url stringclasses 147 values | commit stringclasses 147 values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
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 hxβ.le] at g'_ne gdiff hf' β’ | case inr
f g g' : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (Icc x xβ)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc x xβ)) (uIoo xβ x)
gcont : ContinuousOn g (Icc x xβ)
gdiff : β x_1 β uIoo xβ x, HasDerivAt g (g' x_1) x_1
g'_ne : β x_1 β uIoo xβ x, g' x_1 β 0
hxβ : x < xβ
β’ β x',
β (_ : x' β uIoo xβ x),
f x - taylorWithinEval f n (Icc x xβ) xβ x =
((x - x') ^ n / βn ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc x xβ) x' | case inr
f g g' : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (Icc x xβ)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc x xβ)) (Ioo x xβ)
gcont : ContinuousOn g (Icc x xβ)
gdiff : β x_1 β Ioo x xβ, HasDerivAt g (g' x_1) x_1
g'_ne : β x_1 β Ioo x xβ, g' x_1 β 0
hxβ : x < xβ
β’ β x',
β (_ : x' β Ioo x xβ),
f x - taylorWithinEval f n (Icc x xβ) xβ x =
((x - x') ^ n / βn ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc x xβ) x' | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
f g g' : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (Icc x xβ)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc x xβ)) (uIoo xβ x)
gcont : ContinuousOn g (Icc x xβ)
gdiff : β x_1 β uIoo xβ x, HasDerivAt g (g' x_1) x_1
g'_ne : β x_1 β uIoo xβ x, g' x_1 β 0
hxβ : x < xβ
β’ β x',
β (_ : x' β uIoo xβ x),
f x - taylorWithinEval f n (Icc x xβ) xβ x =
((x - x') ^ n / βn ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc x xβ) x'
TACTIC:
|
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 xβ) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc x xβ) t) hxβ
(continuousOn_taylorWithinEval (uniqueDiffOn_Icc hxβ) hf)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo x hxβ hy hf hf') g g' gcont gdiff with
β¨y, hy, hβ© | case inr
f g g' : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (Icc x xβ)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc x xβ)) (Ioo x xβ)
gcont : ContinuousOn g (Icc x xβ)
gdiff : β x_1 β Ioo x xβ, HasDerivAt g (g' x_1) x_1
g'_ne : β x_1 β Ioo x xβ, g' x_1 β 0
hxβ : x < xβ
β’ β x',
β (_ : x' β Ioo x xβ),
f x - taylorWithinEval f n (Icc x xβ) xβ x =
((x - x') ^ n / βn ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc x xβ) x' | case inr.intro.intro
f g g' : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (Icc x xβ)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc x xβ)) (Ioo x xβ)
gcont : ContinuousOn g (Icc x xβ)
gdiff : β x_1 β Ioo x xβ, HasDerivAt g (g' x_1) x_1
g'_ne : β x_1 β Ioo x xβ, g' x_1 β 0
hxβ : x < xβ
y : β
hy : y β Ioo x xβ
h :
(g xβ - g x) * ((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc x xβ) y =
(taylorWithinEval f n (Icc x xβ) xβ x - taylorWithinEval f n (Icc x xβ) x x) * g' y
β’ β x',
β (_ : x' β Ioo x xβ),
f x - taylorWithinEval f n (Icc x xβ) xβ x =
((x - x') ^ n / βn ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc x xβ) x' | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
f g g' : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (Icc x xβ)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc x xβ)) (Ioo x xβ)
gcont : ContinuousOn g (Icc x xβ)
gdiff : β x_1 β Ioo x xβ, HasDerivAt g (g' x_1) x_1
g'_ne : β x_1 β Ioo x xβ, g' x_1 β 0
hxβ : x < xβ
β’ β x',
β (_ : x' β Ioo x xβ),
f x - taylorWithinEval f n (Icc x xβ) xβ x =
((x - x') ^ n / βn ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc x xβ) x'
TACTIC:
|
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' : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (Icc x xβ)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc x xβ)) (Ioo x xβ)
gcont : ContinuousOn g (Icc x xβ)
gdiff : β x_1 β Ioo x xβ, HasDerivAt g (g' x_1) x_1
g'_ne : β x_1 β Ioo x xβ, g' x_1 β 0
hxβ : x < xβ
y : β
hy : y β Ioo x xβ
h :
(g xβ - g x) * ((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc x xβ) y =
(taylorWithinEval f n (Icc x xβ) xβ x - taylorWithinEval f n (Icc x xβ) x x) * g' y
β’ β x',
β (_ : x' β Ioo x xβ),
f x - taylorWithinEval f n (Icc x xβ) xβ x =
((x - x') ^ n / βn ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc x xβ) x' | case h
f g g' : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (Icc x xβ)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc x xβ)) (Ioo x xβ)
gcont : ContinuousOn g (Icc x xβ)
gdiff : β x_1 β Ioo x xβ, HasDerivAt g (g' x_1) x_1
g'_ne : β x_1 β Ioo x xβ, g' x_1 β 0
hxβ : x < xβ
y : β
hy : y β Ioo x xβ
h :
(g xβ - g x) * ((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc x xβ) y =
(taylorWithinEval f n (Icc x xβ) xβ x - taylorWithinEval f n (Icc x xβ) x x) * g' y
β’ f x - taylorWithinEval f n (Icc x xβ) xβ x =
((x - y) ^ n / βn ! * (g x - g xβ) / g' y) β’ iteratedDerivWithin (n + 1) f (Icc x xβ) y | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.intro
f g g' : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (Icc x xβ)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc x xβ)) (Ioo x xβ)
gcont : ContinuousOn g (Icc x xβ)
gdiff : β x_1 β Ioo x xβ, HasDerivAt g (g' x_1) x_1
g'_ne : β x_1 β Ioo x xβ, g' x_1 β 0
hxβ : x < xβ
y : β
hy : y β Ioo x xβ
h :
(g xβ - g x) * ((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc x xβ) y =
(taylorWithinEval f n (Icc x xβ) xβ x - taylorWithinEval f n (Icc x xβ) x x) * g' y
β’ β x',
β (_ : x' β Ioo x xβ),
f x - taylorWithinEval f n (Icc x xβ) xβ x =
((x - x') ^ n / βn ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc x xβ) x'
TACTIC:
|
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' : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (Icc x xβ)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc x xβ)) (Ioo x xβ)
gcont : ContinuousOn g (Icc x xβ)
gdiff : β x_1 β Ioo x xβ, HasDerivAt g (g' x_1) x_1
g'_ne : β x_1 β Ioo x xβ, g' x_1 β 0
hxβ : x < xβ
y : β
hy : y β Ioo x xβ
h :
(g xβ - g x) * ((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc x xβ) y =
(taylorWithinEval f n (Icc x xβ) xβ x - taylorWithinEval f n (Icc x xβ) x x) * g' y
β’ f x - taylorWithinEval f n (Icc x xβ) xβ x =
((x - y) ^ n / βn ! * (g x - g xβ) / g' y) β’ iteratedDerivWithin (n + 1) f (Icc x xβ) y | case h
f g g' : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (Icc x xβ)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc x xβ)) (Ioo x xβ)
gcont : ContinuousOn g (Icc x xβ)
gdiff : β x_1 β Ioo x xβ, HasDerivAt g (g' x_1) x_1
g'_ne : β x_1 β Ioo x xβ, g' x_1 β 0
hxβ : x < xβ
y : β
hy : y β Ioo x xβ
h :
(g xβ - g x) * ((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc x xβ) y =
(taylorWithinEval f n (Icc x xβ) xβ x - f x) * g' y
β’ f x - taylorWithinEval f n (Icc x xβ) xβ x =
((x - y) ^ n / βn ! * (g x - g xβ) / g' y) β’ iteratedDerivWithin (n + 1) f (Icc x xβ) y | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f g g' : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (Icc x xβ)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc x xβ)) (Ioo x xβ)
gcont : ContinuousOn g (Icc x xβ)
gdiff : β x_1 β Ioo x xβ, HasDerivAt g (g' x_1) x_1
g'_ne : β x_1 β Ioo x xβ, g' x_1 β 0
hxβ : x < xβ
y : β
hy : y β Ioo x xβ
h :
(g xβ - g x) * ((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc x xβ) y =
(taylorWithinEval f n (Icc x xβ) xβ x - taylorWithinEval f n (Icc x xβ) x x) * g' y
β’ f x - taylorWithinEval f n (Icc x xβ) xβ x =
((x - y) ^ n / βn ! * (g x - g xβ) / g' y) β’ iteratedDerivWithin (n + 1) f (Icc x xβ) y
TACTIC:
|
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, β div_left_inj' (g'_ne y hy), mul_div_cancel _ (g'_ne y hy)] at h | case h
f g g' : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (Icc x xβ)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc x xβ)) (Ioo x xβ)
gcont : ContinuousOn g (Icc x xβ)
gdiff : β x_1 β Ioo x xβ, HasDerivAt g (g' x_1) x_1
g'_ne : β x_1 β Ioo x xβ, g' x_1 β 0
hxβ : x < xβ
y : β
hy : y β Ioo x xβ
h :
(g xβ - g x) * ((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc x xβ) y =
(taylorWithinEval f n (Icc x xβ) xβ x - f x) * g' y
β’ f x - taylorWithinEval f n (Icc x xβ) xβ x =
((x - y) ^ n / βn ! * (g x - g xβ) / g' y) β’ iteratedDerivWithin (n + 1) f (Icc x xβ) y | case h
f g g' : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (Icc x xβ)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc x xβ)) (Ioo x xβ)
gcont : ContinuousOn g (Icc x xβ)
gdiff : β x_1 β Ioo x xβ, HasDerivAt g (g' x_1) x_1
g'_ne : β x_1 β Ioo x xβ, g' x_1 β 0
hxβ : x < xβ
y : β
hy : y β Ioo x xβ
h :
((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc x xβ) y * (g xβ - g x) / g' y =
taylorWithinEval f n (Icc x xβ) xβ x - f x
β’ f x - taylorWithinEval f n (Icc x xβ) xβ x =
((x - y) ^ n / βn ! * (g x - g xβ) / g' y) β’ iteratedDerivWithin (n + 1) f (Icc x xβ) y | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f g g' : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (Icc x xβ)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc x xβ)) (Ioo x xβ)
gcont : ContinuousOn g (Icc x xβ)
gdiff : β x_1 β Ioo x xβ, HasDerivAt g (g' x_1) x_1
g'_ne : β x_1 β Ioo x xβ, g' x_1 β 0
hxβ : x < xβ
y : β
hy : y β Ioo x xβ
h :
(g xβ - g x) * ((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc x xβ) y =
(taylorWithinEval f n (Icc x xβ) xβ x - f x) * g' y
β’ f x - taylorWithinEval f n (Icc x xβ) xβ x =
((x - y) ^ n / βn ! * (g x - g xβ) / g' y) β’ iteratedDerivWithin (n + 1) f (Icc x xβ) y
TACTIC:
|
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_unordered | [43, 1] | [71, 7] | rw [β neg_sub, β h] | case h
f g g' : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (Icc x xβ)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc x xβ)) (Ioo x xβ)
gcont : ContinuousOn g (Icc x xβ)
gdiff : β x_1 β Ioo x xβ, HasDerivAt g (g' x_1) x_1
g'_ne : β x_1 β Ioo x xβ, g' x_1 β 0
hxβ : x < xβ
y : β
hy : y β Ioo x xβ
h :
((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc x xβ) y * (g xβ - g x) / g' y =
taylorWithinEval f n (Icc x xβ) xβ x - f x
β’ f x - taylorWithinEval f n (Icc x xβ) xβ x =
((x - y) ^ n / βn ! * (g x - g xβ) / g' y) β’ iteratedDerivWithin (n + 1) f (Icc x xβ) y | case h
f g g' : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (Icc x xβ)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc x xβ)) (Ioo x xβ)
gcont : ContinuousOn g (Icc x xβ)
gdiff : β x_1 β Ioo x xβ, HasDerivAt g (g' x_1) x_1
g'_ne : β x_1 β Ioo x xβ, g' x_1 β 0
hxβ : x < xβ
y : β
hy : y β Ioo x xβ
h :
((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc x xβ) y * (g xβ - g x) / g' y =
taylorWithinEval f n (Icc x xβ) xβ x - f x
β’ -(((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc x xβ) y * (g xβ - g x) / g' y) =
((x - y) ^ n / βn ! * (g x - g xβ) / g' y) β’ iteratedDerivWithin (n + 1) f (Icc x xβ) y | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f g g' : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (Icc x xβ)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc x xβ)) (Ioo x xβ)
gcont : ContinuousOn g (Icc x xβ)
gdiff : β x_1 β Ioo x xβ, HasDerivAt g (g' x_1) x_1
g'_ne : β x_1 β Ioo x xβ, g' x_1 β 0
hxβ : x < xβ
y : β
hy : y β Ioo x xβ
h :
((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc x xβ) y * (g xβ - g x) / g' y =
taylorWithinEval f n (Icc x xβ) xβ x - f x
β’ f x - taylorWithinEval f n (Icc x xβ) xβ x =
((x - y) ^ n / βn ! * (g x - g xβ) / g' y) β’ iteratedDerivWithin (n + 1) f (Icc x xβ) y
TACTIC:
|
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' : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (Icc x xβ)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc x xβ)) (Ioo x xβ)
gcont : ContinuousOn g (Icc x xβ)
gdiff : β x_1 β Ioo x xβ, HasDerivAt g (g' x_1) x_1
g'_ne : β x_1 β Ioo x xβ, g' x_1 β 0
hxβ : x < xβ
y : β
hy : y β Ioo x xβ
h :
((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc x xβ) y * (g xβ - g x) / g' y =
taylorWithinEval f n (Icc x xβ) xβ x - f x
β’ -(((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc x xβ) y * (g xβ - g x) / g' y) =
((x - y) ^ n / βn ! * (g x - g xβ) / g' y) β’ iteratedDerivWithin (n + 1) f (Icc x xβ) y | case h
f g g' : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (Icc x xβ)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc x xβ)) (Ioo x xβ)
gcont : ContinuousOn g (Icc x xβ)
gdiff : β x_1 β Ioo x xβ, HasDerivAt g (g' x_1) x_1
g'_ne : β x_1 β Ioo x xβ, g' x_1 β 0
hxβ : x < xβ
y : β
hy : y β Ioo x xβ
h :
((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc x xβ) y * (g xβ - g x) / g' y =
taylorWithinEval f n (Icc x xβ) xβ x - f x
β’ -((x - y) ^ n * iteratedDerivWithin (n + 1) f (Icc x xβ) y * (g xβ - g x) * (βn ! * g' y)) =
(x - y) ^ n * (g x - g xβ) * iteratedDerivWithin (n + 1) f (Icc x xβ) y * (βn ! * g' y) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f g g' : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (Icc x xβ)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc x xβ)) (Ioo x xβ)
gcont : ContinuousOn g (Icc x xβ)
gdiff : β x_1 β Ioo x xβ, HasDerivAt g (g' x_1) x_1
g'_ne : β x_1 β Ioo x xβ, g' x_1 β 0
hxβ : x < xβ
y : β
hy : y β Ioo x xβ
h :
((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc x xβ) y * (g xβ - g x) / g' y =
taylorWithinEval f n (Icc x xβ) xβ x - f x
β’ -(((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc x xβ) y * (g xβ - g x) / g' y) =
((x - y) ^ n / βn ! * (g x - g xβ) / g' y) β’ iteratedDerivWithin (n + 1) f (Icc x xβ) y
TACTIC:
|
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' : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (Icc x xβ)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc x xβ)) (Ioo x xβ)
gcont : ContinuousOn g (Icc x xβ)
gdiff : β x_1 β Ioo x xβ, HasDerivAt g (g' x_1) x_1
g'_ne : β x_1 β Ioo x xβ, g' x_1 β 0
hxβ : x < xβ
y : β
hy : y β Ioo x xβ
h :
((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc x xβ) y * (g xβ - g x) / g' y =
taylorWithinEval f n (Icc x xβ) xβ x - f x
β’ -((x - y) ^ n * iteratedDerivWithin (n + 1) f (Icc x xβ) y * (g xβ - g x) * (βn ! * g' y)) =
(x - y) ^ n * (g x - g xβ) * iteratedDerivWithin (n + 1) f (Icc x xβ) y * (βn ! * g' y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f g g' : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (Icc x xβ)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc x xβ)) (Ioo x xβ)
gcont : ContinuousOn g (Icc x xβ)
gdiff : β x_1 β Ioo x xβ, HasDerivAt g (g' x_1) x_1
g'_ne : β x_1 β Ioo x xβ, g' x_1 β 0
hxβ : x < xβ
y : β
hy : y β Ioo x xβ
h :
((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc x xβ) y * (g xβ - g x) / g' y =
taylorWithinEval f n (Icc x xβ) xβ x - f x
β’ -((x - y) ^ n * iteratedDerivWithin (n + 1) f (Icc x xβ) y * (g xβ - g x) * (βn ! * g' y)) =
(x - y) ^ n * (g x - g xβ) * iteratedDerivWithin (n + 1) f (Icc x xβ) y * (βn ! * g' y)
TACTIC:
|
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 hxβ.le] at hf hf' gcont β’ | case inl
f g g' : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn g (uIcc xβ x)
gdiff : β x_1 β uIoo xβ x, HasDerivAt g (g' x_1) x_1
g'_ne : β x_1 β uIoo xβ x, g' x_1 β 0
hxβ : xβ < x
β’ β x',
β (_ : x' β uIoo xβ x),
f x - taylorWithinEval f n (uIcc xβ x) xβ x =
((x - x') ^ n / βn ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (uIcc xβ x) x' | case inl
f g g' : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (Icc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (uIoo xβ x)
gcont : ContinuousOn g (Icc xβ x)
gdiff : β x_1 β uIoo xβ x, HasDerivAt g (g' x_1) x_1
g'_ne : β x_1 β uIoo xβ x, g' x_1 β 0
hxβ : xβ < x
β’ β x',
β (_ : x' β uIoo xβ x),
f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / βn ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
f g g' : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn g (uIcc xβ x)
gdiff : β x_1 β uIoo xβ x, HasDerivAt g (g' x_1) x_1
g'_ne : β x_1 β uIoo xβ x, g' x_1 β 0
hxβ : xβ < x
β’ β x',
β (_ : x' β uIoo xβ x),
f x - taylorWithinEval f n (uIcc xβ x) xβ x =
((x - x') ^ n / βn ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (uIcc xβ x) x'
TACTIC:
|
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 hxβ.le] at g'_ne gdiff hf' β’ | case inl
f g g' : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (Icc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (uIoo xβ x)
gcont : ContinuousOn g (Icc xβ x)
gdiff : β x_1 β uIoo xβ x, HasDerivAt g (g' x_1) x_1
g'_ne : β x_1 β uIoo xβ x, g' x_1 β 0
hxβ : xβ < x
β’ β x',
β (_ : x' β uIoo xβ x),
f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / βn ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' | case inl
f g g' : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (Icc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)
gcont : ContinuousOn g (Icc xβ x)
gdiff : β x_1 β Ioo xβ x, HasDerivAt g (g' x_1) x_1
g'_ne : β x_1 β Ioo xβ x, g' x_1 β 0
hxβ : xβ < x
β’ β x',
β (_ : x' β Ioo xβ x),
f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / βn ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
f g g' : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (Icc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (uIoo xβ x)
gcont : ContinuousOn g (Icc xβ x)
gdiff : β x_1 β uIoo xβ x, HasDerivAt g (g' x_1) x_1
g'_ne : β x_1 β uIoo xβ x, g' x_1 β 0
hxβ : xβ < x
β’ β x',
β (_ : x' β uIoo xβ x),
f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / βn ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x'
TACTIC:
|
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 hxβ hf hf' gcont gdiff g'_ne | case inl
f g g' : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (Icc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)
gcont : ContinuousOn g (Icc xβ x)
gdiff : β x_1 β Ioo xβ x, HasDerivAt g (g' x_1) x_1
g'_ne : β x_1 β Ioo xβ x, g' x_1 β 0
hxβ : xβ < x
β’ β x',
β (_ : x' β Ioo xβ x),
f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / βn ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
f g g' : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (Icc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc xβ x)) (Ioo xβ x)
gcont : ContinuousOn g (Icc xβ x)
gdiff : β x_1 β Ioo xβ x, HasDerivAt g (g' x_1) x_1
g'_ne : β x_1 β Ioo xβ x, g' x_1 β 0
hxβ : xβ < x
β’ β x',
β (_ : x' β Ioo xβ x),
f x - taylorWithinEval f n (Icc xβ x) xβ x =
((x - x') ^ n / βn ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc xβ x) x'
TACTIC:
|
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 : β => (x - t) ^ (n + 1)) (uIcc xβ x) := by
refine' Continuous.continuousOn _
sorry | f : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
β’ β x',
β (_ : x' β uIoo xβ x),
f x - taylorWithinEval f n (uIcc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (uIcc xβ x) x' * (x - xβ) ^ (n + 1) / β(n + 1)! | f : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
β’ β x',
β (_ : x' β uIoo xβ x),
f x - taylorWithinEval f n (uIcc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (uIcc xβ x) x' * (x - xβ) ^ (n + 1) / β(n + 1)! | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
β’ β x',
β (_ : x' β uIoo xβ x),
f x - taylorWithinEval f n (uIcc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (uIcc xβ x) x' * (x - xβ) ^ (n + 1) / β(n + 1)!
TACTIC:
|
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' : β y : β, y β uIoo xβ x β -(βn + 1) * (x - y) ^ n β 0 := fun y hy =>
mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy) | f : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
xy_ne : β y β uIoo xβ x, (x - y) ^ n β 0
β’ β x',
β (_ : x' β uIoo xβ x),
f x - taylorWithinEval f n (uIcc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (uIcc xβ x) x' * (x - xβ) ^ (n + 1) / β(n + 1)! | f : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
xy_ne : β y β uIoo xβ x, (x - y) ^ n β 0
hg' : β y β uIoo xβ x, -(βn + 1) * (x - y) ^ n β 0
β’ β x',
β (_ : x' β uIoo xβ x),
f x - taylorWithinEval f n (uIcc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (uIcc xβ x) x' * (x - xβ) ^ (n + 1) / β(n + 1)! | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
xy_ne : β y β uIoo xβ x, (x - y) ^ n β 0
β’ β x',
β (_ : x' β uIoo xβ x),
f x - taylorWithinEval f n (uIcc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (uIcc xβ x) x' * (x - xβ) ^ (n + 1) / β(n + 1)!
TACTIC:
|
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
β¨y, hy, hβ© | f : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
xy_ne : β y β uIoo xβ x, (x - y) ^ n β 0
hg' : β y β uIoo xβ x, -(βn + 1) * (x - y) ^ n β 0
β’ β x',
β (_ : x' β uIoo xβ x),
f x - taylorWithinEval f n (uIcc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (uIcc xβ x) x' * (x - xβ) ^ (n + 1) / β(n + 1)! | case intro.intro
f : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
xy_ne : β y β uIoo xβ x, (x - y) ^ n β 0
hg' : β y β uIoo xβ x, -(βn + 1) * (x - y) ^ n β 0
y : β
hy : y β uIoo xβ x
h :
f x - taylorWithinEval f n (uIcc xβ x) xβ x =
((x - y) ^ n / βn ! * ((x - x) ^ (n + 1) - (x - xβ) ^ (n + 1)) / (-(βn + 1) * (x - y) ^ n)) β’
iteratedDerivWithin (n + 1) f (uIcc xβ x) y
β’ β x',
β (_ : x' β uIoo xβ x),
f x - taylorWithinEval f n (uIcc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (uIcc xβ x) x' * (x - xβ) ^ (n + 1) / β(n + 1)! | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
xy_ne : β y β uIoo xβ x, (x - y) ^ n β 0
hg' : β y β uIoo xβ x, -(βn + 1) * (x - y) ^ n β 0
β’ β x',
β (_ : x' β uIoo xβ x),
f x - taylorWithinEval f n (uIcc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (uIcc xβ x) x' * (x - xβ) ^ (n + 1) / β(n + 1)!
TACTIC:
|
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 : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
xy_ne : β y β uIoo xβ x, (x - y) ^ n β 0
hg' : β y β uIoo xβ x, -(βn + 1) * (x - y) ^ n β 0
y : β
hy : y β uIoo xβ x
h :
f x - taylorWithinEval f n (uIcc xβ x) xβ x =
((x - y) ^ n / βn ! * ((x - x) ^ (n + 1) - (x - xβ) ^ (n + 1)) / (-(βn + 1) * (x - y) ^ n)) β’
iteratedDerivWithin (n + 1) f (uIcc xβ x) y
β’ β x',
β (_ : x' β uIoo xβ x),
f x - taylorWithinEval f n (uIcc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (uIcc xβ x) x' * (x - xβ) ^ (n + 1) / β(n + 1)! | case h
f : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
xy_ne : β y β uIoo xβ x, (x - y) ^ n β 0
hg' : β y β uIoo xβ x, -(βn + 1) * (x - y) ^ n β 0
y : β
hy : y β uIoo xβ x
h :
f x - taylorWithinEval f n (uIcc xβ x) xβ x =
((x - y) ^ n / βn ! * ((x - x) ^ (n + 1) - (x - xβ) ^ (n + 1)) / (-(βn + 1) * (x - y) ^ n)) β’
iteratedDerivWithin (n + 1) f (uIcc xβ x) y
β’ f x - taylorWithinEval f n (uIcc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (uIcc xβ x) y * (x - xβ) ^ (n + 1) / β(n + 1)! | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
xy_ne : β y β uIoo xβ x, (x - y) ^ n β 0
hg' : β y β uIoo xβ x, -(βn + 1) * (x - y) ^ n β 0
y : β
hy : y β uIoo xβ x
h :
f x - taylorWithinEval f n (uIcc xβ x) xβ x =
((x - y) ^ n / βn ! * ((x - x) ^ (n + 1) - (x - xβ) ^ (n + 1)) / (-(βn + 1) * (x - y) ^ n)) β’
iteratedDerivWithin (n + 1) f (uIcc xβ x) y
β’ β x',
β (_ : x' β uIoo xβ x),
f x - taylorWithinEval f n (uIcc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (uIcc xβ x) x' * (x - xβ) ^ (n + 1) / β(n + 1)!
TACTIC:
|
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 : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
xy_ne : β y β uIoo xβ x, (x - y) ^ n β 0
hg' : β y β uIoo xβ x, -(βn + 1) * (x - y) ^ n β 0
y : β
hy : y β uIoo xβ x
h :
f x - taylorWithinEval f n (uIcc xβ x) xβ x =
((x - y) ^ n / βn ! * ((x - x) ^ (n + 1) - (x - xβ) ^ (n + 1)) / (-(βn + 1) * (x - y) ^ n)) β’
iteratedDerivWithin (n + 1) f (uIcc xβ x) y
β’ f x - taylorWithinEval f n (uIcc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (uIcc xβ x) y * (x - xβ) ^ (n + 1) / β(n + 1)! | case h
f : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
xy_ne : β y β uIoo xβ x, (x - y) ^ n β 0
hg' : β y β uIoo xβ x, -(βn + 1) * (x - y) ^ n β 0
y : β
hy : y β uIoo xβ x
h :
f x - taylorWithinEval f n (uIcc xβ x) xβ x =
(-((x - y) ^ n / βn ! * (x - xβ) ^ (n + 1)) / (-(βn + 1) * (x - y) ^ n)) β’
iteratedDerivWithin (n + 1) f (uIcc xβ x) y
β’ f x - taylorWithinEval f n (uIcc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (uIcc xβ x) y * (x - xβ) ^ (n + 1) / β(n + 1)! | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
xy_ne : β y β uIoo xβ x, (x - y) ^ n β 0
hg' : β y β uIoo xβ x, -(βn + 1) * (x - y) ^ n β 0
y : β
hy : y β uIoo xβ x
h :
f x - taylorWithinEval f n (uIcc xβ x) xβ x =
((x - y) ^ n / βn ! * ((x - x) ^ (n + 1) - (x - xβ) ^ (n + 1)) / (-(βn + 1) * (x - y) ^ n)) β’
iteratedDerivWithin (n + 1) f (uIcc xβ x) y
β’ f x - taylorWithinEval f n (uIcc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (uIcc xβ x) y * (x - xβ) ^ (n + 1) / β(n + 1)!
TACTIC:
|
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, β div_neg, neg_mul, neg_neg] | case h
f : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
xy_ne : β y β uIoo xβ x, (x - y) ^ n β 0
hg' : β y β uIoo xβ x, -(βn + 1) * (x - y) ^ n β 0
y : β
hy : y β uIoo xβ x
h :
f x - taylorWithinEval f n (uIcc xβ x) xβ x =
(-((x - y) ^ n / βn ! * (x - xβ) ^ (n + 1)) / (-(βn + 1) * (x - y) ^ n)) β’
iteratedDerivWithin (n + 1) f (uIcc xβ x) y
β’ f x - taylorWithinEval f n (uIcc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (uIcc xβ x) y * (x - xβ) ^ (n + 1) / β(n + 1)! | case h
f : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
xy_ne : β y β uIoo xβ x, (x - y) ^ n β 0
hg' : β y β uIoo xβ x, -(βn + 1) * (x - y) ^ n β 0
y : β
hy : y β uIoo xβ x
h :
f x - taylorWithinEval f n (uIcc xβ x) xβ x =
(-((x - y) ^ n / βn ! * (x - xβ) ^ (n + 1)) / (-(βn + 1) * (x - y) ^ n)) β’
iteratedDerivWithin (n + 1) f (uIcc xβ x) y
β’ ((x - y) ^ n / βn ! * (x - xβ) ^ (n + 1) / ((βn + 1) * (x - y) ^ n)) β’ iteratedDerivWithin (n + 1) f (uIcc xβ x) y =
iteratedDerivWithin (n + 1) f (uIcc xβ x) y * (x - xβ) ^ (n + 1) / β(n + 1)! | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
xy_ne : β y β uIoo xβ x, (x - y) ^ n β 0
hg' : β y β uIoo xβ x, -(βn + 1) * (x - y) ^ n β 0
y : β
hy : y β uIoo xβ x
h :
f x - taylorWithinEval f n (uIcc xβ x) xβ x =
(-((x - y) ^ n / βn ! * (x - xβ) ^ (n + 1)) / (-(βn + 1) * (x - y) ^ n)) β’
iteratedDerivWithin (n + 1) f (uIcc xβ x) y
β’ f x - taylorWithinEval f n (uIcc xβ x) xβ x =
iteratedDerivWithin (n + 1) f (uIcc xβ x) y * (x - xβ) ^ (n + 1) / β(n + 1)!
TACTIC:
|
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 : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
xy_ne : β y β uIoo xβ x, (x - y) ^ n β 0
hg' : β y β uIoo xβ x, -(βn + 1) * (x - y) ^ n β 0
y : β
hy : y β uIoo xβ x
h :
f x - taylorWithinEval f n (uIcc xβ x) xβ x =
(-((x - y) ^ n / βn ! * (x - xβ) ^ (n + 1)) / (-(βn + 1) * (x - y) ^ n)) β’
iteratedDerivWithin (n + 1) f (uIcc xβ x) y
β’ ((x - y) ^ n / βn ! * (x - xβ) ^ (n + 1) / ((βn + 1) * (x - y) ^ n)) β’ iteratedDerivWithin (n + 1) f (uIcc xβ x) y =
iteratedDerivWithin (n + 1) f (uIcc xβ x) y * (x - xβ) ^ (n + 1) / β(n + 1)! | case h
f : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
xy_ne : β y β uIoo xβ x, (x - y) ^ n β 0
hg' : β y β uIoo xβ x, -(βn + 1) * (x - y) ^ n β 0
y : β
hy : y β uIoo xβ x
h :
f x - taylorWithinEval f n (uIcc xβ x) xβ x =
(-((x - y) ^ n / βn ! * (x - xβ) ^ (n + 1)) / (-(βn + 1) * (x - y) ^ n)) β’
iteratedDerivWithin (n + 1) f (uIcc xβ x) y
β’ (x - y) ^ n * (x - xβ) ^ (n + 1) * iteratedDerivWithin (n + 1) f (uIcc xβ x) y * β(n + 1)! =
iteratedDerivWithin (n + 1) f (uIcc xβ x) y * (x - xβ) ^ (n + 1) * (βn ! * ((βn + 1) * (x - y) ^ n)) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
xy_ne : β y β uIoo xβ x, (x - y) ^ n β 0
hg' : β y β uIoo xβ x, -(βn + 1) * (x - y) ^ n β 0
y : β
hy : y β uIoo xβ x
h :
f x - taylorWithinEval f n (uIcc xβ x) xβ x =
(-((x - y) ^ n / βn ! * (x - xβ) ^ (n + 1)) / (-(βn + 1) * (x - y) ^ n)) β’
iteratedDerivWithin (n + 1) f (uIcc xβ x) y
β’ ((x - y) ^ n / βn ! * (x - xβ) ^ (n + 1) / ((βn + 1) * (x - y) ^ n)) β’ iteratedDerivWithin (n + 1) f (uIcc xβ x) y =
iteratedDerivWithin (n + 1) f (uIcc xβ x) y * (x - xβ) ^ (n + 1) / β(n + 1)!
TACTIC:
|
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 : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
xy_ne : β y β uIoo xβ x, (x - y) ^ n β 0
hg' : β y β uIoo xβ x, -(βn + 1) * (x - y) ^ n β 0
y : β
hy : y β uIoo xβ x
h :
f x - taylorWithinEval f n (uIcc xβ x) xβ x =
(-((x - y) ^ n / βn ! * (x - xβ) ^ (n + 1)) / (-(βn + 1) * (x - y) ^ n)) β’
iteratedDerivWithin (n + 1) f (uIcc xβ x) y
β’ (x - y) ^ n * (x - xβ) ^ (n + 1) * iteratedDerivWithin (n + 1) f (uIcc xβ x) y * β(n + 1)! =
iteratedDerivWithin (n + 1) f (uIcc xβ x) y * (x - xβ) ^ (n + 1) * (βn ! * ((βn + 1) * (x - y) ^ n)) | case h
f : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
xy_ne : β y β uIoo xβ x, (x - y) ^ n β 0
hg' : β y β uIoo xβ x, -(βn + 1) * (x - y) ^ n β 0
y : β
hy : y β uIoo xβ x
h :
f x - taylorWithinEval f n (uIcc xβ x) xβ x =
(-((x - y) ^ n / βn ! * (x - xβ) ^ (n + 1)) / (-(βn + 1) * (x - y) ^ n)) β’
iteratedDerivWithin (n + 1) f (uIcc xβ x) y
β’ x * iteratedDerivWithin (1 + n) f (uIcc xβ x) y * β(1 + n)! * (x - y) ^ n * (x - xβ) ^ n -
xβ * iteratedDerivWithin (1 + n) f (uIcc xβ x) y * β(1 + n)! * (x - y) ^ n * (x - xβ) ^ n =
x * iteratedDerivWithin (1 + n) f (uIcc xβ x) y * βn ! * βn * (x - y) ^ n * (x - xβ) ^ n +
x * iteratedDerivWithin (1 + n) f (uIcc xβ x) y * βn ! * (x - y) ^ n * (x - xβ) ^ n +
(-(xβ * iteratedDerivWithin (1 + n) f (uIcc xβ x) y * βn ! * βn * (x - y) ^ n * (x - xβ) ^ n) -
xβ * iteratedDerivWithin (1 + n) f (uIcc xβ x) y * βn ! * (x - y) ^ n * (x - xβ) ^ n) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
xy_ne : β y β uIoo xβ x, (x - y) ^ n β 0
hg' : β y β uIoo xβ x, -(βn + 1) * (x - y) ^ n β 0
y : β
hy : y β uIoo xβ x
h :
f x - taylorWithinEval f n (uIcc xβ x) xβ x =
(-((x - y) ^ n / βn ! * (x - xβ) ^ (n + 1)) / (-(βn + 1) * (x - y) ^ n)) β’
iteratedDerivWithin (n + 1) f (uIcc xβ x) y
β’ (x - y) ^ n * (x - xβ) ^ (n + 1) * iteratedDerivWithin (n + 1) f (uIcc xβ x) y * β(n + 1)! =
iteratedDerivWithin (n + 1) f (uIcc xβ x) y * (x - xβ) ^ (n + 1) * (βn ! * ((βn + 1) * (x - y) ^ n))
TACTIC:
|
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 : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
β’ ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x) | f : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
β’ Continuous fun t => (x - t) ^ (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
β’ ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
TACTIC:
|
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 : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
β’ Continuous fun t => (x - t) ^ (n + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
β’ Continuous fun t => (x - t) ^ (n + 1)
TACTIC:
|
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 : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
β’ β y β uIoo xβ x, (x - y) ^ n β 0 | f : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
y : β
hy : y β uIoo xβ x
β’ (x - y) ^ n β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
β’ β y β uIoo xβ x, (x - y) ^ n β 0
TACTIC:
|
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 : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
y : β
hy : y β uIoo xβ x
β’ (x - y) ^ n β 0 | f : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
y : β
hy : y β uIoo xβ x
β’ x - y β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
y : β
hy : y β uIoo xβ x
β’ (x - y) ^ n β 0
TACTIC:
|
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 : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
y : β
hy : y β uIoo xβ x
β’ x - y β 0 | f : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
y : β
hy : y β uIoo xβ x
β’ x β y | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
y : β
hy : y β uIoo xβ x
β’ x - y β 0
TACTIC:
|
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 xβ x with h h | f : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
y : β
hy : y β uIoo xβ x
β’ x β y | case inl
f : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
y : β
hy : y β uIoo xβ x
h : xβ β€ x
β’ x β y
case inr
f : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
y : β
hy : y β uIoo xβ x
h : x β€ xβ
β’ x β y | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
y : β
hy : y β uIoo xβ x
β’ x β y
TACTIC:
|
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 : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
y : β
hy : y β uIoo xβ x
h : xβ β€ x
β’ x β y | case inl
f : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
y : β
hy : y β Ioo xβ x
h : xβ β€ x
β’ x β y | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
f : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
y : β
hy : y β uIoo xβ x
h : xβ β€ x
β’ x β y
TACTIC:
|
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 : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
y : β
hy : y β Ioo xβ x
h : xβ β€ x
β’ x β y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
f : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
y : β
hy : y β Ioo xβ x
h : xβ β€ x
β’ x β y
TACTIC:
|
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 : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
y : β
hy : y β uIoo xβ x
h : x β€ xβ
β’ x β y | case inr
f : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
y : β
hy : y β Ioo x xβ
h : x β€ xβ
β’ x β y | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
f : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
y : β
hy : y β uIoo xβ x
h : x β€ xβ
β’ x β y
TACTIC:
|
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 : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
y : β
hy : y β Ioo x xβ
h : x β€ xβ
β’ x β y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
f : β β β
x xβ : β
n : β
hx : xβ β x
hf : ContDiffOn β (βn) f (uIcc xβ x)
hf' : DifferentiableOn β (iteratedDerivWithin n f (uIcc xβ x)) (uIoo xβ x)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (uIcc xβ x)
y : β
hy : y β Ioo x xβ
h : x β€ xβ
β’ x β y
TACTIC:
|
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 xβ x with (rfl | hx') | f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
β’ β x',
β (_ : x' β Ioo a b),
x' β x β§
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' x' =
((x - x') ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) x' | case inl
f g g' : β β β
xβ a b : β
n : β
hab : a < b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx : xβ β Icc a b
β’ β x',
β (_ : x' β Ioo a b),
x' β xβ β§
(f xβ - taylorWithinEval f n (Icc a b) xβ xβ) * g' x' =
((xβ - x') ^ n / βn ! * (g xβ - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) x'
case inr
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx' : xβ β x
β’ β x',
β (_ : x' β Ioo a b),
x' β x β§
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' x' =
((x - x') ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) x' | Please generate a tactic in lean4 to solve the state.
STATE:
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
β’ β x',
β (_ : x' β Ioo a b),
x' β x β§
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' x' =
((x - x') ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) x'
TACTIC:
|
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' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx' : xβ β x
β’ β x',
β (_ : x' β Ioo a b),
x' β x β§
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' x' =
((x - x') ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) x' | case inr.inl
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx'β : xβ β x
hx' : xβ < x
β’ β x',
β (_ : x' β Ioo a b),
x' β x β§
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' x' =
((x - x') ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) x'
case inr.inr
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx'β : xβ β x
hx' : x < xβ
β’ β x',
β (_ : x' β Ioo a b),
x' β x β§
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' x' =
((x - x') ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) x' | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx' : xβ β x
β’ β x',
β (_ : x' β Ioo a b),
x' β x β§
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' x' =
((x - x') ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) x'
TACTIC:
|
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' : β β β
xβ a b : β
n : β
hab : a < b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx : xβ β Icc a b
β’ β x',
β (_ : x' β Ioo a b),
x' β xβ β§
(f xβ - taylorWithinEval f n (Icc a b) xβ xβ) * g' x' =
((xβ - x') ^ n / βn ! * (g xβ - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) x' | case inl
f g g' : β β β
xβ a b : β
n : β
hab : a < b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx : xβ β Icc a b
β’ β x' β Ioo a b, x' β xβ | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
f g g' : β β β
xβ a b : β
n : β
hab : a < b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx : xβ β Icc a b
β’ β x',
β (_ : x' β Ioo a b),
x' β xβ β§
(f xβ - taylorWithinEval f n (Icc a b) xβ xβ) * g' x' =
((xβ - x') ^ n / βn ! * (g xβ - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) x'
TACTIC:
|
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 β¨x', hx'β© := ((Ioo_infinite hab).diffβ (Set.finite_singleton xβ)).Nonempty | case inl
f g g' : β β β
xβ a b : β
n : β
hab : a < b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx : xβ β Icc a b
β’ β x' β Ioo a b, x' β xβ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
f g g' : β β β
xβ a b : β
n : β
hab : a < b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx : xβ β Icc a b
β’ β x' β Ioo a b, x' β xβ
TACTIC:
|
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 hβ : Icc xβ x β Icc a b := Icc_subset_Icc hxβ.1 hx.2 | case inr.inl
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx'β : xβ β x
hx' : xβ < x
β’ β x',
β (_ : x' β Ioo a b),
x' β x β§
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' x' =
((x - x') ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) x' | case inr.inl
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx'β : xβ β x
hx' : xβ < x
hβ : Icc xβ x β Icc a b
β’ β x',
β (_ : x' β Ioo a b),
x' β x β§
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' x' =
((x - x') ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) x' | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx'β : xβ β x
hx' : xβ < x
β’ β x',
β (_ : x' β Ioo a b),
x' β x β§
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' x' =
((x - x') ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) x'
TACTIC:
|
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 hβ : Ioo xβ x β Ioo a b := Ioo_subset_Ioo hxβ.1 hx.2 | case inr.inl
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx'β : xβ β x
hx' : xβ < x
hβ : Icc xβ x β Icc a b
β’ β x',
β (_ : x' β Ioo a b),
x' β x β§
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' x' =
((x - x') ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) x' | case inr.inl
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx'β : xβ β x
hx' : xβ < x
hβ : Icc xβ x β Icc a b
hβ : Ioo xβ x β Ioo a b
β’ β x',
β (_ : x' β Ioo a b),
x' β x β§
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' x' =
((x - x') ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) x' | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx'β : xβ β x
hx' : xβ < x
hβ : Icc xβ x β Icc a b
β’ β x',
β (_ : x' β Ioo a b),
x' β x β§
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' x' =
((x - x') ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) x'
TACTIC:
|
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 β¨y, hy, hβ© :=
exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc a b) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) hx'
((continuousOn_taylorWithinEval (uniqueDiffOn_Icc hab) hf).mono hβ)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo _ hab (hβ hy) hf hf') g g' (gcont.mono hβ)
fun y hy => gdiff y (hβ hy) | case inr.inl
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx'β : xβ β x
hx' : xβ < x
hβ : Icc xβ x β Icc a b
hβ : Ioo xβ x β Ioo a b
β’ β x',
β (_ : x' β Ioo a b),
x' β x β§
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' x' =
((x - x') ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) x' | case inr.inl.intro.intro
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx'β : xβ β x
hx' : xβ < x
hβ : Icc xβ x β Icc a b
hβ : Ioo xβ x β Ioo a b
y : β
hy : y β Ioo xβ x
h :
(g x - g xβ) * ((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) y =
(taylorWithinEval f n (Icc a b) x x - taylorWithinEval f n (Icc a b) xβ x) * g' y
β’ β x',
β (_ : x' β Ioo a b),
x' β x β§
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' x' =
((x - x') ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) x' | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx'β : xβ β x
hx' : xβ < x
hβ : Icc xβ x β Icc a b
hβ : Ioo xβ x β Ioo a b
β’ β x',
β (_ : x' β Ioo a b),
x' β x β§
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' x' =
((x - x') ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) x'
TACTIC:
|
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' β¨y, hβ hy, hy.2.Ne, _β© | case inr.inl.intro.intro
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx'β : xβ β x
hx' : xβ < x
hβ : Icc xβ x β Icc a b
hβ : Ioo xβ x β Ioo a b
y : β
hy : y β Ioo xβ x
h :
(g x - g xβ) * ((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) y =
(taylorWithinEval f n (Icc a b) x x - taylorWithinEval f n (Icc a b) xβ x) * g' y
β’ β x',
β (_ : x' β Ioo a b),
x' β x β§
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' x' =
((x - x') ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) x' | case inr.inl.intro.intro
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx'β : xβ β x
hx' : xβ < x
hβ : Icc xβ x β Icc a b
hβ : Ioo xβ x β Ioo a b
y : β
hy : y β Ioo xβ x
h :
(g x - g xβ) * ((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) y =
(taylorWithinEval f n (Icc a b) x x - taylorWithinEval f n (Icc a b) xβ x) * g' y
β’ (f x - taylorWithinEval f n (Icc a b) xβ x) * g' y =
((x - y) ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) y | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl.intro.intro
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx'β : xβ β x
hx' : xβ < x
hβ : Icc xβ x β Icc a b
hβ : Ioo xβ x β Ioo a b
y : β
hy : y β Ioo xβ x
h :
(g x - g xβ) * ((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) y =
(taylorWithinEval f n (Icc a b) x x - taylorWithinEval f n (Icc a b) xβ x) * g' y
β’ β x',
β (_ : x' β Ioo a b),
x' β x β§
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' x' =
((x - x') ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) x'
TACTIC:
|
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' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx'β : xβ β x
hx' : xβ < x
hβ : Icc xβ x β Icc a b
hβ : Ioo xβ x β Ioo a b
y : β
hy : y β Ioo xβ x
h :
(g x - g xβ) * ((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) y =
(taylorWithinEval f n (Icc a b) x x - taylorWithinEval f n (Icc a b) xβ x) * g' y
β’ (f x - taylorWithinEval f n (Icc a b) xβ x) * g' y =
((x - y) ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) y | case inr.inl.intro.intro
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx'β : xβ β x
hx' : xβ < x
hβ : Icc xβ x β Icc a b
hβ : Ioo xβ x β Ioo a b
y : β
hy : y β Ioo xβ x
h :
(g x - g xβ) * ((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) y =
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' y
β’ (f x - taylorWithinEval f n (Icc a b) xβ x) * g' y =
((x - y) ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) y | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl.intro.intro
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx'β : xβ β x
hx' : xβ < x
hβ : Icc xβ x β Icc a b
hβ : Ioo xβ x β Ioo a b
y : β
hy : y β Ioo xβ x
h :
(g x - g xβ) * ((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) y =
(taylorWithinEval f n (Icc a b) x x - taylorWithinEval f n (Icc a b) xβ x) * g' y
β’ (f x - taylorWithinEval f n (Icc a b) xβ x) * g' y =
((x - y) ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) y
TACTIC:
|
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 [β h, n.factorial_ne_zero] | case inr.inl.intro.intro
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx'β : xβ β x
hx' : xβ < x
hβ : Icc xβ x β Icc a b
hβ : Ioo xβ x β Ioo a b
y : β
hy : y β Ioo xβ x
h :
(g x - g xβ) * ((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) y =
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' y
β’ (f x - taylorWithinEval f n (Icc a b) xβ x) * g' y =
((x - y) ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) y | case inr.inl.intro.intro
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx'β : xβ β x
hx' : xβ < x
hβ : Icc xβ x β Icc a b
hβ : Ioo xβ x β Ioo a b
y : β
hy : y β Ioo xβ x
h :
(g x - g xβ) * ((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) y =
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' y
β’ (g x - g xβ) * ((x - y) ^ n * iteratedDerivWithin (n + 1) f (Icc a b) y) =
(x - y) ^ n * (g x - g xβ) * iteratedDerivWithin (n + 1) f (Icc a b) y | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl.intro.intro
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx'β : xβ β x
hx' : xβ < x
hβ : Icc xβ x β Icc a b
hβ : Ioo xβ x β Ioo a b
y : β
hy : y β Ioo xβ x
h :
(g x - g xβ) * ((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) y =
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' y
β’ (f x - taylorWithinEval f n (Icc a b) xβ x) * g' y =
((x - y) ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) y
TACTIC:
|
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' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx'β : xβ β x
hx' : xβ < x
hβ : Icc xβ x β Icc a b
hβ : Ioo xβ x β Ioo a b
y : β
hy : y β Ioo xβ x
h :
(g x - g xβ) * ((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) y =
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' y
β’ (g x - g xβ) * ((x - y) ^ n * iteratedDerivWithin (n + 1) f (Icc a b) y) =
(x - y) ^ n * (g x - g xβ) * iteratedDerivWithin (n + 1) f (Icc a b) y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl.intro.intro
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx'β : xβ β x
hx' : xβ < x
hβ : Icc xβ x β Icc a b
hβ : Ioo xβ x β Ioo a b
y : β
hy : y β Ioo xβ x
h :
(g x - g xβ) * ((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) y =
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' y
β’ (g x - g xβ) * ((x - y) ^ n * iteratedDerivWithin (n + 1) f (Icc a b) y) =
(x - y) ^ n * (g x - g xβ) * iteratedDerivWithin (n + 1) f (Icc a b) y
TACTIC:
|
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 hβ : Icc x xβ β Icc a b := Icc_subset_Icc hx.1 hxβ.2 | case inr.inr
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx'β : xβ β x
hx' : x < xβ
β’ β x',
β (_ : x' β Ioo a b),
x' β x β§
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' x' =
((x - x') ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) x' | case inr.inr
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx'β : xβ β x
hx' : x < xβ
hβ : Icc x xβ β Icc a b
β’ β x',
β (_ : x' β Ioo a b),
x' β x β§
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' x' =
((x - x') ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) x' | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx'β : xβ β x
hx' : x < xβ
β’ β x',
β (_ : x' β Ioo a b),
x' β x β§
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' x' =
((x - x') ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) x'
TACTIC:
|
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 hβ : Ioo x xβ β Ioo a b := Ioo_subset_Ioo hx.1 hxβ.2 | case inr.inr
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx'β : xβ β x
hx' : x < xβ
hβ : Icc x xβ β Icc a b
β’ β x',
β (_ : x' β Ioo a b),
x' β x β§
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' x' =
((x - x') ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) x' | case inr.inr
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx'β : xβ β x
hx' : x < xβ
hβ : Icc x xβ β Icc a b
hβ : Ioo x xβ β Ioo a b
β’ β x',
β (_ : x' β Ioo a b),
x' β x β§
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' x' =
((x - x') ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) x' | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx'β : xβ β x
hx' : x < xβ
hβ : Icc x xβ β Icc a b
β’ β x',
β (_ : x' β Ioo a b),
x' β x β§
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' x' =
((x - x') ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) x'
TACTIC:
|
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 β¨y, hy, hβ© :=
exists_ratio_hasDerivAt_eq_ratio_slope (fun t => taylorWithinEval f n (Icc a b) t x)
(fun t => ((n ! : β)β»ΒΉ * (x - t) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) t) hx'
((continuousOn_taylorWithinEval (uniqueDiffOn_Icc hab) hf).mono hβ)
(fun _ hy => taylorWithinEval_hasDerivAt_Ioo _ hab (hβ hy) hf hf') g g' (gcont.mono hβ)
fun y hy => gdiff y (hβ hy) | case inr.inr
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx'β : xβ β x
hx' : x < xβ
hβ : Icc x xβ β Icc a b
hβ : Ioo x xβ β Ioo a b
β’ β x',
β (_ : x' β Ioo a b),
x' β x β§
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' x' =
((x - x') ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) x' | case inr.inr.intro.intro
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx'β : xβ β x
hx' : x < xβ
hβ : Icc x xβ β Icc a b
hβ : Ioo x xβ β Ioo a b
y : β
hy : y β Ioo x xβ
h :
(g xβ - g x) * ((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) y =
(taylorWithinEval f n (Icc a b) xβ x - taylorWithinEval f n (Icc a b) x x) * g' y
β’ β x',
β (_ : x' β Ioo a b),
x' β x β§
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' x' =
((x - x') ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) x' | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx'β : xβ β x
hx' : x < xβ
hβ : Icc x xβ β Icc a b
hβ : Ioo x xβ β Ioo a b
β’ β x',
β (_ : x' β Ioo a b),
x' β x β§
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' x' =
((x - x') ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) x'
TACTIC:
|
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' β¨y, hβ hy, hy.1.ne', _β© | case inr.inr.intro.intro
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx'β : xβ β x
hx' : x < xβ
hβ : Icc x xβ β Icc a b
hβ : Ioo x xβ β Ioo a b
y : β
hy : y β Ioo x xβ
h :
(g xβ - g x) * ((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) y =
(taylorWithinEval f n (Icc a b) xβ x - taylorWithinEval f n (Icc a b) x x) * g' y
β’ β x',
β (_ : x' β Ioo a b),
x' β x β§
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' x' =
((x - x') ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) x' | case inr.inr.intro.intro
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx'β : xβ β x
hx' : x < xβ
hβ : Icc x xβ β Icc a b
hβ : Ioo x xβ β Ioo a b
y : β
hy : y β Ioo x xβ
h :
(g xβ - g x) * ((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) y =
(taylorWithinEval f n (Icc a b) xβ x - taylorWithinEval f n (Icc a b) x x) * g' y
β’ (f x - taylorWithinEval f n (Icc a b) xβ x) * g' y =
((x - y) ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) y | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.intro.intro
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx'β : xβ β x
hx' : x < xβ
hβ : Icc x xβ β Icc a b
hβ : Ioo x xβ β Ioo a b
y : β
hy : y β Ioo x xβ
h :
(g xβ - g x) * ((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) y =
(taylorWithinEval f n (Icc a b) xβ x - taylorWithinEval f n (Icc a b) x x) * g' y
β’ β x',
β (_ : x' β Ioo a b),
x' β x β§
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' x' =
((x - x') ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) x'
TACTIC:
|
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' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx'β : xβ β x
hx' : x < xβ
hβ : Icc x xβ β Icc a b
hβ : Ioo x xβ β Ioo a b
y : β
hy : y β Ioo x xβ
h :
(g xβ - g x) * ((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) y =
(taylorWithinEval f n (Icc a b) xβ x - taylorWithinEval f n (Icc a b) x x) * g' y
β’ (f x - taylorWithinEval f n (Icc a b) xβ x) * g' y =
((x - y) ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) y | case inr.inr.intro.intro
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx'β : xβ β x
hx' : x < xβ
hβ : Icc x xβ β Icc a b
hβ : Ioo x xβ β Ioo a b
y : β
hy : y β Ioo x xβ
h :
(g xβ - g x) * ((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) y =
(taylorWithinEval f n (Icc a b) xβ x - f x) * g' y
β’ (f x - taylorWithinEval f n (Icc a b) xβ x) * g' y =
((x - y) ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) y | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.intro.intro
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx'β : xβ β x
hx' : x < xβ
hβ : Icc x xβ β Icc a b
hβ : Ioo x xβ β Ioo a b
y : β
hy : y β Ioo x xβ
h :
(g xβ - g x) * ((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) y =
(taylorWithinEval f n (Icc a b) xβ x - taylorWithinEval f n (Icc a b) x x) * g' y
β’ (f x - taylorWithinEval f n (Icc a b) xβ x) * g' y =
((x - y) ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) y
TACTIC:
|
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 [β neg_sub, neg_mul, β h] | case inr.inr.intro.intro
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx'β : xβ β x
hx' : x < xβ
hβ : Icc x xβ β Icc a b
hβ : Ioo x xβ β Ioo a b
y : β
hy : y β Ioo x xβ
h :
(g xβ - g x) * ((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) y =
(taylorWithinEval f n (Icc a b) xβ x - f x) * g' y
β’ (f x - taylorWithinEval f n (Icc a b) xβ x) * g' y =
((x - y) ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) y | case inr.inr.intro.intro
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx'β : xβ β x
hx' : x < xβ
hβ : Icc x xβ β Icc a b
hβ : Ioo x xβ β Ioo a b
y : β
hy : y β Ioo x xβ
h :
(g xβ - g x) * ((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) y =
(taylorWithinEval f n (Icc a b) xβ x - f x) * g' y
β’ -((g xβ - g x) * ((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) y) =
((x - y) ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) y | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.intro.intro
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx'β : xβ β x
hx' : x < xβ
hβ : Icc x xβ β Icc a b
hβ : Ioo x xβ β Ioo a b
y : β
hy : y β Ioo x xβ
h :
(g xβ - g x) * ((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) y =
(taylorWithinEval f n (Icc a b) xβ x - f x) * g' y
β’ (f x - taylorWithinEval f n (Icc a b) xβ x) * g' y =
((x - y) ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) y
TACTIC:
|
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' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx'β : xβ β x
hx' : x < xβ
hβ : Icc x xβ β Icc a b
hβ : Ioo x xβ β Ioo a b
y : β
hy : y β Ioo x xβ
h :
(g xβ - g x) * ((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) y =
(taylorWithinEval f n (Icc a b) xβ x - f x) * g' y
β’ -((g xβ - g x) * ((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) y) =
((x - y) ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) y | case inr.inr.intro.intro
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx'β : xβ β x
hx' : x < xβ
hβ : Icc x xβ β Icc a b
hβ : Ioo x xβ β Ioo a b
y : β
hy : y β Ioo x xβ
h :
(g xβ - g x) * ((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) y =
(taylorWithinEval f n (Icc a b) xβ x - f x) * g' y
β’ -((g xβ - g x) * ((x - y) ^ n * iteratedDerivWithin (n + 1) f (Icc a b) y) * βn !) =
(x - y) ^ n * (g x - g xβ) * iteratedDerivWithin (n + 1) f (Icc a b) y * βn ! | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.intro.intro
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx'β : xβ β x
hx' : x < xβ
hβ : Icc x xβ β Icc a b
hβ : Ioo x xβ β Ioo a b
y : β
hy : y β Ioo x xβ
h :
(g xβ - g x) * ((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) y =
(taylorWithinEval f n (Icc a b) xβ x - f x) * g' y
β’ -((g xβ - g x) * ((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) y) =
((x - y) ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) y
TACTIC:
|
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' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx'β : xβ β x
hx' : x < xβ
hβ : Icc x xβ β Icc a b
hβ : Ioo x xβ β Ioo a b
y : β
hy : y β Ioo x xβ
h :
(g xβ - g x) * ((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) y =
(taylorWithinEval f n (Icc a b) xβ x - f x) * g' y
β’ -((g xβ - g x) * ((x - y) ^ n * iteratedDerivWithin (n + 1) f (Icc a b) y) * βn !) =
(x - y) ^ n * (g x - g xβ) * iteratedDerivWithin (n + 1) f (Icc a b) y * βn ! | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.intro.intro
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
hx'β : xβ β x
hx' : x < xβ
hβ : Icc x xβ β Icc a b
hβ : Ioo x xβ β Ioo a b
y : β
hy : y β Ioo x xβ
h :
(g xβ - g x) * ((βn !)β»ΒΉ * (x - y) ^ n) β’ iteratedDerivWithin (n + 1) f (Icc a b) y =
(taylorWithinEval f n (Icc a b) xβ x - f x) * g' y
β’ -((g xβ - g x) * ((x - y) ^ n * iteratedDerivWithin (n + 1) f (Icc a b) y) * βn !) =
(x - y) ^ n * (g x - g xβ) * iteratedDerivWithin (n + 1) f (Icc a b) y * βn !
TACTIC:
|
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_central | [149, 1] | [162, 19] | obtain β¨y, hy, hyx, hβ© := taylor_mean_remainder_central_aux hab hx hxβ hf hf' gcont gdiff | f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
g'_ne : β y β Ioo a b, g' y β 0
β’ β x',
β (_ : x' β Ioo a b),
f x - taylorWithinEval f n (Icc a b) xβ x =
((x - x') ^ n / βn ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc a b) x' | case intro.intro.intro
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
g'_ne : β y β Ioo a b, g' y β 0
y : β
hy : y β Ioo a b
hyx : y β x
h :
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' y =
((x - y) ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) y
β’ β x',
β (_ : x' β Ioo a b),
f x - taylorWithinEval f n (Icc a b) xβ x =
((x - x') ^ n / βn ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc a b) x' | Please generate a tactic in lean4 to solve the state.
STATE:
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
g'_ne : β y β Ioo a b, g' y β 0
β’ β x',
β (_ : x' β Ioo a b),
f x - taylorWithinEval f n (Icc a b) xβ x =
((x - x') ^ n / βn ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc a b) x'
TACTIC:
|
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/Taylor.lean | taylor_mean_remainder_central | [149, 1] | [162, 19] | refine' β¨y, hy, _β© | case intro.intro.intro
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
g'_ne : β y β Ioo a b, g' y β 0
y : β
hy : y β Ioo a b
hyx : y β x
h :
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' y =
((x - y) ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) y
β’ β x',
β (_ : x' β Ioo a b),
f x - taylorWithinEval f n (Icc a b) xβ x =
((x - x') ^ n / βn ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc a b) x' | case intro.intro.intro
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
g'_ne : β y β Ioo a b, g' y β 0
y : β
hy : y β Ioo a b
hyx : y β x
h :
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' y =
((x - y) ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) y
β’ f x - taylorWithinEval f n (Icc a b) xβ x =
((x - y) ^ n / βn ! * (g x - g xβ) / g' y) β’ iteratedDerivWithin (n + 1) f (Icc a b) y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
g'_ne : β y β Ioo a b, g' y β 0
y : β
hy : y β Ioo a b
hyx : y β x
h :
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' y =
((x - y) ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) y
β’ β x',
β (_ : x' β Ioo a b),
f x - taylorWithinEval f n (Icc a b) xβ x =
((x - x') ^ n / βn ! * (g x - g xβ) / g' x') β’ iteratedDerivWithin (n + 1) f (Icc a b) x'
TACTIC:
|
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' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
g'_ne : β y β Ioo a b, g' y β 0
y : β
hy : y β Ioo a b
hyx : y β x
h :
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' y =
((x - y) ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) y
β’ f x - taylorWithinEval f n (Icc a b) xβ x =
((x - y) ^ n / βn ! * (g x - g xβ) / g' y) β’ iteratedDerivWithin (n + 1) f (Icc a b) y | case intro.intro.intro
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
g'_ne : β y β Ioo a b, g' y β 0
y : β
hy : y β Ioo a b
hyx : y β x
h :
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' y =
(x - y) ^ n / βn ! * (g x - g xβ) * iteratedDerivWithin (n + 1) f (Icc a b) y
β’ f x - taylorWithinEval f n (Icc a b) xβ x =
((x - y) ^ n / βn ! * (g x - g xβ) / g' y) β’ iteratedDerivWithin (n + 1) f (Icc a b) y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
g'_ne : β y β Ioo a b, g' y β 0
y : β
hy : y β Ioo a b
hyx : y β x
h :
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' y =
((x - y) ^ n / βn ! * (g x - g xβ)) β’ iteratedDerivWithin (n + 1) f (Icc a b) y
β’ f x - taylorWithinEval f n (Icc a b) xβ x =
((x - y) ^ n / βn ! * (g x - g xβ) / g' y) β’ iteratedDerivWithin (n + 1) f (Icc a b) y
TACTIC:
|
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, β h, mul_div_cancel] | case intro.intro.intro
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
g'_ne : β y β Ioo a b, g' y β 0
y : β
hy : y β Ioo a b
hyx : y β x
h :
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' y =
(x - y) ^ n / βn ! * (g x - g xβ) * iteratedDerivWithin (n + 1) f (Icc a b) y
β’ f x - taylorWithinEval f n (Icc a b) xβ x =
((x - y) ^ n / βn ! * (g x - g xβ) / g' y) β’ iteratedDerivWithin (n + 1) f (Icc a b) y | case intro.intro.intro.h
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
g'_ne : β y β Ioo a b, g' y β 0
y : β
hy : y β Ioo a b
hyx : y β x
h :
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' y =
(x - y) ^ n / βn ! * (g x - g xβ) * iteratedDerivWithin (n + 1) f (Icc a b) y
β’ g' y β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
g'_ne : β y β Ioo a b, g' y β 0
y : β
hy : y β Ioo a b
hyx : y β x
h :
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' y =
(x - y) ^ n / βn ! * (g x - g xβ) * iteratedDerivWithin (n + 1) f (Icc a b) y
β’ f x - taylorWithinEval f n (Icc a b) xβ x =
((x - y) ^ n / βn ! * (g x - g xβ) / g' y) β’ iteratedDerivWithin (n + 1) f (Icc a b) y
TACTIC:
|
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' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
g'_ne : β y β Ioo a b, g' y β 0
y : β
hy : y β Ioo a b
hyx : y β x
h :
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' y =
(x - y) ^ n / βn ! * (g x - g xβ) * iteratedDerivWithin (n + 1) f (Icc a b) y
β’ g' y β 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.h
f g g' : β β β
xβ x a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn g (Icc a b)
gdiff : β y β Ioo a b, HasDerivAt g (g' y) y
g'_ne : β y β Ioo a b, g' y β 0
y : β
hy : y β Ioo a b
hyx : y β x
h :
(f x - taylorWithinEval f n (Icc a b) xβ x) * g' y =
(x - y) ^ n / βn ! * (g x - g xβ) * iteratedDerivWithin (n + 1) f (Icc a b) y
β’ g' y β 0
TACTIC:
|
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 : β => (x - t) ^ (n + 1)) (Icc a b) := by
refine' Continuous.continuousOn _; continuity | f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
β’ β x',
β (_ : x' β Ioo a b),
f x - taylorWithinEval f n (Icc a b) xβ x =
iteratedDerivWithin (n + 1) f (Icc a b) x' * (x - xβ) ^ (n + 1) / β(n + 1)! | f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
β’ β x',
β (_ : x' β Ioo a b),
f x - taylorWithinEval f n (Icc a b) xβ x =
iteratedDerivWithin (n + 1) f (Icc a b) x' * (x - xβ) ^ (n + 1) / β(n + 1)! | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
β’ β x',
β (_ : x' β Ioo a b),
f x - taylorWithinEval f n (Icc a b) xβ x =
iteratedDerivWithin (n + 1) f (Icc a b) x' * (x - xβ) ^ (n + 1) / β(n + 1)!
TACTIC:
|
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 hxβ hf hf' gcont fun y _ =>
monomial_has_deriv_aux y x _ with
β¨y, hy, hy', hβ© | f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
β’ β x',
β (_ : x' β Ioo a b),
f x - taylorWithinEval f n (Icc a b) xβ x =
iteratedDerivWithin (n + 1) f (Icc a b) x' * (x - xβ) ^ (n + 1) / β(n + 1)! | case intro.intro.intro
f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : β
hy : y β Ioo a b
hy' : y β x
h :
(f x - taylorWithinEval f n (Icc a b) xβ x) * (-(βn + 1) * (x - y) ^ n) =
((x - y) ^ n / βn ! * ((x - x) ^ (n + 1) - (x - xβ) ^ (n + 1))) β’ iteratedDerivWithin (n + 1) f (Icc a b) y
β’ β x',
β (_ : x' β Ioo a b),
f x - taylorWithinEval f n (Icc a b) xβ x =
iteratedDerivWithin (n + 1) f (Icc a b) x' * (x - xβ) ^ (n + 1) / β(n + 1)! | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
β’ β x',
β (_ : x' β Ioo a b),
f x - taylorWithinEval f n (Icc a b) xβ x =
iteratedDerivWithin (n + 1) f (Icc a b) x' * (x - xβ) ^ (n + 1) / β(n + 1)!
TACTIC:
|
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 β 0 := sub_ne_zero_of_ne hy'.symm | case intro.intro.intro
f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : β
hy : y β Ioo a b
hy' : y β x
h :
(f x - taylorWithinEval f n (Icc a b) xβ x) * (-(βn + 1) * (x - y) ^ n) =
((x - y) ^ n / βn ! * ((x - x) ^ (n + 1) - (x - xβ) ^ (n + 1))) β’ iteratedDerivWithin (n + 1) f (Icc a b) y
β’ β x',
β (_ : x' β Ioo a b),
f x - taylorWithinEval f n (Icc a b) xβ x =
iteratedDerivWithin (n + 1) f (Icc a b) x' * (x - xβ) ^ (n + 1) / β(n + 1)! | case intro.intro.intro
f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : β
hy : y β Ioo a b
hy' : y β x
h :
(f x - taylorWithinEval f n (Icc a b) xβ x) * (-(βn + 1) * (x - y) ^ n) =
((x - y) ^ n / βn ! * ((x - x) ^ (n + 1) - (x - xβ) ^ (n + 1))) β’ iteratedDerivWithin (n + 1) f (Icc a b) y
hy_ne : x - y β 0
β’ β x',
β (_ : x' β Ioo a b),
f x - taylorWithinEval f n (Icc a b) xβ x =
iteratedDerivWithin (n + 1) f (Icc a b) x' * (x - xβ) ^ (n + 1) / β(n + 1)! | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : β
hy : y β Ioo a b
hy' : y β x
h :
(f x - taylorWithinEval f n (Icc a b) xβ x) * (-(βn + 1) * (x - y) ^ n) =
((x - y) ^ n / βn ! * ((x - x) ^ (n + 1) - (x - xβ) ^ (n + 1))) β’ iteratedDerivWithin (n + 1) f (Icc a b) y
β’ β x',
β (_ : x' β Ioo a b),
f x - taylorWithinEval f n (Icc a b) xβ x =
iteratedDerivWithin (n + 1) f (Icc a b) x' * (x - xβ) ^ (n + 1) / β(n + 1)!
TACTIC:
|
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 : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : β
hy : y β Ioo a b
hy' : y β x
h :
(f x - taylorWithinEval f n (Icc a b) xβ x) * (-(βn + 1) * (x - y) ^ n) =
((x - y) ^ n / βn ! * ((x - x) ^ (n + 1) - (x - xβ) ^ (n + 1))) β’ iteratedDerivWithin (n + 1) f (Icc a b) y
hy_ne : x - y β 0
β’ β x',
β (_ : x' β Ioo a b),
f x - taylorWithinEval f n (Icc a b) xβ x =
iteratedDerivWithin (n + 1) f (Icc a b) x' * (x - xβ) ^ (n + 1) / β(n + 1)! | case h
f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : β
hy : y β Ioo a b
hy' : y β x
h :
(f x - taylorWithinEval f n (Icc a b) xβ x) * (-(βn + 1) * (x - y) ^ n) =
((x - y) ^ n / βn ! * ((x - x) ^ (n + 1) - (x - xβ) ^ (n + 1))) β’ iteratedDerivWithin (n + 1) f (Icc a b) y
hy_ne : x - y β 0
β’ f x - taylorWithinEval f n (Icc a b) xβ x = iteratedDerivWithin (n + 1) f (Icc a b) y * (x - xβ) ^ (n + 1) / β(n + 1)! | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : β
hy : y β Ioo a b
hy' : y β x
h :
(f x - taylorWithinEval f n (Icc a b) xβ x) * (-(βn + 1) * (x - y) ^ n) =
((x - y) ^ n / βn ! * ((x - x) ^ (n + 1) - (x - xβ) ^ (n + 1))) β’ iteratedDerivWithin (n + 1) f (Icc a b) y
hy_ne : x - y β 0
β’ β x',
β (_ : x' β Ioo a b),
f x - taylorWithinEval f n (Icc a b) xβ x =
iteratedDerivWithin (n + 1) f (Icc a b) x' * (x - xβ) ^ (n + 1) / β(n + 1)!
TACTIC:
|
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 : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : β
hy : y β Ioo a b
hy' : y β x
h :
(f x - taylorWithinEval f n (Icc a b) xβ x) * (-(βn + 1) * (x - y) ^ n) =
((x - y) ^ n / βn ! * ((x - x) ^ (n + 1) - (x - xβ) ^ (n + 1))) β’ iteratedDerivWithin (n + 1) f (Icc a b) y
hy_ne : x - y β 0
β’ f x - taylorWithinEval f n (Icc a b) xβ x = iteratedDerivWithin (n + 1) f (Icc a b) y * (x - xβ) ^ (n + 1) / β(n + 1)! | case h
f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : β
hy : y β Ioo a b
hy' : y β x
h :
(f x - taylorWithinEval f n (Icc a b) xβ x) * (-(βn + 1) * (x - y) ^ n) =
(x - y) ^ n / βn ! * ((x - x) ^ (n + 1) - (x - xβ) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y
hy_ne : x - y β 0
β’ f x - taylorWithinEval f n (Icc a b) xβ x = iteratedDerivWithin (n + 1) f (Icc a b) y * (x - xβ) ^ (n + 1) / β(n + 1)! | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : β
hy : y β Ioo a b
hy' : y β x
h :
(f x - taylorWithinEval f n (Icc a b) xβ x) * (-(βn + 1) * (x - y) ^ n) =
((x - y) ^ n / βn ! * ((x - x) ^ (n + 1) - (x - xβ) ^ (n + 1))) β’ iteratedDerivWithin (n + 1) f (Icc a b) y
hy_ne : x - y β 0
β’ f x - taylorWithinEval f n (Icc a b) xβ x = iteratedDerivWithin (n + 1) f (Icc a b) y * (x - xβ) ^ (n + 1) / β(n + 1)!
TACTIC:
|
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 [β eq_div_iff] at h | case h
f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : β
hy : y β Ioo a b
hy' : y β x
h :
(f x - taylorWithinEval f n (Icc a b) xβ x) * (-(βn + 1) * (x - y) ^ n) =
(x - y) ^ n / βn ! * ((x - x) ^ (n + 1) - (x - xβ) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y
hy_ne : x - y β 0
β’ f x - taylorWithinEval f n (Icc a b) xβ x = iteratedDerivWithin (n + 1) f (Icc a b) y * (x - xβ) ^ (n + 1) / β(n + 1)! | case h
f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : β
hy : y β Ioo a b
hy' : y β x
h :
f x - taylorWithinEval f n (Icc a b) xβ x =
(x - y) ^ n / βn ! * ((x - x) ^ (n + 1) - (x - xβ) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y /
(-(βn + 1) * (x - y) ^ n)
hy_ne : x - y β 0
β’ f x - taylorWithinEval f n (Icc a b) xβ x = iteratedDerivWithin (n + 1) f (Icc a b) y * (x - xβ) ^ (n + 1) / β(n + 1)!
case h
f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : β
hy : y β Ioo a b
hy' : y β x
h :
(f x - taylorWithinEval f n (Icc a b) xβ x) * (-(βn + 1) * (x - y) ^ n) =
(x - y) ^ n / βn ! * ((x - x) ^ (n + 1) - (x - xβ) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y
hy_ne : x - y β 0
β’ -(βn + 1) * (x - y) ^ n β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : β
hy : y β Ioo a b
hy' : y β x
h :
(f x - taylorWithinEval f n (Icc a b) xβ x) * (-(βn + 1) * (x - y) ^ n) =
(x - y) ^ n / βn ! * ((x - x) ^ (n + 1) - (x - xβ) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y
hy_ne : x - y β 0
β’ f x - taylorWithinEval f n (Icc a b) xβ x = iteratedDerivWithin (n + 1) f (Icc a b) y * (x - xβ) ^ (n + 1) / β(n + 1)!
TACTIC:
|
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 : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : β
hy : y β Ioo a b
hy' : y β x
h :
f x - taylorWithinEval f n (Icc a b) xβ x =
(x - y) ^ n / βn ! * ((x - x) ^ (n + 1) - (x - xβ) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y /
(-(βn + 1) * (x - y) ^ n)
hy_ne : x - y β 0
β’ f x - taylorWithinEval f n (Icc a b) xβ x = iteratedDerivWithin (n + 1) f (Icc a b) y * (x - xβ) ^ (n + 1) / β(n + 1)!
case h
f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : β
hy : y β Ioo a b
hy' : y β x
h :
(f x - taylorWithinEval f n (Icc a b) xβ x) * (-(βn + 1) * (x - y) ^ n) =
(x - y) ^ n / βn ! * ((x - x) ^ (n + 1) - (x - xβ) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y
hy_ne : x - y β 0
β’ -(βn + 1) * (x - y) ^ n β 0 | case h
f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : β
hy : y β Ioo a b
hy' : y β x
h :
(f x - taylorWithinEval f n (Icc a b) xβ x) * (-(βn + 1) * (x - y) ^ n) =
(x - y) ^ n / βn ! * ((x - x) ^ (n + 1) - (x - xβ) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y
hy_ne : x - y β 0
β’ -(βn + 1) * (x - y) ^ n β 0
case h
f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : β
hy : y β Ioo a b
hy' : y β x
h :
f x - taylorWithinEval f n (Icc a b) xβ x =
(x - y) ^ n / βn ! * ((x - x) ^ (n + 1) - (x - xβ) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y /
(-(βn + 1) * (x - y) ^ n)
hy_ne : x - y β 0
β’ f x - taylorWithinEval f n (Icc a b) xβ x = iteratedDerivWithin (n + 1) f (Icc a b) y * (x - xβ) ^ (n + 1) / β(n + 1)! | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : β
hy : y β Ioo a b
hy' : y β x
h :
f x - taylorWithinEval f n (Icc a b) xβ x =
(x - y) ^ n / βn ! * ((x - x) ^ (n + 1) - (x - xβ) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y /
(-(βn + 1) * (x - y) ^ n)
hy_ne : x - y β 0
β’ f x - taylorWithinEval f n (Icc a b) xβ x = iteratedDerivWithin (n + 1) f (Icc a b) y * (x - xβ) ^ (n + 1) / β(n + 1)!
case h
f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : β
hy : y β Ioo a b
hy' : y β x
h :
(f x - taylorWithinEval f n (Icc a b) xβ x) * (-(βn + 1) * (x - y) ^ n) =
(x - y) ^ n / βn ! * ((x - x) ^ (n + 1) - (x - xβ) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y
hy_ne : x - y β 0
β’ -(βn + 1) * (x - y) ^ n β 0
TACTIC:
|
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 : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : β
hy : y β Ioo a b
hy' : y β x
h :
f x - taylorWithinEval f n (Icc a b) xβ x =
(x - y) ^ n / βn ! * ((x - x) ^ (n + 1) - (x - xβ) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y /
(-(βn + 1) * (x - y) ^ n)
hy_ne : x - y β 0
β’ f x - taylorWithinEval f n (Icc a b) xβ x = iteratedDerivWithin (n + 1) f (Icc a b) y * (x - xβ) ^ (n + 1) / β(n + 1)! | case h
f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : β
hy : y β Ioo a b
hy' : y β x
h :
f x - taylorWithinEval f n (Icc a b) xβ x =
(x - y) ^ n / βn ! * ((x - x) ^ (n + 1) - (x - xβ) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y /
(-(βn + 1) * (x - y) ^ n)
hy_ne : x - y β 0
β’ (x - y) ^ n * (x - xβ) ^ (n + 1) * iteratedDerivWithin (n + 1) f (Icc a b) y / (βn ! * ((βn + 1) * (x - y) ^ n)) =
iteratedDerivWithin (n + 1) f (Icc a b) y * (x - xβ) ^ (n + 1) / ((βn + 1) * βn !) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : β
hy : y β Ioo a b
hy' : y β x
h :
f x - taylorWithinEval f n (Icc a b) xβ x =
(x - y) ^ n / βn ! * ((x - x) ^ (n + 1) - (x - xβ) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y /
(-(βn + 1) * (x - y) ^ n)
hy_ne : x - y β 0
β’ f x - taylorWithinEval f n (Icc a b) xβ x = iteratedDerivWithin (n + 1) f (Icc a b) y * (x - xβ) ^ (n + 1) / β(n + 1)!
TACTIC:
|
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, β mul_assoc, β div_div, div_eq_iff (pow_ne_zero _ hy_ne), div_mul_eq_mul_div] | case h
f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : β
hy : y β Ioo a b
hy' : y β x
h :
f x - taylorWithinEval f n (Icc a b) xβ x =
(x - y) ^ n / βn ! * ((x - x) ^ (n + 1) - (x - xβ) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y /
(-(βn + 1) * (x - y) ^ n)
hy_ne : x - y β 0
β’ (x - y) ^ n * (x - xβ) ^ (n + 1) * iteratedDerivWithin (n + 1) f (Icc a b) y / (βn ! * ((βn + 1) * (x - y) ^ n)) =
iteratedDerivWithin (n + 1) f (Icc a b) y * (x - xβ) ^ (n + 1) / ((βn + 1) * βn !) | case h
f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : β
hy : y β Ioo a b
hy' : y β x
h :
f x - taylorWithinEval f n (Icc a b) xβ x =
(x - y) ^ n / βn ! * ((x - x) ^ (n + 1) - (x - xβ) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y /
(-(βn + 1) * (x - y) ^ n)
hy_ne : x - y β 0
β’ (x - y) ^ n * (x - xβ) ^ (n + 1) * iteratedDerivWithin (n + 1) f (Icc a b) y / ((βn + 1) * βn !) =
iteratedDerivWithin (n + 1) f (Icc a b) y * (x - xβ) ^ (n + 1) * (x - y) ^ n / ((βn + 1) * βn !) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : β
hy : y β Ioo a b
hy' : y β x
h :
f x - taylorWithinEval f n (Icc a b) xβ x =
(x - y) ^ n / βn ! * ((x - x) ^ (n + 1) - (x - xβ) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y /
(-(βn + 1) * (x - y) ^ n)
hy_ne : x - y β 0
β’ (x - y) ^ n * (x - xβ) ^ (n + 1) * iteratedDerivWithin (n + 1) f (Icc a b) y / (βn ! * ((βn + 1) * (x - y) ^ n)) =
iteratedDerivWithin (n + 1) f (Icc a b) y * (x - xβ) ^ (n + 1) / ((βn + 1) * βn !)
TACTIC:
|
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 : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : β
hy : y β Ioo a b
hy' : y β x
h :
f x - taylorWithinEval f n (Icc a b) xβ x =
(x - y) ^ n / βn ! * ((x - x) ^ (n + 1) - (x - xβ) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y /
(-(βn + 1) * (x - y) ^ n)
hy_ne : x - y β 0
β’ (x - y) ^ n * (x - xβ) ^ (n + 1) * iteratedDerivWithin (n + 1) f (Icc a b) y / ((βn + 1) * βn !) =
iteratedDerivWithin (n + 1) f (Icc a b) y * (x - xβ) ^ (n + 1) * (x - y) ^ n / ((βn + 1) * βn !) | case h.e_a
f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : β
hy : y β Ioo a b
hy' : y β x
h :
f x - taylorWithinEval f n (Icc a b) xβ x =
(x - y) ^ n / βn ! * ((x - x) ^ (n + 1) - (x - xβ) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y /
(-(βn + 1) * (x - y) ^ n)
hy_ne : x - y β 0
β’ (x - y) ^ n * (x - xβ) ^ (n + 1) * iteratedDerivWithin (n + 1) f (Icc a b) y =
iteratedDerivWithin (n + 1) f (Icc a b) y * (x - xβ) ^ (n + 1) * (x - y) ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : β
hy : y β Ioo a b
hy' : y β x
h :
f x - taylorWithinEval f n (Icc a b) xβ x =
(x - y) ^ n / βn ! * ((x - x) ^ (n + 1) - (x - xβ) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y /
(-(βn + 1) * (x - y) ^ n)
hy_ne : x - y β 0
β’ (x - y) ^ n * (x - xβ) ^ (n + 1) * iteratedDerivWithin (n + 1) f (Icc a b) y / ((βn + 1) * βn !) =
iteratedDerivWithin (n + 1) f (Icc a b) y * (x - xβ) ^ (n + 1) * (x - y) ^ n / ((βn + 1) * βn !)
TACTIC:
|
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 : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : β
hy : y β Ioo a b
hy' : y β x
h :
f x - taylorWithinEval f n (Icc a b) xβ x =
(x - y) ^ n / βn ! * ((x - x) ^ (n + 1) - (x - xβ) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y /
(-(βn + 1) * (x - y) ^ n)
hy_ne : x - y β 0
β’ (x - y) ^ n * (x - xβ) ^ (n + 1) * iteratedDerivWithin (n + 1) f (Icc a b) y =
iteratedDerivWithin (n + 1) f (Icc a b) y * (x - xβ) ^ (n + 1) * (x - y) ^ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e_a
f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : β
hy : y β Ioo a b
hy' : y β x
h :
f x - taylorWithinEval f n (Icc a b) xβ x =
(x - y) ^ n / βn ! * ((x - x) ^ (n + 1) - (x - xβ) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y /
(-(βn + 1) * (x - y) ^ n)
hy_ne : x - y β 0
β’ (x - y) ^ n * (x - xβ) ^ (n + 1) * iteratedDerivWithin (n + 1) f (Icc a b) y =
iteratedDerivWithin (n + 1) f (Icc a b) y * (x - xβ) ^ (n + 1) * (x - y) ^ n
TACTIC:
|
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 : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
β’ ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b) | f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
β’ Continuous fun t => (x - t) ^ (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
β’ ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
TACTIC:
|
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 : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : β
hy : y β Ioo a b
hy' : y β x
h :
(f x - taylorWithinEval f n (Icc a b) xβ x) * (-(βn + 1) * (x - y) ^ n) =
(x - y) ^ n / βn ! * ((x - x) ^ (n + 1) - (x - xβ) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y
hy_ne : x - y β 0
β’ -(βn + 1) * (x - y) ^ n β 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : β
hy : y β Ioo a b
hy' : y β x
h :
(f x - taylorWithinEval f n (Icc a b) xβ x) * (-(βn + 1) * (x - y) ^ n) =
(x - y) ^ n / βn ! * ((x - x) ^ (n + 1) - (x - xβ) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y
hy_ne : x - y β 0
β’ -(βn + 1) * (x - y) ^ n β 0
TACTIC:
|
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 : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : β
hy : y β Ioo a b
hy' : y β x
h :
(f x - taylorWithinEval f n (Icc a b) xβ x) * (-(βn + 1) * (x - y) ^ n) =
(x - y) ^ n / βn ! * ((x - x) ^ (n + 1) - (x - xβ) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y
hy_ne : x - y β 0
β’ βn + 1 β 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : β
hy : y β Ioo a b
hy' : y β x
h :
(f x - taylorWithinEval f n (Icc a b) xβ x) * (-(βn + 1) * (x - y) ^ n) =
(x - y) ^ n / βn ! * ((x - x) ^ (n + 1) - (x - xβ) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y
hy_ne : x - y β 0
β’ βn + 1 β 0
TACTIC:
|
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 : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : β
hy : y β Ioo a b
hy' : y β x
h :
(f x - taylorWithinEval f n (Icc a b) xβ x) * (-(βn + 1) * (x - y) ^ n) =
(x - y) ^ n / βn ! * ((x - x) ^ (n + 1) - (x - xβ) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y
hy_ne : x - y β 0
β’ (x - y) ^ n β 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
gcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc a b)
y : β
hy : y β Ioo a b
hy' : y β x
h :
(f x - taylorWithinEval f n (Icc a b) xβ x) * (-(βn + 1) * (x - y) ^ n) =
(x - y) ^ n / βn ! * ((x - x) ^ (n + 1) - (x - xβ) ^ (n + 1)) * iteratedDerivWithin (n + 1) f (Icc a b) y
hy_ne : x - y β 0
β’ (x - y) ^ n β 0
TACTIC:
|
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 hxβ hf hf' continuousOn_id
(fun _ _ => hasDerivAt_id _) fun _ _ => by simp with
β¨y, hy, hβ© | f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
β’ β x',
β (_ : x' β Ioo a b),
f x - taylorWithinEval f n (Icc a b) xβ x =
iteratedDerivWithin (n + 1) f (Icc a b) x' * (x - x') ^ n / βn ! * (x - xβ) | case intro.intro
f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
y : β
hy : y β Ioo a b
h :
f x - taylorWithinEval f n (Icc a b) xβ x =
((x - y) ^ n / βn ! * (id x - id xβ) / 1) β’ iteratedDerivWithin (n + 1) f (Icc a b) y
β’ β x',
β (_ : x' β Ioo a b),
f x - taylorWithinEval f n (Icc a b) xβ x =
iteratedDerivWithin (n + 1) f (Icc a b) x' * (x - x') ^ n / βn ! * (x - xβ) | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
β’ β x',
β (_ : x' β Ioo a b),
f x - taylorWithinEval f n (Icc a b) xβ x =
iteratedDerivWithin (n + 1) f (Icc a b) x' * (x - x') ^ n / βn ! * (x - xβ)
TACTIC:
|
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' β¨y, hy, _β© | case intro.intro
f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
y : β
hy : y β Ioo a b
h :
f x - taylorWithinEval f n (Icc a b) xβ x =
((x - y) ^ n / βn ! * (id x - id xβ) / 1) β’ iteratedDerivWithin (n + 1) f (Icc a b) y
β’ β x',
β (_ : x' β Ioo a b),
f x - taylorWithinEval f n (Icc a b) xβ x =
iteratedDerivWithin (n + 1) f (Icc a b) x' * (x - x') ^ n / βn ! * (x - xβ) | case intro.intro
f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
y : β
hy : y β Ioo a b
h :
f x - taylorWithinEval f n (Icc a b) xβ x =
((x - y) ^ n / βn ! * (id x - id xβ) / 1) β’ iteratedDerivWithin (n + 1) f (Icc a b) y
β’ f x - taylorWithinEval f n (Icc a b) xβ x = iteratedDerivWithin (n + 1) f (Icc a b) y * (x - y) ^ n / βn ! * (x - xβ) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
y : β
hy : y β Ioo a b
h :
f x - taylorWithinEval f n (Icc a b) xβ x =
((x - y) ^ n / βn ! * (id x - id xβ) / 1) β’ iteratedDerivWithin (n + 1) f (Icc a b) y
β’ β x',
β (_ : x' β Ioo a b),
f x - taylorWithinEval f n (Icc a b) xβ x =
iteratedDerivWithin (n + 1) f (Icc a b) x' * (x - x') ^ n / βn ! * (x - xβ)
TACTIC:
|
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 : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
y : β
hy : y β Ioo a b
h :
f x - taylorWithinEval f n (Icc a b) xβ x =
((x - y) ^ n / βn ! * (id x - id xβ) / 1) β’ iteratedDerivWithin (n + 1) f (Icc a b) y
β’ f x - taylorWithinEval f n (Icc a b) xβ x = iteratedDerivWithin (n + 1) f (Icc a b) y * (x - y) ^ n / βn ! * (x - xβ) | case intro.intro
f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
y : β
hy : y β Ioo a b
h :
f x - taylorWithinEval f n (Icc a b) xβ x =
((x - y) ^ n / βn ! * (id x - id xβ) / 1) β’ iteratedDerivWithin (n + 1) f (Icc a b) y
β’ ((x - y) ^ n / βn ! * (id x - id xβ) / 1) β’ iteratedDerivWithin (n + 1) f (Icc a b) y =
iteratedDerivWithin (n + 1) f (Icc a b) y * (x - y) ^ n / βn ! * (x - xβ) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
y : β
hy : y β Ioo a b
h :
f x - taylorWithinEval f n (Icc a b) xβ x =
((x - y) ^ n / βn ! * (id x - id xβ) / 1) β’ iteratedDerivWithin (n + 1) f (Icc a b) y
β’ f x - taylorWithinEval f n (Icc a b) xβ x = iteratedDerivWithin (n + 1) f (Icc a b) y * (x - y) ^ n / βn ! * (x - xβ)
TACTIC:
|
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 : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
y : β
hy : y β Ioo a b
h :
f x - taylorWithinEval f n (Icc a b) xβ x =
((x - y) ^ n / βn ! * (id x - id xβ) / 1) β’ iteratedDerivWithin (n + 1) f (Icc a b) y
β’ ((x - y) ^ n / βn ! * (id x - id xβ) / 1) β’ iteratedDerivWithin (n + 1) f (Icc a b) y =
iteratedDerivWithin (n + 1) f (Icc a b) y * (x - y) ^ n / βn ! * (x - xβ) | case intro.intro
f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
y : β
hy : y β Ioo a b
h :
f x - taylorWithinEval f n (Icc a b) xβ x =
((x - y) ^ n / βn ! * (id x - id xβ) / 1) β’ iteratedDerivWithin (n + 1) f (Icc a b) y
β’ (x - y) ^ n * (x - xβ) * iteratedDerivWithin (n + 1) f (Icc a b) y =
iteratedDerivWithin (n + 1) f (Icc a b) y * (x - y) ^ n * (x - xβ) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
y : β
hy : y β Ioo a b
h :
f x - taylorWithinEval f n (Icc a b) xβ x =
((x - y) ^ n / βn ! * (id x - id xβ) / 1) β’ iteratedDerivWithin (n + 1) f (Icc a b) y
β’ ((x - y) ^ n / βn ! * (id x - id xβ) / 1) β’ iteratedDerivWithin (n + 1) f (Icc a b) y =
iteratedDerivWithin (n + 1) f (Icc a b) y * (x - y) ^ n / βn ! * (x - xβ)
TACTIC:
|
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 : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
y : β
hy : y β Ioo a b
h :
f x - taylorWithinEval f n (Icc a b) xβ x =
((x - y) ^ n / βn ! * (id x - id xβ) / 1) β’ iteratedDerivWithin (n + 1) f (Icc a b) y
β’ (x - y) ^ n * (x - xβ) * iteratedDerivWithin (n + 1) f (Icc a b) y =
iteratedDerivWithin (n + 1) f (Icc a b) y * (x - y) ^ n * (x - xβ) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
y : β
hy : y β Ioo a b
h :
f x - taylorWithinEval f n (Icc a b) xβ x =
((x - y) ^ n / βn ! * (id x - id xβ) / 1) β’ iteratedDerivWithin (n + 1) f (Icc a b) y
β’ (x - y) ^ n * (x - xβ) * iteratedDerivWithin (n + 1) f (Icc a b) y =
iteratedDerivWithin (n + 1) f (Icc a b) y * (x - y) ^ n * (x - xβ)
TACTIC:
|
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 : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
xβΒΉ : β
xβ : xβΒΉ β Ioo a b
β’ 1 β 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
x xβ a b : β
n : β
hab : a < b
hx : x β Icc a b
hxβ : xβ β Icc a b
hf : ContDiffOn β (βn) f (Icc a b)
hf' : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
xβΒΉ : β
xβ : xβΒΉ β Ioo a b
β’ 1 β 0
TACTIC:
|
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 : β β β
a b C x xβ : β
n : β
hab : a β€ b
hf : ContDiffOn β (βn + 1) f (Icc a b)
hx : x β Icc a b
hxβ : xβ β Icc a b
hC : β y β Ioo a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C
β’ βf x - taylorWithinEval f n (Icc a b) xβ xβ β€ C * |x - xβ| ^ (n + 1) / β(n + 1)! | case inl
f : β β β
a C x xβ : β
n : β
hab : a β€ a
hf : ContDiffOn β (βn + 1) f (Icc a a)
hx : x β Icc a a
hxβ : xβ β Icc a a
hC : β y β Ioo a a, βiteratedDerivWithin (n + 1) f (Icc a a) yβ β€ C
β’ βf x - taylorWithinEval f n (Icc a a) xβ xβ β€ C * |x - xβ| ^ (n + 1) / β(n + 1)!
case inr
f : β β β
a b C x xβ : β
n : β
habβ : a β€ b
hf : ContDiffOn β (βn + 1) f (Icc a b)
hx : x β Icc a b
hxβ : xβ β Icc a b
hC : β y β Ioo a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C
hab : a < b
β’ βf x - taylorWithinEval f n (Icc a b) xβ xβ β€ C * |x - xβ| ^ (n + 1) / β(n + 1)! | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
a b C x xβ : β
n : β
hab : a β€ b
hf : ContDiffOn β (βn + 1) f (Icc a b)
hx : x β Icc a b
hxβ : xβ β Icc a b
hC : β y β Ioo a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C
β’ βf x - taylorWithinEval f n (Icc a b) xβ xβ β€ C * |x - xβ| ^ (n + 1) / β(n + 1)!
TACTIC:
|
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 β (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 [β Nat.cast_add_one, Nat.cast_lt]
exact Nat.lt_succ_self _ | case inr
f : β β β
a b C x xβ : β
n : β
habβ : a β€ b
hf : ContDiffOn β (βn + 1) f (Icc a b)
hx : x β Icc a b
hxβ : xβ β Icc a b
hC : β y β Ioo a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C
hab : a < b
β’ βf x - taylorWithinEval f n (Icc a b) xβ xβ β€ C * |x - xβ| ^ (n + 1) / β(n + 1)! | case inr
f : β β β
a b C x xβ : β
n : β
habβ : a β€ b
hf : ContDiffOn β (βn + 1) f (Icc a b)
hx : x β Icc a b
hxβ : xβ β Icc a b
hC : β y β Ioo a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C
hab : a < b
this : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
β’ βf x - taylorWithinEval f n (Icc a b) xβ xβ β€ C * |x - xβ| ^ (n + 1) / β(n + 1)! | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
f : β β β
a b C x xβ : β
n : β
habβ : a β€ b
hf : ContDiffOn β (βn + 1) f (Icc a b)
hx : x β Icc a b
hxβ : xβ β Icc a b
hC : β y β Ioo a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C
hab : a < b
β’ βf x - taylorWithinEval f n (Icc a b) xβ xβ β€ C * |x - xβ| ^ (n + 1) / β(n + 1)!
TACTIC:
|
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 β¨x', hx', hβ© := taylor_mean_remainder_lagrange_central hab hx hxβ hf.of_succ this | case inr
f : β β β
a b C x xβ : β
n : β
habβ : a β€ b
hf : ContDiffOn β (βn + 1) f (Icc a b)
hx : x β Icc a b
hxβ : xβ β Icc a b
hC : β y β Ioo a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C
hab : a < b
this : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
β’ βf x - taylorWithinEval f n (Icc a b) xβ xβ β€ C * |x - xβ| ^ (n + 1) / β(n + 1)! | case inr.intro.intro
f : β β β
a b C x xβ : β
n : β
habβ : a β€ b
hf : ContDiffOn β (βn + 1) f (Icc a b)
hx : x β Icc a b
hxβ : xβ β Icc a b
hC : β y β Ioo a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C
hab : a < b
this : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
x' : β
hx' : x' β Ioo a b
h :
f x - taylorWithinEval f n (Icc a b) xβ x =
iteratedDerivWithin (n + 1) f (Icc a b) x' * (x - xβ) ^ (n + 1) / β(n + 1)!
β’ βf x - taylorWithinEval f n (Icc a b) xβ xβ β€ C * |x - xβ| ^ (n + 1) / β(n + 1)! | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
f : β β β
a b C x xβ : β
n : β
habβ : a β€ b
hf : ContDiffOn β (βn + 1) f (Icc a b)
hx : x β Icc a b
hxβ : xβ β Icc a b
hC : β y β Ioo a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C
hab : a < b
this : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
β’ βf x - taylorWithinEval f n (Icc a b) xβ xβ β€ C * |x - xβ| ^ (n + 1) / β(n + 1)!
TACTIC:
|
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 - xβ) ^ _), β abs_pow] | case inr.intro.intro
f : β β β
a b C x xβ : β
n : β
habβ : a β€ b
hf : ContDiffOn β (βn + 1) f (Icc a b)
hx : x β Icc a b
hxβ : xβ β Icc a b
hC : β y β Ioo a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C
hab : a < b
this : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
x' : β
hx' : x' β Ioo a b
h :
f x - taylorWithinEval f n (Icc a b) xβ x =
iteratedDerivWithin (n + 1) f (Icc a b) x' * (x - xβ) ^ (n + 1) / β(n + 1)!
β’ βf x - taylorWithinEval f n (Icc a b) xβ xβ β€ C * |x - xβ| ^ (n + 1) / β(n + 1)! | case inr.intro.intro
f : β β β
a b C x xβ : β
n : β
habβ : a β€ b
hf : ContDiffOn β (βn + 1) f (Icc a b)
hx : x β Icc a b
hxβ : xβ β Icc a b
hC : β y β Ioo a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C
hab : a < b
this : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
x' : β
hx' : x' β Ioo a b
h :
f x - taylorWithinEval f n (Icc a b) xβ x =
iteratedDerivWithin (n + 1) f (Icc a b) x' * (x - xβ) ^ (n + 1) / β(n + 1)!
β’ βiteratedDerivWithin (n + 1) f (Icc a b) x'β * |(x - xβ) ^ (n + 1)| / β(n + 1)! β€ C * |(x - xβ) ^ (n + 1)| / β(n + 1)! | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.intro
f : β β β
a b C x xβ : β
n : β
habβ : a β€ b
hf : ContDiffOn β (βn + 1) f (Icc a b)
hx : x β Icc a b
hxβ : xβ β Icc a b
hC : β y β Ioo a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C
hab : a < b
this : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
x' : β
hx' : x' β Ioo a b
h :
f x - taylorWithinEval f n (Icc a b) xβ x =
iteratedDerivWithin (n + 1) f (Icc a b) x' * (x - xβ) ^ (n + 1) / β(n + 1)!
β’ βf x - taylorWithinEval f n (Icc a b) xβ xβ β€ C * |x - xβ| ^ (n + 1) / β(n + 1)!
TACTIC:
|
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 : β β β
a b C x xβ : β
n : β
habβ : a β€ b
hf : ContDiffOn β (βn + 1) f (Icc a b)
hx : x β Icc a b
hxβ : xβ β Icc a b
hC : β y β Ioo a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C
hab : a < b
this : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
x' : β
hx' : x' β Ioo a b
h :
f x - taylorWithinEval f n (Icc a b) xβ x =
iteratedDerivWithin (n + 1) f (Icc a b) x' * (x - xβ) ^ (n + 1) / β(n + 1)!
β’ βiteratedDerivWithin (n + 1) f (Icc a b) x'β * |(x - xβ) ^ (n + 1)| / β(n + 1)! β€ C * |(x - xβ) ^ (n + 1)| / β(n + 1)! | case inr.intro.intro
f : β β β
a b C x xβ : β
n : β
habβ : a β€ b
hf : ContDiffOn β (βn + 1) f (Icc a b)
hx : x β Icc a b
hxβ : xβ β Icc a b
hC : β y β Ioo a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C
hab : a < b
this : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
x' : β
hx' : x' β Ioo a b
h :
f x - taylorWithinEval f n (Icc a b) xβ x =
iteratedDerivWithin (n + 1) f (Icc a b) x' * (x - xβ) ^ (n + 1) / β(n + 1)!
β’ βiteratedDerivWithin (n + 1) f (Icc a b) x'β * |(x - xβ) ^ (n + 1)| β€ C * |(x - xβ) ^ (n + 1)| | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.intro
f : β β β
a b C x xβ : β
n : β
habβ : a β€ b
hf : ContDiffOn β (βn + 1) f (Icc a b)
hx : x β Icc a b
hxβ : xβ β Icc a b
hC : β y β Ioo a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C
hab : a < b
this : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
x' : β
hx' : x' β Ioo a b
h :
f x - taylorWithinEval f n (Icc a b) xβ x =
iteratedDerivWithin (n + 1) f (Icc a b) x' * (x - xβ) ^ (n + 1) / β(n + 1)!
β’ βiteratedDerivWithin (n + 1) f (Icc a b) x'β * |(x - xβ) ^ (n + 1)| / β(n + 1)! β€ C * |(x - xβ) ^ (n + 1)| / β(n + 1)!
TACTIC:
|
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 : β β β
a b C x xβ : β
n : β
habβ : a β€ b
hf : ContDiffOn β (βn + 1) f (Icc a b)
hx : x β Icc a b
hxβ : xβ β Icc a b
hC : β y β Ioo a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C
hab : a < b
this : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
x' : β
hx' : x' β Ioo a b
h :
f x - taylorWithinEval f n (Icc a b) xβ x =
iteratedDerivWithin (n + 1) f (Icc a b) x' * (x - xβ) ^ (n + 1) / β(n + 1)!
β’ βiteratedDerivWithin (n + 1) f (Icc a b) x'β * |(x - xβ) ^ (n + 1)| β€ C * |(x - xβ) ^ (n + 1)| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.intro
f : β β β
a b C x xβ : β
n : β
habβ : a β€ b
hf : ContDiffOn β (βn + 1) f (Icc a b)
hx : x β Icc a b
hxβ : xβ β Icc a b
hC : β y β Ioo a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C
hab : a < b
this : DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
x' : β
hx' : x' β Ioo a b
h :
f x - taylorWithinEval f n (Icc a b) xβ x =
iteratedDerivWithin (n + 1) f (Icc a b) x' * (x - xβ) ^ (n + 1) / β(n + 1)!
β’ βiteratedDerivWithin (n + 1) f (Icc a b) x'β * |(x - xβ) ^ (n + 1)| β€ C * |(x - xβ) ^ (n + 1)|
TACTIC:
|
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 hxβ | case inl
f : β β β
a C x xβ : β
n : β
hab : a β€ a
hf : ContDiffOn β (βn + 1) f (Icc a a)
hx : x β Icc a a
hxβ : xβ β Icc a a
hC : β y β Ioo a a, βiteratedDerivWithin (n + 1) f (Icc a a) yβ β€ C
β’ βf x - taylorWithinEval f n (Icc a a) xβ xβ β€ C * |x - xβ| ^ (n + 1) / β(n + 1)! | case inl
f : β β β
a C x xβ : β
n : β
hab : a β€ a
hf : ContDiffOn β (βn + 1) f (Icc a a)
hC : β y β Ioo a a, βiteratedDerivWithin (n + 1) f (Icc a a) yβ β€ C
hx : x = a
hxβ : xβ = a
β’ βf x - taylorWithinEval f n (Icc a a) xβ xβ β€ C * |x - xβ| ^ (n + 1) / β(n + 1)! | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
f : β β β
a C x xβ : β
n : β
hab : a β€ a
hf : ContDiffOn β (βn + 1) f (Icc a a)
hx : x β Icc a a
hxβ : xβ β Icc a a
hC : β y β Ioo a a, βiteratedDerivWithin (n + 1) f (Icc a a) yβ β€ C
β’ βf x - taylorWithinEval f n (Icc a a) xβ xβ β€ C * |x - xβ| ^ (n + 1) / β(n + 1)!
TACTIC:
|
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 hxβ hx | case inl
f : β β β
a C x xβ : β
n : β
hab : a β€ a
hf : ContDiffOn β (βn + 1) f (Icc a a)
hC : β y β Ioo a a, βiteratedDerivWithin (n + 1) f (Icc a a) yβ β€ C
hx : x = a
hxβ : xβ = a
β’ βf x - taylorWithinEval f n (Icc a a) xβ xβ β€ C * |x - xβ| ^ (n + 1) / β(n + 1)! | case inl
f : β β β
C x : β
n : β
hab : x β€ x
hf : ContDiffOn β (βn + 1) f (Icc x x)
hC : β y β Ioo x x, βiteratedDerivWithin (n + 1) f (Icc x x) yβ β€ C
β’ βf x - taylorWithinEval f n (Icc x x) x xβ β€ C * |x - x| ^ (n + 1) / β(n + 1)! | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
f : β β β
a C x xβ : β
n : β
hab : a β€ a
hf : ContDiffOn β (βn + 1) f (Icc a a)
hC : β y β Ioo a a, βiteratedDerivWithin (n + 1) f (Icc a a) yβ β€ C
hx : x = a
hxβ : xβ = a
β’ βf x - taylorWithinEval f n (Icc a a) xβ xβ β€ C * |x - xβ| ^ (n + 1) / β(n + 1)!
TACTIC:
|
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 : β β β
C x : β
n : β
hab : x β€ x
hf : ContDiffOn β (βn + 1) f (Icc x x)
hC : β y β Ioo x x, βiteratedDerivWithin (n + 1) f (Icc x x) yβ β€ C
β’ βf x - taylorWithinEval f n (Icc x x) x xβ β€ C * |x - x| ^ (n + 1) / β(n + 1)! | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
f : β β β
C x : β
n : β
hab : x β€ x
hf : ContDiffOn β (βn + 1) f (Icc x x)
hC : β y β Ioo x x, βiteratedDerivWithin (n + 1) f (Icc x x) yβ β€ C
β’ βf x - taylorWithinEval f n (Icc x x) x xβ β€ C * |x - x| ^ (n + 1) / β(n + 1)!
TACTIC:
|
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 : β β β
a b C x xβ : β
n : β
habβ : a β€ b
hf : ContDiffOn β (βn + 1) f (Icc a b)
hx : x β Icc a b
hxβ : xβ β Icc a b
hC : β y β Ioo a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C
hab : a < b
β’ DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
a b C x xβ : β
n : β
habβ : a β€ b
hf : ContDiffOn β (βn + 1) f (Icc a b)
hx : x β Icc a b
hxβ : xβ β Icc a b
hC : β y β Ioo a b, βiteratedDerivWithin (n + 1) f (Icc a b) yβ β€ C
hab : a < b
β’ DifferentiableOn β (iteratedDerivWithin n f (Icc a b)) (Ioo a b)
TACTIC:
|
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 : β β β
a b xβ : β
n : β
hab : a β€ b
hf : ContDiffOn β (βn + 1) f (Icc a b)
hxβ : xβ β Icc a b
β’ β C, β x β Icc a b, βf x - taylorWithinEval f n (Icc a b) xβ xβ β€ C * |x - xβ| ^ (n + 1) | case inl
f : β β β
a xβ : β
n : β
hab : a β€ a
hf : ContDiffOn β (βn + 1) f (Icc a a)
hxβ : xβ β Icc a a
β’ β C, β x β Icc a a, βf x - taylorWithinEval f n (Icc a a) xβ xβ β€ C * |x - xβ| ^ (n + 1)
case inr
f : β β β
a b xβ : β
n : β
hab : a β€ b
hf : ContDiffOn β (βn + 1) f (Icc a b)
hxβ : xβ β Icc a b
h : a < b
β’ β C, β x β Icc a b, βf x - taylorWithinEval f n (Icc a b) xβ xβ β€ C * |x - xβ| ^ (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
a b xβ : β
n : β
hab : a β€ b
hf : ContDiffOn β (βn + 1) f (Icc a b)
hxβ : xβ β Icc a b
β’ β C, β x β Icc a b, βf x - taylorWithinEval f n (Icc a b) xβ xβ β€ C * |x - xβ| ^ (n + 1)
TACTIC:
|
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 => βiteratedDerivWithin (n + 1) f (Icc a b) yβ) '' Icc a b) | case inr
f : β β β
a b xβ : β
n : β
hab : a β€ b
hf : ContDiffOn β (βn + 1) f (Icc a b)
hxβ : xβ β Icc a b
h : a < b
β’ β C, β x β Icc a b, βf x - taylorWithinEval f n (Icc a b) xβ xβ β€ C * |x - xβ| ^ (n + 1) | case inr
f : β β β
a b xβ : β
n : β
hab : a β€ b
hf : ContDiffOn β (βn + 1) f (Icc a b)
hxβ : xβ β Icc a b
h : a < b
C : Type := Sup β((fun y => βiteratedDerivWithin (n + 1) f (Icc a b) yβ) '' Icc a b)
β’ β C, β x β Icc a b, βf x - taylorWithinEval f n (Icc a b) xβ xβ β€ C * |x - xβ| ^ (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
f : β β β
a b xβ : β
n : β
hab : a β€ b
hf : ContDiffOn β (βn + 1) f (Icc a b)
hxβ : xβ β Icc a b
h : a < b
β’ β C, β x β Icc a b, βf x - taylorWithinEval f n (Icc a b) xβ xβ β€ C * |x - xβ| ^ (n + 1)
TACTIC:
|
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' β¨C / (n + 1)!, fun x hx => _β© | case inr
f : β β β
a b xβ : β
n : β
hab : a β€ b
hf : ContDiffOn β (βn + 1) f (Icc a b)
hxβ : xβ β Icc a b
h : a < b
C : Type := Sup β((fun y => βiteratedDerivWithin (n + 1) f (Icc a b) yβ) '' Icc a b)
β’ β C, β x β Icc a b, βf x - taylorWithinEval f n (Icc a b) xβ xβ β€ C * |x - xβ| ^ (n + 1) | case inr
f : β β β
a b xβ : β
n : β
hab : a β€ b
hf : ContDiffOn β (βn + 1) f (Icc a b)
hxβ : xβ β Icc a b
h : a < b
C : Type := Sup β((fun y => βiteratedDerivWithin (n + 1) f (Icc a b) yβ) '' Icc a b)
x : β
hx : x β Icc a b
β’ βf x - taylorWithinEval f n (Icc a b) xβ xβ β€ sorryAx β true * |x - xβ| ^ (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
f : β β β
a b xβ : β
n : β
hab : a β€ b
hf : ContDiffOn β (βn + 1) f (Icc a b)
hxβ : xβ β Icc a b
h : a < b
C : Type := Sup β((fun y => βiteratedDerivWithin (n + 1) f (Icc a b) yβ) '' Icc a b)
β’ β C, β x β Icc a b, βf x - taylorWithinEval f n (Icc a b) xβ xβ β€ C * |x - xβ| ^ (n + 1)
TACTIC:
|
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' β¨0, fun x hx => _β© | case inl
f : β β β
a xβ : β
n : β
hab : a β€ a
hf : ContDiffOn β (βn + 1) f (Icc a a)
hxβ : xβ β Icc a a
β’ β C, β x β Icc a a, βf x - taylorWithinEval f n (Icc a a) xβ xβ β€ C * |x - xβ| ^ (n + 1) | case inl
f : β β β
a xβ : β
n : β
hab : a β€ a
hf : ContDiffOn β (βn + 1) f (Icc a a)
hxβ : xβ β Icc a a
x : β
hx : x β Icc a a
β’ βf x - taylorWithinEval f n (Icc a a) xβ xβ β€ 0 * |x - xβ| ^ (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
f : β β β
a xβ : β
n : β
hab : a β€ a
hf : ContDiffOn β (βn + 1) f (Icc a a)
hxβ : xβ β Icc a a
β’ β C, β x β Icc a a, βf x - taylorWithinEval f n (Icc a a) xβ xβ β€ C * |x - xβ| ^ (n + 1)
TACTIC:
|
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 hxβ | case inl
f : β β β
a xβ : β
n : β
hab : a β€ a
hf : ContDiffOn β (βn + 1) f (Icc a a)
hxβ : xβ β Icc a a
x : β
hx : x β Icc a a
β’ βf x - taylorWithinEval f n (Icc a a) xβ xβ β€ 0 * |x - xβ| ^ (n + 1) | case inl
f : β β β
a xβ : β
n : β
hab : a β€ a
hf : ContDiffOn β (βn + 1) f (Icc a a)
hxβ : xβ = a
x : β
hx : x = a
β’ βf x - taylorWithinEval f n (Icc a a) xβ xβ β€ 0 * |x - xβ| ^ (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
f : β β β
a xβ : β
n : β
hab : a β€ a
hf : ContDiffOn β (βn + 1) f (Icc a a)
hxβ : xβ β Icc a a
x : β
hx : x β Icc a a
β’ βf x - taylorWithinEval f n (Icc a a) xβ xβ β€ 0 * |x - xβ| ^ (n + 1)
TACTIC:
|
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 [hxβ, hx, taylorWithinEval_self, sub_self, MulZeroClass.zero_mul, norm_zero] | case inl
f : β β β
a xβ : β
n : β
hab : a β€ a
hf : ContDiffOn β (βn + 1) f (Icc a a)
hxβ : xβ = a
x : β
hx : x = a
β’ βf x - taylorWithinEval f n (Icc a a) xβ xβ β€ 0 * |x - xβ| ^ (n + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
f : β β β
a xβ : β
n : β
hab : a β€ a
hf : ContDiffOn β (βn + 1) f (Icc a a)
hxβ : xβ = a
x : β
hx : x = a
β’ βf x - taylorWithinEval f n (Icc a a) xβ xβ β€ 0 * |x - xβ| ^ (n + 1)
TACTIC:
|
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/RamseyPrereq.lean | Fin.ne_zero_iff_eq_one | [23, 1] | [23, 75] | decide | β’ β {x : Fin 2}, x β 0 β x = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
β’ β {x : Fin 2}, x β 0 β x = 1
TACTIC:
|
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/RamseyPrereq.lean | Fin.eq_zero_iff_ne_one | [25, 1] | [25, 75] | decide | β’ β {x : Fin 2}, x = 0 β x β 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
β’ β {x : Fin 2}, x = 0 β x β 1
TACTIC:
|
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 β 1
β’ x = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : Fin 2
hx : x β 1
β’ x = 0
TACTIC:
|
https://github.com/b-mehta/ExponentialRamsey.git | 7e17b629a915a082869f494c8afa56a3e1c7a88d | ExponentialRamsey/Prereq/Mathlib/Analysis/Calculus/IteratedDeriv.lean | iteratedFDerivWithin_nhds | [29, 1] | [31, 82] | rw [β iteratedFDerivWithin_univ, β univ_inter u, iteratedFDerivWithin_inter hu] | π : Type u_1
instββ΄ : NontriviallyNormedField π
F : Type u_2
instβΒ³ : NormedAddCommGroup F
instβΒ² : NormedSpace π F
E : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace π E
u : Set E
x : E
f : E β F
n : β
hu : u β π x
β’ iteratedFDerivWithin π n f u x = iteratedFDeriv π n f x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
π : Type u_1
instββ΄ : NontriviallyNormedField π
F : Type u_2
instβΒ³ : NormedAddCommGroup F
instβΒ² : NormedSpace π F
E : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace π E
u : Set E
x : E
f : E β F
n : β
hu : u β π x
β’ iteratedFDerivWithin π n f u x = iteratedFDeriv π n f x
TACTIC:
|
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] | π : Type u_1
instββ΄ : NontriviallyNormedField π
F : Type u_2
instβΒ³ : NormedAddCommGroup F
instβΒ² : NormedSpace π F
E : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace π E
s : Set π
f : π β F
n : β
hs : IsOpen s
x : π
hx : x β s
β’ iteratedDerivWithin n f s x = iteratedDeriv n f x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
π : Type u_1
instββ΄ : NontriviallyNormedField π
F : Type u_2
instβΒ³ : NormedAddCommGroup F
instβΒ² : NormedSpace π F
E : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace π E
s : Set π
f : π β F
n : β
hs : IsOpen s
x : π
hx : x β s
β’ iteratedDerivWithin n f s x = iteratedDeriv n f x
TACTIC:
|
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] | π : Type u_1
instββ΄ : NontriviallyNormedField π
F : Type u_2
instβΒ³ : NormedAddCommGroup F
instβΒ² : NormedSpace π F
E : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace π E
u : Set π
x : π
f : π β F
n : β
hu : u β π x
β’ iteratedDerivWithin n f u x = iteratedDeriv n f x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
π : Type u_1
instββ΄ : NontriviallyNormedField π
F : Type u_2
instβΒ³ : NormedAddCommGroup F
instβΒ² : NormedSpace π F
E : Type u_3
instβΒΉ : NormedAddCommGroup E
instβ : NormedSpace π E
u : Set π
x : π
f : π β F
n : β
hu : u β π x
β’ iteratedDerivWithin n f u x = iteratedDeriv n f x
TACTIC:
|
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 : β
β’ choose (s + t) s β€ (t + 1) ^ s | s t : β
β’ ascFactorial t s / s ! β€ (t + 1) ^ s | Please generate a tactic in lean4 to solve the state.
STATE:
s t : β
β’ choose (s + t) s β€ (t + 1) ^ s
TACTIC:
|
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 : β
β’ ascFactorial t s / s ! β€ (t + 1) ^ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s t : β
β’ ascFactorial t s / s ! β€ (t + 1) ^ s
TACTIC:
|
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 : β
β’ choose s t β€ (s + 1 - t) ^ t | case inl
s t : β
h : t β€ s
β’ choose s t β€ (s + 1 - t) ^ t
case inr
s t : β
h : s < t
β’ choose s t β€ (s + 1 - t) ^ t | Please generate a tactic in lean4 to solve the state.
STATE:
s t : β
β’ choose s t β€ (s + 1 - t) ^ t
TACTIC:
|
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 : β
h : s < t
β’ choose s t β€ (s + 1 - t) ^ t | case inr
s t : β
h : s < t
β’ 0 β€ (s + 1 - t) ^ t | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
s t : β
h : s < t
β’ choose s t β€ (s + 1 - t) ^ t
TACTIC:
|
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