mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction

Goal state before:
pp: ι : Type u_1
R : Type u_2
inst✝ : CommSemiring R
f : ι → ArithmeticFunction R
hf : ∀ (i : ι), (f i).IsMultiplicative
⊢ (eulerProduct f).IsMultiplicative

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.EGauge

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
inst✝² : NormedDivisionRing 𝕜
inst✝¹ : (i : ι) → AddCommGroup (E i)
inst✝ : (i : ι) → Module 𝕜 (E i)
I : Set ι
hI : I.Finite
U : (i : ι) → Set (E i)
hU : ∀ i ∈ I, Balanced 𝕜 (U i)
x : (i : ι) → E i
hI₀ : I = univ ∨ (∃ i ∈ I, x i ≠ 0) ∨ (𝓝[≠] 0).NeBot
r : ℝ≥0∞
hr : ⨆ i ∈ I, egauge 𝕜 (U i) (x i) < r
c : ι → 𝕜
hc : ∀ i ∈ I, x i ∈ c i • U i
hcr : ∀ i ∈ I, ‖c i‖ₑ < r
hr₀ : 0 < r
hIne : I.Nonempty
i₀ : ι
hi₀I : i₀ ∈ I
hc_max : IsMaxOn (fun x ↦ ‖c x‖ₑ) I i₀
H : c i₀ = 0 ∧ I ≠ univ
hc0 : ∀ i ∈ I, c i = 0
heg0 : ∀ i ∈ I, x i = 0
⊢ ¬∃ i ∈ I, x i ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction

Goal state before:
pp: ι : Type u_1
R : Type u_2
inst✝ : CommSemiring R
q : ι → ℕ
hq : Northcott q
f : ι → PowerSeries R
hf : ∀ (i : ι), PowerSeries.constantCoeff (f i) = 1
n✝ : ℕ
i : ι
n : ℕ
hi : n + 1 + 1 + 1 ≤ q i
⊢ 1 < q i

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction

Goal state before:
pp: ι : Type u_1
R : Type u_2
inst✝ : CommSemiring R
q : ι → ℕ
hq : Northcott q
f : ι → PowerSeries R
hf : ∀ (i : ι), PowerSeries.constantCoeff (f i) = 1
n✝ : ℕ
i : ι
n : ℕ
hi : n + 1 + 1 + 1 ≤ q i
hqi : 1 < q i
⊢ n + 1 + 1 ≠ 1

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction

Goal state before:
pp: ι : Type u_1
R : Type u_2
inst✝ : CommSemiring R
q : ι → ℕ
hq : Northcott q
f : ι → PowerSeries R
hf : ∀ (i : ι), PowerSeries.constantCoeff (f i) = 1
n✝ : ℕ
i : ι
n : ℕ
hi : n + 1 + 1 + 1 ≤ q i
hqi : 1 < q i
k : ℕ
hk : q i ^ k = n + 1 + 1
h : k = 0
⊢ False

Tactic:
by

Goal state after:
Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction

Goal state before:
pp: ι : Type u_1
R : Type u_2
inst✝ : CommSemiring R
q : ι → ℕ
hq : Northcott q
f : ι → PowerSeries R
hf : ∀ (i : ι), PowerSeries.constantCoeff (f i) = 1
n : ℕ
⊢ ∀ᶠ (s : Finset ι) in atTop,
    (∏ i ∈ s, (ofPowerSeries (q i)) (f i)) n = (eulerProduct fun i ↦ (ofPowerSeries (q i)) (f i)) n

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace E
F : Type u_3
inst✝² : AddCommGroup F
inst✝¹ : Module 𝕜 F
inst✝ : TopologicalSpace F
f : E → F
f' : E →L[𝕜] F
x : E
⊢ HasFDerivAt f f' x ↔ (fun x' ↦ f x' - f x - f' (x' - x)) =o[𝕜; 𝓝 x] fun x' ↦ x' - x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace E
F : Type u_3
inst✝² : AddCommGroup F
inst✝¹ : Module 𝕜 F
inst✝ : TopologicalSpace F
f : E → F
f' : E →L[𝕜] F
x : E
s : Set E
⊢ HasFDerivWithinAt f f' s x ↔ (fun x' ↦ f x' - f x - f' (x' - x)) =o[𝕜; 𝓝[s] x] fun x' ↦ x' - x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace E
F : Type u_3
inst✝² : AddCommGroup F
inst✝¹ : Module 𝕜 F
inst✝ : TopologicalSpace F
f : E → F
x : E
s : Set E
h : ¬DifferentiableWithinAt 𝕜 f s x
⊢ fderivWithin 𝕜 f s x = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace E
F : Type u_3
inst✝² : AddCommGroup F
inst✝¹ : Module 𝕜 F
inst✝ : TopologicalSpace F
f : E → F
⊢ fderivWithin 𝕜 f univ = fderiv 𝕜 f

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.TangentCone.Defs

Goal state before:
pp: R : Type u
E : Type v
inst✝² : AddCommGroup E
inst✝¹ : SMul R E
inst✝ : TopologicalSpace E
s : Set E
x y : E
α : Type u_1
l : Filter α
c : α → R
d : α → E
hd₀ : Tendsto d l (𝓝 0)
hds : ∃ᶠ (n : α) in l, x + d n ∈ s
hcd : Tendsto (fun n ↦ c n • d n) l (𝓝 y)
this : Tendsto (fun n ↦ c n • d n) (l ⊓ 𝓟 {y | x + d y ∈ s}) (⊤ • 𝓝[(fun x_1 ↦ x + x_1) ⁻¹' s] 0)
⊢ y ∈ tangentConeAt R s x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.EGauge

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
inst✝² : NormedDivisionRing 𝕜
inst✝¹ : (i : ι) → AddCommGroup (E i)
inst✝ : (i : ι) → Module 𝕜 (E i)
I : Set ι
hI : I.Finite
U : (i : ι) → Set (E i)
hU : ∀ i ∈ I, Balanced 𝕜 (U i)
x : (i : ι) → E i
hI₀ : I = univ ∨ (∃ i ∈ I, x i ≠ 0) ∨ (𝓝[≠] 0).NeBot
r : ℝ≥0∞
hr : ⨆ i ∈ I, egauge 𝕜 (U i) (x i) < r
c : ι → 𝕜
hc : ∀ i ∈ I, x i ∈ c i • U i
hcr : ∀ i ∈ I, ‖c i‖ₑ < r
⊢ ∃ c₀, (c₀ ≠ 0 ∨ I = univ) ∧ (∀ i ∈ I, ‖c i‖ ≤ ‖c₀‖) ∧ ‖c₀‖ₑ < r

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.TangentCone.Defs

Goal state before:
pp: R : Type u
E : Type v
inst✝² : AddCommGroup E
inst✝¹ : SMul R E
inst✝ : TopologicalSpace E
s : Set E
x y : E
α : Type u_1
l : Filter α
c : α → R
d : α → E
hd₀ : Tendsto d l (𝓝 0)
hds : ∃ᶠ (n : α) in l, x + d n ∈ s
hcd : Tendsto (fun n ↦ c n • d n) l (𝓝 y)
⊢ y ∈ tangentConeAt R s x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.EGauge

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
inst✝² : NormedDivisionRing 𝕜
inst✝¹ : (i : ι) → AddCommGroup (E i)
inst✝ : (i : ι) → Module 𝕜 (E i)
I : Set ι
hI : I.Finite
U : (i : ι) → Set (E i)
hU : ∀ i ∈ I, Balanced 𝕜 (U i)
x : (i : ι) → E i
hI₀ : I = univ ∨ (∃ i ∈ I, x i ≠ 0) ∨ (𝓝[≠] 0).NeBot
⊢ egauge 𝕜 (I.pi U) x = ⨆ i ∈ I, egauge 𝕜 (U i) (x i)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.EGauge

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
inst✝³ : NormedDivisionRing 𝕜
inst✝² : (i : ι) → AddCommGroup (E i)
inst✝¹ : (i : ι) → Module 𝕜 (E i)
inst✝ : Finite ι
U : (i : ι) → Set (E i)
hU : ∀ (i : ι), Balanced 𝕜 (U i)
x : (i : ι) → E i
⊢ ⨆ i ∈ univ, egauge 𝕜 (U i) (x i) = ⨆ i, egauge 𝕜 (U i) (x i)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.EGauge

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
r : ℝ≥0
x : E
⊢ ‖x‖ₑ / ↑r ≤ egauge 𝕜 (closedBall 0 ↑r) x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.EGauge

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
⊢ ‖x‖ₑ ≤ egauge 𝕜 (closedBall 0 1) x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.EGauge

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
⊢ ‖x‖ₑ ≤ egauge 𝕜 (ball 0 1) x

Tactic:
by

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

Goal state before:
pp: R : Type u_2
inst✝¹ : CommRing R
P : R⟦X⟧
hP : constantCoeff P = 0
inst✝ : Invertible ((coeff 1) P)
n : ℕ
B : R⟦X⟧
hB : ∑ i, C (P.substInvFun ↑i) * X ^ ↑i = B
hB' : constantCoeff B = 0
k : R
hk : ⅟((coeff 1) P) * (coeff (n + 1 + 1)) (subst B P) = k
i : ℕ
⊢ X ∣ B

Tactic:
by

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

Goal state before:
pp: R : Type u_2
inst✝¹ : CommRing R
P : R⟦X⟧
hP : constantCoeff P = 0
inst✝ : Invertible ((coeff 1) P)
n : ℕ
B : R⟦X⟧
hB : ∑ i, C (P.substInvFun ↑i) * X ^ ↑i = B
hB' : constantCoeff B = 0
k : R
hk : ⅟((coeff 1) P) * (coeff (n + 1 + 1)) (subst B P) = k
x✝ : ℕ
⊢ (x✝ ∈ Function.support fun d ↦ (coeff d) P * if d = 1 then k else 0) → x✝ ∈ {1}

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.TangentCone.Defs

Goal state before:
pp: R : Type u
E : Type v
inst✝² : AddCommGroup E
inst✝¹ : SMul R E
inst✝ : TopologicalSpace E
s : Set E
x y : E
h : y ∈ tangentConeAt R s x
⊢ ∃ α l,
    ∃ (_ : l.NeBot), ∃ c d, Tendsto d l (𝓝 0) ∧ (∀ᶠ (n : α) in l, x + d n ∈ s) ∧ Tendsto (fun n ↦ c n • d n) l (𝓝 y)

Tactic:
by

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

Goal state before:
pp: R : Type u_2
inst✝¹ : CommRing R
P : R⟦X⟧
hP : constantCoeff P = 0
inst✝ : Invertible ((coeff 1) P)
n : ℕ
⊢ (coeff n) (subst (∑ i, C (P.substInvFun ↑i) * X ^ ↑i) P - X) = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.EGauge

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
x : E
r : ℝ≥0
hc : 1 < ‖c‖
h₀ : r ≠ 0 ∨ ‖x‖ ≠ 0
this : NontriviallyNormedField 𝕜 := { toNormedField := inst✝², non_trivial := ⋯ }
hr : 0 < r
hx : ‖x‖ = 0
⊢ ‖x‖ₑ = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.EGauge

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
x : E
r : ℝ≥0
hc : 1 < ‖c‖
h₀ : r ≠ 0 ∨ ‖x‖ ≠ 0
this : NontriviallyNormedField 𝕜 := { toNormedField := inst✝², non_trivial := ⋯ }
hr : 0 < r
hx : ‖x‖ ≠ 0
a : 𝕜
ha₀ : a ≠ 0
har : ‖a • x‖ < ↑r
hainv : ‖a‖⁻¹ ≤ (↑r)⁻¹ * ‖c‖ * ‖x‖
⊢ ↑‖a‖₊⁻¹ ≤ ↑(‖c‖₊ * ‖x‖₊ / r)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.EGauge

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
x : E
r : ℝ≥0
hc : 1 < ‖c‖
h₀ : r ≠ 0 ∨ ‖x‖ ≠ 0
this : NontriviallyNormedField 𝕜 := { toNormedField := inst✝², non_trivial := ⋯ }
hr : 0 < r
hx : ‖x‖ ≠ 0
a : 𝕜
ha₀ : a ≠ 0
har : ‖a • x‖ < ↑r
hainv : ‖a‖⁻¹ ≤ (↑r)⁻¹ * ‖c‖ * ‖x‖
⊢ ↑(‖c‖₊ * ‖x‖₊) / ↑r ≤ ‖c‖ₑ * ‖x‖ₑ / ↑r

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.EGauge

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
x : E
r : ℝ≥0
hc : 1 < ‖c‖
h₀ : r ≠ 0 ∨ ‖x‖ ≠ 0
⊢ egauge 𝕜 (ball 0 ↑r) x ≤ ‖c‖ₑ * ‖x‖ₑ / ↑r

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Convex.EGauge

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
hc : 1 < ‖c‖
x : E
⊢ egauge 𝕜 (ball 0 1) x ≤ ‖c‖ₑ * ‖x‖ₑ

Tactic:
by

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

Goal state before:
pp: R : Type u_2
inst✝¹ : CommRing R
P : R⟦X⟧
hP : constantCoeff P = 0
inst✝ : Invertible ((coeff 1) P)
n : ℕ
this : (coeff n) (subst (∑ i, C (P.substInvFun ↑i) * X ^ ↑i) P) = (coeff n) X
m : ℕ
B : R⟦X⟧
hB : ∑ i, C (P.substInvFun ↑i) * X ^ ↑i = B
⊢ X ^ (n + 1) ∣ mk P.substInvFun - B

Tactic:
by

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

Goal state before:
pp: case hb
R : Type u_2
inst✝¹ : CommRing R
P : R⟦X⟧
hP : constantCoeff P = 0
inst✝ : Invertible ((coeff 1) P)
n : ℕ
this : (coeff n) (subst (∑ i, C (P.substInvFun ↑i) * X ^ ↑i) P) = (coeff n) X
⊢ HasSubst (∑ i, C (P.substInvFun ↑i) * X ^ ↑i)

pp: case hb
R : Type u_2
inst✝¹ : CommRing R
P : R⟦X⟧
hP : constantCoeff P = 0
inst✝ : Invertible ((coeff 1) P)
n : ℕ
this : (coeff n) (subst (∑ i, C (P.substInvFun ↑i) * X ^ ↑i) P) = (coeff n) X
⊢ HasSubst P.substInv

Tactic:
· simp [HasSubst, X, zero_pow_eq, C, substInvFun]

Goal state after:
case hb
R : Type u_2
inst✝¹ : CommRing R
P : R⟦X⟧
hP : constantCoeff P = 0
inst✝ : Invertible ((coeff 1) P)
n : ℕ
this : (coeff n) (subst (∑ i, C (P.substInvFun ↑i) * X ^ ↑i) P) = (coeff n) X
⊢ HasSubst P.substInv
Module: Mathlib.RingTheory.PowerSeries.Substitution

Goal state before:
pp: R : Type u_2
inst✝¹ : CommRing R
P : R⟦X⟧
hP : constantCoeff P = 0
inst✝ : Invertible ((coeff 1) P)
⊢ subst P.substInv P = X

Tactic:
by

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

Goal state before:
pp: R : Type u_2
inst✝¹ : CommRing R
P : R⟦X⟧
inst✝ : Invertible ((coeff 1) P)
⊢ constantCoeff P.substInv = 0

Tactic:
by

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

Goal state before:
pp: R : Type u_2
inst✝¹ : CommRing R
P : R⟦X⟧
inst✝ : Invertible ((coeff 1) P)
⊢ HasSubst P.substInv

Tactic:
by

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

Goal state before:
pp: R : Type u_2
inst✝¹ : CommRing R
P : R⟦X⟧
inst✝ : Invertible ((coeff 1) P)
⊢ (coeff 1) P.substInv = ⅟((coeff 1) P)

Tactic:
by

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

Goal state before:
pp: R : Type u_2
inst✝¹ : CommRing R
P : R⟦X⟧
hP : constantCoeff P = 0
inst✝ : Invertible ((coeff 1) P)
⊢ Invertible ((coeff 1) P.substInv)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.TangentCone.DimOne

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NormedDivisionRing 𝕜
s : Set 𝕜
x : 𝕜
hx : AccPt x (𝓟 s)
y : 𝕜
⊢ Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace]
    fun x_1 ↦ x_1 - x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.TangentCone.DimOne

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NormedDivisionRing 𝕜
s : Set 𝕜
x : 𝕜
hx : AccPt x (𝓟 s)
y : 𝕜
⊢ x - x = 0

Tactic:
by

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

Goal state before:
pp: R : Type u_2
inst✝¹ : CommRing R
P : R⟦X⟧
hP : constantCoeff P = 0
inst✝ : Invertible ((coeff 1) P)
this : Invertible ((coeff 1) P.substInv)
Q : R⟦X⟧ := P.substInv.substInv
hQ : HasSubst Q
eq_aux : subst Q P.substInv = X
h : Q = P
⊢ subst P P.substInv = X

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.TangentCone.DimOne

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NormedDivisionRing 𝕜
s : Set 𝕜
x : 𝕜
hx : AccPt x (𝓟 s)
y z : 𝕜
hz : z ∈ {x}ᶜ
⊢ z - x ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.TangentCone.DimOne

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NormedDivisionRing 𝕜
s : Set 𝕜
x : 𝕜
hx : AccPt x (𝓟 s)
⊢ tangentConeAt 𝕜 s x = univ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.TangentCone.DimOne

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NormedDivisionRing 𝕜
s : Set 𝕜
x : 𝕜
h : AccPt x (𝓟 s)
⊢ Dense ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 s x))

Tactic:
by

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

Goal state before:
pp: R : Type u_2
inst✝¹ : CommRing R
P : R⟦X⟧
hP : constantCoeff P = 0
inst✝ : Invertible ((coeff 1) P)
this : Invertible ((coeff 1) P.substInv)
Q : R⟦X⟧ := P.substInv.substInv
hQ : HasSubst Q
eq_aux : subst Q P.substInv = X
⊢ Q = subst Q (subst P.substInv P)

Tactic:
by

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

Goal state before:
pp: R : Type u_2
inst✝¹ : CommRing R
P : R⟦X⟧
hP : constantCoeff P = 0
inst✝ : Invertible ((coeff 1) P)
this : Invertible ((coeff 1) P.substInv)
Q : R⟦X⟧ := P.substInv.substInv
hQ : HasSubst Q
eq_aux : subst Q P.substInv = X
⊢ subst Q (subst P.substInv P) = P

Tactic:
by

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

Goal state before:
pp: R : Type u_2
inst✝¹ : CommRing R
P : R⟦X⟧
hP : constantCoeff P = 0
inst✝ : Invertible ((coeff 1) P)
⊢ subst P P.substInv = X

Tactic:
by

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

Goal state before:
pp: R : Type u_2
inst✝ : CommRing R
P : R⟦X⟧
hP' : IsUnit ((coeff 1) P)
⊢ constantCoeff (P.substInvOfIsUnit hP') = 0

Tactic:
by

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

Goal state before:
pp: R : Type u_2
inst✝ : CommRing R
P : R⟦X⟧
hP' : IsUnit ((coeff 1) P)
⊢ HasSubst (P.substInvOfIsUnit hP')

Tactic:
by

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

Goal state before:
pp: R : Type u_2
inst✝ : CommRing R
P : R⟦X⟧
hP' : IsUnit ((coeff 1) P)
⊢ (coeff 1) (P.substInvOfIsUnit hP') = ↑hP'.unit⁻¹

Tactic:
by

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

Goal state before:
pp: R : Type u_2
inst✝ : CommRing R
P : R⟦X⟧
hP : constantCoeff P = 0
hP' : IsUnit ((coeff 1) P)
⊢ subst (P.substInvOfIsUnit hP') P = X

Tactic:
by

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

Goal state before:
pp: R : Type u_2
inst✝ : CommRing R
P : R⟦X⟧
hP : constantCoeff P = 0
hP' : IsUnit ((coeff 1) P)
⊢ subst P (P.substInvOfIsUnit hP') = X

Tactic:
by

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

Goal state before:
pp: R : Type u_2
inst✝⁶ : CommRing R
τ : Type u_3
S : Type u_4
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : ℕ → R⟦X⟧
a : MvPowerSeries τ S
inst✝³ : UniformSpace R
inst✝² : DiscreteUniformity R
inst✝¹ : UniformSpace S
inst✝ : DiscreteUniformity S
hx : Summable x
ha : HasSubst a
⊢ subst a (∑' (i : ℕ), x i) = ∑' (i : ℕ), subst a (x i)

Tactic:
by

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

Goal state before:
pp: R : Type u_2
inst✝⁶ : CommRing R
τ : Type u_3
S : Type u_4
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : ℕ → R⟦X⟧
a : MvPowerSeries τ S
inst✝³ : UniformSpace R
inst✝² : DiscreteUniformity R
inst✝¹ : UniformSpace S
inst✝ : DiscreteUniformity S
hx : Summable x
ha : HasSubst a
⊢ Summable fun i ↦ subst a (x i)

Tactic:
by

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

Goal state before:
pp: R : Type u_2
inst✝ : CommRing R
f : R⟦X⟧
e : Fin 2 →₀ ℕ
x✝ : Unit
⊢ MvPowerSeries.constantCoeff (X₀ + X₁) = 0

Tactic:
by

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

Goal state before:
pp: R : Type u_2
inst✝ : CommRing R
f : R⟦X⟧
e : Fin 2 →₀ ℕ
⊢ (MvPowerSeries.coeff e) (subst (X₀ + X₁) f) = ↑((e 0 + e 1).choose (e 0)) * (coeff (e 0 + e 1)) f

Tactic:
by

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

Goal state before:
pp: R : Type u_2
inst✝ : CommRing R
f : R⟦X⟧
e : Fin 2 →₀ ℕ
he : (single 0 (e 0), single 1 (e 1)) ∉ antidiagonal e
⊢ single 0 (e 0) + single 1 (e 1) = e

Tactic:
by

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

Goal state before:
pp: R : Type u_2
inst✝ : CommRing R
f : R⟦X⟧
e : Fin 2 →₀ ℕ
⊢ (MvPowerSeries.coeff e) (subst X₀ f * subst X₁ f) = (coeff (e 0)) f * (coeff (e 1)) f

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.TangentCone.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : AddCommGroup E
inst✝¹ : SMul 𝕜 E
inst✝ : TopologicalSpace E
s t : Set E
x : E
h : s ⊆ t
⊢ tangentConeAt 𝕜 s x ⊆ tangentConeAt 𝕜 t x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.TangentCone.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁶ : AddCommGroup E
inst✝⁵ : SMul 𝕜 E
inst✝⁴ : TopologicalSpace E
s : Set E
x : E
𝕜' : Type u_3
inst✝³ : Monoid 𝕜'
inst✝² : SMul 𝕜 𝕜'
inst✝¹ : MulAction 𝕜' E
inst✝ : IsScalarTower 𝕜 𝕜' E
⊢ tangentConeAt 𝕜 s x ⊆ tangentConeAt 𝕜' s x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.TangentCone.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : AddCommGroup E
inst✝¹ : SMul 𝕜 E
inst✝ : TopologicalSpace E
s : Set E
x : E
ι : Sort u_3
p : ι → Prop
U : ι → Set E
h : (𝓝 0).HasBasis p U
⊢ tangentConeAt 𝕜 s x = ⋂ i, ⋂ (_ : p i), closure[inst✝] (univ • (U i ∩ (fun x_2 ↦ x + x_2) ⁻¹' s))

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.TangentCone.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : SMul 𝕜 E
inst✝¹ : TopologicalSpace E
s t : Set E
x : E
inst✝ : ContinuousAdd E
h : 𝓝[s] x ≤ 𝓝[t] x
⊢ tangentConeAt 𝕜 s x ⊆ tangentConeAt 𝕜 t x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.TangentCone.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : SMul 𝕜 E
inst✝¹ : TopologicalSpace E
s : Set E
x : E
inst✝ : ContinuousAdd E
h : (tangentConeAt 𝕜 s x).Nonempty
⊢ x ∈ closure[inst✝¹] s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.TangentCone.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : AddCommGroup E
inst✝³ : SMul 𝕜 E
inst✝² : TopologicalSpace E
s : Set E
x : E
inst✝¹ : ContinuousAdd E
inst✝ : ContinuousConstSMul 𝕜 E
⊢ tangentConeAt 𝕜 (closure[inst✝²] s) x = tangentConeAt 𝕜 s x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.TangentCone.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : Semiring 𝕜
inst✝¹ : Module 𝕜 E
inst✝ : TopologicalSpace E
s t : Set E
x : E
h : UniqueDiffWithinAt 𝕜 s x
st : s ⊆ t
⊢ UniqueDiffWithinAt 𝕜 t x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.TangentCone.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : AddCommGroup E
inst✝³ : Semiring 𝕜
inst✝² : Module 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousAdd E
s t : Set E
x : E
h : UniqueDiffWithinAt 𝕜 s x
st : 𝓝[s] x ≤ 𝓝[t] x
⊢ UniqueDiffWithinAt 𝕜 t x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.TangentCone.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : AddCommGroup E
inst✝³ : Semiring 𝕜
inst✝² : Module 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousAdd E
s : Set E
x : E
hx : ∃ᶠ (x : E) in 𝓝 x, x ∈ s
⊢ Continuous[inst✝¹, inst✝¹] fun x_1 ↦ x_1 + -x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.TangentCone.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : AddCommGroup E
inst✝³ : Semiring 𝕜
inst✝² : Module 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousAdd E
s : Set E
x : E
hx : ∃ᶠ (x : E) in 𝓝 x, x ∈ s
⊢ x + -x = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.TangentCone.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : AddCommGroup E
inst✝³ : Semiring 𝕜
inst✝² : Module 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousAdd E
s : Set E
x : E
hx : ∃ᶠ (x : E) in 𝓝 x, x ∈ s
⊢ Continuous[inst✝¹, inst✝¹] fun n ↦ n + -x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.TangentCone.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : AddCommGroup E
inst✝³ : Semiring 𝕜
inst✝² : Module 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousAdd E
s : Set E
x : E
hx : x ∈ closure[inst✝¹] s
⊢ 0 ∈ tangentConeAt 𝕜 s x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.TangentCone.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : AddCommGroup E
inst✝⁴ : Semiring 𝕜
inst✝³ : Module 𝕜 E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
s : Set E
x : E
inst✝ : T2Space E
hx : ¬AccPt x (𝓟 s)
y : E
hy : y ∈ tangentConeAt 𝕜 s x
ι : Type (max u_1 u_2)
l : Filter ι
hl : l.NeBot
c : ι → 𝕜
d : ι → E
hd₀ : Tendsto d l (𝓝 0)
hds : ∀ᶠ (n : ι) in l, x + d n ∈ s
hcd : Tendsto (fun n ↦ c n • d n) l (𝓝 y)
⊢ Tendsto (fun x_1 ↦ x + d x_1) l (𝓝 x)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.TangentCone.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : AddCommGroup E
inst✝⁴ : Semiring 𝕜
inst✝³ : Module 𝕜 E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
s : Set E
x : E
inst✝ : T2Space E
hx : ¬AccPt x (𝓟 s)
y : E
hy : y ∈ tangentConeAt 𝕜 s x
ι : Type (max u_1 u_2)
l : Filter ι
hl : l.NeBot
c : ι → 𝕜
d : ι → E
hd₀ : Tendsto d l (𝓝 0)
hds : ∀ᶠ (n : ι) in l, x + d n ∈ s
hcd : Tendsto (fun n ↦ c n • d n) l (𝓝 y)
H₁ : Tendsto (fun x_1 ↦ x + d x_1) l (𝓝 x)
⊢ ∀ᶠ (n : ι) in l, d n = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.TangentCone.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : AddCommGroup E
inst✝⁴ : Semiring 𝕜
inst✝³ : Module 𝕜 E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
s : Set E
x : E
inst✝ : T2Space E
hx : ¬AccPt x (𝓟 s)
y : E
hy : y ∈ tangentConeAt 𝕜 s x
ι : Type (max u_1 u_2)
l : Filter ι
hl : l.NeBot
c : ι → 𝕜
d : ι → E
hd₀ : Tendsto d l (𝓝 0)
hds : ∀ᶠ (n : ι) in l, x + d n ∈ s
hcd : Tendsto (fun n ↦ c n • d n) l (𝓝 y)
H₁ : Tendsto (fun x_1 ↦ x + d x_1) l (𝓝 x)
H₂ : ∀ᶠ (n : ι) in l, d n = 0
n : ι
hn : d n = 0
⊢ c n • d n = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.TangentCone.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : AddCommGroup E
inst✝⁴ : Semiring 𝕜
inst✝³ : Module 𝕜 E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
s : Set E
x : E
inst✝ : T2Space E
hx : ¬AccPt x (𝓟 s)
⊢ tangentConeAt 𝕜 s x ⊆ 0

Tactic:
by

Goal state after: