mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree

Goal state before:
pp: case even.right
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
dm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n :=
  fun {m n} {p q} ↦ natDegree_mul_le_of_le
dp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} ↦ natDegree_pow_le_of_le n
cm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).coeff (m + n) = p.coeff m * q.coeff n :=
  fun {m n} {p q} ↦ coeff_mul_add_eq_of_natDegree_le
cp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ n → (p ^ m).coeff (m * n) = p.coeff n ^ m :=
  fun {m n} {p} ↦ coeff_pow_of_natDegree_le
m : ℕ
h₁ : (W.preΨ' (m + 1)).natDegree ≤ expDegree (m + 1) ∧ (W.preΨ' (m + 1)).coeff (expDegree (m + 1)) = ↑(expCoeff (m + 1))
h₂ : (W.preΨ' (m + 2)).natDegree ≤ expDegree (m + 2) ∧ (W.preΨ' (m + 2)).coeff (expDegree (m + 2)) = ↑(expCoeff (m + 2))
h₃ : (W.preΨ' (m + 3)).natDegree ≤ expDegree (m + 3) ∧ (W.preΨ' (m + 3)).coeff (expDegree (m + 3)) = ↑(expCoeff (m + 3))
h₄ : (W.preΨ' (m + 4)).natDegree ≤ expDegree (m + 4) ∧ (W.preΨ' (m + 4)).coeff (expDegree (m + 4)) = ↑(expCoeff (m + 4))
h₅ : (W.preΨ' (m + 5)).natDegree ≤ expDegree (m + 5) ∧ (W.preΨ' (m + 5)).coeff (expDegree (m + 5)) = ↑(expCoeff (m + 5))
⊢ (W.preΨ' (m + 2) ^ 2).coeff (2 * expDegree (m + 2)) * (W.preΨ' (m + 3)).coeff (expDegree (m + 3)) *
        (W.preΨ' (m + 5)).coeff (expDegree (m + 5)) -
      (W.preΨ' (m + 1) * W.preΨ' (m + 3) * W.preΨ' (m + 4) ^ 2).coeff (expDegree (2 * (m + 3))) =
    ↑(expCoeff (2 * (m + 3)))

Tactic:
cp h₂.1,

Goal state after:
case even.right
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
dm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n :=
  fun {m n} {p q} ↦ natDegree_mul_le_of_le
dp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} ↦ natDegree_pow_le_of_le n
cm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).coeff (m + n) = p.coeff m * q.coeff n :=
  fun {m n} {p q} ↦ coeff_mul_add_eq_of_natDegree_le
cp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ n → (p ^ m).coeff (m * n) = p.coeff n ^ m :=
  fun {m n} {p} ↦ coeff_pow_of_natDegree_le
m : ℕ
h₁ : (W.preΨ' (m + 1)).natDegree ≤ expDegree (m + 1) ∧ (W.preΨ' (m + 1)).coeff (expDegree (m + 1)) = ↑(expCoeff (m + 1))
h₂ : (W.preΨ' (m + 2)).natDegree ≤ expDegree (m + 2) ∧ (W.preΨ' (m + 2)).coeff (expDegree (m + 2)) = ↑(expCoeff (m + 2))
h₃ : (W.preΨ' (m + 3)).natDegree ≤ expDegree (m + 3) ∧ (W.preΨ' (m + 3)).coeff (expDegree (m + 3)) = ↑(expCoeff (m + 3))
h₄ : (W.preΨ' (m + 4)).natDegree ≤ expDegree (m + 4) ∧ (W.preΨ' (m + 4)).coeff (expDegree (m + 4)) = ↑(expCoeff (m + 4))
h₅ : (W.preΨ' (m + 5)).natDegree ≤ expDegree (m + 5) ∧ (W.preΨ' (m + 5)).coeff (expDegree (m + 5)) = ↑(expCoeff (m + 5))
⊢ (W.preΨ' (m + 2)).coeff (expDegree (m + 2)) ^ 2 * (W.preΨ' (m + 3)).coeff (expDegree (m + 3)) *
        (W.preΨ' (m + 5)).coeff (expDegree (m + 5)) -
      (W.preΨ' (m + 1) * W.preΨ' (m + 3) * W.preΨ' (m + 4) ^ 2).coeff (expDegree (2 * (m + 3))) =
    ↑(expCoeff (2 * (m + 3)))
Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree

Goal state before:
pp: case odd.left
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
dm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n :=
  fun {m n} {p q} ↦ natDegree_mul_le_of_le
dp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} ↦ natDegree_pow_le_of_le n
cm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).coeff (m + n) = p.coeff m * q.coeff n :=
  fun {m n} {p q} ↦ coeff_mul_add_eq_of_natDegree_le
cp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ n → (p ^ m).coeff (m * n) = p.coeff n ^ m :=
  fun {m n} {p} ↦ coeff_pow_of_natDegree_le
m : ℕ
h₁ : (W.preΨ' (m + 1)).natDegree ≤ expDegree (m + 1) ∧ (W.preΨ' (m + 1)).coeff (expDegree (m + 1)) = ↑(expCoeff (m + 1))
h₂ : (W.preΨ' (m + 2)).natDegree ≤ expDegree (m + 2) ∧ (W.preΨ' (m + 2)).coeff (expDegree (m + 2)) = ↑(expCoeff (m + 2))
h₃ : (W.preΨ' (m + 3)).natDegree ≤ expDegree (m + 3) ∧ (W.preΨ' (m + 3)).coeff (expDegree (m + 3)) = ↑(expCoeff (m + 3))
h₄ : (W.preΨ' (m + 4)).natDegree ≤ expDegree (m + 4) ∧ (W.preΨ' (m + 4)).coeff (expDegree (m + 4)) = ↑(expCoeff (m + 4))
⊢ ((W.preΨ' (m + 4) * W.preΨ' (m + 2) ^ 3 * if Even m then W.Ψ₂Sq ^ 2 else 1) -
        W.preΨ' (m + 1) * W.preΨ' (m + 3) ^ 3 * if Even m then 1 else W.Ψ₂Sq ^ 2).natDegree ≤
    max (expDegree (m + 4) + 3 * expDegree (m + 2) + if Even m then 2 * 3 else 0) (expDegree (2 * (m + 2) + 1))

Tactic:
(expDegree_rec m).2.2

Goal state after:
case odd.left
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
dm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n :=
  fun {m n} {p q} ↦ natDegree_mul_le_of_le
dp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} ↦ natDegree_pow_le_of_le n
cm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).coeff (m + n) = p.coeff m * q.coeff n :=
  fun {m n} {p q} ↦ coeff_mul_add_eq_of_natDegree_le
cp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ n → (p ^ m).coeff (m * n) = p.coeff n ^ m :=
  fun {m n} {p} ↦ coeff_pow_of_natDegree_le
m : ℕ
h₁ : (W.preΨ' (m + 1)).natDegree ≤ expDegree (m + 1) ∧ (W.preΨ' (m + 1)).coeff (expDegree (m + 1)) = ↑(expCoeff (m + 1))
h₂ : (W.preΨ' (m + 2)).natDegree ≤ expDegree (m + 2) ∧ (W.preΨ' (m + 2)).coeff (expDegree (m + 2)) = ↑(expCoeff (m + 2))
h₃ : (W.preΨ' (m + 3)).natDegree ≤ expDegree (m + 3) ∧ (W.preΨ' (m + 3)).coeff (expDegree (m + 3)) = ↑(expCoeff (m + 3))
h₄ : (W.preΨ' (m + 4)).natDegree ≤ expDegree (m + 4) ∧ (W.preΨ' (m + 4)).coeff (expDegree (m + 4)) = ↑(expCoeff (m + 4))
⊢ ((W.preΨ' (m + 4) * W.preΨ' (m + 2) ^ 3 * if Even m then W.Ψ₂Sq ^ 2 else 1) -
        W.preΨ' (m + 1) * W.preΨ' (m + 3) ^ 3 * if Even m then 1 else W.Ψ₂Sq ^ 2).natDegree ≤
    max (expDegree (m + 4) + 3 * expDegree (m + 2) + if Even m then 2 * 3 else 0)
      (expDegree (m + 1) + 3 * expDegree (m + 3) + if Even m then 0 else 2 * 3)
Module: Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ

Goal state before:
pp: case h.a₆
F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝⁴ : E.IsElliptic
inst✝³ : E'.IsElliptic
inst✝² : CharP F 3
inst✝¹ : E.IsCharThreeJNeZeroNF
inst✝ : E'.IsCharThreeJNeZeroNF
heq : E.j = E'.j
ha₂ : E.a₂ ≠ 0
ha₆ : E.a₆ ≠ 0
ha₂' : E'.a₂ ≠ 0
ha₆' : E'.a₆ ≠ 0
this : NeZero 2
u : F
hu : u ^ 2 = E.a₂ / E'.a₂
hu0 : u ≠ 0
⊢ ({ u := Units.mk0 u hu0, r := 0, s := 0, t := 0 } • E).a₆ = E'.a₆

Tactic:
j_of_isCharThreeJNeZeroNF_of_char_three,

Goal state after:
case h.a₆
F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝⁴ : E.IsElliptic
inst✝³ : E'.IsElliptic
inst✝² : CharP F 3
inst✝¹ : E.IsCharThreeJNeZeroNF
inst✝ : E'.IsCharThreeJNeZeroNF
ha₂ : E.a₂ ≠ 0
ha₆ : E.a₆ ≠ 0
ha₂' : E'.a₂ ≠ 0
ha₆' : E'.a₆ ≠ 0
this : NeZero 2
u : F
hu : u ^ 2 = E.a₂ / E'.a₂
hu0 : u ≠ 0
heq : -E.a₂ ^ 3 / E.a₆ = -E'.a₂ ^ 3 / E'.a₆
⊢ ({ u := Units.mk0 u hu0, r := 0, s := 0, t := 0 } • E).a₆ = E'.a₆
Module: Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ

Goal state before:
pp: F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E✝ E'✝ : WeierstrassCurve F
inst✝⁴ : E✝.IsElliptic
inst✝³ : E'✝.IsElliptic
p : ℕ
inst✝² : CharP F p
hchar2 : 2 ≠ 0
hchar3 : 3 ≠ 0
this✝³ : NeZero 2
this✝² : NeZero 4
this✝¹ : NeZero 6
this✝ : Invertible 2 := invertibleOfNonzero hchar2
this : Invertible 3 := invertibleOfNonzero hchar3
E : WeierstrassCurve F
inst✝¹ : E.IsElliptic
h✝¹ : E.IsShortNF
E' : WeierstrassCurve F
inst✝ : E'.IsElliptic
h✝ : E'.IsShortNF
heq : E.a₄ ^ 3 * E'.a₆ ^ 2 = E'.a₄ ^ 3 * E.a₆ ^ 2
ha₄ : E.a₄ = 0
ha₆ : E.a₆ ≠ 0
⊢ E'.a₄ = 0

Tactic:
rw [ha₄, zero_pow three_ne_zero, zero_mul, zero_eq_mul] at heq

Goal state after:
F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E✝ E'✝ : WeierstrassCurve F
inst✝⁴ : E✝.IsElliptic
inst✝³ : E'✝.IsElliptic
p : ℕ
inst✝² : CharP F p
hchar2 : 2 ≠ 0
hchar3 : 3 ≠ 0
this✝³ : NeZero 2
this✝² : NeZero 4
this✝¹ : NeZero 6
this✝ : Invertible 2 := invertibleOfNonzero hchar2
this : Invertible 3 := invertibleOfNonzero hchar3
E : WeierstrassCurve F
inst✝¹ : E.IsElliptic
h✝¹ : E.IsShortNF
E' : WeierstrassCurve F
inst✝ : E'.IsElliptic
h✝ : E'.IsShortNF
heq : E'.a₄ ^ 3 = 0 ∨ E.a₆ ^ 2 = 0
ha₄ : E.a₄ = 0
ha₆ : E.a₆ ≠ 0
⊢ E'.a₄ = 0
Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree

Goal state before:
pp: case nat
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
n : ℕ
⊢ (W.preΨ ↑n).natDegree ≤ ((↑n).natAbs ^ 2 - if Even ↑n then 4 else 1) / 2

Tactic:
| nat n =>

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree

Goal state before:
pp: case nat
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
n : ℕ
⊢ (W.preΨ ↑n).coeff (((↑n).natAbs ^ 2 - if Even ↑n then 4 else 1) / 2) = ↑(if Even ↑n then ↑n / 2 else ↑n)

Tactic:
| nat n =>

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree

Goal state before:
pp: case nat
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
n : ℕ
h : ↑↑n ≠ 0
⊢ (W.preΨ ↑n).coeff (((↑n).natAbs ^ 2 - if Even ↑n then 4 else 1) / 2) ≠ 0

Tactic:
| nat n =>

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree

Goal state before:
pp: case nat
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
n : ℕ
hn : 2 < (↑n).natAbs
h : ↑↑n ≠ 0
⊢ 0 < (W.preΨ ↑n).natDegree

Tactic:
| nat n =>

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree

Goal state before:
pp: case nat
R : Type u
inst✝¹ : CommRing R
W : WeierstrassCurve R
inst✝ : Nontrivial R
n : ℕ
h : ↑↑n ≠ 0
⊢ W.preΨ ↑n ≠ 0

Tactic:
| nat n =>

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree

Goal state before:
pp: case nat
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
n : ℕ
⊢ (W.ΨSq ↑n).natDegree ≤ (↑n).natAbs ^ 2 - 1

Tactic:
| nat n =>

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree

Goal state before:
pp: case nat
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
n : ℕ
⊢ (W.ΨSq ↑n).coeff ((↑n).natAbs ^ 2 - 1) = ↑↑n ^ 2

Tactic:
| nat n =>

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

Goal state before:
pp: case refine_1
F : Type u
inst✝¹ : Field F
W : Jacobian F
K : Type v
inst✝ : Field K
f : F →+* K
P Q : Fin 3 → F
hP : W.Nonsingular P
hQ : W.Nonsingular Q
h : ⇑f ∘ P ≈ ⇑f ∘ Q
⊢ P ≈ Q

Tactic:
by_cases hz : f (P z) = 0
    · exact equiv_of_Z_eq_zero hP hQ ((map_eq_zero_iff f f.injective).mp hz) <|
        (map_eq_zero_iff f f.injective).mp <| (Z_eq_zero_of_equiv h).mp hz
    · refine equiv_of_X_eq_of_Y_eq ((map_ne_zero_iff f f.injective).mp hz)
        ((map_ne_zero_iff f f.injective).mp <| hz.comp (Z_eq_zero_of_equiv h).mpr) ?_ ?_
      all_goals apply f.injective; map_simp
      exacts [X_eq_of_equiv h, Y_eq_of_equiv h]

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

Goal state before:
pp: R : Type r
inst✝¹⁰ : CommRing R
W' : Jacobian R
S : Type s
inst✝⁹ : CommRing S
A : Type u
inst✝⁸ : CommRing A
B : Type v
inst✝⁷ : CommRing B
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
⊢ (W'⁄B).polynomialY = (MvPolynomial.map ↑f) (W'⁄A).polynomialY

Tactic:
rw [← map_polynomialY, map_baseChange]

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

Goal state before:
pp: F : Type u
inst✝ : Field F
W : Jacobian F
P : Fin 3 → F
hP : W.Nonsingular P
hPz : P z = 0
⊢ W.neg P = -(P y / P x) • ![1, 1, 0]

Tactic:
have hX {n : ℕ} : IsUnit <| P x ^ n := (isUnit_X_of_Z_eq_zero hP hPz).pow n
  erw [neg_of_Z_eq_zero' hPz, smul_fin3, neg_sq, div_pow, (equation_of_Z_eq_zero hPz).mp hP.left,
    pow_succ, hX.mul_div_cancel_left, mul_one, Odd.neg_pow <| by decide, div_pow, pow_succ,
    (equation_of_Z_eq_zero hPz).mp hP.left, hX.mul_div_cancel_left, mul_one, mul_zero]

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree

Goal state before:
pp: case nat
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
n : ℕ
⊢ (W.Φ ↑n).natDegree ≤ (↑n).natAbs ^ 2

Tactic:
| nat n =>

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree

Goal state before:
pp: case nat
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
n : ℕ
⊢ (W.Φ ↑n).coeff ((↑n).natAbs ^ 2) = 1

Tactic:
| nat n =>

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree

Goal state before:
pp: R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
n : ℤ
⊢ (W.Φ n).coeff (n.natAbs ^ 2) = 1

Tactic:
by
  induction n using Int.negInduction with
  | nat n => exact (W.natDegree_coeff_Φ_ofNat n).right
  | neg ih => rw [Φ_neg, Int.natAbs_neg, ih]

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

Goal state before:
pp: 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]

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

Goal state before:
pp: case neg
F : Type u
inst✝ : Field F
W : Jacobian F
P : Fin 3 → F
Q✝ : PointClass F
hP : W.Nonsingular P
hPz : P z = 0
Q : Fin 3 → F
hQ : W.NonsingularLift ⟦Q⟧
hQz : ¬Q z = 0
⊢ ⟦W.add P Q⟧ = ⟦Q⟧

Tactic:
add_of_Z_eq_zero_left hP.left hPz hQz,

Goal state after:
case neg
F : Type u
inst✝ : Field F
W : Jacobian F
P : Fin 3 → F
Q✝ : PointClass F
hP : W.Nonsingular P
hPz : P z = 0
Q : Fin 3 → F
hQ : W.NonsingularLift ⟦Q⟧
hQz : ¬Q z = 0
⊢ ⟦(P x * Q z) • Q⟧ = ⟦Q⟧
Module: Mathlib.Topology.Algebra.UniformFilterBasis

Goal state before:
pp: G : Type u_1
inst✝ : AddCommGroup G
B : AddGroupFilterBasis G
F : Filter G
this✝ : UniformSpace G := B.uniformSpace
this : IsUniformAddGroup G
⊢ F ×ˢ F ≤ uniformity G ↔ ∀ U ∈ B, ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, y - x ∈ U

Tactic:
rw [uniformity_eq_comap_nhds_zero G, ← map_le_iff_le_comap]

Goal state after:
G : Type u_1
inst✝ : AddCommGroup G
B : AddGroupFilterBasis G
F : Filter G
this✝ : UniformSpace G := B.uniformSpace
this : IsUniformAddGroup G
⊢ map (fun x ↦ x.2 - x.1) (F ×ˢ F) ≤ nhds 0 ↔ ∀ U ∈ B, ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, y - x ∈ U
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:
add_of_Z_eq_zero_left hP.left hPz hQz,

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 ((P x * Q z) • 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 P + toAffine W Q

Tactic:
by_cases hxy : P x * Q z ^ 2 = Q x * P z ^ 2 ∧ P y * Q z ^ 3 = W.negY Q * P z ^ 3
      · rw [toAffine_of_Z_ne_zero hP hPz, toAffine_of_Z_ne_zero hQ hQz, Affine.Point.add_of_Y_eq
            ((X_eq_iff hPz hQz).mp hxy.left) ((Y_eq_iff' hPz hQz).mp hxy.right)]
        by_cases hy : P y * Q z ^ 3 = Q y * P z ^ 3
        · rw [add_of_Y_eq hPz hQz hxy.left hy hxy.right,
            toAffine_smul _ <| isUnit_dblU_of_Y_eq hP hPz hQz hxy.left hy hxy.right, toAffine_zero]
        · rw [add_of_Y_ne hP.left hQ.left hPz hQz hxy.left hy,
            toAffine_smul _ <| isUnit_addU_of_Y_ne hPz hQz hy, toAffine_zero]
      · have := toAffine_add_of_Z_ne_zero hP hQ hPz hQz hxy
        by_cases hx : P x * Q z ^ 2 = Q x * P z ^ 2
        · 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]
        · rwa [add_of_X_ne hP.left hQ.left hPz hQz hx, toAffine_smul _ <| isUnit_addZ_of_X_ne hx]

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

Goal state before:
pp: case neg
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
hw : w ≤ᵥ 0
hz : ¬z ≤ᵥ 0
⊢ (x * ↑v ≤ᵥ z * ↑s) = (y * ↑u ≤ᵥ w * ↑t)

Tactic:
absurd hz
      apply vle_mul_cancel u.prop
      simpa using h₃.trans (mul_vle_mul_left hw v)

Goal state after:
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
_i : Valued R Γ₀
r : (ofClass v).ValueGroup₀
hr : r ≠ 0
⊢ ∀ x ∈ {x | v.restrict x < r}, {x | v.restrict x < r} ∈ 𝓝 x

Tactic:
intro x hx

Goal state after:
case inr
R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
_i : Valued R Γ₀
r : (ofClass v).ValueGroup₀
hr : r ≠ 0
x : R
hx : x ∈ {x | v.restrict x < r}
⊢ {x | v.restrict x < r} ∈ 𝓝 x
Module: Mathlib.Topology.Algebra.Valued.ValuationTopology

Goal state before:
pp: R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
_i : Valued R Γ₀
r : (ofClass v).ValueGroup₀
hr : r ≠ 0
⊢ ∀ x ∈ {x | v.restrict x ≤ r}, {x | v.restrict x ≤ r} ∈ 𝓝 x

Tactic:
intro x hx

Goal state after:
R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
_i : Valued R Γ₀
r : (ofClass v).ValueGroup₀
hr : r ≠ 0
x : R
hx : x ∈ {x | v.restrict x ≤ r}
⊢ {x | v.restrict x ≤ r} ∈ 𝓝 x
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: K : Type u_2
inst✝¹ : DivisionRing K
inst✝ : ValuativeRel K
a : K
⊢ a = 0 → a =ᵥ 0

Tactic:
simp +contextual

Goal state after:
Module: Mathlib.Topology.Algebra.Valued.ValuationTopology

Goal state before:
pp: R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
_i : Valued R Γ₀
r : (ofClass v).ValueGroup₀
⊢ ∀ x ∈ {x | v.restrict x ≤ r}ᶜ, {x | v.restrict x ≤ r}ᶜ ∈ 𝓝 x

Tactic:
intro x hx

Goal state after:
R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
_i : Valued R Γ₀
r : (ofClass v).ValueGroup₀
x : R
hx : x ∈ {x | v.restrict x ≤ r}ᶜ
⊢ {x | v.restrict x ≤ r}ᶜ ∈ 𝓝 x
Module: Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology

Goal state before:
pp: R : Type u_1
inst✝⁵ : Ring R
inst✝⁴ : ValuativeRel R
Γ₀ : Type u_3
inst✝³ : LinearOrderedCommGroupWithZero Γ₀
_u : UniformSpace R
inst✝² : IsUniformAddGroup R
inst✝¹ : IsValuativeTopology R
v : Valuation R Γ₀
inst✝ : v.Compatible
⊢ _u = IsTopologicalAddGroup.rightUniformSpace R

Tactic:
by
  refine UniformSpace.ext (v.hasBasis_uniformity.eq_of_same_basis ?_)
  convert! v.subgroups_basis.hasBasis_nhds_zero.comap _
  simp [restrict_lt_iff_lt_embedding, sub_eq_add_neg]

Goal state after:
Module: Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology

Goal state before:
pp: R : Type u_1
inst✝⁴ : Ring R
inst✝³ : ValuativeRel R
Γ₀ : Type u_3
inst✝² : LinearOrderedCommGroupWithZero Γ₀
_t : TopologicalSpace R
inst✝¹ : IsValuativeTopology R
v : Valuation R Γ₀
inst✝ : v.Compatible
⊢ _t = ⋯.topology

Tactic:
let u := IsTopologicalAddGroup.rightUniformSpace R

Goal state after:
R : Type u_1
inst✝⁴ : Ring R
inst✝³ : ValuativeRel R
Γ₀ : Type u_3
inst✝² : LinearOrderedCommGroupWithZero Γ₀
_t : TopologicalSpace R
inst✝¹ : IsValuativeTopology R
v : Valuation R Γ₀
inst✝ : v.Compatible
u : UniformSpace R := IsTopologicalAddGroup.rightUniformSpace R
⊢ _t = ⋯.topology
Module: Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology

Goal state before:
pp: case inr
R : Type u_1
inst✝⁴ : Ring R
inst✝³ : ValuativeRel R
Γ₀ : Type u_3
inst✝² : LinearOrderedCommGroupWithZero Γ₀
_t : TopologicalSpace R
inst✝¹ : IsValuativeTopology R
v : Valuation R Γ₀
inst✝ : v.Compatible
r : (ofClass v).ValueGroup₀
hr : r ≠ 0
⊢ ∀ x ∈ {x | v.restrict x < r}, {x | v.restrict x < r} ∈ 𝓝 x

Tactic:
intro x hx

Goal state after:
case inr
R : Type u_1
inst✝⁴ : Ring R
inst✝³ : ValuativeRel R
Γ₀ : Type u_3
inst✝² : LinearOrderedCommGroupWithZero Γ₀
_t : TopologicalSpace R
inst✝¹ : IsValuativeTopology R
v : Valuation R Γ₀
inst✝ : v.Compatible
r : (ofClass v).ValueGroup₀
hr : r ≠ 0
x : R
hx : x ∈ {x | v.restrict x < r}
⊢ {x | v.restrict x < r} ∈ 𝓝 x
Module: Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology

Goal state before:
pp: R : Type u_1
inst✝⁴ : Ring R
inst✝³ : ValuativeRel R
Γ₀ : Type u_3
inst✝² : LinearOrderedCommGroupWithZero Γ₀
_t : TopologicalSpace R
inst✝¹ : IsValuativeTopology R
v : Valuation R Γ₀
inst✝ : v.Compatible
r : (ofClass v).ValueGroup₀
hr : r ≠ 0
⊢ ∀ x ∈ {x | v.restrict x ≤ r}, {x | v.restrict x ≤ r} ∈ 𝓝 x

Tactic:
intro x hx

Goal state after:
R : Type u_1
inst✝⁴ : Ring R
inst✝³ : ValuativeRel R
Γ₀ : Type u_3
inst✝² : LinearOrderedCommGroupWithZero Γ₀
_t : TopologicalSpace R
inst✝¹ : IsValuativeTopology R
v : Valuation R Γ₀
inst✝ : v.Compatible
r : (ofClass v).ValueGroup₀
hr : r ≠ 0
x : R
hx : x ∈ {x | v.restrict x ≤ r}
⊢ {x | v.restrict x ≤ r} ∈ 𝓝 x
Module: Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology

Goal state before:
pp: R : Type u_1
inst✝⁴ : Ring R
inst✝³ : ValuativeRel R
Γ₀ : Type u_3
inst✝² : LinearOrderedCommGroupWithZero Γ₀
_t : TopologicalSpace R
inst✝¹ : IsValuativeTopology R
v : Valuation R Γ₀
inst✝ : v.Compatible
r : (ofClass v).ValueGroup₀
⊢ ∀ x ∈ {x | v.restrict x ≤ r}ᶜ, {x | v.restrict x ≤ r}ᶜ ∈ 𝓝 x

Tactic:
intro x hx

Goal state after:
R : Type u_1
inst✝⁴ : Ring R
inst✝³ : ValuativeRel R
Γ₀ : Type u_3
inst✝² : LinearOrderedCommGroupWithZero Γ₀
_t : TopologicalSpace R
inst✝¹ : IsValuativeTopology R
v : Valuation R Γ₀
inst✝ : v.Compatible
r : (ofClass v).ValueGroup₀
x : R
hx : x ∈ {x | v.restrict x ≤ r}ᶜ
⊢ {x | v.restrict x ≤ r}ᶜ ∈ 𝓝 x
Module: Mathlib.Topology.Algebra.UniformField

Goal state before:
pp: K : Type u_1
inst✝² : Field K
inst✝¹ : UniformSpace K
inst✝ : CompletableTopField K
x : hat K
h : x ≠ 0
y : hat K
y_ne : y ∈ {0}ᶜ
⊢ y ∈ {x | (fun x ↦ ∃ c, Tendsto (fun x ↦ ↑x⁻¹) (Filter.comap coe' (𝓝 x)) (𝓝 c)) x}

Tactic:
mem_compl_singleton_iff

Goal state after:
K : Type u_1
inst✝² : Field K
inst✝¹ : UniformSpace K
inst✝ : CompletableTopField K
x : hat K
h : x ≠ 0
y : hat K
y_ne : y ≠ 0
⊢ y ∈ {x | (fun x ↦ ∃ c, Tendsto (fun x ↦ ↑x⁻¹) (Filter.comap coe' (𝓝 x)) (𝓝 c)) x}
Module: Mathlib.Topology.Algebra.UniformField

Goal state before:
pp: case a
K : Type u_1
inst✝² : Field K
inst✝¹ : UniformSpace K
inst✝ : CompletableTopField K
x : hat K
h : x ≠ 0
y : hat K
y_ne : y ≠ 0
this✝ : (fun x ↦ ↑x⁻¹) = (fun y ↦ ↑y) ∘ fun x ↦ x⁻¹
this : (Filter.comap coe' (𝓝 y)).NeBot
⊢ Filter.comap (fun p ↦ (↑p.1, ↑p.2)) (uniformity (hat K)) ≤ uniformity K

Tactic:
rw [Completion.comap_coe_eq_uniformity]

Goal state after:
Module: Mathlib.Topology.Algebra.UniformField

Goal state before:
pp: case neg
K : Type u_1
inst✝³ : Field K
inst✝² : UniformSpace K
inst✝¹ : IsTopologicalDivisionRing K
inst✝ : CompletableTopField K
x : K
h : ¬x = 0
⊢ (↑x)⁻¹ = ↑x⁻¹

Tactic:
conv_lhs => dsimp [Inv.inv]
    rw [if_neg]
    · exact hatInv_extends h
    · exact fun H => h (isDenseEmbedding_coe.injective H)

Goal state after: