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

Goal state before:
pp: F : Type u
inst✝¹ : Field F
W : Affine F
inst✝ : DecidableEq F
x₁ x₂ y₁ y₂ : F
h₁ : W.Equation x₁ y₁
h₂ : W.Equation x₂ y₂
hxy : ¬(x₁ = x₂ ∧ y₁ = W.negY x₂ y₂)
hy₂ : y₂ = eval x₂ (linePolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂))
⊢ XYIdeal W x₂ (C y₂) = XYIdeal W x₂ (linePolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂))

Tactic:
nth_rw 1 [hy₂]

Goal state after:
F : Type u
inst✝¹ : Field F
W : Affine F
inst✝ : DecidableEq F
x₁ x₂ y₁ y₂ : F
h₁ : W.Equation x₁ y₁
h₂ : W.Equation x₂ y₂
hxy : ¬(x₁ = x₂ ∧ y₁ = W.negY x₂ y₂)
hy₂ : y₂ = eval x₂ (linePolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂))
⊢ XYIdeal W x₂ (C (eval x₂ (linePolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂)))) =
    XYIdeal W x₂ (linePolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂))
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Formula

Goal state before:
pp: case pos
F : Type u
inst✝¹ : Field F
W : Affine F
inst✝ : DecidableEq F
x₁ x₂ y₁ y₂ : F
h₁ : W.Nonsingular x₁ y₁
h₂ : W.Nonsingular x₂ y₂
hxy : ¬(x₁ = x₂ ∧ y₁ = W.negY x₂ y₂)
hx₁ : ¬W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₁
hx₂ : W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₂
hx : x₁ = x₂
⊢ W.Nonsingular (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂)) (W.negAddY x₁ x₂ y₁ (W.slope x₁ x₂ y₁ y₂))

Tactic:
subst hx

Goal state after:
case pos
F : Type u
inst✝¹ : Field F
W : Affine F
inst✝ : DecidableEq F
x₁ y₁ y₂ : F
h₁ : W.Nonsingular x₁ y₁
h₂ : W.Nonsingular x₁ y₂
hxy : ¬(x₁ = x₁ ∧ y₁ = W.negY x₁ y₂)
hx₁ : ¬W.addX x₁ x₁ (W.slope x₁ x₁ y₁ y₂) = x₁
hx₂ : W.addX x₁ x₁ (W.slope x₁ x₁ y₁ y₂) = x₁
⊢ W.Nonsingular (W.addX x₁ x₁ (W.slope x₁ x₁ y₁ y₂)) (W.negAddY x₁ x₁ y₁ (W.slope x₁ x₁ y₁ y₂))
Module: Mathlib.NumberTheory.EllipticDivisibilitySequence

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
b c d : R
k : ℤ
⊢ (preNormEDS (b ^ 4) c d k * if Even k then b else 1) * complEDS₂ b c d k =
    preNormEDS (b ^ 4) c d (2 * k) * if Even (2 * k) then b else 1

Tactic:
mul_right_comm,

Goal state after:
R : Type u_1
inst✝ : CommRing R
b c d : R
k : ℤ
⊢ (preNormEDS (b ^ 4) c d k * complEDS₂ b c d k * if Even k then b else 1) =
    preNormEDS (b ^ 4) c d (2 * k) * if Even (2 * k) then b else 1
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point

Goal state before:
pp: case some.some.mp
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 → x₁ = x₂ ∧ y₁ = W.negY x₂ y₂

Tactic:
contrapose

Goal state after:
case some.some.mp
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
⊢ ¬(x₁ = x₂ ∧ y₁ = W.negY x₂ y₂) → ¬some x₁ y₁ h✝¹ + some x₂ y₂ h✝ = 0
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 2
inst✝¹ : E.IsCharTwoJEqZeroNF
inst✝ : E'.IsCharTwoJEqZeroNF
ha₃ : E.a₃ ≠ 0
ha₃' : E'.a₃ ≠ 0
⊢ 3 ≠ 0

Tactic:
rw [show (3 : F) = 1 by linear_combination CharP.cast_eq_zero F 2]

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 2
inst✝¹ : E.IsCharTwoJEqZeroNF
inst✝ : E'.IsCharTwoJEqZeroNF
ha₃ : E.a₃ ≠ 0
ha₃' : E'.a₃ ≠ 0
⊢ 1 ≠ 0
Module: Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ

Goal state before:
pp: case pos
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
⊢ ∃ C, C • E = E'

