ≠ 0 ∨ (MvPolynomial.eval P) W'.polynomialY ≠ 0 ∨ (MvPolynomial.eval P) W'.polynomialZ ≠ 0))

theorem WeierstrassCurve.Projective.nonsingular_iff {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] (P : Fin 3 → R) : W'.Nonsingular P ↔ W'.Equation P ∧ (W'.a₁ * P 1 * P 2 - (3 * P 0 ^ 2 + 2 * W'.a₂ * P 0 * P 2 + W'.a₄ * P 2 ^ 2) ≠ 0 ∨ 2 * P 1 * P 2 + W'.a₁ * P 0 * P 2 + W'.a₃ * P 2 ^ 2 ≠ 0 ∨ P 1 ^ 2 + W'.a₁ * P 0 * P 1 + 2 * W'.a₃ * P 1 * P 2 - (W'.a₂ * P 0 ^ 2 + 2 * W'.a₄ * P 0 * P 2 + 3 * W'.a₆ * P 2 ^ 2) ≠ 0)

theorem WeierstrassCurve.Projective.nonsingular_smul {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] (P : Fin 3 → R) {u : R} (hu : IsUnit u) : W'.Nonsingular (u • P) ↔ W'.Nonsingular P

theorem WeierstrassCurve.Projective.nonsingular_of_equiv {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] {P : Fin 3 → R} {Q : Fin 3 → R} (h : P ≈ Q) : W'.Nonsingular P ↔ W'.Nonsingular Q

theorem WeierstrassCurve.Projective.nonsingular_of_Z_eq_zero {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] {P : Fin 3 → R} (hPz : P 2 = 0) : W'.Nonsingular P ↔ W'.Equation P ∧ (3 * P 0 ^ 2 ≠ 0 ∨ P 1 ^ 2 + W'.a₁ * P 0 * P 1 - W'.a₂ * P 0 ^ 2 ≠ 0)

theorem WeierstrassCurve.Projective.nonsingular_zero {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] [Nontrivial R] : W'.Nonsingular ![0, 1, 0]

theorem WeierstrassCurve.Projective.nonsingular_some {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] (X : R) (Y : R) : W'.Nonsingular ![X, Y, 1] ↔ W'.toAffine.Nonsingular X Y

theorem WeierstrassCurve.Projective.nonsingular_of_Z_ne_zero {F : Type v} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} (hPz : P 2 ≠ 0) : W.Nonsingular P ↔ W.toAffine.Nonsingular (P 0 / P 2) (P 1 / P 2)

theorem WeierstrassCurve.Projective.nonsingular_iff_of_Z_ne_zero {F : Type v} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} (hPz : P 2 ≠ 0) : W.Nonsingular P ↔ W.Equation P ∧ ((MvPolynomial.eval P) W.polynomialX ≠ 0 ∨ (MvPolynomial.eval P) W.polynomialY ≠ 0)

theorem WeierstrassCurve.Projective.Y_ne_zero_of_Z_eq_zero {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] [NoZeroDivisors R] {P : Fin 3 → R} (hP : W'.Nonsingular P) (hPz : P 2 = 0) : P 1 ≠ 0

theorem WeierstrassCurve.Projective.isUnit_Y_of_Z_eq_zero {F : Type v} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} (hP : W.Nonsingular P) (hPz : P 2 = 0) : IsUnit (P 1)

theorem WeierstrassCurve.Projective.equiv_of_Z_eq_zero {F : Type v} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} {Q : Fin 3 → F} (hP : W.Nonsingular P) (hQ : W.Nonsingular Q) (hPz : P 2 = 0) (hQz : Q 2 = 0) : P ≈ Q

theorem WeierstrassCurve.Projective.equiv_zero_of_Z_eq_zero {F : Type v} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} (hP : W.Nonsingular P) (hPz : P 2 = 0) : P ≈ ![0, 1, 0]

def WeierstrassCurve.Projective.NonsingularLift {R : Type u} (W' : WeierstrassCurve.Projective R) [CommRing R] (P : WeierstrassCurve.Projective.PointClass R) : Prop

The proposition that a point class on W' is nonsingular. If P is a point representative, then W.NonsingularLift ⟦P⟧ is definitionally equivalent to W.Nonsingular P.

Equations

• W'.NonsingularLift P = Quotient.lift W'.Nonsingular ⋯ P

theorem WeierstrassCurve.Projective.nonsingularLift_iff {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] (P : Fin 3 → R) : W'.NonsingularLift ⟦P⟧ ↔ W'.Nonsingular P

theorem WeierstrassCurve.Projective.nonsingularLift_zero {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] [Nontrivial R] : W'.NonsingularLift ⟦![0, 1, 0]⟧

theorem WeierstrassCurve.Projective.nonsingularLift_some {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] (X : R) (Y : R) : W'.NonsingularLift ⟦![X, Y, 1]⟧ ↔ W'.toAffine.Nonsingular X Y

@[deprecated WeierstrassCurve.Projective.equation_smul]

theorem WeierstrassCurve.Projective.equation_smul_iff {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] (P : Fin 3 → R) {u : R} (hu : IsUnit u) : W'.Equation (u • P) ↔ W'.Equation P

Alias of WeierstrassCurve.Projective.equation_smul.

@[deprecated WeierstrassCurve.Projective.nonsingularLift_zero]

theorem WeierstrassCurve.Projective.nonsingularLift_zero' {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] [Nontrivial R] : W'.NonsingularLift ⟦![0, 1, 0]⟧

Alias of WeierstrassCurve.Projective.nonsingularLift_zero.

@[deprecated WeierstrassCurve.Projective.nonsingular_of_Z_ne_zero]

theorem WeierstrassCurve.Projective.nonsingular_affine_of_Z_ne_zero {F : Type v} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} (hPz : P 2 ≠ 0) : W.Nonsingular P ↔ W.toAffine.Nonsingular (P 0 / P 2) (P 1 / P 2)

Alias of WeierstrassCurve.Projective.nonsingular_of_Z_ne_zero.

@[deprecated WeierstrassCurve.Projective.nonsingular_of_Z_ne_zero]

theorem WeierstrassCurve.Projective.nonsingular_iff_affine_of_Z_ne_zero {F : Type v} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} (hPz : P 2 ≠ 0) : W.Nonsingular P ↔ W.toAffine.Nonsingular (P 0 / P 2) (P 1 / P 2)

Alias of WeierstrassCurve.Projective.nonsingular_of_Z_ne_zero.

@[deprecated WeierstrassCurve.Projective.nonsingular_of_Z_ne_zero]

theorem WeierstrassCurve.Projective.nonsingular_of_affine_of_Z_ne_zero {F : Type v} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} (hPz : P 2 ≠ 0) : W.Nonsingular P ↔ W.toAffine.Nonsingular (P 0 / P 2) (P 1 / P 2)

Alias of WeierstrassCurve.Projective.nonsingular_of_Z_ne_zero.

@[deprecated WeierstrassCurve.Projective.nonsingular_smul]

theorem WeierstrassCurve.Projective.nonsingular_smul_iff {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] (P : Fin 3 → R) {u : R} (hu : IsUnit u) : W'.Nonsingular (u • P) ↔ W'.Nonsingular P

Alias of WeierstrassCurve.Projective.nonsingular_smul.

@[deprecated WeierstrassCurve.Projective.nonsingular_zero]

theorem WeierstrassCurve.Projective.nonsingular_zero' {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] [Nontrivial R] : W'.Nonsingular ![0, 1, 0]

Alias of WeierstrassCurve.Projective.nonsingular_zero.

Negation formulae

def WeierstrassCurve.Projective.negY {R : Type u} (W' : WeierstrassCurve.Projective R) [CommRing R] (P : Fin 3 → R) : R

The $Y$-coordinate of a representative of -P for a point P.

Equations

• W'.negY P = -P 1 - W'.a₁ * P 0 - W'.a₃ * P 2

theorem WeierstrassCurve.Projective.negY_eq {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] (X : R) (Y : R) (Z : R) : W'.negY ![X, Y, Z] = -Y - W'.a₁ * X - W'.a₃ * Z

theorem WeierstrassCurve.Projective.negY_smul {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] (P : Fin 3 → R) (u : R) : W'.negY (u • P) = u * W'.negY P

theorem WeierstrassCurve.Projective.negY_of_Z_eq_zero {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] [NoZeroDivisors R] {P : Fin 3 → R} (hP : W'.Equation P) (hPz : P 2 = 0) : W'.negY P = -P 1

theorem WeierstrassCurve.Projective.negY_of_Z_ne_zero {F : Type v} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} (hPz : P 2 ≠ 0) : W.negY P / P 2 = W.toAffine.negY (P 0 / P 2) (P 1 / P 2)

theorem WeierstrassCurve.Projective.Y_sub_Y_mul_Y_sub_negY {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] {P : Fin 3 → R} {Q : Fin 3 → R} (hP : W'.Equation P) (hQ : W'.Equation Q) (hx : P 0 * Q 2 = Q 0 * P 2) : P 2 * Q 2 * (P 1 * Q 2 - Q 1 * P 2) * (P 1 * Q 2 - W'.negY Q * P 2) = 0

theorem WeierstrassCurve.Projective.Y_eq_of_Y_ne {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] [NoZeroDivisors R] {P : Fin 3 → R} {Q : Fin 3 → R} (hP : W'.Equation P) (hQ : W'.Equation Q) (hPz : P 2 ≠ 0) (hQz : Q 2 ≠ 0) (hx : P 0 * Q 2 = Q 0 * P 2) (hy : P 1 * Q 2 ≠ Q 1 * P 2) : P 1 * Q 2 = W'.negY Q * P 2

theorem WeierstrassCurve.Projective.Y_eq_of_Y_ne' {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] [NoZeroDivisors R] {P : Fin 3 → R} {Q : Fin 3 → R} (hP : W'.Equation P) (hQ : W'.Equation Q) (hPz : P 2 ≠ 0) (hQz : Q 2 ≠ 0) (hx : P 0 * Q 2 = Q 0 * P 2) (hy : P 1 * Q 2 ≠ W'.negY Q * P 2) : P 1 * Q 2 = Q 1 * P 2

theorem WeierstrassCurve.Projective.Y_eq_iff' {F : Type v} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} {Q : Fin 3 → F} (hPz : P 2 ≠ 0) (hQz : Q 2 ≠ 0) : P 1 * Q 2 = W.negY Q * P 2 ↔ P 1 / P 2 = W.toAffine.negY (Q 0 / Q 2) (Q 1 / Q 2)

theorem WeierstrassCurve.Projective.Y_sub_Y_add_Y_sub_negY {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] {P : Fin 3 → R} {Q : Fin 3 → R} (hx : P 0 * Q 2 = Q 0 * P 2) : P 1 * Q 2 - Q 1 * P 2 + (P 1 * Q 2 - W'.negY Q * P 2) = (P 1 - W'.negY P) * Q 2

theorem WeierstrassCurve.Projective.Y_ne_negY_of_Y_ne {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] [NoZeroDivisors R] {P : Fin 3 → R} {Q : Fin 3 → R} (hP : W'.Equation P) (hQ : W'.Equation Q) (hPz : P 2 ≠ 0) (hQz : Q 2 ≠ 0) (hx : P 0 * Q 2 = Q 0 * P 2) (hy : P 1 * Q 2 ≠ Q 1 * P 2) : P 1 ≠ W'.negY P

theorem WeierstrassCurve.Projective.Y_ne_negY_of_Y_ne' {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] [NoZeroDivisors R] {P : Fin 3 → R} {Q : Fin 3 → R} (hP : W'.Equation P) (hQ : W'.Equation Q) (hPz : P 2 ≠ 0) (hQz : Q 2 ≠ 0) (hx : P 0 * Q 2 = Q 0 * P 2) (hy : P 1 * Q 2 ≠ W'.negY Q * P 2) : P 1 ≠ W'.negY P

theorem WeierstrassCurve.Projective.Y_eq_negY_of_Y_eq {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] [NoZeroDivisors R] {P : Fin 3 → R} {Q : Fin 3 → R} (hQz : Q 2 ≠ 0) (hx : P 0 * Q 2 = Q 0 * P 2) (hy : P 1 * Q 2 = Q 1 * P 2) (hy' : P 1 * Q 2 = W'.negY Q * P 2) : P 1 = W'.negY P

theorem WeierstrassCurve.Projective.nonsingular_iff_of_Y_eq_negY {F : Type v} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} (hPz : P 2 ≠ 0) (hy : P 1 = W.negY P) : W.Nonsingular P ↔ W.Equation P ∧ (MvPolynomial.eval P) W.polynomialX ≠ 0

Doubling formulae

noncomputable def WeierstrassCurve.Projective.dblU {F : Type v} [Field F] (W : WeierstrassCurve.Projective F) (P : Fin 3 → F) : F

The unit associated to the doubling of a 2-torsion point P. More specifically, the unit u such that W.add P P = u • ![0, 1, 0] where P = W.neg P.

Equations

• W.dblU P = (MvPolynomial.eval P) W.polynomialX ^ 3 / P 2 ^ 2

theorem WeierstrassCurve.Projective.dblU_eq {F : Type v} [Field F] {W : WeierstrassCurve.Projective F} (P : Fin 3 → F) : W.dblU P = (W.a₁ * P 1 * P 2 - (3 * P 0 ^ 2 + 2 * W.a₂ * P 0 * P 2 + W.a₄ * P 2 ^ 2)) ^ 3 / P 2 ^ 2

theorem WeierstrassCurve.Projective.dblU_smul {F : Type v} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} (hPz : P 2 ≠ 0) {u : F} (hu : u ≠ 0) : W.dblU (u • P) = u ^ 4 * W.dblU P

theorem WeierstrassCurve.Projective.dblU_of_Z_eq_zero {F : Type v} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} (hPz : P 2 = 0) : W.dblU P = 0

theorem WeierstrassCurve.Projective.dblU_ne_zero_of_Y_eq {F : Type v} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} {Q : Fin 3 → F} (hP : W.Nonsingular P) (hPz : P 2 ≠ 0) (hQz : Q 2 ≠ 0) (hx : P 0 * Q 2 = Q 0 * P 2) (hy : P 1 * Q 2 = Q 1 * P 2) (hy' : P 1 * Q 2 = W.negY Q * P 2) : W.dblU P ≠ 0

theorem WeierstrassCurve.Projective.isUnit_dblU_of_Y_eq {F : Type v} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} {Q : Fin 3 → F} (hP : W.Nonsingular P) (hPz : P 2 ≠ 0) (hQz : Q 2 ≠ 0) (hx : P 0 * Q 2 = Q 0 * P 2) (hy : P 1 * Q 2 = Q 1 * P 2) (hy' : P 1 * Q 2 = W.negY Q * P 2) : IsUnit (W.dblU P)

def WeierstrassCurve.Projective.dblZ {R : Type u} (W' : WeierstrassCurve.Projective R) [CommRing R] (P : Fin 3 → R) : R

The $Z$-coordinate of a representative of 2 • P for a point P.

Equations

• W'.dblZ P = P 2 * (P 1 - W'.negY P) ^ 3

theorem WeierstrassCurve.Projective.dblZ_smul {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] (P : Fin 3 → R) (u : R) : W'.dblZ (u • P) = u ^ 4 * W'.dblZ P

theorem WeierstrassCurve.Projective.dblZ_of_Z_eq_zero {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] {P : Fin 3 → R} (hPz : P 2 = 0) : W'.dblZ P = 0

theorem WeierstrassCurve.Projective.dblZ_of_Y_eq {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] [NoZeroDivisors R] {P : Fin 3 → R} {Q : Fin 3 → R} (hQz : Q 2 ≠ 0) (hx : P 0 * Q 2 = Q 0 * P 2) (hy : P 1 * Q 2 = Q 1 * P 2) (hy' : P 1 * Q 2 = W'.negY Q * P 2) : W'.dblZ P = 0

theorem WeierstrassCurve.Projective.dblZ_ne_zero_of_Y_ne {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] [NoZeroDivisors R] {P : Fin 3 → R} {Q : Fin 3 → R} (hP : W'.Equation P) (hQ : W'.Equation Q) (hPz : P 2 ≠ 0) (hQz : Q 2 ≠ 0) (hx : P 0 * Q 2 = Q 0 * P 2) (hy : P 1 * Q 2 ≠ Q 1 * P 2) : W'.dblZ P ≠ 0

theorem WeierstrassCurve.Projective.isUnit_dblZ_of_Y_ne {F : Type v} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} {Q : Fin 3 → F} (hP : W.Equation P) (hQ : W.Equation Q) (hPz : P 2 ≠ 0) (hQz : Q 2 ≠ 0) (hx : P 0 * Q 2 = Q 0 * P 2) (hy : P 1 * Q 2 ≠ Q 1 * P 2) : IsUnit (W.dblZ P)

theorem WeierstrassCurve.Projective.dblZ_ne_zero_of_Y_ne' {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] [NoZeroDivisors R] {P : Fin 3 → R} {Q : Fin 3 → R} (hP : W'.Equation P) (hQ : W'.Equation Q) (hPz : P 2 ≠ 0) (hQz : Q 2 ≠ 0) (hx : P 0 * Q 2 = Q 0 * P 2) (hy : P 1 * Q 2 ≠ W'.negY Q * P 2) : W'.dblZ P ≠ 0

theorem WeierstrassCurve.Projective.isUnit_dblZ_of_Y_ne' {F : Type v} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} {Q : Fin 3 → F} (hP : W.Equation P) (hQ : W.Equation Q) (hPz : P 2 ≠ 0) (hQz : Q 2 ≠ 0) (hx : P 0 * Q 2 = Q 0 * P 2) (hy : P 1 * Q 2 ≠ W.negY Q * P 2) : IsUnit (W.dblZ P)

noncomputable def WeierstrassCurve.Projective.dblX {R : Type u} (W' : WeierstrassCurve.Projective R) [CommRing R] (P : Fin 3 → R) : R

The $X$-coordinate of a representative of 2 • P for a point P.

Equations

• One or more equations did not get rendered due to their size.

theorem WeierstrassCurve.Projective.dblX_eq' {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] {P : Fin 3 → R} (hP : W'.Equation P) : W'.dblX P * P 2 = ((MvPolynomial.eval P) W'.polynomialX ^ 2 - W'.a₁ * (MvPolynomial.eval P) W'.polynomialX * P 2 * (P 1 - W'.negY P) - W'.a₂ * P 2 ^ 2 * (P 1 - W'.negY P) ^ 2 - 2 * P 0 * P 2 * (P 1 - W'.negY P) ^ 2) * (P 1 - W'.negY P)

theorem WeierstrassCurve.Projective.dblX_eq {F : Type v} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} (hP : W.Equation P) (hPz : P 2 ≠ 0) : W.dblX P = ((MvPolynomial.eval P) W.polynomialX ^ 2 - W.a₁ * (MvPolynomial.eval P) W.polynomialX * P 2 * (P 1 - W.negY P) - W.a₂ * P 2 ^ 2 * (P 1 - W.negY P) ^ 2 - 2 * P 0 * P 2 * (P 1 - W.negY P) ^ 2) * (P 1 - W.negY P) / P 2

theorem WeierstrassCurve.Projective.dblX_smul {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] (P : Fin 3 → R) (u : R) : W'.dblX (u • P) = u ^ 4 * W'.dblX P

theorem WeierstrassCurve.Projective.dblX_of_Z_eq_zero {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] [NoZeroDivisors R] {P : Fin 3 → R} (hP : W'.Equation P) (hPz : P 2 = 0) : W'.dblX P = 0

theorem WeierstrassCurve.Projective.dblX_of_Y_eq {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] [NoZeroDivisors R] {P : Fin 3 → R} {Q : Fin 3 → R} (hP : W'.Equation P) (hPz : P 2 ≠ 0) (hQz : Q 2 ≠ 0) (hx : P 0 * Q 2 = Q 0 * P 2) (hy : P 1 * Q 2 = Q 1 * P 2) (hy' : P 1 * Q 2 = W'.negY Q * P 2) : W'.dblX P = 0

theorem WeierstrassCurve.Projective.dblX_of_Z_ne_zero {F : Type v} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} {Q : Fin 3 → F} (hP : W.Equation P) (hQ : W.Equation Q) (hPz : P 2 ≠ 0) (hQz : Q 2 ≠ 0) (hx : P 0 * Q 2 = Q 0 * P 2) (hy : P 1 * Q 2 ≠ W.negY Q * P 2) : W.dblX P / W.dblZ P = W.toAffine.addX (P 0 / P 2) (Q 0 / Q 2) (W.toAffine.slope (P 0 / P 2) (Q 0 / Q 2) (P 1 / P 2) (Q 1 / Q 2))

noncomputable def WeierstrassCurve.Projective.negDblY {R : Type u} (W' : WeierstrassCurve.Projective R) [CommRing R] (P : Fin 3 → R) : R

The $Y$-coordinate of a representative of -(2 • P) for a point P.

Equations

• One or more equations did not get rendered due to their size.

theorem WeierstrassCurve.Projective.negDblY_eq' {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] {P : Fin 3 → R} (hP : W'.Equation P) : W'.negDblY P * P 2 ^ 2 = -(MvPolynomial.eval P) W'.polynomialX * ((MvPolynomial.eval P) W'.polynomialX ^ 2 - W'.a₁ * (MvPolynomial.eval P) W'.polynomialX * P 2 * (P 1 - W'.negY P) - W'.a₂ * P 2 ^ 2 * (P 1 - W'.negY P) ^ 2 - 2 * P 0 * P 2 * (P 1 - W'.negY P) ^ 2 - P 0 * P 2 * (P 1 - W'.negY P) ^ 2) + P 1 * P 2 ^ 2 * (P 1 - W'.negY P) ^ 3

theorem WeierstrassCurve.Projective.negDblY_eq {F : Type v} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} (hP : W.Equation P) (hPz : P 2 ≠ 0) : W.negDblY P = (-(MvPolynomial.eval P) W.polynomialX * ((MvPolynomial.eval P) W.polynomialX ^ 2 - W.a₁ * (MvPolynomial.eval P) W.polynomialX * P 2 * (P 1 - W.negY P) - W.a₂ * P 2 ^ 2 * (P 1 - W.negY P) ^ 2 - 2 * P 0 * P 2 * (P 1 - W.negY P) ^ 2 - P 0 * P 2 * (P 1 - W.negY P) ^ 2) + P 1 * P 2 ^ 2 * (P 1 - W.negY P) ^ 3) / P 2 ^ 2

theorem WeierstrassCurve.Projective.negDblY_smul {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] (P : Fin 3 → R) (u : R) : W'.negDblY (u • P) = u ^ 4 * W'.negDblY P

theorem WeierstrassCurve.Projective.negDblY_of_Z_eq_zero {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] [NoZeroDivisors R] {P : Fin 3 → R} (hP : W'.Equation P) (hPz : P 2 = 0) : W'.negDblY P = -P 1 ^ 4

theorem WeierstrassCurve.Projective.negDblY_of_Y_eq' {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] [NoZeroDivisors R] {P : Fin 3 → R} {Q : Fin 3 → R} (hP : W'.Equation P) (hQz : Q 2 ≠ 0) (hx : P 0 * Q 2 = Q 0 * P 2) (hy : P 1 * Q 2 = Q 1 * P 2) (hy' : P 1 * Q 2 = W'.negY Q * P 2) : W'.negDblY P * P 2 ^ 2 = -(MvPolynomial.eval P) W'.polynomialX ^ 3

theorem WeierstrassCurve.Projective.negDblY_of_Y_eq {F : Type v} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} {Q : Fin 3 → F} (hP : W.Equation P) (hPz : P 2 ≠ 0) (hQz : Q 2 ≠ 0) (hx : P 0 * Q 2 = Q 0 * P 2) (hy : P 1 * Q 2 = Q 1 * P 2) (hy' : P 1 * Q 2 = W.negY Q * P 2) : W.negDblY P = -W.dblU P

theorem WeierstrassCurve.Projective.negDblY_of_Z_ne_zero {F : Type v} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} {Q : Fin 3 → F} (hP : W.Equation P) (hQ : W.Equation Q) (hPz : P 2 ≠ 0) (hQz : Q 2 ≠ 0) (hx : P 0 * Q 2 = Q 0 * P 2) (hy : P 1 * Q 2 ≠ W.negY Q * P 2) : W.negDblY P / W.dblZ P = W.toAffine.negAddY (P 0 / P 2) (Q 0 / Q 2) (P 1 / P 2) (W.toAffine.slope (P 0 / P 2) (Q 0 / Q 2) (P 1 / P 2) (Q 1 / Q 2))

noncomputable def WeierstrassCurve.Projective.dblY {R : Type u} (W' : WeierstrassCurve.Projective R) [CommRing R] (P : Fin 3 → R) : R

The $Y$-coordinate of a representative of 2 • P for a point P.

Equations

• W'.dblY P = W'.negY ![W'.dblX P, W'.negDblY P, W'.dblZ P]

theorem WeierstrassCurve.Projective.dblY_smul {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] (P : Fin 3 → R) (u : R) : W'.dblY (u • P) = u ^ 4 * W'.dblY P

theorem WeierstrassCurve.Projective.dblY_of_Z_eq_zero {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] [NoZeroDivisors R] {P : Fin 3 → R} (hP : W'.Equation P) (hPz : P 2 = 0) : W'.dblY P = P 1 ^ 4

theorem WeierstrassCurve.Projective.dblY_of_Y_eq' {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] [NoZeroDivisors R] {P : Fin 3 → R} {Q : Fin 3 → R} (hP : W'.Equation P) (hPz : P 2 ≠ 0) (hQz : Q 2 ≠ 0) (hx : P 0 * Q 2 = Q 0 * P 2) (hy : P 1 * Q 2 = Q 1 * P 2) (hy' : P 1 * Q 2 = W'.negY Q * P 2) : W'.dblY P * P 2 ^ 2 = (MvPolynomial.eval P) W'.polynomialX ^ 3

theorem WeierstrassCurve.Projective.dblY_of_Y_eq {F : Type v} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} {Q : Fin 3 → F} (hP : W.Equation P) (hPz : P 2 ≠ 0) (hQz : Q 2 ≠ 0) (hx : P 0 * Q 2 = Q 0 * P 2) (hy : P 1 * Q 2 = Q 1 * P 2) (hy' : P 1 * Q 2 = W.negY Q * P 2) : W.dblY P = W.dblU P

theorem WeierstrassCurve.Projective.dblY_of_Z_ne_zero {F : Type v} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} {Q : Fin 3 → F} (hP : W.Equation P) (hQ : W.Equation Q) (hPz : P 2 ≠ 0) (hQz : Q 2 ≠ 0) (hx : P 0 * Q 2 = Q 0 * P 2) (hy : P 1 * Q 2 ≠ W.negY Q * P 2) : W.dblY P / W.dblZ P = W.toAffine.addY (P 0 / P 2) (Q 0 / Q 2) (P 1 / P 2) (W.toAffine.slope (P 0 / P 2) (Q 0 / Q 2) (P 1 / P 2) (Q 1 / Q 2))

noncomputable def WeierstrassCurve.Projective.dblXYZ {R : Type u} (W' : WeierstrassCurve.Projective R) [CommRing R] (P : Fin 3 → R) : Fin 3 → R

The coordinates of a representative of 2 • P for a point P.

Equations

• W'.dblXYZ P = ![W'.dblX P, W'.dblY P, W'.dblZ P]

theorem WeierstrassCurve.Projective.dblXYZ_X {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] (P : Fin 3 → R) : W'.dblXYZ P 0 = W'.dblX P

theorem WeierstrassCurve.Projective.dblXYZ_Y {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] (P : Fin 3 → R) : W'.dblXYZ P 1 = W'.dblY P

theorem WeierstrassCurve.Projective.dblXYZ_Z {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] (P : Fin 3 → R) : W'.dblXYZ P 2 = W'.dblZ P

theorem WeierstrassCurve.Projective.dblXYZ_smul {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] (P : Fin 3 → R) (u : R) : W'.dblXYZ (u • P) = u ^ 4 • W'.dblXYZ P

theorem WeierstrassCurve.Projective.dblXYZ_of_Z_eq_zero {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] [NoZeroDivisors R] {P : Fin 3 → R} (hP : W'.Equation P) (hPz : P 2 = 0) : W'.dblXYZ P = P 1 ^ 4 • ![0, 1, 0]

theorem WeierstrassCurve.Projective.dblXYZ_of_Y_eq {F : Type v} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} {Q : Fin 3 → F} (hP : W.Equation P) (hPz : P 2 ≠ 0) (hQz : Q 2 ≠ 0) (hx : P 0 * Q 2 = Q 0 * P 2) (hy : P 1 * Q 2 = Q 1 * P 2) (hy' : P 1 * Q 2 = W.negY Q * P 2) : W.dblXYZ P = W.dblU P • ![0, 1, 0]

theorem WeierstrassCurve.Projective.dblXYZ_of_Z_ne_zero {F : Type v} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} {Q : Fin 3 → F} (hP : W.Equation P) (hQ : W.Equation Q) (hPz : P 2 ≠ 0) (hQz : Q 2 ≠ 0) (hx : P 0 * Q 2 = Q 0 * P 2) (hy : P 1 * Q 2 ≠ W.negY Q * P 2) : W.dblXYZ P = W.dblZ P • ![W.toAffine.addX (P 0 / P 2) (Q 0 / Q 2) (W.toAffine.slope (P 0 / P 2) (Q 0 / Q 2) (P 1 / P 2) (Q 1 / Q 2)), W.toAffine.addY (P 0 / P 2) (Q 0 / Q 2) (P 1 / P 2) (W.toAffine.slope (P 0 / P 2) (Q 0 / Q 2) (P 1 / P 2) (Q 1 / Q 2)), 1]

Addition formulae

def WeierstrassCurve.Projective.addU {F : Type v} [Field F] (P : Fin 3 → F) (Q : Fin 3 → F) : F

The unit associated to the addition of a non-2-torsion point P with its negation. More specifically, the unit u such that W.add P Q = u • ![0, 1, 0] where P x / P z = Q x / Q z but P ≠ W.neg P.

Equations

• WeierstrassCurve.Projective.addU P Q = -(P 1 * Q 2 - Q 1 * P 2) ^ 3 / (P 2 * Q 2)

theorem WeierstrassCurve.Projective.addU_smul {F : Type v} [Field F] {P : Fin 3 → F} {Q : Fin 3 → F} (hPz : P 2 ≠ 0) (hQz : Q 2 ≠ 0) {u : F} {v : F} (hu : u ≠ 0) (hv : v ≠ 0) : WeierstrassCurve.Projective.addU (u • P) (v • Q) = (u * v) ^ 2 * WeierstrassCurve.Projective.addU P Q

theorem WeierstrassCurve.Projective.addU_of_Z_eq_zero_left {F : Type v} [Field F] {P : Fin 3 → F} {Q : Fin 3 → F} (hPz : P 2 = 0) : WeierstrassCurve.Projective.addU P Q = 0

theorem WeierstrassCurve.Projective.addU_of_Z_eq_zero_right {F : Type v} [Field F] {P : Fin 3 → F} {Q : Fin 3 → F} (hQz : Q 2 = 0) : WeierstrassCurve.Projective.addU P Q = 0

theorem WeierstrassCurve.Projective.addU_ne_zero_of_Y_ne {F : Type v} [Field F] {P : Fin 3 → F} {Q : Fin 3 → F} (hPz : P 2 ≠ 0) (hQz : Q 2 ≠ 0) (hy : P 1 * Q 2 ≠ Q 1 * P 2) : WeierstrassCurve.Projective.addU P Q ≠ 0

theorem WeierstrassCurve.Projective.isUnit_addU_of_Y_ne {F : Type v} [Field F] {P : Fin 3 → F} {Q : Fin 3 → F} (hPz : P 2 ≠ 0) (hQz : Q 2 ≠ 0) (hy : P 1 * Q 2 ≠ Q 1 * P 2) : IsUnit (WeierstrassCurve.Projective.addU P Q)

def WeierstrassCurve.Projective.addZ {R : Type u} (W' : WeierstrassCurve.Projective R) [CommRing R] (P : Fin 3 → R) (Q : Fin 3 → R) : R

The $Z$-coordinate of a representative of P + Q for two distinct points P and Q. Note that this returns the value 0 if the representatives of P and Q are equal.

Equations

• One or more equations did not get rendered due to their size.

theorem WeierstrassCurve.Projective.addZ_eq' {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] {P : Fin 3 → R} {Q : Fin 3 → R} (hP : W'.Equation P) (hQ : W'.Equation Q) : W'.addZ P Q * (P 2 * Q 2) = (P 0 * Q 2 - Q 0 * P 2) ^ 3

theorem WeierstrassCurve.Projective.addZ_eq {F : Type v} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} {Q : Fin 3 → F} (hP : W.Equation P) (hQ : W.Equation Q) (hPz : P 2 ≠ 0) (hQz : Q 2 ≠ 0) : W.addZ P Q = (P 0 * Q 2 - Q 0 * P 2) ^ 3 / (P 2 * Q 2)

theorem WeierstrassCurve.Projective.addZ_smul {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] (P : Fin 3 → R) (Q : Fin 3 → R) (u : R) (v : R) : W'.addZ (u • P) (v • Q) = (u * v) ^ 2 * W'.addZ P Q

theorem WeierstrassCurve.Projective.addZ_self {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] (P : Fin 3 → R) : W'.addZ P P = 0

theorem WeierstrassCurve.Projective.addZ_of_Z_eq_zero_left {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] [NoZeroDivisors R] {P : Fin 3 → R} {Q : Fin 3 → R} (hP : W'.Equation P) (hPz : P 2 = 0) : W'.addZ P Q = P 1 ^ 2 * Q 2 * Q 2

theorem WeierstrassCurve.Projective.addZ_of_Z_eq_zero_right {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] [NoZeroDivisors R] {P : Fin 3 → R} {Q : Fin 3 → R} (hQ : W'.Equation Q) (hQz : Q 2 = 0) : W'.addZ P Q = -(Q 1 ^ 2 * P 2) * P 2

theorem WeierstrassCurve.Projective.addZ_of_X_eq {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] [NoZeroDivisors R] {P : Fin 3 → R} {Q : Fin 3 → R} (hP : W'.Equation P) (hQ : W'.Equation Q) (hPz : P 2 ≠ 0) (hQz : Q 2 ≠ 0) (hx : P 0 * Q 2 = Q 0 * P 2) : W'.addZ P Q = 0

theorem WeierstrassCurve.Projective.addZ_ne_zero_of_X_ne {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] [NoZeroDivisors R] {P : Fin 3 → R} {Q : Fin 3 → R} (hP : W'.Equation P) (hQ : W'.Equation Q) (hx : P 0 * Q 2 ≠ Q 0 * P 2) : W'.addZ P Q ≠ 0

theorem WeierstrassCurve.Projective.isUnit_addZ_of_X_ne {F : Type v} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} {Q : Fin 3 → F} (hP : W.Equation P) (hQ : W.Equation Q) (hx : P 0 * Q 2 ≠ Q 0 * P 2) : IsUnit (W.addZ P Q)

def WeierstrassCurve.Projective.addX {R : Type u} (W' : WeierstrassCurve.Projective R) [CommRing R] (P : Fin 3 → R) (Q : Fin 3 → R) : R

The $X$-coordinate of a representative of P + Q for two distinct points P and Q. Note that this returns the value 0 if the representatives of P and Q are equal.

Equations

• One or more equations did not get rendered due to their size.

theorem WeierstrassCurve.Projective.addX_eq' {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] {P : Fin 3 → R} {Q : Fin 3 → R} (hP : W'.Equation P) (hQ : W'.Equation Q) : W'.addX P Q * (P 2 * Q 2) ^ 2 = ((P 1 * Q 2 - Q 1 * P 2) ^ 2 * P 2 * Q 2 + W'.a₁ * (P 1 * Q 2 - Q 1 * P 2) * P 2 * Q 2 * (P 0 * Q 2 - Q 0 * P 2) - W'.a₂ * P 2 * Q 2 * (P 0 * Q 2 - Q 0 * P 2) ^ 2 - P 0 * Q 2 * (P 0 * Q 2 - Q 0 * P 2) ^ 2 - Q 0 * P 2 * (P 0 * Q 2 - Q 0 * P 2) ^ 2) * (P 0 * Q 2 - Q 0 * P 2)

theorem WeierstrassCurve.Projective.addX_eq {F : Type v} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} {Q : Fin 3 → F} (hP : W.Equation P) (hQ : W.Equation Q) (hPz : P 2 ≠ 0) (hQz : Q 2 ≠ 0) : W.addX P Q = ((P 1 * Q 2 - Q 1 * P 2) ^ 2 * P 2 * Q 2 + W.a₁ * (P 1 * Q 2 - Q 1 * P 2) * P 2 * Q 2 * (P 0 * Q 2 - Q 0 * P 2) - W.a₂ * P 2 * Q 2 * (P 0 * Q 2 - Q 0 * P 2) ^ 2 - P 0 * Q 2 * (P 0 * Q 2 - Q 0 * P 2) ^ 2 - Q 0 * P 2 * (P 0 * Q 2 - Q 0 * P 2) ^ 2) * (P 0 * Q 2 - Q 0 * P 2) / (P 2 * Q 2) ^ 2

theorem WeierstrassCurve.Projective.addX_smul {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] (P : Fin 3 → R) (Q : Fin 3 → R) (u : R) (v : R) : W'.addX (u • P) (v • Q) = (u * v) ^ 2 * W'.addX P Q

theorem WeierstrassCurve.Projective.addX_self {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] (P : Fin 3 → R) : W'.addX P P = 0

theorem WeierstrassCurve.Projective.addX_of_Z_eq_zero_left {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] [NoZeroDivisors R] {P : Fin 3 → R} {Q : Fin 3 → R} (hP : W'.Equation P) (hPz : P 2 = 0) : W'.addX P Q = P 1 ^ 2 * Q 2 * Q 0

theorem WeierstrassCurve.Projective.addX_of_Z_eq_zero_right {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] [NoZeroDivisors R] {P : Fin 3 → R} {Q : Fin 3 → R} (hQ : W'.Equation Q) (hQz : Q 2 = 0) : W'.addX P Q = -(Q 1 ^ 2 * P 2) * P 0

theorem WeierstrassCurve.Projective.addX_of_X_eq {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] [NoZeroDivisors R] {P : Fin 3 → R} {Q : Fin 3 → R} (hP : W'.Equation P) (hQ : W'.Equation Q) (hPz : P 2 ≠ 0) (hQz : Q 2 ≠ 0) (hx : P 0 * Q 2 = Q 0 * P 2) : W'.addX P Q = 0

theorem WeierstrassCurve.Projective.addX_of_Z_ne_zero {F : Type v} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} {Q : Fin 3 → F} (hP : W.Equation P) (hQ : W.Equation Q) (hPz : P 2 ≠ 0) (hQz : Q 2 ≠ 0) (hx : P 0 * Q 2 ≠ Q 0 * P 2) : W.addX P Q / W.addZ P Q = W.toAffine.addX (P 0 / P 2) (Q 0 / Q 2) (W.toAffine.slope (P 0 / P 2) (Q 0 / Q 2) (P 1 / P 2) (Q 1 / Q 2))

def WeierstrassCurve.Projective.negAddY {R : Type u} (W' : WeierstrassCurve.Projective R) [CommRing R] (P : Fin 3 → R) (Q : Fin 3 → R) : R

The $Y$-coordinate of a representative of -(P + Q) for two distinct points P and Q. Note that this returns the value 0 if the representatives of P and Q are equal.

Equations

• One or more equations did not get rendered due to their size.

theorem WeierstrassCurve.Projective.negAddY_eq' {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] {P : Fin 3 → R} {Q : Fin 3 → R} (hP : W'.Equation P) (hQ : W'.Equation Q) : W'.negAddY P Q * (P 2 * Q 2) ^ 2 = (P 1 * Q 2 - Q 1 * P 2) * ((P 1 * Q 2 - Q 1 * P 2) ^ 2 * P 2 * Q 2 + W'.a₁ * (P 1 * Q 2 - Q 1 * P 2) * P 2 * Q 2 * (P 0 * Q 2 - Q 0 * P 2) - W'.a₂ * P 2 * Q 2 * (P 0 * Q 2 - Q 0 * P 2) ^ 2 - P 0 * Q 2 * (P 0 * Q 2 - Q 0 * P 2) ^ 2 - Q 0 * P 2 * (P 0 * Q 2 - Q 0 * P 2) ^ 2 - P 0 * Q 2 * (P 0 * Q 2 - Q 0 * P 2) ^ 2) + P 1 * Q 2 * (P 0 * Q 2 - Q 0 * P 2) ^ 3

theorem WeierstrassCurve.Projective.negAddY_eq {F : Type v} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} {Q : Fin 3 → F} (hP : W.Equation P) (hQ : W.Equation Q) (hPz : P 2 ≠ 0) (hQz : Q 2 ≠ 0) : W.negAddY P Q = ((P 1 * Q 2 - Q 1 * P 2) * ((P 1 * Q 2 - Q 1 * P 2) ^ 2 * P 2 * Q 2 + W.a₁ * (P 1 * Q 2 - Q 1 * P 2) * P 2 * Q 2 * (P 0 * Q 2 - Q 0 * P 2) - W.a₂ * P 2 * Q 2 * (P 0 * Q 2 - Q 0 * P 2) ^ 2 - P 0 * Q 2 * (P 0 * Q 2 - Q 0 * P 2) ^ 2 - Q 0 * P 2 * (P 0 * Q 2 - Q 0 * P 2) ^ 2 - P 0 * Q 2 * (P 0 * Q 2 - Q 0 * P 2) ^ 2) + P 1 * Q 2 * (P 0 * Q 2 - Q 0 * P 2) ^ 3) / (P 2 * Q 2) ^ 2

theorem WeierstrassCurve.Projective.negAddY_smul {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] (P : Fin 3 → R) (Q : Fin 3 → R) (u : R) (v : R) : W'.negAddY (u • P) (v • Q) = (u * v) ^ 2 * W'.negAddY P Q

theorem WeierstrassCurve.Projective.negAddY_self {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] (P : Fin 3 → R) : W'.negAddY P P = 0

theorem WeierstrassCurve.Projective.negAddY_of_Z_eq_zero_left {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] [NoZeroDivisors R] {P : Fin 3 → R} {Q : Fin 3 → R} (hP : W'.Equation P) (hPz : P 2 = 0) : W'.negAddY P Q = P 1 ^ 2 * Q 2 * W'.negY Q

theorem WeierstrassCurve.Projective.negAddY_of_Z_eq_zero_right {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] [NoZeroDivisors R] {P : Fin 3 → R} {Q : Fin 3 → R} (hQ : W'.Equation Q) (hQz : Q 2 = 0) : W'.negAddY P Q = -(Q 1 ^ 2 * P 2) * W'.negY P

theorem WeierstrassCurve.Projective.negAddY_of_X_eq' {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] {P : Fin 3 → R} {Q : Fin 3 → R} (hP : W'.Equation P) (hQ : W'.Equation Q) (hx : P 0 * Q 2 = Q 0 * P 2) : W'.negAddY P Q * (P 2 * Q 2) ^ 2 = (P 1 * Q 2 - Q 1 * P 2) ^ 3 * (P 2 * Q 2)

theorem WeierstrassCurve.Projective.negAddY_of_X_eq {F : Type v} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} {Q : Fin 3 → F} (hP : W.Equation P) (hQ : W.Equation Q) (hPz : P 2 ≠ 0) (hQz : Q 2 ≠ 0) (hx : P 0 * Q 2 = Q 0 * P 2) : W.negAddY P Q = -WeierstrassCurve.Projective.addU P Q

theorem WeierstrassCurve.Projective.negAddY_of_Z_ne_zero {F : Type v} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} {Q : Fin 3 → F} (hP : W.Equation P) (hQ : W.Equation Q) (hPz : P 2 ≠ 0) (hQz : Q 2 ≠ 0) (hx : P 0 * Q 2 ≠ Q 0 * P 2) : W.negAddY P Q / W.addZ P Q = W.toAffine.negAddY (P 0 / P 2) (Q 0 / Q 2) (P 1 / P 2) (W.toAffine.slope (P 0 / P 2) (Q 0 / Q 2) (P 1 / P 2) (Q 1 / Q 2))

def WeierstrassCurve.Projective.addY {R : Type u} (W' : WeierstrassCurve.Projective R) [CommRing R] (P : Fin 3 → R) (Q : Fin 3 → R) : R

The $Y$-coordinate of a representative of P + Q for two distinct points P and Q. Note that this returns the value 0 if the representatives of P and Q are equal.

Equations

• W'.addY P Q = W'.negY ![W'.addX P Q, W'.negAddY P Q, W'.addZ P Q]

theorem WeierstrassCurve.Projective.addY_smul {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] (P : Fin 3 → R) (Q : Fin 3 → R) (u : R) (v : R) : W'.addY (u • P) (v • Q) = (u * v) ^ 2 * W'.addY P Q

theorem WeierstrassCurve.Projective.addY_self {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] (P : Fin 3 → R) : W'.addY P P = 0

theorem WeierstrassCurve.Projective.addY_of_Z_eq_zero_left {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] [NoZeroDivisors R] {P : Fin 3 → R} {Q : Fin 3 → R} (hP : W'.Equation P) (hPz : P 2 = 0) : W'.addY P Q = P 1 ^ 2 * Q 2 * Q 1

theorem WeierstrassCurve.Projective.addY_of_Z_eq_zero_right {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] [NoZeroDivisors R] {P : Fin 3 → R} {Q : Fin 3 → R} (hQ : W'.Equation Q) (hQz : Q 2 = 0) : W'.addY P Q = -(Q 1 ^ 2 * P 2) * P 1

theorem WeierstrassCurve.Projective.addY_of_X_eq' {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] [NoZeroDivisors R] {P : Fin 3 → R} {Q : Fin 3 → R} (hP : W'.Equation P) (hQ : W'.Equation Q) (hPz : P 2 ≠ 0) (hQz : Q 2 ≠ 0) (hx : P 0 * Q 2 = Q 0 * P 2) : W'.addY P Q * (P 2 * Q 2) ^ 3 = -(P 1 * Q 2 - Q 1 * P 2) ^ 3 * (P 2 * Q 2) ^ 2

theorem WeierstrassCurve.Projective.addY_of_X_eq {F : Type v} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} {Q : Fin 3 → F} (hP : W.Equation P) (hQ : W.Equation Q) (hPz : P 2 ≠ 0) (hQz : Q 2 ≠ 0) (hx : P 0 * Q 2 = Q 0 * P 2) : W.addY P Q = WeierstrassCurve.Projective.addU P Q

theorem WeierstrassCurve.Projective.addY_of_Z_ne_zero {F : Type v} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} {Q : Fin 3 → F} (hP : W.Equation P) (hQ : W.Equation Q) (hPz : P 2 ≠ 0) (hQz : Q 2 ≠ 0) (hx : P 0 * Q 2 ≠ Q 0 * P 2) : W.addY P Q / W.addZ P Q = W.toAffine.addY (P 0 / P 2) (Q 0 / Q 2) (P 1 / P 2) (W.toAffine.slope (P 0 / P 2) (Q 0 / Q 2) (P 1 / P 2) (Q 1 / Q 2))

noncomputable def WeierstrassCurve.Projective.addXYZ {R : Type u} (W' : WeierstrassCurve.Projective R) [CommRing R] (P : Fin 3 → R) (Q : Fin 3 → R) : Fin 3 → R

The coordinates of a representative of P + Q for two distinct points P and Q. Note that this returns the value ![0, 0, 0] if the representatives of P and Q are equal.

Equations

• W'.addXYZ P Q = ![W'.addX P Q, W'.addY P Q, W'.addZ P Q]

theorem WeierstrassCurve.Projective.addXYZ_X {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] (P : Fin 3 → R) (Q : Fin 3 → R) : W'.addXYZ P Q 0 = W'.addX P Q

theorem WeierstrassCurve.Projective.addXYZ_Y {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] (P : Fin 3 → R) (Q : Fin 3 → R) : W'.addXYZ P Q 1 = W'.addY P Q

theorem WeierstrassCurve.Projective.addXYZ_Z {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] (P : Fin 3 → R) (Q : Fin 3 → R) : W'.addXYZ P Q 2 = W'.addZ P Q

theorem WeierstrassCurve.Projective.addXYZ_smul {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] (P : Fin 3 → R) (Q : Fin 3 → R) (u : R) (v : R) : W'.addXYZ (u • P) (v • Q) = (u * v) ^ 2 • W'.addXYZ P Q

theorem WeierstrassCurve.Projective.addXYZ_self {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] (P : Fin 3 → R) : W'.addXYZ P P = ![0, 0, 0]

theorem WeierstrassCurve.Projective.addXYZ_of_Z_eq_zero_left {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] [NoZeroDivisors R] {P : Fin 3 → R} {Q : Fin 3 → R} (hP : W'.Equation P) (hPz : P 2 = 0) : W'.addXYZ P Q = (P 1 ^ 2 * Q 2) • Q

theorem WeierstrassCurve.Projective.addXYZ_of_Z_eq_zero_right {R : Type u} {W' : WeierstrassCurve.Projective R} [CommRing R] [NoZeroDivisors R] {P : Fin 3 → R} {Q : Fin 3 → R} (hQ : W'.Equation Q) (hQz : Q 2 = 0) : W'.addXYZ P Q = -(Q 1 ^ 2 * P 2) • P

theorem WeierstrassCurve.Projective.addXYZ_of_X_eq {F : Type v} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} {Q : Fin 3 → F} (hP : W.Equation P) (hQ : W.Equation Q) (hPz : P 2 ≠ 0) (hQz : Q 2 ≠ 0) (hx : P 0 * Q 2 = Q 0 * P 2) : W.addXYZ P Q = WeierstrassCurve.Projective.addU P Q • ![0, 1, 0]

theorem WeierstrassCurve.Projective.addXYZ_of_Z_ne_zero {F : Type v} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} {Q : Fin 3 → F} (hP : W.Equation P) (hQ : W.Equation Q) (hPz : P 2 ≠ 0) (hQz : Q 2 ≠ 0) (hx : P 0 * Q 2 ≠ Q 0 * P 2) : W.addXYZ P Q = W.addZ P Q • ![W.toAffine.addX (P 0 / P 2) (Q 0 / Q 2) (W.toAffine.slope (P 0 / P 2) (Q 0 / Q 2) (P 1 / P 2) (Q 1 / Q 2)), W.toAffine.addY (P 0 / P 2) (Q 0 / Q 2) (P 1 / P 2) (W.toAffine.slope (P 0 / P 2) (Q 0 / Q 2) (P 1 / P 2) (Q 1 / Q 2)), 1]