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

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

Tactic:
addMap_eq,

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

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

Tactic:
addMap_eq,

Goal state after:
case pos
F : Type u
inst✝ : Field F
W : Jacobian F
P✝ : PointClass F
Q : Fin 3 → F
hQ : W.Nonsingular Q
hQz : Q z = 0
P : Fin 3 → F
hP : W.NonsingularLift ⟦P⟧
hPz : P z = 0
⊢ ⟦W.add P Q⟧ = ⟦P⟧
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Point

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

Tactic:
addMap_eq,

Goal state after:
case neg
F : Type u
inst✝ : Field F
W : Jacobian F
P✝ : PointClass F
Q : Fin 3 → F
hQ : W.Nonsingular Q
hQz : Q z = 0
P : Fin 3 → F
hP : W.NonsingularLift ⟦P⟧
hPz : ¬P z = 0
⊢ ⟦W.add P Q⟧ = ⟦P⟧
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.Equation Q
hPz : P z ≠ 0
hQz : Q z ≠ 0
hx : P x * Q z ^ 2 = Q x * P z ^ 2
hy' : P y * Q z ^ 3 = W.negY Q * P z ^ 3
hy : P y * Q z ^ 3 = Q y * P z ^ 3
⊢ W.addMap ⟦P⟧ ⟦Q⟧ = ⟦![1, 1, 0]⟧

Tactic:
addMap_eq,

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.Equation Q
hPz : P z ≠ 0
hQz : Q z ≠ 0
hx : P x * Q z ^ 2 = Q x * P z ^ 2
hy' : P y * Q z ^ 3 = W.negY Q * P z ^ 3
hy : P y * Q z ^ 3 = Q y * P z ^ 3
⊢ ⟦W.add P Q⟧ = ⟦![1, 1, 0]⟧
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.Equation Q
hPz : P z ≠ 0
hQz : Q z ≠ 0
hx : P x * Q z ^ 2 = Q x * P z ^ 2
hy' : P y * Q z ^ 3 = W.negY Q * P z ^ 3
hy : ¬P y * Q z ^ 3 = Q y * P z ^ 3
⊢ W.addMap ⟦P⟧ ⟦Q⟧ = ⟦![1, 1, 0]⟧

Tactic:
addMap_eq,

Goal state after:
case neg
F : Type u
inst✝ : Field F
W : Jacobian F
P Q : Fin 3 → F
hP : W.Nonsingular P
hQ : W.Equation Q
hPz : P z ≠ 0
hQz : Q z ≠ 0
hx : P x * Q z ^ 2 = Q x * P z ^ 2
hy' : P y * Q z ^ 3 = W.negY Q * P z ^ 3
hy : ¬P y * Q z ^ 3 = Q y * P z ^ 3
⊢ ⟦W.add P Q⟧ = ⟦![1, 1, 0]⟧
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.Equation P
hQ : W.Equation Q
hPz : P z ≠ 0
hQz : Q z ≠ 0
hxy : ¬(P x * Q z ^ 2 = Q x * P z ^ 2 ∧ P y * Q z ^ 3 = W.negY Q * P z ^ 3)
hx : P x * Q z ^ 2 = Q x * P z ^ 2
⊢ W.addMap ⟦P⟧ ⟦Q⟧ =
    ⟦![W.toAffine.addX (P x / P z ^ 2) (Q x / Q z ^ 2)
          (W.toAffine.slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3)),
        W.toAffine.addY (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3)
          (W.toAffine.slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3)),
        1]⟧

Tactic:
addMap_eq,

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.Equation P
hQ : W.Equation Q
hPz : P z ≠ 0
hQz : Q z ≠ 0
hxy : ¬(P x * Q z ^ 2 = Q x * P z ^ 2 ∧ P y * Q z ^ 3 = W.negY Q * P z ^ 3)
hx : P x * Q z ^ 2 = Q x * P z ^ 2
⊢ ⟦W.add P Q⟧ =
    ⟦![W.toAffine.addX (P x / P z ^ 2) (Q x / Q z ^ 2)
          (W.toAffine.slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3)),
        W.toAffine.addY (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3)
          (W.toAffine.slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3)),
        1]⟧
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.Equation P
hQ : W.Equation Q
hPz : P z ≠ 0
hQz : Q z ≠ 0
hxy : ¬(P x * Q z ^ 2 = Q x * P z ^ 2 ∧ P y * Q z ^ 3 = W.negY Q * P z ^ 3)
hx : ¬P x * Q z ^ 2 = Q x * P z ^ 2
⊢ W.addMap ⟦P⟧ ⟦Q⟧ =
    ⟦![W.toAffine.addX (P x / P z ^ 2) (Q x / Q z ^ 2)
          (W.toAffine.slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3)),
        W.toAffine.addY (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3)
          (W.toAffine.slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3)),
        1]⟧

Tactic:
addMap_eq,

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.Equation P
hQ : W.Equation Q
hPz : P z ≠ 0
hQz : Q z ≠ 0
hxy : ¬(P x * Q z ^ 2 = Q x * P z ^ 2 ∧ P y * Q z ^ 3 = W.negY Q * P z ^ 3)
hx : ¬P x * Q z ^ 2 = Q x * P z ^ 2
⊢ ⟦W.add P Q⟧ =
    ⟦![W.toAffine.addX (P x / P z ^ 2) (Q x / Q z ^ 2)
          (W.toAffine.slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3)),
        W.toAffine.addY (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3)
          (W.toAffine.slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3)),
        1]⟧
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x y z : R
hx : 0 <ᵥ x
⊢ x * y ≤ᵥ x * z ↔ y ≤ᵥ z

Tactic:
by
  refine ⟨fun h ↦ ?_ , fun h ↦ ?_⟩
  · grw [veq_mul_comm, veq_mul_comm (x := x)] at h
    rwa [mul_vle_mul_iff_left hx] at h
  · grw [veq_mul_comm, veq_mul_comm (x := x)]
    rwa [mul_vle_mul_iff_left hx]

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

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x y z w : R
⊢ x * y * (z * w) =ᵥ x * z * (y * w)

Tactic:
← mul_assoc,

Goal state after:
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x y z w : R
⊢ x * y * z * w =ᵥ x * z * (y * w)
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: case pos
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x✝ x' y✝ y' z✝ x y z w : R
t s u v : ↥(posSubmonoid R)
h₁ : x * ↑t ≤ᵥ y * ↑s
h₂ : y * ↑s ≤ᵥ x * ↑t
h₃ : z * ↑u ≤ᵥ w * ↑v
h₄ : w * ↑v ≤ᵥ z * ↑u
hw : w ≤ᵥ 0
hz : z ≤ᵥ 0
⊢ (x * ↑v ≤ᵥ z * ↑s) = (y * ↑u ≤ᵥ w * ↑t)

Tactic:
refine propext ⟨fun h => vle_trans ?_ (zero_vle _), fun h => vle_trans ?_ (zero_vle _)⟩

Goal state after:
case pos.refine_1
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x✝ x' y✝ y' z✝ x y z w : R
t s u v : ↥(posSubmonoid R)
h₁ : x * ↑t ≤ᵥ y * ↑s
h₂ : y * ↑s ≤ᵥ x * ↑t
h₃ : z * ↑u ≤ᵥ w * ↑v
h₄ : w * ↑v ≤ᵥ z * ↑u
hw : w ≤ᵥ 0
hz : z ≤ᵥ 0
h : x * ↑v ≤ᵥ z * ↑s
⊢ y * ↑u ≤ᵥ 0

case pos.refine_2
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x✝ x' y✝ y' z✝ x y z w : R
t s u v : ↥(posSubmonoid R)
h₁ : x * ↑t ≤ᵥ y * ↑s
h₂ : y * ↑s ≤ᵥ x * ↑t
h₃ : z * ↑u ≤ᵥ w * ↑v
h₄ : w * ↑v ≤ᵥ z * ↑u
hw : w ≤ᵥ 0
hz : z ≤ᵥ 0
h : y * ↑u ≤ᵥ w * ↑t
⊢ x * ↑v ≤ᵥ 0
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: case neg
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x✝ x' y✝ y' z✝ x y z w : R
t s u v : ↥(posSubmonoid R)
h₁ : x * ↑t ≤ᵥ y * ↑s
h₂ : y * ↑s ≤ᵥ x * ↑t
h₃ : z * ↑u ≤ᵥ w * ↑v
h₄ : w * ↑v ≤ᵥ z * ↑u
hw : w ≤ᵥ 0
hz : ¬z ≤ᵥ 0
⊢ z * ↑u ≤ᵥ 0 * ↑u

Tactic:
simpa using h₃.trans (mul_vle_mul_left hw v)

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

Goal state before:
pp: case neg.refine_2
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x✝ x' y✝ y' z✝ x y z w : R
t s u v : ↥(posSubmonoid R)
h₁ : x * ↑t ≤ᵥ y * ↑s
h₂ : y * ↑s ≤ᵥ x * ↑t
h₃ : z * ↑u ≤ᵥ w * ↑v
h₄ : w * ↑v ≤ᵥ z * ↑u
hw : ¬w ≤ᵥ 0
hz : ¬z ≤ᵥ 0
h : y * ↑u ≤ᵥ w * ↑t
⊢ x * ↑v * ↑t ≤ᵥ z * ↑s * ↑t

Tactic:
apply vle_mul_cancel hw

Goal state after:
case neg.refine_2
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x✝ x' y✝ y' z✝ x y z w : R
t s u v : ↥(posSubmonoid R)
h₁ : x * ↑t ≤ᵥ y * ↑s
h₂ : y * ↑s ≤ᵥ x * ↑t
h₃ : z * ↑u ≤ᵥ w * ↑v
h₄ : w * ↑v ≤ᵥ z * ↑u
hw : ¬w ≤ᵥ 0
hz : ¬z ≤ᵥ 0
h : y * ↑u ≤ᵥ w * ↑t
⊢ x * ↑v * ↑t * w ≤ᵥ z * ↑s * ↑t * w
Module: Mathlib.Topology.Algebra.Valued.ValuationTopology

Goal state before:
pp: case mpr
R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
_i : Valued R Γ₀
F : Filter R
⊢ (∀ (γ : (ofClass v).ValueGroup₀ˣ), ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, v.restrict (y - x) < ↑γ) →
    ∀ (U : Set R), (∃ i, U = ↑(v.ltAddSubgroup ((Units.map ↑embedding) i))) → ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, y - x ∈ U

Tactic:
· rintro h - ⟨γ, rfl⟩
    simp_rw [restrict_lt_iff_lt_embedding] at h
    exact h γ

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

Goal state before:
pp: R✝ : Type u_1
inst✝⁵ : Semiring R✝
inst✝⁴ : ValuativeRel R✝
x✝² x' y y' z : R✝
R : Type u_2
inst✝³ : Ring R
inst✝² : ValuativeRel R
x : R
S : Type u_3
Γ : Type u_4
inst✝¹ : Ring S
inst✝ : LinearOrderedCommGroupWithZero Γ
v : Valuation S Γ
x✝¹ y✝ : S
x✝ : v x✝¹ ≤ v y✝
h : S
⊢ v (x✝¹ * h) ≤ v (y✝ * h)

Tactic:
simp only [map_mul]; gcongr

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

Goal state before:
pp: case mpr
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
⊢ (∀ (γ : (ofClass v).ValueGroup₀ˣ), ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, v.restrict (y - x) < ↑γ) →
    ∀ (U : Set R), (∃ i, U = ↑(v.ltAddSubgroup ((Units.map ↑embedding) i))) → ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, y - x ∈ U

Tactic:
· rintro h - ⟨γ, rfl⟩
    simp_rw [restrict_lt_iff_lt_embedding] at h
    exact h γ

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

Goal state before:
pp: α : Type u_1
inst✝⁹ : UniformSpace α
inst✝⁸ : Semiring α
β : Type u_2
inst✝⁷ : UniformSpace β
inst✝⁶ : Semiring β
inst✝⁵ : IsTopologicalSemiring β
γ : Type u_3
inst✝⁴ : UniformSpace γ
inst✝³ : Semiring γ
inst✝² : IsTopologicalSemiring γ
inst✝¹ : T2Space γ
inst✝ : CompleteSpace γ
i : α →+* β
f : α →+* γ
ue : IsUniformInducing ⇑i
dr : DenseRange ⇑i
hf : UniformContinuous ⇑f
⊢ ⋯.extend (⇑f) 1 = 1

Tactic:
convert! IsDenseInducing.extend_eq (ue.isDenseInducing dr) hf.continuous 1
    exacts [i.map_one.symm, f.map_one.symm]

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

Goal state before:
pp: α : Type u_1
inst✝⁹ : UniformSpace α
inst✝⁸ : Semiring α
β : Type u_2
inst✝⁷ : UniformSpace β
inst✝⁶ : Semiring β
inst✝⁵ : IsTopologicalSemiring β
γ : Type u_3
inst✝⁴ : UniformSpace γ
inst✝³ : Semiring γ
inst✝² : IsTopologicalSemiring γ
inst✝¹ : T2Space γ
inst✝ : CompleteSpace γ
i : α →+* β
f : α →+* γ
ue : IsUniformInducing ⇑i
dr : DenseRange ⇑i
hf : UniformContinuous ⇑f
⊢ ⋯.extend (⇑f) 1 = 1

Tactic:
by
    convert! IsDenseInducing.extend_eq (ue.isDenseInducing dr) hf.continuous 1
    exacts [i.map_one.symm, f.map_one.symm]

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

Goal state before:
pp: case e'_2
α : Type u_1
inst✝⁹ : UniformSpace α
inst✝⁸ : Semiring α
β : Type u_2
inst✝⁷ : UniformSpace β
inst✝⁶ : Semiring β
inst✝⁵ : IsTopologicalSemiring β
γ : Type u_3
inst✝⁴ : UniformSpace γ
inst✝³ : Semiring γ
inst✝² : IsTopologicalSemiring γ
inst✝¹ : T2Space γ
inst✝ : CompleteSpace γ
i : α →+* β
f : α →+* γ
ue : IsUniformInducing ⇑i
dr : DenseRange ⇑i
hf : UniformContinuous ⇑f
⊢ 0 = i 0

Tactic:
simp only [map_zero]

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

Goal state before:
pp: case e'_3
α : Type u_1
inst✝⁹ : UniformSpace α
inst✝⁸ : Semiring α
β : Type u_2
inst✝⁷ : UniformSpace β
inst✝⁶ : Semiring β
inst✝⁵ : IsTopologicalSemiring β
γ : Type u_3
inst✝⁴ : UniformSpace γ
inst✝³ : Semiring γ
inst✝² : IsTopologicalSemiring γ
inst✝¹ : T2Space γ
inst✝ : CompleteSpace γ
i : α →+* β
f : α →+* γ
ue : IsUniformInducing ⇑i
dr : DenseRange ⇑i
hf : UniformContinuous ⇑f
⊢ 0 = f 0

Tactic:
simp only [map_zero]

Goal state after:
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 ^ (-m) * ↑r, ⋯⟩

Tactic:
← mul_assoc,

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 ^ (-m) * ↑r
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 * ((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:
← mul_assoc,

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 * ((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.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'
hxc : closure {x} = closure {x'}
hyc : closure {y} = closure {y'}
⊢ ∀ (a : G) (b : a ∈ zpowers x'), ⟨a, ⋯⟩ ∈ zpowers ⟨x', ⋯⟩

Tactic:
mem_zpowers_iff,

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'
y' : G' := max y y⁻¹
hy' : y' = max y y⁻¹
ypos : 1 < y'
hxc : closure {x} = closure {x'}
hyc : closure {y} = closure {y'}
⊢ ∀ (a : G) (b : ∃ k, x' ^ k = a), ∃ k, ⟨x', ⋯⟩ ^ k = ⟨a, ⋯⟩
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'
hxc : closure {x} = closure {x'}
hyc : closure {y} = closure {y'}
Hx : ∀ (z : ↥(closure {x})), z ∈ zpowers ⟨x', ⋯⟩
⊢ ∀ (a : G') (b : a ∈ zpowers y'), ⟨a, ⋯⟩ ∈ zpowers ⟨y', ⋯⟩

Tactic:
mem_zpowers_iff,

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'
y' : G' := max y y⁻¹
hy' : y' = max y y⁻¹
ypos : 1 < y'
hxc : closure {x} = closure {x'}
hyc : closure {y} = closure {y'}
Hx : ∀ (z : ↥(closure {x})), z ∈ zpowers ⟨x', ⋯⟩
⊢ ∀ (a : G') (b : ∃ k, y' ^ k = a), ∃ k, ⟨y', ⋯⟩ ^ k = ⟨a, ⋯⟩
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
⊢ (ofClass v) (a * z₀⁻¹) = 1

Tactic:
apply hV

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
⊢ ↑(a * z₀⁻¹) ∈ 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
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
⊢ (fun x ↦ x * x₀) y * (x₀⁻¹ * y₀⁻¹) ∈ V

Tactic:
← mul_assoc,

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
⊢ (fun x ↦ x * x₀) y * x₀⁻¹ * 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
x y : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ (⇑embedding ∘ extension) (x + y) ≤ max ((⇑embedding ∘ extension) x) ((⇑embedding ∘ extension) y)

Tactic:
simp_rw [le_max_iff, Function.comp_apply]

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)
⊢ embedding (extension (x + y)) ≤ embedding (extension x) ∨ embedding (extension (x + y)) ≤ embedding (extension y)
Module: Mathlib.RingTheory.PowerSeries.Basic

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
p : R⟦X⟧
n d : ℕ
⊢ ((coeff (d, n).1) p * if (d, n).2 = n then 1 else 0) = (coeff d) p

pp: case h₀
R : Type u_1
inst✝ : Semiring R
p : R⟦X⟧
n d : ℕ
⊢ ∀ b ∈ antidiagonal (d + n), b ≠ (d, n) → (coeff b.1) p * (coeff b.2) (X ^ n) = 0

pp: case h₁
R : Type u_1
inst✝ : Semiring R
p : R⟦X⟧
n d : ℕ
⊢ (d, n) ∉ antidiagonal (d + n) → (coeff (d, n).1) p * (coeff (d, n).2) (X ^ n) = 0

Tactic:
if_pos rfl,

Goal state after:
R : Type u_1
inst✝ : Semiring R
p : R⟦X⟧
n d : ℕ
⊢ (coeff (d, n).1) p * 1 = (coeff d) p

case h₀
R : Type u_1
inst✝ : Semiring R
p : R⟦X⟧
n d : ℕ
⊢ ∀ b ∈ antidiagonal (d + n), b ≠ (d, n) → (coeff b.1) p * (coeff b.2) (X ^ n) = 0

case h₁
R : Type u_1
inst✝ : Semiring R
p : R⟦X⟧
n d : ℕ
⊢ (d, n) ∉ antidiagonal (d + n) → (coeff (d, n).1) p * (coeff (d, n).2) (X ^ n) = 0
Module: Mathlib.RingTheory.PowerSeries.Basic

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
p : R⟦X⟧
n d : ℕ
⊢ (if (n, d).1 = n then 1 else 0) * (coeff (n, d).2) p = (coeff d) p

pp: case h₀
R : Type u_1
inst✝ : Semiring R
p : R⟦X⟧
n d : ℕ
⊢ ∀ b ∈ antidiagonal (d + n), b ≠ (n, d) → (coeff b.1) (X ^ n) * (coeff b.2) p = 0

pp: case h₁
R : Type u_1
inst✝ : Semiring R
p : R⟦X⟧
n d : ℕ
⊢ (n, d) ∉ antidiagonal (d + n) → (coeff (n, d).1) (X ^ n) * (coeff (n, d).2) p = 0

Tactic:
if_pos rfl,

Goal state after:
R : Type u_1
inst✝ : Semiring R
p : R⟦X⟧
n d : ℕ
⊢ 1 * (coeff (n, d).2) p = (coeff d) p

case h₀
R : Type u_1
inst✝ : Semiring R
p : R⟦X⟧
n d : ℕ
⊢ ∀ b ∈ antidiagonal (d + n), b ≠ (n, d) → (coeff b.1) (X ^ n) * (coeff b.2) p = 0

case h₁
R : Type u_1
inst✝ : Semiring R
p : R⟦X⟧
n d : ℕ
⊢ (n, d) ∉ antidiagonal (d + n) → (coeff (n, d).1) (X ^ n) * (coeff (n, d).2) p = 0
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: case pos
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
x y : R
hx : x = 0
⊢ (if 0 = 0 then 0 else exp (-↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {0})).factors))) =
    (if 0 = 0 then 0 else exp (-↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {0})).factors))) *
      if y = 0 then 0 else exp (-↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {y})).factors))

Tactic:
if_pos rfl,

Goal state after:
case pos
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
x y : R
hx : x = 0
⊢ 0 = 0 * if y = 0 then 0 else exp (-↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {y})).factors))
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: case pos
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
x y : R
hx : ¬x = 0
hy : y = 0
⊢ (if 0 = 0 then 0 else exp (-↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {0})).factors))) =
    (if x = 0 then 0 else exp (-↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {x})).factors))) *
      if 0 = 0 then 0 else exp (-↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {0})).factors))

Tactic:
if_pos rfl,

Goal state after:
case pos
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
x y : R
hx : ¬x = 0
hy : y = 0
⊢ 0 = (if x = 0 then 0 else exp (-↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {x})).factors))) * 0
Module: Mathlib.RingTheory.MvPowerSeries.Basic

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
⊢ (monomial m) a * (monomial n) b = (monomial (m + n)) (a * b)

Tactic:
ext k
  simp only [coeff_mul_monomial, coeff_monomial]
  split_ifs with h₁ h₂ h₃ h₃ h₂ <;> try rfl
  · rw [← h₂, tsub_add_cancel_of_le h₁] at h₃
    exact (h₃ rfl).elim
  · rw [h₃, add_tsub_cancel_right] at h₂
    exact (h₂ rfl).elim
  · exact zero_mul b
  · rw [h₂] at h₁
    exact (h₁ <| le_add_left le_rfl).elim

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝¹ : Semiring R
inst✝ : Nontrivial R
s t : σ
h : (if single s 1 = single s 1 then 1 else 0) = (coeff (single s 1)) (X t)
⊢ s = t

Tactic:
if_pos rfl,

Goal state after:
σ : Type u_1
R : Type u_2
inst✝¹ : Semiring R
inst✝ : Nontrivial R
s t : σ
h : 1 = (coeff (single s 1)) (X t)
⊢ s = t