Tactic:
have ha₆ := E.Δ'.ne_zero
    rw [coe_Δ', Δ_of_isShortNF, ha₄, zero_pow three_ne_zero, mul_zero, zero_add, ← mul_assoc,
      mul_ne_zero_iff, pow_ne_zero_iff two_ne_zero] at ha₆
    replace ha₆ := ha₆.2
    have ha₄' : E'.a₄ = 0 := by
      rw [ha₄, zero_pow three_ne_zero, zero_mul, zero_eq_mul] at heq
      exact (pow_eq_zero_iff three_ne_zero).1 <| heq.resolve_right <| pow_ne_zero 2 ha₆
    have ha₆' := E'.Δ'.ne_zero
    rw [coe_Δ', Δ_of_isShortNF, ha₄', zero_pow three_ne_zero, mul_zero, zero_add, ← mul_assoc,
      mul_ne_zero_iff, pow_ne_zero_iff two_ne_zero] at ha₆'
    replace ha₆' := ha₆'.2
    obtain ⟨u, hu⟩ := IsSepClosed.exists_pow_nat_eq (E.a₆ / E'.a₆) 6
    have hu0 : u ≠ 0 := by
      rw [← pow_ne_zero_iff (Nat.succ_ne_zero 5), hu, div_ne_zero_iff]
      exact ⟨ha₆, ha₆'⟩
    use ⟨Units.mk0 u hu0, 0, 0, 0⟩
    ext
    · simp [variableChange_a₁]
    · simp [variableChange_a₂]
    · simp [variableChange_a₃]
    · simp [ha₄, ha₄', variableChange_a₄]
    · simp_rw [variableChange_a₆, a₁_of_isShortNF, a₂_of_isShortNF, a₃_of_isShortNF,
        ha₄, Units.val_inv_eq_inv_val, Units.val_mk0, inv_pow, inv_mul_eq_div, hu]
      simp [field]

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

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

Tactic:
simpa only [preΨ_neg, natDegree_neg]
        using! ih n (by rwa [← Int.natAbs_neg]) <| neg_ne_zero.mp <| by exact_mod_cast h

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

Goal state before:
pp: R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
n : ℤ
h : ↑n ≠ 0
⊢ (W.preΨ n).leadingCoeff = ↑(if Even n then n / 2 else n)

Tactic:
rw [leadingCoeff, W.natDegree_preΨ h, coeff_preΨ]

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

Goal state before:
pp: A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
⊢ {U | ∃ i, U = ↑(B i)}.Nonempty

Tactic:
inhabit ι

Goal state after:
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
inhabited_h : Inhabited ι
⊢ {U | ∃ i, U = ↑(B i)}.Nonempty
Module: Mathlib.Topology.Algebra.Nonarchimedean.Bases

Goal state before:
pp: case h.right
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
i : ι
hi : {b | b - a ∈ B i} ⊆ s
⊢ (fun y ↦ a + y) '' ↑(B i) ⊆ s

Tactic:
· rw [image_subset_iff]
        rintro b b_in
        apply hi
        simpa using b_in

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

Goal state before:
pp: A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
i : ι
x✝ : TopologicalSpace A := hB.topology
⊢ IsOpen[hB.topology] (B i).carrier

Tactic:
rw [isOpen_iff_mem_nhds]

Goal state after:
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
i : ι
x✝ : TopologicalSpace A := hB.topology
⊢ ∀ x ∈ (B i).carrier, (B i).carrier ∈ 𝓝 x
Module: Mathlib.Topology.Algebra.Nonarchimedean.Bases

Goal state before:
pp: case h
ι : Type u_1
R : Type u_2
A : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
B : ι → Submodule R A
hB : SubmodulesRingBasis B
a : A
i j : ι
hj : a • B j ≤ B i
b : A
b_in : b ∈ B j
⊢ b ∈ (fun y ↦ a * y) ⁻¹' ↑(B i).toAddSubgroup

Tactic:
exact hj ⟨b, b_in, rfl⟩

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

Goal state before:
pp: 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
⊢ {U | ∃ i, U = ↑(B i)}.Nonempty

Tactic:
inhabit ι

Goal state after:
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
inhabited_h : Inhabited ι
⊢ {U | ∃ i, U = ↑(B i)}.Nonempty
Module: Mathlib.Topology.Algebra.Nonarchimedean.Bases

Goal state before:
pp: 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
i : ι
x✝ : TopologicalSpace M := hB.topology
this : TopologicalSpace M := hB.topology
⊢ IsOpen[hB.topology] (B i).toAddSubgroup.carrier

