mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Calculus.Deriv.Comp

Goal state before:
pp: case pos
𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
x : 𝕜
s : Set 𝕜
𝕜' : Type u_1
inst✝³ : NontriviallyNormedField 𝕜'
inst✝² : NormedAlgebra 𝕜 𝕜'
inst✝¹ : NormedSpace 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
t' : Set 𝕜'
h : 𝕜 → 𝕜'
g₁ : 𝕜' → F
hg : DifferentiableWithinAt 𝕜' g₁ t' (h x)
hh : DifferentiableWithinAt 𝕜 h s x
hs : MapsTo h s t'
hsx : UniqueDiffWithinAt 𝕜 s x
⊢ derivWithin (g₁ ∘ h) s x = derivWithin h s x • derivWithin g₁ t' (h x)

pp: case neg
𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
x : 𝕜
s : Set 𝕜
𝕜' : Type u_1
inst✝³ : NontriviallyNormedField 𝕜'
inst✝² : NormedAlgebra 𝕜 𝕜'
inst✝¹ : NormedSpace 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
t' : Set 𝕜'
h : 𝕜 → 𝕜'
g₁ : 𝕜' → F
hg : DifferentiableWithinAt 𝕜' g₁ t' (h x)
hh : DifferentiableWithinAt 𝕜 h s x
hs : MapsTo h s t'
hsx : ¬UniqueDiffWithinAt 𝕜 s x
⊢ derivWithin (g₁ ∘ h) s x = derivWithin h s x • derivWithin g₁ t' (h x)

Tactic:
·

Goal state after:
case pos
𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
x : 𝕜
s : Set 𝕜
𝕜' : Type u_1
inst✝³ : NontriviallyNormedField 𝕜'
inst✝² : NormedAlgebra 𝕜 𝕜'
inst✝¹ : NormedSpace 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
t' : Set 𝕜'
h : 𝕜 → 𝕜'
g₁ : 𝕜' → F
hg : DifferentiableWithinAt 𝕜' g₁ t' (h x)
hh : DifferentiableWithinAt 𝕜 h s x
hs : MapsTo h s t'
hsx : UniqueDiffWithinAt 𝕜 s x
⊢ derivWithin (g₁ ∘ h) s x = derivWithin h s x • derivWithin g₁ t' (h x)
Module: Mathlib.Analysis.Calculus.Deriv.Comp

Goal state before:
pp: case neg
𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
x : 𝕜
s : Set 𝕜
𝕜' : Type u_1
inst✝³ : NontriviallyNormedField 𝕜'
inst✝² : NormedAlgebra 𝕜 𝕜'
inst✝¹ : NormedSpace 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
t' : Set 𝕜'
h : 𝕜 → 𝕜'
g₁ : 𝕜' → F
hg : DifferentiableWithinAt 𝕜' g₁ t' (h x)
hh : DifferentiableWithinAt 𝕜 h s x
hs : MapsTo h s t'
hsx : ¬UniqueDiffWithinAt 𝕜 s x
⊢ derivWithin (g₁ ∘ h) s x = derivWithin h s x • derivWithin g₁ t' (h x)

Tactic:
·

Goal state after:
case neg
𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
x : 𝕜
s : Set 𝕜
𝕜' : Type u_1
inst✝³ : NontriviallyNormedField 𝕜'
inst✝² : NormedAlgebra 𝕜 𝕜'
inst✝¹ : NormedSpace 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
t' : Set 𝕜'
h : 𝕜 → 𝕜'
g₁ : 𝕜' → F
hg : DifferentiableWithinAt 𝕜' g₁ t' (h x)
hh : DifferentiableWithinAt 𝕜 h s x
hs : MapsTo h s t'
hsx : ¬UniqueDiffWithinAt 𝕜 s x
⊢ derivWithin (g₁ ∘ h) s x = derivWithin h s x • derivWithin g₁ t' (h x)
Module: Mathlib.Analysis.Calculus.Deriv.Inv

Goal state before:
pp: case pos
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
𝕜' : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
c : 𝕜 → 𝕜'
hc : DifferentiableWithinAt 𝕜 c s x
hx : c x ≠ 0
hsx : UniqueDiffWithinAt 𝕜 s x
⊢ derivWithin (fun x ↦ (c x)⁻¹) s x = -derivWithin c s x / c x ^ 2

pp: case neg
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
𝕜' : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
c : 𝕜 → 𝕜'
hc : DifferentiableWithinAt 𝕜 c s x
hx : c x ≠ 0
hsx : ¬UniqueDiffWithinAt 𝕜 s x
⊢ derivWithin (fun x ↦ (c x)⁻¹) s x = -derivWithin c s x / c x ^ 2

Tactic:
·

Goal state after:
case pos
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
𝕜' : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
c : 𝕜 → 𝕜'
hc : DifferentiableWithinAt 𝕜 c s x
hx : c x ≠ 0
hsx : UniqueDiffWithinAt 𝕜 s x
⊢ derivWithin (fun x ↦ (c x)⁻¹) s x = -derivWithin c s x / c x ^ 2
Module: Mathlib.Analysis.Calculus.Deriv.Inv

Goal state before:
pp: case neg
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
𝕜' : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
c : 𝕜 → 𝕜'
hc : DifferentiableWithinAt 𝕜 c s x
hx : c x ≠ 0
hsx : ¬UniqueDiffWithinAt 𝕜 s x
⊢ derivWithin (fun x ↦ (c x)⁻¹) s x = -derivWithin c s x / c x ^ 2

Tactic:
·

Goal state after:
case neg
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
𝕜' : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
c : 𝕜 → 𝕜'
hc : DifferentiableWithinAt 𝕜 c s x
hx : c x ≠ 0
hsx : ¬UniqueDiffWithinAt 𝕜 s x
⊢ derivWithin (fun x ↦ (c x)⁻¹) s x = -derivWithin c s x / c x ^ 2
Module: Mathlib.Analysis.Calculus.Deriv.Inv

Goal state before:
pp: case e'_8
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
𝕜' : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
c d : 𝕜 → 𝕜'
c' d' : 𝕜'
hc : HasDerivWithinAt c c' s x
hd : HasDerivWithinAt d d' s x
hx : d x ≠ 0
x✝ : 𝕜
⊢ c x✝ / d x✝ = c x✝ * ((fun y ↦ y⁻¹) ∘ d) x✝

pp: case e'_9
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
𝕜' : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
c d : 𝕜 → 𝕜'
c' d' : 𝕜'
hc : HasDerivWithinAt c c' s x
hd : HasDerivWithinAt d d' s x
hx : d x ≠ 0
⊢ (c' * d x - c x * d') / d x ^ 2 = c' * ((fun y ↦ y⁻¹) ∘ d) x + c x * (-(d x ^ 2)⁻¹ * d')

Tactic:
·

Goal state after:
case e'_8
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
𝕜' : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
c d : 𝕜 → 𝕜'
c' d' : 𝕜'
hc : HasDerivWithinAt c c' s x
hd : HasDerivWithinAt d d' s x
hx : d x ≠ 0
x✝ : 𝕜
⊢ c x✝ / d x✝ = c x✝ * ((fun y ↦ y⁻¹) ∘ d) x✝
Module: Mathlib.Analysis.Calculus.Deriv.Inv

Goal state before:
pp: case e'_9
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
𝕜' : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
c d : 𝕜 → 𝕜'
c' d' : 𝕜'
hc : HasDerivWithinAt c c' s x
hd : HasDerivWithinAt d d' s x
hx : d x ≠ 0
⊢ (c' * d x - c x * d') / d x ^ 2 = c' * ((fun y ↦ y⁻¹) ∘ d) x + c x * (-(d x ^ 2)⁻¹ * d')

Tactic:
·

Goal state after:
case e'_9
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
𝕜' : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
c d : 𝕜 → 𝕜'
c' d' : 𝕜'
hc : HasDerivWithinAt c c' s x
hd : HasDerivWithinAt d d' s x
hx : d x ≠ 0
⊢ (c' * d x - c x * d') / d x ^ 2 = c' * ((fun y ↦ y⁻¹) ∘ d) x + c x * (-(d x ^ 2)⁻¹ * d')
Module: Mathlib.Analysis.Calculus.Deriv.Inv

Goal state before:
pp: case e'_8
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
𝕜' : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
c d : 𝕜 → 𝕜'
c' d' : 𝕜'
hc : HasStrictDerivAt c c' x
hd : HasStrictDerivAt d d' x
hx : d x ≠ 0
x✝ : 𝕜
⊢ c x✝ / d x✝ = c x✝ * (Inv.inv ∘ d) x✝

pp: case e'_9
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
𝕜' : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
c d : 𝕜 → 𝕜'
c' d' : 𝕜'
hc : HasStrictDerivAt c c' x
hd : HasStrictDerivAt d d' x
hx : d x ≠ 0
⊢ (c' * d x - c x * d') / d x ^ 2 = c' * (Inv.inv ∘ d) x + c x * (-(d x ^ 2)⁻¹ * d')

Tactic:
·

Goal state after:
case e'_8
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
𝕜' : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
c d : 𝕜 → 𝕜'
c' d' : 𝕜'
hc : HasStrictDerivAt c c' x
hd : HasStrictDerivAt d d' x
hx : d x ≠ 0
x✝ : 𝕜
⊢ c x✝ / d x✝ = c x✝ * (Inv.inv ∘ d) x✝
Module: Mathlib.Analysis.Analytic.IsolatedZeros

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
p : FormalMultilinearSeries 𝕜 𝕜 E
hp : HasFPowerSeriesAt f p z₀
h : p = 0
⊢ (∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = 0) ∨ ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z ≠ 0

pp: case neg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
p : FormalMultilinearSeries 𝕜 𝕜 E
hp : HasFPowerSeriesAt f p z₀
h : ¬p = 0
⊢ (∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = 0) ∨ ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z ≠ 0

Tactic:
·

Goal state after:
case pos
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
p : FormalMultilinearSeries 𝕜 𝕜 E
hp : HasFPowerSeriesAt f p z₀
h : p = 0
⊢ (∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = 0) ∨ ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z ≠ 0
Module: Mathlib.Analysis.Calculus.Deriv.Inv

Goal state before:
pp: case e'_9
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
𝕜' : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
c d : 𝕜 → 𝕜'
c' d' : 𝕜'
hc : HasStrictDerivAt c c' x
hd : HasStrictDerivAt d d' x
hx : d x ≠ 0
⊢ (c' * d x - c x * d') / d x ^ 2 = c' * (Inv.inv ∘ d) x + c x * (-(d x ^ 2)⁻¹ * d')

Tactic:
·

Goal state after:
case e'_9
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
𝕜' : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
c d : 𝕜 → 𝕜'
c' d' : 𝕜'
hc : HasStrictDerivAt c c' x
hd : HasStrictDerivAt d d' x
hx : d x ≠ 0
⊢ (c' * d x - c x * d') / d x ^ 2 = c' * (Inv.inv ∘ d) x + c x * (-(d x ^ 2)⁻¹ * d')
Module: Mathlib.Analysis.Analytic.IsolatedZeros

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
p : FormalMultilinearSeries 𝕜 𝕜 E
hp : HasFPowerSeriesAt f p z₀
h : ¬p = 0
⊢ (∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = 0) ∨ ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z ≠ 0

Tactic:
·

Goal state after:
case neg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
p : FormalMultilinearSeries 𝕜 𝕜 E
hp : HasFPowerSeriesAt f p z₀
h : ¬p = 0
⊢ (∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = 0) ∨ ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z ≠ 0
Module: Mathlib.Analysis.Analytic.IsolatedZeros

Goal state before:
pp: case inr
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
m n : ℤ
hm : ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z = (z - z₀) ^ m • g z
hn : ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z = (z - z₀) ^ n • g z
this :
  ∀ {m n : ℤ},
    (∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z = (z - z₀) ^ m • g z) →
      (∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z = (z - z₀) ^ n • g z) → n ≤ m → m = n
h_le : m < n
⊢ m = n

pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
m n : ℤ
hm : ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z = (z - z₀) ^ m • g z
hn : ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z = (z - z₀) ^ n • g z
h_le : n ≤ m
⊢ m = n

Tactic:
·

Goal state after:
case inr
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
m n : ℤ
hm : ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z = (z - z₀) ^ m • g z
hn : ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z = (z - z₀) ^ n • g z
this :
  ∀ {m n : ℤ},
    (∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z = (z - z₀) ^ m • g z) →
      (∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z = (z - z₀) ^ n • g z) → n ≤ m → m = n
h_le : m < n
⊢ m = n
Module: Mathlib.Analysis.Calculus.Deriv.Inv

Goal state before:
pp: case pos
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
𝕜' : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
c d : 𝕜 → 𝕜'
hc : DifferentiableWithinAt 𝕜 c s x
hd : DifferentiableWithinAt 𝕜 d s x
hx : d x ≠ 0
hsx : UniqueDiffWithinAt 𝕜 s x
⊢ derivWithin (fun x ↦ c x / d x) s x = (derivWithin c s x * d x - c x * derivWithin d s x) / d x ^ 2

pp: case neg
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
𝕜' : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
c d : 𝕜 → 𝕜'
hc : DifferentiableWithinAt 𝕜 c s x
hd : DifferentiableWithinAt 𝕜 d s x
hx : d x ≠ 0
hsx : ¬UniqueDiffWithinAt 𝕜 s x
⊢ derivWithin (fun x ↦ c x / d x) s x = (derivWithin c s x * d x - c x * derivWithin d s x) / d x ^ 2

Tactic:
·

Goal state after:
case pos
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
𝕜' : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
c d : 𝕜 → 𝕜'
hc : DifferentiableWithinAt 𝕜 c s x
hd : DifferentiableWithinAt 𝕜 d s x
hx : d x ≠ 0
hsx : UniqueDiffWithinAt 𝕜 s x
⊢ derivWithin (fun x ↦ c x / d x) s x = (derivWithin c s x * d x - c x * derivWithin d s x) / d x ^ 2
Module: Mathlib.Analysis.Calculus.Deriv.Inv

Goal state before:
pp: case neg
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
𝕜' : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
c d : 𝕜 → 𝕜'
hc : DifferentiableWithinAt 𝕜 c s x
hd : DifferentiableWithinAt 𝕜 d s x
hx : d x ≠ 0
hsx : ¬UniqueDiffWithinAt 𝕜 s x
⊢ derivWithin (fun x ↦ c x / d x) s x = (derivWithin c s x * d x - c x * derivWithin d s x) / d x ^ 2

Tactic:
·

Goal state after:
case neg
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
𝕜' : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
c d : 𝕜 → 𝕜'
hc : DifferentiableWithinAt 𝕜 c s x
hd : DifferentiableWithinAt 𝕜 d s x
hx : d x ≠ 0
hsx : ¬UniqueDiffWithinAt 𝕜 s x
⊢ derivWithin (fun x ↦ c x / d x) s x = (derivWithin c s x * d x - c x * derivWithin d s x) / d x ^ 2
Module: Mathlib.Analysis.Calculus.Deriv.Comp

Goal state before:
pp: case pos
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
𝕜' : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
s' : Set 𝕜'
h : 𝕜 → 𝕜'
h₂ : 𝕜' → 𝕜'
hh₂ : DifferentiableWithinAt 𝕜' h₂ s' (h x)
hh : DifferentiableWithinAt 𝕜 h s x
hs : MapsTo h s s'
hsx : UniqueDiffWithinAt 𝕜 s x
⊢ derivWithin (h₂ ∘ h) s x = derivWithin h₂ s' (h x) * derivWithin h s x

pp: case neg
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
𝕜' : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
s' : Set 𝕜'
h : 𝕜 → 𝕜'
h₂ : 𝕜' → 𝕜'
hh₂ : DifferentiableWithinAt 𝕜' h₂ s' (h x)
hh : DifferentiableWithinAt 𝕜 h s x
hs : MapsTo h s s'
hsx : ¬UniqueDiffWithinAt 𝕜 s x
⊢ derivWithin (h₂ ∘ h) s x = derivWithin h₂ s' (h x) * derivWithin h s x

Tactic:
·

Goal state after:
case pos
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
𝕜' : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
s' : Set 𝕜'
h : 𝕜 → 𝕜'
h₂ : 𝕜' → 𝕜'
hh₂ : DifferentiableWithinAt 𝕜' h₂ s' (h x)
hh : DifferentiableWithinAt 𝕜 h s x
hs : MapsTo h s s'
hsx : UniqueDiffWithinAt 𝕜 s x
⊢ derivWithin (h₂ ∘ h) s x = derivWithin h₂ s' (h x) * derivWithin h s x
Module: Mathlib.Analysis.Calculus.Deriv.Comp

Goal state before:
pp: case neg
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
𝕜' : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
s' : Set 𝕜'
h : 𝕜 → 𝕜'
h₂ : 𝕜' → 𝕜'
hh₂ : DifferentiableWithinAt 𝕜' h₂ s' (h x)
hh : DifferentiableWithinAt 𝕜 h s x
hs : MapsTo h s s'
hsx : ¬UniqueDiffWithinAt 𝕜 s x
⊢ derivWithin (h₂ ∘ h) s x = derivWithin h₂ s' (h x) * derivWithin h s x

Tactic:
·

Goal state after:
case neg
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
𝕜' : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
s' : Set 𝕜'
h : 𝕜 → 𝕜'
h₂ : 𝕜' → 𝕜'
hh₂ : DifferentiableWithinAt 𝕜' h₂ s' (h x)
hh : DifferentiableWithinAt 𝕜 h s x
hs : MapsTo h s s'
hsx : ¬UniqueDiffWithinAt 𝕜 s x
⊢ derivWithin (h₂ ∘ h) s x = derivWithin h₂ s' (h x) * derivWithin h s x
Module: Mathlib.Analysis.Calculus.Deriv.ZPow

Goal state before:
pp: case inl
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
m : ℤ
x : 𝕜
h : x ≠ 0 ∨ 0 ≤ m
this : ∀ (m : ℤ), 0 < m → HasStrictDerivAt (fun x ↦ x ^ m) (↑m * x ^ (m - 1)) x
hm : m < 0
⊢ HasStrictDerivAt (fun x ↦ x ^ m) (↑m * x ^ (m - 1)) x

pp: case inr.inl
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
m : ℤ
x : 𝕜
h : x ≠ 0 ∨ 0 ≤ m
this : ∀ (m : ℤ), 0 < m → HasStrictDerivAt (fun x ↦ x ^ m) (↑m * x ^ (m - 1)) x
hm : m = 0
⊢ HasStrictDerivAt (fun x ↦ x ^ m) (↑m * x ^ (m - 1)) x

pp: case inr.inr
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
m : ℤ
x : 𝕜
h : x ≠ 0 ∨ 0 ≤ m
this : ∀ (m : ℤ), 0 < m → HasStrictDerivAt (fun x ↦ x ^ m) (↑m * x ^ (m - 1)) x
hm : 0 < m
⊢ HasStrictDerivAt (fun x ↦ x ^ m) (↑m * x ^ (m - 1)) x

Tactic:
·

Goal state after:
case inl
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
m : ℤ
x : 𝕜
h : x ≠ 0 ∨ 0 ≤ m
this : ∀ (m : ℤ), 0 < m → HasStrictDerivAt (fun x ↦ x ^ m) (↑m * x ^ (m - 1)) x
hm : m < 0
⊢ HasStrictDerivAt (fun x ↦ x ^ m) (↑m * x ^ (m - 1)) x
Module: Mathlib.Analysis.Calculus.Deriv.Shift

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
a : 𝕜
s : Set 𝕜
x : 𝕜
⊢ derivWithin (fun x ↦ f (x - a)) s x = derivWithin f (-a +ᵥ s) (x - a)

Tactic:
simp_rw [sub_eq_add_neg, derivWithin_comp_add_const]

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.ZPow

Goal state before:
pp: case inr.inl
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
m : ℤ
x : 𝕜
h : x ≠ 0 ∨ 0 ≤ m
this : ∀ (m : ℤ), 0 < m → HasStrictDerivAt (fun x ↦ x ^ m) (↑m * x ^ (m - 1)) x
hm : m = 0
⊢ HasStrictDerivAt (fun x ↦ x ^ m) (↑m * x ^ (m - 1)) x

pp: case inr.inr
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
m : ℤ
x : 𝕜
h : x ≠ 0 ∨ 0 ≤ m
this : ∀ (m : ℤ), 0 < m → HasStrictDerivAt (fun x ↦ x ^ m) (↑m * x ^ (m - 1)) x
hm : 0 < m
⊢ HasStrictDerivAt (fun x ↦ x ^ m) (↑m * x ^ (m - 1)) x

Tactic:
·

Goal state after:
case inr.inl
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
m : ℤ
x : 𝕜
h : x ≠ 0 ∨ 0 ≤ m
this : ∀ (m : ℤ), 0 < m → HasStrictDerivAt (fun x ↦ x ^ m) (↑m * x ^ (m - 1)) x
hm : m = 0
⊢ HasStrictDerivAt (fun x ↦ x ^ m) (↑m * x ^ (m - 1)) x
Module: Mathlib.Analysis.Calculus.Deriv.ZPow

Goal state before:
pp: case inr.inr
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
m : ℤ
x : 𝕜
h : x ≠ 0 ∨ 0 ≤ m
this : ∀ (m : ℤ), 0 < m → HasStrictDerivAt (fun x ↦ x ^ m) (↑m * x ^ (m - 1)) x
hm : 0 < m
⊢ HasStrictDerivAt (fun x ↦ x ^ m) (↑m * x ^ (m - 1)) x

Tactic:
·

Goal state after:
case inr.inr
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
m : ℤ
x : 𝕜
h : x ≠ 0 ∨ 0 ≤ m
this : ∀ (m : ℤ), 0 < m → HasStrictDerivAt (fun x ↦ x ^ m) (↑m * x ^ (m - 1)) x
hm : 0 < m
⊢ HasStrictDerivAt (fun x ↦ x ^ m) (↑m * x ^ (m - 1)) x
Module: Mathlib.Analysis.Calculus.Deriv.ZPow

Goal state before:
pp: case pos
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
m : ℤ
x : 𝕜
H : x ≠ 0 ∨ 0 ≤ m
⊢ deriv (fun x ↦ x ^ m) x = ↑m * x ^ (m - 1)

pp: case neg
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
m : ℤ
x : 𝕜
H : ¬(x ≠ 0 ∨ 0 ≤ m)
⊢ deriv (fun x ↦ x ^ m) x = ↑m * x ^ (m - 1)

Tactic:
·

Goal state after:
case pos
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
m : ℤ
x : 𝕜
H : x ≠ 0 ∨ 0 ≤ m
⊢ deriv (fun x ↦ x ^ m) x = ↑m * x ^ (m - 1)
Module: Mathlib.Analysis.Calculus.Deriv.ZPow

Goal state before:
pp: case neg
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
m : ℤ
x : 𝕜
H : ¬(x ≠ 0 ∨ 0 ≤ m)
⊢ deriv (fun x ↦ x ^ m) x = ↑m * x ^ (m - 1)

Tactic:
·

Goal state after:
case neg
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
m : ℤ
x : 𝕜
H : ¬(x ≠ 0 ∨ 0 ≤ m)
⊢ deriv (fun x ↦ x ^ m) x = ↑m * x ^ (m - 1)
Module: Mathlib.Analysis.Calculus.Deriv.ZPow

Goal state before:
pp: case inl
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
n : ℕ
x : 𝕜
k : ℕ
hkn : k ≤ n
⊢ (∏ x ∈ Finset.range k, (↑n - ↑x)) * x ^ (↑n - ↑k) = (∏ x ∈ Finset.range k, (↑n - ↑x)) * x ^ ↑(n - k)

pp: case inr
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
n : ℕ
x : 𝕜
k : ℕ
hnk : n < k
⊢ (∏ x ∈ Finset.range k, (↑n - ↑x)) * x ^ (↑n - ↑k) = (∏ x ∈ Finset.range k, (↑n - ↑x)) * x ^ ↑(n - k)

Tactic:
·

Goal state after:
case inl
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
n : ℕ
x : 𝕜
k : ℕ
hkn : k ≤ n
⊢ (∏ x ∈ Finset.range k, (↑n - ↑x)) * x ^ (↑n - ↑k) = (∏ x ∈ Finset.range k, (↑n - ↑x)) * x ^ ↑(n - k)
Module: Mathlib.Analysis.Calculus.Deriv.Comp

Goal state before:
pp: case pos
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type w
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → F
x : 𝕜
s : Set 𝕜
l : F → E
t : Set F
hl : DifferentiableWithinAt 𝕜 l t (f x)
hf : DifferentiableWithinAt 𝕜 f s x
hs : MapsTo f s t
hsx : UniqueDiffWithinAt 𝕜 s x
⊢ derivWithin (l ∘ f) s x = (fderivWithin 𝕜 l t (f x)) (derivWithin f s x)

pp: case neg
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type w
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → F
x : 𝕜
s : Set 𝕜
l : F → E
t : Set F
hl : DifferentiableWithinAt 𝕜 l t (f x)
hf : DifferentiableWithinAt 𝕜 f s x
hs : MapsTo f s t
hsx : ¬UniqueDiffWithinAt 𝕜 s x
⊢ derivWithin (l ∘ f) s x = (fderivWithin 𝕜 l t (f x)) (derivWithin f s x)

Tactic:
·

Goal state after:
case pos
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type w
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → F
x : 𝕜
s : Set 𝕜
l : F → E
t : Set F
hl : DifferentiableWithinAt 𝕜 l t (f x)
hf : DifferentiableWithinAt 𝕜 f s x
hs : MapsTo f s t
hsx : UniqueDiffWithinAt 𝕜 s x
⊢ derivWithin (l ∘ f) s x = (fderivWithin 𝕜 l t (f x)) (derivWithin f s x)
Module: Mathlib.Analysis.Calculus.Deriv.ZPow

Goal state before:
pp: case inr
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
n : ℕ
x : 𝕜
k : ℕ
hnk : n < k
⊢ (∏ x ∈ Finset.range k, (↑n - ↑x)) * x ^ (↑n - ↑k) = (∏ x ∈ Finset.range k, (↑n - ↑x)) * x ^ ↑(n - k)

Tactic:
·

Goal state after:
case inr
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
n : ℕ
x : 𝕜
k : ℕ
hnk : n < k
⊢ (∏ x ∈ Finset.range k, (↑n - ↑x)) * x ^ (↑n - ↑k) = (∏ x ∈ Finset.range k, (↑n - ↑x)) * x ^ ↑(n - k)
Module: Mathlib.Analysis.Calculus.Deriv.Comp

Goal state before:
pp: case neg
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type w
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → F
x : 𝕜
s : Set 𝕜
l : F → E
t : Set F
hl : DifferentiableWithinAt 𝕜 l t (f x)
hf : DifferentiableWithinAt 𝕜 f s x
hs : MapsTo f s t
hsx : ¬UniqueDiffWithinAt 𝕜 s x
⊢ derivWithin (l ∘ f) s x = (fderivWithin 𝕜 l t (f x)) (derivWithin f s x)

Tactic:
·

Goal state after:
case neg
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type w
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → F
x : 𝕜
s : Set 𝕜
l : F → E
t : Set F
hl : DifferentiableWithinAt 𝕜 l t (f x)
hf : DifferentiableWithinAt 𝕜 f s x
hs : MapsTo f s t
hsx : ¬UniqueDiffWithinAt 𝕜 s x
⊢ derivWithin (l ∘ f) s x = (fderivWithin 𝕜 l t (f x)) (derivWithin f s x)