mathlib-initiative/mathlib-tactics huggingface dataset:
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 : ↑((weight fun x ↦ 1) d) < g.order
⊢ (coeff d) ((1 - g) * f) = (coeff d) f

Tactic:
]

Goal state after:
σ : Type u_1
R : Type u_3
inst✝ : Ring R
f g : MvPowerSeries σ R
d : σ →₀ ℕ
h : ↑((weight fun x ↦ 1) d) < g.order
⊢ (coeff d) ((1 - g) * f) = (coeff d) f
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, ↑((weight fun x ↦ 1) d) < (g i).order) → (coeff d) (f * ∏ i ∈ s, (1 - g i)) = (coeff d) f

Tactic:
]

Goal state after:
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, ↑((weight fun x ↦ 1) d) < (g i).order) → (coeff d) (f * ∏ i ∈ s, (1 - g i)) = (coeff d) f
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
⊢ (∀ x ∈ ↑p.support, x ∈ {f | degree f < n}) ↔ ∀ b ∈ p.support, (b.sum fun x e ↦ e) < n

Tactic:
]

Goal state after:
σ : Type u_1
R : Type u_2
n : ℕ
inst✝¹ : Finite σ
inst✝ : CommSemiring R
p : MvPolynomial σ R
h : n ≠ 0
⊢ (∀ x ∈ ↑p.support, x ∈ {f | degree f < n}) ↔ ∀ b ∈ p.support, (b.sum fun x e ↦ e) < n
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:
refine eq_of_le_of_ge ?_ ?_
  · exact le_order fun d hd => by simp [coeff_of_lt_order hd, ← p.coeff_toSubring T hp]
  · exact le_order fun d hd => by exact_mod_cast (coeff_toSubring p T hp) ▸ (coeff_of_lt_order hd)

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)
⊢ (∃ t ∈ 𝓝 0, toCompletion ⁻¹' t ⊆ s) ↔ ∃ γ, {x | (valuation K v).restrict x < ↑γ} ⊆ s

Tactic:
]

Goal state after:
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)
⊢ (∃ t ∈ 𝓝 0, toCompletion ⁻¹' t ⊆ s) ↔ ∃ γ, {x | (valuation K v).restrict x < ↑γ} ⊆ s
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
⊢ ¬(weight w) d = p → (coeff d) (f + g) = 0

Tactic:
]

Goal state after:
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
⊢ ¬(weight w) d = p → (coeff d) (f + g) = 0
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
hd : ¬(weight w) d = p
⊢ 0 = 0

Tactic:
]

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
⊢ ¬(weight w) d = p + q → (coeff d) (f * g) = 0

Tactic:
]

Goal state after:
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
⊢ ¬(weight w) d = p + q → (coeff d) (f * g) = 0
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
⊢ ∑ p ∈ Finset.antidiagonal d, (coeff p.1) f * (coeff p.2) g = 0

Tactic:
]

Goal state after:
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
⊢ ∑ p ∈ Finset.antidiagonal d, (coeff p.1) f * (coeff p.2) g = 0
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 = 0

Tactic:
]

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
⊢ (coeff x.1) f * (coeff x.2) g = 0

Tactic:
]

Goal state after:
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
⊢ (coeff x.1) f * (coeff x.2) g = 0
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: case inl
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
hp : (weight w) x.1 ≠ p
⊢ 0 = 0

Tactic:
]

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
n : ℕ
hn : a ^ n = 0
i : ℕ
hi : i ≥ n
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: case inr
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
hq : (weight w) x.2 ≠ q
⊢ 0 = 0

Tactic:
]

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
⊢ ¬((weight w) x.1 = p ∧ (weight w) x.2 = q)

Tactic:
]

Goal state after:
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
⊢ ¬((weight w) x.1 = p ∧ (weight w) x.2 = q)
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: case refine_1
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)
x✝ : ∃ t ∈ 𝓝 0, toCompletion ⁻¹' t ⊆ s
t : Set (HeightOneSpectrum.valuation K v).Completion
ht : t ∈ 𝓝 0
hts : toCompletion ⁻¹' t ⊆ s
δ : (ofClass Valued.v).ValueGroup₀ˣ
hδ : {x | Valued.v.restrict x < ↑δ} ⊆ t
x : adicCompletion K v
hx : (valuation K v).restrict x < ↑((Units.mapEquiv (valueGroupOrderIso K v).symm.toMulEquiv) δ)
⊢ Valued.v.restrict x.toCompletion < ↑δ

Tactic:
]

Goal state after:
case refine_1
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)
x✝ : ∃ t ∈ 𝓝 0, toCompletion ⁻¹' t ⊆ s
t : Set (HeightOneSpectrum.valuation K v).Completion
ht : t ∈ 𝓝 0
hts : toCompletion ⁻¹' t ⊆ s
δ : (ofClass Valued.v).ValueGroup₀ˣ
hδ : {x | Valued.v.restrict x < ↑δ} ⊆ t
x : adicCompletion K v
hx : (valuation K v).restrict x < ↑((Units.mapEquiv (valueGroupOrderIso K v).symm.toMulEquiv) δ)
⊢ Valued.v.restrict x.toCompletion < ↑δ
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
hp : (weight w) x.1 = p
hq : (weight w) x.2 = q
⊢ p + q = p + q

Tactic:
]

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

Goal state before:
pp: case neg
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
p : ℕ
f g : MvPowerSeries σ R
d : σ →₀ ℕ
h : ¬(weight w) d = p
⊢ 0 = 0

Tactic:
]

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
d : σ →₀ ℕ
⊢ (if (weight w) d = p then (coeff d) f else 0) = (coeff d) 0

Tactic:
]

Goal state after:
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
d : σ →₀ ℕ
⊢ (if (weight w) d = p then (coeff d) f else 0) = (coeff d) 0
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: case pos
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
d : σ →₀ ℕ
hd : (weight w) d = p
⊢ (coeff d) f = 0

Tactic:
]

Goal state after:
case pos
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
d : σ →₀ ℕ
hd : (weight w) d = p
⊢ (coeff d) f = 0
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: case pos
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
d : σ →₀ ℕ
hd : (weight w) d = p
⊢ ↑p < weightedOrder w f

Tactic:
]

Goal state after:
case pos
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
d : σ →₀ ℕ
hd : (weight w) d = p
⊢ ↑p < weightedOrder w f
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: case refine_2.refine_1
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)
x✝ : ∃ γ, {x | (valuation K v).restrict x < ↑γ} ⊆ s
γ : (ofClass (valuation K v)).ValueGroup₀ˣ
hγ : {x | (valuation K v).restrict x < ↑γ} ⊆ s
⊢ ∃ γ_1,
    {x | Valued.v.restrict x < ↑γ_1} ⊆
      {y | Valued.v.restrict y < ↑((Units.mapEquiv (valueGroupOrderIso K v).toMulEquiv) γ)}

Tactic:
]

Goal state after:
case refine_2.refine_1
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)
x✝ : ∃ γ, {x | (valuation K v).restrict x < ↑γ} ⊆ s
γ : (ofClass (valuation K v)).ValueGroup₀ˣ
hγ : {x | (valuation K v).restrict x < ↑γ} ⊆ s
⊢ ∃ γ_1,
    {x | Valued.v.restrict x < ↑γ_1} ⊆
      {y | Valued.v.restrict y < ↑((Units.mapEquiv (valueGroupOrderIso K v).toMulEquiv) γ)}
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: case neg
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
d : σ →₀ ℕ
hd : ¬(weight w) d = p
⊢ 0 = 0

Tactic:
]

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
⊢ ↑p = ↑p

Tactic:
]

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' : ∀ (n : σ →₀ ℕ), (coeff n) ((weightedHomogeneousComponent w p) f) = (coeff n) 0
d : σ →₀ ℕ
hd : ¬(coeff d) f = 0 ∧ (weight w) d = p
⊢ (coeff d) f = 0

Tactic:
]

Goal state after:
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' : ∀ (n : σ →₀ ℕ), (coeff n) ((weightedHomogeneousComponent w p) f) = (coeff n) 0
d : σ →₀ ℕ
hd : ¬(coeff d) f = 0 ∧ (weight w) d = p
⊢ (coeff d) f = 0
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
⊢ ¬(weight w) d = p → (coeff d) ((weightedHomogeneousComponent w p) f) = 0

Tactic:
]

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

Tactic:
]

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

Goal state before:
pp: case mp
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
⊢ (coeff d) f = if (weight w) d = p then (coeff d) f else 0

Tactic:
]

Goal state after:
case mp
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
⊢ (coeff d) f = if (weight w) d = p then (coeff d) f else 0
Module: Mathlib.RingTheory.MvPowerSeries.Order

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

Tactic:
]

Goal state after:
case mpr
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f : MvPowerSeries σ R
p : ℕ
hf : f = (weightedHomogeneousComponent w p) f
⊢ IsWeightedHomogeneous w ((weightedHomogeneousComponent w p) f) p
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: case refine_2.refine_2
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)
x✝ : ∃ γ, {x | (valuation K v).restrict x < ↑γ} ⊆ s
γ : (ofClass (valuation K v)).ValueGroup₀ˣ
hγ : {x | (valuation K v).restrict x < ↑γ} ⊆ s
x : adicCompletion K v
hx : x ∈ toCompletion ⁻¹' {y | Valued.v.restrict y < ↑((Units.mapEquiv (valueGroupOrderIso K v).toMulEquiv) γ)}
⊢ Valued.v.restrict x.toCompletion < (valueGroupOrderIso K v) ↑γ

Tactic:
]

Goal state after:
case refine_2.refine_2
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)
x✝ : ∃ γ, {x | (valuation K v).restrict x < ↑γ} ⊆ s
γ : (ofClass (valuation K v)).ValueGroup₀ˣ
hγ : {x | (valuation K v).restrict x < ↑γ} ⊆ s
x : adicCompletion K v
hx : x ∈ toCompletion ⁻¹' {y | Valued.v.restrict y < ↑((Units.mapEquiv (valueGroupOrderIso K v).toMulEquiv) γ)}
⊢ Valued.v.restrict x.toCompletion < (valueGroupOrderIso K v) ↑γ
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 : ↑p ≤ weightedOrder w f
hg : ↑q ≤ weightedOrder w g
d : σ →₀ ℕ
⊢ (if (weight w) d = p + q then (coeff d) (f * g) else 0) =
    (coeff d) ((weightedHomogeneousComponent w p) f * (weightedHomogeneousComponent w q) g)

Tactic:
]

Goal state after:
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f g : MvPowerSeries σ R
p q : ℕ
hf : ↑p ≤ weightedOrder w f
hg : ↑q ≤ weightedOrder w g
d : σ →₀ ℕ
⊢ (if (weight w) d = p + q then (coeff d) (f * g) else 0) =
    (coeff d) ((weightedHomogeneousComponent w p) f * (weightedHomogeneousComponent w q) g)
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: case pos
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f g : MvPowerSeries σ R
p q : ℕ
hf : ↑p ≤ weightedOrder w f
hg : ↑q ≤ weightedOrder w g
d : σ →₀ ℕ
hd : (weight w) d = p + q
x : (σ →₀ ℕ) × (σ →₀ ℕ)
hx : x.1 + x.2 = d
⊢ (coeff x.1) f * (coeff x.2) g =
    (coeff x.1) ((weightedHomogeneousComponent w p) f) * (coeff x.2) ((weightedHomogeneousComponent w q) g)

Tactic:
]

Goal state after:
case pos
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f g : MvPowerSeries σ R
p q : ℕ
hf : ↑p ≤ weightedOrder w f
hg : ↑q ≤ weightedOrder w g
d : σ →₀ ℕ
hd : (weight w) d = p + q
x : (σ →₀ ℕ) × (σ →₀ ℕ)
hx : x.1 + x.2 = d
⊢ (coeff x.1) f * (coeff x.2) g =
    (coeff x.1) ((weightedHomogeneousComponent w p) f) * (coeff x.2) ((weightedHomogeneousComponent w q) g)
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: case pos
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f g : MvPowerSeries σ R
p q : ℕ
hf : ↑p ≤ weightedOrder w f
hg : ↑q ≤ weightedOrder w g
d : σ →₀ ℕ
x : (σ →₀ ℕ) × (σ →₀ ℕ)
hd : (weight w) x.1 + (weight w) x.2 = p + q
hx : x.1 + x.2 = d
⊢ (coeff x.1) f * (coeff x.2) g =
    (coeff x.1) ((weightedHomogeneousComponent w p) f) * (coeff x.2) ((weightedHomogeneousComponent w q) g)

Tactic:
]

Goal state after:
case pos
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f g : MvPowerSeries σ R
p q : ℕ
hf : ↑p ≤ weightedOrder w f
hg : ↑q ≤ weightedOrder w g
d : σ →₀ ℕ
x : (σ →₀ ℕ) × (σ →₀ ℕ)
hd : (weight w) x.1 + (weight w) x.2 = p + q
hx : x.1 + x.2 = d
⊢ (coeff x.1) f * (coeff x.2) g =
    (coeff x.1) ((weightedHomogeneousComponent w p) f) * (coeff x.2) ((weightedHomogeneousComponent w q) g)
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: case pos.inl
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f g : MvPowerSeries σ R
p q : ℕ
hf : ↑p ≤ weightedOrder w f
hg : ↑q ≤ weightedOrder w g
d : σ →₀ ℕ
x : (σ →₀ ℕ) × (σ →₀ ℕ)
hd : (weight w) x.1 + (weight w) x.2 = p + q
hx : x.1 + x.2 = d
h : (weight w) x.1 = p ∧ (weight w) x.2 = q
⊢ (coeff x.1) f * (coeff x.2) g = (coeff x.1) f * (coeff x.2) g

Tactic:
]

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

Goal state before:
pp: case pos.inr.inl
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f g : MvPowerSeries σ R
p q : ℕ
hf : ↑p ≤ weightedOrder w f
hg : ↑q ≤ weightedOrder w g
d : σ →₀ ℕ
x : (σ →₀ ℕ) × (σ →₀ ℕ)
hd : (weight w) x.1 + (weight w) x.2 = p + q
hx : x.1 + x.2 = d
h : (weight w) x.1 < p
⊢ (coeff x.1) f * (coeff x.2) g = 0

Tactic:
]

Goal state after:
case pos.inr.inl
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f g : MvPowerSeries σ R
p q : ℕ
hf : ↑p ≤ weightedOrder w f
hg : ↑q ≤ weightedOrder w g
d : σ →₀ ℕ
x : (σ →₀ ℕ) × (σ →₀ ℕ)
hd : (weight w) x.1 + (weight w) x.2 = p + q
hx : x.1 + x.2 = d
h : (weight w) x.1 < p
⊢ (coeff x.1) f * (coeff x.2) g = 0
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: case pos.inr.inl
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f g : MvPowerSeries σ R
p q : ℕ
hf : ↑p ≤ weightedOrder w f
hg : ↑q ≤ weightedOrder w g
d : σ →₀ ℕ
x : (σ →₀ ℕ) × (σ →₀ ℕ)
hd : (weight w) x.1 + (weight w) x.2 = p + q
hx : x.1 + x.2 = d
h : ↑((weight w) x.1) < ↑p
⊢ (coeff x.1) f * (coeff x.2) g = 0

Tactic:
]

Goal state after:
case pos.inr.inl
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f g : MvPowerSeries σ R
p q : ℕ
hf : ↑p ≤ weightedOrder w f
hg : ↑q ≤ weightedOrder w g
d : σ →₀ ℕ
x : (σ →₀ ℕ) × (σ →₀ ℕ)
hd : (weight w) x.1 + (weight w) x.2 = p + q
hx : x.1 + x.2 = d
h : ↑((weight w) x.1) < ↑p
⊢ (coeff x.1) f * (coeff x.2) g = 0
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: case pos.inr.inl
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f g : MvPowerSeries σ R
p q : ℕ
hf : ↑p ≤ weightedOrder w f
hg : ↑q ≤ weightedOrder w g
d : σ →₀ ℕ
x : (σ →₀ ℕ) × (σ →₀ ℕ)
hd : (weight w) x.1 + (weight w) x.2 = p + q
hx : x.1 + x.2 = d
h : ↑((weight w) x.1) < ↑p
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: case pos.inr.inr
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f g : MvPowerSeries σ R
p q : ℕ
hf : ↑p ≤ weightedOrder w f
hg : ↑q ≤ weightedOrder w g
d : σ →₀ ℕ
x : (σ →₀ ℕ) × (σ →₀ ℕ)
hd : (weight w) x.1 + (weight w) x.2 = p + q
hx : x.1 + x.2 = d
h : (weight w) x.2 < q
⊢ (coeff x.1) f * (coeff x.2) g = 0

Tactic:
]

Goal state after:
case pos.inr.inr
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f g : MvPowerSeries σ R
p q : ℕ
hf : ↑p ≤ weightedOrder w f
hg : ↑q ≤ weightedOrder w g
d : σ →₀ ℕ
x : (σ →₀ ℕ) × (σ →₀ ℕ)
hd : (weight w) x.1 + (weight w) x.2 = p + q
hx : x.1 + x.2 = d
h : (weight w) x.2 < q
⊢ (coeff x.1) f * (coeff x.2) g = 0
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
n : (WithZero (Multiplicative ℤ))ˣ
x✝ : ∃ a, ¬Valued.v a = 0 ∧ ∃ x, Valued.v a * ↑n = Valued.v x
a : adicCompletion K v
ha0 : ¬Valued.v a = 0
x : adicCompletion K v
hx : Valued.v a * ↑n = Valued.v x
b : K
hb : (valuation K v) b = Valued.v a
y : K
hy : (valuation K v) y = Valued.v x
⊢ ¬Valued.v a = 0

Tactic:
]

Goal state after:
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
n : (WithZero (Multiplicative ℤ))ˣ
x✝ : ∃ a, ¬Valued.v a = 0 ∧ ∃ x, Valued.v a * ↑n = Valued.v x
a : adicCompletion K v
ha0 : ¬Valued.v a = 0
x : adicCompletion K v
hx : Valued.v a * ↑n = Valued.v x
b : K
hb : (valuation K v) b = Valued.v a
y : K
hy : (valuation K v) y = Valued.v x
⊢ ¬Valued.v a = 0
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: case pos.inr.inr
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f g : MvPowerSeries σ R
p q : ℕ
hf : ↑p ≤ weightedOrder w f
hg : ↑q ≤ weightedOrder w g
d : σ →₀ ℕ
x : (σ →₀ ℕ) × (σ →₀ ℕ)
hd : (weight w) x.1 + (weight w) x.2 = p + q
hx : x.1 + x.2 = d
h : ↑((weight w) x.2) < ↑q
⊢ (coeff x.1) f * (coeff x.2) g = 0

Tactic:
]

Goal state after:
case pos.inr.inr
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f g : MvPowerSeries σ R
p q : ℕ
hf : ↑p ≤ weightedOrder w f
hg : ↑q ≤ weightedOrder w g
d : σ →₀ ℕ
x : (σ →₀ ℕ) × (σ →₀ ℕ)
hd : (weight w) x.1 + (weight w) x.2 = p + q
hx : x.1 + x.2 = d
h : ↑((weight w) x.2) < ↑q
⊢ (coeff x.1) f * (coeff x.2) g = 0
Module: Mathlib.RingTheory.MvPowerSeries.PiTopology

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝¹ : TopologicalSpace R
inst✝ : Semiring R
ι : Type u_3
f : ι → MvPowerSeries σ R
u : Filter ι
g : MvPowerSeries σ R
⊢ (∀ (i : σ →₀ ℕ), Tendsto (fun x ↦ f x i) u (𝓝 (g i))) ↔
    ∀ (d : σ →₀ ℕ), Tendsto (fun i ↦ (coeff d) (f i)) u (𝓝 ((coeff d) g))

Tactic:
]

Goal state after:
σ : Type u_1
R : Type u_2
inst✝¹ : TopologicalSpace R
inst✝ : Semiring R
ι : Type u_3
f : ι → MvPowerSeries σ R
u : Filter ι
g : MvPowerSeries σ R
⊢ (∀ (i : σ →₀ ℕ), Tendsto (fun x ↦ f x i) u (𝓝 (g i))) ↔
    ∀ (d : σ →₀ ℕ), Tendsto (fun i ↦ (coeff d) (f i)) u (𝓝 ((coeff d) g))
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: case pos.inr.inr
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f g : MvPowerSeries σ R
p q : ℕ
hf : ↑p ≤ weightedOrder w f
hg : ↑q ≤ weightedOrder w g
d : σ →₀ ℕ
x : (σ →₀ ℕ) × (σ →₀ ℕ)
hd : (weight w) x.1 + (weight w) x.2 = p + q
hx : x.1 + x.2 = d
h : ↑((weight w) x.2) < ↑q
⊢ 0 = 0

Tactic:
]

Goal state after: