mathlib-initiative/mathlib-tactics huggingface dataset:
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
⊢ ∑ᶠ (d : ℕ), (coeff d) P • (coeff n) (P.substInv ^ d) =
    ∑ᶠ (d : ℕ), (coeff d) P • (coeff n) ((∑ i, C (P.substInvFun ↑i) * X ^ ↑i) ^ 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
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:
]

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
⊢ ∑ᶠ (d : ℕ), (coeff d) P • (coeff n) (P.substInv ^ d) =
    ∑ᶠ (d : ℕ), (coeff d) P • (coeff n) ((∑ i, C (P.substInvFun ↑i) * X ^ ↑i) ^ 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
this : (coeff n) (subst (∑ i, C (P.substInvFun ↑i) * X ^ ↑i) P) = (coeff n) X
⊢ HasSubst (∑ i, C (P.substInvFun ↑i) * X ^ ↑i)

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.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 ≤ r⁻¹ * ‖c‖₊ * ‖x‖₊

Tactic:
]

Goal state after:
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 ≤ r⁻¹ * ‖c‖₊ * ‖x‖₊
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
⊢ ∀ m < n + 1, (coeff m) (mk P.substInvFun - 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
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
⊢ ∀ m < n + 1, (coeff m) (mk P.substInvFun - B) = 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)
this : Invertible ((coeff 1) P.substInv)
Q : R⟦X⟧ := P.substInv.substInv
hQ : HasSubst Q
eq_aux : subst Q P.substInv = X
⊢ Q = Q

Tactic:
]

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)
this : Invertible ((coeff 1) P) := hP'.invertible
⊢ ⅟((coeff 1) P) = ↑hP'.unit⁻¹

Tactic:
]

Goal state after:
R : Type u_2
inst✝ : CommRing R
P : R⟦X⟧
hP' : IsUnit ((coeff 1) P)
this : Invertible ((coeff 1) P) := hP'.invertible
⊢ ⅟((coeff 1) P) = ↑hP'.unit⁻¹
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)
this : Invertible ((coeff 1) P) := hP'.invertible
⊢ X = X

Tactic:
]

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
⊢ ∑' (i : ℕ), (aeval ⋯) (x i) = ∑' (i : ℕ), (aeval ⋯) (x i)

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
⊢ Continuous ⇑(aeval ⋯)

Tactic:
]

Goal state after:
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
⊢ Continuous ⇑(aeval ⋯)
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 ↦ (aeval ⋯) (x i)

Tactic:
]

Goal state after:
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 ↦ (aeval ⋯) (x i)
Module: Mathlib.RingTheory.PowerSeries.Substitution

Goal state before:
pp: R : Type u_2
inst✝ : CommRing R
f : R⟦X⟧
e : Fin 2 →₀ ℕ
⊢ ∑ᶠ (d : Unit →₀ ℕ), (MvPowerSeries.coeff d) f • (MvPowerSeries.coeff e) (d.prod fun s e ↦ (X₀ + X₁) ^ e) =
    ↑((e 0 + e 1).choose (e 0)) * (coeff (e 0 + e 1)) f

Tactic:
]

Goal state after:
R : Type u_2
inst✝ : CommRing R
f : R⟦X⟧
e : Fin 2 →₀ ℕ
⊢ ∑ᶠ (d : Unit →₀ ℕ), (MvPowerSeries.coeff d) f • (MvPowerSeries.coeff e) (d.prod fun s e ↦ (X₀ + X₁) ^ e) =
    ↑((e 0 + e 1).choose (e 0)) * (coeff (e 0 + e 1)) f
Module: Mathlib.RingTheory.PowerSeries.Substitution

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

pp: R : Type u_2
inst✝ : CommRing R
f : R⟦X⟧
e : Fin 2 →₀ ℕ
⊢ ∀ (x : Unit →₀ ℕ),
    x ≠ single () (e 0 + e 1) →
      (MvPowerSeries.coeff x) f * MvPolynomial.coeff e ((MvPolynomial.X 0 + MvPolynomial.X 1) ^ x PUnit.unit) = 0

Tactic:
]

Goal state after:
R : Type u_2
inst✝ : CommRing R
f : R⟦X⟧
e : Fin 2 →₀ ℕ
⊢ MvPolynomial.coeff e ((MvPolynomial.X 0 + MvPolynomial.X 1) ^ (single () (e 0 + e 1)) PUnit.unit) *
      (MvPowerSeries.coeff (single () (e 0 + e 1))) f =
    ↑((e 0 + e 1).choose (e 0)) * (coeff (e 0 + e 1)) f

R : Type u_2
inst✝ : CommRing R
f : R⟦X⟧
e : Fin 2 →₀ ℕ
⊢ ∀ (x : Unit →₀ ℕ),
    x ≠ single () (e 0 + e 1) →
      (MvPowerSeries.coeff x) f * MvPolynomial.coeff e ((MvPolynomial.X 0 + MvPolynomial.X 1) ^ x PUnit.unit) = 0
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 (single 0 (e 0), single 1 (e 1)).1) (subst X₀ f) *
      (MvPowerSeries.coeff (single 0 (e 0), single 1 (e 1)).2) (subst X₁ f) =
    (coeff (e 0)) f * (coeff (e 1)) f

pp: R : Type u_2
inst✝ : CommRing R
f : R⟦X⟧
e : Fin 2 →₀ ℕ
⊢ ∀ b ∈ antidiagonal e,
    b ≠ (single 0 (e 0), single 1 (e 1)) →
      (MvPowerSeries.coeff b.1) (subst X₀ f) * (MvPowerSeries.coeff b.2) (subst X₁ f) = 0

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

Tactic:
]

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

R : Type u_2
inst✝ : CommRing R
f : R⟦X⟧
e : Fin 2 →₀ ℕ
⊢ ∀ b ∈ antidiagonal e,
    b ≠ (single 0 (e 0), single 1 (e 1)) →
      (MvPowerSeries.coeff b.1) (subst X₀ f) * (MvPowerSeries.coeff b.2) (subst X₁ f) = 0

R : Type u_2
inst✝ : CommRing R
f : R⟦X⟧
e : Fin 2 →₀ ℕ
⊢ (single 0 (e 0), single 1 (e 1)) ∉ antidiagonal e →
    (MvPowerSeries.coeff (single 0 (e 0), single 1 (e 1)).1) (subst X₀ f) *
        (MvPowerSeries.coeff (single 0 (e 0), single 1 (e 1)).2) (subst X₁ f) =
      0
Module: Mathlib.RingTheory.PowerSeries.Substitution

Goal state before:
pp: R : Type u_2
inst✝ : CommRing R
f : R⟦X⟧
e : Fin 2 →₀ ℕ
b : (Fin 2 →₀ ℕ) × (Fin 2 →₀ ℕ)
hb : b ∈ antidiagonal e
hb' : b ≠ (single 0 (e 0), single 1 (e 1))
hmul_ne_zero : ¬(MvPowerSeries.coeff b.1) (subst X₀ f) * (MvPowerSeries.coeff b.2) (subst X₁ f) = 0
h0 : b.1 = single 0 (b.1 0) ∧ ¬(coeff (b.1 0)) f = 0
h1 : b.2 = single 1 (b.2 1) ∧ ¬(coeff (b.2 1)) f = 0
⊢ single 0 (b.1 0) =
      (single 0 ((single 0 (b.1 0) + single 1 (b.2 1)) 0), single 1 ((single 0 (b.1 0) + single 1 (b.2 1)) 1)).1 ∧
    single 1 (b.2 1) =
      (single 0 ((single 0 (b.1 0) + single 1 (b.2 1)) 0), single 1 ((single 0 (b.1 0) + single 1 (b.2 1)) 1)).2

Tactic:
]

Goal state after:
R : Type u_2
inst✝ : CommRing R
f : R⟦X⟧
e : Fin 2 →₀ ℕ
b : (Fin 2 →₀ ℕ) × (Fin 2 →₀ ℕ)
hb : b ∈ antidiagonal e
hb' : b ≠ (single 0 (e 0), single 1 (e 1))
hmul_ne_zero : ¬(MvPowerSeries.coeff b.1) (subst X₀ f) * (MvPowerSeries.coeff b.2) (subst X₁ f) = 0
h0 : b.1 = single 0 (b.1 0) ∧ ¬(coeff (b.1 0)) f = 0
h1 : b.2 = single 1 (b.2 1) ∧ ¬(coeff (b.2 1)) f = 0
⊢ single 0 (b.1 0) =
      (single 0 ((single 0 (b.1 0) + single 1 (b.2 1)) 0), single 1 ((single 0 (b.1 0) + single 1 (b.2 1)) 1)).1 ∧
    single 1 (b.2 1) =
      (single 0 ((single 0 (b.1 0) + single 1 (b.2 1)) 0), single 1 ((single 0 (b.1 0) + single 1 (b.2 1)) 1)).2
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
y : E
hy : ClusterPt y (⊤ • 𝓝[(fun x_1 ↦ x + x_1) ⁻¹' s] 0)
⊢ (⊤ • 1) • 𝓝[(fun x_1 ↦ x + x_1) ⁻¹' s] 0 ≤ ⊤ • 𝓝[(fun x_1 ↦ x + x_1) ⁻¹' s] 0

Tactic:
]

Goal state after:
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
y : E
hy : ClusterPt y (⊤ • 𝓝[(fun x_1 ↦ x + x_1) ⁻¹' s] 0)
⊢ (⊤ • 1) • 𝓝[(fun x_1 ↦ x + x_1) ⁻¹' s] 0 ≤ ⊤ • 𝓝[(fun x_1 ↦ x + x_1) ⁻¹' s] 0
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
y : E
hy : ClusterPt y (⊤ • 𝓝[(fun x_1 ↦ x + x_1) ⁻¹' s] 0)
⊢ ⊤ • 𝓝[(fun x_1 ↦ x + x_1) ⁻¹' s] 0 ≤ ⊤ • 𝓝[(fun x_1 ↦ x + x_1) ⁻¹' s] 0

Tactic:
]

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

Goal state before:
pp: case h
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
y : E
hy : ClusterPt y (⊤ • 𝓝[(fun x_1 ↦ x + x_1) ⁻¹' s] 0)
⊢ (fun x_1 ↦ x + x_1) ⁻¹' t ∈ 𝓝[(fun x_1 ↦ x + x_1) ⁻¹' s] 0

Tactic:
]

Goal state after:
case h
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
y : E
hy : ClusterPt y (⊤ • 𝓝[(fun x_1 ↦ x + x_1) ⁻¹' s] 0)
⊢ (fun x_1 ↦ x + x_1) ⁻¹' t ∈ 𝓝[(fun x_1 ↦ x + x_1) ⁻¹' s] 0
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
U : Set E
hU : 0 ∈ U ∧ IsOpen[inst✝²] U
⊢ closure[inst✝²] (univ • (U ∩ (fun x_1 ↦ x + x_1) ⁻¹' s)) ⊆ closure[inst✝²] (univ • (U ∩ (fun x_1 ↦ x + x_1) ⁻¹' s))

Tactic:
]

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 : Dense ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 s x)) ∧ x ∈ closure[inst✝] s
st : s ⊆ t
⊢ Dense ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 t x)) ∧ x ∈ closure[inst✝] t

Tactic:
]

Goal state after:
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 : Dense ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 s x)) ∧ x ∈ closure[inst✝] s
st : s ⊆ t
⊢ Dense ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 t x)) ∧ x ∈ closure[inst✝] t
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 : Dense ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 s x)) ∧ x ∈ closure[inst✝] s
st : s ⊆ t
⊢ Dense ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 s x)) ∧ x ∈ closure[inst✝] s

Tactic:
]

Goal state after:
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 : Dense ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 s x)) ∧ x ∈ closure[inst✝] s
st : s ⊆ t
⊢ Dense ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 s x)) ∧ x ∈ closure[inst✝] s
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
st : 𝓝[s] x ≤ 𝓝[t] x
h : Dense ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 s x)) ∧ (𝓝[s] x).NeBot
⊢ Dense ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 t x)) ∧ (𝓝[t] x).NeBot

Tactic:
]

Goal state after:
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
st : 𝓝[s] x ≤ 𝓝[t] x
h : Dense ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 s x)) ∧ (𝓝[s] x).NeBot
⊢ Dense ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 t x)) ∧ (𝓝[t] x).NeBot
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
⊢ 0 ∈ tangentConeAt 𝕜 s x

Tactic:
]

Goal state after:
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
⊢ 0 ∈ tangentConeAt 𝕜 s x
Module: Mathlib.Analysis.Calculus.TangentCone.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁶ : DivisionSemiring 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace 𝕜
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousSMul 𝕜 E
s : Set E
x y : E
α : Type u_3
l : Filter α
inst✝ : l.NeBot
c : α → 𝕜
hc₀ : Tendsto c l (𝓝 0) ∧ ∀ᶠ (n : α) in l, c n ∈ {0}ᶜ
hmem : ∀ᶠ (n : α) in l, x + c n • y ∈ s
⊢ y ∈ tangentConeAt 𝕜 s x

Tactic:
]

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : DivisionSemiring 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace 𝕜
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousSMul 𝕜 E
s : Set E
x y : E
α : Type u_3
l : Filter α
inst✝ : l.NeBot
c : α → 𝕜
hc₀ : Tendsto c l (𝓝 0) ∧ ∀ᶠ (n : α) in l, c n ∈ {0}ᶜ
hmem : ∀ᶠ (n : α) in l, x + c n • y ∈ s
⊢ y ∈ tangentConeAt 𝕜 s x
Module: Mathlib.Analysis.Calculus.FDeriv.Congr

Goal state before:
pp: case inr
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : TopologicalSpace E
F : Type u_3
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : TopologicalSpace F
f : E → F
f' : E →L[𝕜] F
x : E
s t : Set E
inst✝ : T1Space E
y : E
h : s =ᶠ[𝓝 x] t
hne : x ≠ y
⊢ HasFDerivWithinAt f f' s x ↔ HasFDerivWithinAt f f' t x

Tactic:
]

Goal state after:
case inr
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : TopologicalSpace E
F : Type u_3
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : TopologicalSpace F
f : E → F
f' : E →L[𝕜] F
x : E
s t : Set E
inst✝ : T1Space E
y : E
h : s =ᶠ[𝓝 x] t
hne : x ≠ y
⊢ HasFDerivWithinAt f f' s x ↔ HasFDerivWithinAt f f' t x
Module: Mathlib.Analysis.Calculus.FDeriv.Congr

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 f₁ : E → F
x : E
h : f₁ =ᶠ[𝓝 x] f
⊢ fderivWithin 𝕜 f univ x = fderivWithin 𝕜 f univ x

Tactic:
]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁷ : DivisionSemiring 𝕜
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : TopologicalSpace 𝕜
inst✝³ : TopologicalSpace E
inst✝² : ContinuousSMul 𝕜 E
s : Set E
x : E
inst✝¹ : (𝓝[≠] 0).NeBot
inst✝ : ContinuousAdd E
h : s ∈ 𝓝 x
⊢ univ = univ

Tactic:
]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁶ : DivisionSemiring 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace E
inst✝² : TopologicalSpace 𝕜
inst✝¹ : (𝓝[≠] 0).NeBot
inst✝ : ContinuousSMul 𝕜 E
x : E
⊢ Dense ↑(Submodule.span 𝕜 univ) ∧ x ∈ closure[inst✝³] univ

Tactic:
]

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : DivisionSemiring 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace E
inst✝² : TopologicalSpace 𝕜
inst✝¹ : (𝓝[≠] 0).NeBot
inst✝ : ContinuousSMul 𝕜 E
x : E
⊢ Dense ↑(Submodule.span 𝕜 univ) ∧ x ∈ closure[inst✝³] univ
Module: Mathlib.Analysis.Calculus.TangentCone.Real

Goal state before:
pp: E : Type u_1
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul ℝ E
s : Set E
x y : E
h : openSegment ℝ x y ⊆ s
a : ℝ≥0
ha : a ∈ Ioo 0 1
⊢ x + id a • (y - x) ∈ ⇑(AffineMap.lineMap x y) '' Ioo 0 1

Tactic:
]

Goal state after:
E : Type u_1
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul ℝ E
s : Set E
x y : E
h : openSegment ℝ x y ⊆ s
a : ℝ≥0
ha : a ∈ Ioo 0 1
⊢ x + id a • (y - x) ∈ ⇑(AffineMap.lineMap x y) '' Ioo 0 1
Module: Mathlib.Analysis.Calculus.TangentCone.Real

Goal state before:
pp: E : Type u_1
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousSMul ℝ E
s : Set E
x : E
inst✝ : IsTopologicalAddGroup E
conv : Convex ℝ s
hx : x ∈ closure[inst✝²] s
y : E
hy : y ∈ interior s
⊢ tangentConeAt ℝ s x ∈ 𝓝 (y - x)

Tactic:
]

Goal state after:
E : Type u_1
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousSMul ℝ E
s : Set E
x : E
inst✝ : IsTopologicalAddGroup E
conv : Convex ℝ s
hx : x ∈ closure[inst✝²] s
y : E
hy : y ∈ interior s
⊢ tangentConeAt ℝ s x ∈ 𝓝 (y - x)
Module: Mathlib.Analysis.Calculus.FDeriv.Const

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
x : E
s : Set E
c : F
⊢ 0 = 0

pp: case hc
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
x : E
s : Set E
c : F
⊢ HasFDerivWithinAt (fun x ↦ c) 0 s x

Tactic:
]

Goal state after:
case hc
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
x : E
s : Set E
c : F
⊢ HasFDerivWithinAt (fun x ↦ c) 0 s x
Module: Mathlib.Analysis.Calculus.FDeriv.Const

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
s : Set E
c : F
x✝¹ x✝ : E
⊢ 0 x✝ = 0 x✝

Tactic:
]

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

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
x : E
c : F
⊢ 0 = 0

Tactic:
]

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

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
c : F
⊢ 0 = 0

Tactic:
]

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

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
⊢ ¬ClusterPt x (𝓟 ({x}ᶜ ∩ {x}))

Tactic:
]

Goal state after:
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
⊢ ¬ClusterPt x (𝓟 ({x}ᶜ ∩ {x}))
Module: Mathlib.Analysis.Calculus.FDeriv.Const

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
h : Subsingleton E
f : E → F
x : E
⊢ HasFDerivWithinAt f 0 {x} x

Tactic:
]

Goal state after:
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
h : Subsingleton E
f : E → F
x : E
⊢ HasFDerivWithinAt f 0 {x} x
Module: Mathlib.Analysis.Calculus.FDeriv.Const

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
a✝ : Nontrivial E
hf : ¬DifferentiableWithinAt 𝕜 f s x
⊢ ¬Injective ⇑0

Tactic:
]

Goal state after:
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
a✝ : Nontrivial E
hf : ¬DifferentiableWithinAt 𝕜 f s x
⊢ ¬Injective ⇑0