mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
x : K
hv : (valuation K v) x ≤ 1
γ : (WithZero (Multiplicative ℤ))ˣ
⊢ ∃ a, (valuation K v) ((algebraMap R K) a - x) < ↑γ

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
w : σ → ℕ
R : Type u_4
ι : Type u_5
inst✝ : CommRing R
d : σ →₀ ℕ
s : Finset ι
f : MvPowerSeries σ R
g : ι → MvPowerSeries σ R
h : ∀ i ∈ s, ↑((weight w) d) < weightedOrder w (g i)
⊢ (coeff d) (f * ∏ i ∈ s, (1 - g i)) = (coeff d) f

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
w : σ → ℕ
R : Type u_3
inst✝ : Ring R
f : MvPowerSeries σ R
h : weightedOrder w (-f) ≠ weightedOrder w f
⊢ f = 0

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
w : σ → ℕ
R : Type u_3
inst✝ : Ring R
f : MvPowerSeries σ R
⊢ weightedOrder w (-f) = weightedOrder w f

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
⊢ (ofClass (valuation K v)).valueGroup = (ofClass Valued.v).valueGroup

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
w : σ → ℕ
R : Type u_3
inst✝ : Ring R
p : MvPowerSeries σ R
T : Subring R
hp : ∀ (n : σ →₀ ℕ), (coeff n) p ∈ T
d : σ →₀ ℕ
hd : ↑((weight w) d) < weightedOrder w (p.toSubring T hp)
⊢ (coeff d) p = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: R : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : IsDedekindDomain R
K : Type u_2
S : Type u_3
inst✝³ : Field K
inst✝² : CommSemiring S
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
⊢ ↑(ofClass (valuation K v)).valueGroup = ↑(ofClass Valued.v).valueGroup

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
w : σ → ℕ
R : Type u_3
inst✝ : Ring R
p : MvPowerSeries σ R
T : Subring R
hp : ∀ (n : σ →₀ ℕ), (coeff n) p ∈ T
d : σ →₀ ℕ
hd : ↑((weight w) d) < weightedOrder w p
⊢ (coeff d) (p.toSubring T hp) = 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Instances.ENat

Goal state before:
pp: n : ℕ
⊢ 𝓝 ↑n = pure ↑n

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Instances.ENat

Goal state before:
pp: n : ℕ∞
h : n ≠ ⊤
⊢ 𝓝 n = pure n

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
w : σ → ℕ
R : Type u_3
inst✝ : Ring R
p : MvPowerSeries σ R
T : Subring R
hp : ∀ (n : σ →₀ ℕ), (coeff n) p ∈ T
⊢ weightedOrder w (p.toSubring T hp) = weightedOrder w p

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Instances.ENat

Goal state before:
pp: x : ℕ∞
hx : x ≠ ⊤
⊢ IsOpen {x}

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Instances.ENat

Goal state before:
pp: x : ℕ∞
s : Set ℕ∞
hx : x ≠ ⊤
⊢ s ∈ 𝓝 x ↔ x ∈ s

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
n : ℕ
inst✝¹ : Finite σ
inst✝ : CommSemiring R
p : MvPowerSeries σ R
x : σ →₀ ℕ
⊢ MvPolynomial.coeff x ((truncTotal n) p) = if degree x < n then (coeff x) p else 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: R : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : IsDedekindDomain R
K : Type u_2
S : Type u_3
inst✝³ : Field K
inst✝² : CommSemiring S
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
x : (ofClass (valuation K v)).ValueGroup₀
⊢ (map' ↑(valueGroupEquiv K v).symm) ((map' ↑(valueGroupEquiv K v)) x) = x

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
n : ℕ
inst✝¹ : Finite σ
inst✝ : CommSemiring R
p : MvPowerSeries σ R
⊢ MvPolynomial.constantCoeff ((truncTotal n) p) = if 0 < n then constantCoeff p else 0

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
f : MvPowerSeries σ R
⊢ f ≠ 0 ↔ ∃ n d, (coeff d) f ≠ 0 ∧ degree d = n

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
n : ℕ
inst✝¹ : Finite σ
inst✝ : CommSemiring R
p : MvPowerSeries σ R
⊢ ↑((truncTotal n) p) = ∑ i ∈ range n, (homogeneousComponent i) p

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Instances.ENat

Goal state before:
pp: α : Type u_1
l : Filter α
f : α → ℕ∞
⊢ Tendsto f l (𝓝 ⊤) ↔ ∀ (n : ℕ), ∀ᶠ (a : α) in l, ↑n < f a

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Instances.ENat

Goal state before:
pp: n a : ℕ
ha : n + 1 ≤ a
⊢ ↑n < ↑a

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Instances.ENat

Goal state before:
pp: ⊢ Tendsto Nat.cast atTop (𝓝 ⊤)

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
n : ℕ
inst✝¹ : Finite σ
inst✝ : CommSemiring R
h : n ≠ 0
⊢ 0 ∈ ⋯.toFinset

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
f : MvPowerSeries σ R
h : ↑f.order.toNat = f.order
⊢ ∃ d, (coeff d) f ≠ 0 ∧ ↑(degree d) = f.order

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Instances.ENat

Goal state before:
pp: ⊢ ContinuousAdd ℕ∞

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
n : ℕ
inst✝¹ : Finite σ
inst✝ : CommSemiring R
p q : MvPowerSeries σ R
x : σ →₀ ℕ
hx : degree x < n
x✝¹ x✝ : σ →₀ ℕ
h : x✝ ≤ x✝¹
⊢ x✝¹ ∈ ↑⋯.toFinset → x✝ ∈ ↑⋯.toFinset

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
n : ℕ
inst✝¹ : Finite σ
inst✝ : CommSemiring R
p q : MvPowerSeries σ R
x : σ →₀ ℕ
hx : degree x < n
⊢ x ∈ ⋯.toFinset

Tactic:
by

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

Goal state before:
pp: case zero
σ : Type u_1
R : Type u_2
n : ℕ
inst✝¹ : Finite σ
inst✝ : CommSemiring R
p : MvPowerSeries σ R
x : σ →₀ ℕ
h : degree x < n
⊢ MvPolynomial.coeff x ((truncTotal n) p ^ 0) = (coeff x) (p ^ 0)

Tactic:
grind [coeff_one, MvPolynomial.coeff_one]

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
f : MvPowerSeries σ R
d : σ →₀ ℕ
h : (coeff d) f ≠ 0
⊢ f.order ≤ ↑(degree d)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: R : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : IsDedekindDomain R
K : Type u_2
S : Type u_3
inst✝³ : Field K
inst✝² : CommSemiring S
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
y : (ofClass Valued.v).ValueGroup₀
⊢ (map' ↑(valueGroupEquiv K v)) ((map' ↑(valueGroupEquiv K v).symm) y) = y

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Instances.ENat

Goal state before:
pp: a : ℕ∞
⊢ ContinuousAt (Function.uncurry fun x1 x2 ↦ x1 * x2) (a, ⊤)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Instances.ENat

Goal state before:
pp: key : ∀ (a : ℕ∞), ContinuousAt (Function.uncurry fun x1 x2 ↦ x1 * x2) (a, ⊤)
a b : ℕ
⊢ ContinuousAt (fun p ↦ p.1 * p.2) (↑a, ↑b)

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
f : MvPowerSeries σ R
d : σ →₀ ℕ
h : ↑(degree d) < f.order
⊢ (coeff d) f = 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Instances.ENat

Goal state before:
pp: a✝ b : ℕ∞
h : a✝ ≠ ⊤ ∨ b ≠ ⊤
a : ℕ
x✝ : ↑a ≠ ⊤ ∨ ⊤ ≠ ⊤
this : ∀ᶠ (b : ℕ∞) in 𝓝 ⊤, ↑a - b = 0
⊢ ContinuousAt (Function.uncurry fun x1 x2 ↦ x1 - x2) (↑a, ⊤)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: R : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : IsDedekindDomain R
K : Type u_2
S : Type u_3
inst✝³ : Field K
inst✝² : CommSemiring S
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
⊢ ∀ (x y : (ofClass (valuation K v)).ValueGroup₀),
    (map' ↑(valueGroupEquiv K v)) (x * y) = (map' ↑(valueGroupEquiv K v)) x * (map' ↑(valueGroupEquiv K v)) y

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
f : MvPowerSeries σ R
n : ℕ
h : ∀ (d : σ →₀ ℕ), degree d < n → (coeff d) f = 0
⊢ ↑n ≤ f.order

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Instances.ENat

Goal state before:
pp: a b✝ : ℕ∞
h : a ≠ ⊤ ∨ b✝ ≠ ⊤
b : ℕ
x✝ : ⊤ ≠ ⊤ ∨ ↑b ≠ ⊤
this : ∀ (n : ℕ), ∀ᶠ (a : ℕ∞) in 𝓝 ⊤, ↑b + ↑n < a
⊢ ContinuousAt (Function.uncurry fun x1 x2 ↦ x1 - x2) (⊤, ↑b)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Instances.ENat

Goal state before:
pp: a b : ℕ∞
h : a ≠ ⊤ ∨ b ≠ ⊤
⊢ ContinuousAt (Function.uncurry fun x1 x2 ↦ x1 - x2) (a, b)

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
f : MvPowerSeries σ R
n : ℕ∞
h : ∀ (d : σ →₀ ℕ), ↑(degree d) < n → (coeff d) f = 0
⊢ n ≤ f.order

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
f : MvPowerSeries σ R
n : ℕ
⊢ f.order = ↑n ↔ (∃ d, (coeff d) f ≠ 0 ∧ degree d = n) ∧ ∀ (d : σ →₀ ℕ), degree d < n → (coeff d) f = 0

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝¹ : Semiring R
d : σ →₀ ℕ
a : R
inst✝ : Decidable (a = 0)
⊢ ((monomial d) a).order = if a = 0 then ⊤ else ↑(degree d)

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
d : σ →₀ ℕ
a : R
h : a ≠ 0
⊢ ((monomial d) a).order = ↑(degree d)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: R : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : IsDedekindDomain R
K : Type u_2
S : Type u_3
inst✝³ : Field K
inst✝² : CommSemiring S
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
a b : (ofClass (valuation K v)).ValueGroup₀
⊢ (map' ↑(valueGroupEquiv K v)) a ≤ (map' ↑(valueGroupEquiv K v)) b ↔ a ≤ b

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
n : ℕ
inst✝¹ : Finite σ
inst✝ : CommSemiring R
p : MvPowerSeries σ R
k : ℕ
ih : ∀ m ≤ k, ∀ {x : σ →₀ ℕ}, degree x < n → MvPolynomial.coeff x ((truncTotal n) p ^ m) = (coeff x) (p ^ m)
x : σ →₀ ℕ
h : degree x < n
x✝ : (σ →₀ ℕ) × (σ →₀ ℕ)
a✝ : x✝ ∈ antidiagonal x
⊢ degree x✝.1 < n

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
a : ↥(ofClass (valuation K v)).valueGroup
⊢ (valueGroupOrderIso K v) ↑a = ↑((valueGroupEquiv K v) a)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
g : (ofClass (valuation K v)).ValueGroup₀
⊢ embedding ((valueGroupOrderIso K v) g) = embedding g

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
f : MvPowerSeries σ R
⊢ 1 ≤ f.order ↔ constantCoeff f = 0

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
n : ℕ
inst✝¹ : Finite σ
inst✝ : CommSemiring R
p : MvPowerSeries σ R
k : ℕ
ih : ∀ m ≤ k, ∀ {x : σ →₀ ℕ}, degree x < n → MvPolynomial.coeff x ((truncTotal n) p ^ m) = (coeff x) (p ^ m)
x : σ →₀ ℕ
h : degree x < n
x✝ : (σ →₀ ℕ) × (σ →₀ ℕ)
a✝ : x✝ ∈ antidiagonal x
⊢ degree x✝.2 < n

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
f : MvPowerSeries σ R
⊢ f.order ≠ 0 ↔ constantCoeff f = 0

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
f : MvPowerSeries σ R
n : ℕ
hf : constantCoeff f = 0
⊢ ↑n ≤ (f ^ n).order

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
n m : ℕ
inst✝¹ : Finite σ
inst✝ : CommSemiring R
p : MvPowerSeries σ R
x : σ →₀ ℕ
h : degree x < n
⊢ MvPolynomial.coeff x ((truncTotal n) p ^ m) = (coeff x) (p ^ m)

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝ : Ring R
f g : MvPowerSeries σ R
d : σ →₀ ℕ
h : ↑(degree d) < g.order
⊢ (coeff d) (f * (1 - g)) = (coeff d) f

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
n m : ℕ
inst✝¹ : Finite σ
inst✝ : CommSemiring R
p : MvPowerSeries σ R
⊢ (truncTotal n) (p ^ m) = (truncTotal n) (↑((truncTotal n) p) ^ m)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
x : adicCompletion K v
⊢ (valueGroupOrderIso K v) ((valuation K v).restrict x) = Valued.v.restrict x.toCompletion

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
n : ℕ
inst✝¹ : Finite σ
inst✝ : CommSemiring R
p : MvPowerSeries σ R
h : n ≠ 0
⊢ ((truncTotal n) p).totalDegree < n

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝ : Ring R
f g : MvPowerSeries σ R
d : σ →₀ ℕ
h : ↑(degree d) < g.order
⊢ (coeff d) ((1 - g) * f) = (coeff d) f

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
n : ℕ
inst✝¹ : Finite σ
inst✝ : CommSemiring R
p : MvPolynomial σ R
h : n ≠ 0
⊢ (truncTotal n) ↑p = p ↔ p.totalDegree < n

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
R : Type u_4
ι : Type u_5
inst✝ : CommRing R
d : σ →₀ ℕ
s : Finset ι
f : MvPowerSeries σ R
g : ι → MvPowerSeries σ R
⊢ (∀ i ∈ s, ↑(degree d) < (g i).order) → (coeff d) (f * ∏ i ∈ s, (1 - g i)) = (coeff d) f

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝ : Ring R
p : MvPowerSeries σ R
T : Subring R
hp : ∀ (n : σ →₀ ℕ), (coeff n) p ∈ T
d : σ →₀ ℕ
hd : ↑(degree d) < (p.toSubring T hp).order
⊢ (coeff d) p = 0

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝ : Ring R
p : MvPowerSeries σ R
T : Subring R
hp : ∀ (n : σ →₀ ℕ), (coeff n) p ∈ T
d : σ →₀ ℕ
hd : ↑(degree d) < p.order
⊢ (coeff d) (p.toSubring T hp) = 0

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝ : Ring R
p : MvPowerSeries σ R
T : Subring R
hp : ∀ (n : σ →₀ ℕ), (coeff n) p ∈ T
⊢ (p.toSubring T hp).order = p.order

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f : MvPowerSeries σ R
p : ℕ
hf : IsWeightedHomogeneous w f p
d : σ →₀ ℕ
hd : (weight w) d ≠ p
⊢ (coeff d) f = 0

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f g : MvPowerSeries σ R
p : ℕ
hf : IsWeightedHomogeneous w f p
hg : IsWeightedHomogeneous w g p
d : σ →₀ ℕ
⊢ (coeff d) (f + g) ≠ 0 → (weight w) d = p

Tactic:
by

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

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁵ : TopologicalSpace R
inst✝⁴ : MonoidWithZero R
inst✝³ : MonoidWithZero S
inst✝² : TopologicalSpace S
F : Type u_3
inst✝¹ : FunLike F R S
inst✝ : MonoidWithZeroHomClass F R S
φ : F
hφ : Continuous[inst✝⁵, inst✝²] ⇑φ
a : R
ha : IsTopologicallyNilpotent a
⊢ IsTopologicallyNilpotent (φ a)

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝¹ : TopologicalSpace R
inst✝ : MonoidWithZero R
x✝ : ℕ
hi : x✝ ≥ 1
⊢ 0 ^ x✝ = 0

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝¹ : TopologicalSpace R
inst✝ : MonoidWithZero R
a : R
ha : IsNilpotent a
⊢ IsTopologicallyNilpotent a

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝² : TopologicalSpace R
inst✝¹ : Ring R
inst✝ : IsLinearTopology Rᵐᵒᵖ R
a b : R
ha : IsTopologicallyNilpotent a
hab : Commute a b
⊢ IsTopologicallyNilpotent (a * b)

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝² : TopologicalSpace R
inst✝¹ : Ring R
inst✝ : IsLinearTopology R R
a b : R
hb : IsTopologicallyNilpotent b
hab : Commute a b
⊢ IsTopologicallyNilpotent (a * b)

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f g : MvPowerSeries σ R
p q : ℕ
hf : IsWeightedHomogeneous w f p
hg : IsWeightedHomogeneous w g q
d : σ →₀ ℕ
hd : ¬(weight w) d = p + q
x : (σ →₀ ℕ) × (σ →₀ ℕ)
hx : x.1 + x.2 = d
this : (weight w) x.1 ≠ p ∨ (weight w) x.2 ≠ q
⊢ (coeff x.1) f * (coeff x.2) g = 0

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝² : TopologicalSpace R
inst✝¹ : Ring R
inst✝ : IsLinearTopology R R
a b : R
ha : IsTopologicallyNilpotent a
hb : IsTopologicallyNilpotent b
h : Commute a b
⊢ IsTopologicallyNilpotent (a + b)

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝² : TopologicalSpace R
inst✝¹ : CommRing R
inst✝ : IsLinearTopology R R
a : R
⊢ a ∈ topologicalNilradical R ↔ IsTopologicallyNilpotent a

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f g : MvPowerSeries σ R
p q : ℕ
hf : IsWeightedHomogeneous w f p
hg : IsWeightedHomogeneous w g q
d : σ →₀ ℕ
⊢ (coeff d) (f * g) ≠ 0 → (weight w) d = p + q

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
p : ℕ
f g : MvPowerSeries σ R
⊢ (fun d ↦ if (weight w) d = p then (coeff d) (f + g) else 0) =
    (fun d ↦ if (weight w) d = p then (coeff d) f else 0) + fun d ↦ if (weight w) d = p then (coeff d) g else 0

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
p : ℕ
a : R
f : MvPowerSeries σ R
⊢ (fun d ↦ if (weight w) d = p then (coeff d) (a • f) else 0) =
    (RingHom.id R) a • fun d ↦ if (weight w) d = p then (coeff d) f else 0

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f : MvPowerSeries σ R
p : ℕ
hf : ↑p < weightedOrder w f
⊢ (weightedHomogeneousComponent w p) f = 0

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f : MvPowerSeries σ R
p : ℕ
hf : ↑p = weightedOrder w f
hf' : (weightedHomogeneousComponent w p) f = 0
⊢ ↑(weightedOrder w f).toNat = weightedOrder w f

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f : MvPowerSeries σ R
p : ℕ
hf : ↑p = weightedOrder w f
⊢ (weightedHomogeneousComponent w p) f ≠ 0

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f : MvPowerSeries σ R
p : ℕ
d : σ →₀ ℕ
⊢ (coeff d) ((weightedHomogeneousComponent w p) f) ≠ 0 → (weight w) d = p

Tactic:
by

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f : MvPowerSeries σ R
p : ℕ
⊢ IsWeightedHomogeneous w f p ↔ f = (weightedHomogeneousComponent w p) f

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: R : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : IsDedekindDomain R
K : Type u_2
S : Type u_3
inst✝³ : Field K
inst✝² : CommSemiring S
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
s : Set (adicCompletion K v)
⊢ s ∈ 𝓝 0 ↔ ∃ γ, {x | (valuation K v).restrict x < ↑γ} ⊆ s

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
k : K
⊢ Valued.v { toCompletion := ↑((WithVal.equiv (HeightOneSpectrum.valuation K v)).symm k) } =
    (HeightOneSpectrum.valuation K v) k

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
x y : adicCompletion K v
h : x.toCompletion = y.toCompletion
⊢ x = y

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
⊢ { toCompletion := ↑((WithVal.equiv (HeightOneSpectrum.valuation K v)).symm 0) } = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
⊢ { toCompletion := ↑((WithVal.equiv (HeightOneSpectrum.valuation K v)).symm 1) } = 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
x y : K
⊢ { toCompletion := ↑((WithVal.equiv (HeightOneSpectrum.valuation K v)).symm (x + y)) } =
    { toCompletion := ↑((WithVal.equiv (HeightOneSpectrum.valuation K v)).symm x) } +
      { toCompletion := ↑((WithVal.equiv (HeightOneSpectrum.valuation K v)).symm y) }

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
x y : K
⊢ { toCompletion := ↑((WithVal.equiv (HeightOneSpectrum.valuation K v)).symm (x * y)) } =
    { toCompletion := ↑((WithVal.equiv (HeightOneSpectrum.valuation K v)).symm x) } *
      { toCompletion := ↑((WithVal.equiv (HeightOneSpectrum.valuation K v)).symm y) }

Tactic:
by

Goal state after: