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https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_unordered | [43, 1] | [71, 7] | rw [uIoo_of_ge hx₀.le] at g'_ne gdiff hf' ⊢ | case inr
f g g' : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (Icc x x₀)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x x₀)) (uIoo x₀ x)
gcont : ContinuousOn g (Icc x x₀)
gdiff : ∀ x_1 ∈ uIoo x₀ x, HasDerivAt g (g' x_1) x_1
g'_ne : ∀ x_1 ∈ uIoo x₀ x, g' x_1 ≠ 0
hx₀ : x < x₀
⊢ ∃ x',
∃ (_ : x' ∈ uIoo x₀ x),
f x - taylorWithinEval f n (Icc x x₀) x₀ x =
((x - x') ^ n / ↑n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x x₀) x' | case inr
f g g' : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (Icc x x₀)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x x₀)) (Ioo x x₀)
gcont : ContinuousOn g (Icc x x₀)
gdiff : ∀ x_1 ∈ Ioo x x₀, HasDerivAt g (g' x_1) x_1
g'_ne : ∀ x_1 ∈ Ioo x x₀, g' x_1 ≠ 0
hx₀ : x < x₀
⊢ ∃ x',
∃ (_ : x' ∈ Ioo x x₀),
f x - taylorWithinEval f n (Icc x x₀) x₀ x =
((x - x') ^ n / ↑n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x x₀) x' |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_unordered | [43, 1] | [71, 7] | rcases exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc x x₀) t x)
(fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc x x₀) t) hx₀
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hx₀) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hx₀ hy hf hf') g g' gcont gdiff with
⟨y, hy, h⟩ | case inr
f g g' : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (Icc x x₀)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x x₀)) (Ioo x x₀)
gcont : ContinuousOn g (Icc x x₀)
gdiff : ∀ x_1 ∈ Ioo x x₀, HasDerivAt g (g' x_1) x_1
g'_ne : ∀ x_1 ∈ Ioo x x₀, g' x_1 ≠ 0
hx₀ : x < x₀
⊢ ∃ x',
∃ (_ : x' ∈ Ioo x x₀),
f x - taylorWithinEval f n (Icc x x₀) x₀ x =
((x - x') ^ n / ↑n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x x₀) x' | case inr.intro.intro
f g g' : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (Icc x x₀)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x x₀)) (Ioo x x₀)
gcont : ContinuousOn g (Icc x x₀)
gdiff : ∀ x_1 ∈ Ioo x x₀, HasDerivAt g (g' x_1) x_1
g'_ne : ∀ x_1 ∈ Ioo x x₀, g' x_1 ≠ 0
hx₀ : x < x₀
y : ℝ
hy : y ∈ Ioo x x₀
h :
(g x₀ - g x) * ((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc x x₀) y =
(taylorWithinEval f n (Icc x x₀) x₀ x - taylorWithinEval f n (Icc x x₀) x x) * g' y
⊢ ∃ x',
∃ (_ : x' ∈ Ioo x x₀),
f x - taylorWithinEval f n (Icc x x₀) x₀ x =
((x - x') ^ n / ↑n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x x₀) x' |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_unordered | [43, 1] | [71, 7] | use y, hy | case inr.intro.intro
f g g' : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (Icc x x₀)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x x₀)) (Ioo x x₀)
gcont : ContinuousOn g (Icc x x₀)
gdiff : ∀ x_1 ∈ Ioo x x₀, HasDerivAt g (g' x_1) x_1
g'_ne : ∀ x_1 ∈ Ioo x x₀, g' x_1 ≠ 0
hx₀ : x < x₀
y : ℝ
hy : y ∈ Ioo x x₀
h :
(g x₀ - g x) * ((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc x x₀) y =
(taylorWithinEval f n (Icc x x₀) x₀ x - taylorWithinEval f n (Icc x x₀) x x) * g' y
⊢ ∃ x',
∃ (_ : x' ∈ Ioo x x₀),
f x - taylorWithinEval f n (Icc x x₀) x₀ x =
((x - x') ^ n / ↑n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x x₀) x' | case h
f g g' : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (Icc x x₀)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x x₀)) (Ioo x x₀)
gcont : ContinuousOn g (Icc x x₀)
gdiff : ∀ x_1 ∈ Ioo x x₀, HasDerivAt g (g' x_1) x_1
g'_ne : ∀ x_1 ∈ Ioo x x₀, g' x_1 ≠ 0
hx₀ : x < x₀
y : ℝ
hy : y ∈ Ioo x x₀
h :
(g x₀ - g x) * ((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc x x₀) y =
(taylorWithinEval f n (Icc x x₀) x₀ x - taylorWithinEval f n (Icc x x₀) x x) * g' y
⊢ f x - taylorWithinEval f n (Icc x x₀) x₀ x =
((x - y) ^ n / ↑n ! * (g x - g x₀) / g' y) • iteratedDerivWithin (n + 1) f (Icc x x₀) y |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_unordered | [43, 1] | [71, 7] | simp only [taylorWithinEval_self] at h | case h
f g g' : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (Icc x x₀)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x x₀)) (Ioo x x₀)
gcont : ContinuousOn g (Icc x x₀)
gdiff : ∀ x_1 ∈ Ioo x x₀, HasDerivAt g (g' x_1) x_1
g'_ne : ∀ x_1 ∈ Ioo x x₀, g' x_1 ≠ 0
hx₀ : x < x₀
y : ℝ
hy : y ∈ Ioo x x₀
h :
(g x₀ - g x) * ((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc x x₀) y =
(taylorWithinEval f n (Icc x x₀) x₀ x - taylorWithinEval f n (Icc x x₀) x x) * g' y
⊢ f x - taylorWithinEval f n (Icc x x₀) x₀ x =
((x - y) ^ n / ↑n ! * (g x - g x₀) / g' y) • iteratedDerivWithin (n + 1) f (Icc x x₀) y | case h
f g g' : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (Icc x x₀)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x x₀)) (Ioo x x₀)
gcont : ContinuousOn g (Icc x x₀)
gdiff : ∀ x_1 ∈ Ioo x x₀, HasDerivAt g (g' x_1) x_1
g'_ne : ∀ x_1 ∈ Ioo x x₀, g' x_1 ≠ 0
hx₀ : x < x₀
y : ℝ
hy : y ∈ Ioo x x₀
h :
(g x₀ - g x) * ((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc x x₀) y =
(taylorWithinEval f n (Icc x x₀) x₀ x - f x) * g' y
⊢ f x - taylorWithinEval f n (Icc x x₀) x₀ x =
((x - y) ^ n / ↑n ! * (g x - g x₀) / g' y) • iteratedDerivWithin (n + 1) f (Icc x x₀) y |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_unordered | [43, 1] | [71, 7] | rw [mul_comm, ← div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h | case h
f g g' : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (Icc x x₀)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x x₀)) (Ioo x x₀)
gcont : ContinuousOn g (Icc x x₀)
gdiff : ∀ x_1 ∈ Ioo x x₀, HasDerivAt g (g' x_1) x_1
g'_ne : ∀ x_1 ∈ Ioo x x₀, g' x_1 ≠ 0
hx₀ : x < x₀
y : ℝ
hy : y ∈ Ioo x x₀
h :
(g x₀ - g x) * ((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc x x₀) y =
(taylorWithinEval f n (Icc x x₀) x₀ x - f x) * g' y
⊢ f x - taylorWithinEval f n (Icc x x₀) x₀ x =
((x - y) ^ n / ↑n ! * (g x - g x₀) / g' y) • iteratedDerivWithin (n + 1) f (Icc x x₀) y | case h
f g g' : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (Icc x x₀)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x x₀)) (Ioo x x₀)
gcont : ContinuousOn g (Icc x x₀)
gdiff : ∀ x_1 ∈ Ioo x x₀, HasDerivAt g (g' x_1) x_1
g'_ne : ∀ x_1 ∈ Ioo x x₀, g' x_1 ≠ 0
hx₀ : x < x₀
y : ℝ
hy : y ∈ Ioo x x₀
h :
((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc x x₀) y * (g x₀ - g x) / g' y =
taylorWithinEval f n (Icc x x₀) x₀ x - f x
⊢ f x - taylorWithinEval f n (Icc x x₀) x₀ x =
((x - y) ^ n / ↑n ! * (g x - g x₀) / g' y) • iteratedDerivWithin (n + 1) f (Icc x x₀) y |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_unordered | [43, 1] | [71, 7] | rw [← neg_sub, ← h] | case h
f g g' : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (Icc x x₀)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x x₀)) (Ioo x x₀)
gcont : ContinuousOn g (Icc x x₀)
gdiff : ∀ x_1 ∈ Ioo x x₀, HasDerivAt g (g' x_1) x_1
g'_ne : ∀ x_1 ∈ Ioo x x₀, g' x_1 ≠ 0
hx₀ : x < x₀
y : ℝ
hy : y ∈ Ioo x x₀
h :
((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc x x₀) y * (g x₀ - g x) / g' y =
taylorWithinEval f n (Icc x x₀) x₀ x - f x
⊢ f x - taylorWithinEval f n (Icc x x₀) x₀ x =
((x - y) ^ n / ↑n ! * (g x - g x₀) / g' y) • iteratedDerivWithin (n + 1) f (Icc x x₀) y | case h
f g g' : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (Icc x x₀)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x x₀)) (Ioo x x₀)
gcont : ContinuousOn g (Icc x x₀)
gdiff : ∀ x_1 ∈ Ioo x x₀, HasDerivAt g (g' x_1) x_1
g'_ne : ∀ x_1 ∈ Ioo x x₀, g' x_1 ≠ 0
hx₀ : x < x₀
y : ℝ
hy : y ∈ Ioo x x₀
h :
((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc x x₀) y * (g x₀ - g x) / g' y =
taylorWithinEval f n (Icc x x₀) x₀ x - f x
⊢ -(((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc x x₀) y * (g x₀ - g x) / g' y) =
((x - y) ^ n / ↑n ! * (g x - g x₀) / g' y) • iteratedDerivWithin (n + 1) f (Icc x x₀) y |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_unordered | [43, 1] | [71, 7] | field_simp [g'_ne y hy, n.factorial_ne_zero] | case h
f g g' : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (Icc x x₀)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x x₀)) (Ioo x x₀)
gcont : ContinuousOn g (Icc x x₀)
gdiff : ∀ x_1 ∈ Ioo x x₀, HasDerivAt g (g' x_1) x_1
g'_ne : ∀ x_1 ∈ Ioo x x₀, g' x_1 ≠ 0
hx₀ : x < x₀
y : ℝ
hy : y ∈ Ioo x x₀
h :
((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc x x₀) y * (g x₀ - g x) / g' y =
taylorWithinEval f n (Icc x x₀) x₀ x - f x
⊢ -(((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc x x₀) y * (g x₀ - g x) / g' y) =
((x - y) ^ n / ↑n ! * (g x - g x₀) / g' y) • iteratedDerivWithin (n + 1) f (Icc x x₀) y | case h
f g g' : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (Icc x x₀)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x x₀)) (Ioo x x₀)
gcont : ContinuousOn g (Icc x x₀)
gdiff : ∀ x_1 ∈ Ioo x x₀, HasDerivAt g (g' x_1) x_1
g'_ne : ∀ x_1 ∈ Ioo x x₀, g' x_1 ≠ 0
hx₀ : x < x₀
y : ℝ
hy : y ∈ Ioo x x₀
h :
((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc x x₀) y * (g x₀ - g x) / g' y =
taylorWithinEval f n (Icc x x₀) x₀ x - f x
⊢ -((x - y) ^ n * iteratedDerivWithin (n + 1) f (Icc x x₀) y * (g x₀ - g x) * (↑n ! * g' y)) =
(x - y) ^ n * (g x - g x₀) * iteratedDerivWithin (n + 1) f (Icc x x₀) y * (↑n ! * g' y) |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_unordered | [43, 1] | [71, 7] | ring | case h
f g g' : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (Icc x x₀)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x x₀)) (Ioo x x₀)
gcont : ContinuousOn g (Icc x x₀)
gdiff : ∀ x_1 ∈ Ioo x x₀, HasDerivAt g (g' x_1) x_1
g'_ne : ∀ x_1 ∈ Ioo x x₀, g' x_1 ≠ 0
hx₀ : x < x₀
y : ℝ
hy : y ∈ Ioo x x₀
h :
((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc x x₀) y * (g x₀ - g x) / g' y =
taylorWithinEval f n (Icc x x₀) x₀ x - f x
⊢ -((x - y) ^ n * iteratedDerivWithin (n + 1) f (Icc x x₀) y * (g x₀ - g x) * (↑n ! * g' y)) =
(x - y) ^ n * (g x - g x₀) * iteratedDerivWithin (n + 1) f (Icc x x₀) y * (↑n ! * g' y) | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_unordered | [43, 1] | [71, 7] | rw [uIcc_of_le hx₀.le] at hf hf' gcont ⊢ | case inl
f g g' : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (uIcc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (uIcc x₀ x)) (uIoo x₀ x)
gcont : ContinuousOn g (uIcc x₀ x)
gdiff : ∀ x_1 ∈ uIoo x₀ x, HasDerivAt g (g' x_1) x_1
g'_ne : ∀ x_1 ∈ uIoo x₀ x, g' x_1 ≠ 0
hx₀ : x₀ < x
⊢ ∃ x',
∃ (_ : x' ∈ uIoo x₀ x),
f x - taylorWithinEval f n (uIcc x₀ x) x₀ x =
((x - x') ^ n / ↑n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (uIcc x₀ x) x' | case inl
f g g' : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (Icc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (uIoo x₀ x)
gcont : ContinuousOn g (Icc x₀ x)
gdiff : ∀ x_1 ∈ uIoo x₀ x, HasDerivAt g (g' x_1) x_1
g'_ne : ∀ x_1 ∈ uIoo x₀ x, g' x_1 ≠ 0
hx₀ : x₀ < x
⊢ ∃ x',
∃ (_ : x' ∈ uIoo x₀ x),
f x - taylorWithinEval f n (Icc x₀ x) x₀ x =
((x - x') ^ n / ↑n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_unordered | [43, 1] | [71, 7] | rw [uIoo_of_le hx₀.le] at g'_ne gdiff hf' ⊢ | case inl
f g g' : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (Icc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (uIoo x₀ x)
gcont : ContinuousOn g (Icc x₀ x)
gdiff : ∀ x_1 ∈ uIoo x₀ x, HasDerivAt g (g' x_1) x_1
g'_ne : ∀ x_1 ∈ uIoo x₀ x, g' x_1 ≠ 0
hx₀ : x₀ < x
⊢ ∃ x',
∃ (_ : x' ∈ uIoo x₀ x),
f x - taylorWithinEval f n (Icc x₀ x) x₀ x =
((x - x') ^ n / ↑n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' | case inl
f g g' : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (Icc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)
gcont : ContinuousOn g (Icc x₀ x)
gdiff : ∀ x_1 ∈ Ioo x₀ x, HasDerivAt g (g' x_1) x_1
g'_ne : ∀ x_1 ∈ Ioo x₀ x, g' x_1 ≠ 0
hx₀ : x₀ < x
⊢ ∃ x',
∃ (_ : x' ∈ Ioo x₀ x),
f x - taylorWithinEval f n (Icc x₀ x) x₀ x =
((x - x') ^ n / ↑n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_unordered | [43, 1] | [71, 7] | exact taylor_mean_remainder hx₀ hf hf' gcont gdiff g'_ne | case inl
f g g' : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (Icc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)
gcont : ContinuousOn g (Icc x₀ x)
gdiff : ∀ x_1 ∈ Ioo x₀ x, HasDerivAt g (g' x_1) x_1
g'_ne : ∀ x_1 ∈ Ioo x₀ x, g' x_1 ≠ 0
hx₀ : x₀ < x
⊢ ∃ x',
∃ (_ : x' ∈ Ioo x₀ x),
f x - taylorWithinEval f n (Icc x₀ x) x₀ x =
((x - x') ^ n / ↑n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc x₀ x) x' | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_lagrange_unordered | [73, 1] | [103, 10] | have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (uIcc x₀ x) := by
refine' Continuous.continuousOn _
sorry | f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (uIcc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (uIcc x₀ x)) (uIoo x₀ x)
⊢ ∃ x',
∃ (_ : x' ∈ uIoo x₀ x),
f x - taylorWithinEval f n (uIcc x₀ x) x₀ x =
iteratedDerivWithin (n + 1) f (uIcc x₀ x) x' * (x - x₀) ^ (n + 1) / ↑(n + 1)! | f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (uIcc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (uIcc x₀ x)) (uIoo x₀ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc x₀ x)
⊢ ∃ x',
∃ (_ : x' ∈ uIoo x₀ x),
f x - taylorWithinEval f n (uIcc x₀ x) x₀ x =
iteratedDerivWithin (n + 1) f (uIcc x₀ x) x' * (x - x₀) ^ (n + 1) / ↑(n + 1)! |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_lagrange_unordered | [73, 1] | [103, 10] | have hg' : ∀ y : ℝ, y ∈ uIoo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0 := fun y hy =>
mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy) | f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (uIcc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (uIcc x₀ x)) (uIoo x₀ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc x₀ x)
xy_ne : ∀ y ∈ uIoo x₀ x, (x - y) ^ n ≠ 0
⊢ ∃ x',
∃ (_ : x' ∈ uIoo x₀ x),
f x - taylorWithinEval f n (uIcc x₀ x) x₀ x =
iteratedDerivWithin (n + 1) f (uIcc x₀ x) x' * (x - x₀) ^ (n + 1) / ↑(n + 1)! | f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (uIcc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (uIcc x₀ x)) (uIoo x₀ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc x₀ x)
xy_ne : ∀ y ∈ uIoo x₀ x, (x - y) ^ n ≠ 0
hg' : ∀ y ∈ uIoo x₀ x, -(↑n + 1) * (x - y) ^ n ≠ 0
⊢ ∃ x',
∃ (_ : x' ∈ uIoo x₀ x),
f x - taylorWithinEval f n (uIcc x₀ x) x₀ x =
iteratedDerivWithin (n + 1) f (uIcc x₀ x) x' * (x - x₀) ^ (n + 1) / ↑(n + 1)! |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_lagrange_unordered | [73, 1] | [103, 10] | rcases taylor_mean_remainder_unordered hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _)
hg' with
⟨y, hy, h⟩ | f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (uIcc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (uIcc x₀ x)) (uIoo x₀ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc x₀ x)
xy_ne : ∀ y ∈ uIoo x₀ x, (x - y) ^ n ≠ 0
hg' : ∀ y ∈ uIoo x₀ x, -(↑n + 1) * (x - y) ^ n ≠ 0
⊢ ∃ x',
∃ (_ : x' ∈ uIoo x₀ x),
f x - taylorWithinEval f n (uIcc x₀ x) x₀ x =
iteratedDerivWithin (n + 1) f (uIcc x₀ x) x' * (x - x₀) ^ (n + 1) / ↑(n + 1)! | case intro.intro
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (uIcc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (uIcc x₀ x)) (uIoo x₀ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc x₀ x)
xy_ne : ∀ y ∈ uIoo x₀ x, (x - y) ^ n ≠ 0
hg' : ∀ y ∈ uIoo x₀ x, -(↑n + 1) * (x - y) ^ n ≠ 0
y : ℝ
hy : y ∈ uIoo x₀ x
h :
f x - taylorWithinEval f n (uIcc x₀ x) x₀ x =
((x - y) ^ n / ↑n ! * ((x - x) ^ (n + 1) - (x - x₀) ^ (n + 1)) / (-(↑n + 1) * (x - y) ^ n)) •
iteratedDerivWithin (n + 1) f (uIcc x₀ x) y
⊢ ∃ x',
∃ (_ : x' ∈ uIoo x₀ x),
f x - taylorWithinEval f n (uIcc x₀ x) x₀ x =
iteratedDerivWithin (n + 1) f (uIcc x₀ x) x' * (x - x₀) ^ (n + 1) / ↑(n + 1)! |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_lagrange_unordered | [73, 1] | [103, 10] | use y, hy | case intro.intro
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (uIcc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (uIcc x₀ x)) (uIoo x₀ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc x₀ x)
xy_ne : ∀ y ∈ uIoo x₀ x, (x - y) ^ n ≠ 0
hg' : ∀ y ∈ uIoo x₀ x, -(↑n + 1) * (x - y) ^ n ≠ 0
y : ℝ
hy : y ∈ uIoo x₀ x
h :
f x - taylorWithinEval f n (uIcc x₀ x) x₀ x =
((x - y) ^ n / ↑n ! * ((x - x) ^ (n + 1) - (x - x₀) ^ (n + 1)) / (-(↑n + 1) * (x - y) ^ n)) •
iteratedDerivWithin (n + 1) f (uIcc x₀ x) y
⊢ ∃ x',
∃ (_ : x' ∈ uIoo x₀ x),
f x - taylorWithinEval f n (uIcc x₀ x) x₀ x =
iteratedDerivWithin (n + 1) f (uIcc x₀ x) x' * (x - x₀) ^ (n + 1) / ↑(n + 1)! | case h
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (uIcc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (uIcc x₀ x)) (uIoo x₀ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc x₀ x)
xy_ne : ∀ y ∈ uIoo x₀ x, (x - y) ^ n ≠ 0
hg' : ∀ y ∈ uIoo x₀ x, -(↑n + 1) * (x - y) ^ n ≠ 0
y : ℝ
hy : y ∈ uIoo x₀ x
h :
f x - taylorWithinEval f n (uIcc x₀ x) x₀ x =
((x - y) ^ n / ↑n ! * ((x - x) ^ (n + 1) - (x - x₀) ^ (n + 1)) / (-(↑n + 1) * (x - y) ^ n)) •
iteratedDerivWithin (n + 1) f (uIcc x₀ x) y
⊢ f x - taylorWithinEval f n (uIcc x₀ x) x₀ x =
iteratedDerivWithin (n + 1) f (uIcc x₀ x) y * (x - x₀) ^ (n + 1) / ↑(n + 1)! |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_lagrange_unordered | [73, 1] | [103, 10] | simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h | case h
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (uIcc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (uIcc x₀ x)) (uIoo x₀ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc x₀ x)
xy_ne : ∀ y ∈ uIoo x₀ x, (x - y) ^ n ≠ 0
hg' : ∀ y ∈ uIoo x₀ x, -(↑n + 1) * (x - y) ^ n ≠ 0
y : ℝ
hy : y ∈ uIoo x₀ x
h :
f x - taylorWithinEval f n (uIcc x₀ x) x₀ x =
((x - y) ^ n / ↑n ! * ((x - x) ^ (n + 1) - (x - x₀) ^ (n + 1)) / (-(↑n + 1) * (x - y) ^ n)) •
iteratedDerivWithin (n + 1) f (uIcc x₀ x) y
⊢ f x - taylorWithinEval f n (uIcc x₀ x) x₀ x =
iteratedDerivWithin (n + 1) f (uIcc x₀ x) y * (x - x₀) ^ (n + 1) / ↑(n + 1)! | case h
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (uIcc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (uIcc x₀ x)) (uIoo x₀ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc x₀ x)
xy_ne : ∀ y ∈ uIoo x₀ x, (x - y) ^ n ≠ 0
hg' : ∀ y ∈ uIoo x₀ x, -(↑n + 1) * (x - y) ^ n ≠ 0
y : ℝ
hy : y ∈ uIoo x₀ x
h :
f x - taylorWithinEval f n (uIcc x₀ x) x₀ x =
(-((x - y) ^ n / ↑n ! * (x - x₀) ^ (n + 1)) / (-(↑n + 1) * (x - y) ^ n)) •
iteratedDerivWithin (n + 1) f (uIcc x₀ x) y
⊢ f x - taylorWithinEval f n (uIcc x₀ x) x₀ x =
iteratedDerivWithin (n + 1) f (uIcc x₀ x) y * (x - x₀) ^ (n + 1) / ↑(n + 1)! |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_lagrange_unordered | [73, 1] | [103, 10] | rw [h, neg_div, ← div_neg, neg_mul, neg_neg] | case h
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (uIcc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (uIcc x₀ x)) (uIoo x₀ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc x₀ x)
xy_ne : ∀ y ∈ uIoo x₀ x, (x - y) ^ n ≠ 0
hg' : ∀ y ∈ uIoo x₀ x, -(↑n + 1) * (x - y) ^ n ≠ 0
y : ℝ
hy : y ∈ uIoo x₀ x
h :
f x - taylorWithinEval f n (uIcc x₀ x) x₀ x =
(-((x - y) ^ n / ↑n ! * (x - x₀) ^ (n + 1)) / (-(↑n + 1) * (x - y) ^ n)) •
iteratedDerivWithin (n + 1) f (uIcc x₀ x) y
⊢ f x - taylorWithinEval f n (uIcc x₀ x) x₀ x =
iteratedDerivWithin (n + 1) f (uIcc x₀ x) y * (x - x₀) ^ (n + 1) / ↑(n + 1)! | case h
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (uIcc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (uIcc x₀ x)) (uIoo x₀ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc x₀ x)
xy_ne : ∀ y ∈ uIoo x₀ x, (x - y) ^ n ≠ 0
hg' : ∀ y ∈ uIoo x₀ x, -(↑n + 1) * (x - y) ^ n ≠ 0
y : ℝ
hy : y ∈ uIoo x₀ x
h :
f x - taylorWithinEval f n (uIcc x₀ x) x₀ x =
(-((x - y) ^ n / ↑n ! * (x - x₀) ^ (n + 1)) / (-(↑n + 1) * (x - y) ^ n)) •
iteratedDerivWithin (n + 1) f (uIcc x₀ x) y
⊢ ((x - y) ^ n / ↑n ! * (x - x₀) ^ (n + 1) / ((↑n + 1) * (x - y) ^ n)) • iteratedDerivWithin (n + 1) f (uIcc x₀ x) y =
iteratedDerivWithin (n + 1) f (uIcc x₀ x) y * (x - x₀) ^ (n + 1) / ↑(n + 1)! |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_lagrange_unordered | [73, 1] | [103, 10] | field_simp [n.cast_add_one_ne_zero, n.factorial_ne_zero, xy_ne y hy] | case h
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (uIcc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (uIcc x₀ x)) (uIoo x₀ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc x₀ x)
xy_ne : ∀ y ∈ uIoo x₀ x, (x - y) ^ n ≠ 0
hg' : ∀ y ∈ uIoo x₀ x, -(↑n + 1) * (x - y) ^ n ≠ 0
y : ℝ
hy : y ∈ uIoo x₀ x
h :
f x - taylorWithinEval f n (uIcc x₀ x) x₀ x =
(-((x - y) ^ n / ↑n ! * (x - x₀) ^ (n + 1)) / (-(↑n + 1) * (x - y) ^ n)) •
iteratedDerivWithin (n + 1) f (uIcc x₀ x) y
⊢ ((x - y) ^ n / ↑n ! * (x - x₀) ^ (n + 1) / ((↑n + 1) * (x - y) ^ n)) • iteratedDerivWithin (n + 1) f (uIcc x₀ x) y =
iteratedDerivWithin (n + 1) f (uIcc x₀ x) y * (x - x₀) ^ (n + 1) / ↑(n + 1)! | case h
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (uIcc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (uIcc x₀ x)) (uIoo x₀ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc x₀ x)
xy_ne : ∀ y ∈ uIoo x₀ x, (x - y) ^ n ≠ 0
hg' : ∀ y ∈ uIoo x₀ x, -(↑n + 1) * (x - y) ^ n ≠ 0
y : ℝ
hy : y ∈ uIoo x₀ x
h :
f x - taylorWithinEval f n (uIcc x₀ x) x₀ x =
(-((x - y) ^ n / ↑n ! * (x - x₀) ^ (n + 1)) / (-(↑n + 1) * (x - y) ^ n)) •
iteratedDerivWithin (n + 1) f (uIcc x₀ x) y
⊢ (x - y) ^ n * (x - x₀) ^ (n + 1) * iteratedDerivWithin (n + 1) f (uIcc x₀ x) y * ↑(n + 1)! =
iteratedDerivWithin (n + 1) f (uIcc x₀ x) y * (x - x₀) ^ (n + 1) * (↑n ! * ((↑n + 1) * (x - y) ^ n)) |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_lagrange_unordered | [73, 1] | [103, 10] | ring_nf | case h
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (uIcc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (uIcc x₀ x)) (uIoo x₀ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc x₀ x)
xy_ne : ∀ y ∈ uIoo x₀ x, (x - y) ^ n ≠ 0
hg' : ∀ y ∈ uIoo x₀ x, -(↑n + 1) * (x - y) ^ n ≠ 0
y : ℝ
hy : y ∈ uIoo x₀ x
h :
f x - taylorWithinEval f n (uIcc x₀ x) x₀ x =
(-((x - y) ^ n / ↑n ! * (x - x₀) ^ (n + 1)) / (-(↑n + 1) * (x - y) ^ n)) •
iteratedDerivWithin (n + 1) f (uIcc x₀ x) y
⊢ (x - y) ^ n * (x - x₀) ^ (n + 1) * iteratedDerivWithin (n + 1) f (uIcc x₀ x) y * ↑(n + 1)! =
iteratedDerivWithin (n + 1) f (uIcc x₀ x) y * (x - x₀) ^ (n + 1) * (↑n ! * ((↑n + 1) * (x - y) ^ n)) | case h
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (uIcc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (uIcc x₀ x)) (uIoo x₀ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc x₀ x)
xy_ne : ∀ y ∈ uIoo x₀ x, (x - y) ^ n ≠ 0
hg' : ∀ y ∈ uIoo x₀ x, -(↑n + 1) * (x - y) ^ n ≠ 0
y : ℝ
hy : y ∈ uIoo x₀ x
h :
f x - taylorWithinEval f n (uIcc x₀ x) x₀ x =
(-((x - y) ^ n / ↑n ! * (x - x₀) ^ (n + 1)) / (-(↑n + 1) * (x - y) ^ n)) •
iteratedDerivWithin (n + 1) f (uIcc x₀ x) y
⊢ x * iteratedDerivWithin (1 + n) f (uIcc x₀ x) y * ↑(1 + n)! * (x - y) ^ n * (x - x₀) ^ n -
x₀ * iteratedDerivWithin (1 + n) f (uIcc x₀ x) y * ↑(1 + n)! * (x - y) ^ n * (x - x₀) ^ n =
x * iteratedDerivWithin (1 + n) f (uIcc x₀ x) y * ↑n ! * ↑n * (x - y) ^ n * (x - x₀) ^ n +
x * iteratedDerivWithin (1 + n) f (uIcc x₀ x) y * ↑n ! * (x - y) ^ n * (x - x₀) ^ n +
(-(x₀ * iteratedDerivWithin (1 + n) f (uIcc x₀ x) y * ↑n ! * ↑n * (x - y) ^ n * (x - x₀) ^ n) -
x₀ * iteratedDerivWithin (1 + n) f (uIcc x₀ x) y * ↑n ! * (x - y) ^ n * (x - x₀) ^ n) |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_lagrange_unordered | [73, 1] | [103, 10] | refine' Continuous.continuousOn _ | f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (uIcc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (uIcc x₀ x)) (uIoo x₀ x)
⊢ ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc x₀ x) | f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (uIcc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (uIcc x₀ x)) (uIoo x₀ x)
⊢ Continuous fun t => (x - t) ^ (n + 1) |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_lagrange_unordered | [73, 1] | [103, 10] | sorry | f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (uIcc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (uIcc x₀ x)) (uIoo x₀ x)
⊢ Continuous fun t => (x - t) ^ (n + 1) | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_lagrange_unordered | [73, 1] | [103, 10] | intro y hy | f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (uIcc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (uIcc x₀ x)) (uIoo x₀ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc x₀ x)
⊢ ∀ y ∈ uIoo x₀ x, (x - y) ^ n ≠ 0 | f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (uIcc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (uIcc x₀ x)) (uIoo x₀ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc x₀ x)
y : ℝ
hy : y ∈ uIoo x₀ x
⊢ (x - y) ^ n ≠ 0 |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_lagrange_unordered | [73, 1] | [103, 10] | refine' pow_ne_zero _ _ | f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (uIcc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (uIcc x₀ x)) (uIoo x₀ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc x₀ x)
y : ℝ
hy : y ∈ uIoo x₀ x
⊢ (x - y) ^ n ≠ 0 | f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (uIcc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (uIcc x₀ x)) (uIoo x₀ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc x₀ x)
y : ℝ
hy : y ∈ uIoo x₀ x
⊢ x - y ≠ 0 |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_lagrange_unordered | [73, 1] | [103, 10] | rw [sub_ne_zero] | f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (uIcc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (uIcc x₀ x)) (uIoo x₀ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc x₀ x)
y : ℝ
hy : y ∈ uIoo x₀ x
⊢ x - y ≠ 0 | f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (uIcc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (uIcc x₀ x)) (uIoo x₀ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc x₀ x)
y : ℝ
hy : y ∈ uIoo x₀ x
⊢ x ≠ y |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_lagrange_unordered | [73, 1] | [103, 10] | cases' le_total x₀ x with h h | f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (uIcc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (uIcc x₀ x)) (uIoo x₀ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc x₀ x)
y : ℝ
hy : y ∈ uIoo x₀ x
⊢ x ≠ y | case inl
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (uIcc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (uIcc x₀ x)) (uIoo x₀ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc x₀ x)
y : ℝ
hy : y ∈ uIoo x₀ x
h : x₀ ≤ x
⊢ x ≠ y
case inr
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (uIcc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (uIcc x₀ x)) (uIoo x₀ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc x₀ x)
y : ℝ
hy : y ∈ uIoo x₀ x
h : x ≤ x₀
⊢ x ≠ y |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_lagrange_unordered | [73, 1] | [103, 10] | rw [uIoo_of_le h] at hy | case inl
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (uIcc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (uIcc x₀ x)) (uIoo x₀ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc x₀ x)
y : ℝ
hy : y ∈ uIoo x₀ x
h : x₀ ≤ x
⊢ x ≠ y | case inl
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (uIcc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (uIcc x₀ x)) (uIoo x₀ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc x₀ x)
y : ℝ
hy : y ∈ Ioo x₀ x
h : x₀ ≤ x
⊢ x ≠ y |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_lagrange_unordered | [73, 1] | [103, 10] | exact hy.2.ne' | case inl
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (uIcc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (uIcc x₀ x)) (uIoo x₀ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc x₀ x)
y : ℝ
hy : y ∈ Ioo x₀ x
h : x₀ ≤ x
⊢ x ≠ y | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_lagrange_unordered | [73, 1] | [103, 10] | rw [uIoo_of_ge h] at hy | case inr
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (uIcc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (uIcc x₀ x)) (uIoo x₀ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc x₀ x)
y : ℝ
hy : y ∈ uIoo x₀ x
h : x ≤ x₀
⊢ x ≠ y | case inr
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (uIcc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (uIcc x₀ x)) (uIoo x₀ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc x₀ x)
y : ℝ
hy : y ∈ Ioo x x₀
h : x ≤ x₀
⊢ x ≠ y |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_lagrange_unordered | [73, 1] | [103, 10] | exact hy.1.ne | case inr
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hx : x₀ ≠ x
hf : ContDiffOn ℝ (↑n) f (uIcc x₀ x)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (uIcc x₀ x)) (uIoo x₀ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc x₀ x)
y : ℝ
hy : y ∈ Ioo x x₀
h : x ≤ x₀
⊢ x ≠ y | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_central_aux | [106, 1] | [147, 9] | rcases eq_or_ne x₀ x with (rfl | hx') | f g g' : ℝ → ℝ
x₀ x a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
⊢ ∃ x',
∃ (_ : x' ∈ Ioo a b),
x' ≠ x ∧
(f x - taylorWithinEval f n (Icc a b) x₀ x) * g' x' =
((x - x') ^ n / ↑n ! * (g x - g x₀)) • iteratedDerivWithin (n + 1) f (Icc a b) x' | case inl
f g g' : ℝ → ℝ
x₀ a b : ℝ
n : ℕ
hab : a < b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
hx : x₀ ∈ Icc a b
⊢ ∃ x',
∃ (_ : x' ∈ Ioo a b),
x' ≠ x₀ ∧
(f x₀ - taylorWithinEval f n (Icc a b) x₀ x₀) * g' x' =
((x₀ - x') ^ n / ↑n ! * (g x₀ - g x₀)) • iteratedDerivWithin (n + 1) f (Icc a b) x'
case inr
f g g' : ℝ → ℝ
x₀ x a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
hx' : x₀ ≠ x
⊢ ∃ x',
∃ (_ : x' ∈ Ioo a b),
x' ≠ x ∧
(f x - taylorWithinEval f n (Icc a b) x₀ x) * g' x' =
((x - x') ^ n / ↑n ! * (g x - g x₀)) • iteratedDerivWithin (n + 1) f (Icc a b) x' |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_central_aux | [106, 1] | [147, 9] | rcases Ne.lt_or_lt hx' with (hx' | hx') | case inr
f g g' : ℝ → ℝ
x₀ x a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
hx' : x₀ ≠ x
⊢ ∃ x',
∃ (_ : x' ∈ Ioo a b),
x' ≠ x ∧
(f x - taylorWithinEval f n (Icc a b) x₀ x) * g' x' =
((x - x') ^ n / ↑n ! * (g x - g x₀)) • iteratedDerivWithin (n + 1) f (Icc a b) x' | case inr.inl
f g g' : ℝ → ℝ
x₀ x a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
hx'✝ : x₀ ≠ x
hx' : x₀ < x
⊢ ∃ x',
∃ (_ : x' ∈ Ioo a b),
x' ≠ x ∧
(f x - taylorWithinEval f n (Icc a b) x₀ x) * g' x' =
((x - x') ^ n / ↑n ! * (g x - g x₀)) • iteratedDerivWithin (n + 1) f (Icc a b) x'
case inr.inr
f g g' : ℝ → ℝ
x₀ x a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
hx'✝ : x₀ ≠ x
hx' : x < x₀
⊢ ∃ x',
∃ (_ : x' ∈ Ioo a b),
x' ≠ x ∧
(f x - taylorWithinEval f n (Icc a b) x₀ x) * g' x' =
((x - x') ^ n / ↑n ! * (g x - g x₀)) • iteratedDerivWithin (n + 1) f (Icc a b) x' |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_central_aux | [106, 1] | [147, 9] | simp only [sub_self, taylorWithinEval_self, MulZeroClass.mul_zero, zero_div, zero_smul,
eq_self_iff_true, exists_prop, and_true_iff, MulZeroClass.zero_mul] | case inl
f g g' : ℝ → ℝ
x₀ a b : ℝ
n : ℕ
hab : a < b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
hx : x₀ ∈ Icc a b
⊢ ∃ x',
∃ (_ : x' ∈ Ioo a b),
x' ≠ x₀ ∧
(f x₀ - taylorWithinEval f n (Icc a b) x₀ x₀) * g' x' =
((x₀ - x') ^ n / ↑n ! * (g x₀ - g x₀)) • iteratedDerivWithin (n + 1) f (Icc a b) x' | case inl
f g g' : ℝ → ℝ
x₀ a b : ℝ
n : ℕ
hab : a < b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
hx : x₀ ∈ Icc a b
⊢ ∃ x' ∈ Ioo a b, x' ≠ x₀ |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_central_aux | [106, 1] | [147, 9] | obtain ⟨x', hx'⟩ := ((Ioo_infinite hab).diffₓ (Set.finite_singleton x₀)).Nonempty | case inl
f g g' : ℝ → ℝ
x₀ a b : ℝ
n : ℕ
hab : a < b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
hx : x₀ ∈ Icc a b
⊢ ∃ x' ∈ Ioo a b, x' ≠ x₀ | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_central_aux | [106, 1] | [147, 9] | have h₁ : Icc x₀ x ⊆ Icc a b := Icc_subset_Icc hx₀.1 hx.2 | case inr.inl
f g g' : ℝ → ℝ
x₀ x a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
hx'✝ : x₀ ≠ x
hx' : x₀ < x
⊢ ∃ x',
∃ (_ : x' ∈ Ioo a b),
x' ≠ x ∧
(f x - taylorWithinEval f n (Icc a b) x₀ x) * g' x' =
((x - x') ^ n / ↑n ! * (g x - g x₀)) • iteratedDerivWithin (n + 1) f (Icc a b) x' | case inr.inl
f g g' : ℝ → ℝ
x₀ x a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
hx'✝ : x₀ ≠ x
hx' : x₀ < x
h₁ : Icc x₀ x ⊆ Icc a b
⊢ ∃ x',
∃ (_ : x' ∈ Ioo a b),
x' ≠ x ∧
(f x - taylorWithinEval f n (Icc a b) x₀ x) * g' x' =
((x - x') ^ n / ↑n ! * (g x - g x₀)) • iteratedDerivWithin (n + 1) f (Icc a b) x' |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_central_aux | [106, 1] | [147, 9] | have h₂ : Ioo x₀ x ⊆ Ioo a b := Ioo_subset_Ioo hx₀.1 hx.2 | case inr.inl
f g g' : ℝ → ℝ
x₀ x a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
hx'✝ : x₀ ≠ x
hx' : x₀ < x
h₁ : Icc x₀ x ⊆ Icc a b
⊢ ∃ x',
∃ (_ : x' ∈ Ioo a b),
x' ≠ x ∧
(f x - taylorWithinEval f n (Icc a b) x₀ x) * g' x' =
((x - x') ^ n / ↑n ! * (g x - g x₀)) • iteratedDerivWithin (n + 1) f (Icc a b) x' | case inr.inl
f g g' : ℝ → ℝ
x₀ x a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
hx'✝ : x₀ ≠ x
hx' : x₀ < x
h₁ : Icc x₀ x ⊆ Icc a b
h₂ : Ioo x₀ x ⊆ Ioo a b
⊢ ∃ x',
∃ (_ : x' ∈ Ioo a b),
x' ≠ x ∧
(f x - taylorWithinEval f n (Icc a b) x₀ x) * g' x' =
((x - x') ^ n / ↑n ! * (g x - g x₀)) • iteratedDerivWithin (n + 1) f (Icc a b) x' |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_central_aux | [106, 1] | [147, 9] | obtain ⟨y, hy, h⟩ :=
exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc a b) t x)
(fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) hx'
((continuousOn_taylorWithinEval (uniqueDiffOn_Icc hab) hf).mono h₁)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo _ hab (h₂ hy) hf hf') g g' (gcont.mono h₁)
fun y hy => gdiff y (h₂ hy) | case inr.inl
f g g' : ℝ → ℝ
x₀ x a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
hx'✝ : x₀ ≠ x
hx' : x₀ < x
h₁ : Icc x₀ x ⊆ Icc a b
h₂ : Ioo x₀ x ⊆ Ioo a b
⊢ ∃ x',
∃ (_ : x' ∈ Ioo a b),
x' ≠ x ∧
(f x - taylorWithinEval f n (Icc a b) x₀ x) * g' x' =
((x - x') ^ n / ↑n ! * (g x - g x₀)) • iteratedDerivWithin (n + 1) f (Icc a b) x' | case inr.inl.intro.intro
f g g' : ℝ → ℝ
x₀ x a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
hx'✝ : x₀ ≠ x
hx' : x₀ < x
h₁ : Icc x₀ x ⊆ Icc a b
h₂ : Ioo x₀ x ⊆ Ioo a b
y : ℝ
hy : y ∈ Ioo x₀ x
h :
(g x - g x₀) * ((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y =
(taylorWithinEval f n (Icc a b) x x - taylorWithinEval f n (Icc a b) x₀ x) * g' y
⊢ ∃ x',
∃ (_ : x' ∈ Ioo a b),
x' ≠ x ∧
(f x - taylorWithinEval f n (Icc a b) x₀ x) * g' x' =
((x - x') ^ n / ↑n ! * (g x - g x₀)) • iteratedDerivWithin (n + 1) f (Icc a b) x' |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_central_aux | [106, 1] | [147, 9] | refine' ⟨y, h₂ hy, hy.2.Ne, _⟩ | case inr.inl.intro.intro
f g g' : ℝ → ℝ
x₀ x a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
hx'✝ : x₀ ≠ x
hx' : x₀ < x
h₁ : Icc x₀ x ⊆ Icc a b
h₂ : Ioo x₀ x ⊆ Ioo a b
y : ℝ
hy : y ∈ Ioo x₀ x
h :
(g x - g x₀) * ((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y =
(taylorWithinEval f n (Icc a b) x x - taylorWithinEval f n (Icc a b) x₀ x) * g' y
⊢ ∃ x',
∃ (_ : x' ∈ Ioo a b),
x' ≠ x ∧
(f x - taylorWithinEval f n (Icc a b) x₀ x) * g' x' =
((x - x') ^ n / ↑n ! * (g x - g x₀)) • iteratedDerivWithin (n + 1) f (Icc a b) x' | case inr.inl.intro.intro
f g g' : ℝ → ℝ
x₀ x a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
hx'✝ : x₀ ≠ x
hx' : x₀ < x
h₁ : Icc x₀ x ⊆ Icc a b
h₂ : Ioo x₀ x ⊆ Ioo a b
y : ℝ
hy : y ∈ Ioo x₀ x
h :
(g x - g x₀) * ((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y =
(taylorWithinEval f n (Icc a b) x x - taylorWithinEval f n (Icc a b) x₀ x) * g' y
⊢ (f x - taylorWithinEval f n (Icc a b) x₀ x) * g' y =
((x - y) ^ n / ↑n ! * (g x - g x₀)) • iteratedDerivWithin (n + 1) f (Icc a b) y |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_central_aux | [106, 1] | [147, 9] | simp only [taylorWithinEval_self] at h | case inr.inl.intro.intro
f g g' : ℝ → ℝ
x₀ x a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
hx'✝ : x₀ ≠ x
hx' : x₀ < x
h₁ : Icc x₀ x ⊆ Icc a b
h₂ : Ioo x₀ x ⊆ Ioo a b
y : ℝ
hy : y ∈ Ioo x₀ x
h :
(g x - g x₀) * ((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y =
(taylorWithinEval f n (Icc a b) x x - taylorWithinEval f n (Icc a b) x₀ x) * g' y
⊢ (f x - taylorWithinEval f n (Icc a b) x₀ x) * g' y =
((x - y) ^ n / ↑n ! * (g x - g x₀)) • iteratedDerivWithin (n + 1) f (Icc a b) y | case inr.inl.intro.intro
f g g' : ℝ → ℝ
x₀ x a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
hx'✝ : x₀ ≠ x
hx' : x₀ < x
h₁ : Icc x₀ x ⊆ Icc a b
h₂ : Ioo x₀ x ⊆ Ioo a b
y : ℝ
hy : y ∈ Ioo x₀ x
h :
(g x - g x₀) * ((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y =
(f x - taylorWithinEval f n (Icc a b) x₀ x) * g' y
⊢ (f x - taylorWithinEval f n (Icc a b) x₀ x) * g' y =
((x - y) ^ n / ↑n ! * (g x - g x₀)) • iteratedDerivWithin (n + 1) f (Icc a b) y |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_central_aux | [106, 1] | [147, 9] | field_simp [← h, n.factorial_ne_zero] | case inr.inl.intro.intro
f g g' : ℝ → ℝ
x₀ x a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
hx'✝ : x₀ ≠ x
hx' : x₀ < x
h₁ : Icc x₀ x ⊆ Icc a b
h₂ : Ioo x₀ x ⊆ Ioo a b
y : ℝ
hy : y ∈ Ioo x₀ x
h :
(g x - g x₀) * ((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y =
(f x - taylorWithinEval f n (Icc a b) x₀ x) * g' y
⊢ (f x - taylorWithinEval f n (Icc a b) x₀ x) * g' y =
((x - y) ^ n / ↑n ! * (g x - g x₀)) • iteratedDerivWithin (n + 1) f (Icc a b) y | case inr.inl.intro.intro
f g g' : ℝ → ℝ
x₀ x a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
hx'✝ : x₀ ≠ x
hx' : x₀ < x
h₁ : Icc x₀ x ⊆ Icc a b
h₂ : Ioo x₀ x ⊆ Ioo a b
y : ℝ
hy : y ∈ Ioo x₀ x
h :
(g x - g x₀) * ((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y =
(f x - taylorWithinEval f n (Icc a b) x₀ x) * g' y
⊢ (g x - g x₀) * ((x - y) ^ n * iteratedDerivWithin (n + 1) f (Icc a b) y) =
(x - y) ^ n * (g x - g x₀) * iteratedDerivWithin (n + 1) f (Icc a b) y |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_central_aux | [106, 1] | [147, 9] | ring | case inr.inl.intro.intro
f g g' : ℝ → ℝ
x₀ x a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
hx'✝ : x₀ ≠ x
hx' : x₀ < x
h₁ : Icc x₀ x ⊆ Icc a b
h₂ : Ioo x₀ x ⊆ Ioo a b
y : ℝ
hy : y ∈ Ioo x₀ x
h :
(g x - g x₀) * ((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y =
(f x - taylorWithinEval f n (Icc a b) x₀ x) * g' y
⊢ (g x - g x₀) * ((x - y) ^ n * iteratedDerivWithin (n + 1) f (Icc a b) y) =
(x - y) ^ n * (g x - g x₀) * iteratedDerivWithin (n + 1) f (Icc a b) y | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_central_aux | [106, 1] | [147, 9] | have h₁ : Icc x x₀ ⊆ Icc a b := Icc_subset_Icc hx.1 hx₀.2 | case inr.inr
f g g' : ℝ → ℝ
x₀ x a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
hx'✝ : x₀ ≠ x
hx' : x < x₀
⊢ ∃ x',
∃ (_ : x' ∈ Ioo a b),
x' ≠ x ∧
(f x - taylorWithinEval f n (Icc a b) x₀ x) * g' x' =
((x - x') ^ n / ↑n ! * (g x - g x₀)) • iteratedDerivWithin (n + 1) f (Icc a b) x' | case inr.inr
f g g' : ℝ → ℝ
x₀ x a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
hx'✝ : x₀ ≠ x
hx' : x < x₀
h₁ : Icc x x₀ ⊆ Icc a b
⊢ ∃ x',
∃ (_ : x' ∈ Ioo a b),
x' ≠ x ∧
(f x - taylorWithinEval f n (Icc a b) x₀ x) * g' x' =
((x - x') ^ n / ↑n ! * (g x - g x₀)) • iteratedDerivWithin (n + 1) f (Icc a b) x' |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_central_aux | [106, 1] | [147, 9] | have h₂ : Ioo x x₀ ⊆ Ioo a b := Ioo_subset_Ioo hx.1 hx₀.2 | case inr.inr
f g g' : ℝ → ℝ
x₀ x a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
hx'✝ : x₀ ≠ x
hx' : x < x₀
h₁ : Icc x x₀ ⊆ Icc a b
⊢ ∃ x',
∃ (_ : x' ∈ Ioo a b),
x' ≠ x ∧
(f x - taylorWithinEval f n (Icc a b) x₀ x) * g' x' =
((x - x') ^ n / ↑n ! * (g x - g x₀)) • iteratedDerivWithin (n + 1) f (Icc a b) x' | case inr.inr
f g g' : ℝ → ℝ
x₀ x a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
hx'✝ : x₀ ≠ x
hx' : x < x₀
h₁ : Icc x x₀ ⊆ Icc a b
h₂ : Ioo x x₀ ⊆ Ioo a b
⊢ ∃ x',
∃ (_ : x' ∈ Ioo a b),
x' ≠ x ∧
(f x - taylorWithinEval f n (Icc a b) x₀ x) * g' x' =
((x - x') ^ n / ↑n ! * (g x - g x₀)) • iteratedDerivWithin (n + 1) f (Icc a b) x' |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_central_aux | [106, 1] | [147, 9] | obtain ⟨y, hy, h⟩ :=
exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc a b) t x)
(fun t => ((n ! : ℝ)⁻¹ * (x - t) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) t) hx'
((continuousOn_taylorWithinEval (uniqueDiffOn_Icc hab) hf).mono h₁)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo _ hab (h₂ hy) hf hf') g g' (gcont.mono h₁)
fun y hy => gdiff y (h₂ hy) | case inr.inr
f g g' : ℝ → ℝ
x₀ x a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
hx'✝ : x₀ ≠ x
hx' : x < x₀
h₁ : Icc x x₀ ⊆ Icc a b
h₂ : Ioo x x₀ ⊆ Ioo a b
⊢ ∃ x',
∃ (_ : x' ∈ Ioo a b),
x' ≠ x ∧
(f x - taylorWithinEval f n (Icc a b) x₀ x) * g' x' =
((x - x') ^ n / ↑n ! * (g x - g x₀)) • iteratedDerivWithin (n + 1) f (Icc a b) x' | case inr.inr.intro.intro
f g g' : ℝ → ℝ
x₀ x a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
hx'✝ : x₀ ≠ x
hx' : x < x₀
h₁ : Icc x x₀ ⊆ Icc a b
h₂ : Ioo x x₀ ⊆ Ioo a b
y : ℝ
hy : y ∈ Ioo x x₀
h :
(g x₀ - g x) * ((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y =
(taylorWithinEval f n (Icc a b) x₀ x - taylorWithinEval f n (Icc a b) x x) * g' y
⊢ ∃ x',
∃ (_ : x' ∈ Ioo a b),
x' ≠ x ∧
(f x - taylorWithinEval f n (Icc a b) x₀ x) * g' x' =
((x - x') ^ n / ↑n ! * (g x - g x₀)) • iteratedDerivWithin (n + 1) f (Icc a b) x' |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_central_aux | [106, 1] | [147, 9] | refine' ⟨y, h₂ hy, hy.1.ne', _⟩ | case inr.inr.intro.intro
f g g' : ℝ → ℝ
x₀ x a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
hx'✝ : x₀ ≠ x
hx' : x < x₀
h₁ : Icc x x₀ ⊆ Icc a b
h₂ : Ioo x x₀ ⊆ Ioo a b
y : ℝ
hy : y ∈ Ioo x x₀
h :
(g x₀ - g x) * ((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y =
(taylorWithinEval f n (Icc a b) x₀ x - taylorWithinEval f n (Icc a b) x x) * g' y
⊢ ∃ x',
∃ (_ : x' ∈ Ioo a b),
x' ≠ x ∧
(f x - taylorWithinEval f n (Icc a b) x₀ x) * g' x' =
((x - x') ^ n / ↑n ! * (g x - g x₀)) • iteratedDerivWithin (n + 1) f (Icc a b) x' | case inr.inr.intro.intro
f g g' : ℝ → ℝ
x₀ x a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
hx'✝ : x₀ ≠ x
hx' : x < x₀
h₁ : Icc x x₀ ⊆ Icc a b
h₂ : Ioo x x₀ ⊆ Ioo a b
y : ℝ
hy : y ∈ Ioo x x₀
h :
(g x₀ - g x) * ((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y =
(taylorWithinEval f n (Icc a b) x₀ x - taylorWithinEval f n (Icc a b) x x) * g' y
⊢ (f x - taylorWithinEval f n (Icc a b) x₀ x) * g' y =
((x - y) ^ n / ↑n ! * (g x - g x₀)) • iteratedDerivWithin (n + 1) f (Icc a b) y |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_central_aux | [106, 1] | [147, 9] | simp only [taylorWithinEval_self] at h | case inr.inr.intro.intro
f g g' : ℝ → ℝ
x₀ x a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
hx'✝ : x₀ ≠ x
hx' : x < x₀
h₁ : Icc x x₀ ⊆ Icc a b
h₂ : Ioo x x₀ ⊆ Ioo a b
y : ℝ
hy : y ∈ Ioo x x₀
h :
(g x₀ - g x) * ((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y =
(taylorWithinEval f n (Icc a b) x₀ x - taylorWithinEval f n (Icc a b) x x) * g' y
⊢ (f x - taylorWithinEval f n (Icc a b) x₀ x) * g' y =
((x - y) ^ n / ↑n ! * (g x - g x₀)) • iteratedDerivWithin (n + 1) f (Icc a b) y | case inr.inr.intro.intro
f g g' : ℝ → ℝ
x₀ x a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
hx'✝ : x₀ ≠ x
hx' : x < x₀
h₁ : Icc x x₀ ⊆ Icc a b
h₂ : Ioo x x₀ ⊆ Ioo a b
y : ℝ
hy : y ∈ Ioo x x₀
h :
(g x₀ - g x) * ((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y =
(taylorWithinEval f n (Icc a b) x₀ x - f x) * g' y
⊢ (f x - taylorWithinEval f n (Icc a b) x₀ x) * g' y =
((x - y) ^ n / ↑n ! * (g x - g x₀)) • iteratedDerivWithin (n + 1) f (Icc a b) y |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_central_aux | [106, 1] | [147, 9] | rw [← neg_sub, neg_mul, ← h] | case inr.inr.intro.intro
f g g' : ℝ → ℝ
x₀ x a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
hx'✝ : x₀ ≠ x
hx' : x < x₀
h₁ : Icc x x₀ ⊆ Icc a b
h₂ : Ioo x x₀ ⊆ Ioo a b
y : ℝ
hy : y ∈ Ioo x x₀
h :
(g x₀ - g x) * ((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y =
(taylorWithinEval f n (Icc a b) x₀ x - f x) * g' y
⊢ (f x - taylorWithinEval f n (Icc a b) x₀ x) * g' y =
((x - y) ^ n / ↑n ! * (g x - g x₀)) • iteratedDerivWithin (n + 1) f (Icc a b) y | case inr.inr.intro.intro
f g g' : ℝ → ℝ
x₀ x a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
hx'✝ : x₀ ≠ x
hx' : x < x₀
h₁ : Icc x x₀ ⊆ Icc a b
h₂ : Ioo x x₀ ⊆ Ioo a b
y : ℝ
hy : y ∈ Ioo x x₀
h :
(g x₀ - g x) * ((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y =
(taylorWithinEval f n (Icc a b) x₀ x - f x) * g' y
⊢ -((g x₀ - g x) * ((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y) =
((x - y) ^ n / ↑n ! * (g x - g x₀)) • iteratedDerivWithin (n + 1) f (Icc a b) y |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_central_aux | [106, 1] | [147, 9] | field_simp [n.factorial_ne_zero] | case inr.inr.intro.intro
f g g' : ℝ → ℝ
x₀ x a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
hx'✝ : x₀ ≠ x
hx' : x < x₀
h₁ : Icc x x₀ ⊆ Icc a b
h₂ : Ioo x x₀ ⊆ Ioo a b
y : ℝ
hy : y ∈ Ioo x x₀
h :
(g x₀ - g x) * ((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y =
(taylorWithinEval f n (Icc a b) x₀ x - f x) * g' y
⊢ -((g x₀ - g x) * ((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y) =
((x - y) ^ n / ↑n ! * (g x - g x₀)) • iteratedDerivWithin (n + 1) f (Icc a b) y | case inr.inr.intro.intro
f g g' : ℝ → ℝ
x₀ x a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
hx'✝ : x₀ ≠ x
hx' : x < x₀
h₁ : Icc x x₀ ⊆ Icc a b
h₂ : Ioo x x₀ ⊆ Ioo a b
y : ℝ
hy : y ∈ Ioo x x₀
h :
(g x₀ - g x) * ((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y =
(taylorWithinEval f n (Icc a b) x₀ x - f x) * g' y
⊢ -((g x₀ - g x) * ((x - y) ^ n * iteratedDerivWithin (n + 1) f (Icc a b) y) * ↑n !) =
(x - y) ^ n * (g x - g x₀) * iteratedDerivWithin (n + 1) f (Icc a b) y * ↑n ! |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_central_aux | [106, 1] | [147, 9] | ring | case inr.inr.intro.intro
f g g' : ℝ → ℝ
x₀ x a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
hx'✝ : x₀ ≠ x
hx' : x < x₀
h₁ : Icc x x₀ ⊆ Icc a b
h₂ : Ioo x x₀ ⊆ Ioo a b
y : ℝ
hy : y ∈ Ioo x x₀
h :
(g x₀ - g x) * ((↑n !)⁻¹ * (x - y) ^ n) • iteratedDerivWithin (n + 1) f (Icc a b) y =
(taylorWithinEval f n (Icc a b) x₀ x - f x) * g' y
⊢ -((g x₀ - g x) * ((x - y) ^ n * iteratedDerivWithin (n + 1) f (Icc a b) y) * ↑n !) =
(x - y) ^ n * (g x - g x₀) * iteratedDerivWithin (n + 1) f (Icc a b) y * ↑n ! | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_central | [149, 1] | [162, 19] | obtain ⟨y, hy, hyx, h⟩ := taylor_mean_remainder_central_aux hab hx hx₀ hf hf' gcont gdiff | f g g' : ℝ → ℝ
x₀ x a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
g'_ne : ∀ y ∈ Ioo a b, g' y ≠ 0
⊢ ∃ x',
∃ (_ : x' ∈ Ioo a b),
f x - taylorWithinEval f n (Icc a b) x₀ x =
((x - x') ^ n / ↑n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc a b) x' | case intro.intro.intro
f g g' : ℝ → ℝ
x₀ x a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
g'_ne : ∀ y ∈ Ioo a b, g' y ≠ 0
y : ℝ
hy : y ∈ Ioo a b
hyx : y ≠ x
h :
(f x - taylorWithinEval f n (Icc a b) x₀ x) * g' y =
((x - y) ^ n / ↑n ! * (g x - g x₀)) • iteratedDerivWithin (n + 1) f (Icc a b) y
⊢ ∃ x',
∃ (_ : x' ∈ Ioo a b),
f x - taylorWithinEval f n (Icc a b) x₀ x =
((x - x') ^ n / ↑n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc a b) x' |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_central | [149, 1] | [162, 19] | refine' ⟨y, hy, _⟩ | case intro.intro.intro
f g g' : ℝ → ℝ
x₀ x a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
g'_ne : ∀ y ∈ Ioo a b, g' y ≠ 0
y : ℝ
hy : y ∈ Ioo a b
hyx : y ≠ x
h :
(f x - taylorWithinEval f n (Icc a b) x₀ x) * g' y =
((x - y) ^ n / ↑n ! * (g x - g x₀)) • iteratedDerivWithin (n + 1) f (Icc a b) y
⊢ ∃ x',
∃ (_ : x' ∈ Ioo a b),
f x - taylorWithinEval f n (Icc a b) x₀ x =
((x - x') ^ n / ↑n ! * (g x - g x₀) / g' x') • iteratedDerivWithin (n + 1) f (Icc a b) x' | case intro.intro.intro
f g g' : ℝ → ℝ
x₀ x a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
g'_ne : ∀ y ∈ Ioo a b, g' y ≠ 0
y : ℝ
hy : y ∈ Ioo a b
hyx : y ≠ x
h :
(f x - taylorWithinEval f n (Icc a b) x₀ x) * g' y =
((x - y) ^ n / ↑n ! * (g x - g x₀)) • iteratedDerivWithin (n + 1) f (Icc a b) y
⊢ f x - taylorWithinEval f n (Icc a b) x₀ x =
((x - y) ^ n / ↑n ! * (g x - g x₀) / g' y) • iteratedDerivWithin (n + 1) f (Icc a b) y |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_central | [149, 1] | [162, 19] | rw [smul_eq_mul] at h | case intro.intro.intro
f g g' : ℝ → ℝ
x₀ x a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
g'_ne : ∀ y ∈ Ioo a b, g' y ≠ 0
y : ℝ
hy : y ∈ Ioo a b
hyx : y ≠ x
h :
(f x - taylorWithinEval f n (Icc a b) x₀ x) * g' y =
((x - y) ^ n / ↑n ! * (g x - g x₀)) • iteratedDerivWithin (n + 1) f (Icc a b) y
⊢ f x - taylorWithinEval f n (Icc a b) x₀ x =
((x - y) ^ n / ↑n ! * (g x - g x₀) / g' y) • iteratedDerivWithin (n + 1) f (Icc a b) y | case intro.intro.intro
f g g' : ℝ → ℝ
x₀ x a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
g'_ne : ∀ y ∈ Ioo a b, g' y ≠ 0
y : ℝ
hy : y ∈ Ioo a b
hyx : y ≠ x
h :
(f x - taylorWithinEval f n (Icc a b) x₀ x) * g' y =
(x - y) ^ n / ↑n ! * (g x - g x₀) * iteratedDerivWithin (n + 1) f (Icc a b) y
⊢ f x - taylorWithinEval f n (Icc a b) x₀ x =
((x - y) ^ n / ↑n ! * (g x - g x₀) / g' y) • iteratedDerivWithin (n + 1) f (Icc a b) y |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_central | [149, 1] | [162, 19] | rw [smul_eq_mul, div_mul_eq_mul_div, ← h, mul_div_cancel] | case intro.intro.intro
f g g' : ℝ → ℝ
x₀ x a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
g'_ne : ∀ y ∈ Ioo a b, g' y ≠ 0
y : ℝ
hy : y ∈ Ioo a b
hyx : y ≠ x
h :
(f x - taylorWithinEval f n (Icc a b) x₀ x) * g' y =
(x - y) ^ n / ↑n ! * (g x - g x₀) * iteratedDerivWithin (n + 1) f (Icc a b) y
⊢ f x - taylorWithinEval f n (Icc a b) x₀ x =
((x - y) ^ n / ↑n ! * (g x - g x₀) / g' y) • iteratedDerivWithin (n + 1) f (Icc a b) y | case intro.intro.intro.h
f g g' : ℝ → ℝ
x₀ x a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
g'_ne : ∀ y ∈ Ioo a b, g' y ≠ 0
y : ℝ
hy : y ∈ Ioo a b
hyx : y ≠ x
h :
(f x - taylorWithinEval f n (Icc a b) x₀ x) * g' y =
(x - y) ^ n / ↑n ! * (g x - g x₀) * iteratedDerivWithin (n + 1) f (Icc a b) y
⊢ g' y ≠ 0 |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_central | [149, 1] | [162, 19] | exact g'_ne _ hy | case intro.intro.intro.h
f g g' : ℝ → ℝ
x₀ x a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : ∀ y ∈ Ioo a b, HasDerivAt g (g' y) y
g'_ne : ∀ y ∈ Ioo a b, g' y ≠ 0
y : ℝ
hy : y ∈ Ioo a b
hyx : y ≠ x
h :
(f x - taylorWithinEval f n (Icc a b) x₀ x) * g' y =
(x - y) ^ n / ↑n ! * (g x - g x₀) * iteratedDerivWithin (n + 1) f (Icc a b) y
⊢ g' y ≠ 0 | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_lagrange_central | [164, 1] | [186, 10] | have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc a b) := by
refine' Continuous.continuousOn _; continuity | f : ℝ → ℝ
x x₀ a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
⊢ ∃ x',
∃ (_ : x' ∈ Ioo a b),
f x - taylorWithinEval f n (Icc a b) x₀ x =
iteratedDerivWithin (n + 1) f (Icc a b) x' * (x - x₀) ^ (n + 1) / ↑(n + 1)! | f : ℝ → ℝ
x x₀ a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
⊢ ∃ x',
∃ (_ : x' ∈ Ioo a b),
f x - taylorWithinEval f n (Icc a b) x₀ x =
iteratedDerivWithin (n + 1) f (Icc a b) x' * (x - x₀) ^ (n + 1) / ↑(n + 1)! |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_lagrange_central | [164, 1] | [186, 10] | rcases taylor_mean_remainder_central_aux hab hx hx₀ hf hf' gcont fun y _ =>
monomial_has_deriv_aux y x _ with
⟨y, hy, hy', h⟩ | f : ℝ → ℝ
x x₀ a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
⊢ ∃ x',
∃ (_ : x' ∈ Ioo a b),
f x - taylorWithinEval f n (Icc a b) x₀ x =
iteratedDerivWithin (n + 1) f (Icc a b) x' * (x - x₀) ^ (n + 1) / ↑(n + 1)! | case intro.intro.intro
f : ℝ → ℝ
x x₀ a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : ℝ
hy : y ∈ Ioo a b
hy' : y ≠ x
h :
(f x - taylorWithinEval f n (Icc a b) x₀ x) * (-(↑n + 1) * (x - y) ^ n) =
((x - y) ^ n / ↑n ! * ((x - x) ^ (n + 1) - (x - x₀) ^ (n + 1))) • iteratedDerivWithin (n + 1) f (Icc a b) y
⊢ ∃ x',
∃ (_ : x' ∈ Ioo a b),
f x - taylorWithinEval f n (Icc a b) x₀ x =
iteratedDerivWithin (n + 1) f (Icc a b) x' * (x - x₀) ^ (n + 1) / ↑(n + 1)! |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_lagrange_central | [164, 1] | [186, 10] | have hy_ne : x - y ≠ 0 := sub_ne_zero_of_ne hy'.symm | case intro.intro.intro
f : ℝ → ℝ
x x₀ a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : ℝ
hy : y ∈ Ioo a b
hy' : y ≠ x
h :
(f x - taylorWithinEval f n (Icc a b) x₀ x) * (-(↑n + 1) * (x - y) ^ n) =
((x - y) ^ n / ↑n ! * ((x - x) ^ (n + 1) - (x - x₀) ^ (n + 1))) • iteratedDerivWithin (n + 1) f (Icc a b) y
⊢ ∃ x',
∃ (_ : x' ∈ Ioo a b),
f x - taylorWithinEval f n (Icc a b) x₀ x =
iteratedDerivWithin (n + 1) f (Icc a b) x' * (x - x₀) ^ (n + 1) / ↑(n + 1)! | case intro.intro.intro
f : ℝ → ℝ
x x₀ a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : ℝ
hy : y ∈ Ioo a b
hy' : y ≠ x
h :
(f x - taylorWithinEval f n (Icc a b) x₀ x) * (-(↑n + 1) * (x - y) ^ n) =
((x - y) ^ n / ↑n ! * ((x - x) ^ (n + 1) - (x - x₀) ^ (n + 1))) • iteratedDerivWithin (n + 1) f (Icc a b) y
hy_ne : x - y ≠ 0
⊢ ∃ x',
∃ (_ : x' ∈ Ioo a b),
f x - taylorWithinEval f n (Icc a b) x₀ x =
iteratedDerivWithin (n + 1) f (Icc a b) x' * (x - x₀) ^ (n + 1) / ↑(n + 1)! |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_lagrange_central | [164, 1] | [186, 10] | use y, hy | case intro.intro.intro
f : ℝ → ℝ
x x₀ a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : ℝ
hy : y ∈ Ioo a b
hy' : y ≠ x
h :
(f x - taylorWithinEval f n (Icc a b) x₀ x) * (-(↑n + 1) * (x - y) ^ n) =
((x - y) ^ n / ↑n ! * ((x - x) ^ (n + 1) - (x - x₀) ^ (n + 1))) • iteratedDerivWithin (n + 1) f (Icc a b) y
hy_ne : x - y ≠ 0
⊢ ∃ x',
∃ (_ : x' ∈ Ioo a b),
f x - taylorWithinEval f n (Icc a b) x₀ x =
iteratedDerivWithin (n + 1) f (Icc a b) x' * (x - x₀) ^ (n + 1) / ↑(n + 1)! | case h
f : ℝ → ℝ
x x₀ a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : ℝ
hy : y ∈ Ioo a b
hy' : y ≠ x
h :
(f x - taylorWithinEval f n (Icc a b) x₀ x) * (-(↑n + 1) * (x - y) ^ n) =
((x - y) ^ n / ↑n ! * ((x - x) ^ (n + 1) - (x - x₀) ^ (n + 1))) • iteratedDerivWithin (n + 1) f (Icc a b) y
hy_ne : x - y ≠ 0
⊢ f x - taylorWithinEval f n (Icc a b) x₀ x = iteratedDerivWithin (n + 1) f (Icc a b) y * (x - x₀) ^ (n + 1) / ↑(n + 1)! |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_lagrange_central | [164, 1] | [186, 10] | dsimp at h | case h
f : ℝ → ℝ
x x₀ a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : ℝ
hy : y ∈ Ioo a b
hy' : y ≠ x
h :
(f x - taylorWithinEval f n (Icc a b) x₀ x) * (-(↑n + 1) * (x - y) ^ n) =
((x - y) ^ n / ↑n ! * ((x - x) ^ (n + 1) - (x - x₀) ^ (n + 1))) • iteratedDerivWithin (n + 1) f (Icc a b) y
hy_ne : x - y ≠ 0
⊢ f x - taylorWithinEval f n (Icc a b) x₀ x = iteratedDerivWithin (n + 1) f (Icc a b) y * (x - x₀) ^ (n + 1) / ↑(n + 1)! | case h
f : ℝ → ℝ
x x₀ a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : ℝ
hy : y ∈ Ioo a b
hy' : y ≠ x
h :
(f x - taylorWithinEval f n (Icc a b) x₀ x) * (-(↑n + 1) * (x - y) ^ n) =
(x - y) ^ n / ↑n ! * ((x - x) ^ (n + 1) - (x - x₀) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y
hy_ne : x - y ≠ 0
⊢ f x - taylorWithinEval f n (Icc a b) x₀ x = iteratedDerivWithin (n + 1) f (Icc a b) y * (x - x₀) ^ (n + 1) / ↑(n + 1)! |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_lagrange_central | [164, 1] | [186, 10] | rw [← eq_div_iff] at h | case h
f : ℝ → ℝ
x x₀ a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : ℝ
hy : y ∈ Ioo a b
hy' : y ≠ x
h :
(f x - taylorWithinEval f n (Icc a b) x₀ x) * (-(↑n + 1) * (x - y) ^ n) =
(x - y) ^ n / ↑n ! * ((x - x) ^ (n + 1) - (x - x₀) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y
hy_ne : x - y ≠ 0
⊢ f x - taylorWithinEval f n (Icc a b) x₀ x = iteratedDerivWithin (n + 1) f (Icc a b) y * (x - x₀) ^ (n + 1) / ↑(n + 1)! | case h
f : ℝ → ℝ
x x₀ a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : ℝ
hy : y ∈ Ioo a b
hy' : y ≠ x
h :
f x - taylorWithinEval f n (Icc a b) x₀ x =
(x - y) ^ n / ↑n ! * ((x - x) ^ (n + 1) - (x - x₀) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y /
(-(↑n + 1) * (x - y) ^ n)
hy_ne : x - y ≠ 0
⊢ f x - taylorWithinEval f n (Icc a b) x₀ x = iteratedDerivWithin (n + 1) f (Icc a b) y * (x - x₀) ^ (n + 1) / ↑(n + 1)!
case h
f : ℝ → ℝ
x x₀ a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : ℝ
hy : y ∈ Ioo a b
hy' : y ≠ x
h :
(f x - taylorWithinEval f n (Icc a b) x₀ x) * (-(↑n + 1) * (x - y) ^ n) =
(x - y) ^ n / ↑n ! * ((x - x) ^ (n + 1) - (x - x₀) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y
hy_ne : x - y ≠ 0
⊢ -(↑n + 1) * (x - y) ^ n ≠ 0 |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_lagrange_central | [164, 1] | [186, 10] | swap | case h
f : ℝ → ℝ
x x₀ a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : ℝ
hy : y ∈ Ioo a b
hy' : y ≠ x
h :
f x - taylorWithinEval f n (Icc a b) x₀ x =
(x - y) ^ n / ↑n ! * ((x - x) ^ (n + 1) - (x - x₀) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y /
(-(↑n + 1) * (x - y) ^ n)
hy_ne : x - y ≠ 0
⊢ f x - taylorWithinEval f n (Icc a b) x₀ x = iteratedDerivWithin (n + 1) f (Icc a b) y * (x - x₀) ^ (n + 1) / ↑(n + 1)!
case h
f : ℝ → ℝ
x x₀ a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : ℝ
hy : y ∈ Ioo a b
hy' : y ≠ x
h :
(f x - taylorWithinEval f n (Icc a b) x₀ x) * (-(↑n + 1) * (x - y) ^ n) =
(x - y) ^ n / ↑n ! * ((x - x) ^ (n + 1) - (x - x₀) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y
hy_ne : x - y ≠ 0
⊢ -(↑n + 1) * (x - y) ^ n ≠ 0 | case h
f : ℝ → ℝ
x x₀ a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : ℝ
hy : y ∈ Ioo a b
hy' : y ≠ x
h :
(f x - taylorWithinEval f n (Icc a b) x₀ x) * (-(↑n + 1) * (x - y) ^ n) =
(x - y) ^ n / ↑n ! * ((x - x) ^ (n + 1) - (x - x₀) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y
hy_ne : x - y ≠ 0
⊢ -(↑n + 1) * (x - y) ^ n ≠ 0
case h
f : ℝ → ℝ
x x₀ a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : ℝ
hy : y ∈ Ioo a b
hy' : y ≠ x
h :
f x - taylorWithinEval f n (Icc a b) x₀ x =
(x - y) ^ n / ↑n ! * ((x - x) ^ (n + 1) - (x - x₀) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y /
(-(↑n + 1) * (x - y) ^ n)
hy_ne : x - y ≠ 0
⊢ f x - taylorWithinEval f n (Icc a b) x₀ x = iteratedDerivWithin (n + 1) f (Icc a b) y * (x - x₀) ^ (n + 1) / ↑(n + 1)! |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_lagrange_central | [164, 1] | [186, 10] | simp only [h, sub_self, zero_pow' _ (Nat.succ_ne_zero n), zero_sub, mul_neg, neg_mul,
Nat.factorial_succ, Nat.cast_add_one, neg_div_neg_eq, Nat.cast_mul, field_simps] | case h
f : ℝ → ℝ
x x₀ a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : ℝ
hy : y ∈ Ioo a b
hy' : y ≠ x
h :
f x - taylorWithinEval f n (Icc a b) x₀ x =
(x - y) ^ n / ↑n ! * ((x - x) ^ (n + 1) - (x - x₀) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y /
(-(↑n + 1) * (x - y) ^ n)
hy_ne : x - y ≠ 0
⊢ f x - taylorWithinEval f n (Icc a b) x₀ x = iteratedDerivWithin (n + 1) f (Icc a b) y * (x - x₀) ^ (n + 1) / ↑(n + 1)! | case h
f : ℝ → ℝ
x x₀ a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : ℝ
hy : y ∈ Ioo a b
hy' : y ≠ x
h :
f x - taylorWithinEval f n (Icc a b) x₀ x =
(x - y) ^ n / ↑n ! * ((x - x) ^ (n + 1) - (x - x₀) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y /
(-(↑n + 1) * (x - y) ^ n)
hy_ne : x - y ≠ 0
⊢ (x - y) ^ n * (x - x₀) ^ (n + 1) * iteratedDerivWithin (n + 1) f (Icc a b) y / (↑n ! * ((↑n + 1) * (x - y) ^ n)) =
iteratedDerivWithin (n + 1) f (Icc a b) y * (x - x₀) ^ (n + 1) / ((↑n + 1) * ↑n !) |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_lagrange_central | [164, 1] | [186, 10] | rw [mul_left_comm, ← mul_assoc, ← div_div, div_eq_iff (pow_ne_zero _ hy_ne), div_mul_eq_mul_div] | case h
f : ℝ → ℝ
x x₀ a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : ℝ
hy : y ∈ Ioo a b
hy' : y ≠ x
h :
f x - taylorWithinEval f n (Icc a b) x₀ x =
(x - y) ^ n / ↑n ! * ((x - x) ^ (n + 1) - (x - x₀) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y /
(-(↑n + 1) * (x - y) ^ n)
hy_ne : x - y ≠ 0
⊢ (x - y) ^ n * (x - x₀) ^ (n + 1) * iteratedDerivWithin (n + 1) f (Icc a b) y / (↑n ! * ((↑n + 1) * (x - y) ^ n)) =
iteratedDerivWithin (n + 1) f (Icc a b) y * (x - x₀) ^ (n + 1) / ((↑n + 1) * ↑n !) | case h
f : ℝ → ℝ
x x₀ a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : ℝ
hy : y ∈ Ioo a b
hy' : y ≠ x
h :
f x - taylorWithinEval f n (Icc a b) x₀ x =
(x - y) ^ n / ↑n ! * ((x - x) ^ (n + 1) - (x - x₀) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y /
(-(↑n + 1) * (x - y) ^ n)
hy_ne : x - y ≠ 0
⊢ (x - y) ^ n * (x - x₀) ^ (n + 1) * iteratedDerivWithin (n + 1) f (Icc a b) y / ((↑n + 1) * ↑n !) =
iteratedDerivWithin (n + 1) f (Icc a b) y * (x - x₀) ^ (n + 1) * (x - y) ^ n / ((↑n + 1) * ↑n !) |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_lagrange_central | [164, 1] | [186, 10] | congr 1 | case h
f : ℝ → ℝ
x x₀ a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : ℝ
hy : y ∈ Ioo a b
hy' : y ≠ x
h :
f x - taylorWithinEval f n (Icc a b) x₀ x =
(x - y) ^ n / ↑n ! * ((x - x) ^ (n + 1) - (x - x₀) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y /
(-(↑n + 1) * (x - y) ^ n)
hy_ne : x - y ≠ 0
⊢ (x - y) ^ n * (x - x₀) ^ (n + 1) * iteratedDerivWithin (n + 1) f (Icc a b) y / ((↑n + 1) * ↑n !) =
iteratedDerivWithin (n + 1) f (Icc a b) y * (x - x₀) ^ (n + 1) * (x - y) ^ n / ((↑n + 1) * ↑n !) | case h.e_a
f : ℝ → ℝ
x x₀ a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : ℝ
hy : y ∈ Ioo a b
hy' : y ≠ x
h :
f x - taylorWithinEval f n (Icc a b) x₀ x =
(x - y) ^ n / ↑n ! * ((x - x) ^ (n + 1) - (x - x₀) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y /
(-(↑n + 1) * (x - y) ^ n)
hy_ne : x - y ≠ 0
⊢ (x - y) ^ n * (x - x₀) ^ (n + 1) * iteratedDerivWithin (n + 1) f (Icc a b) y =
iteratedDerivWithin (n + 1) f (Icc a b) y * (x - x₀) ^ (n + 1) * (x - y) ^ n |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_lagrange_central | [164, 1] | [186, 10] | ring_nf | case h.e_a
f : ℝ → ℝ
x x₀ a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : ℝ
hy : y ∈ Ioo a b
hy' : y ≠ x
h :
f x - taylorWithinEval f n (Icc a b) x₀ x =
(x - y) ^ n / ↑n ! * ((x - x) ^ (n + 1) - (x - x₀) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y /
(-(↑n + 1) * (x - y) ^ n)
hy_ne : x - y ≠ 0
⊢ (x - y) ^ n * (x - x₀) ^ (n + 1) * iteratedDerivWithin (n + 1) f (Icc a b) y =
iteratedDerivWithin (n + 1) f (Icc a b) y * (x - x₀) ^ (n + 1) * (x - y) ^ n | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_lagrange_central | [164, 1] | [186, 10] | refine' Continuous.continuousOn _ | f : ℝ → ℝ
x x₀ a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
⊢ ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b) | f : ℝ → ℝ
x x₀ a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
⊢ Continuous fun t => (x - t) ^ (n + 1) |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_lagrange_central | [164, 1] | [186, 10] | exact mul_ne_zero (neg_ne_zero.2 (by positivity)) (by positivity) | case h
f : ℝ → ℝ
x x₀ a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : ℝ
hy : y ∈ Ioo a b
hy' : y ≠ x
h :
(f x - taylorWithinEval f n (Icc a b) x₀ x) * (-(↑n + 1) * (x - y) ^ n) =
(x - y) ^ n / ↑n ! * ((x - x) ^ (n + 1) - (x - x₀) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y
hy_ne : x - y ≠ 0
⊢ -(↑n + 1) * (x - y) ^ n ≠ 0 | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_lagrange_central | [164, 1] | [186, 10] | positivity | f : ℝ → ℝ
x x₀ a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : ℝ
hy : y ∈ Ioo a b
hy' : y ≠ x
h :
(f x - taylorWithinEval f n (Icc a b) x₀ x) * (-(↑n + 1) * (x - y) ^ n) =
(x - y) ^ n / ↑n ! * ((x - x) ^ (n + 1) - (x - x₀) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y
hy_ne : x - y ≠ 0
⊢ ↑n + 1 ≠ 0 | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_lagrange_central | [164, 1] | [186, 10] | positivity | f : ℝ → ℝ
x x₀ a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : ℝ
hy : y ∈ Ioo a b
hy' : y ≠ x
h :
(f x - taylorWithinEval f n (Icc a b) x₀ x) * (-(↑n + 1) * (x - y) ^ n) =
(x - y) ^ n / ↑n ! * ((x - x) ^ (n + 1) - (x - x₀) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y
hy_ne : x - y ≠ 0
⊢ (x - y) ^ n ≠ 0 | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_cauchy_central | [188, 1] | [202, 7] | rcases taylor_mean_remainder_central hab hx hx₀ hf hf' continuousOn_id
(fun _ _ => hasDerivAt_id _) fun _ _ => by simp with
⟨y, hy, h⟩ | f : ℝ → ℝ
x x₀ a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
⊢ ∃ x',
∃ (_ : x' ∈ Ioo a b),
f x - taylorWithinEval f n (Icc a b) x₀ x =
iteratedDerivWithin (n + 1) f (Icc a b) x' * (x - x') ^ n / ↑n ! * (x - x₀) | case intro.intro
f : ℝ → ℝ
x x₀ a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
y : ℝ
hy : y ∈ Ioo a b
h :
f x - taylorWithinEval f n (Icc a b) x₀ x =
((x - y) ^ n / ↑n ! * (id x - id x₀) / 1) • iteratedDerivWithin (n + 1) f (Icc a b) y
⊢ ∃ x',
∃ (_ : x' ∈ Ioo a b),
f x - taylorWithinEval f n (Icc a b) x₀ x =
iteratedDerivWithin (n + 1) f (Icc a b) x' * (x - x') ^ n / ↑n ! * (x - x₀) |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_cauchy_central | [188, 1] | [202, 7] | refine' ⟨y, hy, _⟩ | case intro.intro
f : ℝ → ℝ
x x₀ a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
y : ℝ
hy : y ∈ Ioo a b
h :
f x - taylorWithinEval f n (Icc a b) x₀ x =
((x - y) ^ n / ↑n ! * (id x - id x₀) / 1) • iteratedDerivWithin (n + 1) f (Icc a b) y
⊢ ∃ x',
∃ (_ : x' ∈ Ioo a b),
f x - taylorWithinEval f n (Icc a b) x₀ x =
iteratedDerivWithin (n + 1) f (Icc a b) x' * (x - x') ^ n / ↑n ! * (x - x₀) | case intro.intro
f : ℝ → ℝ
x x₀ a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
y : ℝ
hy : y ∈ Ioo a b
h :
f x - taylorWithinEval f n (Icc a b) x₀ x =
((x - y) ^ n / ↑n ! * (id x - id x₀) / 1) • iteratedDerivWithin (n + 1) f (Icc a b) y
⊢ f x - taylorWithinEval f n (Icc a b) x₀ x = iteratedDerivWithin (n + 1) f (Icc a b) y * (x - y) ^ n / ↑n ! * (x - x₀) |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_cauchy_central | [188, 1] | [202, 7] | rw [h] | case intro.intro
f : ℝ → ℝ
x x₀ a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
y : ℝ
hy : y ∈ Ioo a b
h :
f x - taylorWithinEval f n (Icc a b) x₀ x =
((x - y) ^ n / ↑n ! * (id x - id x₀) / 1) • iteratedDerivWithin (n + 1) f (Icc a b) y
⊢ f x - taylorWithinEval f n (Icc a b) x₀ x = iteratedDerivWithin (n + 1) f (Icc a b) y * (x - y) ^ n / ↑n ! * (x - x₀) | case intro.intro
f : ℝ → ℝ
x x₀ a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
y : ℝ
hy : y ∈ Ioo a b
h :
f x - taylorWithinEval f n (Icc a b) x₀ x =
((x - y) ^ n / ↑n ! * (id x - id x₀) / 1) • iteratedDerivWithin (n + 1) f (Icc a b) y
⊢ ((x - y) ^ n / ↑n ! * (id x - id x₀) / 1) • iteratedDerivWithin (n + 1) f (Icc a b) y =
iteratedDerivWithin (n + 1) f (Icc a b) y * (x - y) ^ n / ↑n ! * (x - x₀) |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_cauchy_central | [188, 1] | [202, 7] | field_simp [n.factorial_ne_zero] | case intro.intro
f : ℝ → ℝ
x x₀ a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
y : ℝ
hy : y ∈ Ioo a b
h :
f x - taylorWithinEval f n (Icc a b) x₀ x =
((x - y) ^ n / ↑n ! * (id x - id x₀) / 1) • iteratedDerivWithin (n + 1) f (Icc a b) y
⊢ ((x - y) ^ n / ↑n ! * (id x - id x₀) / 1) • iteratedDerivWithin (n + 1) f (Icc a b) y =
iteratedDerivWithin (n + 1) f (Icc a b) y * (x - y) ^ n / ↑n ! * (x - x₀) | case intro.intro
f : ℝ → ℝ
x x₀ a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
y : ℝ
hy : y ∈ Ioo a b
h :
f x - taylorWithinEval f n (Icc a b) x₀ x =
((x - y) ^ n / ↑n ! * (id x - id x₀) / 1) • iteratedDerivWithin (n + 1) f (Icc a b) y
⊢ (x - y) ^ n * (x - x₀) * iteratedDerivWithin (n + 1) f (Icc a b) y =
iteratedDerivWithin (n + 1) f (Icc a b) y * (x - y) ^ n * (x - x₀) |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_cauchy_central | [188, 1] | [202, 7] | ring | case intro.intro
f : ℝ → ℝ
x x₀ a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
y : ℝ
hy : y ∈ Ioo a b
h :
f x - taylorWithinEval f n (Icc a b) x₀ x =
((x - y) ^ n / ↑n ! * (id x - id x₀) / 1) • iteratedDerivWithin (n + 1) f (Icc a b) y
⊢ (x - y) ^ n * (x - x₀) * iteratedDerivWithin (n + 1) f (Icc a b) y =
iteratedDerivWithin (n + 1) f (Icc a b) y * (x - y) ^ n * (x - x₀) | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_cauchy_central | [188, 1] | [202, 7] | simp | f : ℝ → ℝ
x x₀ a b : ℝ
n : ℕ
hab : a < b
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hf : ContDiffOn ℝ (↑n) f (Icc a b)
hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
x✝¹ : ℝ
x✝ : x✝¹ ∈ Ioo a b
⊢ 1 ≠ 0 | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_bound_central | [204, 1] | [223, 61] | rcases eq_or_lt_of_le hab with (rfl | hab) | f : ℝ → ℝ
a b C x x₀ : ℝ
n : ℕ
hab : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hC : ∀ y ∈ Ioo a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C
⊢ ‖f x - taylorWithinEval f n (Icc a b) x₀ x‖ ≤ C * |x - x₀| ^ (n + 1) / ↑(n + 1)! | case inl
f : ℝ → ℝ
a C x x₀ : ℝ
n : ℕ
hab : a ≤ a
hf : ContDiffOn ℝ (↑n + 1) f (Icc a a)
hx : x ∈ Icc a a
hx₀ : x₀ ∈ Icc a a
hC : ∀ y ∈ Ioo a a, ‖iteratedDerivWithin (n + 1) f (Icc a a) y‖ ≤ C
⊢ ‖f x - taylorWithinEval f n (Icc a a) x₀ x‖ ≤ C * |x - x₀| ^ (n + 1) / ↑(n + 1)!
case inr
f : ℝ → ℝ
a b C x x₀ : ℝ
n : ℕ
hab✝ : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hC : ∀ y ∈ Ioo a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C
hab : a < b
⊢ ‖f x - taylorWithinEval f n (Icc a b) x₀ x‖ ≤ C * |x - x₀| ^ (n + 1) / ↑(n + 1)! |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_bound_central | [204, 1] | [223, 61] | have : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b) :=
by
refine'
(hf.differentiable_on_iterated_deriv_within _ (uniqueDiffOn_Icc hab)).mono Ioo_subset_Icc_self
rw [← Nat.cast_add_one, Nat.cast_lt]
exact Nat.lt_succ_self _ | case inr
f : ℝ → ℝ
a b C x x₀ : ℝ
n : ℕ
hab✝ : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hC : ∀ y ∈ Ioo a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C
hab : a < b
⊢ ‖f x - taylorWithinEval f n (Icc a b) x₀ x‖ ≤ C * |x - x₀| ^ (n + 1) / ↑(n + 1)! | case inr
f : ℝ → ℝ
a b C x x₀ : ℝ
n : ℕ
hab✝ : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hC : ∀ y ∈ Ioo a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C
hab : a < b
this : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
⊢ ‖f x - taylorWithinEval f n (Icc a b) x₀ x‖ ≤ C * |x - x₀| ^ (n + 1) / ↑(n + 1)! |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_bound_central | [204, 1] | [223, 61] | obtain ⟨x', hx', h⟩ := taylor_mean_remainder_lagrange_central hab hx hx₀ hf.of_succ this | case inr
f : ℝ → ℝ
a b C x x₀ : ℝ
n : ℕ
hab✝ : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hC : ∀ y ∈ Ioo a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C
hab : a < b
this : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
⊢ ‖f x - taylorWithinEval f n (Icc a b) x₀ x‖ ≤ C * |x - x₀| ^ (n + 1) / ↑(n + 1)! | case inr.intro.intro
f : ℝ → ℝ
a b C x x₀ : ℝ
n : ℕ
hab✝ : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hC : ∀ y ∈ Ioo a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C
hab : a < b
this : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
x' : ℝ
hx' : x' ∈ Ioo a b
h :
f x - taylorWithinEval f n (Icc a b) x₀ x =
iteratedDerivWithin (n + 1) f (Icc a b) x' * (x - x₀) ^ (n + 1) / ↑(n + 1)!
⊢ ‖f x - taylorWithinEval f n (Icc a b) x₀ x‖ ≤ C * |x - x₀| ^ (n + 1) / ↑(n + 1)! |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_bound_central | [204, 1] | [223, 61] | rw [h, norm_div, norm_mul, Real.norm_coe_nat, Real.norm_eq_abs ((x - x₀) ^ _), ← abs_pow] | case inr.intro.intro
f : ℝ → ℝ
a b C x x₀ : ℝ
n : ℕ
hab✝ : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hC : ∀ y ∈ Ioo a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C
hab : a < b
this : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
x' : ℝ
hx' : x' ∈ Ioo a b
h :
f x - taylorWithinEval f n (Icc a b) x₀ x =
iteratedDerivWithin (n + 1) f (Icc a b) x' * (x - x₀) ^ (n + 1) / ↑(n + 1)!
⊢ ‖f x - taylorWithinEval f n (Icc a b) x₀ x‖ ≤ C * |x - x₀| ^ (n + 1) / ↑(n + 1)! | case inr.intro.intro
f : ℝ → ℝ
a b C x x₀ : ℝ
n : ℕ
hab✝ : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hC : ∀ y ∈ Ioo a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C
hab : a < b
this : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
x' : ℝ
hx' : x' ∈ Ioo a b
h :
f x - taylorWithinEval f n (Icc a b) x₀ x =
iteratedDerivWithin (n + 1) f (Icc a b) x' * (x - x₀) ^ (n + 1) / ↑(n + 1)!
⊢ ‖iteratedDerivWithin (n + 1) f (Icc a b) x'‖ * |(x - x₀) ^ (n + 1)| / ↑(n + 1)! ≤ C * |(x - x₀) ^ (n + 1)| / ↑(n + 1)! |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_bound_central | [204, 1] | [223, 61] | refine' div_le_div_of_le (Nat.cast_nonneg _) _ | case inr.intro.intro
f : ℝ → ℝ
a b C x x₀ : ℝ
n : ℕ
hab✝ : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hC : ∀ y ∈ Ioo a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C
hab : a < b
this : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
x' : ℝ
hx' : x' ∈ Ioo a b
h :
f x - taylorWithinEval f n (Icc a b) x₀ x =
iteratedDerivWithin (n + 1) f (Icc a b) x' * (x - x₀) ^ (n + 1) / ↑(n + 1)!
⊢ ‖iteratedDerivWithin (n + 1) f (Icc a b) x'‖ * |(x - x₀) ^ (n + 1)| / ↑(n + 1)! ≤ C * |(x - x₀) ^ (n + 1)| / ↑(n + 1)! | case inr.intro.intro
f : ℝ → ℝ
a b C x x₀ : ℝ
n : ℕ
hab✝ : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hC : ∀ y ∈ Ioo a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C
hab : a < b
this : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
x' : ℝ
hx' : x' ∈ Ioo a b
h :
f x - taylorWithinEval f n (Icc a b) x₀ x =
iteratedDerivWithin (n + 1) f (Icc a b) x' * (x - x₀) ^ (n + 1) / ↑(n + 1)!
⊢ ‖iteratedDerivWithin (n + 1) f (Icc a b) x'‖ * |(x - x₀) ^ (n + 1)| ≤ C * |(x - x₀) ^ (n + 1)| |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_bound_central | [204, 1] | [223, 61] | exact mul_le_mul_of_nonneg_right (hC _ hx') (abs_nonneg _) | case inr.intro.intro
f : ℝ → ℝ
a b C x x₀ : ℝ
n : ℕ
hab✝ : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hC : ∀ y ∈ Ioo a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C
hab : a < b
this : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
x' : ℝ
hx' : x' ∈ Ioo a b
h :
f x - taylorWithinEval f n (Icc a b) x₀ x =
iteratedDerivWithin (n + 1) f (Icc a b) x' * (x - x₀) ^ (n + 1) / ↑(n + 1)!
⊢ ‖iteratedDerivWithin (n + 1) f (Icc a b) x'‖ * |(x - x₀) ^ (n + 1)| ≤ C * |(x - x₀) ^ (n + 1)| | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_bound_central | [204, 1] | [223, 61] | simp only [Icc_self, mem_singleton_iff] at hx hx₀ | case inl
f : ℝ → ℝ
a C x x₀ : ℝ
n : ℕ
hab : a ≤ a
hf : ContDiffOn ℝ (↑n + 1) f (Icc a a)
hx : x ∈ Icc a a
hx₀ : x₀ ∈ Icc a a
hC : ∀ y ∈ Ioo a a, ‖iteratedDerivWithin (n + 1) f (Icc a a) y‖ ≤ C
⊢ ‖f x - taylorWithinEval f n (Icc a a) x₀ x‖ ≤ C * |x - x₀| ^ (n + 1) / ↑(n + 1)! | case inl
f : ℝ → ℝ
a C x x₀ : ℝ
n : ℕ
hab : a ≤ a
hf : ContDiffOn ℝ (↑n + 1) f (Icc a a)
hC : ∀ y ∈ Ioo a a, ‖iteratedDerivWithin (n + 1) f (Icc a a) y‖ ≤ C
hx : x = a
hx₀ : x₀ = a
⊢ ‖f x - taylorWithinEval f n (Icc a a) x₀ x‖ ≤ C * |x - x₀| ^ (n + 1) / ↑(n + 1)! |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_bound_central | [204, 1] | [223, 61] | substs hx₀ hx | case inl
f : ℝ → ℝ
a C x x₀ : ℝ
n : ℕ
hab : a ≤ a
hf : ContDiffOn ℝ (↑n + 1) f (Icc a a)
hC : ∀ y ∈ Ioo a a, ‖iteratedDerivWithin (n + 1) f (Icc a a) y‖ ≤ C
hx : x = a
hx₀ : x₀ = a
⊢ ‖f x - taylorWithinEval f n (Icc a a) x₀ x‖ ≤ C * |x - x₀| ^ (n + 1) / ↑(n + 1)! | case inl
f : ℝ → ℝ
C x : ℝ
n : ℕ
hab : x ≤ x
hf : ContDiffOn ℝ (↑n + 1) f (Icc x x)
hC : ∀ y ∈ Ioo x x, ‖iteratedDerivWithin (n + 1) f (Icc x x) y‖ ≤ C
⊢ ‖f x - taylorWithinEval f n (Icc x x) x x‖ ≤ C * |x - x| ^ (n + 1) / ↑(n + 1)! |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_bound_central | [204, 1] | [223, 61] | rw [taylorWithinEval_self, sub_self, sub_self, abs_zero, zero_pow Nat.succ_pos',
MulZeroClass.mul_zero, zero_div, norm_zero] | case inl
f : ℝ → ℝ
C x : ℝ
n : ℕ
hab : x ≤ x
hf : ContDiffOn ℝ (↑n + 1) f (Icc x x)
hC : ∀ y ∈ Ioo x x, ‖iteratedDerivWithin (n + 1) f (Icc x x) y‖ ≤ C
⊢ ‖f x - taylorWithinEval f n (Icc x x) x x‖ ≤ C * |x - x| ^ (n + 1) / ↑(n + 1)! | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_bound_central | [204, 1] | [223, 61] | refine'
(hf.differentiable_on_iterated_deriv_within _ (uniqueDiffOn_Icc hab)).mono Ioo_subset_Icc_self | f : ℝ → ℝ
a b C x x₀ : ℝ
n : ℕ
hab✝ : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
hx : x ∈ Icc a b
hx₀ : x₀ ∈ Icc a b
hC : ∀ y ∈ Ioo a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C
hab : a < b
⊢ DifferentiableOn ℝ (iteratedDerivWithin n f (Icc a b)) (Ioo a b) | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | exists_taylor_mean_remainder_bound_central | [225, 1] | [239, 82] | rcases eq_or_lt_of_le hab with (rfl | h) | f : ℝ → ℝ
a b x₀ : ℝ
n : ℕ
hab : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
hx₀ : x₀ ∈ Icc a b
⊢ ∃ C, ∀ x ∈ Icc a b, ‖f x - taylorWithinEval f n (Icc a b) x₀ x‖ ≤ C * |x - x₀| ^ (n + 1) | case inl
f : ℝ → ℝ
a x₀ : ℝ
n : ℕ
hab : a ≤ a
hf : ContDiffOn ℝ (↑n + 1) f (Icc a a)
hx₀ : x₀ ∈ Icc a a
⊢ ∃ C, ∀ x ∈ Icc a a, ‖f x - taylorWithinEval f n (Icc a a) x₀ x‖ ≤ C * |x - x₀| ^ (n + 1)
case inr
f : ℝ → ℝ
a b x₀ : ℝ
n : ℕ
hab : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
hx₀ : x₀ ∈ Icc a b
h : a < b
⊢ ∃ C, ∀ x ∈ Icc a b, ‖f x - taylorWithinEval f n (Icc a b) x₀ x‖ ≤ C * |x - x₀| ^ (n + 1) |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | exists_taylor_mean_remainder_bound_central | [225, 1] | [239, 82] | let C := Sup ((fun y => ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖) '' Icc a b) | case inr
f : ℝ → ℝ
a b x₀ : ℝ
n : ℕ
hab : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
hx₀ : x₀ ∈ Icc a b
h : a < b
⊢ ∃ C, ∀ x ∈ Icc a b, ‖f x - taylorWithinEval f n (Icc a b) x₀ x‖ ≤ C * |x - x₀| ^ (n + 1) | case inr
f : ℝ → ℝ
a b x₀ : ℝ
n : ℕ
hab : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
hx₀ : x₀ ∈ Icc a b
h : a < b
C : Type := Sup ↑((fun y => ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖) '' Icc a b)
⊢ ∃ C, ∀ x ∈ Icc a b, ‖f x - taylorWithinEval f n (Icc a b) x₀ x‖ ≤ C * |x - x₀| ^ (n + 1) |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | exists_taylor_mean_remainder_bound_central | [225, 1] | [239, 82] | refine' ⟨C / (n + 1)!, fun x hx => _⟩ | case inr
f : ℝ → ℝ
a b x₀ : ℝ
n : ℕ
hab : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
hx₀ : x₀ ∈ Icc a b
h : a < b
C : Type := Sup ↑((fun y => ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖) '' Icc a b)
⊢ ∃ C, ∀ x ∈ Icc a b, ‖f x - taylorWithinEval f n (Icc a b) x₀ x‖ ≤ C * |x - x₀| ^ (n + 1) | case inr
f : ℝ → ℝ
a b x₀ : ℝ
n : ℕ
hab : a ≤ b
hf : ContDiffOn ℝ (↑n + 1) f (Icc a b)
hx₀ : x₀ ∈ Icc a b
h : a < b
C : Type := Sup ↑((fun y => ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖) '' Icc a b)
x : ℝ
hx : x ∈ Icc a b
⊢ ‖f x - taylorWithinEval f n (Icc a b) x₀ x‖ ≤ sorryAx ℝ true * |x - x₀| ^ (n + 1) |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | exists_taylor_mean_remainder_bound_central | [225, 1] | [239, 82] | refine' ⟨0, fun x hx => _⟩ | case inl
f : ℝ → ℝ
a x₀ : ℝ
n : ℕ
hab : a ≤ a
hf : ContDiffOn ℝ (↑n + 1) f (Icc a a)
hx₀ : x₀ ∈ Icc a a
⊢ ∃ C, ∀ x ∈ Icc a a, ‖f x - taylorWithinEval f n (Icc a a) x₀ x‖ ≤ C * |x - x₀| ^ (n + 1) | case inl
f : ℝ → ℝ
a x₀ : ℝ
n : ℕ
hab : a ≤ a
hf : ContDiffOn ℝ (↑n + 1) f (Icc a a)
hx₀ : x₀ ∈ Icc a a
x : ℝ
hx : x ∈ Icc a a
⊢ ‖f x - taylorWithinEval f n (Icc a a) x₀ x‖ ≤ 0 * |x - x₀| ^ (n + 1) |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | exists_taylor_mean_remainder_bound_central | [225, 1] | [239, 82] | rw [Icc_self, mem_singleton_iff] at hx hx₀ | case inl
f : ℝ → ℝ
a x₀ : ℝ
n : ℕ
hab : a ≤ a
hf : ContDiffOn ℝ (↑n + 1) f (Icc a a)
hx₀ : x₀ ∈ Icc a a
x : ℝ
hx : x ∈ Icc a a
⊢ ‖f x - taylorWithinEval f n (Icc a a) x₀ x‖ ≤ 0 * |x - x₀| ^ (n + 1) | case inl
f : ℝ → ℝ
a x₀ : ℝ
n : ℕ
hab : a ≤ a
hf : ContDiffOn ℝ (↑n + 1) f (Icc a a)
hx₀ : x₀ = a
x : ℝ
hx : x = a
⊢ ‖f x - taylorWithinEval f n (Icc a a) x₀ x‖ ≤ 0 * |x - x₀| ^ (n + 1) |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | exists_taylor_mean_remainder_bound_central | [225, 1] | [239, 82] | rw [hx₀, hx, taylorWithinEval_self, sub_self, MulZeroClass.zero_mul, norm_zero] | case inl
f : ℝ → ℝ
a x₀ : ℝ
n : ℕ
hab : a ≤ a
hf : ContDiffOn ℝ (↑n + 1) f (Icc a a)
hx₀ : x₀ = a
x : ℝ
hx : x = a
⊢ ‖f x - taylorWithinEval f n (Icc a a) x₀ x‖ ≤ 0 * |x - x₀| ^ (n + 1) | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/RamseyPrereq.lean | Fin.ne_zero_iff_eq_one | [23, 1] | [23, 75] | decide | ⊢ ∀ {x : Fin 2}, x ≠ 0 ↔ x = 1 | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/RamseyPrereq.lean | Fin.eq_zero_iff_ne_one | [25, 1] | [25, 75] | decide | ⊢ ∀ {x : Fin 2}, x = 0 ↔ x ≠ 1 | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/RamseyPrereq.lean | Fin.fin_two_eq_zero_of_ne_one | [27, 1] | [28, 31] | rwa [Fin.eq_zero_iff_ne_one] | x : Fin 2
hx : x ≠ 1
⊢ x = 0 | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/IteratedDeriv.lean | iteratedFDerivWithin_nhds | [29, 1] | [31, 82] | rw [← iteratedFDerivWithin_univ, ← univ_inter u, iteratedFDerivWithin_inter hu] | 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type u_2
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
u : Set E
x : E
f : E → F
n : ℕ
hu : u ∈ 𝓝 x
⊢ iteratedFDerivWithin 𝕜 n f u x = iteratedFDeriv 𝕜 n f x | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/IteratedDeriv.lean | iteratedDerivWithin_of_isOpen | [33, 1] | [35, 82] | rw [iteratedDerivWithin, iteratedDeriv, iteratedFDerivWithin_of_isOpen _ hs hx] | 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type u_2
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set 𝕜
f : 𝕜 → F
n : ℕ
hs : IsOpen s
x : 𝕜
hx : x ∈ s
⊢ iteratedDerivWithin n f s x = iteratedDeriv n f x | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/IteratedDeriv.lean | iteratedDerivWithin_nhds | [37, 1] | [39, 72] | rw [iteratedDerivWithin, iteratedDeriv, iteratedFDerivWithin_nhds hu] | 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type u_2
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
u : Set 𝕜
x : 𝕜
f : 𝕜 → F
n : ℕ
hu : u ∈ 𝓝 x
⊢ iteratedDerivWithin n f u x = iteratedDeriv n f x | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Data/Nat/Choose/Basic.lean | Nat.choose_add_le_pow_left | [18, 1] | [20, 54] | rw [add_comm, choose_eq_asc_factorial_div_factorial] | s t : ℕ
⊢ choose (s + t) s ≤ (t + 1) ^ s | s t : ℕ
⊢ ascFactorial t s / s ! ≤ (t + 1) ^ s |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Data/Nat/Choose/Basic.lean | Nat.choose_add_le_pow_left | [18, 1] | [20, 54] | exact Nat.div_le_of_le_mul asc_le_pow_mul_factorial | s t : ℕ
⊢ ascFactorial t s / s ! ≤ (t + 1) ^ s | no goals |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Data/Nat/Choose/Basic.lean | Nat.choose_le_pow_left | [22, 1] | [28, 17] | cases' le_or_lt t s with h h | s t : ℕ
⊢ choose s t ≤ (s + 1 - t) ^ t | case inl
s t : ℕ
h : t ≤ s
⊢ choose s t ≤ (s + 1 - t) ^ t
case inr
s t : ℕ
h : s < t
⊢ choose s t ≤ (s + 1 - t) ^ t |
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Data/Nat/Choose/Basic.lean | Nat.choose_le_pow_left | [22, 1] | [28, 17] | rw [choose_eq_zero_of_lt h] | case inr
s t : ℕ
h : s < t
⊢ choose s t ≤ (s + 1 - t) ^ t | case inr
s t : ℕ
h : s < t
⊢ 0 ≤ (s + 1 - t) ^ t |
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