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

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝ : CommRing R
a : σ → R
f : MvPowerSeries σ R
n : σ →₀ ℕ
⊢ (n.prod fun s m ↦ a s ^ m) * (coeff n) f = (coeff n) f * (coeff n) (n.prod fun s e ↦ (a s • X s) ^ e)

pp: σ : Type u_1
R : Type u_3
inst✝ : CommRing R
a : σ → R
f : MvPowerSeries σ R
n : σ →₀ ℕ
⊢ ∀ b ∈ ⋯.toFinset, b ≠ n → (coeff b) f * (coeff n) (b.prod fun s e ↦ (a s • X s) ^ e) = 0

pp: σ : Type u_1
R : Type u_3
inst✝ : CommRing R
a : σ → R
f : MvPowerSeries σ R
n : σ →₀ ℕ
⊢ n ∉ ⋯.toFinset → (coeff n) f * (coeff n) (n.prod fun s e ↦ (a s • X s) ^ e) = 0

Tactic:
]

Goal state after:
σ : Type u_1
R : Type u_3
inst✝ : CommRing R
a : σ → R
f : MvPowerSeries σ R
n : σ →₀ ℕ
⊢ (n.prod fun s m ↦ a s ^ m) * (coeff n) f = (coeff n) f * (coeff n) (n.prod fun s e ↦ (a s • X s) ^ e)

σ : Type u_1
R : Type u_3
inst✝ : CommRing R
a : σ → R
f : MvPowerSeries σ R
n : σ →₀ ℕ
⊢ ∀ b ∈ ⋯.toFinset, b ≠ n → (coeff b) f * (coeff n) (b.prod fun s e ↦ (a s • X s) ^ e) = 0

σ : Type u_1
R : Type u_3
inst✝ : CommRing R
a : σ → R
f : MvPowerSeries σ R
n : σ →₀ ℕ
⊢ n ∉ ⋯.toFinset → (coeff n) f * (coeff n) (n.prod fun s e ↦ (a s • X s) ^ e) = 0
Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction

Goal state before:
pp: ι : Type u_1
R : Type u_2
inst✝ : CommSemiring R
f : ι → ArithmeticFunction R
this✝ : UniformSpace R := ⊥
this : IsUniformInducing DFunLike.coe
n : ℕ
s : Set ι
hs : s ∈ cofinite
hs' : ∀ y ∈ s, ∀ i ∈ Set.Iic n, (f y) i = 1 i
t : Finset ι := ⋯.toFinset
u v : Finset ι
hu : ∀ i ∈ u \ t, i ∈ s
hv : ∀ i ∈ v \ t, i ∈ s
i : ι
w : Finset ι
hi : i ∉ w
hw' : (∀ i ∈ w, i ∈ s) → ∀ k ≤ n, (∏ x ∈ w, f x) k = 1 k
hw : ∀ i_1 ∈ insert i w, i_1 ∈ s
k : ℕ
hk : k ≤ n
⊢ ∑ x ∈ k.divisorsAntidiagonal, (f i) x.1 * (∏ x ∈ w, f x) x.2 = ∑ x ∈ k.divisorsAntidiagonal, 1 x.1 * 1 x.2

Tactic:
]

Goal state after:
ι : Type u_1
R : Type u_2
inst✝ : CommSemiring R
f : ι → ArithmeticFunction R
this✝ : UniformSpace R := ⊥
this : IsUniformInducing DFunLike.coe
n : ℕ
s : Set ι
hs : s ∈ cofinite
hs' : ∀ y ∈ s, ∀ i ∈ Set.Iic n, (f y) i = 1 i
t : Finset ι := ⋯.toFinset
u v : Finset ι
hu : ∀ i ∈ u \ t, i ∈ s
hv : ∀ i ∈ v \ t, i ∈ s
i : ι
w : Finset ι
hi : i ∉ w
hw' : (∀ i ∈ w, i ∈ s) → ∀ k ≤ n, (∏ x ∈ w, f x) k = 1 k
hw : ∀ i_1 ∈ insert i w, i_1 ∈ s
k : ℕ
hk : k ≤ n
⊢ ∑ x ∈ k.divisorsAntidiagonal, (f i) x.1 * (∏ x ∈ w, f x) x.2 = ∑ x ∈ k.divisorsAntidiagonal, 1 x.1 * 1 x.2
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝ : CommRing R
a : σ → R
f : MvPowerSeries σ R
n b : σ →₀ ℕ
hb : b ∈ ⋯.toFinset
hbn : b ≠ n
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: ι : Type u_1
R : Type u_2
inst✝ : CommSemiring R
f : ι → ArithmeticFunction R
this✝ : UniformSpace R := ⊥
this : IsUniformInducing DFunLike.coe
n : ℕ
s : Set ι
hs : s ∈ cofinite
hs' : ∀ y ∈ s, ∀ i ∈ Set.Iic n, (f y) i = 1 i
t : Finset ι := ⋯.toFinset
u v : Finset ι
hu : ∀ i ∈ u \ t, i ∈ s
hv : ∀ i ∈ v \ t, i ∈ s
i : ι
w : Finset ι
hi : i ∉ w
hw' : (∀ i ∈ w, i ∈ s) → ∀ k ≤ n, (∏ x ∈ w, f x) k = 1 k
hw : ∀ i_1 ∈ insert i w, i_1 ∈ s
k : ℕ
hk : k ≤ n
j : ℕ × ℕ
hj : j ∈ k.divisorsAntidiagonal
h1 : (f i) j.1 = 1 j.1
h2 : (∏ x ∈ w, f x) j.2 = 1 j.2
⊢ 1 j.1 * 1 j.2 = 1 j.1 * 1 j.2

Tactic:
]

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

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝⁶ : CommRing R
τ : Type u_4
S : Type u_5
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
a : σ → MvPowerSeries τ S
x : ℕ → MvPowerSeries σ R
inst✝³ : UniformSpace R
inst✝² : DiscreteUniformity R
inst✝¹ : UniformSpace S
inst✝ : DiscreteUniformity S
hx : Summable x
ha : HasSubst a
⊢ ∑' (i : ℕ), (aeval ⋯) (x i) = ∑' (i : ℕ), (aeval ⋯) (x i)

Tactic:
]

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

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝⁶ : CommRing R
τ : Type u_4
S : Type u_5
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
a : σ → MvPowerSeries τ S
x : ℕ → MvPowerSeries σ R
inst✝³ : UniformSpace R
inst✝² : DiscreteUniformity R
inst✝¹ : UniformSpace S
inst✝ : DiscreteUniformity S
hx : Summable x
ha : HasSubst a
⊢ Summable fun i ↦ (aeval ⋯) (x i)

Tactic:
]

Goal state after:
σ : Type u_1
R : Type u_3
inst✝⁶ : CommRing R
τ : Type u_4
S : Type u_5
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
a : σ → MvPowerSeries τ S
x : ℕ → MvPowerSeries σ R
inst✝³ : UniformSpace R
inst✝² : DiscreteUniformity R
inst✝¹ : UniformSpace S
inst✝ : DiscreteUniformity S
hx : Summable x
ha : HasSubst a
⊢ Summable fun i ↦ (aeval ⋯) (x i)
Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction

Goal state before:
pp: case pos.refine_1
ι : Type u_1
R : Type u_2
inst✝ : CommSemiring R
f : ι → ArithmeticFunction R
hf : ∀ (i : ι), (f i).IsMultiplicative
hf' : Multipliable f
h✝ : ∀ (s : Finset ι), (∏ b ∈ s, f b).IsMultiplicative
h : ∀ (x : Finset ι), (∏ b ∈ x, f b) 1 = 1
key : (∏' (b : ι), f b) 1 = 1
⊢ (eulerProduct f) 1 = 1

Tactic:
]

Goal state after:
case pos.refine_1
ι : Type u_1
R : Type u_2
inst✝ : CommSemiring R
f : ι → ArithmeticFunction R
hf : ∀ (i : ι), (f i).IsMultiplicative
hf' : Multipliable f
h✝ : ∀ (s : Finset ι), (∏ b ∈ s, f b).IsMultiplicative
h : ∀ (x : Finset ι), (∏ b ∈ x, f b) 1 = 1
key : (∏' (b : ι), f b) 1 = 1
⊢ (eulerProduct f) 1 = 1
Module: Mathlib.RingTheory.PowerSeries.Substitution

Goal state before:
pp: case e_a.succ.succ.h₀.succ.succ
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 j : ℕ
hj : j + 1 + 1 ∈ Finset.range (i + 2 + 1)
hj' : j + 1 + 1 ≠ 0
⊢ (coeff (n + 2))
      (X ^ ((n + 2) * (j + 1 + 1)) * (-C k) ^ (j + 1 + 1) * B ^ (i + 2 - (j + 1 + 1)) * ↑((i + 2).choose (j + 1 + 1))) =
    0

Tactic:
]

Goal state after:
case e_a.succ.succ.h₀.succ.succ
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 j : ℕ
hj : j + 1 + 1 ∈ Finset.range (i + 2 + 1)
hj' : j + 1 + 1 ≠ 0
⊢ (coeff (n + 2))
      (X ^ ((n + 2) * (j + 1 + 1)) * (-C k) ^ (j + 1 + 1) * B ^ (i + 2 - (j + 1 + 1)) * ↑((i + 2).choose (j + 1 + 1))) =
    0
Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction

Goal state before:
pp: case neg
ι : Type u_1
R : Type u_2
inst✝ : CommSemiring R
f : ι → ArithmeticFunction R
hf : ∀ (i : ι), (f i).IsMultiplicative
hf' : ¬Multipliable f
⊢ IsMultiplicative 1

Tactic:
]

Goal state after:
case neg
ι : Type u_1
R : Type u_2
inst✝ : CommSemiring R
f : ι → ArithmeticFunction R
hf : ∀ (i : ι), (f i).IsMultiplicative
hf' : ¬Multipliable f
⊢ IsMultiplicative 1
Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction

Goal state before:
pp: case succ.succ
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
⊢ 0 = 0

pp: case succ.succ.hb
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
⊢ ¬∃ a, q i ^ a = n + 1 + 1

Tactic:
]

Goal state after:
case succ.succ.hb
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
⊢ ¬∃ a, q i ^ a = n + 1 + 1
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✝¹ x✝ : E
⊢ (fderivWithin 𝕜 f univ x✝¹) x✝ = (fderivWithin 𝕜 f univ x✝¹) x✝

Tactic:
]

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
hds : (l ⊓ 𝓟 {x_1 | x + d x_1 ∈ s}).NeBot
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:
]

Goal state after:
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
hds : (l ⊓ 𝓟 {x_1 | x + d x_1 ∈ s}).NeBot
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
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
hds : (l ⊓ 𝓟 {x_1 | x + d x_1 ∈ s}).NeBot
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 ∈ {y | ClusterPt y (⊤ • 𝓝[(fun x_1 ↦ x + x_1) ⁻¹' s] 0)}

Tactic:
]

Goal state after:
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
hds : (l ⊓ 𝓟 {x_1 | x + d x_1 ∈ s}).NeBot
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 ∈ {y | ClusterPt y (⊤ • 𝓝[(fun x_1 ↦ x + x_1) ⁻¹' s] 0)}
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)
⊢ Tendsto (fun n ↦ c n • d n) (l ⊓ 𝓟 {y | x + d y ∈ s}) (map (fun p ↦ p.1 • p.2) (⊤ ×ˢ 𝓝[(fun x_1 ↦ x + x_1) ⁻¹' s] 0))

Tactic:
]

Goal state after:
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)
⊢ Tendsto (fun n ↦ c n • d n) (l ⊓ 𝓟 {y | x + d y ∈ s}) (map (fun p ↦ p.1 • p.2) (⊤ ×ˢ 𝓝[(fun x_1 ↦ x + x_1) ⁻¹' s] 0))
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
⊢ (∑ᶠ (i : ℕ), (coeff i) P * (coeff (n + 1 + 1)) (B ^ i) - ∑ᶠ (i : ℕ), (coeff i) P * if i = 1 then k else 0) = 0

pp: case hf
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
⊢ Function.HasFiniteSupport fun d ↦ (coeff d) P * (coeff (n + 1 + 1)) (B ^ d)

pp: case hg
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
⊢ Function.HasFiniteSupport fun d ↦ (coeff d) P * if d = 1 then k else 0

Tactic:
]

Goal state after:
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
⊢ (∑ᶠ (i : ℕ), (coeff i) P * (coeff (n + 1 + 1)) (B ^ i) - ∑ᶠ (i : ℕ), (coeff i) P * if i = 1 then k else 0) = 0

case hf
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
⊢ Function.HasFiniteSupport fun d ↦ (coeff d) P * (coeff (n + 1 + 1)) (B ^ d)

case hg
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
⊢ Function.HasFiniteSupport fun d ↦ (coeff d) P * if d = 1 then k else 0
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
⊢ (∑ᶠ (i : ℕ), (coeff i) P * (coeff (n + 1 + 1)) (B ^ i) - if 1 = 1 then (coeff 1) P * k else 0) = 0

pp: case ha
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 ≠ 1 → (if x = 1 then (coeff x) P * k else 0) = 0

Tactic:
]

Goal state after:
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
⊢ (∑ᶠ (i : ℕ), (coeff i) P * (coeff (n + 1 + 1)) (B ^ i) - if 1 = 1 then (coeff 1) P * k else 0) = 0

case ha
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 ≠ 1 → (if x = 1 then (coeff x) P * k else 0) = 0
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 : 𝕜
hc₀ : c₀ ≠ 0 ∨ I = univ
hc₀I : ∀ i ∈ I, ‖c i‖ ≤ ‖c₀‖
hc₀r : ‖c₀‖ₑ < r
⊢ x ∈ I.pi (c₀ • U)

Tactic:
]

Goal state after:
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 : 𝕜
hc₀ : c₀ ≠ 0 ∨ I = univ
hc₀I : ∀ i ∈ I, ‖c i‖ ≤ ‖c₀‖
hc₀r : ‖c₀‖ₑ < r
⊢ x ∈ I.pi (c₀ • U)
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
⊢ ∑ᶠ (i : ℕ), (coeff i) P * (coeff (n + 1 + 1)) (B ^ i) = ∑ᶠ (d : ℕ), (coeff d) P • (coeff (n + 1 + 1)) (B ^ d)

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
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
⊢ HasSubst B

Tactic:
]

Goal state after:
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
⊢ ∑ᶠ (i : ℕ), (coeff i) P * (coeff (n + 1 + 1)) (B ^ i) = ∑ᶠ (d : ℕ), (coeff d) P • (coeff (n + 1 + 1)) (B ^ d)

case hb
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
⊢ HasSubst B
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 : (comap (fun p ↦ p.1 • p.2) (𝓝 y) ⊓ ⊤ ×ˢ 𝓝[(fun x_1 ↦ x + x_1) ⁻¹' s] 0).NeBot
⊢ ∃ 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:
]

Goal state after:
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 : (comap (fun p ↦ p.1 • p.2) (𝓝 y) ⊓ ⊤ ×ˢ 𝓝[(fun x_1 ↦ x + x_1) ⁻¹' s] 0).NeBot
⊢ ∃ 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)
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
⊢ ∀ (c : 𝕜), x ∈ c • closedBall 0 ↑r → ‖x‖ₑ / ↑r ≤ ‖c‖ₑ

Tactic:
]

Goal state after:
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
⊢ ∀ (c : 𝕜), x ∈ c • closedBall 0 ↑r → ‖x‖ₑ / ↑r ≤ ‖c‖ₑ
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
c : 𝕜
y : E
hy : ‖y‖₊ ≤ r
⊢ ‖(fun x ↦ c • x) y‖ₑ / ↑r ≤ ‖c‖ₑ

Tactic:
]

Goal state after:
𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
r : ℝ≥0
c : 𝕜
y : E
hy : ‖y‖₊ ≤ r
⊢ ‖(fun x ↦ c • x) y‖ₑ / ↑r ≤ ‖c‖ₑ
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
c : 𝕜
y : E
hy : ‖y‖₊ ≤ r
⊢ ‖c‖ₑ * ‖y‖ₑ / ↑r ≤ ‖c‖ₑ

Tactic:
]

Goal state after:
𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
r : ℝ≥0
c : 𝕜
y : E
hy : ‖y‖₊ ≤ r
⊢ ‖c‖ₑ * ‖y‖ₑ / ↑r ≤ ‖c‖ₑ
Module: Mathlib.Analysis.Convex.EGauge

Goal state before:
pp: case h₂
𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
r : ℝ≥0
c : 𝕜
y : E
hy : ‖y‖₊ ≤ r
⊢ ‖y‖₊ ≤ r

Tactic:
]

Goal state after:
case h₂
𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
r : ℝ≥0
c : 𝕜
y : E
hy : ‖y‖₊ ≤ r
⊢ ‖y‖₊ ≤ r
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
⊢ constantCoeff B = 0

Tactic:
]

Goal state after:
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
⊢ constantCoeff B = 0
Module: Mathlib.Analysis.Calculus.TangentCone.Defs

Goal state before:
pp: case refine_2
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 : (comap (fun p ↦ p.1 • p.2) (𝓝 y) ⊓ ⊤ ×ˢ 𝓝[(fun x_1 ↦ x + x_1) ⁻¹' s] 0).NeBot
⊢ ∀ᶠ (b : E) in 𝓝[(fun x_1 ↦ x + x_1) ⁻¹' s] 0, ∀ (a : R × E), a.2 = b → x + a.2 ∈ s

Tactic:
]

Goal state after:
case refine_2
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 : (comap (fun p ↦ p.1 • p.2) (𝓝 y) ⊓ ⊤ ×ˢ 𝓝[(fun x_1 ↦ x + x_1) ⁻¹' s] 0).NeBot
⊢ ∀ᶠ (b : E) in 𝓝[(fun x_1 ↦ x + x_1) ⁻¹' s] 0, ∀ (a : R × E), a.2 = b → x + a.2 ∈ s
Module: Mathlib.Analysis.Convex.EGauge

Goal state before:
pp: case inl
𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
x : E
hc : 1 < ‖c‖
this : NontriviallyNormedField 𝕜 := { toNormedField := inst✝², non_trivial := ⋯ }
h₀ : 0 ≠ 0 ∨ ‖x‖ ≠ 0
⊢ egauge 𝕜 (ball 0 ↑0) x ≤ ∞

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
hc : 1 < ‖c‖
this : NontriviallyNormedField 𝕜 := { toNormedField := inst✝², non_trivial := ⋯ }
h₀ : 0 ≠ 0 ∨ ‖x‖ ≠ 0
⊢ ‖c‖ₑ ≠ 0

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
hc : 1 < ‖c‖
this : NontriviallyNormedField 𝕜 := { toNormedField := inst✝², non_trivial := ⋯ }
h₀ : 0 ≠ 0 ∨ ‖x‖ ≠ 0
⊢ ‖x‖ₑ ≠ 0

Tactic:
]

Goal state after:
case inl
𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
x : E
hc : 1 < ‖c‖
this : NontriviallyNormedField 𝕜 := { toNormedField := inst✝², non_trivial := ⋯ }
h₀ : 0 ≠ 0 ∨ ‖x‖ ≠ 0
⊢ egauge 𝕜 (ball 0 ↑0) x ≤ ∞

𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
x : E
hc : 1 < ‖c‖
this : NontriviallyNormedField 𝕜 := { toNormedField := inst✝², non_trivial := ⋯ }
h₀ : 0 ≠ 0 ∨ ‖x‖ ≠ 0
⊢ ‖c‖ₑ ≠ 0

𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
x : E
hc : 1 < ‖c‖
this : NontriviallyNormedField 𝕜 := { toNormedField := inst✝², non_trivial := ⋯ }
h₀ : 0 ≠ 0 ∨ ‖x‖ ≠ 0
⊢ ‖x‖ₑ ≠ 0
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
⊢ (coeff n) (subst P.substInv P) = (coeff n) X

Tactic:
]

Goal state after:
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
⊢ (coeff n) (subst P.substInv P) = (coeff n) X