Theorem: WeierstrassCurve.Affine.nonsingular_negAdd
File: Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean

tactic: by_cases hx₁ : W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₁
state_before: R : Type u
inst✝¹ : CommRing R
W✝ : Affine R
F : Type u
inst✝ : Field F
W : Affine F
x₁ x₂ y₁ y₂ : F
h₁ : W.Nonsingular x₁ y₁
h₂ : W.Nonsingular x₂ y₂
hxy : x₁ = x₂ → y₁ ≠ W.negY x₂ y₂
⊢ W.Nonsingular (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂)) (W.negAddY x₁ x₂ y₁ (W.slope x₁ x₂ y₁ y₂))
state_after: case pos
R : Type u
inst✝¹ : CommRing R
W✝ : Affine R
F : Type u
inst✝ : Field F
W : Affine F
x₁ x₂ y₁ y₂ : F
h₁ : W.Nonsingular x₁ y₁
h₂ : W.Nonsingular x₂ y₂
hxy : x₁ = x₂ → y₁ ≠ W.negY x₂ y₂
hx₁ : W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₁
⊢ W.Nonsingular (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂)) (W.negAddY x₁ x₂ y₁ (W.slope x₁ x₂ y₁ y₂))

case neg
R : Type u
inst✝¹ : CommRing R
W✝ : Affine R
F : Type u
inst✝ : Field F
W : Affine F
x₁ x₂ y₁ y₂ : F
h₁ : W.Nonsingular x₁ y₁
h₂ : W.Nonsingular x₂ y₂
hxy : x₁ = x₂ → y₁ ≠ W.negY x₂ y₂
hx₁ : ¬W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₁
⊢ W.Nonsingular (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂)) (W.negAddY x₁ x₂ y₁ (W.slope x₁ x₂ y₁ y₂))

tactic: rwa [negAddY, hx₁, sub_self, mul_zero, zero_add]
state_before: case pos
R : Type u
inst✝¹ : CommRing R
W✝ : Affine R
F : Type u
inst✝ : Field F
W : Affine F
x₁ x₂ y₁ y₂ : F
h₁ : W.Nonsingular x₁ y₁
h₂ : W.Nonsingular x₂ y₂
hxy : x₁ = x₂ → y₁ ≠ W.negY x₂ y₂
hx₁ : W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₁
⊢ W.Nonsingular (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂)) (W.negAddY x₁ x₂ y₁ (W.slope x₁ x₂ y₁ y₂))
state_after: no goals

tactic: by_cases hx₂ : W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₂
state_before: case neg
R : Type u
inst✝¹ : CommRing R
W✝ : Affine R
F : Type u
inst✝ : Field F
W : Affine F
x₁ x₂ y₁ y₂ : F
h₁ : W.Nonsingular x₁ y₁
h₂ : W.Nonsingular x₂ y₂
hxy : x₁ = x₂ → y₁ ≠ W.negY x₂ y₂
hx₁ : ¬W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₁
⊢ W.Nonsingular (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂)) (W.negAddY x₁ x₂ y₁ (W.slope x₁ x₂ y₁ y₂))
state_after: case pos
R : Type u
inst✝¹ : CommRing R
W✝ : Affine R
F : Type u
inst✝ : Field F
W : Affine F
x₁ x₂ y₁ y₂ : F
h₁ : W.Nonsingular x₁ y₁
h₂ : W.Nonsingular x₂ y₂
hxy : x₁ = x₂ → y₁ ≠ W.negY x₂ y₂
hx₁ : ¬W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₁
hx₂ : W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₂
⊢ W.Nonsingular (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂)) (W.negAddY x₁ x₂ y₁ (W.slope x₁ x₂ y₁ y₂))

case neg
R : Type u
inst✝¹ : CommRing R
W✝ : Affine R
F : Type u
inst✝ : Field F
W : Affine F
x₁ x₂ y₁ y₂ : F
h₁ : W.Nonsingular x₁ y₁
h₂ : W.Nonsingular x₂ y₂
hxy : x₁ = x₂ → y₁ ≠ W.negY x₂ y₂
hx₁ : ¬W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₁
hx₂ : ¬W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₂
⊢ W.Nonsingular (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂)) (W.negAddY x₁ x₂ y₁ (W.slope x₁ x₂ y₁ y₂))

tactic: by_cases hx : x₁ = x₂
state_before: case pos
R : Type u
inst✝¹ : CommRing R
W✝ : Affine R
F : Type u
inst✝ : Field F
W : Affine F
x₁ x₂ y₁ y₂ : F
h₁ : W.Nonsingular x₁ y₁
h₂ : W.Nonsingular x₂ y₂
hxy : x₁ = x₂ → y₁ ≠ W.negY x₂ y₂
hx₁ : ¬W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₁
hx₂ : W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₂
⊢ W.Nonsingular (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂)) (W.negAddY x₁ x₂ y₁ (W.slope x₁ x₂ y₁ y₂))
state_after: case pos
R : Type u
inst✝¹ : CommRing R
W✝ : Affine R
F : Type u
inst✝ : Field F
W : Affine F
x₁ x₂ y₁ y₂ : F
h₁ : W.Nonsingular x₁ y₁
h₂ : W.Nonsingular x₂ y₂
hxy : x₁ = x₂ → y₁ ≠ W.negY x₂ y₂
hx₁ : ¬W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₁
hx₂ : W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₂
hx : x₁ = x₂
⊢ W.Nonsingular (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂)) (W.negAddY x₁ x₂ y₁ (W.slope x₁ x₂ y₁ y₂))

case neg
R : Type u
inst✝¹ : CommRing R
W✝ : Affine R
F : Type u
inst✝ : Field F
W : Affine F
x₁ x₂ y₁ y₂ : F
h₁ : W.Nonsingular x₁ y₁
h₂ : W.Nonsingular x₂ y₂
hxy : x₁ = x₂ → y₁ ≠ W.negY x₂ y₂
hx₁ : ¬W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₁
hx₂ : W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₂
hx : ¬x₁ = x₂
⊢ W.Nonsingular (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂)) (W.negAddY x₁ x₂ y₁ (W.slope x₁ x₂ y₁ y₂))

tactic: subst hx
state_before: case pos
R : Type u
inst✝¹ : CommRing R
W✝ : Affine R
F : Type u
inst✝ : Field F
W : Affine F
x₁ x₂ y₁ y₂ : F
h₁ : W.Nonsingular x₁ y₁
h₂ : W.Nonsingular x₂ y₂
hxy : x₁ = x₂ → y₁ ≠ W.negY x₂ y₂
hx₁ : ¬W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₁
hx₂ : W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₂
hx : x₁ = x₂
⊢ W.Nonsingular (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂)) (W.negAddY x₁ x₂ y₁ (W.slope x₁ x₂ y₁ y₂))
state_after: case pos
R : Type u
inst✝¹ : CommRing R
W✝ : Affine R
F : Type u
inst✝ : Field F
W : Affine F
x₁ y₁ y₂ : F
h₁ : W.Nonsingular x₁ y₁
h₂ : W.Nonsingular x₁ y₂
hxy : x₁ = x₁ → y₁ ≠ W.negY x₁ y₂
hx₁ : ¬W.addX x₁ x₁ (W.slope x₁ x₁ y₁ y₂) = x₁
hx₂ : W.addX x₁ x₁ (W.slope x₁ x₁ y₁ y₂) = x₁
⊢ W.Nonsingular (W.addX x₁ x₁ (W.slope x₁ x₁ y₁ y₂)) (W.negAddY x₁ x₁ y₁ (W.slope x₁ x₁ y₁ y₂))

tactic: contradiction
state_before: case pos
R : Type u
inst✝¹ : CommRing R
W✝ : Affine R
F : Type u
inst✝ : Field F
W : Affine F
x₁ y₁ y₂ : F
h₁ : W.Nonsingular x₁ y₁
h₂ : W.Nonsingular x₁ y₂
hxy : x₁ = x₁ → y₁ ≠ W.negY x₁ y₂
hx₁ : ¬W.addX x₁ x₁ (W.slope x₁ x₁ y₁ y₂) = x₁
hx₂ : W.addX x₁ x₁ (W.slope x₁ x₁ y₁ y₂) = x₁
⊢ W.Nonsingular (W.addX x₁ x₁ (W.slope x₁ x₁ y₁ y₂)) (W.negAddY x₁ x₁ y₁ (W.slope x₁ x₁ y₁ y₂))
state_after: no goals

tactic: rwa [negAddY, ← neg_sub, mul_neg, hx₂, slope_of_X_ne hx,
  div_mul_cancel₀ _ <| sub_ne_zero_of_ne hx, neg_sub, sub_add_cancel]
state_before: case neg
R : Type u
inst✝¹ : CommRing R
W✝ : Affine R
F : Type u
inst✝ : Field F
W : Affine F
x₁ x₂ y₁ y₂ : F
h₁ : W.Nonsingular x₁ y₁
h₂ : W.Nonsingular x₂ y₂
hxy : x₁ = x₂ → y₁ ≠ W.negY x₂ y₂
hx₁ : ¬W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₁
hx₂ : W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₂
hx : ¬x₁ = x₂
⊢ W.Nonsingular (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂)) (W.negAddY x₁ x₂ y₁ (W.slope x₁ x₂ y₁ y₂))
state_after: no goals

tactic: apply nonsingular_negAdd_of_eval_derivative_ne_zero <| equation_negAdd h₁.1 h₂.1 hxy
state_before: case neg
R : Type u
inst✝¹ : CommRing R
W✝ : Affine R
F : Type u
inst✝ : Field F
W : Affine F
x₁ x₂ y₁ y₂ : F
h₁ : W.Nonsingular x₁ y₁
h₂ : W.Nonsingular x₂ y₂
hxy : x₁ = x₂ → y₁ ≠ W.negY x₂ y₂
hx₁ : ¬W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₁
hx₂ : ¬W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₂
⊢ W.Nonsingular (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂)) (W.negAddY x₁ x₂ y₁ (W.slope x₁ x₂ y₁ y₂))
state_after: case neg
R : Type u
inst✝¹ : CommRing R
W✝ : Affine R
F : Type u
inst✝ : Field F
W : Affine F
x₁ x₂ y₁ y₂ : F
h₁ : W.Nonsingular x₁ y₁
h₂ : W.Nonsingular x₂ y₂
hxy : x₁ = x₂ → y₁ ≠ W.negY x₂ y₂
hx₁ : ¬W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₁
hx₂ : ¬W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₂
⊢ eval (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂)) (derivative (W.addPolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂))) ≠ 0

tactic: rw [derivative_addPolynomial_slope h₁.left h₂.left hxy]
state_before: case neg
R : Type u
inst✝¹ : CommRing R
W✝ : Affine R
F : Type u
inst✝ : Field F
W : Affine F
x₁ x₂ y₁ y₂ : F
h₁ : W.Nonsingular x₁ y₁
h₂ : W.Nonsingular x₂ y₂
hxy : x₁ = x₂ → y₁ ≠ W.negY x₂ y₂
hx₁ : ¬W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₁
hx₂ : ¬W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₂
⊢ eval (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂)) (derivative (W.addPolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂))) ≠ 0
state_after: case neg
R : Type u
inst✝¹ : CommRing R
W✝ : Affine R
F : Type u
inst✝ : Field F
W : Affine F
x₁ x₂ y₁ y₂ : F
h₁ : W.Nonsingular x₁ y₁
h₂ : W.Nonsingular x₂ y₂
hxy : x₁ = x₂ → y₁ ≠ W.negY x₂ y₂
hx₁ : ¬W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₁
hx₂ : ¬W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₂
⊢ eval (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂))
      (-((X - C x₁) * (X - C x₂) + (X - C x₁) * (X - C (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂))) +
          (X - C x₂) * (X - C (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂))))) ≠
    0

tactic: eval_simp
state_before: case neg
R : Type u
inst✝¹ : CommRing R
W✝ : Affine R
F : Type u
inst✝ : Field F
W : Affine F
x₁ x₂ y₁ y₂ : F
h₁ : W.Nonsingular x₁ y₁
h₂ : W.Nonsingular x₂ y₂
hxy : x₁ = x₂ → y₁ ≠ W.negY x₂ y₂
hx₁ : ¬W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₁
hx₂ : ¬W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₂
⊢ eval (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂))
      (-((X - C x₁) * (X - C x₂) + (X - C x₁) * (X - C (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂))) +
          (X - C x₂) * (X - C (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂))))) ≠
    0
state_after: case neg
R : Type u
inst✝¹ : CommRing R
W✝ : Affine R
F : Type u
inst✝ : Field F
W : Affine F
x₁ x₂ y₁ y₂ : F
h₁ : W.Nonsingular x₁ y₁
h₂ : W.Nonsingular x₂ y₂
hxy : x₁ = x₂ → y₁ ≠ W.negY x₂ y₂
hx₁ : ¬W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₁
hx₂ : ¬W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₂
⊢ -((W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) - x₁) * (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) - x₂) +
          (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) - x₁) *
            (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) - W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂)) +
        (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) - x₂) *
          (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) - W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂))) ≠
    0

tactic: simpa only [neg_ne_zero, sub_self, mul_zero, add_zero] using
  mul_ne_zero (sub_ne_zero_of_ne hx₁) (sub_ne_zero_of_ne hx₂)
state_before: case neg
R : Type u
inst✝¹ : CommRing R
W✝ : Affine R
F : Type u
inst✝ : Field F
W : Affine F
x₁ x₂ y₁ y₂ : F
h₁ : W.Nonsingular x₁ y₁
h₂ : W.Nonsingular x₂ y₂
hxy : x₁ = x₂ → y₁ ≠ W.negY x₂ y₂
hx₁ : ¬W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₁
hx₂ : ¬W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₂
⊢ -((W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) - x₁) * (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) - x₂) +
          (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) - x₁) *
            (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) - W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂)) +
        (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) - x₂) *
          (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) - W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂))) ≠
    0
state_after: no goals