-- Mathlib/AlgebraicGeometry/EllipticCurve/Projective.lean
lemma toAffine_add {P Q : Fin 3 → F} (hP : W.Nonsingular P) (hQ : W.Nonsingular Q) :
    toAffine W (W.add P Q) = toAffine W P + toAffine W Q

Proof state:
case neg
F : Type u
inst✝ : Field F
W : Projective 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:
by_cases hQz : Q z = 0

Result:
case pos
F : Type u
inst✝ : Field F
W : Projective 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

case neg
F : Type u
inst✝ : Field F
W : Projective 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

Proof state:
F : Type u
inst✝ : Field F
W : Projective F
P Q : Fin 3 → F
hP : W.Nonsingular P
hQ : W.Nonsingular Q
⊢ toAffine W (W.add P Q) = toAffine W P + toAffine W Q

Tactic:
by_cases hPz : P z = 0

Result:
case pos
F : Type u
inst✝ : Field F
W : Projective 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

case neg
F : Type u
inst✝ : Field F
W : Projective 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

Proof state:
case pos
F : Type u
inst✝ : Field F
W : Projective 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 = Q x * P z ∧ P y * Q z = W.negY Q * P z
hy : P y * Q z = Q y * P z
⊢ toAffine W (W.add P Q) = 0

Tactic:
rw [add_of_Y_eq hP.left hPz hQz hxy.left hy hxy.right,
  toAffine_smul _ <| isUnit_dblU_of_Y_eq hP hPz hQz hxy.left hy hxy.right, toAffine_zero]

Result:
no goals

Proof state:
case pos
F : Type u
inst✝ : Field F
W : Projective 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 = Q x * P z ∧ P y * Q z = W.negY Q * P z)
this :
  toAffine W
      ![(Projective.toAffine W).addX (P x / P z) (Q x / Q z)
          ((Projective.toAffine W).slope (P x / P z) (Q x / Q z) (P y / P z) (Q y / Q z)),
        (Projective.toAffine W).addY (P x / P z) (Q x / Q z) (P y / P z)
          ((Projective.toAffine W).slope (P x / P z) (Q x / Q z) (P y / P z) (Q y / Q z)),
        1] =
    toAffine W P + toAffine W Q
hx : P x * Q z = Q x * P z
⊢ 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 hQz hx <| not_and.mp hxy hx]

Result:
no goals

Proof state:
case pos
F : Type u
inst✝ : Field F
W : Projective 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 Q

Tactic:
by_cases hQz : Q z = 0

Result:
case pos
F : Type u
inst✝ : Field F
W : Projective 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

case neg
F : Type u
inst✝ : Field F
W : Projective 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

Proof state:
case pos
F : Type u
inst✝ : Field F
W : Projective 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:
rw [add_of_Z_eq_zero_right hQ.left hPz hQz,
  toAffine_smul _ (((isUnit_Y_of_Z_eq_zero hQ hQz).pow 2).mul <| Ne.isUnit hPz).neg,
  toAffine_of_Z_eq_zero hQz, add_zero]

Result:
no goals