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

Goal state before:
pp: E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
f : E → F
x y : E
n✝ : ℕ
inst✝ : CompleteSpace F
n : ℕ
ih :
  f (x + y) =
    (∑ k ∈ Finset.range (n + 1), (↑k !)⁻¹ • (iteratedFDeriv ℝ k f x) fun x ↦ y) +
      (↑n !)⁻¹ • ∫ (t : ℝ) in 0..1, (1 - t) ^ n • (iteratedFDeriv ℝ (n + 1) f (x + t • y)) fun x ↦ y
hf : ∀ t ∈ Set.uIcc 0 1, ContDiffAt ℝ (↑(n + 1) + 1) f (x + t • y)
u : ℕ → ℝ → ℝ := fun k t ↦ (↑k !)⁻¹ * (1 - t) ^ k
hu : ∀ (t : ℝ), HasDerivAt (u (n + 1)) (-u n t) t
hu' : Continuous (u n)
v : ℕ → ℝ → F := fun k t ↦ (iteratedFDeriv ℝ k f (x + t • y)) fun x ↦ y
hv : ∀ t ∈ Set.uIcc 0 1, HasDerivAt (v (n + 1)) (v (n + 1 + 1) t) t
t : ℝ
ht : t ∈ Set.uIcc 0 1
h_cont : ContinuousAt (iteratedFDeriv ℝ (n + 1 + 1) f) (x + t • y)
⊢ ContinuousAt (fun t ↦ x + t • y) t

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Analysis.Calculus.TaylorIntegral

Goal state before:
pp: E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
f : E → F
x y : E
n✝ : ℕ
inst✝ : CompleteSpace F
n : ℕ
ih :
  f (x + y) =
    (∑ k ∈ Finset.range (n + 1), (↑k !)⁻¹ • (iteratedFDeriv ℝ k f x) fun x ↦ y) +
      (↑n !)⁻¹ • ∫ (t : ℝ) in 0..1, (1 - t) ^ n • (iteratedFDeriv ℝ (n + 1) f (x + t • y)) fun x ↦ y
hf : ∀ t ∈ Set.uIcc 0 1, ContDiffAt ℝ (↑(n + 1) + 1) f (x + t • y)
u : ℕ → ℝ → ℝ := fun k t ↦ (↑k !)⁻¹ * (1 - t) ^ k
hu : ∀ (t : ℝ), HasDerivAt (u (n + 1)) (-u n t) t
hu' : Continuous (u n)
v : ℕ → ℝ → F := fun k t ↦ (iteratedFDeriv ℝ k f (x + t • y)) fun x ↦ y
hv : ∀ t ∈ Set.uIcc 0 1, HasDerivAt (v (n + 1)) (v (n + 1 + 1) t) t
t : ℝ
ht : t ∈ Set.uIcc 0 1
h_cont : ContinuousAt (iteratedFDeriv ℝ (n + 1 + 1) f) (x + t • y)
⊢ ContinuousWithinAt (v (n + 1 + 1)) (Set.uIcc 0 1) t

Tactic:
exact (h_cont.comp (x := t) (by fun_prop)).continuousWithinAt.eval_const _

Goal state after:
Module: Mathlib.Analysis.Calculus.Taylor

Goal state before:
pp: case inr.hf
f : ℝ → ℝ
x x₀ : ℝ
n : ℕ
hf₁ : ContDiffOn ℝ (↑n) f [[x₀, x]]
hf₂ : AbsolutelyContinuousOnInterval (iteratedDerivWithin n f [[x₀, x]]) x₀ x
this : x₀ ≠ x
k : ℕ
hk : k ≤ n
⊢ ContDiffOn ℝ 1 (fun t ↦ (x - t) ^ k / ↑k !) [[x₀, x]]

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Analysis.Calculus.Taylor

Goal state before:
pp: case inr.hu
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : ℝ → F
x x₀ : ℝ
n : ℕ
hf : ContDiffOn ℝ (↑(n + 1)) f [[x₀, x]]
this✝ : x₀ ≠ x
this : UniqueDiffOn ℝ [[x₀, x]]
k : ℕ
hk : k ≤ n
⊢ ContinuousOn (fun t ↦ (x - t) ^ k / ↑k !) [[x₀, x]]

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Analysis.Calculus.Taylor

Goal state before:
pp: case inr.huu'
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : ℝ → F
x x₀ : ℝ
n : ℕ
hf : ContDiffOn ℝ (↑(n + 1)) f [[x₀, x]]
this✝ : x₀ ≠ x
this : UniqueDiffOn ℝ [[x₀, x]]
k : ℕ
hk : k ≤ n
t : ℝ
ht : t ∈ Ioo (min x₀ x) (max x₀ x)
⊢ DifferentiableAt ℝ (fun t ↦ (x - t) ^ k / ↑k !) t

Tactic:
fun_prop

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.AbsolutelyContinuousFun

Goal state before:
pp: F✝ : Type u_2
inst✝³ : NormedAddCommGroup F✝
inst✝² : NormedSpace ℝ F✝
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
a b : ℝ
hf : AbsolutelyContinuousOnInterval f a b
hab : a ≤ b
d : ℝ
hd : a ≤ d ∧ d ≤ b
r : ℝ
hr : r > 0
had : a ≤ d
hdb : d < b
hf₀ : ∀ᵐ (x : ℝ), x ∈ Ioo d b → HasDerivAt f 0 x
hfdb' : 0 < r / (b - d)
u : Set (ℝ × ℝ)
hu₁ : ∀ z ∈ u, (d < z.1 ∧ z.1 < z.2 ∧ z.2 < b) ∧ dist (slope f z.1 z.2) 0 < r / (b - d)
hu₂ : u.PairwiseDisjoint fun z ↦ Icc z.1 z.2
hu₃ : HasSum (fun z ↦ (↑z).2 - (↑z).1) (b - d)
g : ↑u → ℝ := fun z ↦ dist (f (↑z).1) (f (↑z).2)
g_nonneg : 0 ≤ g
s : Finset ↑u
z : ↑u
hz : z ∈ s
slope_bound : dist (slope f (↑z).1 (↑z).2) 0 ≤ r / (b - d)
this : 0 < (↑z).2 - (↑z).1
⊢ ‖f (↑z).1 - f (↑z).2‖ ≤ ‖f (↑z).2 - f (↑z).1‖

Tactic:
norm_sub_rev

Goal state after:
F✝ : Type u_2
inst✝³ : NormedAddCommGroup F✝
inst✝² : NormedSpace ℝ F✝
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
a b : ℝ
hf : AbsolutelyContinuousOnInterval f a b
hab : a ≤ b
d : ℝ
hd : a ≤ d ∧ d ≤ b
r : ℝ
hr : r > 0
had : a ≤ d
hdb : d < b
hf₀ : ∀ᵐ (x : ℝ), x ∈ Ioo d b → HasDerivAt f 0 x
hfdb' : 0 < r / (b - d)
u : Set (ℝ × ℝ)
hu₁ : ∀ z ∈ u, (d < z.1 ∧ z.1 < z.2 ∧ z.2 < b) ∧ dist (slope f z.1 z.2) 0 < r / (b - d)
hu₂ : u.PairwiseDisjoint fun z ↦ Icc z.1 z.2
hu₃ : HasSum (fun z ↦ (↑z).2 - (↑z).1) (b - d)
g : ↑u → ℝ := fun z ↦ dist (f (↑z).1) (f (↑z).2)
g_nonneg : 0 ≤ g
s : Finset ↑u
z : ↑u
hz : z ∈ s
slope_bound : dist (slope f (↑z).1 (↑z).2) 0 ≤ r / (b - d)
this : 0 < (↑z).2 - (↑z).1
⊢ ‖f (↑z).2 - f (↑z).1‖ ≤ ‖f (↑z).2 - f (↑z).1‖
Module: Mathlib.Analysis.Calculus.Taylor

Goal state before:
pp: case inr.hu'
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : ℝ → F
x x₀ : ℝ
n : ℕ
hf : ContDiffOn ℝ (↑(n + 1)) f [[x₀, x]]
this✝ : x₀ ≠ x
this : UniqueDiffOn ℝ [[x₀, x]]
k : ℕ
hk : k ≤ n
⊢ ContinuousOn (deriv fun t ↦ (x - t) ^ k / ↑k !) [[x₀, x]]

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Analysis.Calculus.Taylor

Goal state before:
pp: case inr.hv'.hf
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : ℝ → F
x x₀ : ℝ
n : ℕ
hf : ContDiffOn ℝ (↑(n + 1)) f [[x₀, x]]
this✝ : x₀ ≠ x
this : UniqueDiffOn ℝ [[x₀, x]]
k : ℕ
hk : k ≤ n
⊢ IntervalIntegrable (iteratedDerivWithin (k + 1) f [[x₀, x]]) MeasureTheory.volume x₀ x

pp: case inr.hv'.h
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : ℝ → F
x x₀ : ℝ
n : ℕ
hf : ContDiffOn ℝ (↑(n + 1)) f [[x₀, x]]
this✝ : x₀ ≠ x
this : UniqueDiffOn ℝ [[x₀, x]]
k : ℕ
hk : k ≤ n
⊢ iteratedDerivWithin (k + 1) f [[x₀, x]] =ᵐ[MeasureTheory.volume.restrict (Ι x₀ x)]
    deriv fun t ↦ iteratedDerivWithin k f [[x₀, x]] t

Tactic:
· exact hf.continuousOn_iteratedDerivWithin (by norm_cast; omega) this |>.intervalIntegrable

Goal state after:
case inr.hv'.h
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : ℝ → F
x x₀ : ℝ
n : ℕ
hf : ContDiffOn ℝ (↑(n + 1)) f [[x₀, x]]
this✝ : x₀ ≠ x
this : UniqueDiffOn ℝ [[x₀, x]]
k : ℕ
hk : k ≤ n
⊢ iteratedDerivWithin (k + 1) f [[x₀, x]] =ᵐ[MeasureTheory.volume.restrict (Ι x₀ x)]
    deriv fun t ↦ iteratedDerivWithin k f [[x₀, x]] t
Module: Mathlib.Analysis.Convex.StrictConvexSpace

Goal state before:
pp: E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : StrictConvexSpace ℝ E
x y : E
h : ¬SameRay ℝ x y
⊢ ‖x‖ - ‖y‖ < ‖x - y‖

Tactic:
← sub_add_cancel x y

Goal state after:
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : StrictConvexSpace ℝ E
x y : E
h : ¬SameRay ℝ (x - y + y) y
⊢ ‖x - y + y‖ - ‖y‖ < ‖x - y‖
Module: Mathlib.Analysis.Convex.StrictConvexSpace

Goal state before:
pp: E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : StrictConvexSpace ℝ E
x y : E
h : ¬SameRay ℝ x y
⊢ ‖y‖ - ‖x‖ < ‖x - y‖

Tactic:
norm_sub_rev

Goal state after:
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : StrictConvexSpace ℝ E
x y : E
h : ¬SameRay ℝ x y
⊢ ‖y‖ - ‖x‖ < ‖y - x‖
Module: Mathlib.Analysis.Complex.AbelLimit

Goal state before:
pp: case right
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n ↦ ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z ↦ ∑' (n : ℕ), f n * z ^ n
hm : ∀ ε > 0, ∃ N, ∀ n ≥ N, ‖∑ i ∈ range n, f i - l‖ < ε
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
p :
  ∀ ε > 0,
    ∃ N, ∀ n ≥ N, ‖(1 - z) * ∑ i ∈ range n, (l - ∑ j ∈ range (i + 1), f j) * z ^ i - (l - ∑' (n : ℕ), f n * z ^ n)‖ < ε
⊢ ‖∑' (n : ℕ), f n * z ^ n - l‖ < ε

Tactic:
norm_sub_rev

Goal state after:
case right
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n ↦ ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z ↦ ∑' (n : ℕ), f n * z ^ n
hm : ∀ ε > 0, ∃ N, ∀ n ≥ N, ‖∑ i ∈ range n, f i - l‖ < ε
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
p :
  ∀ ε > 0,
    ∃ N, ∀ n ≥ N, ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range n, (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε
⊢ ‖∑' (n : ℕ), f n * z ^ n - l‖ < ε
Module: Mathlib.Analysis.Complex.AbelLimit

Goal state before:
pp: f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n ↦ ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z ↦ ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
this : ‖(1 - z) * ∑ i ∈ range (max B₁ B₂), (l - s (i + 1)) * z ^ i‖ < ε / 2
⊢ ‖∑' (n : ℕ), f n * z ^ n - l‖ = ‖l - g z‖

Tactic:
norm_sub_rev

Goal state after:
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n ↦ ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z ↦ ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
this : ‖(1 - z) * ∑ i ∈ range (max B₁ B₂), (l - s (i + 1)) * z ^ i‖ < ε / 2
⊢ ‖l - ∑' (n : ℕ), f n * z ^ n‖ = ‖l - g z‖
Module: Mathlib.Analysis.Complex.AbelLimit

Goal state before:
pp: case hbc
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n ↦ ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z ↦ ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
S₁ : ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4
i : ℕ
hi : i ∈ Ico B₁ (max B₁ B₂)
this : ‖∑ i ∈ range (i + 1), f i - l‖ < ε / 4 / M
⊢ ‖l - s (i + 1)‖ ≤ ε / 4 / M

Tactic:
norm_sub_rev

Goal state after:
case hbc
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n ↦ ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z ↦ ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
S₁ : ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4
i : ℕ
hi : i ∈ Ico B₁ (max B₁ B₂)
this : ‖l - ∑ i ∈ range (i + 1), f i‖ < ε / 4 / M
⊢ ‖l - s (i + 1)‖ ≤ ε / 4 / M
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f w = 0
hxw : x < w
hwy : w < y
⊢ ∃ a ∈ Ioo x w, (f w - f x) / (w - x) < deriv f a

Tactic:
by
      apply StrictMonoOn.exists_slope_lt_deriv_aux _ hxw _ _
      · exact hf.mono (Icc_subset_Icc le_rfl hwy.le)
      · exact hf'_mono.mono (Ioo_subset_Ioo le_rfl hwy.le)
      · intro z hz
        rw [← hw]
        apply ne_of_lt
        exact hf'_mono ⟨hz.1, hz.2.trans hwy⟩ ⟨hxw, hwy⟩ hz.2

Goal state after:
Module: Mathlib.Analysis.Complex.AbsMax

Goal state before:
pp: E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℂ E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : StrictConvexSpace ℝ F
f : E → F
s : Set E
z w : E
hd : DiffContOnCl ℂ f s
hz : IsMaxOn (norm ∘ f) s z
hsub : ball z (dist w z) ⊆ s
H₁ : ‖f w‖ = ‖f z‖
H₂ : ‖f w + f z‖ = ‖f z + f z‖
⊢ ‖f w + f z‖ = ‖f w‖ + ‖f z‖

Tactic:
simp only [H₂, SameRay.rfl.norm_add, H₁]

Goal state after:
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: case neg
x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f w = 0
hxw : x < w
hwy : w < y
a : ℝ
ha : deriv f a < (f w - f x) / (w - x)
hxa : x < a
haw : a < w
b : ℝ
hb : deriv f b < (f y - f w) / (y - w)
hwb : w < b
hby : b < y
⊢ deriv f a < (f y - f x) / (y - x)

Tactic:
simp only [lt_div_iff₀, hxy, hxw, hwy, sub_pos] at ha hb ⊢

Goal state after:
case neg
x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f w = 0
hxw : x < w
hwy : w < y
a : ℝ
hxa : x < a
haw : a < w
b : ℝ
hwb : w < b
hby : b < y
ha : deriv f a * (w - x) < f w - f x
hb : deriv f b * (y - w) < f y - f w
⊢ deriv f a * (y - x) < f y - f x
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf' : DifferentiableOn ℝ f (interior D)
hf'' : DifferentiableOn ℝ (deriv f) (interior D)
hf''_nonneg : ∀ x ∈ interior D, 0 ≤ deriv^[2] f x
⊢ DifferentiableOn ℝ (deriv f) (interior (interior D))

Tactic:
rwa [interior_interior]

Goal state after:
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf' : DifferentiableOn ℝ f (interior D)
hf'' : DifferentiableOn ℝ (deriv f) (interior D)
hf''_nonneg : ∀ x ∈ interior D, 0 ≤ deriv^[2] f x
⊢ ∀ x ∈ interior (interior D), 0 ≤ deriv (deriv f) x

Tactic:
rwa [interior_interior]

Goal state after:
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf' : DifferentiableOn ℝ f (interior D)
hf'' : DifferentiableOn ℝ (deriv f) (interior D)
hf''_nonpos : ∀ x ∈ interior D, deriv^[2] f x ≤ 0
⊢ DifferentiableOn ℝ (deriv f) (interior (interior D))

Tactic:
rwa [interior_interior]

Goal state after:
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf' : DifferentiableOn ℝ f (interior D)
hf'' : DifferentiableOn ℝ (deriv f) (interior D)
hf''_nonpos : ∀ x ∈ interior D, deriv^[2] f x ≤ 0
⊢ ∀ x ∈ interior (interior D), deriv (deriv f) x ≤ 0

Tactic:
rwa [interior_interior]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: ⊢ π / 4 ∈ Ioo (-(π / 2)) (π / 2)

Tactic:
constructor <;> linarith [pi_pos]

Goal state after:
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf'' : ∀ x ∈ interior D, 0 < deriv^[2] f x
⊢ ∀ x ∈ interior (interior D), 0 < deriv (deriv f) x

Tactic:
rwa [interior_interior]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds

Goal state before:
pp: x : ℝ
hx : 0 < x
f : ℝ → ℝ := fun t ↦ sin t - (t - t ^ 3 / 6)
hderiv : ∀ (t : ℝ), deriv f t = cos t - 1 + t ^ 2 / 2
⊢ ContinuousOn f (Ici 0)

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: D : Set ℝ
hD : Convex ℝ D
f : ℝ → ℝ
hf : ContinuousOn f D
hf'' : ∀ x ∈ interior D, deriv^[2] f x < 0
⊢ ∀ x ∈ interior (interior D), deriv (deriv f) x < 0

Tactic:
rwa [interior_interior]

Goal state after:
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: case refine_2
𝕜 : Type u_1
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
x : 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
hfc : ConvexOn 𝕜 s f
hxs : x ∈ interior s
y : 𝕜
hyx : x < y
hys : y ∈ s
y' : 𝕜
x✝ : y' ∈ slope f x '' {y | y ∈ s ∧ y < x}
z : 𝕜
hz : z ∈ {y | y ∈ s ∧ y < x}
hz' : slope f x z = y'
⊢ y ∈ s \ {x}

Tactic:
· simp [hys, hyx.ne']

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: case hx₁
x y : ℝ
h : x * y < 1
⊢ -(π / 2) < arctan x + arctan y

Tactic:
neg_lt,

Goal state after:
case hx₁
x y : ℝ
h : x * y < 1
⊢ -(arctan x + arctan y) < π / 2
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds

Goal state before:
pp: case e'_4
x : ℝ
hx1 : -(3 * π / 2) ≤ x
hx2 : x ≤ 3 * π / 2
hx3 : x ≠ 0
this : ∀ {y : ℝ}, 0 < y → y ≤ 3 * π / 2 → cos y < 1 / √(y ^ 2 + 1)
h✝ : x < 0
⊢ 1 / √(x ^ 2 + 1) = 1 / √((-x) ^ 2 + 1)

Tactic:
rw [neg_sq]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds

Goal state before:
pp: case inr
x : ℝ
hx1 : -(3 * π / 2) ≤ x
hx2 : x ≤ 3 * π / 2
hx3 : x ≠ 0
y : ℝ
hy1 : 0 < y
hy2 : y ≤ 3 * π / 2
hy3 : 0 < y ^ 2 + 1
hy1' : π / 2 ≤ y
⊢ cos y ≤ 0

Tactic:
exact cos_nonpos_of_pi_div_two_le_of_le hy1' (by linarith [pi_pos])

Goal state after:
Module: Mathlib.Analysis.Complex.BorelCaratheodory

Goal state before:
pp: M : ℝ
z w : ℂ
x✝¹ : M ≠ 0
x✝ : 2 * ↑M - z ≠ 0
h : w = z / (2 * ↑M - z)
⊢ z = z * ↑M * (↑M)⁻¹

Tactic:
rw [mul_inv_cancel_right₀]

Goal state after:
case h
M : ℝ
z w : ℂ
x✝¹ : M ≠ 0
x✝ : 2 * ↑M - z ≠ 0
h : w = z / (2 * ↑M - z)
⊢ ↑M ≠ 0
Module: Mathlib.Analysis.Complex.BorelCaratheodory

Goal state before:
pp: f : ℂ → ℂ
M R : ℝ
z : ℂ
hM : 0 < M
hf : DifferentiableOn ℂ f (ball 0 R)
hf₁ : Set.MapsTo f (ball 0 R) {z | z.re ≤ M}
hR : 0 < R
hz : z ∈ ball 0 R
⊢ DifferentiableOn ℂ (fun {z} ↦ f z - f 0) (ball 0 R)

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
⊢ π / 2 ∈ Icc 0 π

Tactic:
constructor <;> linarith [pi_pos]

Goal state after:
Module: Mathlib.Analysis.Complex.CoveringMap

Goal state before:
pp: ⊢ Continuous fun z ↦ ⟨cexp z, ⋯⟩

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Analysis.Complex.CoveringMap

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : ProperSpace 𝕜
n : ℕ
hn : ↑n ≠ 0
surj : Function.Surjective fun x ↦ x ^ n
this : NeZero n
⊢ Continuous[instTopologicalSpaceSubtype, instTopologicalSpaceSubtype] ({0}ᶜ.restrictPreimage fun x ↦ x ^ n)

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Ball.Action

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
E : Type u_3
inst✝⁴ : NormedField 𝕜
inst✝³ : NormedField 𝕜'
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜' E
r : ℝ
⊢ Continuous fun p ↦ ↑p.1 • ↑p.2

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Ball.Action

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
E : Type u_3
inst✝⁴ : NormedField 𝕜
inst✝³ : NormedField 𝕜'
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜' E
r : ℝ
⊢ Continuous fun p ↦ ↑(inclusion ⋯ p.1) • ↑p.2

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Analysis.Complex.BranchLogRoot

Goal state before:
pp: case refine_1
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : LocallyPathConnectedSpace X
U : Set X
hUc : IsSimplyConnected U
hUo : IsOpen U
g : X → ℂ
hgc : ContinuousOn g U
hU₀ : 0 ∉ g '' U
n : ℕ
hn : n ≠ 0
f : X → ℂ
hfc : ContinuousOn f U
hf : EqOn (cexp ∘ f) g U
⊢ ContinuousOn (fun x ↦ cexp (f x / ↑n)) U

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Analysis.Complex.Schwarz

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℂ F
f : E → F
c z : E
R₁ R₂ : ℝ
n : ℕ
hd : DifferentiableOn ℂ f (ball c R₁)
h_maps : MapsTo f (ball c R₁) (closedBall (f c) R₂)
hn : (fun x ↦ f x - f ((lineMap c z) 0)) =o[𝓝 ((lineMap c z) 0)] fun w ↦ ‖w - (lineMap c z) 0‖ ^ n
hz : z ∈ ball c R₁
hR₁ : 0 < R₁
hR₂ : 0 ≤ R₂
hfne : f z ≠ f c
hne : z ≠ c
g : StrongDual ℂ F
hg : ‖g‖ = 1
hgf : g (f z - f c) = ↑‖f z - f c‖
h : ℂ → ℂ := ⇑g ∘ f ∘ ⇑(lineMap c z)
hmaps_line : MapsTo (⇑(lineMap c z)) (ball 0 (R₁ / dist z c)) (ball c R₁)
hmaps : MapsTo h (ball 0 (R₁ / dist z c)) (closedBall (h 0) R₂)
hdiff : DifferentiableOn ℂ h (ball 0 (R₁ / dist z c))
⊢ Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ⇑(lineMap c z)

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Analysis.Complex.UnitDisc.Basic

Goal state before:
pp: n : ℕ+
⊢ Continuous fun x ↦ ↑x ^ ↑n

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Analysis.Complex.UnitDisc.Basic

Goal state before:
pp: n : ℕ
⊢ Continuous fun x ↦ ↑x ^ n

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Analysis.Meromorphic.Divisor

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f₁ f₂ : 𝕜 → E
z : 𝕜
U : Set 𝕜
hf₁ : MeromorphicOn f₁ U
hf₂ : MeromorphicOn f₂ U
h₁z : z ∈ U
h₃ : meromorphicOrderAt (f₁ + f₂) z ≠ ⊤
hz : z ∈ U
⊢ min (meromorphicOrderAt f₁ z).untop₀ (meromorphicOrderAt f₂ z).untop₀ ≤ (divisor (f₁ + f₂) U) z

Tactic:
divisor_apply (hf₁.add hf₂) hz

Goal state after:
case neg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f₁ f₂ : 𝕜 → E
z : 𝕜
U : Set 𝕜
hf₁ : MeromorphicOn f₁ U
hf₂ : MeromorphicOn f₂ U
h₁z : z ∈ U
h₃ : meromorphicOrderAt (f₁ + f₂) z ≠ ⊤
hz : z ∈ U
⊢ min (meromorphicOrderAt f₁ z).untop₀ (meromorphicOrderAt f₂ z).untop₀ ≤ (meromorphicOrderAt (f₁ + f₂) z).untop₀
Module: Mathlib.Analysis.Meromorphic.NormalForm

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
n : ℤ
g : 𝕜 → E
h₁g : AnalyticAt 𝕜 g x
h₂g : g x ≠ 0
h₃g : f =ᶠ[𝓝 x] (fun x_1 ↦ x_1 - x) ^ n • g
⊢ MeromorphicAt ((fun x_1 ↦ x_1 - x) ^ n • g) x

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Analysis.Meromorphic.NormalForm

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
n : ℤ
g : 𝕜 → E
h₁g : AnalyticAt 𝕜 g x
h₂g : g x ≠ 0
h₃g : f =ᶠ[𝓝 x] (fun x_1 ↦ x_1 - x) ^ n • g
hf : MeromorphicAt f x
this : meromorphicOrderAt f x = ↑n
hn : 0 ≤ n
⊢ AnalyticAt 𝕜 (fun x_1 ↦ x_1 - x) x

Tactic:
fun_prop

Goal state after:
Module: Mathlib.Analysis.Meromorphic.NormalForm

Goal state before:
pp: case mp.inr
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
n : ℤ
g : 𝕜 → E
h₁g : AnalyticAt 𝕜 g x
h₂g : g x ≠ 0
h₃g : f =ᶠ[𝓝 x] (fun x_1 ↦ x_1 - x) ^ n • g
⊢ AnalyticAt 𝕜 f x ∨ MeromorphicAt f x ∧ meromorphicOrderAt f x < 0 ∧ f x = 0

Tactic:
· have hf : MeromorphicAt f x := by
        apply MeromorphicAt.congr _ (h₃g.filter_mono nhdsWithin_le_nhds).symm
        fun_prop
      have : meromorphicOrderAt f x = n := by
        rw [meromorphicOrderAt_eq_int_iff hf]
        use g, h₁g, h₂g
        exact eventually_nhdsWithin_of_eventually_nhds h₃g
      by_cases! hn : 0 ≤ n
      · left
        rw [analyticAt_congr h₃g]
        apply (AnalyticAt.zpow_nonneg (by fun_prop) hn).smul h₁g
      · right
        use hf
        simp [this, WithTop.coe_lt_zero.2 hn, h₃g.eq_of_nhds,
          zero_zpow n hn.ne]

Goal state after:
Module: Mathlib.Analysis.Meromorphic.NormalForm

Goal state before:
pp: case mpr
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
hf : MeromorphicNFAt f x
h : f x ≠ 0
⊢ meromorphicOrderAt f x = 0

Tactic:
rcases id hf with h₁ | ⟨n, g, h₁g, h₂g, h₃g⟩

Goal state after:
case mpr.inl
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
hf : MeromorphicNFAt f x
h : f x ≠ 0
h₁ : f =ᶠ[𝓝 x] 0
⊢ meromorphicOrderAt f x = 0

case mpr.inr
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
hf : MeromorphicNFAt f x
h : f x ≠ 0
n : ℤ
g : 𝕜 → E
h₁g : AnalyticAt 𝕜 g x
h₂g : g x ≠ 0
h₃g : f =ᶠ[𝓝 x] (fun x_1 ↦ x_1 - x) ^ n • g
⊢ meromorphicOrderAt f x = 0