mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Point

Goal state before:
pp: case pos
F : Type u
inst✝¹ : Field F
W : Jacobian F
inst✝ : DecidableEq F
P Q : Fin 3 → F
hP : W.Nonsingular P
hQ : W.Nonsingular Q
hPz : P z = 0
hQz : Q z = 0
⊢ toAffine W (W.add P Q) = toAffine W Q

pp: case neg
F : Type u
inst✝¹ : Field F
W : Jacobian F
inst✝ : DecidableEq F
P Q : Fin 3 → F
hP : W.Nonsingular P
hQ : W.Nonsingular Q
hPz : P z = 0
hQz : ¬Q z = 0
⊢ toAffine W (W.add P Q) = toAffine W Q

Tactic:
·

Goal state after:
case pos
F : Type u
inst✝¹ : Field F
W : Jacobian F
inst✝ : DecidableEq F
P Q : Fin 3 → F
hP : W.Nonsingular P
hQ : W.Nonsingular Q
hPz : P z = 0
hQz : Q z = 0
⊢ toAffine W (W.add P Q) = toAffine W Q
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Point

Goal state before:
pp: case neg
F : Type u
inst✝¹ : Field F
W : Jacobian F
inst✝ : DecidableEq F
P Q : Fin 3 → F
hP : W.Nonsingular P
hQ : W.Nonsingular Q
hPz : P z = 0
hQz : ¬Q z = 0
⊢ toAffine W (W.add P Q) = toAffine W Q

Tactic:
·

Goal state after:
case neg
F : Type u
inst✝¹ : Field F
W : Jacobian F
inst✝ : DecidableEq F
P Q : Fin 3 → F
hP : W.Nonsingular P
hQ : W.Nonsingular Q
hPz : P z = 0
hQz : ¬Q z = 0
⊢ toAffine W (W.add P Q) = toAffine W Q
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Point

Goal state before:
pp: case neg
F : Type u
inst✝¹ : Field F
W : Jacobian F
inst✝ : DecidableEq F
P Q : Fin 3 → F
hP : W.Nonsingular P
hQ : W.Nonsingular Q
hPz : ¬P z = 0
⊢ toAffine W (W.add P Q) = toAffine W P + toAffine W Q

Tactic:
·

Goal state after:
case neg
F : Type u
inst✝¹ : Field F
W : Jacobian F
inst✝ : DecidableEq F
P Q : Fin 3 → F
hP : W.Nonsingular P
hQ : W.Nonsingular Q
hPz : ¬P z = 0
⊢ toAffine W (W.add P Q) = toAffine W P + toAffine W Q
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Point

Goal state before:
pp: case pos
F : Type u
inst✝¹ : Field F
W : Jacobian F
inst✝ : DecidableEq F
P Q : Fin 3 → F
hP : W.Nonsingular P
hQ : W.Nonsingular Q
hPz : ¬P z = 0
hQz : Q z = 0
⊢ toAffine W (W.add P Q) = toAffine W P + toAffine W Q

pp: case neg
F : Type u
inst✝¹ : Field F
W : Jacobian F
inst✝ : DecidableEq F
P Q : Fin 3 → F
hP : W.Nonsingular P
hQ : W.Nonsingular Q
hPz : ¬P z = 0
hQz : ¬Q z = 0
⊢ toAffine W (W.add P Q) = toAffine W P + toAffine W Q

Tactic:
·

Goal state after:
case pos
F : Type u
inst✝¹ : Field F
W : Jacobian F
inst✝ : DecidableEq F
P Q : Fin 3 → F
hP : W.Nonsingular P
hQ : W.Nonsingular Q
hPz : ¬P z = 0
hQz : Q z = 0
⊢ toAffine W (W.add P Q) = toAffine W P + toAffine W Q
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Point

Goal state before:
pp: case neg
F : Type u
inst✝¹ : Field F
W : Jacobian F
inst✝ : DecidableEq F
P Q : Fin 3 → F
hP : W.Nonsingular P
hQ : W.Nonsingular Q
hPz : ¬P z = 0
hQz : ¬Q z = 0
⊢ toAffine W (W.add P Q) = toAffine W P + toAffine W Q

