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

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : Nontrivial R
n : ℕ
⊢ ↑n = ↑n

pp: case h
R : Type u_1
inst✝¹ : Semiring R
inst✝ : Nontrivial R
n : ℕ
⊢ 1 ≠ 0

Tactic:
]

Goal state after:
case h
R : Type u_1
inst✝¹ : Semiring R
inst✝ : Nontrivial R
n : ℕ
⊢ 1 ≠ 0
Module: Mathlib.RingTheory.PowerSeries.Order

Goal state before:
pp: case pos
R : Type u_1
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
φ ψ : R⟦X⟧
h : φ = 0 ∨ ψ = 0
⊢ (φ * ψ).order ≤ φ.order + ψ.order

Tactic:
rcases h with h | h <;> simp [h]

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

Goal state before:
pp: case neg
R : Type u_1
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
φ ψ : R⟦X⟧
h : φ ≠ 0 ∧ ψ ≠ 0
⊢ (φ * ψ).order ≤ ↑(φ.order.toNat + ψ.order.toNat)

Tactic:
]

Goal state after:
case neg
R : Type u_1
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
φ ψ : R⟦X⟧
h : φ ≠ 0 ∧ ψ ≠ 0
⊢ (φ * ψ).order ≤ ↑(φ.order.toNat + ψ.order.toNat)
Module: Mathlib.RingTheory.PowerSeries.Order

Goal state before:
pp: case neg
R : Type u_1
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
φ ψ : R⟦X⟧
h : φ ≠ 0 ∧ ψ ≠ 0
⊢ (coeff (φ.order.toNat, ψ.order.toNat).1) φ * (coeff (φ.order.toNat, ψ.order.toNat).2) ψ ≠ 0

pp: case neg
R : Type u_1
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
φ ψ : R⟦X⟧
h : φ ≠ 0 ∧ ψ ≠ 0
⊢ ∀ b ∈ antidiagonal (φ.order.toNat + ψ.order.toNat),
    b ≠ (φ.order.toNat, ψ.order.toNat) → (coeff b.1) φ * (coeff b.2) ψ = 0

Tactic:
]

Goal state after:
case neg
R : Type u_1
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
φ ψ : R⟦X⟧
h : φ ≠ 0 ∧ ψ ≠ 0
⊢ (coeff (φ.order.toNat, ψ.order.toNat).1) φ * (coeff (φ.order.toNat, ψ.order.toNat).2) ψ ≠ 0

case neg
R : Type u_1
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
φ ψ : R⟦X⟧
h : φ ≠ 0 ∧ ψ ≠ 0
⊢ ∀ b ∈ antidiagonal (φ.order.toNat + ψ.order.toNat),
    b ≠ (φ.order.toNat, ψ.order.toNat) → (coeff b.1) φ * (coeff b.2) ψ = 0
Module: Mathlib.RingTheory.MvPowerSeries.Evaluation

Goal state before:
pp: σ : Type u_1
S : Type u_3
inst✝³ : CommRing S
inst✝² : TopologicalSpace S
inst✝¹ : ContinuousAdd S
inst✝ : IsLinearTopology S S
a b : σ → S
ha : HasEval a
hb : HasEval b
⊢ Tendsto (a + b) cofinite (𝓝 (0 + 0))

Tactic:
]

Goal state after:
σ : Type u_1
S : Type u_3
inst✝³ : CommRing S
inst✝² : TopologicalSpace S
inst✝¹ : ContinuousAdd S
inst✝ : IsLinearTopology S S
a b : σ → S
ha : HasEval a
hb : HasEval b
⊢ Tendsto (a + b) cofinite (𝓝 (0 + 0))
Module: Mathlib.RingTheory.MvPowerSeries.Evaluation

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : UniformSpace R
S : Type u_3
inst✝⁴ : CommRing S
inst✝³ : UniformSpace S
φ : R →+* S
a : σ → S
inst✝² : IsUniformAddGroup R
inst✝¹ : IsUniformAddGroup S
inst✝ : IsLinearTopology S S
hφ : Continuous[inst✝⁵.toTopologicalSpace, inst✝³.toTopologicalSpace] ⇑φ
ha : HasEval a
⊢ ∀ (i : Ideal S), ↑i ∈ 𝓝 0 → ∀ᶠ (x : MvPolynomial σ R) in 𝓝 0, (eval₂Hom φ a) x ∈ ↑i

Tactic:
]

Goal state after:
σ : Type u_1
R : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : UniformSpace R
S : Type u_3
inst✝⁴ : CommRing S
inst✝³ : UniformSpace S
φ : R →+* S
a : σ → S
inst✝² : IsUniformAddGroup R
inst✝¹ : IsUniformAddGroup S
inst✝ : IsLinearTopology S S
hφ : Continuous[inst✝⁵.toTopologicalSpace, inst✝³.toTopologicalSpace] ⇑φ
ha : HasEval a
⊢ ∀ (i : Ideal S), ↑i ∈ 𝓝 0 → ∀ᶠ (x : MvPolynomial σ R) in 𝓝 0, (eval₂Hom φ a) x ∈ ↑i
Module: Mathlib.RingTheory.MvPowerSeries.Evaluation

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : UniformSpace R
S : Type u_3
inst✝⁴ : CommRing S
inst✝³ : UniformSpace S
φ : R →+* S
a : σ → S
inst✝² : IsUniformAddGroup R
inst✝¹ : IsUniformAddGroup S
inst✝ : IsLinearTopology S S
hφ : Continuous[inst✝⁵.toTopologicalSpace, inst✝³.toTopologicalSpace] ⇑φ
ha : HasEval a
I : Ideal S
⊢ ↑I ∈ 𝓝 0 → ∀ᶠ (x : MvPolynomial σ R) in 𝓝 0, (eval₂Hom φ a) x ∈ ↑I

Tactic:
hI

Goal state after:
σ : Type u_1
R : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : UniformSpace R
S : Type u_3
inst✝⁴ : CommRing S
inst✝³ : UniformSpace S
φ : R →+* S
a : σ → S
inst✝² : IsUniformAddGroup R
inst✝¹ : IsUniformAddGroup S
inst✝ : IsLinearTopology S S
hφ : Continuous[inst✝⁵.toTopologicalSpace, inst✝³.toTopologicalSpace] ⇑φ
ha : HasEval a
I : Ideal S
hI : ↑I ∈ 𝓝 0
⊢ ∀ᶠ (x : MvPolynomial σ R) in 𝓝 0, (eval₂Hom φ a) x ∈ ↑I
Module: Mathlib.RingTheory.PowerSeries.PiTopology

Goal state before:
pp: case inr
R : Type u_1
inst✝³ : TopologicalSpace R
inst✝² : CommSemiring R
ι : Type u_2
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrderBot ι
f : ι → R⟦X⟧
hempty : Nonempty ι
n : ℕ
h : ∀ (n : ℕ), ∃ a, ∀ (b : ι), a ≤ b → ↑n < (f b).order
i : ι
hi : ∀ (b : ι), i ≤ b → ↑n < (f b).order
s : Finset ι
hs : ¬↑s ⊆ Set.Iio i
x : ι
hxs : x ∈ ↑s
hxi : i ≤ x
⊢ (coeff n) (∏ i ∈ s, f i) = 0

Tactic:
]

Goal state after:
case inr
R : Type u_1
inst✝³ : TopologicalSpace R
inst✝² : CommSemiring R
ι : Type u_2
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrderBot ι
f : ι → R⟦X⟧
hempty : Nonempty ι
n : ℕ
h : ∀ (n : ℕ), ∃ a, ∀ (b : ι), a ≤ b → ↑n < (f b).order
i : ι
hi : ∀ (b : ι), i ≤ b → ↑n < (f b).order
s : Finset ι
hs : ¬↑s ⊆ Set.Iio i
x : ι
hxs : x ∈ ↑s
hxi : i ≤ x
⊢ (coeff n) (∏ i ∈ s, f i) = 0
Module: Mathlib.RingTheory.PowerSeries.Order

Goal state before:
pp: case neg.inr.inl
R : Type u_1
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
φ ψ : R⟦X⟧
h✝ : φ ≠ 0 ∧ ψ ≠ 0
ij : ℕ × ℕ
hij : ij ∈ antidiagonal (φ.order.toNat + ψ.order.toNat)
h : ij ≠ (φ.order.toNat, ψ.order.toNat)
h' : ij.1 < φ.order.toNat
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: case neg.inr.inr
R : Type u_1
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
φ ψ : R⟦X⟧
h✝ : φ ≠ 0 ∧ ψ ≠ 0
ij : ℕ × ℕ
hij : ij ∈ antidiagonal (φ.order.toNat + ψ.order.toNat)
h : ij ≠ (φ.order.toNat, ψ.order.toNat)
h' : ij.2 < ψ.order.toNat
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.PowerSeries.PiTopology

Goal state before:
pp: R : Type u_2
inst✝¹ : CommRing R
inst✝ : TopologicalSpace R
a✝ : Nontrivial R
n m : ℕ
hm : n ≤ m
⊢ ↑n < ↑(m + 1)

Tactic:
]

Goal state after:
R : Type u_2
inst✝¹ : CommRing R
inst✝ : TopologicalSpace R
a✝ : Nontrivial R
n m : ℕ
hm : n ≤ m
⊢ ↑n < ↑(m + 1)
Module: Mathlib.RingTheory.PowerSeries.PiTopology

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : TopologicalSpace R
f : R⟦X⟧
⊢ HasSum ((fun d ↦ (monomial d) ((coeff d) f)) ∘ ⇑(Finsupp.uniqueLinearEquiv ℕ ℕ ()).toEquiv) f

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹ : Semiring R
inst✝ : TopologicalSpace R
f : R⟦X⟧
⊢ HasSum ((fun d ↦ (monomial d) ((coeff d) f)) ∘ ⇑(Finsupp.uniqueLinearEquiv ℕ ℕ ()).toEquiv) f
Module: Mathlib.RingTheory.PowerSeries.Order

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
f g : R⟦X⟧
h : f ≠ 0 ∧ g ≠ 0
⊢ X ^ (f.order.toNat + g.order.toNat) * (f * g).divXPowOrder =
    X ^ (f.order.toNat + g.order.toNat) * (f * g).divXPowOrder

Tactic:
]

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

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
f g : R⟦X⟧
h : f ≠ 0 ∧ g ≠ 0
⊢ f.divXPowOrder * g.divXPowOrder * X ^ (g.order.toNat + f.order.toNat) =
    f.divXPowOrder * g.divXPowOrder * X ^ (g.order.toNat + f.order.toNat)

Tactic:
]

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

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
f g : R⟦X⟧
h : f ≠ 0 ∧ g ≠ 0
⊢ f.divXPowOrder * g.divXPowOrder * X ^ (f.order.toNat + g.order.toNat) =
    f.divXPowOrder * g.divXPowOrder * X ^ (f.order.toNat + g.order.toNat)

Tactic:
]

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Ring R
Jd : TwoSidedIdeal R × (σ →₀ ℕ)
x✝ y✝ : MvPowerSeries σ R
hf : x✝ ∈ {f | ∀ e ≤ Jd.2, (coeff e) f ∈ Jd.1}
hg : y✝ ∈ {f | ∀ e ≤ Jd.2, (coeff e) f ∈ Jd.1}
e : σ →₀ ℕ
he : e ≤ Jd.2
⊢ (coeff e) x✝ + (coeff e) y✝ ∈ Jd.1

Tactic:
]

Goal state after:
σ : Type u_1
R : Type u_2
inst✝ : Ring R
Jd : TwoSidedIdeal R × (σ →₀ ℕ)
x✝ y✝ : MvPowerSeries σ R
hf : x✝ ∈ {f | ∀ e ≤ Jd.2, (coeff e) f ∈ Jd.1}
hg : y✝ ∈ {f | ∀ e ≤ Jd.2, (coeff e) f ∈ Jd.1}
e : σ →₀ ℕ
he : e ≤ Jd.2
⊢ (coeff e) x✝ + (coeff e) y✝ ∈ Jd.1
Module: Mathlib.RingTheory.MvPowerSeries.Evaluation

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : UniformSpace R
S : Type u_3
inst✝⁴ : CommRing S
inst✝³ : UniformSpace S
φ : R →+* S
a : σ → S
inst✝² : IsUniformAddGroup R
inst✝¹ : IsUniformAddGroup S
inst✝ : IsLinearTopology S S
hφ : Continuous[inst✝⁵.toTopologicalSpace, inst✝³.toTopologicalSpace] ⇑φ
ha : HasEval a
I : Ideal S
hI : ↑I ∈ 𝓝 0
n : σ → ℕ := fun s ↦ sInf {n | a s ^ n.succ ∈ I}
hn_ne : ∀ (s : σ), {n | a s ^ n.succ ∈ I}.Nonempty
hn : (Function.support n).Finite
n₀ : σ →₀ ℕ := Finsupp.ofSupportFinite n hn
D : Set (σ →₀ ℕ) := Iic n₀
hD : D.Finite
this : ∀ᶠ (x : MvPolynomial σ R) in 𝓝 0, ∀ i ∈ D, φ (MvPolynomial.coeff i x) ∈ I
⊢ ∀ᶠ (x : MvPolynomial σ R) in 𝓝 0, (eval₂Hom φ a) x ∈ ↑I

Tactic:
]

Goal state after:
σ : Type u_1
R : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : UniformSpace R
S : Type u_3
inst✝⁴ : CommRing S
inst✝³ : UniformSpace S
φ : R →+* S
a : σ → S
inst✝² : IsUniformAddGroup R
inst✝¹ : IsUniformAddGroup S
inst✝ : IsLinearTopology S S
hφ : Continuous[inst✝⁵.toTopologicalSpace, inst✝³.toTopologicalSpace] ⇑φ
ha : HasEval a
I : Ideal S
hI : ↑I ∈ 𝓝 0
n : σ → ℕ := fun s ↦ sInf {n | a s ^ n.succ ∈ I}
hn_ne : ∀ (s : σ), {n | a s ^ n.succ ∈ I}.Nonempty
hn : (Function.support n).Finite
n₀ : σ →₀ ℕ := Finsupp.ofSupportFinite n hn
D : Set (σ →₀ ℕ) := Iic n₀
hD : D.Finite
this : ∀ᶠ (x : MvPolynomial σ R) in 𝓝 0, ∀ i ∈ D, φ (MvPolynomial.coeff i x) ∈ I
⊢ ∀ᶠ (x : MvPolynomial σ R) in 𝓝 0, (eval₂Hom φ a) x ∈ ↑I
Module: Mathlib.RingTheory.MvPowerSeries.Evaluation

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : UniformSpace R
S : Type u_3
inst✝⁴ : CommRing S
inst✝³ : UniformSpace S
φ : R →+* S
a : σ → S
inst✝² : IsUniformAddGroup R
inst✝¹ : IsUniformAddGroup S
inst✝ : IsLinearTopology S S
hφ : Continuous[inst✝⁵.toTopologicalSpace, inst✝³.toTopologicalSpace] ⇑φ
ha : HasEval a
I : Ideal S
hI : ↑I ∈ 𝓝 0
n : σ → ℕ := fun s ↦ sInf {n | a s ^ n.succ ∈ I}
hn_ne : ∀ (s : σ), {n | a s ^ n.succ ∈ I}.Nonempty
hn : (Function.support n).Finite
n₀ : σ →₀ ℕ := Finsupp.ofSupportFinite n hn
D : Set (σ →₀ ℕ) := Iic n₀
hD : D.Finite
this : ∀ᶠ (x : MvPolynomial σ R) in 𝓝 0, ∀ i ∈ D, φ (MvPolynomial.coeff i x) ∈ I
p : MvPolynomial σ R
hp : ∀ i ∈ D, φ (MvPolynomial.coeff i p) ∈ I
⊢ ∑ d ∈ p.support, φ (MvPolynomial.coeff d p) * ∏ i ∈ d.support, a i ^ d i ∈ I

Tactic:
]

Goal state after:
σ : Type u_1
R : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : UniformSpace R
S : Type u_3
inst✝⁴ : CommRing S
inst✝³ : UniformSpace S
φ : R →+* S
a : σ → S
inst✝² : IsUniformAddGroup R
inst✝¹ : IsUniformAddGroup S
inst✝ : IsLinearTopology S S
hφ : Continuous[inst✝⁵.toTopologicalSpace, inst✝³.toTopologicalSpace] ⇑φ
ha : HasEval a
I : Ideal S
hI : ↑I ∈ 𝓝 0
n : σ → ℕ := fun s ↦ sInf {n | a s ^ n.succ ∈ I}
hn_ne : ∀ (s : σ), {n | a s ^ n.succ ∈ I}.Nonempty
hn : (Function.support n).Finite
n₀ : σ →₀ ℕ := Finsupp.ofSupportFinite n hn
D : Set (σ →₀ ℕ) := Iic n₀
hD : D.Finite
this : ∀ᶠ (x : MvPolynomial σ R) in 𝓝 0, ∀ i ∈ D, φ (MvPolynomial.coeff i x) ∈ I
p : MvPolynomial σ R
hp : ∀ i ∈ D, φ (MvPolynomial.coeff i p) ∈ I
⊢ ∑ d ∈ p.support, φ (MvPolynomial.coeff d p) * ∏ i ∈ d.support, a i ^ d i ∈ I
Module: Mathlib.RingTheory.MvPowerSeries.LinearTopology

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Ring R
Jd : TwoSidedIdeal R × (σ →₀ ℕ)
f g : MvPowerSeries σ R
hg : g ∈ {f | ∀ e ≤ Jd.2, (coeff e) f ∈ Jd.1}
e : σ →₀ ℕ
he : e ≤ Jd.2
⊢ ∑ p ∈ Finset.antidiagonal e, (coeff p.1) f * (coeff p.2) g ∈ Jd.1

Tactic:
]

Goal state after:
σ : Type u_1
R : Type u_2
inst✝ : Ring R
Jd : TwoSidedIdeal R × (σ →₀ ℕ)
f g : MvPowerSeries σ R
hg : g ∈ {f | ∀ e ≤ Jd.2, (coeff e) f ∈ Jd.1}
e : σ →₀ ℕ
he : e ≤ Jd.2
⊢ ∑ p ∈ Finset.antidiagonal e, (coeff p.1) f * (coeff p.2) g ∈ Jd.1
Module: Mathlib.RingTheory.MvPowerSeries.LinearTopology

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Ring R
Jd : TwoSidedIdeal R × (σ →₀ ℕ)
f g : MvPowerSeries σ R
hf : f ∈ {f | ∀ e ≤ Jd.2, (coeff e) f ∈ Jd.1}
e : σ →₀ ℕ
he : e ≤ Jd.2
⊢ ∑ p ∈ Finset.antidiagonal e, (coeff p.1) f * (coeff p.2) g ∈ Jd.1

Tactic:
]

Goal state after:
σ : Type u_1
R : Type u_2
inst✝ : Ring R
Jd : TwoSidedIdeal R × (σ →₀ ℕ)
f g : MvPowerSeries σ R
hf : f ∈ {f | ∀ e ≤ Jd.2, (coeff e) f ∈ Jd.1}
e : σ →₀ ℕ
he : e ≤ Jd.2
⊢ ∑ p ∈ Finset.antidiagonal e, (coeff p.1) f * (coeff p.2) g ∈ Jd.1
Module: Mathlib.RingTheory.MvPowerSeries.LinearTopology

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Ring R
J K : TwoSidedIdeal R
d e : σ →₀ ℕ
hK : K ≠ ⊤
h : ∀ (a : MvPowerSeries σ R), (∀ e ≤ d, (coeff e) a ∈ J) → ∀ e_1 ≤ e, (coeff e_1) a ∈ K
x : R
hx : x ∈ ↑J
d' : σ →₀ ℕ
⊢ (if d' = 0 then x else 0) ∈ J

Tactic:
]

Goal state after:
σ : Type u_1
R : Type u_2
inst✝ : Ring R
J K : TwoSidedIdeal R
d e : σ →₀ ℕ
hK : K ≠ ⊤
h : ∀ (a : MvPowerSeries σ R), (∀ e ≤ d, (coeff e) a ∈ J) → ∀ e_1 ≤ e, (coeff e_1) a ∈ K
x : R
hx : x ∈ ↑J
d' : σ →₀ ℕ
⊢ (if d' = 0 then x else 0) ∈ J
Module: Mathlib.RingTheory.MvPowerSeries.LinearTopology

Goal state before:
pp: case mp.right
σ : Type u_1
R : Type u_2
inst✝ : Ring R
J K : TwoSidedIdeal R
d e : σ →₀ ℕ
hK : K ≠ ⊤
h : ∀ (a : MvPowerSeries σ R), (∀ e ≤ d, (coeff e) a ∈ J) → ∀ e_1 ≤ e, (coeff e_1) a ∈ K
h' : ¬e ≤ d
⊢ ⊤ ≤ K

Tactic:
]

Goal state after:
case mp.right
σ : Type u_1
R : Type u_2
inst✝ : Ring R
J K : TwoSidedIdeal R
d e : σ →₀ ℕ
hK : K ≠ ⊤
h : ∀ (a : MvPowerSeries σ R), (∀ e ≤ d, (coeff e) a ∈ J) → ∀ e_1 ≤ e, (coeff e_1) a ∈ K
h' : ¬e ≤ d
⊢ ⊤ ≤ K
Module: Mathlib.RingTheory.MvPowerSeries.LinearTopology

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Ring R
J K : TwoSidedIdeal R
d e : σ →₀ ℕ
hK : K ≠ ⊤
h : ∀ (a : MvPowerSeries σ R), (∀ e ≤ d, (coeff e) a ∈ J) → ∀ e_1 ≤ e, (coeff e_1) a ∈ K
h' : ¬e ≤ d
x : R
a✝ : x ∈ ⊤
d' : σ →₀ ℕ
hd'_le : d' ≤ d
⊢ (if d' = e then x else 0) ∈ J

Tactic:
]

Goal state after:
σ : Type u_1
R : Type u_2
inst✝ : Ring R
J K : TwoSidedIdeal R
d e : σ →₀ ℕ
hK : K ≠ ⊤
h : ∀ (a : MvPowerSeries σ R), (∀ e ≤ d, (coeff e) a ∈ J) → ∀ e_1 ≤ e, (coeff e_1) a ∈ K
h' : ¬e ≤ d
x : R
a✝ : x ∈ ⊤
d' : σ →₀ ℕ
hd'_le : d' ≤ d
⊢ (if d' = e then x else 0) ∈ J
Module: Mathlib.RingTheory.MvPowerSeries.LinearTopology

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝³ : Ring R
inst✝² : TopologicalSpace R
inst✝¹ : IsLinearTopology R R
inst✝ : IsLinearTopology Rᵐᵒᵖ R
⊢ (Filter.pi fun i ↦ 𝓝 (0 i)).HasBasis (fun Id ↦ ↑Id.1 ∈ 𝓝 0) fun Id ↦ ↑(basis σ R Id)

Tactic:
]

Goal state after:
σ : Type u_1
R : Type u_2
inst✝³ : Ring R
inst✝² : TopologicalSpace R
inst✝¹ : IsLinearTopology R R
inst✝ : IsLinearTopology Rᵐᵒᵖ R
⊢ (Filter.pi fun i ↦ 𝓝 (0 i)).HasBasis (fun Id ↦ ↑Id.1 ∈ 𝓝 0) fun Id ↦ ↑(basis σ R Id)
Module: Mathlib.RingTheory.MvPowerSeries.Evaluation

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝³ : CommRing R
inst✝² : UniformSpace R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : UniformSpace S
φ : R →+* S
a : σ → S
f : MvPolynomial σ R
this : ∃ p, ↑p = ↑f
⊢ MvPolynomial.eval₂ φ a this.choose = MvPolynomial.eval₂ φ a f

Tactic:
]

Goal state after:
σ : Type u_1
R : Type u_2
inst✝³ : CommRing R
inst✝² : UniformSpace R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : UniformSpace S
φ : R →+* S
a : σ → S
f : MvPolynomial σ R
this : ∃ p, ↑p = ↑f
⊢ MvPolynomial.eval₂ φ a this.choose = MvPolynomial.eval₂ φ a f
Module: Mathlib.RingTheory.MvPowerSeries.Evaluation

Goal state before:
pp: case e_p
σ : Type u_1
R : Type u_2
inst✝³ : CommRing R
inst✝² : UniformSpace R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : UniformSpace S
φ : R →+* S
a : σ → S
f : MvPolynomial σ R
this : ∃ p, ↑p = ↑f
⊢ ↑f = ↑f

Tactic:
]

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

Goal state before:
pp: case refine_1
σ : Type u_1
R : Type u_2
inst✝³ : Ring R
inst✝² : TopologicalSpace R
inst✝¹ : IsLinearTopology R R
inst✝ : IsLinearTopology Rᵐᵒᵖ R
D : Set (σ →₀ ℕ)
I : TwoSidedIdeal R
hD : (D, I).1.Finite
hI : ↑(D, I).2 ∈ 𝓝 0
f : (σ →₀ ℕ) → R
hf : ∀ e ≤ (I, hD.toFinset.sup id).2, (coeff e) f ∈ (I, hD.toFinset.sup id).1
d : σ →₀ ℕ
hd : d ∈ (D, I).1
⊢ f d ∈ (fun x ↦ ↑(D, I).2) d

Tactic:
]

Goal state after:
case refine_1
σ : Type u_1
R : Type u_2
inst✝³ : Ring R
inst✝² : TopologicalSpace R
inst✝¹ : IsLinearTopology R R
inst✝ : IsLinearTopology Rᵐᵒᵖ R
D : Set (σ →₀ ℕ)
I : TwoSidedIdeal R
hD : (D, I).1.Finite
hI : ↑(D, I).2 ∈ 𝓝 0
f : (σ →₀ ℕ) → R
hf : ∀ e ≤ (I, hD.toFinset.sup id).2, (coeff e) f ∈ (I, hD.toFinset.sup id).1
d : σ →₀ ℕ
hd : d ∈ (D, I).1
⊢ f d ∈ (fun x ↦ ↑(D, I).2) d
Module: Mathlib.RingTheory.MvPowerSeries.Evaluation

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝³ : CommRing R
inst✝² : UniformSpace R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : UniformSpace S
φ : R →+* S
a : σ → S
r : R
⊢ φ r = φ r

Tactic:
]

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

Goal state before:
pp: case refine_2
σ : Type u_1
R : Type u_2
inst✝³ : Ring R
inst✝² : TopologicalSpace R
inst✝¹ : IsLinearTopology R R
inst✝ : IsLinearTopology Rᵐᵒᵖ R
I : TwoSidedIdeal R
d : σ →₀ ℕ
⊢ ↑(I, d).1 ∈ 𝓝 0 → ∃ i, (i.1.Finite ∧ ↑i.2 ∈ 𝓝 0) ∧ (i.1.pi fun x ↦ ↑i.2) ⊆ ↑(basis σ R (I, d))

Tactic:
hI

Goal state after:
case refine_2
σ : Type u_1
R : Type u_2
inst✝³ : Ring R
inst✝² : TopologicalSpace R
inst✝¹ : IsLinearTopology R R
inst✝ : IsLinearTopology Rᵐᵒᵖ R
I : TwoSidedIdeal R
d : σ →₀ ℕ
hI : ↑(I, d).1 ∈ 𝓝 0
⊢ ∃ i, (i.1.Finite ∧ ↑i.2 ∈ 𝓝 0) ∧ (i.1.pi fun x ↦ ↑i.2) ⊆ ↑(basis σ R (I, d))
Module: Mathlib.RingTheory.MvPowerSeries.Evaluation

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝³ : CommRing R
inst✝² : UniformSpace R
S : Type u_3
inst✝¹ : CommRing S
inst✝ : UniformSpace S
φ : R →+* S
a : σ → S
s : σ
⊢ a s = a s

Tactic:
]

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

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝² : CommRing R
inst✝¹ : TopologicalSpace R
inst✝ : IsLinearTopology R R
f : MvPowerSeries σ R
hf : IsTopologicallyNilpotent (constantCoeff f)
d : σ →₀ ℕ
I : Ideal R
⊢ ↑I ∈ 𝓝 0 → ∀ᶠ (x : ℕ) in atTop, (coeff d) (f ^ x) ∈ ↑I

Tactic:
hI

Goal state after:
σ : Type u_1
R : Type u_3
inst✝² : CommRing R
inst✝¹ : TopologicalSpace R
inst✝ : IsLinearTopology R R
f : MvPowerSeries σ R
hf : IsTopologicallyNilpotent (constantCoeff f)
d : σ →₀ ℕ
I : Ideal R
hI : ↑I ∈ 𝓝 0
⊢ ∀ᶠ (x : ℕ) in atTop, (coeff d) (f ^ x) ∈ ↑I
Module: Mathlib.RingTheory.PowerSeries.Evaluation

Goal state before:
pp: S : Type u_2
inst✝¹ : CommRing S
inst✝ : TopologicalSpace S
⊢ MvPowerSeries.HasEval fun x ↦ 0

Tactic:
]

Goal state after:
S : Type u_2
inst✝¹ : CommRing S
inst✝ : TopologicalSpace S
⊢ MvPowerSeries.HasEval fun x ↦ 0
Module: Mathlib.RingTheory.PowerSeries.Evaluation

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : TopologicalSpace R
⊢ MvPowerSeries.HasEval fun x ↦ X

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : TopologicalSpace R
⊢ MvPowerSeries.HasEval fun x ↦ X
Module: Mathlib.RingTheory.MvPowerSeries.Evaluation

Goal state before:
pp: case neg
σ : Type u_1
R : Type u_2
inst✝¹⁰ : CommRing R
inst✝⁹ : UniformSpace R
S : Type u_3
inst✝⁸ : CommRing S
inst✝⁷ : UniformSpace S
φ : R →+* S
a : σ → S
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : IsUniformAddGroup R
inst✝⁴ : IsUniformAddGroup S
inst✝³ : CompleteSpace S
inst✝² : T2Space S
inst✝¹ : IsTopologicalRing S
inst✝ : IsLinearTopology S S
hφ : Continuous[inst✝⁹.toTopologicalSpace, inst✝⁷.toTopologicalSpace] ⇑φ
ha : HasEval a
f : MvPowerSeries σ R
h : ¬∃ p, ↑p = f
⊢ (eval₂Hom hφ ha) f = (eval₂Hom hφ ha) f

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.PowerSeries.Evaluation

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
S : Type u_2
inst✝² : CommRing S
φ : R →+* S
a : S
inst✝¹ : UniformSpace R
inst✝ : UniformSpace S
f : Polynomial R
⊢ eval₂ φ a ↑((MvPolynomial.uniqueAlgEquiv R Unit) ((MvPolynomial.uniqueAlgEquiv R Unit).symm f)) =
    Polynomial.eval₂ φ a ((MvPolynomial.uniqueAlgEquiv R Unit) ((MvPolynomial.uniqueAlgEquiv R Unit).symm f))

Tactic:
]

Goal state after:
R : Type u_1
inst✝³ : CommRing R
S : Type u_2
inst✝² : CommRing S
φ : R →+* S
a : S
inst✝¹ : UniformSpace R
inst✝ : UniformSpace S
f : Polynomial R
⊢ eval₂ φ a ↑((MvPolynomial.uniqueAlgEquiv R Unit) ((MvPolynomial.uniqueAlgEquiv R Unit).symm f)) =
    Polynomial.eval₂ φ a ((MvPolynomial.uniqueAlgEquiv R Unit) ((MvPolynomial.uniqueAlgEquiv R Unit).symm f))
Module: Mathlib.RingTheory.PowerSeries.Evaluation

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
S : Type u_2
inst✝² : CommRing S
φ : R →+* S
a : S
inst✝¹ : UniformSpace R
inst✝ : UniformSpace S
f : Polynomial R
⊢ MvPolynomial.eval₂ φ (fun x ↦ a) ((MvPolynomial.uniqueAlgEquiv R Unit).symm f) =
    MvPolynomial.eval₂ φ (fun x ↦ a) ((MvPolynomial.uniqueAlgEquiv R Unit).symm f)

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.PowerSeries.Evaluation

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
S : Type u_2
inst✝² : CommRing S
φ : R →+* S
a : S
inst✝¹ : UniformSpace R
inst✝ : UniformSpace S
r : R
⊢ φ r = φ r

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.PowerSeries.Evaluation

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
S : Type u_2
inst✝² : CommRing S
φ : R →+* S
a : S
inst✝¹ : UniformSpace R
inst✝ : UniformSpace S
⊢ a = a

Tactic:
]

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝¹⁰ : CommRing R
inst✝⁹ : UniformSpace R
S : Type u_3
inst✝⁸ : CommRing S
inst✝⁷ : UniformSpace S
φ : R →+* S
a : σ → S
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : IsUniformAddGroup R
inst✝⁴ : IsUniformAddGroup S
inst✝³ : CompleteSpace S
inst✝² : T2Space S
inst✝¹ : IsTopologicalRing S
inst✝ : IsLinearTopology S S
hφ : Continuous[inst✝⁹.toTopologicalSpace, inst✝⁷.toTopologicalSpace] ⇑φ
ha : HasEval a
⊢ UniformContinuous ⇑(eval₂Hom hφ ha)

Tactic:
]

Goal state after:
σ : Type u_1
R : Type u_2
inst✝¹⁰ : CommRing R
inst✝⁹ : UniformSpace R
S : Type u_3
inst✝⁸ : CommRing S
inst✝⁷ : UniformSpace S
φ : R →+* S
a : σ → S
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : IsUniformAddGroup R
inst✝⁴ : IsUniformAddGroup S
inst✝³ : CompleteSpace S
inst✝² : T2Space S
inst✝¹ : IsTopologicalRing S
inst✝ : IsLinearTopology S S
hφ : Continuous[inst✝⁹.toTopologicalSpace, inst✝⁷.toTopologicalSpace] ⇑φ
ha : HasEval a
⊢ UniformContinuous ⇑(eval₂Hom hφ ha)