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

Goal state before:
pp: case some.some
F : Type u
inst✝¹ : Field F
W : Affine F
inst✝ : DecidableEq F
x₁ y₁ : F
h✝¹ : W.Nonsingular x₁ y₁
x₂ y₂ : F
h✝ : W.Nonsingular x₂ y₂
⊢ some x₁ y₁ h✝¹ + some x₂ y₂ h✝ = 0 ↔ some x₁ y₁ h✝¹ = -some x₂ y₂ h✝

Tactic:
neg_some,

Goal state after:
case some.some
F : Type u
inst✝¹ : Field F
W : Affine F
inst✝ : DecidableEq F
x₁ y₁ : F
h✝¹ : W.Nonsingular x₁ y₁
x₂ y₂ : F
h✝ : W.Nonsingular x₂ y₂
⊢ some x₁ y₁ h✝¹ + some x₂ y₂ h✝ = 0 ↔ some x₁ y₁ h✝¹ = some x₂ (W.negY x₂ y₂) ⋯
Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree

Goal state before:
pp: R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
n : ℕ
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
⊢ (W.preΨ' n).natDegree ≤ expDegree n ∧ (W.preΨ' n).coeff (expDegree n) = ↑(expCoeff n)

Tactic:
let cp {m n p} : _ → (p ^ m : R[X]).coeff (m * n) = _ := coeff_pow_of_natDegree_le

Goal state after:
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
n : ℕ
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
⊢ (W.preΨ' n).natDegree ≤ expDegree n ∧ (W.preΨ' n).coeff (expDegree n) = ↑(expCoeff n)
Module: Mathlib.AlgebraicGeometry.EllipticCurve.NormalForms

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
W : WeierstrassCurve R
inst✝¹ : W.IsCharTwoJNeZeroNF
inst✝ : CharP R 2
⊢ W.c₆ = 1

Tactic:
rw [c₆_of_isCharTwoJNeZeroNF, b₂_of_isCharTwoJNeZeroNF_of_char_two]
  linear_combination (-1 - 432 * W.a₆) * CharP.cast_eq_zero R 2

Goal state after:
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 := ⋯
dp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := ⋯
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 := ⋯
cp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ n → (p ^ m).coeff (m * n) = p.coeff n ^ m := ⋯
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' (2 * (m + 3))).coeff (expDegree (2 * (m + 3))) = ↑(expCoeff (2 * (m + 3)))

Tactic:
nth_rw 1 [preΨ'_even, coeff_sub, (expDegree_rec m).1.1, cm (dm (dp h₂.1) h₃.1) h₅.1,
        cm (dp h₂.1) h₃.1, cp h₂.1, h₂.2, h₃.2, h₅.2, (expDegree_rec m).1.2,
        cm (dm h₁.1 h₃.1) (dp h₄.1), cm h₁.1 h₃.1, h₁.2, cp h₄.1, h₃.2, h₄.2, (expCoeff_rec m).1]
      norm_cast

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ

Goal state before:
pp: F : Type u_1
inst✝⁴ : Field F
inst✝³ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝² : E.IsElliptic
inst✝¹ : E'.IsElliptic
inst✝ : CharP F 2
heq : E.j = E'.j
⊢ ∃ C, C • E = E'

Tactic:
by
  obtain ⟨C, _ | _⟩ := E.exists_variableChange_isCharTwoNF
  · obtain ⟨C', _ | _⟩ := E'.exists_variableChange_isCharTwoNF
    · simp_rw [← variableChange_j E C, ← variableChange_j E' C',
        j_of_isCharTwoJNeZeroNF_of_char_two, one_div, inv_inj] at heq
      obtain ⟨C'', hC⟩ := exists_variableChange_of_char_two_of_j_ne_zero _ _ heq
      use C'⁻¹ * C'' * C
      rw [mul_smul, mul_smul, hC, ← mul_smul, inv_mul_cancel, one_smul]
    · have h := (C • E).j_ne_zero_of_isCharTwoJNeZeroNF_of_char_two
      rw [variableChange_j, heq, ← variableChange_j E' C',
        j_of_isCharTwoJEqZeroNF_of_char_two] at h
      exact False.elim (h rfl)
  · obtain ⟨C', _ | _⟩ := E'.exists_variableChange_isCharTwoNF
    · have h := (C' • E').j_ne_zero_of_isCharTwoJNeZeroNF_of_char_two
      rw [variableChange_j, ← heq, ← variableChange_j E C,
        j_of_isCharTwoJEqZeroNF_of_char_two] at h
      exact False.elim (h rfl)
    · obtain ⟨C'', hC⟩ := exists_variableChange_of_char_two_of_j_eq_zero (C • E) (C' • E')
      use C'⁻¹ * C'' * C
      rw [mul_smul, mul_smul, hC, ← mul_smul, inv_mul_cancel, one_smul]

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ

Goal state before:
pp: F : Type u_1
inst✝⁴ : Field F
inst✝³ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝² : E.IsElliptic
inst✝¹ : E'.IsElliptic
p : ℕ
inst✝ : CharP F p
heq : E.j = E'.j
hchar2 : 2 ≠ 0
hchar3 : 3 ≠ 0
this✝² : NeZero 2
this✝¹ : NeZero 4
this✝ : NeZero 6
this : Invertible 2 := invertibleOfNonzero hchar2
⊢ ∃ C, C • E = E'

Tactic:
let : Invertible (3 : F) := invertibleOfNonzero hchar3

Goal state after:
F : Type u_1
inst✝⁴ : Field F
inst✝³ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝² : E.IsElliptic
inst✝¹ : E'.IsElliptic
p : ℕ
inst✝ : CharP F p
heq : E.j = E'.j
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
⊢ ∃ C, C • E = E'
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic

Goal state before:
pp: 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
⊢ P ≈ Q

Tactic:
have hPx : IsUnit <| P x := isUnit_X_of_Z_eq_zero hP hPz
  have hPy : IsUnit <| P y := isUnit_Y_of_Z_eq_zero hP hPz
  have hQx : IsUnit <| Q x := isUnit_X_of_Z_eq_zero hQ hQz
  have hQy : IsUnit <| Q y := isUnit_Y_of_Z_eq_zero hQ hQz
  simp only [nonsingular_of_Z_eq_zero, equation_of_Z_eq_zero, hPz, hQz] at hP hQ
  use (hPy.unit / hPx.unit) * (hQx.unit / hQy.unit)
  simp only [Units.smul_def, smul_fin3, Units.val_mul, Units.val_div_eq_div_val, IsUnit.unit_spec,
    mul_pow, div_pow, hQz, mul_zero]
  conv_rhs => rw [← fin3_def P, hPz]
  congr! 2
  · rw [hP.left, pow_succ, (hPx.pow 2).mul_div_cancel_left, hQ.left, pow_succ _ 2,
      (hQx.pow 2).div_mul_cancel_left, hQx.inv_mul_cancel_right]
  · rw [← hP.left, pow_succ, (hPy.pow 2).mul_div_cancel_left, ← hQ.left, pow_succ _ 2,
      (hQy.pow 2).div_mul_cancel_left, hQy.inv_mul_cancel_right]

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

Goal state before:
pp: R : Type r
inst✝¹ : CommRing R
W' : Jacobian R
S : Type s
inst✝ : CommRing S
f : R →+* S
⊢ (W'.map f).polynomial = (MvPolynomial.map f) W'.polynomial

Tactic:
by
  simp only [polynomial]
  map_simp

Goal state after:
Module: Mathlib.Topology.Algebra.Nonarchimedean.Basic

Goal state before:
pp: G : Type u_1
inst✝⁵ : Group G
inst✝⁴ : TopologicalSpace G
inst✝³ : NonarchimedeanGroup G
K : Type u_3
inst✝² : Group K
inst✝¹ : TopologicalSpace K
inst✝ : NonarchimedeanGroup K
U : Set (G × K)
U₁ : Set G
hU₁ : U₁ ∈ 𝓝 1.1
U₂ : Set K
hU₂ : U₂ ∈ 𝓝 1.2
h : U₁ ×ˢ U₂ ⊆ U
⊢ ∃ V W, ↑V ×ˢ ↑W ⊆ U

Tactic:
obtain ⟨V, hV⟩ := is_nonarchimedean _ hU₁

Goal state after:
G : Type u_1
inst✝⁵ : Group G
inst✝⁴ : TopologicalSpace G
inst✝³ : NonarchimedeanGroup G
K : Type u_3
inst✝² : Group K
inst✝¹ : TopologicalSpace K
inst✝ : NonarchimedeanGroup K
U : Set (G × K)
U₁ : Set G
hU₁ : U₁ ∈ 𝓝 1.1
U₂ : Set K
hU₂ : U₂ ∈ 𝓝 1.2
h : U₁ ×ˢ U₂ ⊆ U
V : OpenSubgroup G
hV : ↑V ⊆ U₁
⊢ ∃ V W, ↑V ×ˢ ↑W ⊆ U
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Point

Goal state before:
pp: 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

Goal state after:
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)
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
⊢ W.Nonsingular (W.add P Q)

Tactic:
by_cases hQz : Q z = 0

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

Goal state before:
pp: 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⟧
⊢ W.addMap ⟦P⟧ ⟦Q⟧ = ⟦Q⟧

Tactic:
by_cases hQz : Q z = 0

Goal state after:
case pos
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.addMap ⟦P⟧ ⟦Q⟧ = ⟦Q⟧

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.addMap ⟦P⟧ ⟦Q⟧ = ⟦Q⟧
Module: Mathlib.Topology.Algebra.Nonarchimedean.Bases

Goal state before:
pp: case h.right
ι : Type u_1
R : Type u_2
A : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
M : Type u_4
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : TopologicalSpace R
inst✝ : Nonempty ι
B : ι → Submodule R M
hB : SubmodulesBasis B
x₀ : R
i : ι
m : M
⊢ m ∈ ↑(B i) → m ∈ (fun x ↦ x₀ • x) ⁻¹' ↑(B i)

Tactic:
exact (B i).smul_mem _

Goal state after:
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
⊢ toAffine W (W.add P Q) = toAffine W Q

Tactic:
by_cases hQz : Q z = 0

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

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:
by_cases hQz : Q z = 0

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

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.Topology.Algebra.ValuativeRel.ValuativeTopology

Goal state before:
pp: R : Type u_1
inst✝⁷ : Ring R
inst✝⁶ : ValuativeRel R
K : Type u_2
inst✝⁵ : DivisionRing K
inst✝⁴ : ValuativeRel K
Γ₀ : Type u_3
inst✝³ : LinearOrderedCommGroupWithZero Γ₀
inst✝² : TopologicalSpace R
v : Valuation R Γ₀
inst✝¹ : v.Compatible
inst✝ : IsValuativeTopology R
cts_add : ContinuousConstVAdd R R
⊢ IsTopologicalAddGroup R

Tactic:
have basis := (valuation R).hasBasis_nhds_zero

Goal state after:
R : Type u_1
inst✝⁷ : Ring R
inst✝⁶ : ValuativeRel R
K : Type u_2
inst✝⁵ : DivisionRing K
inst✝⁴ : ValuativeRel K
Γ₀ : Type u_3
inst✝³ : LinearOrderedCommGroupWithZero Γ₀
inst✝² : TopologicalSpace R
v : Valuation R Γ₀
inst✝¹ : v.Compatible
inst✝ : IsValuativeTopology R
cts_add : ContinuousConstVAdd R R
basis : (𝓝 0).HasBasis (fun x ↦ True) fun γ ↦ {x | (valuation R).restrict x < ↑γ}
⊢ IsTopologicalAddGroup R
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: case more
R : Type u_2
inst✝¹ : Ring R
inst✝ : ValuativeRel R
a b : R
hab : a <ᵥ b
n✝ : ℕ
a✝¹ : n✝ ≠ 0 → a ^ n✝ <ᵥ b ^ n✝
a✝ : n✝ + 1 ≠ 0 → a ^ (n✝ + 1) <ᵥ b ^ (n✝ + 1)
hn : n✝ + 2 ≠ 0
⊢ a ^ (n✝ + 2) <ᵥ b ^ (n✝ + 2)

Tactic:
simp_all [pow_succ, mul_vlt_mul]

Goal state after:
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
F : Filter R
⊢ Cauchy F ↔ F.NeBot ∧ ∀ (γ : (ofClass v).ValueGroup₀ˣ), ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, v.restrict (y - x) < ↑γ

Tactic:
v.toUniformSpace_eq,

Goal state after:
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
F : Filter R
⊢ Cauchy F ↔ F.NeBot ∧ ∀ (γ : (ofClass v).ValueGroup₀ˣ), ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, v.restrict (y - x) < ↑γ
Module: Mathlib.Topology.Algebra.UniformField

Goal state before:
pp: K✝ : Type u_1
inst✝⁴ : Field K✝
inst✝³ : UniformSpace K✝
L : Type u_2
inst✝² : Field L
inst✝¹ : UniformSpace L
inst✝ : CompletableTopField L
K : Subfield L
F : Filter ↥K
inf_F : 𝓝 0 ⊓ F = ⊥
i : ↥K →+* L := K.subtype
F_cau : Cauchy (map (⇑i) F)
hi : IsUniformInducing ⇑i
⊢ Cauchy (map (⇑i) (map (fun x ↦ x⁻¹) F))

Tactic:
map_comm (show (i ∘ fun x => x⁻¹) = (fun x => x⁻¹) ∘ i by ext; rfl)

Goal state after:
K✝ : Type u_1
inst✝⁴ : Field K✝
inst✝³ : UniformSpace K✝
L : Type u_2
inst✝² : Field L
inst✝¹ : UniformSpace L
inst✝ : CompletableTopField L
K : Subfield L
F : Filter ↥K
inf_F : 𝓝 0 ⊓ F = ⊥
i : ↥K →+* L := K.subtype
F_cau : Cauchy (map (⇑i) F)
hi : IsUniformInducing ⇑i
⊢ Cauchy (map (fun x ↦ x⁻¹) (map (⇑i) F))
Module: Mathlib.RingTheory.PrincipalIdealDomainOfPrime

Goal state before:
pp: case refine_2
R : Type u_1
inst✝ : CommSemiring R
I : Ideal R
a x : R
hx : I ⊔ span {a} = span {x}
y : R
hy : Submodule.colon I ↑(span {a}) = span {y}
⊢ I * span {y} ≤ I ∧ span {a * y} ≤ I

Tactic:
mul_comm a,

Goal state after:
case refine_2
R : Type u_1
inst✝ : CommSemiring R
I : Ideal R
a x : R
hx : I ⊔ span {a} = span {x}
y : R
hy : Submodule.colon I ↑(span {a}) = span {y}
⊢ I * span {y} ≤ I ∧ span {y * a} ≤ I
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W' : Jacobian R
P Q : Fin 3 → R
hQ : W'.Equation Q
hQz : Q z = 0
⊢ -W'.negAddY P Q - W'.a₁ * W'.addX P Q * addZ P Q - W'.a₃ * addZ P Q ^ 3 = (-(Q x * P z)) ^ 3 * P y

Tactic:
negAddY_of_Z_eq_zero_right hQ hQz,

Goal state after:
R : Type r
inst✝ : CommRing R
W' : Jacobian R
P Q : Fin 3 → R
hQ : W'.Equation Q
hQz : Q z = 0
⊢ -((-(Q x * P z)) ^ 3 * W'.negY P) - W'.a₁ * W'.addX P Q * addZ P Q - W'.a₃ * addZ P Q ^ 3 = (-(Q x * P z)) ^ 3 * P y
Module: Mathlib.RingTheory.Valuation.Discrete.Basic

Goal state before:
pp: Γ : Type u_1
inst✝² : LinearOrderedCommGroupWithZero Γ
A : Type u_2
inst✝¹ : Ring A
v : Valuation A Γ
inst✝ : v.IsRankOneDiscrete
⊢ zpowers ↑(generator' v) = Subgroup.map (MonoidWithZeroHom.ofClass v).valueGroup.subtype ⊤

Tactic:
← MonoidHom.range_eq_map,

Goal state after:
Γ : Type u_1
inst✝² : LinearOrderedCommGroupWithZero Γ
A : Type u_2
inst✝¹ : Ring A
v : Valuation A Γ
inst✝ : v.IsRankOneDiscrete
⊢ zpowers ↑(generator' v) = (MonoidWithZeroHom.ofClass v).valueGroup.subtype.range
Module: Mathlib.Topology.Algebra.Valued.WithVal

Goal state before:
pp: case h
R : Type u_4
Γ₀ : Type u_5
Γ₀' : Type u_6
inst✝² : Ring R
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ₀'
v : Valuation R Γ₀
w : Valuation R Γ₀'
hval : Valued R Γ₀'
hv : Valued.v = w
h : w.IsEquiv v
γ : (ofClass Valued.v).ValueGroup₀ˣ
r s : WithVal v
hr₀ : 0 < Valued.v r
hs₀ : 0 < Valued.v s
hr : Valued.v.restrict r / Valued.v.restrict s = ↑γ
h' : w.restrict.IsEquiv v.restrict
x : R
hx : Valued.v.restrict x < Valued.v.restrict r.ofVal / Valued.v.restrict s.ofVal
⊢ 0 < Valued.v.restrict s.ofVal

Tactic:
restrict_pos_iff,

Goal state after:
case h
R : Type u_4
Γ₀ : Type u_5
Γ₀' : Type u_6
inst✝² : Ring R
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ₀'
v : Valuation R Γ₀
w : Valuation R Γ₀'
hval : Valued R Γ₀'
hv : Valued.v = w
h : w.IsEquiv v
γ : (ofClass Valued.v).ValueGroup₀ˣ
r s : WithVal v
hr₀ : 0 < Valued.v r
hs₀ : 0 < Valued.v s
hr : Valued.v.restrict r / Valued.v.restrict s = ↑γ
h' : w.restrict.IsEquiv v.restrict
x : R
hx : Valued.v.restrict x < Valued.v.restrict r.ofVal / Valued.v.restrict s.ofVal
⊢ 0 < Valued.v s.ofVal
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
F : Filter K
hF : Cauchy F
h0 : 𝓝 0 ⊓ F = ⊥
⊢ ∃ γ₀, ∃ M ∈ F, ∀ x ∈ M, ↑γ₀ ≤ v.restrict x

Tactic:
rcases Filter.inf_eq_bot_iff.mp h0 with ⟨U, U_in, M, M_in, H⟩

Goal state after:
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
F : Filter K
hF : Cauchy F
h0 : 𝓝 0 ⊓ F = ⊥
U : Set K
U_in : U ∈ 𝓝 0
M : Set K
M_in : M ∈ F
H : U ∩ M = ∅
⊢ ∃ γ₀, ∃ M ∈ F, ∀ x ∈ M, ↑γ₀ ≤ v.restrict x
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: case mp
G : Type u_1
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
inst✝ : MulArchimedean G
b : G
n : ℤ
h : IsLeast {y | y ∈ closure {b ^ n} ∧ 1 < y} b
this✝ : closure {b ^ n} = closure {b}
this : (∃ n_1, (b ^ n) ^ n_1 = b) ∧ 1 < b
m : ℤ
hm : (b ^ n) ^ m = b
key : m = 1 ∧ n = 1 ∨ m = -1 ∧ n = -1
⊢ |b ^ n|ₘ = b ∧ 1 < b

Tactic:
rcases key with ⟨rfl, rfl⟩ | ⟨rfl, rfl⟩

Goal state after:
case mp.inl
G : Type u_1
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
inst✝ : MulArchimedean G
b : G
h : IsLeast {y | y ∈ closure {b ^ 1} ∧ 1 < y} b
this✝ : closure {b ^ 1} = closure {b}
this : (∃ n, (b ^ 1) ^ n = b) ∧ 1 < b
hm : (b ^ 1) ^ 1 = b
⊢ |b ^ 1|ₘ = b ∧ 1 < b

case mp.inr
G : Type u_1
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
inst✝ : MulArchimedean G
b : G
h : IsLeast {y | y ∈ closure {b ^ (-1)} ∧ 1 < y} b
this✝ : closure {b ^ (-1)} = closure {b}
this : (∃ n, (b ^ (-1)) ^ n = b) ∧ 1 < b
hm : (b ^ (-1)) ^ (-1) = b
⊢ |b ^ (-1)|ₘ = b ∧ 1 < b
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: case inr
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x₀ : hat K
h : x₀ ≠ 0
preimage_one : ⇑v ⁻¹' {1} ∈ 𝓝 1
V : Set (hat K)
V_in : V ∈ 𝓝 1
hV : ∀ (x : K), ↑x ∈ V → v x = 1
V' : Set (hat K)
V'_in : V' ∈ 𝓝 1
zeroV' : 0 ∉ V'
hV' : ∀ x ∈ V', ∀ y ∈ V', x * y⁻¹ ∈ V
nhds_right : (fun x ↦ x * x₀) '' V' ∈ 𝓝 x₀
z₀ : K
y₀ : hat K
y₀_in : y₀ ∈ V'
hz₀ : ↑z₀ = y₀ * x₀
z₀_ne : z₀ ≠ 0
vz₀_ne : v.restrict z₀ ≠ 0
⊢ ∃ c, Tendsto (⇑v.restrict) (Filter.comap Completion.coe' (𝓝 x₀)) (𝓝 c)

Tactic:
refine ⟨v.restrict z₀, ?_⟩

Goal state after:
case inr
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x₀ : hat K
h : x₀ ≠ 0
preimage_one : ⇑v ⁻¹' {1} ∈ 𝓝 1
V : Set (hat K)
V_in : V ∈ 𝓝 1
hV : ∀ (x : K), ↑x ∈ V → v x = 1
V' : Set (hat K)
V'_in : V' ∈ 𝓝 1
zeroV' : 0 ∉ V'
hV' : ∀ x ∈ V', ∀ y ∈ V', x * y⁻¹ ∈ V
nhds_right : (fun x ↦ x * x₀) '' V' ∈ 𝓝 x₀
z₀ : K
y₀ : hat K
y₀_in : y₀ ∈ V'
hz₀ : ↑z₀ = y₀ * x₀
z₀_ne : z₀ ≠ 0
vz₀_ne : v.restrict z₀ ≠ 0
⊢ Tendsto (⇑v.restrict) (Filter.comap Completion.coe' (𝓝 x₀)) (𝓝 (v.restrict z₀))
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: G✝ : Type u_1
inst✝⁷ : CommGroup G✝
inst✝⁶ : LinearOrder G✝
inst✝⁵ : IsOrderedMonoid G✝
inst✝⁴ : MulArchimedean G✝
G : Type u_2
inst✝³ : AddCommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedAddMonoid G
inst✝ : Archimedean G
x : G
h : IsLeast {y | 0 < y} x
this : ⊤ = AddSubgroup.closure {x}
e : G ≃+o ↥⊤ := { toAddEquiv := AddSubsemigroup.topEquiv.symm, map_le_map_iff' := ⋯ }
e' : ↥⊤ ≃+o ↥(AddSubgroup.closure {x}) := { toAddEquiv := AddEquiv.subsemigroupCongr ⋯, map_le_map_iff' := ⋯ }
g : ↥⊤ ≃+o ℤ := { toAddEquiv := AddSubsemigroup.topEquiv, map_le_map_iff' := ⋯ }
g' : ↥(AddSubgroup.closure {1}) ≃+o ↥⊤ :=
  { toAddEquiv := AddEquiv.subsemigroupCongr int_orderAddMonoidIso_of_isLeast_pos._proof_2, map_le_map_iff' := ⋯ }
⊢ x = 0 ↔ 1 = 0

Tactic:
by simp [h.left.ne']

Goal state after:
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: case refine_1
G : Type u_2
inst✝² : LinearOrderedCommGroupWithZero G
inst✝¹ : Nontrivial Gˣ
inst✝ : MulArchimedean G
f : G ≃*o WithZero (Multiplicative ℤ)
⊢ Gˣ ≃*o Multiplicative ℤ

pp: case refine_2
G : Type u_2
inst✝² : LinearOrderedCommGroupWithZero G
inst✝¹ : Nontrivial Gˣ
inst✝ : MulArchimedean G
f : Gˣ ≃*o Multiplicative ℤ
⊢ G ≃*o WithZero (Multiplicative ℤ)

Tactic:
· refine ⟨MulEquiv.withZero.symm (withZeroUnitsEquiv.trans f), ?_⟩
    intros
    simp only [MulEquiv.withZero, withZeroUnitsEquiv, MulEquiv.trans_apply,
      MulEquiv.coe_mk, Equiv.coe_fn_mk, recZeroCoe_coe, OrderMonoidIso.coe_mulEquiv,
      MulEquiv.symm_trans_apply, MulEquiv.symm_mk, Equiv.coe_fn_symm_mk, map_eq_zero, coe_ne_zero,
      ↓reduceDIte, unzero_coe, MulEquiv.toEquiv_eq_coe, Equiv.toFun_as_coe, EquivLike.coe_coe]
    rw [← Units.val_le_val, ← map_le_map_iff f, ← coe_le_coe, coe_unzero, coe_unzero]

Goal state after:
case refine_2
G : Type u_2
inst✝² : LinearOrderedCommGroupWithZero G
inst✝¹ : Nontrivial Gˣ
inst✝ : MulArchimedean G
f : Gˣ ≃*o Multiplicative ℤ
⊢ G ≃*o WithZero (Multiplicative ℤ)
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x : K
⊢ Tendsto (⇑v.restrict) (Filter.comap Completion.coe' (𝓝 ↑x)) (𝓝 (v.restrict x))

Tactic:
← Completion.isDenseInducing_coe.nhds_eq_comap

Goal state after:
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x : K
⊢ Tendsto (⇑v.restrict) (𝓝 x) (𝓝 (v.restrict x))
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: case e'_2
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
r s : K
hr : r ≠ 0
hs : s ≠ 0
hrs : v s / v r ≠ 0
⊢ {x | v x * v r < v s} = {x | v x < ↑(Units.mk0 (v s / v r) hrs)}

Tactic:
simp [← lt_div_iff₀, zero_lt_iff, hr]

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

Goal state before:
pp: case e'_3
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
r s : K
hr : r ≠ 0
hs : s ≠ 0
hrs : v s / v r ≠ 0
x✝ : hat K
⊢ extensionValuation x✝ * v r < v s ↔ extensionValuation x✝ < ↑(Units.mk0 (v s / v r) hrs)

Tactic:
simp [← lt_div_iff₀, zero_lt_iff, hr]

Goal state after:
Module: Mathlib.RingTheory.PowerSeries.Basic

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
⊢ (MvPowerSeries.coeff 0) X = 0

Tactic:
X,

Goal state after:
R : Type u_1
inst✝ : Semiring R
⊢ (MvPowerSeries.coeff 0) (MvPowerSeries.X ()) = 0