Tactic:
rw [isOpen_iff_mem_nhds]

Goal state after:
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
i : ι
x✝ : TopologicalSpace M := hB.topology
this : TopologicalSpace M := hB.topology
⊢ ∀ x ∈ (B i).toAddSubgroup.carrier, (B i).toAddSubgroup.carrier ∈ 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
⊢ IsOpen[_i.toTopologicalSpace] {x | v.restrict x < r}

Tactic:
rw [isOpen_iff_mem_nhds]

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 ∈ {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
⊢ IsOpen[_i.toTopologicalSpace] {x | v.restrict x ≤ r}

Tactic:
rw [isOpen_iff_mem_nhds]

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 ∈ {x | v.restrict x ≤ r}, {x | v.restrict x ≤ r} ∈ 𝓝 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:
by rw [← coe_zero, ← coe_smul, MulActionWithZero.smul_zero r]

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
r : (ofClass v).ValueGroup₀
⊢ IsOpen[_t] {x | v.restrict x < r}

Tactic:
rw [isOpen_iff_mem_nhds]

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 ∈ {x | v.restrict x < r}, {x | v.restrict x < r} ∈ 𝓝 x
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
⊢ 1 * ↑a = ↑a

Tactic:
rw [← coe_one, ← coe_mul, one_mul]

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
r : (ofClass v).ValueGroup₀
hr : r ≠ 0
⊢ IsOpen[_t] {x | v.restrict x ≤ r}

Tactic:
rw [isOpen_iff_mem_nhds]

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 ∈ {x | v.restrict x ≤ r}, {x | v.restrict x ≤ r} ∈ 𝓝 x
Module: Mathlib.RingTheory.Valuation.Discrete.Basic

Goal state before:
pp: Γ : Type u_1
inst✝² : LinearOrderedCommGroupWithZero Γ
K : Type u_3
inst✝¹ : Field K
w : Valuation K Γ
inst✝ : w.IsRankOneDiscrete
⊢ ↑(generator w) ∈ range ⇑w

Tactic:
apply sdiff_subset
  rw [← generator_zpowers_eq_range]
  exact ⟨generator w, by simp⟩

Goal state after:
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
hpow : v (↑π.val ^ (-m) * ↑r) = 1
a : ↥v.valuationSubring := ⟨↑π.val ^ (-m) * ↑r, ⋯⟩
ha : a = ⟨↑π.val ^ (-m) * ↑r, ⋯⟩
⊢ ¬↑π.val ^ ↑n = 0

Tactic:
zpow_natCast,

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
hpow : v (↑π.val ^ (-m) * ↑r) = 1
a : ↥v.valuationSubring := ⟨↑π.val ^ (-m) * ↑r, ⋯⟩
ha : a = ⟨↑π.val ^ (-m) * ↑r, ⋯⟩
⊢ ¬↑π.val ^ n = 0
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
⊢ ![(-(Q x * P z)) ^ 2 * P x, W'.addY P Q, addZ P Q] = -(Q x * P z) • P

Tactic:
addY_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)) ^ 2 * P x, (-(Q x * P z)) ^ 3 * P y, addZ P Q] = -(Q x * P z) • P
Module: Mathlib.RingTheory.Valuation.Discrete.Basic

Goal state before:
pp: case h
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
hpow : v (↑π.val ^ (-m) * ↑r) = 1
a : ↥v.valuationSubring := ⟨↑π.val ^ (-m) * ↑r, ⋯⟩
ha : a = ⟨↑π.val ^ (-m) * ↑r, ⋯⟩
ha₀ : ↑a ≠ 0
h_unit_a : IsUnit a
⊢ ↑r = ↑π.val ^ n * (↑π.val ^ ↑n)⁻¹ * ↑r

Tactic:
zpow_natCast,

Goal state after:
case h
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
hpow : v (↑π.val ^ (-m) * ↑r) = 1
a : ↥v.valuationSubring := ⟨↑π.val ^ (-m) * ↑r, ⋯⟩
ha : a = ⟨↑π.val ^ (-m) * ↑r, ⋯⟩
ha₀ : ↑a ≠ 0
h_unit_a : IsUnit a
⊢ ↑r = ↑π.val ^ n * (↑π.val ^ n)⁻¹ * ↑r
Module: Mathlib.Topology.Algebra.Valued.WithVal

Goal state before:
pp: 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 : v.IsEquiv w
⊢ Filter.Tendsto (⇑(equiv v)) (nhds 0) (nhds 0)

Tactic:
(Valued.hasBasis_nhds_zero _ _).tendsto_iff
    (Valued.hasBasis_nhds_zero _ _),

Goal state after:
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 : v.IsEquiv w
⊢ ∀ (ib : (ofClass Valued.v).ValueGroup₀ˣ),
    True → ∃ ia, True ∧ ∀ x ∈ {x | Valued.v.restrict x < ↑ia}, (equiv v) x ∈ {x | Valued.v.restrict x < ↑ib}
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula

Goal state before:
pp: R : Type r
S : Type s
inst✝¹ : CommRing R
inst✝ : CommRing S
W' : Jacobian R
f : R →+* S
P : Fin 3 → R
⊢ (W'.map f).negDblY (⇑f ∘ P) = f (W'.negDblY P)

Tactic:
simp only [negDblY, map_dblU, map_dblX, map_negY]
  simp

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

Goal state before:
pp: 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
⊢ Filter.Tendsto (⇑(equiv v).symm) (nhds 0) (nhds 0)

Tactic:
(Valued.hasBasis_nhds_zero _ _).tendsto_iff
    (Valued.hasBasis_nhds_zero _ _),

Goal state after:
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
⊢ ∀ (ib : (ofClass Valued.v).ValueGroup₀ˣ),
    True → ∃ ia, True ∧ ∀ x ∈ {x | Valued.v.restrict x < ↑ia}, (equiv v).symm x ∈ {x | Valued.v.restrict x < ↑ib}
Module: Mathlib.Topology.Algebra.Valued.WithVal

Goal state before:
pp: 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'
h : v.IsEquiv w
h_val : Valued.v.IsEquiv Valued.v
h_res : v.restrict.IsEquiv w.restrict
⊢ Filter.Tendsto (⇑(RingHom.id R)) (nhds 0) (nhds 0)

Tactic:
(Valued.hasBasis_nhds_zero _ _).tendsto_iff
    (Valued.hasBasis_nhds_zero _ _),

Goal state after:
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'
h : v.IsEquiv w
h_val : Valued.v.IsEquiv Valued.v
h_res : v.restrict.IsEquiv w.restrict
⊢ ∀ (ib : (ofClass Valued.v).ValueGroup₀ˣ),
    True → ∃ ia, True ∧ ∀ x ∈ {x | Valued.v.restrict x < ↑ia}, (RingHom.id R) x ∈ {x | Valued.v.restrict x < ↑ib}
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)) /
      (valuation K (maximalIdeal A)) ((algebraMap A K) (π ^ m)) ≤
    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_div₀,

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) / (algebraMap A K) (π ^ m)) ≤ 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.Topology.Algebra.Valued.WithVal

Goal state before:
pp: case hp
Γ₀ : Type u_5
Γ₀' : Type u_6
inst✝² : LinearOrderedCommGroupWithZero Γ₀
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀'
K : Type u_7
inst✝ : Field K
v : Valuation K Γ₀
w : Valuation K Γ₀'
h : v.IsEquiv w
h1 : ∀ (x : UniformSpace.Completion (WithVal v)), Valued.v x ≤ 1 ↔ Valued.v.restrict x ≤ 1
⊢ IsClosed {a | Valued.v.restrict a ≤ 1 ↔ Valued.v ((mapEquiv h.uniformEquiv) a) ≤ 1}

Tactic:
convert!
      (mapEquiv h.uniformEquiv).toHomeomorph.isClosed_setOfPred_iff
        (Valued.isClopen_closedBall _ one_ne_zero) (Valued.isClopen_closedBall _ one_ne_zero)

Goal state after:
Γ₀ : Type u_5
Γ₀' : Type u_6
inst✝² : LinearOrderedCommGroupWithZero Γ₀
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀'
K : Type u_7
inst✝ : Field K
v : Valuation K Γ₀
w : Valuation K Γ₀'
h : v.IsEquiv w
h1 : ∀ (x : UniformSpace.Completion (WithVal v)), Valued.v x ≤ 1 ↔ Valued.v.restrict x ≤ 1
x✝ : UniformSpace.Completion (WithVal v)
⊢ Valued.v ((mapEquiv h.uniformEquiv) x✝) ≤ 1 ↔ Valued.v.restrict ((mapEquiv h.uniformEquiv).toHomeomorph x✝) ≤ 1
Module: Mathlib.RingTheory.Valuation.Discrete.IsDiscreteValuationRing

Goal state before:
pp: case pos
A : Type u_1
inst✝² : CommRing A
inst✝¹ : IsDomain A
inst✝ : IsDiscreteValuationRing A
x : A
hx : x = 0
⊢ (maximalIdeal A).intValuation x = (ENat.recTopCoe 0 (fun x ↦ ↑(ofAdd ↑x)) ((addVal A) x))⁻¹

pp: case neg
A : Type u_1
inst✝² : CommRing A
inst✝¹ : IsDomain A
inst✝ : IsDiscreteValuationRing A
x : A
hx : ¬x = 0
⊢ (maximalIdeal A).intValuation x = (ENat.recTopCoe 0 (fun x ↦ ↑(ofAdd ↑x)) ((addVal A) x))⁻¹

Tactic:
· simp [hx]

Goal state after:
case neg
A : Type u_1
inst✝² : CommRing A
inst✝¹ : IsDomain A
inst✝ : IsDiscreteValuationRing A
x : A
hx : ¬x = 0
⊢ (maximalIdeal A).intValuation x = (ENat.recTopCoe 0 (fun x ↦ ↑(ofAdd ↑x)) ((addVal A) x))⁻¹
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Reduction

Goal state before:
pp: R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsDomain R
inst✝¹ : ValuationRing R
inst✝ : IsFractionRing R K
W : WeierstrassCurve K
l₀ : List K := [W.a₁, W.a₂, W.a₃, W.a₄, W.a₆]
l : List (ValueGroup R K) := List.map (fun a ↦ (valuation R K) a) l₀
lmax : ValueGroup R K := List.maximum_of_length_pos ⋯
hlmax_mem : lmax ∈ l
hlmax : ∀ v ∈ l, v ≤ lmax
hlmax_le_1 : ¬lmax ≤ 1
hlmax_ge_1 : lmax ≥ 1
a : K
ha : (valuation R K) a = lmax
⊢ a ≠ 0

Tactic:
by_contra ha₀

Goal state after:
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsDomain R
inst✝¹ : ValuationRing R
inst✝ : IsFractionRing R K
W : WeierstrassCurve K
l₀ : List K := [W.a₁, W.a₂, W.a₃, W.a₄, W.a₆]
l : List (ValueGroup R K) := List.map (fun a ↦ (valuation R K) a) l₀
lmax : ValueGroup R K := List.maximum_of_length_pos ⋯
hlmax_mem : lmax ∈ l
hlmax : ∀ v ∈ l, v ≤ lmax
hlmax_le_1 : ¬lmax ≤ 1
hlmax_ge_1 : lmax ≥ 1
a : K
ha : (valuation R K) a = lmax
ha₀ : a = 0
⊢ False
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: case refine_1
G : Type u_2
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
inst✝ : Nontrivial G
g : G
h : {x | x ≤ g}.WellFoundedOn fun x1 x2 ↦ x1 > x2
x✝ : G
⊢ x✝ ∈ {x | g⁻¹ ≤ x} → (fun x ↦ x⁻¹) x✝ ∈ {x | x ≤ g}

Tactic:
simp [Function.onFun, inv_le']

Goal state after:
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: case refine_2
G : Type u_2
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
inst✝ : Nontrivial G
g : G
h : {x | x ≤ g}.WellFoundedOn fun x1 x2 ↦ x1 > x2
a✝ : G
⊢ a✝ ∈ {x | g⁻¹ ≤ x} → ∀ b ∈ {x | g⁻¹ ≤ x}, a✝ < b → Function.onFun (fun x1 x2 ↦ x1 > x2) (fun x ↦ x⁻¹) a✝ b

Tactic:
simp [Function.onFun, inv_le']

Goal state after:
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: case refine_3
G : Type u_2
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
inst✝ : Nontrivial G
g : G
h : {x | g⁻¹ ≤ x}.WellFoundedOn fun x1 x2 ↦ x1 < x2
x✝ : G
⊢ x✝ ∈ {x | x ≤ g} → (fun x ↦ x⁻¹) x✝ ∈ {x | g⁻¹ ≤ x}

Tactic:
simp [Function.onFun, inv_le']

Goal state after:
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: case refine_4
G : Type u_2
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
inst✝ : Nontrivial G
g : G
h : {x | g⁻¹ ≤ x}.WellFoundedOn fun x1 x2 ↦ x1 < x2
a✝ : G
⊢ a✝ ∈ {x | x ≤ g} → ∀ b ∈ {x | x ≤ g}, a✝ > b → Function.onFun (fun x1 x2 ↦ x1 < x2) (fun x ↦ x⁻¹) a✝ b

Tactic:
simp [Function.onFun, inv_le']

Goal state after: