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

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W : WeierstrassCurve R
n : ℤ
⊢ (mk W) (C X) * (mk W) (C (W.ΨSq n)) -
      (mk W) (C (W.preΨ (n + 1)) * if Even (n + 1) then W.ψ₂ else 1) *
        (mk W) (C (W.preΨ (n - 1)) * if Even (n - 1) then W.ψ₂ else 1) =
    (mk W) (C X) * (mk W) (C (W.ΨSq n)) -
      (mk W) (C (W.preΨ (n + 1))) * (mk W) (C (W.preΨ (n - 1))) * (mk W) (C (if Even n then 1 else W.Ψ₂Sq))

Tactic:
map_mul,

Goal state after:
R : Type r
inst✝ : CommRing R
W : WeierstrassCurve R
n : ℤ
⊢ (mk W) (C X) * (mk W) (C (W.ΨSq n)) -
      (mk W) (C (W.preΨ (n + 1))) * (mk W) (if Even (n + 1) then W.ψ₂ else 1) *
        ((mk W) (C (W.preΨ (n - 1))) * (mk W) (if Even (n - 1) then W.ψ₂ else 1)) =
    (mk W) (C X) * (mk W) (C (W.ΨSq n)) -
      (mk W) (C (W.preΨ (n + 1))) * (mk W) (C (W.preΨ (n - 1))) * (mk W) (C (if Even n then 1 else W.Ψ₂Sq))
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
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
⊢ u ≠ 0

Tactic:
rw [← pow_ne_zero_iff two_ne_zero, hu, div_ne_zero_iff]

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
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
⊢ E.a₂ ≠ 0 ∧ E'.a₂ ≠ 0
Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree

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

Tactic:
rw [ΨSq_neg, Int.natAbs_neg, ← Int.cast_pow, neg_sq, Int.cast_pow, ih]

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
hX : ∀ {n : ℕ}, IsUnit (P x ^ n)
⊢ W.neg P = -(P y / P x) • ![1, 1, 0]

Tactic:
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.Jacobian.Formula

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W' : Jacobian R
P : Fin 3 → R
u : R
⊢ ![(u ^ 4) ^ 2 * W'.dblX P, (u ^ 4) ^ 3 * W'.dblY P, u ^ 4 * W'.dblZ P] =
    ![(u ^ 4) ^ 2 * W'.dblX P, (u ^ 4) ^ 3 * W'.dblY P, u ^ 4 * W'.dblXYZ P z]

Tactic:
dblXYZ_Z

Goal state after:
R : Type r
inst✝ : CommRing R
W' : Jacobian R
P : Fin 3 → R
u : R
⊢ ![(u ^ 4) ^ 2 * W'.dblX P, (u ^ 4) ^ 3 * W'.dblY P, u ^ 4 * W'.dblZ P] =
    ![(u ^ 4) ^ 2 * W'.dblX P, (u ^ 4) ^ 3 * W'.dblY P, u ^ 4 * W'.dblZ P]
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
hQz : Q z = 0
⊢ toAffine W (W.add P Q) = toAffine W P + toAffine W Q

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:
· rw [add_of_Z_eq_zero_right hQ.left hPz hQz,
        toAffine_smul _ ((isUnit_X_of_Z_eq_zero hQ hQz).mul <| Ne.isUnit hPz).neg,
        toAffine_of_Z_eq_zero hQz, add_zero]

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 (W.add P Q) = toAffine W P + toAffine W Q
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: case left
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x x' y y' z r : R
u : ↥(posSubmonoid R)
s : R
v : ↥(posSubmonoid R)
t : R
w : ↥(posSubmonoid R)
h1 : r * ↑v ≤ᵥ s * ↑u
h2 : s * ↑u ≤ᵥ r * ↑v
h3 : s * ↑w ≤ᵥ t * ↑v
h4 : t * ↑v ≤ᵥ s * ↑w
this : r * ↑v * ↑w ≤ᵥ s * ↑u * ↑w
⊢ r * ↑w ≤ᵥ t * ↑u

Tactic:
veq_mul_right_comm (x := s)

Goal state after:
case left
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x x' y y' z r : R
u : ↥(posSubmonoid R)
s : R
v : ↥(posSubmonoid R)
t : R
w : ↥(posSubmonoid R)
h1 : r * ↑v ≤ᵥ s * ↑u
h2 : s * ↑u ≤ᵥ r * ↑v
h3 : s * ↑w ≤ᵥ t * ↑v
h4 : t * ↑v ≤ᵥ s * ↑w
this : r * ↑v * ↑w ≤ᵥ s * ↑w * ↑u
⊢ r * ↑w ≤ᵥ t * ↑u
Module: Mathlib.Topology.Algebra.Valued.ValuationTopology

Goal state before:
pp: case h
R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation R Γ₀
x : R
γ : (ofClass v).ValueGroup₀ˣ
γx : Γ₀ˣ
Hx : v x = ↑γx
u : (ofClass v).ValueGroup₀ˣ := Units.mk0 ((restrict₀ (ofClass v)) x) ⋯
hu_def : u = Units.mk0 ((restrict₀ (ofClass v)) x) ⋯
hu : embedding ↑u⁻¹ = ↑γx⁻¹
y : R
vy_lt : v y < embedding (↑γ * ↑u⁻¹)
⊢ v y * ↑γx < ↑((Units.map ↑embedding) γ)

Tactic:
map_mul,

Goal state after:
case h
R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation R Γ₀
x : R
γ : (ofClass v).ValueGroup₀ˣ
γx : Γ₀ˣ
Hx : v x = ↑γx
u : (ofClass v).ValueGroup₀ˣ := Units.mk0 ((restrict₀ (ofClass v)) x) ⋯
hu_def : u = Units.mk0 ((restrict₀ (ofClass v)) x) ⋯
hu : embedding ↑u⁻¹ = ↑γx⁻¹
y : R
vy_lt : v y < embedding ↑γ * embedding ↑u⁻¹
⊢ v y * ↑γx < ↑((Units.map ↑embedding) γ)
Module: Mathlib.Topology.Algebra.WithZeroTopology

Goal state before:
pp: case inr
α : Type u_1
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
γ γ₁ γ₂ : Γ₀
l : Filter α
f : α → Γ₀
x y : Γ₀
hle : x ≤ y
hx : x ≠ 0
hy : y ≠ 0
⊢ Tendsto (fun p ↦ p.1 * p.2) (𝓝 x ×ˢ 𝓝 y) (𝓝 ((x, y).1 * (x, y).2))

Tactic:
nhds_of_ne_zero hx,

Goal state after:
case inr
α : Type u_1
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
γ γ₁ γ₂ : Γ₀
l : Filter α
f : α → Γ₀
x y : Γ₀
hle : x ≤ y
hx : x ≠ 0
hy : y ≠ 0
⊢ Tendsto (fun p ↦ p.1 * p.2) (pure x ×ˢ 𝓝 y) (𝓝 ((x, y).1 * (x, y).2))
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x y : R
⊢ x <ᵥ y ↔ x ≤ᵥ y ∧ ¬y ≤ᵥ x

Tactic:
have := vle_total x y

Goal state after:
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x y : R
this : x ≤ᵥ y ∨ y ≤ᵥ x
⊢ x <ᵥ y ↔ x ≤ᵥ y ∧ ¬y ≤ᵥ x
Module: Mathlib.Topology.Algebra.GroupCompletion

Goal state before:
pp: M : Type u_1
R : Type u_2
α : Type u_3
β : Type u_4
inst✝⁴ : UniformSpace α
inst✝³ : MonoidWithZero M
inst✝² : Zero α
inst✝¹ : MulActionWithZero M α
inst✝ : UniformContinuousConstSMul M α
r : M
⊢ r • 0 = 0

Tactic:
← coe_zero,

Goal state after:
M : Type u_1
R : Type u_2
α : Type u_3
β : Type u_4
inst✝⁴ : UniformSpace α
inst✝³ : MonoidWithZero M
inst✝² : Zero α
inst✝¹ : MulActionWithZero M α
inst✝ : UniformContinuousConstSMul M α
r : M
⊢ r • ↑0 = ↑0
Module: Mathlib.Topology.Algebra.GroupCompletion

Goal state before:
pp: M : Type u_1
R : Type u_2
α : Type u_3
β : Type u_4
inst✝⁴ : UniformSpace α
inst✝³ : MonoidWithZero M
inst✝² : Zero α
inst✝¹ : MulActionWithZero M α
inst✝ : UniformContinuousConstSMul M α
a : α
⊢ ↑0 = 0

Tactic:
coe_zero

Goal state after:
M : Type u_1
R : Type u_2
α : Type u_3
β : Type u_4
inst✝⁴ : UniformSpace α
inst✝³ : MonoidWithZero M
inst✝² : Zero α
inst✝¹ : MulActionWithZero M α
inst✝ : UniformContinuousConstSMul M α
a : α
⊢ 0 = 0
Module: Mathlib.Topology.Algebra.GroupCompletion

Goal state before:
pp: M : Type u_1
R : Type u_2
α : Type u_3
β : Type u_4
inst✝² : UniformSpace α
inst✝¹ : AddGroup α
inst✝ : IsUniformAddGroup α
a✝ : Completion α
a : α
⊢ 0 + ↑a = ↑a

Tactic:
← coe_zero,

Goal state after:
M : Type u_1
R : Type u_2
α : Type u_3
β : Type u_4
inst✝² : UniformSpace α
inst✝¹ : AddGroup α
inst✝ : IsUniformAddGroup α
a✝ : Completion α
a : α
⊢ ↑0 + ↑a = ↑a
Module: Mathlib.Topology.Algebra.GroupCompletion

Goal state before:
pp: M : Type u_1
R : Type u_2
α : Type u_3
β : Type u_4
inst✝² : UniformSpace α
inst✝¹ : AddGroup α
inst✝ : IsUniformAddGroup α
a✝ : Completion α
a : α
⊢ ↑a + 0 = ↑a

Tactic:
← coe_zero,

Goal state after:
M : Type u_1
R : Type u_2
α : Type u_3
β : Type u_4
inst✝² : UniformSpace α
inst✝¹ : AddGroup α
inst✝ : IsUniformAddGroup α
a✝ : Completion α
a : α
⊢ ↑a + ↑0 = ↑a
Module: Mathlib.Topology.Algebra.GroupCompletion

Goal state before:
pp: M : Type u_1
R : Type u_2
α : Type u_3
β : Type u_4
inst✝² : UniformSpace α
inst✝¹ : AddGroup α
inst✝ : IsUniformAddGroup α
a✝ : Completion α
a : α
⊢ ↑(0 • a) = 0

Tactic:
← coe_zero,

Goal state after:
M : Type u_1
R : Type u_2
α : Type u_3
β : Type u_4
inst✝² : UniformSpace α
inst✝¹ : AddGroup α
inst✝ : IsUniformAddGroup α
a✝ : Completion α
a : α
⊢ ↑(0 • a) = ↑0
Module: Mathlib.Topology.Algebra.GroupCompletion

Goal state before:
pp: M✝ : Type u_1
R : Type u_2
α : Type u_3
β : Type u_4
inst✝⁵ : UniformSpace α
inst✝⁴ : AddGroup α
inst✝³ : IsUniformAddGroup α
M : Type ?u.13
inst✝² : Monoid M
inst✝¹ : DistribMulAction M α
inst✝ : UniformContinuousConstSMul M α
r : M
⊢ r • 0 = 0

Tactic:
← coe_zero,

Goal state after:
M✝ : Type u_1
R : Type u_2
α : Type u_3
β : Type u_4
inst✝⁵ : UniformSpace α
inst✝⁴ : AddGroup α
inst✝³ : IsUniformAddGroup α
M : Type ?u.13
inst✝² : Monoid M
inst✝¹ : DistribMulAction M α
inst✝ : UniformContinuousConstSMul M α
r : M
⊢ r • ↑0 = ↑0
Module: Mathlib.Topology.Algebra.GroupCompletion

Goal state before:
pp: M : Type u_1
R : Type u_2
α : Type u_3
β : Type u_4
inst✝⁷ : UniformSpace α
inst✝⁶ : AddGroup α
inst✝⁵ : IsUniformAddGroup α
inst✝⁴ : UniformSpace β
inst✝³ : AddGroup β
inst✝² : IsUniformAddGroup β
inst✝¹ : CompleteSpace β
inst✝ : T0Space β
f : α →+ β
hf✝ : Continuous ⇑f
hf : UniformContinuous ⇑f
⊢ Completion.extension (⇑f) 0 = 0

Tactic:
← coe_zero,

Goal state after:
M : Type u_1
R : Type u_2
α : Type u_3
β : Type u_4
inst✝⁷ : UniformSpace α
inst✝⁶ : AddGroup α
inst✝⁵ : IsUniformAddGroup α
inst✝⁴ : UniformSpace β
inst✝³ : AddGroup β
inst✝² : IsUniformAddGroup β
inst✝¹ : CompleteSpace β
inst✝ : T0Space β
f : α →+ β
hf✝ : Continuous ⇑f
hf : UniformContinuous ⇑f
⊢ Completion.extension ⇑f ↑0 = 0
Module: Mathlib.Topology.Algebra.UniformRing

Goal state before:
pp: α : Type u_1
inst✝³ : Ring α
inst✝² : UniformSpace α
inst✝¹ : IsTopologicalRing α
inst✝ : IsUniformAddGroup α
a✝ : Completion α
a : α
⊢ 0 * ↑a = 0

Tactic:
← coe_zero,

Goal state after:
α : Type u_1
inst✝³ : Ring α
inst✝² : UniformSpace α
inst✝¹ : IsTopologicalRing α
inst✝ : IsUniformAddGroup α
a✝ : Completion α
a : α
⊢ ↑0 * ↑a = ↑0
Module: Mathlib.Topology.Algebra.UniformRing

Goal state before:
pp: α : Type u_1
inst✝³ : Ring α
inst✝² : UniformSpace α
inst✝¹ : IsTopologicalRing α
inst✝ : IsUniformAddGroup α
a✝ : Completion α
a : α
⊢ ↑a * 0 = 0

Tactic:
← coe_zero,

Goal state after:
α : Type u_1
inst✝³ : Ring α
inst✝² : UniformSpace α
inst✝¹ : IsTopologicalRing α
inst✝ : IsUniformAddGroup α
a✝ : Completion α
a : α
⊢ ↑a * ↑0 = ↑0
Module: Mathlib.Topology.Algebra.UniformRing

Goal state before:
pp: α : Type u_1
inst✝⁹ : Ring α
inst✝⁸ : UniformSpace α
inst✝⁷ : IsTopologicalRing α
inst✝⁶ : IsUniformAddGroup α
β : Type u
inst✝⁵ : UniformSpace β
inst✝⁴ : Ring β
inst✝³ : IsUniformAddGroup β
inst✝² : IsTopologicalRing β
f : α →+* β
hf✝ : Continuous ⇑f
inst✝¹ : CompleteSpace β
inst✝ : T0Space β
hf' : Continuous ⇑↑f
hf : UniformContinuous ⇑f
⊢ Completion.extension (⇑f) 0 = 0

Tactic:
← coe_zero,

Goal state after:
α : Type u_1
inst✝⁹ : Ring α
inst✝⁸ : UniformSpace α
inst✝⁷ : IsTopologicalRing α
inst✝⁶ : IsUniformAddGroup α
β : Type u
inst✝⁵ : UniformSpace β
inst✝⁴ : Ring β
inst✝³ : IsUniformAddGroup β
inst✝² : IsTopologicalRing β
f : α →+* β
hf✝ : Continuous ⇑f
inst✝¹ : CompleteSpace β
inst✝ : T0Space β
hf' : Continuous ⇑↑f
hf : UniformContinuous ⇑f
⊢ Completion.extension ⇑f ↑0 = 0
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
⊢ IsClosed[_t] {x | v.restrict x ≤ 1}

Tactic:
exact isClosed_closedBall _

Goal state after:
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
⊢ Cauchy (Filter.comap coe' (𝓝 y))

Tactic:
apply cauchy_nhds.comap

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

Goal state before:
pp: Γ : Type u_1
inst✝¹ : LinearOrderedCommGroupWithZero Γ
K : Type u_2
inst✝ : Field K
v : Valuation K Γ
hv : v.IsRankOneDiscrete
π₁ π₂ : ↥v.valuationSubring
h1 : v.IsUniformizer ↑π₁
u : (↥v.valuationSubring)ˣ
hu : π₁ * ↑u = π₂
this : v ↑↑u = 1
⊢ v (↑π₁ * ↑↑u) = ↑(generator v)

Tactic:
map_mul,

Goal state after:
Γ : Type u_1
inst✝¹ : LinearOrderedCommGroupWithZero Γ
K : Type u_2
inst✝ : Field K
v : Valuation K Γ
hv : v.IsRankOneDiscrete
π₁ π₂ : ↥v.valuationSubring
h1 : v.IsUniformizer ↑π₁
u : (↥v.valuationSubring)ˣ
hu : π₁ * ↑u = π₂
this : v ↑↑u = 1
⊢ v ↑π₁ * v ↑↑u = ↑(generator v)
Module: Mathlib.RingTheory.Valuation.Discrete.Basic

Goal state before:
pp: Γ : Type u_1
inst✝¹ : LinearOrderedCommGroupWithZero Γ
K : Type u_2
inst✝ : Field K
v : Valuation K Γ
hv : v.IsRankOneDiscrete
r : ↥v.valuationSubring
hr : r ≠ 0
π : v.Uniformizer
hr₀ : v ↑r ≠ 0
vr : Γˣ := ⋯
hvr_def : vr = Units.mk0 (v ↑r) hr₀
⊢ ↑vr ∈ range ⇑(MonoidWithZeroHom.ofClass v)

Tactic:
rw [hvr_def, Units.val_mk0 hr₀]

Goal state after:
Γ : Type u_1
inst✝¹ : LinearOrderedCommGroupWithZero Γ
K : Type u_2
inst✝ : Field K
v : Valuation K Γ
hv : v.IsRankOneDiscrete
r : ↥v.valuationSubring
hr : r ≠ 0
π : v.Uniformizer
hr₀ : v ↑r ≠ 0
vr : Γˣ := Units.mk0 (v ↑r) hr₀
hvr_def : vr = Units.mk0 (v ↑r) hr₀
⊢ v ↑r ∈ range ⇑(MonoidWithZeroHom.ofClass v)
Module: Mathlib.RingTheory.Valuation.Discrete.Basic

Goal state before:
pp: Γ : Type u_1
inst✝¹ : LinearOrderedCommGroupWithZero Γ
K : Type u_2
inst✝ : Field K
v : Valuation K Γ
hv : v.IsRankOneDiscrete
r : ↥v.valuationSubring
hr : r ≠ 0
π : v.Uniformizer
hr₀ : v ↑r ≠ 0
vr : Γˣ := Units.mk0 (v ↑r) hr₀
hvr_def : vr = Units.mk0 (v ↑r) hr₀
m : ℤ
hm : Units.mk0 (v ↑π.val) ⋯ ^ m = vr
hm' : v ↑π.val ^ m = v ↑r
hm₀ : 0 ≤ m
n : ℕ
hn : m = ↑n
⊢ v (↑π.val ^ (-m) * ↑r) = 1

Tactic:
map_mul,

Goal state after:
Γ : Type u_1
inst✝¹ : LinearOrderedCommGroupWithZero Γ
K : Type u_2
inst✝ : Field K
v : Valuation K Γ
hv : v.IsRankOneDiscrete
r : ↥v.valuationSubring
hr : r ≠ 0
π : v.Uniformizer
hr₀ : v ↑r ≠ 0
vr : Γˣ := Units.mk0 (v ↑r) hr₀
hvr_def : vr = Units.mk0 (v ↑r) hr₀
m : ℤ
hm : Units.mk0 (v ↑π.val) ⋯ ^ m = vr
hm' : v ↑π.val ^ m = v ↑r
hm₀ : 0 ≤ m
n : ℕ
hn : m = ↑n
⊢ v (↑π.val ^ (-m)) * v ↑r = 1
Module: Mathlib.RingTheory.Valuation.Discrete.IsDiscreteValuationRing

Goal state before:
pp: case neg
A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : IsDiscreteValuationRing A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
x : K
π : A
hπ : Irreducible π
n : ℕ
u : Aˣ
ha : ¬↑u * π ^ n = 0
m : ℕ
w : Aˣ
H :
  (valuation K (maximalIdeal A)) ((algebraMap A K) (π ^ n) * ((algebraMap A K) (π ^ m))⁻¹) *
      (valuation K (maximalIdeal A)) ((algebraMap A K) ↑w)⁻¹ ≤
    1
h_frac : (algebraMap A K) (↑u * π ^ n) / (algebraMap A K) (↑w * π ^ m) = x
hb : π ^ m ∈ nonZeroDivisors A
⊢ ∃ a, (algebraMap A K) a = x

Tactic:
map_mul,

Goal state after:
case neg
A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : IsDiscreteValuationRing A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
x : K
π : A
hπ : Irreducible π
n : ℕ
u : Aˣ
ha : ¬↑u * π ^ n = 0
m : ℕ
w : Aˣ
H :
  (valuation K (maximalIdeal A)) ((algebraMap A K) (π ^ n)) *
        (valuation K (maximalIdeal A)) ((algebraMap A K) (π ^ m))⁻¹ *
      (valuation K (maximalIdeal A)) ((algebraMap A K) ↑w)⁻¹ ≤
    1
h_frac : (algebraMap A K) (↑u * π ^ n) / (algebraMap A K) (↑w * π ^ m) = x
hb : π ^ m ∈ nonZeroDivisors A
⊢ ∃ a, (algebraMap A K) a = x
Module: Mathlib.Order.Interval.Finset.DenselyOrdered

Goal state before:
pp: X : Type u_1
inst✝¹ : LinearOrder X
inst✝ : LocallyFiniteOrder X
s : Set (WithBot X)
H : DenselyOrdered ↑s
⊢ s.Subsingleton

Tactic:
← Set.subsingleton_coe,

Goal state after:
X : Type u_1
inst✝¹ : LinearOrder X
inst✝ : LocallyFiniteOrder X
s : Set (WithBot X)
H : DenselyOrdered ↑s
⊢ Subsingleton ↑s
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: case h.right.y_ne
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
F : Filter K
h0 : 𝓝 0 ⊓ F = ⊥
γ₀ : (MonoidWithZeroHom.ofClass v).ValueGroup₀ˣ
M₀ : Set K
M₀_in : M₀ ∈ F
γ : (MonoidWithZeroHom.ofClass v).ValueGroup₀ˣ
M₁ : Set K
M₁_in : M₁ ∈ F
x : K
x_in₁ : x ∈ M₁
y : K
H₁ : v.restrict (y - x) < ↑(min (γ * γ₀ * γ₀) γ₀)
left✝ : y ∈ M₀
y_in₁ : y ∈ M₁
x_in₀ : ↑γ₀ ≤ v.restrict x
⊢ x ≠ 0

pp: case h
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
F : Filter K
h0 : 𝓝 0 ⊓ F = ⊥
γ₀ : (MonoidWithZeroHom.ofClass v).ValueGroup₀ˣ
M₀ : Set K
M₀_in : M₀ ∈ F
γ : (MonoidWithZeroHom.ofClass v).ValueGroup₀ˣ
M₁ : Set K
M₁_in : M₁ ∈ F
x : K
x_in₁ : x ∈ M₁
y : K
H₁ : v.restrict (y - x) < ↑(min (γ * γ₀ * γ₀) γ₀)
left✝ : y ∈ M₀
y_in₁ : y ∈ M₁
x_in₀ : ↑γ₀ ≤ v.restrict x
⊢ v.restrict (y - x) < min (↑γ * (v.restrict x * v.restrict x)) (v.restrict x)

Tactic:
· have : (v.restrict x) ≠ 0 := by
            intro h
            rw [h] at x_in₀
            simp at x_in₀
          exact (Valuation.ne_zero_iff _).mp this

Goal state after:
case h
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
F : Filter K
h0 : 𝓝 0 ⊓ F = ⊥
γ₀ : (MonoidWithZeroHom.ofClass v).ValueGroup₀ˣ
M₀ : Set K
M₀_in : M₀ ∈ F
γ : (MonoidWithZeroHom.ofClass v).ValueGroup₀ˣ
M₁ : Set K
M₁_in : M₁ ∈ F
x : K
x_in₁ : x ∈ M₁
y : K
H₁ : v.restrict (y - x) < ↑(min (γ * γ₀ * γ₀) γ₀)
left✝ : y ∈ M₀
y_in₁ : y ∈ M₁
x_in₀ : ↑γ₀ ≤ v.restrict x
⊢ v.restrict (y - x) < min (↑γ * (v.restrict x * v.restrict x)) (v.restrict x)
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
U : Set K
hU : IsClosed[hv.toTopologicalSpace] U
hU' : U.Nonempty
⊢ IsClosed (⇑v.restrict '' U)

Tactic:
simp only [← isOpen_compl_iff, isOpen_iff_mem_nhds, mem_compl_iff, mem_nhds, subset_compl_comm,
    compl_ofPred, not_lt] at hU

Goal state after:
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
U : Set K
hU' : U.Nonempty
hU : ∀ x ∉ U, ∃ γ, U ⊆ {a | ↑γ ≤ v.restrict (a - x)}
⊢ IsClosed (⇑v.restrict '' U)
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
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
a : K
y : hat K
y_in : y ∈ V'
ha : ↑a = (fun x ↦ x * x₀) y
this : ↑z₀⁻¹ = (↑z₀)⁻¹
⊢ ↑(a * z₀⁻¹) ∈ V

Tactic:
rw [Completion.coe_mul, this, ha, hz₀, mul_inv, mul_comm y₀⁻¹, ← mul_assoc, mul_assoc y,
        mul_inv_cancel₀ h, mul_one]

Goal state after:
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
a : K
y : hat K
y_in : y ∈ V'
ha : ↑a = (fun x ↦ x * x₀) y
this : ↑z₀⁻¹ = (↑z₀)⁻¹
⊢ y * y₀⁻¹ ∈ V
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
γ : Γ₀ˣ
⊢ closure (Completion.coe' '' {x | v x < ↑γ}) = {x | extensionValuation x < ↑γ}

Tactic:
ext x
  set γ₀' := extension x with hγ₀'_def
  set γ₀ := extensionValuation x with hγ₀_def
  have heq : γ₀ = embedding γ₀' := rfl
  suffices γ₀ ≠ 0 → (x ∈ closure ((↑) '' { x : K | v x < (γ : Γ₀) }) ↔ γ₀ < (γ : Γ₀)) by
    rcases eq_or_ne γ₀ 0 with h | h
    · simp only [(Valuation.zero_iff _).mp h, mem_ofPred_eq, Valuation.map_zero, Units.zero_lt,
        iff_true]
      apply subset_closure
      exact ⟨0, by simp only [mem_ofPred_eq, Valuation.map_zero, Units.zero_lt, true_and]; rfl⟩
    · exact this h
  intro h
  have h' : γ₀' ≠ 0 := by simpa only [heq, map_ne_zero] using h
  have hγ₀ : extension ⁻¹' {γ₀'} ∈ 𝓝 x :=
    continuous_extension.continuousAt.preimage_mem_nhds
      (WithZeroTopology.singleton_mem_nhds_of_ne_zero h')
  rw [mem_closure_iff_nhds']
  refine ⟨fun hx => ?_, fun hx s hs => ?_⟩
  · obtain ⟨⟨-, y, hy₁ : v y < (γ : Γ₀), rfl⟩, hy₂⟩ := hx _ hγ₀
    replace hy₂ : v y = γ₀ := by
      simp only [mem_preimage, extension_extends, mem_singleton_iff, v.restrict_def] at hy₂
      apply_fun embedding at hy₂
      simpa [heq] using hy₂
    rwa [← hy₂]
  · obtain ⟨y, hy₁, hy₂⟩ := Completion.denseRange_coe.mem_nhds (inter_mem hγ₀ hs)
    replace hy₁ : v y = γ₀ := by
      simp only [mem_preimage, extension_extends, mem_singleton_iff, v.restrict_def] at hy₁
      apply_fun embedding at hy₁
      simpa [heq] using hy₁
    rw [← hy₁] at hx
    exact ⟨⟨y, ⟨y, hx, rfl⟩⟩, hy₂⟩

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝¹ : Semiring R
m : ℕ
inst✝ : NeZero m
n : σ →₀ ℕ
a : R
i : σ
H : n + single i m = 0
⊢ False

Tactic:
by simpa [NeZero.ne] using congr($(H) i)

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

Goal state before:
pp: case refine_1
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
a b : (ofClass v).ValueGroup₀
hab : valueGroup₀_hom_extensionValuation a = valueGroup₀_hom_extensionValuation b
x : K := ⋯.choose
hx_def : x = ⋯.choose
y : K := ⋯.choose
hy_def : y = ⋯.choose
hx : embedding ((restrict₀ (ofClass v)) ⋯.choose) = embedding a
hy : embedding ((restrict₀ (ofClass v)) ⋯.choose) = embedding b
⊢ embedding a = embedding b

Tactic:
simp only [← hx_def, embedding_restrict₀, coe_ofClass, ← hy_def] at hx hy

Goal state after:
case refine_1
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
a b : (ofClass v).ValueGroup₀
hab : valueGroup₀_hom_extensionValuation a = valueGroup₀_hom_extensionValuation b
x : K := ⋯.choose
hx_def : x = ⋯.choose
y : K := ⋯.choose
hy_def : y = ⋯.choose
hx : v x = embedding a
hy : v y = embedding b
⊢ embedding a = embedding b