-- Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean
lemma nonsingular_add {P Q : Fin 3 → F} (hP : W.Nonsingular P) (hQ : W.Nonsingular Q) :
    W.Nonsingular <| W.add P Q

Proof state:
case pos
F : Type u
inst✝ : Field F
W : Jacobian F
P Q : Fin 3 → F
hP : W.Nonsingular P
hQ : W.Nonsingular Q
hPz : ¬P z = 0
hQz : Q z = 0
⊢ W.Nonsingular (W.add P Q)

Tactic:
simpa only [add_of_Z_eq_zero_right hQ.left hPz hQz,
  nonsingular_smul _ ((isUnit_X_of_Z_eq_zero hQ hQz).mul <| Ne.isUnit hPz).neg]

Result:
no goals

Proof state:
case neg
F : Type u
inst✝ : Field F
W : Jacobian 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)
⊢ W.Nonsingular (W.add P Q)

Tactic:
have := nonsingular_add_of_Z_ne_zero hP hQ hPz hQz hxy

Result:
case neg
F : Type u
inst✝ : Field F
W : Jacobian 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 :
  W.Nonsingular
    ![(toAffine W).addX (P x / P z ^ 2) (Q x / Q z ^ 2)
        ((toAffine W).slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3)),
      (toAffine W).addY (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3)
        ((toAffine W).slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3)),
      1]
⊢ W.Nonsingular (W.add P Q)

Proof state:
case neg
F : Type u
inst✝ : Field F
W : Jacobian 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 :
  W.Nonsingular
    ![(toAffine W).addX (P x / P z ^ 2) (Q x / Q z ^ 2)
        ((toAffine W).slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3)),
      (toAffine W).addY (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3)
        ((toAffine W).slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3)),
      1]
⊢ W.Nonsingular (W.add P Q)

Tactic:
by_cases hx : P x * Q z ^ 2 = Q x * P z ^ 2

Result:
case pos
F : Type u
inst✝ : Field F
W : Jacobian 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 :
  W.Nonsingular
    ![(toAffine W).addX (P x / P z ^ 2) (Q x / Q z ^ 2)
        ((toAffine W).slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3)),
      (toAffine W).addY (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3)
        ((toAffine W).slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3)),
      1]
hx : P x * Q z ^ 2 = Q x * P z ^ 2
⊢ W.Nonsingular (W.add P Q)

case neg
F : Type u
inst✝ : Field F
W : Jacobian 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 :
  W.Nonsingular
    ![(toAffine W).addX (P x / P z ^ 2) (Q x / Q z ^ 2)
        ((toAffine W).slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3)),
      (toAffine W).addY (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3)
        ((toAffine W).slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3)),
      1]
hx : ¬P x * Q z ^ 2 = Q x * P z ^ 2
⊢ W.Nonsingular (W.add P Q)

Proof state:
case neg
F : Type u
inst✝ : Field F
W : Jacobian 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 :
  W.Nonsingular
    ![(toAffine W).addX (P x / P z ^ 2) (Q x / Q z ^ 2)
        ((toAffine W).slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3)),
      (toAffine W).addY (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3)
        ((toAffine W).slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3)),
      1]
hx : ¬P x * Q z ^ 2 = Q x * P z ^ 2
⊢ W.Nonsingular (W.add P Q)

Tactic:
simpa only [add_of_X_ne hP.left hQ.left hPz hQz hx,
  nonsingular_smul _ <| isUnit_addZ_of_X_ne hx]

Result:
no goals

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

Tactic:
by_cases hQz : Q z = 0

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

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

Proof state:
case pos
F : Type u
inst✝ : Field F
W : Jacobian F
P Q : Fin 3 → F
hP : W.Nonsingular P
hQ : W.Nonsingular Q
hPz : P z = 0
⊢ W.Nonsingular (W.add P Q)

Tactic:
by_cases hQz : Q z = 0

Result:
case pos
F : Type u
inst✝ : Field F
W : Jacobian F
P Q : Fin 3 → F
hP : W.Nonsingular P
hQ : W.Nonsingular Q
hPz : P z = 0
hQz : Q z = 0
⊢ W.Nonsingular (W.add P Q)

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

Proof state:
case neg
F : Type u
inst✝ : Field F
W : Jacobian 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
⊢ W.Nonsingular (W.add P Q)

Tactic:
simp only [add_of_Y_ne hP.left hQ.left hPz hQz hxy.left hy,
  nonsingular_smul _ <| isUnit_addU_of_Y_ne hPz hQz hy, nonsingular_zero]

Result:
no goals

Proof state:
case pos
F : Type u
inst✝ : Field F
W : Jacobian 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
⊢ W.Nonsingular (W.add P Q)

Tactic:
simp only [add_of_Y_eq hPz hQz hxy.left hy hxy.right, nonsingular_smul _ <|
    isUnit_dblU_of_Y_eq hP hPz hQz hxy.left hy hxy.right, nonsingular_zero]

Result:
no goals

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

Tactic:
simpa only [add_of_Z_eq_zero_left hP.left hPz hQz,
  nonsingular_smul _ <| (isUnit_X_of_Z_eq_zero hP hPz).mul <| Ne.isUnit hQz]

Result:
no goals

Proof state:
case pos
F : Type u
inst✝ : Field F
W : Jacobian 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 :
  W.Nonsingular
    ![(toAffine W).addX (P x / P z ^ 2) (Q x / Q z ^ 2)
        ((toAffine W).slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3)),
      (toAffine W).addY (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3)
        ((toAffine W).slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3)),
      1]
hx : P x * Q z ^ 2 = Q x * P z ^ 2
⊢ W.Nonsingular (W.add P Q)

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

Result:
no goals