Tactic:
·

Goal state after:
case neg
F : Type u
inst✝¹ : Field F
W : Jacobian F
inst✝ : DecidableEq F
P Q : Fin 3 → F
hP : W.Nonsingular P
hQ : W.Nonsingular Q
hPz : ¬P z = 0
hQz : ¬Q z = 0
⊢ toAffine W (W.add P Q) = toAffine W P + toAffine W Q
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Point

Goal state before:
pp: case pos
F : Type u
inst✝¹ : Field F
W : Jacobian F
inst✝ : DecidableEq F
P Q : Fin 3 → F
hP : W.Nonsingular P
hQ : W.Nonsingular Q
hPz : ¬P z = 0
hQz : ¬Q z = 0
hxy : P x * Q z ^ 2 = Q x * P z ^ 2 ∧ P y * Q z ^ 3 = W.negY Q * P z ^ 3
⊢ toAffine W (W.add P Q) = toAffine W P + toAffine W Q

pp: case neg
F : Type u
inst✝¹ : Field F
W : Jacobian F
inst✝ : DecidableEq F
P Q : Fin 3 → F
hP : W.Nonsingular P
hQ : W.Nonsingular Q
hPz : ¬P z = 0
hQz : ¬Q z = 0
hxy : ¬(P x * Q z ^ 2 = Q x * P z ^ 2 ∧ P y * Q z ^ 3 = W.negY Q * P z ^ 3)
⊢ toAffine W (W.add P Q) = toAffine W P + toAffine W Q

Tactic:
·

Goal state after:
case pos
F : Type u
inst✝¹ : Field F
W : Jacobian F
inst✝ : DecidableEq F
P Q : Fin 3 → F
hP : W.Nonsingular P
hQ : W.Nonsingular Q
hPz : ¬P z = 0
hQz : ¬Q z = 0
hxy : P x * Q z ^ 2 = Q x * P z ^ 2 ∧ P y * Q z ^ 3 = W.negY Q * P z ^ 3
⊢ toAffine W (W.add P Q) = toAffine W P + toAffine W Q
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Point

Goal state before:
pp: case pos
F : Type u
inst✝¹ : Field F
W : Jacobian F
inst✝ : DecidableEq F
P Q : Fin 3 → F
hP : W.Nonsingular P
hQ : W.Nonsingular Q
hPz : ¬P z = 0
hQz : ¬Q z = 0
hxy : P x * Q z ^ 2 = Q x * P z ^ 2 ∧ P y * Q z ^ 3 = W.negY Q * P z ^ 3
hy : P y * Q z ^ 3 = Q y * P z ^ 3
⊢ toAffine W (W.add P Q) = 0

pp: case neg
F : Type u
inst✝¹ : Field F
W : Jacobian F
inst✝ : DecidableEq F
P Q : Fin 3 → F
hP : W.Nonsingular P
hQ : W.Nonsingular Q
hPz : ¬P z = 0
hQz : ¬Q z = 0
hxy : P x * Q z ^ 2 = Q x * P z ^ 2 ∧ P y * Q z ^ 3 = W.negY Q * P z ^ 3
hy : ¬P y * Q z ^ 3 = Q y * P z ^ 3
⊢ toAffine W (W.add P Q) = 0

Tactic:
·

Goal state after:
case pos
F : Type u
inst✝¹ : Field F
W : Jacobian F
inst✝ : DecidableEq F
P Q : Fin 3 → F
hP : W.Nonsingular P
hQ : W.Nonsingular Q
hPz : ¬P z = 0
hQz : ¬Q z = 0
hxy : P x * Q z ^ 2 = Q x * P z ^ 2 ∧ P y * Q z ^ 3 = W.negY Q * P z ^ 3
hy : P y * Q z ^ 3 = Q y * P z ^ 3
⊢ toAffine W (W.add P Q) = 0
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Point

Goal state before:
pp: case neg
F : Type u
inst✝¹ : Field F
W : Jacobian F
inst✝ : DecidableEq F
P Q : Fin 3 → F
hP : W.Nonsingular P
hQ : W.Nonsingular Q
hPz : ¬P z = 0
hQz : ¬Q z = 0
hxy : P x * Q z ^ 2 = Q x * P z ^ 2 ∧ P y * Q z ^ 3 = W.negY Q * P z ^ 3
hy : ¬P y * Q z ^ 3 = Q y * P z ^ 3
⊢ toAffine W (W.add P Q) = 0

Tactic:
·

Goal state after:
case neg
F : Type u
inst✝¹ : Field F
W : Jacobian F
inst✝ : DecidableEq F
P Q : Fin 3 → F
hP : W.Nonsingular P
hQ : W.Nonsingular Q
hPz : ¬P z = 0
hQz : ¬Q z = 0
hxy : P x * Q z ^ 2 = Q x * P z ^ 2 ∧ P y * Q z ^ 3 = W.negY Q * P z ^ 3
hy : ¬P y * Q z ^ 3 = Q y * P z ^ 3
⊢ toAffine W (W.add P Q) = 0
Module: Mathlib.Topology.Algebra.Valued.ValuationTopology

Goal state before:
pp: case inl
K : Type u
inst✝² : DivisionRing K
Γ₀ : Type v
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : MulArchimedean Γ₀
v : Valuation K Γ₀
x : K
hx : v x ≠ 0
r : Γ₀ˣ
hr : ↑r ≠ 0
h : ∀ (x : K), v x ≠ 0 → ↑r < v x
H : Units.mk0 (v x) hx < 1
⊢ v x = 1

pp: case inr.inl
K : Type u
inst✝² : DivisionRing K
Γ₀ : Type v
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : MulArchimedean Γ₀
v : Valuation K Γ₀
x : K
hx : v x ≠ 0
r : Γ₀ˣ
hr : ↑r ≠ 0
h : ∀ (x : K), v x ≠ 0 → ↑r < v x
H : Units.mk0 (v x) hx = 1
⊢ v x = 1

pp: case inr.inr
K : Type u
inst✝² : DivisionRing K
Γ₀ : Type v
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : MulArchimedean Γ₀
v : Valuation K Γ₀
x : K
hx : v x ≠ 0
r : Γ₀ˣ
hr : ↑r ≠ 0
h : ∀ (x : K), v x ≠ 0 → ↑r < v x
H : 1 < Units.mk0 (v x) hx
⊢ v x = 1

Tactic:
·

Goal state after:
case inl
K : Type u
inst✝² : DivisionRing K
Γ₀ : Type v
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : MulArchimedean Γ₀
v : Valuation K Γ₀
x : K
hx : v x ≠ 0
r : Γ₀ˣ
hr : ↑r ≠ 0
h : ∀ (x : K), v x ≠ 0 → ↑r < v x
H : Units.mk0 (v x) hx < 1
⊢ v x = 1
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: case refine_1
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x y z : R
hx : 0 <ᵥ x
h : x * y ≤ᵥ x * z
⊢ y ≤ᵥ z

pp: case refine_2
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x y z : R
hx : 0 <ᵥ x
h : y ≤ᵥ z
⊢ x * y ≤ᵥ x * z

Tactic:
·

Goal state after:
case refine_1
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x y z : R
hx : 0 <ᵥ x
h : x * y ≤ᵥ x * z
⊢ y ≤ᵥ z
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: case refine_2
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x y z : R
hx : 0 <ᵥ x
h : y ≤ᵥ z
⊢ x * y ≤ᵥ x * z

Tactic:
·

Goal state after:
case refine_2
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x y z : R
hx : 0 <ᵥ x
h : y ≤ᵥ z
⊢ x * y ≤ᵥ x * z
Module: Mathlib.Topology.Algebra.Valued.ValuationTopology

Goal state before:
pp: case inr.inl
K : Type u
inst✝² : DivisionRing K
Γ₀ : Type v
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : MulArchimedean Γ₀
v : Valuation K Γ₀
x : K
hx : v x ≠ 0
r : Γ₀ˣ
hr : ↑r ≠ 0
h : ∀ (x : K), v x ≠ 0 → ↑r < v x
H : Units.mk0 (v x) hx = 1
⊢ v x = 1

pp: case inr.inr
K : Type u
inst✝² : DivisionRing K
Γ₀ : Type v
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : MulArchimedean Γ₀
v : Valuation K Γ₀
x : K
hx : v x ≠ 0
r : Γ₀ˣ
hr : ↑r ≠ 0
h : ∀ (x : K), v x ≠ 0 → ↑r < v x
H : 1 < Units.mk0 (v x) hx
⊢ v x = 1

Tactic:
·

Goal state after:
case inr.inl
K : Type u
inst✝² : DivisionRing K
Γ₀ : Type v
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : MulArchimedean Γ₀
v : Valuation K Γ₀
x : K
hx : v x ≠ 0
r : Γ₀ˣ
hr : ↑r ≠ 0
h : ∀ (x : K), v x ≠ 0 → ↑r < v x
H : Units.mk0 (v x) hx = 1
⊢ v x = 1
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Point

Goal state before:
pp: case neg
F : Type u
inst✝¹ : Field F
W : Jacobian F
inst✝ : DecidableEq F
P Q : Fin 3 → F
hP : W.Nonsingular P
hQ : W.Nonsingular Q
hPz : ¬P z = 0
hQz : ¬Q z = 0
hxy : ¬(P x * Q z ^ 2 = Q x * P z ^ 2 ∧ P y * Q z ^ 3 = W.negY Q * P z ^ 3)
⊢ toAffine W (W.add P Q) = toAffine W P + toAffine W Q

Tactic:
·

Goal state after:
case neg
F : Type u
inst✝¹ : Field F
W : Jacobian F
inst✝ : DecidableEq F
P Q : Fin 3 → F
hP : W.Nonsingular P
hQ : W.Nonsingular Q
hPz : ¬P z = 0
hQz : ¬Q z = 0
hxy : ¬(P x * Q z ^ 2 = Q x * P z ^ 2 ∧ P y * Q z ^ 3 = W.negY Q * P z ^ 3)
⊢ toAffine W (W.add P Q) = toAffine W P + toAffine W Q
Module: Mathlib.Topology.Algebra.Valued.ValuationTopology

Goal state before:
pp: case inr.inr
K : Type u
inst✝² : DivisionRing K
Γ₀ : Type v
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : MulArchimedean Γ₀
v : Valuation K Γ₀
x : K
hx : v x ≠ 0
r : Γ₀ˣ
hr : ↑r ≠ 0
h : ∀ (x : K), v x ≠ 0 → ↑r < v x
H : 1 < Units.mk0 (v x) hx
⊢ v x = 1

Tactic:
·

Goal state after:
case inr.inr
K : Type u
inst✝² : DivisionRing K
Γ₀ : Type v
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : MulArchimedean Γ₀
v : Valuation K Γ₀
x : K
hx : v x ≠ 0
r : Γ₀ˣ
hr : ↑r ≠ 0
h : ∀ (x : K), v x ≠ 0 → ↑r < v x
H : 1 < Units.mk0 (v x) hx
⊢ v x = 1
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: case left
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x x' y y' z r : R
u : ↥(posSubmonoid R)
s : R
v : ↥(posSubmonoid R)
t : R
w : ↥(posSubmonoid R)
h1 : r * ↑v ≤ᵥ s * ↑u
h2 : s * ↑u ≤ᵥ r * ↑v
h3 : s * ↑w ≤ᵥ t * ↑v
h4 : t * ↑v ≤ᵥ s * ↑w
⊢ r * ↑w ≤ᵥ t * ↑u

pp: case right
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x x' y y' z r : R
u : ↥(posSubmonoid R)
s : R
v : ↥(posSubmonoid R)
t : R
w : ↥(posSubmonoid R)
h1 : r * ↑v ≤ᵥ s * ↑u
h2 : s * ↑u ≤ᵥ r * ↑v
h3 : s * ↑w ≤ᵥ t * ↑v
h4 : t * ↑v ≤ᵥ s * ↑w
⊢ t * ↑u ≤ᵥ r * ↑w

Tactic:
·

Goal state after:
case left
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x x' y y' z r : R
u : ↥(posSubmonoid R)
s : R
v : ↥(posSubmonoid R)
t : R
w : ↥(posSubmonoid R)
h1 : r * ↑v ≤ᵥ s * ↑u
h2 : s * ↑u ≤ᵥ r * ↑v
h3 : s * ↑w ≤ᵥ t * ↑v
h4 : t * ↑v ≤ᵥ s * ↑w
⊢ r * ↑w ≤ᵥ t * ↑u
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Point

Goal state before:
pp: case pos
F : Type u
inst✝¹ : Field F
W : Jacobian F
inst✝ : DecidableEq F
P Q : Fin 3 → F
hP : W.Nonsingular P
hQ : W.Nonsingular Q
hPz : ¬P z = 0
hQz : ¬Q z = 0
hxy : ¬(P x * Q z ^ 2 = Q x * P z ^ 2 ∧ P y * Q z ^ 3 = W.negY Q * P z ^ 3)
this :
  toAffine W
      ![W.toAffine.addX (P x / P z ^ 2) (Q x / Q z ^ 2)
          (W.toAffine.slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3)),
        W.toAffine.addY (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3)
          (W.toAffine.slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3)),
        1] =
    toAffine W P + toAffine W Q
hx : P x * Q z ^ 2 = Q x * P z ^ 2
⊢ toAffine W (W.add P Q) = toAffine W P + toAffine W Q

pp: case neg
F : Type u
inst✝¹ : Field F
W : Jacobian F
inst✝ : DecidableEq F
P Q : Fin 3 → F
hP : W.Nonsingular P
hQ : W.Nonsingular Q
hPz : ¬P z = 0
hQz : ¬Q z = 0
hxy : ¬(P x * Q z ^ 2 = Q x * P z ^ 2 ∧ P y * Q z ^ 3 = W.negY Q * P z ^ 3)
this :
  toAffine W
      ![W.toAffine.addX (P x / P z ^ 2) (Q x / Q z ^ 2)
          (W.toAffine.slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3)),
        W.toAffine.addY (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3)
          (W.toAffine.slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3)),
        1] =
    toAffine W P + toAffine W Q
hx : ¬P x * Q z ^ 2 = Q x * P z ^ 2
⊢ toAffine W (W.add P Q) = toAffine W P + toAffine W Q

Tactic:
·

Goal state after:
case pos
F : Type u
inst✝¹ : Field F
W : Jacobian F
inst✝ : DecidableEq F
P Q : Fin 3 → F
hP : W.Nonsingular P
hQ : W.Nonsingular Q
hPz : ¬P z = 0
hQz : ¬Q z = 0
hxy : ¬(P x * Q z ^ 2 = Q x * P z ^ 2 ∧ P y * Q z ^ 3 = W.negY Q * P z ^ 3)
this :
  toAffine W
      ![W.toAffine.addX (P x / P z ^ 2) (Q x / Q z ^ 2)
          (W.toAffine.slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3)),
        W.toAffine.addY (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3)
          (W.toAffine.slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3)),
        1] =
    toAffine W P + toAffine W Q
hx : P x * Q z ^ 2 = Q x * P z ^ 2
⊢ toAffine W (W.add P Q) = toAffine W P + toAffine W Q
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: case right
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x x' y y' z r : R
u : ↥(posSubmonoid R)
s : R
v : ↥(posSubmonoid R)
t : R
w : ↥(posSubmonoid R)
h1 : r * ↑v ≤ᵥ s * ↑u
h2 : s * ↑u ≤ᵥ r * ↑v
h3 : s * ↑w ≤ᵥ t * ↑v
h4 : t * ↑v ≤ᵥ s * ↑w
⊢ t * ↑u ≤ᵥ r * ↑w

Tactic:
·

Goal state after:
case right
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x x' y y' z r : R
u : ↥(posSubmonoid R)
s : R
v : ↥(posSubmonoid R)
t : R
w : ↥(posSubmonoid R)
h1 : r * ↑v ≤ᵥ s * ↑u
h2 : s * ↑u ≤ᵥ r * ↑v
h3 : s * ↑w ≤ᵥ t * ↑v
h4 : t * ↑v ≤ᵥ s * ↑w
⊢ t * ↑u ≤ᵥ r * ↑w
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Point

Goal state before:
pp: case pos
F : Type u
inst✝¹ : Field F
W : Jacobian F
inst✝ : DecidableEq F
P Q : Fin 3 → F
hP : W.Nonsingular P
hQ : W.Nonsingular Q
hPz : ¬P z = 0
hQz : ¬Q z = 0
hxy : ¬(P x * Q z ^ 2 = Q x * P z ^ 2 ∧ P y * Q z ^ 3 = W.negY Q * P z ^ 3)
this :
  toAffine W
      ![W.toAffine.addX (P x / P z ^ 2) (Q x / Q z ^ 2)
          (W.toAffine.slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3)),
        W.toAffine.addY (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3)
          (W.toAffine.slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3)),
        1] =
    toAffine W P + toAffine W Q
hx : P x * Q z ^ 2 = Q x * P z ^ 2
⊢ toAffine W (W.add P Q) = toAffine W P + toAffine W Q

Tactic:
rwa [add_of_Y_ne' hP.left hQ.left hPz hQz hx <| not_and.mp hxy hx,
            toAffine_smul _ <| isUnit_dblZ_of_Y_ne' hP.left hQ.left hPz hx <| not_and.mp hxy hx]

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Point

Goal state before:
pp: case neg
F : Type u
inst✝¹ : Field F
W : Jacobian F
inst✝ : DecidableEq F
P Q : Fin 3 → F
hP : W.Nonsingular P
hQ : W.Nonsingular Q
hPz : ¬P z = 0
hQz : ¬Q z = 0
hxy : ¬(P x * Q z ^ 2 = Q x * P z ^ 2 ∧ P y * Q z ^ 3 = W.negY Q * P z ^ 3)
this :
  toAffine W
      ![W.toAffine.addX (P x / P z ^ 2) (Q x / Q z ^ 2)
          (W.toAffine.slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3)),
        W.toAffine.addY (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3)
          (W.toAffine.slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3)),
        1] =
    toAffine W P + toAffine W Q
hx : ¬P x * Q z ^ 2 = Q x * P z ^ 2
⊢ toAffine W (W.add P Q) = toAffine W P + toAffine W Q

Tactic:
·

Goal state after:
case neg
F : Type u
inst✝¹ : Field F
W : Jacobian F
inst✝ : DecidableEq F
P Q : Fin 3 → F
hP : W.Nonsingular P
hQ : W.Nonsingular Q
hPz : ¬P z = 0
hQz : ¬Q z = 0
hxy : ¬(P x * Q z ^ 2 = Q x * P z ^ 2 ∧ P y * Q z ^ 3 = W.negY Q * P z ^ 3)
this :
  toAffine W
      ![W.toAffine.addX (P x / P z ^ 2) (Q x / Q z ^ 2)
          (W.toAffine.slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3)),
        W.toAffine.addY (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3)
          (W.toAffine.slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3)),
        1] =
    toAffine W P + toAffine W Q
hx : ¬P x * Q z ^ 2 = Q x * P z ^ 2
⊢ toAffine W (W.add P Q) = toAffine W P + toAffine W Q
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: case h₁
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x✝ x' y✝ y' z✝ x y z w : R
t s u v : ↥(posSubmonoid R)
h₁ : x * ↑t ≤ᵥ y * ↑s
h₂ : y * ↑s ≤ᵥ x * ↑t
h₃ : z * ↑u ≤ᵥ w * ↑v
h₄ : w * ↑v ≤ᵥ z * ↑u
⊢ x * z * ↑(t * u) ≤ᵥ y * w * ↑(s * v)

pp: case h₂
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x✝ x' y✝ y' z✝ x y z w : R
t s u v : ↥(posSubmonoid R)
h₁ : x * ↑t ≤ᵥ y * ↑s
h₂ : y * ↑s ≤ᵥ x * ↑t
h₃ : z * ↑u ≤ᵥ w * ↑v
h₄ : w * ↑v ≤ᵥ z * ↑u
⊢ y * w * ↑(s * v) ≤ᵥ x * z * ↑(t * u)

Tactic:
·

Goal state after:
case h₁
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x✝ x' y✝ y' z✝ x y z w : R
t s u v : ↥(posSubmonoid R)
h₁ : x * ↑t ≤ᵥ y * ↑s
h₂ : y * ↑s ≤ᵥ x * ↑t
h₃ : z * ↑u ≤ᵥ w * ↑v
h₄ : w * ↑v ≤ᵥ z * ↑u
⊢ x * z * ↑(t * u) ≤ᵥ y * w * ↑(s * v)
Module: Mathlib.Topology.Algebra.Valued.ValuationTopology

Goal state before:
pp: case inl
R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation R Γ₀
x : R
γ : (ofClass v).ValueGroup₀ˣ
Hx : v x = 0
⊢ ∃ j,
    ↑(v.ltAddSubgroup ((Units.map ↑embedding) j)) ⊆
      (fun x_1 ↦ x * x_1) ⁻¹' ↑(v.ltAddSubgroup ((Units.map ↑embedding) γ))

pp: case inr
R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation R Γ₀
x : R
γ : (ofClass v).ValueGroup₀ˣ
γx : Γ₀ˣ
Hx : v x = ↑γx
⊢ ∃ j,
    ↑(v.ltAddSubgroup ((Units.map ↑embedding) j)) ⊆
      (fun x_1 ↦ x * x_1) ⁻¹' ↑(v.ltAddSubgroup ((Units.map ↑embedding) γ))

Tactic:
·

Goal state after:
case inl
R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation R Γ₀
x : R
γ : (ofClass v).ValueGroup₀ˣ
Hx : v x = 0
⊢ ∃ j,
    ↑(v.ltAddSubgroup ((Units.map ↑embedding) j)) ⊆
      (fun x_1 ↦ x * x_1) ⁻¹' ↑(v.ltAddSubgroup ((Units.map ↑embedding) γ))
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: case h₂
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x✝ x' y✝ y' z✝ x y z w : R
t s u v : ↥(posSubmonoid R)
h₁ : x * ↑t ≤ᵥ y * ↑s
h₂ : y * ↑s ≤ᵥ x * ↑t
h₃ : z * ↑u ≤ᵥ w * ↑v
h₄ : w * ↑v ≤ᵥ z * ↑u
⊢ y * w * ↑(s * v) ≤ᵥ x * z * ↑(t * u)

Tactic:
·

Goal state after:
case h₂
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x✝ x' y✝ y' z✝ x y z w : R
t s u v : ↥(posSubmonoid R)
h₁ : x * ↑t ≤ᵥ y * ↑s
h₂ : y * ↑s ≤ᵥ x * ↑t
h₃ : z * ↑u ≤ᵥ w * ↑v
h₄ : w * ↑v ≤ᵥ z * ↑u
⊢ y * w * ↑(s * v) ≤ᵥ x * z * ↑(t * u)
Module: Mathlib.Topology.Algebra.Valued.ValuationTopology

Goal state before:
pp: case inr
R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation R Γ₀
x : R
γ : (ofClass v).ValueGroup₀ˣ
γx : Γ₀ˣ
Hx : v x = ↑γx
⊢ ∃ j,
    ↑(v.ltAddSubgroup ((Units.map ↑embedding) j)) ⊆
      (fun x_1 ↦ x * x_1) ⁻¹' ↑(v.ltAddSubgroup ((Units.map ↑embedding) γ))

Tactic:
·

Goal state after:
case inr
R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation R Γ₀
x : R
γ : (ofClass v).ValueGroup₀ˣ
γx : Γ₀ˣ
Hx : v x = ↑γx
⊢ ∃ j,
    ↑(v.ltAddSubgroup ((Units.map ↑embedding) j)) ⊆
      (fun x_1 ↦ x * x_1) ⁻¹' ↑(v.ltAddSubgroup ((Units.map ↑embedding) γ))
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: case mk.mk.h₁
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x x' y y' z x✝¹ : R
y✝¹ : ↥(posSubmonoid R)
x✝ : R
y✝ : ↥(posSubmonoid R)
⊢ x✝¹ * x✝ * ↑(y✝ * y✝¹) ≤ᵥ x✝ * x✝¹ * ↑(y✝¹ * y✝)

Tactic:
·

Goal state after:
case mk.mk.h₁
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x x' y y' z x✝¹ : R
y✝¹ : ↥(posSubmonoid R)
x✝ : R
y✝ : ↥(posSubmonoid R)
⊢ x✝¹ * x✝ * ↑(y✝ * y✝¹) ≤ᵥ x✝ * x✝¹ * ↑(y✝¹ * y✝)
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: case h₂
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x x' y y' z x✝¹ : R
y✝¹ : ↥(posSubmonoid R)
x✝ : R
y✝ : ↥(posSubmonoid R)
⊢ x✝ * x✝¹ * ↑(y✝¹ * y✝) ≤ᵥ x✝¹ * x✝ * ↑(y✝ * y✝¹)

Tactic:
·

Goal state after:
case h₂
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x x' y y' z x✝¹ : R
y✝¹ : ↥(posSubmonoid R)
x✝ : R
y✝ : ↥(posSubmonoid R)
⊢ x✝ * x✝¹ * ↑(y✝¹ * y✝) ≤ᵥ x✝¹ * x✝ * ↑(y✝ * y✝¹)
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x x' y y' z x✝¹ : R
x✝ : ↥(posSubmonoid R)
⊢ 0 * ValueGroupWithZero.mk x✝¹ x✝ = 0

Tactic:
rw [← ValueGroupWithZero.mk_zero 1, ValueGroupWithZero.mk_mul_mk]
    simp

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x x' y y' z x✝¹ : R
x✝ : ↥(posSubmonoid R)
⊢ ValueGroupWithZero.mk x✝¹ x✝ * 0 = 0

Tactic:
rw [← ValueGroupWithZero.mk_zero 1, ValueGroupWithZero.mk_mul_mk]
    simp

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Point

Goal state before:
pp: case pos
R : Type r
S : Type s
A F : Type u
B K : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : CommRing A
inst✝³ : CommRing B
inst✝² : Field F
inst✝¹ : Field K
W' : Jacobian R
W : Jacobian F
inst✝ : DecidableEq F
point✝ : PointClass F
P : Fin 3 → F
hP : W.NonsingularLift (Quot.mk (⇑(MulAction.orbitRel Fˣ (Fin 3 → F))) P)
hPz : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ fromAffine { point := Quot.mk (⇑(MulAction.orbitRel Fˣ (Fin 3 → F))) P, nonsingular := hP }.toAffineLift =
    { point := Quot.mk (⇑(MulAction.orbitRel Fˣ (Fin 3 → F))) P, nonsingular := hP }

pp: case neg
R : Type r
S : Type s
A F : Type u
B K : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : CommRing A
inst✝³ : CommRing B
inst✝² : Field F
inst✝¹ : Field K
W' : Jacobian R
W : Jacobian F
inst✝ : DecidableEq F
point✝ : PointClass F
P : Fin 3 → F
hP : W.NonsingularLift (Quot.mk (⇑(MulAction.orbitRel Fˣ (Fin 3 → F))) P)
hPz : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ fromAffine { point := Quot.mk (⇑(MulAction.orbitRel Fˣ (Fin 3 → F))) P, nonsingular := hP }.toAffineLift =
    { point := Quot.mk (⇑(MulAction.orbitRel Fˣ (Fin 3 → F))) P, nonsingular := hP }

Tactic:
·

Goal state after:
case pos
R : Type r
S : Type s
A F : Type u
B K : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : CommRing A
inst✝³ : CommRing B
inst✝² : Field F
inst✝¹ : Field K
W' : Jacobian R
W : Jacobian F
inst✝ : DecidableEq F
point✝ : PointClass F
P : Fin 3 → F
hP : W.NonsingularLift (Quot.mk (⇑(MulAction.orbitRel Fˣ (Fin 3 → F))) P)
hPz : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ fromAffine { point := Quot.mk (⇑(MulAction.orbitRel Fˣ (Fin 3 → F))) P, nonsingular := hP }.toAffineLift =
    { point := Quot.mk (⇑(MulAction.orbitRel Fˣ (Fin 3 → F))) P, nonsingular := hP }