mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.MvPowerSeries.Trunc

Goal state before:
pp: case neg
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) = if degree x < n then (coeff x) p else 0

Tactic:
·

Goal state after:
case neg
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) = if degree x < n then (coeff x) p else 0
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: case refine_2
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 ≤ weightedOrder w (p.toSubring T hp)

Tactic:
·

Goal state after:
case refine_2
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 ≤ weightedOrder w (p.toSubring T hp)
Module: Mathlib.Topology.Instances.ENat

Goal state before:
pp: x : ℕ∞
hx : x ≠ ⊤
⊢ 𝓝 x = pure x

Tactic:
ENat.nhds_eq_pure hx

Goal state after:
x : ℕ∞
hx : x ≠ ⊤
⊢ pure x = pure x
Module: Mathlib.Topology.Instances.ENat

Goal state before:
pp: case inl
⊢ ContinuousAt (Function.uncurry fun x1 x2 ↦ x1 * x2) (0, ⊤)

pp: case inr
a : ℕ∞
ha : 0 < a
⊢ ContinuousAt (Function.uncurry fun x1 x2 ↦ x1 * x2) (a, ⊤)

Tactic:
·

Goal state after:
case inl
⊢ ContinuousAt (Function.uncurry fun x1 x2 ↦ x1 * x2) (0, ⊤)
Module: Mathlib.Topology.Instances.ENat

Goal state before:
pp: case inr
a : ℕ∞
ha : 0 < a
⊢ ContinuousAt (Function.uncurry fun x1 x2 ↦ x1 * x2) (a, ⊤)

Tactic:
·

Goal state after:
case inr
a : ℕ∞
ha : 0 < a
⊢ ContinuousAt (Function.uncurry fun x1 x2 ↦ x1 * x2) (a, ⊤)
Module: Mathlib.RingTheory.MvPowerSeries.Trunc

Goal state before:
pp: case ind.a.e'_5
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
⊢ MvPolynomial.coeff x✝.1 ((truncTotal n) p ^ k) = (coeff x✝.1) (p ^ k)

pp: case ind.a.e'_6
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
⊢ MvPolynomial.coeff x✝.2 ((truncTotal n) p) = (coeff x✝.2) p

Tactic:
·

Goal state after:
case ind.a.e'_5
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
⊢ MvPolynomial.coeff x✝.1 ((truncTotal n) p ^ k) = (coeff x✝.1) (p ^ k)
Module: Mathlib.RingTheory.MvPowerSeries.Order

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

pp: case mpr
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
f : MvPowerSeries σ R
⊢ constantCoeff f = 0 → 1 ≤ f.order

Tactic:
·

Goal state after:
case mp
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
f : MvPowerSeries σ R
⊢ 1 ≤ f.order → constantCoeff f = 0
Module: Mathlib.RingTheory.MvPowerSeries.Order

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

Tactic:
·

Goal state after:
case mpr
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
f : MvPowerSeries σ R
⊢ constantCoeff f = 0 → 1 ≤ f.order
Module: Mathlib.RingTheory.MvPowerSeries.Trunc

Goal state before:
pp: case ind.a.e'_6
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
⊢ MvPolynomial.coeff x✝.2 ((truncTotal n) p) = (coeff x✝.2) p

Tactic:
·

Goal state after:
case ind.a.e'_6
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
⊢ MvPolynomial.coeff x✝.2 ((truncTotal n) p) = (coeff x✝.2) p
Module: Mathlib.RingTheory.MvPowerSeries.Trunc

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

pp: case neg
σ : Type u_1
R : Type u_2
n m : ℕ
inst✝¹ : Finite σ
inst✝ : CommSemiring R
p : MvPowerSeries σ R
d : σ →₀ ℕ
hd : ¬degree d < n
⊢ MvPolynomial.coeff d ((truncTotal n) (p ^ m)) = MvPolynomial.coeff d ((truncTotal n) (↑((truncTotal n) p) ^ m))

Tactic:
·

Goal state after:
case pos
σ : Type u_1
R : Type u_2
n m : ℕ
inst✝¹ : Finite σ
inst✝ : CommSemiring R
p : MvPowerSeries σ R
d : σ →₀ ℕ
hd : degree d < n
⊢ MvPolynomial.coeff d ((truncTotal n) (p ^ m)) = MvPolynomial.coeff d ((truncTotal n) (↑((truncTotal n) p) ^ m))
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: case refine_1
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

pp: case refine_2
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.order ≤ (p.toSubring T hp).order

Tactic:
·

Goal state after:
case refine_1
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
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: case refine_2
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.order ≤ (p.toSubring T hp).order

Tactic:
·

Goal state after:
case refine_2
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.order ≤ (p.toSubring T hp).order
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
⊢ ∃ γ, {x | (valuation K v).restrict x < ↑γ} ⊆ s

pp: case 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
⊢ ∃ t ∈ 𝓝 0, toCompletion ⁻¹' t ⊆ s

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
⊢ ∃ γ, {x | (valuation K v).restrict x < ↑γ} ⊆ s
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
⊢ (coeff x.1) f * (coeff x.2) g = 0

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
⊢ (coeff x.1) f * (coeff x.2) g = 0

Tactic:
·

Goal state after:
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
⊢ (coeff x.1) f * (coeff x.2) g = 0
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
⊢ (coeff x.1) f * (coeff x.2) g = 0

Tactic:
·

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

Goal state before:
pp: case 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
⊢ ∃ t ∈ 𝓝 0, toCompletion ⁻¹' t ⊆ s

Tactic:
·

Goal state after:
case 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
⊢ ∃ t ∈ 𝓝 0, toCompletion ⁻¹' t ⊆ s
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
p : ℕ
f g : MvPowerSeries σ R
d : σ →₀ ℕ
h : (weight w) d = p
⊢ f d + g d = f d + g d

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 + 0

Tactic:
·

Goal state after:
case pos
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
⊢ f d + g d = f d + g d
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 + 0

Tactic:
·

Goal state after:
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 + 0
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
⊢ {y | Valued.v.restrict y < ↑((Units.mapEquiv (valueGroupOrderIso K v).toMulEquiv) γ)} ∈ 𝓝 0

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)}
⊢ x ∈ {x | (valuation K v).restrict x < ↑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
⊢ {y | Valued.v.restrict y < ↑((Units.mapEquiv (valueGroupOrderIso K v).toMulEquiv) γ)} ∈ 𝓝 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 = (coeff d) 0

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 = (coeff d) 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 = (coeff d) 0
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 = (coeff d) 0

Tactic:
·

Goal state after:
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 = (coeff d) 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
hf : ↑p = weightedOrder w f
⊢ (weightedHomogeneousComponent w p) f ≠ 0

Tactic:
intro hf'
  obtain ⟨d, hd⟩ := f.exists_coeff_ne_zero_and_weightedOrder w (by rw [← hf, toNat_natCast])
  simp only [ne_eq, ← hf, Nat.cast_inj] at hd
  apply hd.1
  rw [MvPowerSeries.ext_iff] at hf'
  specialize hf' d
  simp only [coeff_weightedHomogeneousComponent, coeff_zero, ite_eq_right_iff] at hf'
  exact hf' hd.2

Goal state after:
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)}
⊢ x ∈ {x | (valuation K v).restrict x < ↑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)}
⊢ x ∈ {x | (valuation K v).restrict x < ↑γ}
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
⊢ IsWeightedHomogeneous w f p → f = (weightedHomogeneousComponent w p) f

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

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
⊢ IsWeightedHomogeneous w f p → f = (weightedHomogeneousComponent w p) f
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 : IsWeightedHomogeneous w f p
d : σ →₀ ℕ
hd : (weight w) d = p
⊢ (coeff d) f = (coeff d) f

pp: case neg
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:
·

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 : IsWeightedHomogeneous w f p
d : σ →₀ ℕ
hd : (weight w) d = p
⊢ (coeff d) f = (coeff d) f
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 : IsWeightedHomogeneous w f p
d : σ →₀ ℕ
hd : ¬(weight w) d = p
⊢ (coeff d) f = 0

Tactic:
·

Goal state after:
case neg
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
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
⊢ f = (weightedHomogeneousComponent w p) f → IsWeightedHomogeneous w 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
⊢ f = (weightedHomogeneousComponent w p) f → IsWeightedHomogeneous w f p
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
⊢ (coeff d) (f * g) = (coeff d) ((weightedHomogeneousComponent w p) f * (weightedHomogeneousComponent w q) g)

pp: case neg
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
⊢ 0 = (coeff d) ((weightedHomogeneousComponent w p) f * (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
⊢ (coeff d) (f * g) = (coeff d) ((weightedHomogeneousComponent w p) f * (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 =
    (if (weight w) x.1 = p then (coeff x.1) f else 0) * if (weight w) x.2 = q then (coeff x.2) g else 0

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 =
    (if (weight w) x.1 = p then (coeff x.1) f else 0) * if (weight w) x.2 = q then (coeff x.2) g else 0

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 =
    (if (weight w) x.1 = p then (coeff x.1) f else 0) * if (weight w) x.2 = q then (coeff x.2) g else 0

Tactic:
·

Goal state after:
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 =
    (if (weight w) x.1 = p then (coeff x.1) f else 0) * if (weight w) x.2 = q then (coeff x.2) g else 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 =
    (if (weight w) x.1 = p then (coeff x.1) f else 0) * if (weight w) x.2 = q then (coeff x.2) g else 0

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 =
    (if (weight w) x.1 = p then (coeff x.1) f else 0) * if (weight w) x.2 = q then (coeff x.2) g else 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 =
    (if (weight w) x.1 = p then (coeff x.1) f else 0) * if (weight w) x.2 = q then (coeff x.2) g else 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 =
    (if (weight w) x.1 = p then (coeff x.1) f else 0) * if (weight w) x.2 = q then (coeff x.2) g else 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 =
    (if (weight w) x.1 = p then (coeff x.1) f else 0) * if (weight w) x.2 = q then (coeff x.2) g else 0
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 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
⊢ 0 = (coeff d) ((weightedHomogeneousComponent w p) f * (weightedHomogeneousComponent w q) g)

Tactic:
·

Goal state after:
case neg
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
⊢ 0 = (coeff d) ((weightedHomogeneousComponent w p) f * (weightedHomogeneousComponent w q) g)