mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.PowerSeries.Basic

Goal state before:
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:
·

Goal state after:
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.MvPowerSeries.Basic

Goal state before:
pp: case refine_1
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
φ : MvPowerSeries σ R
a : R
n : σ →₀ ℕ
h : ∀ (n_1 : σ →₀ ℕ), (coeff n_1) (φ * (monomial n) a) = (coeff n_1) ((monomial n) a * φ)
m : σ →₀ ℕ
⊢ Commute ((coeff m) φ) a

pp: case refine_2
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
φ : MvPowerSeries σ R
a : R
n : σ →₀ ℕ
h : ∀ (m : σ →₀ ℕ), Commute ((coeff m) φ) a
m : σ →₀ ℕ
⊢ (coeff m) (φ * (monomial n) a) = (coeff m) ((monomial n) a * φ)

Tactic:
·

Goal state after:
case refine_1
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
φ : MvPowerSeries σ R
a : R
n : σ →₀ ℕ
h : ∀ (n_1 : σ →₀ ℕ), (coeff n_1) (φ * (monomial n) a) = (coeff n_1) ((monomial n) a * φ)
m : σ →₀ ℕ
⊢ Commute ((coeff m) φ) a
Module: Mathlib.RingTheory.MvPowerSeries.Basic

Goal state before:
pp: case refine_2
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
φ : MvPowerSeries σ R
a : R
n : σ →₀ ℕ
h : ∀ (m : σ →₀ ℕ), Commute ((coeff m) φ) a
m : σ →₀ ℕ
⊢ (coeff m) (φ * (monomial n) a) = (coeff m) ((monomial n) a * φ)

Tactic:
·

Goal state after:
case refine_2
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
φ : MvPowerSeries σ R
a : R
n : σ →₀ ℕ
h : ∀ (m : σ →₀ ℕ), Commute ((coeff m) φ) a
m : σ →₀ ℕ
⊢ (coeff m) (φ * (monomial n) a) = (coeff m) ((monomial n) a * φ)
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
a b : (ofClass v).ValueGroup₀
x : K := ⋯.choose
hx_def : x = ⋯.choose
y : K := ⋯.choose
hy_def : y = ⋯.choose
xy : K := ⋯.choose
hxy_def : xy = ⋯.choose
hxy : v xy = embedding a * embedding b
hx : v x = embedding a
hy : v y = embedding b
hx0 : ¬x = 0
hy0 : ¬y = 0
⊢ ↑⟨Units.mk0 (v xy) ⋯, ⋯⟩ = ↑⟨Units.mk0 (v x) ⋯, ⋯⟩ * ↑⟨Units.mk0 (v y) ⋯, ⋯⟩

pp: case neg.hnc
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
a b : (ofClass v).ValueGroup₀
x : K := ⋯.choose
hx_def : x = ⋯.choose
y : K := ⋯.choose
hy_def : y = ⋯.choose
xy : K := ⋯.choose
hxy_def : xy = ⋯.choose
hxy : v xy = embedding a * embedding b
hx : v x = embedding a
hy : v y = embedding b
hx0 : ¬x = 0
hy0 : ¬y = 0
⊢ ¬extensionValuation ↑x = 0

pp: case neg.hnc
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
a b : (ofClass v).ValueGroup₀
x : K := ⋯.choose
hx_def : x = ⋯.choose
y : K := ⋯.choose
hy_def : y = ⋯.choose
xy : K := ⋯.choose
hxy_def : xy = ⋯.choose
hxy : v xy = embedding a * embedding b
hx : v x = embedding a
hy : v y = embedding b
hx0 : ¬x = 0
hy0 : ¬y = 0
⊢ ¬extensionValuation ↑y = 0

pp: case neg.hnc
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
a b : (ofClass v).ValueGroup₀
x : K := ⋯.choose
hx_def : x = ⋯.choose
y : K := ⋯.choose
hy_def : y = ⋯.choose
xy : K := ⋯.choose
hxy_def : xy = ⋯.choose
hxy : v xy = embedding a * embedding b
hx : v x = embedding a
hy : v y = embedding b
hx0 : ¬x = 0
hy0 : ¬y = 0
⊢ ¬extensionValuation ↑xy = 0

Tactic:
·

Goal state after:
case neg
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
a b : (ofClass v).ValueGroup₀
x : K := ⋯.choose
hx_def : x = ⋯.choose
y : K := ⋯.choose
hy_def : y = ⋯.choose
xy : K := ⋯.choose
hxy_def : xy = ⋯.choose
hxy : v xy = embedding a * embedding b
hx : v x = embedding a
hy : v y = embedding b
hx0 : ¬x = 0
hy0 : ¬y = 0
⊢ ↑⟨Units.mk0 (v xy) ⋯, ⋯⟩ = ↑⟨Units.mk0 (v x) ⋯, ⋯⟩ * ↑⟨Units.mk0 (v y) ⋯, ⋯⟩
Module: Mathlib.RingTheory.PowerSeries.Basic

Goal state before:
pp: case pos
R : Type u_1
inst✝ : Semiring R
p : R⟦X⟧
n d : ℕ
h : n ≤ d
⊢ (coeff d) (p * X ^ n) = (coeff (d - n)) p

pp: case neg
R : Type u_1
inst✝ : Semiring R
p : R⟦X⟧
n d : ℕ
h : ¬n ≤ d
⊢ (coeff d) (p * X ^ n) = 0

Tactic:
·

Goal state after:
case pos
R : Type u_1
inst✝ : Semiring R
p : R⟦X⟧
n d : ℕ
h : n ≤ d
⊢ (coeff d) (p * X ^ n) = (coeff (d - n)) p
Module: Mathlib.RingTheory.PowerSeries.Basic

Goal state before:
pp: case neg
R : Type u_1
inst✝ : Semiring R
p : R⟦X⟧
n d : ℕ
h : ¬n ≤ d
⊢ (coeff d) (p * X ^ n) = 0

Tactic:
·

Goal state after:
case neg
R : Type u_1
inst✝ : Semiring R
p : R⟦X⟧
n d : ℕ
h : ¬n ≤ d
⊢ (coeff d) (p * X ^ n) = 0
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
a b : (ofClass v).ValueGroup₀
x : K := ⋯.choose
hx_def : x = ⋯.choose
y : K := ⋯.choose
hy_def : y = ⋯.choose
xy : K := ⋯.choose
hxy_def : xy = ⋯.choose
hxy : v xy = embedding a * embedding b
hx : v x = embedding a
hy : v y = embedding b
hx0 : ¬x = 0
hy0 : ¬y = 0
⊢ Units.mk0 (v xy) ⋯ = Units.mk0 (v x * v y) ?neg.hxy✝

pp: case neg.hxy
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
a b : (ofClass v).ValueGroup₀
x : K := ⋯.choose
hx_def : x = ⋯.choose
y : K := ⋯.choose
hy_def : y = ⋯.choose
xy : K := ⋯.choose
hxy_def : xy = ⋯.choose
hxy : v xy = embedding a * embedding b
hx : v x = embedding a
hy : v y = embedding b
hx0 : ¬x = 0
hy0 : ¬y = 0
⊢ v x * v y ≠ 0

Tactic:
·

Goal state after:
case neg
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
a b : (ofClass v).ValueGroup₀
x : K := ⋯.choose
hx_def : x = ⋯.choose
y : K := ⋯.choose
hy_def : y = ⋯.choose
xy : K := ⋯.choose
hxy_def : xy = ⋯.choose
hxy : v xy = embedding a * embedding b
hx : v x = embedding a
hy : v y = embedding b
hx0 : ¬x = 0
hy0 : ¬y = 0
⊢ Units.mk0 (v xy) ⋯ = Units.mk0 (v x * v y) ?neg.hxy✝
Module: Mathlib.RingTheory.PowerSeries.Basic

Goal state before:
pp: case pos
R : Type u_1
inst✝ : Semiring R
p : R⟦X⟧
n d : ℕ
h : n ≤ d
⊢ (coeff d) (X ^ n * p) = (coeff (d - n)) p

pp: case neg
R : Type u_1
inst✝ : Semiring R
p : R⟦X⟧
n d : ℕ
h : ¬n ≤ d
⊢ (coeff d) (X ^ n * p) = 0

Tactic:
·

Goal state after:
case pos
R : Type u_1
inst✝ : Semiring R
p : R⟦X⟧
n d : ℕ
h : n ≤ d
⊢ (coeff d) (X ^ n * p) = (coeff (d - n)) p
Module: Mathlib.RingTheory.PowerSeries.Basic

Goal state before:
pp: case neg
R : Type u_1
inst✝ : Semiring R
p : R⟦X⟧
n d : ℕ
h : ¬n ≤ d
⊢ (coeff d) (X ^ n * p) = 0

Tactic:
·

Goal state after:
case neg
R : Type u_1
inst✝ : Semiring R
p : R⟦X⟧
n d : ℕ
h : ¬n ≤ d
⊢ (coeff d) (X ^ n * p) = 0
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: case neg.hxy
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
a b : (ofClass v).ValueGroup₀
x : K := ⋯.choose
hx_def : x = ⋯.choose
y : K := ⋯.choose
hy_def : y = ⋯.choose
xy : K := ⋯.choose
hxy_def : xy = ⋯.choose
hxy : v xy = embedding a * embedding b
hx : v x = embedding a
hy : v y = embedding b
hx0 : ¬x = 0
hy0 : ¬y = 0
⊢ v x * v y ≠ 0

Tactic:
·

Goal state after:
case neg.hxy
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
a b : (ofClass v).ValueGroup₀
x : K := ⋯.choose
hx_def : x = ⋯.choose
y : K := ⋯.choose
hy_def : y = ⋯.choose
xy : K := ⋯.choose
hxy_def : xy = ⋯.choose
hxy : v xy = embedding a * embedding b
hx : v x = embedding a
hy : v y = embedding b
hx0 : ¬x = 0
hy0 : ¬y = 0
⊢ v x * v y ≠ 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 x * y = 0 then 0 else exp (-↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {x * y})).factors))) =
    (if x = 0 then 0 else exp (-↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {x})).factors))) *
      if y = 0 then 0 else exp (-↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {y})).factors))

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

Tactic:
·

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
⊢ (if x * y = 0 then 0 else exp (-↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {x * y})).factors))) =
    (if x = 0 then 0 else exp (-↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {x})).factors))) *
      if y = 0 then 0 else exp (-↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {y})).factors))
Module: Mathlib.RingTheory.PowerSeries.Basic

Goal state before:
pp: case zero
R : Type u_1
inst✝ : Semiring R
φ : R⟦X⟧
⊢ (coeff 0) φ = (coeff 0) ((mk fun p ↦ (coeff (p + 1)) φ) * X + C (constantCoeff φ))

pp: case succ
R : Type u_1
inst✝ : Semiring R
φ : R⟦X⟧
n : ℕ
⊢ (coeff (n + 1)) φ = (coeff (n + 1)) ((mk fun p ↦ (coeff (p + 1)) φ) * X + C (constantCoeff φ))

Tactic:
·

Goal state after:
case zero
R : Type u_1
inst✝ : Semiring R
φ : R⟦X⟧
⊢ (coeff 0) φ = (coeff 0) ((mk fun p ↦ (coeff (p + 1)) φ) * X + C (constantCoeff φ))
Module: Mathlib.RingTheory.PowerSeries.Basic

Goal state before:
pp: case succ
R : Type u_1
inst✝ : Semiring R
φ : R⟦X⟧
n : ℕ
⊢ (coeff (n + 1)) φ = (coeff (n + 1)) ((mk fun p ↦ (coeff (p + 1)) φ) * X + C (constantCoeff φ))

Tactic:
·

Goal state after:
case succ
R : Type u_1
inst✝ : Semiring R
φ : R⟦X⟧
n : ℕ
⊢ (coeff (n + 1)) φ = (coeff (n + 1)) ((mk fun p ↦ (coeff (p + 1)) φ) * X + C (constantCoeff φ))
Module: Mathlib.RingTheory.PowerSeries.Basic

Goal state before:
pp: case zero
R : Type u_1
inst✝ : Semiring R
φ : R⟦X⟧
⊢ (coeff 0) φ = (coeff 0) ((X * mk fun p ↦ (coeff (p + 1)) φ) + C (constantCoeff φ))

pp: case succ
R : Type u_1
inst✝ : Semiring R
φ : R⟦X⟧
n : ℕ
⊢ (coeff (n + 1)) φ = (coeff (n + 1)) ((X * mk fun p ↦ (coeff (p + 1)) φ) + C (constantCoeff φ))

Tactic:
·

Goal state after:
case zero
R : Type u_1
inst✝ : Semiring R
φ : R⟦X⟧
⊢ (coeff 0) φ = (coeff 0) ((X * mk fun p ↦ (coeff (p + 1)) φ) + C (constantCoeff φ))
Module: Mathlib.RingTheory.PowerSeries.Basic

Goal state before:
pp: case succ
R : Type u_1
inst✝ : Semiring R
φ : R⟦X⟧
n : ℕ
⊢ (coeff (n + 1)) φ = (coeff (n + 1)) ((X * mk fun p ↦ (coeff (p + 1)) φ) + C (constantCoeff φ))

Tactic:
·

Goal state after:
case succ
R : Type u_1
inst✝ : Semiring R
φ : R⟦X⟧
n : ℕ
⊢ (coeff (n + 1)) φ = (coeff (n + 1)) ((X * mk fun p ↦ (coeff (p + 1)) φ) + C (constantCoeff φ))
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

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

Tactic:
·

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

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

Tactic:
·

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
⊢ (if x * y = 0 then 0 else exp (-↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {x * y})).factors))) =
    (if x = 0 then 0 else exp (-↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {x})).factors))) *
      if y = 0 then 0 else exp (-↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {y})).factors))
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: case neg.hnc
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
a b : (ofClass v).ValueGroup₀
x : K := ⋯.choose
hx_def : x = ⋯.choose
y : K := ⋯.choose
hy_def : y = ⋯.choose
xy : K := ⋯.choose
hxy_def : xy = ⋯.choose
hxy : v xy = embedding a * embedding b
hx : v x = embedding a
hy : v y = embedding b
hx0 : ¬x = 0
hy0 : ¬y = 0
⊢ ¬extensionValuation ↑x = 0

pp: case neg.hnc
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
a b : (ofClass v).ValueGroup₀
x : K := ⋯.choose
hx_def : x = ⋯.choose
y : K := ⋯.choose
hy_def : y = ⋯.choose
xy : K := ⋯.choose
hxy_def : xy = ⋯.choose
hxy : v xy = embedding a * embedding b
hx : v x = embedding a
hy : v y = embedding b
hx0 : ¬x = 0
hy0 : ¬y = 0
⊢ ¬extensionValuation ↑y = 0

pp: case neg.hnc
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
a b : (ofClass v).ValueGroup₀
x : K := ⋯.choose
hx_def : x = ⋯.choose
y : K := ⋯.choose
hy_def : y = ⋯.choose
xy : K := ⋯.choose
hxy_def : xy = ⋯.choose
hxy : v xy = embedding a * embedding b
hx : v x = embedding a
hy : v y = embedding b
hx0 : ¬x = 0
hy0 : ¬y = 0
⊢ ¬extensionValuation ↑xy = 0

Tactic:
·

Goal state after:
case neg.hnc
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
a b : (ofClass v).ValueGroup₀
x : K := ⋯.choose
hx_def : x = ⋯.choose
y : K := ⋯.choose
hy_def : y = ⋯.choose
xy : K := ⋯.choose
hxy_def : xy = ⋯.choose
hxy : v xy = embedding a * embedding b
hx : v x = embedding a
hy : v y = embedding b
hx0 : ¬x = 0
hy0 : ¬y = 0
⊢ ¬extensionValuation ↑x = 0
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

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

Tactic:
·

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

Goal state before:
pp: case neg
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
h₁ : n ≤ k
h₂ : k - n = m
h₃ : ¬k = m + n
⊢ a * b = 0

pp: case pos
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
h₁ : n ≤ k
h₂ : ¬k - n = m
h₃ : k = m + n
⊢ 0 * b = a * b

pp: case neg
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
h₁ : n ≤ k
h₂ : ¬k - n = m
h₃ : ¬k = m + n
⊢ 0 * b = 0

pp: case pos
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
h₁ : ¬n ≤ k
h₂ : k = m + n
⊢ 0 = a * b

Tactic:
·

Goal state after:
case neg
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
h₁ : n ≤ k
h₂ : k - n = m
h₃ : ¬k = m + n
⊢ a * b = 0
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: case neg.hnc
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
a b : (ofClass v).ValueGroup₀
x : K := ⋯.choose
hx_def : x = ⋯.choose
y : K := ⋯.choose
hy_def : y = ⋯.choose
xy : K := ⋯.choose
hxy_def : xy = ⋯.choose
hxy : v xy = embedding a * embedding b
hx : v x = embedding a
hy : v y = embedding b
hx0 : ¬x = 0
hy0 : ¬y = 0
⊢ ¬extensionValuation ↑y = 0

pp: case neg.hnc
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
a b : (ofClass v).ValueGroup₀
x : K := ⋯.choose
hx_def : x = ⋯.choose
y : K := ⋯.choose
hy_def : y = ⋯.choose
xy : K := ⋯.choose
hxy_def : xy = ⋯.choose
hxy : v xy = embedding a * embedding b
hx : v x = embedding a
hy : v y = embedding b
hx0 : ¬x = 0
hy0 : ¬y = 0
⊢ ¬extensionValuation ↑xy = 0

Tactic:
·

Goal state after:
case neg.hnc
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
a b : (ofClass v).ValueGroup₀
x : K := ⋯.choose
hx_def : x = ⋯.choose
y : K := ⋯.choose
hy_def : y = ⋯.choose
xy : K := ⋯.choose
hxy_def : xy = ⋯.choose
hxy : v xy = embedding a * embedding b
hx : v x = embedding a
hy : v y = embedding b
hx0 : ¬x = 0
hy0 : ¬y = 0
⊢ ¬extensionValuation ↑y = 0
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: case neg.hnc
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
a b : (ofClass v).ValueGroup₀
x : K := ⋯.choose
hx_def : x = ⋯.choose
y : K := ⋯.choose
hy_def : y = ⋯.choose
xy : K := ⋯.choose
hxy_def : xy = ⋯.choose
hxy : v xy = embedding a * embedding b
hx : v x = embedding a
hy : v y = embedding b
hx0 : ¬x = 0
hy0 : ¬y = 0
⊢ ¬extensionValuation ↑xy = 0

Tactic:
·

Goal state after:
case neg.hnc
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
a b : (ofClass v).ValueGroup₀
x : K := ⋯.choose
hx_def : x = ⋯.choose
y : K := ⋯.choose
hy_def : y = ⋯.choose
xy : K := ⋯.choose
hxy_def : xy = ⋯.choose
hxy : v xy = embedding a * embedding b
hx : v x = embedding a
hy : v y = embedding b
hx0 : ¬x = 0
hy0 : ¬y = 0
⊢ ¬extensionValuation ↑xy = 0
Module: Mathlib.RingTheory.MvPowerSeries.Basic

Goal state before:
pp: case pos
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
h₁ : n ≤ k
h₂ : ¬k - n = m
h₃ : k = m + n
⊢ 0 * b = a * b

pp: case neg
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
h₁ : n ≤ k
h₂ : ¬k - n = m
h₃ : ¬k = m + n
⊢ 0 * b = 0

pp: case pos
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
h₁ : ¬n ≤ k
h₂ : k = m + n
⊢ 0 = a * b

Tactic:
·

Goal state after:
case pos
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
h₁ : n ≤ k
h₂ : ¬k - n = m
h₃ : k = m + n
⊢ 0 * b = a * b
Module: Mathlib.RingTheory.MvPowerSeries.Basic

Goal state before:
pp: case neg
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
h₁ : n ≤ k
h₂ : ¬k - n = m
h₃ : ¬k = m + n
⊢ 0 * b = 0

pp: case pos
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
h₁ : ¬n ≤ k
h₂ : k = m + n
⊢ 0 = a * b

Tactic:
·

Goal state after:
case neg
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
h₁ : n ≤ k
h₂ : ¬k - n = m
h₃ : ¬k = m + n
⊢ 0 * b = 0
Module: Mathlib.RingTheory.MvPowerSeries.Basic

Goal state before:
pp: case pos
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
h₁ : ¬n ≤ k
h₂ : k = m + n
⊢ 0 = a * b

Tactic:
·

Goal state after:
case pos
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
h₁ : ¬n ≤ k
h₂ : k = m + n
⊢ 0 = a * b
Module: Mathlib.RingTheory.PowerSeries.Basic

Goal state before:
pp: case e'_2.mp
R : Type u_1
inst✝ : Semiring R
n : ℕ
φ : R⟦X⟧
h : ∀ m < n, (coeff m) φ = 0
m : Unit →₀ ℕ
hm : m () < n
⊢ (MvPowerSeries.coeff m) φ = 0

pp: case e'_2.mpr
R : Type u_1
inst✝ : Semiring R
n : ℕ
φ : R⟦X⟧
h : ∀ (m : Unit →₀ ℕ), m () < n → (MvPowerSeries.coeff m) φ = 0
m : ℕ
hm : m < n
⊢ (coeff m) φ = 0

Tactic:
·

Goal state after:
case e'_2.mp
R : Type u_1
inst✝ : Semiring R
n : ℕ
φ : R⟦X⟧
h : ∀ m < n, (coeff m) φ = 0
m : Unit →₀ ℕ
hm : m () < n
⊢ (MvPowerSeries.coeff m) φ = 0
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
⊢ Function.Injective ⇑valueGroup₀_hom_extensionValuation

pp: case refine_2
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
⊢ Function.Surjective ⇑valueGroup₀_hom_extensionValuation

Tactic:
·

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
⊢ Function.Injective ⇑valueGroup₀_hom_extensionValuation
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
⊢ v.intValuationDef (x + y) ≤ max (v.intValuationDef x) (v.intValuationDef y)

pp: case neg
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
x y : R
hx : ¬x = 0
⊢ v.intValuationDef (x + y) ≤ max (v.intValuationDef x) (v.intValuationDef y)

Tactic:
·

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
⊢ v.intValuationDef (x + y) ≤ max (v.intValuationDef x) (v.intValuationDef y)
Module: Mathlib.RingTheory.PowerSeries.Basic

Goal state before:
pp: case e'_2.mpr
R : Type u_1
inst✝ : Semiring R
n : ℕ
φ : R⟦X⟧
h : ∀ (m : Unit →₀ ℕ), m () < n → (MvPowerSeries.coeff m) φ = 0
m : ℕ
hm : m < n
⊢ (coeff m) φ = 0

Tactic:
·

Goal state after:
case e'_2.mpr
R : Type u_1
inst✝ : Semiring R
n : ℕ
φ : R⟦X⟧
h : ∀ (m : Unit →₀ ℕ), m () < n → (MvPowerSeries.coeff m) φ = 0
m : ℕ
hm : m < n
⊢ (coeff m) φ = 0
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: case neg
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
x y : R
hx : ¬x = 0
⊢ v.intValuationDef (x + y) ≤ max (v.intValuationDef x) (v.intValuationDef y)

Tactic:
·

Goal state after:
case neg
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
x y : R
hx : ¬x = 0
⊢ v.intValuationDef (x + y) ≤ max (v.intValuationDef x) (v.intValuationDef y)
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
⊢ v.intValuationDef (x + y) ≤ max (v.intValuationDef x) (v.intValuationDef y)

pp: case neg
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
x y : R
hx : ¬x = 0
hy : ¬y = 0
⊢ v.intValuationDef (x + y) ≤ max (v.intValuationDef x) (v.intValuationDef y)

Tactic:
·

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
⊢ v.intValuationDef (x + y) ≤ max (v.intValuationDef x) (v.intValuationDef y)