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

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W' : Jacobian R
P✝ : Fin 3 → R
u✝ : R
hu✝ : IsUnit u✝
u : R
hu : IsUnit u
P : Fin 3 → R
hP✝ : W'.Nonsingular (u • P)
hP : W'.Equation (u • P)
hP' :
  W'.a₁ * P y * P z - (3 * P x ^ 2 + 2 * W'.a₂ * P x * P z ^ 2 + W'.a₄ * P z ^ 4) = 0 ∧
    2 * P y + W'.a₁ * P x * P z + W'.a₃ * P z ^ 3 = 0 ∧
      W'.a₁ * P x * P y + 3 * W'.a₃ * P y * P z ^ 2 -
          (2 * W'.a₂ * P x ^ 2 * P z + 4 * W'.a₄ * P x * P z ^ 3 + 6 * W'.a₆ * P z ^ 5) =
        0
⊢ W'.a₁ * (u ^ 3 * P y) * (u * P z) -
        (3 * (u ^ 2 * P x) ^ 2 + 2 * W'.a₂ * (u ^ 2 * P x) * (u * P z) ^ 2 + W'.a₄ * (u * P z) ^ 4) =
      0 ∧
    2 * (u ^ 3 * P y) + W'.a₁ * (u ^ 2 * P x) * (u * P z) + W'.a₃ * (u * P z) ^ 3 = 0 ∧
      W'.a₁ * (u ^ 2 * P x) * (u ^ 3 * P y) + 3 * W'.a₃ * (u ^ 3 * P y) * (u * P z) ^ 2 -
          (2 * W'.a₂ * (u ^ 2 * P x) ^ 2 * (u * P z) + 4 * W'.a₄ * (u ^ 2 * P x) * (u * P z) ^ 3 +
            6 * W'.a₆ * (u * P z) ^ 5) =
        0

Tactic:
exact ⟨by linear_combination (norm := ring1) u ^ 4 * hP'.left,
      by linear_combination (norm := ring1) u ^ 3 * hP'.right.left,
      by linear_combination (norm := ring1) u ^ 5 * hP'.right.right⟩

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

Goal state before:
pp: case neg
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
ha₄' : E'.a₄ ≠ 0
ha₆' : E'.a₆ ≠ 0
u : F
hu : u ^ 2 = E.a₆ / E'.a₆ / (E.a₄ / E'.a₄)
hu4 : u ^ 4 = E.a₄ / E'.a₄
⊢ ∃ C, C • E = E'

Tactic:
have hu6 : u ^ 6 = E.a₆ / E'.a₆ := by
    rw [pow_mul u 2 3, hu]
    simp [field]
    linear_combination -heq

Goal state after:
case neg
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
ha₄' : E'.a₄ ≠ 0
ha₆' : E'.a₆ ≠ 0
u : F
hu : u ^ 2 = E.a₆ / E'.a₆ / (E.a₄ / E'.a₄)
hu4 : u ^ 4 = E.a₄ / E'.a₄
hu6 : u ^ 6 = E.a₆ / E'.a₆
⊢ ∃ C, C • E = E'
Module: Mathlib.RingTheory.Valuation.ExtendToLocalization

Goal state before:
pp: A : Type u_1
inst✝⁴ : CommRing A
Γ : Type u_2
inst✝³ : LinearOrderedCommGroupWithZero Γ
v : Valuation A Γ
S : Submonoid A
hS : S ≤ v.supp.primeCompl
B : Type u_3
inst✝² : CommRing B
inst✝¹ : Algebra A B
inst✝ : IsLocalization S B
f : S.LocalizationMap B := IsLocalization.toLocalizationMap S B
h : ∀ (s : ↥S), IsUnit (↑v.toMonoidWithZeroHom ↑s) := fun s ↦ isUnit_iff_ne_zero.mpr (hS s.property)
⊢ (↑(f.lift h)).toFun 0 = 0

Tactic:
convert! f.lift_eq (P := Γ) _ 0 <;> simp [f]

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 : OpenSubgroup G
hV : ↑V ⊆ U₁
⊢ ∃ V W, ↑V ×ˢ ↑W ⊆ U

Tactic:
obtain ⟨W, hW⟩ := is_nonarchimedean _ hU₂

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
W : OpenSubgroup K
hW : ↑W ⊆ U₂
⊢ ∃ V W, ↑V ×ˢ ↑W ⊆ U
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
inst✝² : TopologicalSpace R
v : Valuation R Γ₀
inst✝¹ : v.Compatible
inst✝ : IsTopologicalAddGroup R
H : ∀ {s : Set R}, s ∈ 𝓝 0 ↔ ∃ γ, {z | v.restrict z < ↑γ} ⊆ s
s : Set R
x : R
⊢ ∀ (γ : (ofClass v).ValueGroup₀ˣ), x +ᵥ {z | v.restrict z < ↑γ} = {a | v.restrict (-x + a) < ↑γ}

Tactic:
simp [Set.ext_iff, mem_vadd_set_iff_neg_vadd_mem]

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

Goal state before:
pp: R : Type u_1
inst✝² : Semiring R
inst✝¹ : ValuativeRel R
inst✝ : IsNontrivial R
⊢ ∃ γ, 0 < γ ∧ γ < 1

Tactic:
obtain ⟨γ, h0, h1⟩ := IsNontrivial.condition (R := R)
  obtain h1 | h1 := lt_or_lt_iff_ne.mpr h1
  · exact ⟨γ, zero_lt_iff.mpr h0, h1⟩
  · exact ⟨γ⁻¹, by simpa [zero_lt_iff], by simp [inv_lt_one_iff₀, h0, h1]⟩

Goal state after:
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 : AddCommGroup α
inst✝ : IsUniformAddGroup α
a b : Completion α
x y : α
⊢ ↑x + ↑y = ↑y + ↑x

Tactic:
change (x : Completion α) + ↑y = ↑y + ↑x
        rw [← coe_add, ← coe_add, add_comm]

Goal state after:
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 : AddCommGroup α
inst✝³ : IsUniformAddGroup α
inst✝² : Semiring R
inst✝¹ : Module R α
inst✝ : UniformContinuousConstSMul R α
a b : R
x : α
⊢ (a + b) • ↑x = a • ↑x + b • ↑x

Tactic:
rw [← coe_smul, add_smul, coe_add, coe_smul, coe_smul]

Goal state after:
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:
rw [← coe_zero, ← coe_mul, mul_zero]

Goal state after:
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
F_cau : Cauchy F
inf_F : 𝓝 0 ⊓ F = ⊥
⊢ Cauchy (map (fun x ↦ x⁻¹) F)

Tactic:
let i : K →+* L := K.subtype

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
F_cau : Cauchy F
inf_F : 𝓝 0 ⊓ F = ⊥
i : ↥K →+* L := K.subtype
⊢ Cauchy (map (fun x ↦ x⁻¹) F)
Module: Mathlib.RingTheory.Valuation.Discrete.Basic

Goal state before:
pp: Γ : Type u_1
inst✝³ : LinearOrderedCommGroupWithZero Γ
K : Type u_2
inst✝² : Field K
v : Valuation K Γ
inst✝¹ : IsCyclic ↥(MonoidWithZeroHom.ofClass v).valueGroup
inst✝ : Nontrivial ↥(MonoidWithZeroHom.ofClass v).valueGroup
g : Γˣ := Subgroup.genLTOne (MonoidWithZeroHom.ofClass v).valueGroup
hg : g = Subgroup.genLTOne (MonoidWithZeroHom.ofClass v).valueGroup
⊢ ↑g ∈ range ⇑(MonoidWithZeroHom.ofClass v) \ {0}

Tactic:
by
    rw [← valueGroup_eq_range, hg]
    exact mem_image_of_mem Units.val (valueGroup (.ofClass v)).genLTOne_mem

Goal state after:
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'.addXYZ P Q = -(Q x * P z) • P

Tactic:
by
  rw [addXYZ, addX_of_Z_eq_zero_right hQz, addY_of_Z_eq_zero_right hQ hQz,
    addZ_of_Z_eq_zero_right hQz, smul_fin3]

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
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
⊢ ∃ ia, ∀ x ∈ {x | Valued.v.restrict x < ↑ia}, (equiv v).symm x ∈ {x | Valued.v.restrict x < ↑γ}

Tactic:
use .mk0 ((Valued.v.restrict ((WithVal.equiv v) r)) /
    (Valued.v.restrict ((WithVal.equiv v) s))) (by
    simp only [equiv_apply, restrict_def, ne_eq, div_eq_zero_iff, restrict₀_eq_zero_iff, hv,
      MonoidWithZeroHom.coe_ofClass, not_or, (eq_zero h (r := r.ofVal)).ne,
      (eq_zero h (r := s.ofVal)).ne]
    exact ⟨hr₀.ne', hs₀.ne'⟩)

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 ∈ {x | Valued.v.restrict x < ↑(Units.mk0 (Valued.v.restrict ((equiv v) r) / Valued.v.restrict ((equiv v) s)) ⋯)},
    (equiv v).symm x ∈ {x | Valued.v.restrict x < ↑γ}
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula

Goal state before:
pp: R : Type r
S : Type s
A : Type u
B : Type v
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
W' : Jacobian R
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
P Q : Fin 3 → A
⊢ (W'⁄B).addY (⇑f ∘ P) (⇑f ∘ Q) = f ((W'⁄A).addY P Q)

Tactic:
by
  rw [← RingHom.coe_coe, ← map_addY, map_baseChange]

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'
h : v.IsEquiv w
h_val : Valued.v.IsEquiv Valued.v
h_res : v.restrict.IsEquiv w.restrict
x : (ofClass Valued.v).ValueGroup₀ˣ
⊢ ∃ ia, ∀ x_1 ∈ {x | Valued.v.restrict x < ↑ia}, (RingHom.id R) x_1 ∈ {x_2 | Valued.v.restrict x_2 < ↑x}

Tactic:
let u := WithZero.unzero (Units.ne_zero x)

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
x : (ofClass Valued.v).ValueGroup₀ˣ
u : ↥(ofClass Valued.v).valueGroup := WithZero.unzero ⋯
⊢ ∃ ia, ∀ x_1 ∈ {x | Valued.v.restrict x < ↑ia}, (RingHom.id R) x_1 ∈ {x_2 | Valued.v.restrict x_2 < ↑x}
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: K : Type u_1
inst✝¹ : DivisionRing K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
⊢ Continuous[hv.toTopologicalSpace, _] ⇑v.restrict

Tactic:
continuous_iff_continuousAt

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

Goal state before:
pp: K : Type u_1
inst✝¹ : DivisionRing K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
hsurj : Function.Surjective ⇑v
⊢ Continuous[hv.toTopologicalSpace, topologicalSpace] ⇑v

Tactic:
continuous_iff_continuousAt

Goal state after:
K : Type u_1
inst✝¹ : DivisionRing K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
hsurj : Function.Surjective ⇑v
⊢ ∀ (x : K), ContinuousAt (⇑v) x
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: e : ℤ ≃+o ℤᵒᵈ
H : (↑e).trans { toEquiv := toDual, map_add' := ⋯ } = AddEquiv.refl ℤ
⊢ False

Tactic:
by
      replace H : e 1 = 1 := congr($H 1)
      have h1 : 0 < e 1 := by
        rw [← map_zero e, map_lt_map_iff]
        simp
      simp [H, ← ofDual_lt_ofDual] at h1

Goal state after:
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: e : ℤ ≃+o ℤᵒᵈ
⊢ ↑e = (AddEquiv.neg ℤ).trans { toEquiv := toDual, map_add' := ⋯ }

Tactic:
refine Int.addEquiv_eq_refl_or_neg ((e : ℤ ≃+ ℤᵒᵈ).trans ⟨toDual, toDual_add⟩)
        |>.resolve_left fun H => by
      replace H : e 1 = 1 := congr($H 1)
      have h1 : 0 < e 1 := by
        rw [← map_zero e, map_lt_map_iff]
        simp
      simp [H, ← ofDual_lt_ofDual] at h1

Goal state after:
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: G : Type u_1
G' : Type u_2
inst✝⁵ : CommGroup G
inst✝⁴ : LinearOrder G
inst✝³ : IsOrderedMonoid G
inst✝² : CommGroup G'
inst✝¹ : LinearOrder G'
inst✝ : IsOrderedMonoid G'
x : G
y : G'
hxy : x = 1 ↔ y = 1
hx : ¬x = 1
x' : G := max x x⁻¹
hx' : x' = max x x⁻¹
⊢ ↥(closure {x}) ≃*o ↥(closure {y})

Tactic:
have xpos : 1 < x' := by
      simp [hx', eq_comm, hx]

Goal state after:
G : Type u_1
G' : Type u_2
inst✝⁵ : CommGroup G
inst✝⁴ : LinearOrder G
inst✝³ : IsOrderedMonoid G
inst✝² : CommGroup G'
inst✝¹ : LinearOrder G'
inst✝ : IsOrderedMonoid G'
x : G
y : G'
hxy : x = 1 ↔ y = 1
hx : ¬x = 1
x' : G := max x x⁻¹
hx' : x' = max x x⁻¹
xpos : 1 < x'
⊢ ↥(closure {x}) ≃*o ↥(closure {y})
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: G : Type u_1
G' : Type u_2
inst✝⁵ : CommGroup G
inst✝⁴ : LinearOrder G
inst✝³ : IsOrderedMonoid G
inst✝² : CommGroup G'
inst✝¹ : LinearOrder G'
inst✝ : IsOrderedMonoid G'
x : G
y : G'
hxy : x = 1 ↔ y = 1
hx : ¬x = 1
x' : G := max x x⁻¹
hx' : x' = max x x⁻¹
xpos : 1 < x'
y' : G' := max y y⁻¹
hy' : y' = max y y⁻¹
ypos : 1 < y'
⊢ closure {x} = closure {x'}

Tactic:
rcases max_cases x x⁻¹

Goal state after:
case inl
G : Type u_1
G' : Type u_2
inst✝⁵ : CommGroup G
inst✝⁴ : LinearOrder G
inst✝³ : IsOrderedMonoid G
inst✝² : CommGroup G'
inst✝¹ : LinearOrder G'
inst✝ : IsOrderedMonoid G'
x : G
y : G'
hxy : x = 1 ↔ y = 1
hx : ¬x = 1
x' : G := max x x⁻¹
hx' : x' = max x x⁻¹
xpos : 1 < x'
y' : G' := max y y⁻¹
hy' : y' = max y y⁻¹
ypos : 1 < y'
h✝ : max x x⁻¹ = x ∧ x⁻¹ ≤ x
⊢ closure {x} = closure {x'}

case inr
G : Type u_1
G' : Type u_2
inst✝⁵ : CommGroup G
inst✝⁴ : LinearOrder G
inst✝³ : IsOrderedMonoid G
inst✝² : CommGroup G'
inst✝¹ : LinearOrder G'
inst✝ : IsOrderedMonoid G'
x : G
y : G'
hxy : x = 1 ↔ y = 1
hx : ¬x = 1
x' : G := max x x⁻¹
hx' : x' = max x x⁻¹
xpos : 1 < x'
y' : G' := max y y⁻¹
hy' : y' = max y y⁻¹
ypos : 1 < y'
h✝ : max x x⁻¹ = x⁻¹ ∧ x < x⁻¹
⊢ closure {x} = closure {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
x y : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ embedding (extension (x * y)) = embedding (extension x * extension y)

Tactic:
embedding_strictMono.injective.eq_iff

Goal state after:
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x y : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ extension (x * y) = extension x * extension y
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
i : ℕ := List.idxOf lmax l
hi : i < l.length
⊢ (valuation R K) l₀[i] = l[i]

Tactic:
simp [l]

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

Goal state before:
pp: case h.h₁
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 := ⋯
l : List (ValueGroup R K) := ⋯
lmax : ValueGroup R K := ⋯
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
⊢ ∃ r₁, (algebraMap R K) r₁ = ({ u := Units.mk0 a ha₀, r := 0, s := 0, t := 0 } • W).a₁

Tactic:
apply (mem_integer_iff _ _ _).mp

Goal state after:
case h.h₁
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 := ⋯
l : List (ValueGroup R K) := ⋯
lmax : ValueGroup R K := ⋯
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
⊢ ({ u := Units.mk0 a ha₀, r := 0, s := 0, t := 0 } • W).a₁ ∈ (valuation R K).integer
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Reduction

Goal state before:
pp: case h.h₂
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 := ⋯
l : List (ValueGroup R K) := ⋯
lmax : ValueGroup R K := ⋯
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
⊢ ∃ r₂, (algebraMap R K) r₂ = ({ u := Units.mk0 a ha₀, r := 0, s := 0, t := 0 } • W).a₂

Tactic:
apply (mem_integer_iff _ _ _).mp

Goal state after:
case h.h₂
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 := ⋯
l : List (ValueGroup R K) := ⋯
lmax : ValueGroup R K := ⋯
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
⊢ ({ u := Units.mk0 a ha₀, r := 0, s := 0, t := 0 } • W).a₂ ∈ (valuation R K).integer
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 : hat K
h : x ≠ 0
hr : ∀ (r : K), embedding ((restrict₀ (ofClass v)) r) = v r
a b : (ofClass v).ValueGroup₀
⊢ embedding a = embedding b ↔ a = b

Tactic:
embedding_strictMono.injective.eq_iff

Goal state after:
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x : hat K
h : x ≠ 0
hr : ∀ (r : K), embedding ((restrict₀ (ofClass v)) r) = v r
a b : (ofClass v).ValueGroup₀
⊢ a = b ↔ a = b
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Reduction

Goal state before:
pp: case h.h₃
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 := ⋯
l : List (ValueGroup R K) := ⋯
lmax : ValueGroup R K := ⋯
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
⊢ ∃ r₃, (algebraMap R K) r₃ = ({ u := Units.mk0 a ha₀, r := 0, s := 0, t := 0 } • W).a₃

Tactic:
apply (mem_integer_iff _ _ _).mp

Goal state after:
case h.h₃
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 := ⋯
l : List (ValueGroup R K) := ⋯
lmax : ValueGroup R K := ⋯
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
⊢ ({ u := Units.mk0 a ha₀, r := 0, s := 0, t := 0 } • W).a₃ ∈ (valuation R K).integer
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Reduction

Goal state before:
pp: case h.h₄
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 := ⋯
l : List (ValueGroup R K) := ⋯
lmax : ValueGroup R K := ⋯
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
⊢ ∃ r₄, (algebraMap R K) r₄ = ({ u := Units.mk0 a ha₀, r := 0, s := 0, t := 0 } • W).a₄

Tactic:
apply (mem_integer_iff _ _ _).mp

Goal state after:
case h.h₄
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 := ⋯
l : List (ValueGroup R K) := ⋯
lmax : ValueGroup R K := ⋯
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
⊢ ({ u := Units.mk0 a ha₀, r := 0, s := 0, t := 0 } • W).a₄ ∈ (valuation R K).integer
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Reduction

Goal state before:
pp: case h.h₆
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 := ⋯
l : List (ValueGroup R K) := ⋯
lmax : ValueGroup R K := ⋯
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
⊢ ∃ r₆, (algebraMap R K) r₆ = ({ u := Units.mk0 a ha₀, r := 0, s := 0, t := 0 } • W).a₆

Tactic:
apply (mem_integer_iff _ _ _).mp

Goal state after:
case h.h₆
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 := ⋯
l : List (ValueGroup R K) := ⋯
lmax : ValueGroup R K := ⋯
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
⊢ ({ u := Units.mk0 a ha₀, r := 0, s := 0, t := 0 } • W).a₆ ∈ (valuation R K).integer
Module: Mathlib.RingTheory.PowerSeries.Basic

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
n : ℕ
⊢ (coeff n) X = if n = 1 then 1 else 0

Tactic:
X_eq,

Goal state after:
R : Type u_1
inst✝ : Semiring R
n : ℕ
⊢ (coeff n) ((monomial 1) 1) = if n = 1 then 1 else 0
Module: Mathlib.RingTheory.MvPowerSeries.Basic

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
φ₁ φ₂ φ₃ : MvPowerSeries σ R
n : σ →₀ ℕ
⊢ (coeff n) (φ₁ * (φ₂ + φ₃)) = (coeff n) (φ₁ * φ₂ + φ₁ * φ₃)

Tactic:
simp only [coeff_mul, mul_add, Finset.sum_add_distrib, map_add]

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

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝² : CommSemiring R
ι : Type u_4
inst✝¹ : DecidableEq ι
inst✝ : DecidableEq σ
f : ι → MvPowerSeries σ R
d : σ →₀ ℕ
s : Finset ι
⊢ (coeff d) (∏ j ∈ s, f j) = ∑ l ∈ s.finsuppAntidiag d, ∏ i ∈ s, (coeff (l i)) (f i)

Tactic:
induction s using Finset.induction_on generalizing d with

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

Goal state before:
pp: case neg
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
s : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
this : (𝓝 0).HasBasis (fun x ↦ True) fun γ ↦ {x | extensionValuation x < ↑((Units.map ↑embedding) γ)}
x : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
γ : (ofClass v).ValueGroup₀ˣ
hx0 : ¬x = 0
y : K := ⋯.choose
hy_def : y = ⋯.choose
hy : v y = embedding ↑γ
⊢ ↑⟨Units.mk0 (embedding (extension x)) ⋯, ⋯⟩ < ↑⟨Units.mk0 ((ofClass extensionValuation) ↑y) ?neg.hnc✝, ⋯⟩ ↔
    extension x < ↑γ

Tactic:
simp only [coe_ofClass, extensionValuation_toFun, extension_extends,
              Valuation.embedding_restrict, WithZero.coe_lt_coe, Subtype.mk_lt_mk,
              ← Units.val_lt_val, Units.val_mk0]
              convert embedding_strictMono (f := (.ofClass hv.v)).lt_iff_lt

Goal state after: