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

Goal state before:
pp: case e'_9
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
k : ℕ
s t : Set ℝ
ht : UniqueDiffWithinAt ℝ t y
hs : s ∈ 𝓝[t] y
hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y
this : HasDerivWithinAt (fun t ↦ ((↑k + 1) * ↑k !)⁻¹ * (x - t) ^ (k + 1)) (-((↑k !)⁻¹ * (x - y) ^ k)) t y
⊢ (((↑k + 1) * ↑k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y -
      ((↑k !)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y =
    (((↑k + 1) * ↑k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y +
      -((↑k !)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y

Tactic:
field_simp

Goal state after:
case e'_9
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
k : ℕ
s t : Set ℝ
ht : UniqueDiffWithinAt ℝ t y
hs : s ∈ 𝓝[t] y
hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y
this : HasDerivWithinAt (fun t ↦ ((↑k + 1) * ↑k !)⁻¹ * (x - t) ^ (k + 1)) (-((↑k !)⁻¹ * (x - y) ^ k)) t y
⊢ ((x - y) ^ (k + 1) / ((↑k + 1) * ↑k !)) • iteratedDerivWithin (k + 2) f s y -
      ((x - y) ^ k / ↑k !) • iteratedDerivWithin (k + 1) f s y =
    ((x - y) ^ (k + 1) / ((↑k + 1) * ↑k !)) • iteratedDerivWithin (k + 2) f s y +
      -((x - y) ^ k / ↑k !) • iteratedDerivWithin (k + 1) f s y
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
t : ℝ
⊢ -((↑n !)⁻¹ * (1 - t) ^ n) = (↑(n + 1)!)⁻¹ * ((↑n + 1) * (1 - t) ^ n * -1)

Tactic:
field_simp

Goal state after:
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
t : ℝ
⊢ -((1 - t) ^ n * ↑(n + 1)!) = -(↑n ! * (1 - t) ^ n * (↑n + 1))
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
t : ℝ
⊢ (1 - t) ^ n * ↑(n + 1)! = ↑n ! * (1 - t) ^ n * (↑n + 1)

Tactic:
Nat.factorial_succ

Goal state after:
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
t : ℝ
⊢ (1 - t) ^ n * ↑((n + 1) * n !) = ↑n ! * (1 - t) ^ n * (↑n + 1)
Module: Mathlib.Analysis.Calculus.Taylor

Goal state before:
pp: case e'_2.x.e_f.e_a
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
x x₀ : ℝ
this✝ : x₀ ≠ x
n : ℕ
ih :
  f x - taylorWithinEval f n [[x₀, x]] x₀ x =
    ∫ (t : ℝ) in x₀..x, ((x - t) ^ n / ↑n !) • iteratedDerivWithin (n + 1) f [[x₀, x]] t
hf :
  ∀ k ≤ n + 1,
    let u := fun t ↦ (x - t) ^ k / ↑k !;
    let v := fun t ↦ iteratedDerivWithin k f [[x₀, x]] t;
    ∫ (t : ℝ) in x₀..x, u t • deriv v t = u x • v x - u x₀ • v x₀ - ∫ (t : ℝ) in x₀..x, deriv u t • v t
this :
  let u := fun t ↦ (x - t) ^ (n + 1) / ↑(n + 1)!;
  let v := fun t ↦ iteratedDerivWithin (n + 1) f [[x₀, x]] t;
  ∫ (t : ℝ) in x₀..x, u t • deriv v t = u x • v x - u x₀ • v x₀ - ∫ (t : ℝ) in x₀..x, deriv u t • v t
t : ℝ
⊢ (x - t) ^ n / ↑n ! = -(↑(n + 1) * (x - t) ^ (n + 1 - 1) * -(fun x ↦ 1) t / ↑(n.succ * n !))

Tactic:
field_simp

Goal state after:
case e'_2.x.e_f.e_a
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
x x₀ : ℝ
this✝ : x₀ ≠ x
n : ℕ
ih :
  f x - taylorWithinEval f n [[x₀, x]] x₀ x =
    ∫ (t : ℝ) in x₀..x, ((x - t) ^ n / ↑n !) • iteratedDerivWithin (n + 1) f [[x₀, x]] t
hf :
  ∀ k ≤ n + 1,
    let u := fun t ↦ (x - t) ^ k / ↑k !;
    let v := fun t ↦ iteratedDerivWithin k f [[x₀, x]] t;
    ∫ (t : ℝ) in x₀..x, u t • deriv v t = u x • v x - u x₀ • v x₀ - ∫ (t : ℝ) in x₀..x, deriv u t • v t
this :
  let u := fun t ↦ (x - t) ^ (n + 1) / ↑(n + 1)!;
  let v := fun t ↦ iteratedDerivWithin (n + 1) f [[x₀, x]] t;
  ∫ (t : ℝ) in x₀..x, u t • deriv v t = u x • v x - u x₀ • v x₀ - ∫ (t : ℝ) in x₀..x, deriv u t • v t
t : ℝ
⊢ (x - t) ^ n * ↑(n.succ * n !) = ↑n ! * ↑(n + 1) * (x - t) ^ (n + 1 - 1)
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
hf₀ : ∀ᵐ (x : ℝ), x ∈ Icc a b → HasDerivAt f 0 x
hab : a ≤ b
d : ℝ
hd : d ∈ Icc a b
r : ℝ
⊢ r > 0 → dist (f d) (f b) ≤ r

Tactic:
hr

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
hf₀ : ∀ᵐ (x : ℝ), x ∈ Icc a b → HasDerivAt f 0 x
hab : a ≤ b
d : ℝ
hd : d ∈ Icc a b
r : ℝ
hr : r > 0
⊢ dist (f d) (f b) ≤ r
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‖ ≤ |(↑z).2 - (↑z).1| * (|(↑z).2 - (↑z).1|⁻¹ * ‖f (↑z).2 - f (↑z).1‖)

Tactic:
field_simp

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).1 - f (↑z).2‖ ≤ ‖f (↑z).2 - f (↑z).1‖
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
this✝ : ∀ z ∈ s, g z ≤ r / (b - d) * ((↑z).2 - (↑z).1)
this : ∑ z ∈ s, ((↑z).2 - (↑z).1) ≤ b - d
⊢ r / (b - d) * (b - d) ≤ r

Tactic:
field_simp

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
this✝ : ∀ z ∈ s, g z ≤ r / (b - d) * ((↑z).2 - (↑z).1)
this : ∑ z ∈ s, ((↑z).2 - (↑z).1) ≤ b - d
⊢ 1 ≤ 1
Module: Mathlib.Analysis.Convex.StrictConvexSpace

Goal state before:
pp: E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
h : ∀ (x y : E), ‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ∃ a b, a + b = 1 ∧ ‖a • x + b • y‖ < 1
x : E
hx : ‖x‖ = 1
y : E
hy : ‖y‖ = 1
hne : x ≠ y
a b : ℝ
hab : a + b = 1
hlt : ‖a • x + b • y‖ < 1
⊢ ∃ c, (AffineMap.lineMap x y) c ∈ interior (closedBall 0 1)

Tactic:
use b

Goal state after:
case h
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
h : ∀ (x y : E), ‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ∃ a b, a + b = 1 ∧ ‖a • x + b • y‖ < 1
x : E
hx : ‖x‖ = 1
y : E
hy : ‖y‖ = 1
hne : x ≠ y
a b : ℝ
hab : a + b = 1
hlt : ‖a • x + b • y‖ < 1
⊢ (AffineMap.lineMap x y) b ∈ interior (closedBall 0 1)
Module: Mathlib.MeasureTheory.Function.AbsolutelyContinuous

Goal state before:
pp: case inr.x
F : Type u_2
inst✝ : SeminormedAddCommGroup F
a✝ b✝ : ℝ
f : ℝ → F
a b : ℝ
hf :
  ∀ ε > 0,
    ∃ δ > 0,
      ∀ E ∈ disjWithin a b,
        ∑ i ∈ Finset.range E.1, dist (E.2 i).1 (E.2 i).2 < δ →
          ∑ i ∈ Finset.range E.1, dist (f (E.2 i).1) (f (E.2 i).2) < ε
hab₀ : a ≤ b
hab : a < b
δ : ℝ
hδ₁ : δ > 0
hδ₂ :
  ∀ E ∈ disjWithin a b,
    ∑ i ∈ Finset.range E.1, dist (E.2 i).1 (E.2 i).2 < δ → ∑ i ∈ Finset.range E.1, dist (f (E.2 i).1) (f (E.2 i).2) < 1
hab₁ : 0 < b - a
n : ℕ
hn : 1 / (↑n + 1) < δ / (b - a)
δ' : ℝ := ⋯
hδ₃ : δ' < δ
h_mono : Monotone fun i ↦ a + ↑i * δ'
v_sum : eVariationOn f (Icc a b) = ∑ i ∈ Finset.range (n + 1), eVariationOn f (Icc (a + ↑i * δ') (a + (↑i + 1) * δ'))
v_each : ∀ (x y : ℝ), a ≤ x → x ≤ y → y < x + δ → y ≤ b → eVariationOn f (Icc x y) ≤ 1
i : ℕ
hi : i < n + 1
⊢ a ≤ a + ↑i * δ'

Tactic:
convert! h_mono (show 0 ≤ i by lia)

Goal state after:
case e'_3
F : Type u_2
inst✝ : SeminormedAddCommGroup F
a✝ b✝ : ℝ
f : ℝ → F
a b : ℝ
hf :
  ∀ ε > 0,
    ∃ δ > 0,
      ∀ E ∈ disjWithin a b,
        ∑ i ∈ Finset.range E.1, dist (E.2 i).1 (E.2 i).2 < δ →
          ∑ i ∈ Finset.range E.1, dist (f (E.2 i).1) (f (E.2 i).2) < ε
hab₀ : a ≤ b
hab : a < b
δ : ℝ
hδ₁ : δ > 0
hδ₂ :
  ∀ E ∈ disjWithin a b,
    ∑ i ∈ Finset.range E.1, dist (E.2 i).1 (E.2 i).2 < δ → ∑ i ∈ Finset.range E.1, dist (f (E.2 i).1) (f (E.2 i).2) < 1
hab₁ : 0 < b - a
n : ℕ
hn : 1 / (↑n + 1) < δ / (b - a)
δ' : ℝ := (b - a) / (↑n + 1)
hδ₃ : δ' < δ
h_mono : Monotone fun i ↦ a + ↑i * δ'
v_sum : eVariationOn f (Icc a b) = ∑ i ∈ Finset.range (n + 1), eVariationOn f (Icc (a + ↑i * δ') (a + (↑i + 1) * δ'))
v_each : ∀ (x y : ℝ), a ≤ x → x ≤ y → y < x + δ → y ≤ b → eVariationOn f (Icc x y) ≤ 1
i : ℕ
hi : i < n + 1
⊢ a = (fun i ↦ a + ↑i * δ') 0
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.AbsolutelyContinuousFun

Goal state before:
pp: case hg
f g : ℝ → ℝ
a b : ℝ
hf : AbsolutelyContinuousOnInterval f a b
hg : AbsolutelyContinuousOnInterval g a b
⊢ IntervalIntegrable (fun x ↦ deriv f x * g x) volume a b

Tactic:
· exact hf.intervalIntegrable_deriv.mul_continuousOn hg.continuousOn

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

Goal state before:
pp: F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → F
z w : ℂ
r : ℝ := dist w z
hd : DiffContOnCl ℂ f (ball z r)
hz : IsMaxOn (norm ∘ f) (closedBall z r) z
hw : w ∈ closedBall z r
hw_lt : ‖f w‖ < ‖f z‖
hr : 0 < r
this : ‖∮ (ζ : ℂ) in C(z, r), (ζ - z)⁻¹ • f ζ‖ < 2 * π * ‖f z‖
⊢ False

Tactic:
refine this.ne ?_

Goal state after:
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → F
z w : ℂ
r : ℝ := dist w z
hd : DiffContOnCl ℂ f (ball z r)
hz : IsMaxOn (norm ∘ f) (closedBall z r) z
hw : w ∈ closedBall z r
hw_lt : ‖f w‖ < ‖f z‖
hr : 0 < r
this : ‖∮ (ζ : ℂ) in C(z, r), (ζ - z)⁻¹ • f ζ‖ < 2 * π * ‖f z‖
⊢ ‖∮ (ζ : ℂ) in C(z, r), (ζ - z)⁻¹ • f ζ‖ = 2 * π * ‖f z‖
Module: Mathlib.Analysis.Convex.SpecificFunctions.Deriv

Goal state before:
pp: case e'_9
x : ℝ
hx : x ≠ 0
⊢ (2 + log x) / (2 * √x) = 1 / (2 * √x) * log x + √x * x⁻¹

Tactic:
rw [add_div, div_mul_cancel_left₀ two_ne_zero, ← div_eq_mul_inv, sqrt_div_self', add_comm,
    one_div, one_div, ← div_eq_inv_mul]

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

Goal state before:
pp: θ : ℂ
⊢ sin θ = 0 ↔ ∃ k, θ = ↑k * ↑π

Tactic:
rw [← Complex.cos_sub_pi_div_two, cos_eq_zero_iff]
  constructor
  · rintro ⟨k, hk⟩
    use k + 1
    simp [eq_add_of_sub_eq hk]
    ring
  · rintro ⟨k, rfl⟩
    use k - 1
    simp
    ring

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

Goal state before:
pp: case hx₁
⊢ -(π / 2) < π / 3

Tactic:
field_simp

Goal state after:
case hx₁
⊢ -3 < 2
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: case hx₂
⊢ π / 3 < π / 2

Tactic:
field_simp

Goal state after:
case hx₂
⊢ 2 < 3
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: case hx₁
⊢ -(π / 2) < π / 6

Tactic:
field_simp

Goal state after:
case hx₁
⊢ -6 < 2
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: case hx₂
⊢ π / 6 < π / 2

Tactic:
field_simp

Goal state after:
case hx₂
⊢ 2 < 6
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : Field 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
x : 𝕜
hfc : ConcaveOn 𝕜 s f
hx : x ∈ s
⊢ AntitoneOn ((-slope (-f)) x) (s \ {x})

Tactic:
exact (ConvexOn.slope_mono hfc.neg hx).neg

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

Goal state before:
pp: case inr
x : ℝ
h1 : 0 ≤ x
h2 : x < π / 2
h1' : 0 < x
⊢ x ≤ tan x

Tactic:
exact le_of_lt (lt_tan h1' h2)

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
⊢ angle x (-y) = π - angle x y

Tactic:
angle_neg_right

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
⊢ π - angle x y = π - angle x y
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
r : ℝ
hr : r < 0
⊢ π - angle x y = angle x (-y)

Tactic:
angle_neg_right

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
r : ℝ
hr : r < 0
⊢ π - angle x y = π - angle x y
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic

Goal state before:
pp: case inr.inr
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
⊢ √((1 - ⟪x, y⟫ / (‖x‖ * ‖y‖) * (⟪x, y⟫ / (‖x‖ * ‖y‖))) * (‖x‖ * ‖y‖) ^ 2) =
    √(‖x‖ * ‖x‖ * (‖y‖ * ‖y‖) - ⟪x, y⟫ * ⟪x, y⟫)

Tactic:
field_simp

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 = 2 * ↑M * (z / (2 * ↑M - z)) / (1 + z / (2 * ↑M - z))

Tactic:
field_simp

Goal state after:
M : ℝ
z w : ℂ
x✝¹ : M ≠ 0
x✝ : 2 * ↑M - z ≠ 0
h : w = z / (2 * ↑M - z)
⊢ z = z * 2 * ↑M / (2 * ↑M - z + z)
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
hx : x ≠ 0
hy : x + y ≠ 0
⊢ √(⟪x, x⟫ * ⟪x + y, x + y⟫ - ⟪x, x + y⟫ * ⟪x, x + y⟫) / (‖x‖ * ‖x + y‖) =
    √(⟪x, x⟫ * ⟪y, y⟫ - ⟪x, y⟫ * ⟪x, y⟫) / (‖x‖ * ‖x + y‖)

Tactic:
field_simp

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : x + y ≠ 0
⊢ √(⟪x, x⟫ * ⟪x + y, x + y⟫ - ⟪x, x + y⟫ ^ 2) = √(⟪x, x⟫ * ⟪y, y⟫ - ⟪x, y⟫ ^ 2)
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
hf₂ : f 0 = 0
w : ℂ := f z / (2 * ↑M - f z)
hzR : ‖z‖ < R
hwR : ‖f z / (2 * ↑M - f z)‖ ≤ ‖z‖ / R
h_denom : 2 * ↑M - f z ≠ 0
⊢ 2 * M * (‖z‖ / R / (1 - ‖z‖ / R)) = 2 * M * ‖z‖ / (R - ‖z‖)

Tactic:
field_simp

Goal state after:
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
hfz : ‖f z - f 0‖ ≤ 2 * (M + ‖f 0‖) * ‖z‖ / (R - ‖z‖)
h_denom_ne : R - ‖z‖ ≠ 0
⊢ 2 * (M + ‖f 0‖) * ‖z‖ / (R - ‖z‖) + ‖f 0‖ = 2 * M * ‖z‖ / (R - ‖z‖) + ‖f 0‖ * (R + ‖z‖) / (R - ‖z‖)

Tactic:
field_simp

Goal state after:
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
hfz : ‖f z - f 0‖ ≤ 2 * (M + ‖f 0‖) * ‖z‖ / (R - ‖z‖)
h_denom_ne : R - ‖z‖ ≠ 0
⊢ 2 * (M + ‖f 0‖) * ‖z‖ + ‖f 0‖ * (R - ‖z‖) = 2 * M * ‖z‖ + ‖f 0‖ * (R + ‖z‖)
Module: Mathlib.Analysis.Meromorphic.Divisor

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
U : Set 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f₁ f₂ : 𝕜 → E
hf₁ : MeromorphicOn f₁ U
h₂ : Set.EqOn f₁ f₂ Uᶜ
x : 𝕜
hx : x ∈ U
h₁ : ∀ x ∈ U, (U \ {x | f₁ x = f₂ x})ᶜ ∈ 𝓝[≠] x
⊢ f₁ =ᶠ[𝓝[≠] x] f₂

Tactic:
filter_upwards [h₁ x hx] with a ha

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
U : Set 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f₁ f₂ : 𝕜 → E
hf₁ : MeromorphicOn f₁ U
h₂ : Set.EqOn f₁ f₂ Uᶜ
x : 𝕜
hx : x ∈ U
h₁ : ∀ x ∈ U, (U \ {x | f₁ x = f₂ x})ᶜ ∈ 𝓝[≠] x
a : 𝕜
ha : a ∈ (U \ {x | f₁ x = f₂ x})ᶜ
⊢ f₁ a = f₂ a
Module: Mathlib.Analysis.Meromorphic.NormalForm

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
ι : Type u_3
s : Finset ι
f : ι → 𝕜 → 𝕜
h₁f : ∀ i ∈ s, MeromorphicNFAt (f i) x
h₂f : {σ | σ ∈ s ∧ f σ x = 0}.Subsingleton
h₃f : ∀ {τ : ι}, τ ∈ s → τ ∉ {σ ∈ s | f σ x = 0} → AnalyticAt 𝕜 (f τ) x
τ : ι
h₁τ : τ ∈ s
h₂τ : f τ x = 0
μ : ι
hμ : μ ∈ s.erase τ
⊢ f μ x ≠ 0

Tactic:
by_contra

Goal state after:
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
ι : Type u_3
s : Finset ι
f : ι → 𝕜 → 𝕜
h₁f : ∀ i ∈ s, MeromorphicNFAt (f i) x
h₂f : {σ | σ ∈ s ∧ f σ x = 0}.Subsingleton
h₃f : ∀ {τ : ι}, τ ∈ s → τ ∉ {σ ∈ s | f σ x = 0} → AnalyticAt 𝕜 (f τ) x
τ : ι
h₁τ : τ ∈ s
h₂τ : f τ x = 0
μ : ι
hμ : μ ∈ s.erase τ
this : f μ x = 0
⊢ False
Module: Mathlib.Analysis.Meromorphic.TrailingCoefficient

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : DecidableEq 𝕜
x y : 𝕜
⊢ meromorphicTrailingCoeffAt (fun x ↦ x - y) x = if x = y then 1 else x - y

Tactic:
by
  by_cases h : x = y
  · simp_all only [sub_self, ite_true]
    apply AnalyticAt.meromorphicTrailingCoeffAt_of_ne_zero_of_eq_nhdsNE (n := 1) (by fun_prop)
      (by apply one_ne_zero)
    simp
  · simp_all only [ite_false]
    apply AnalyticAt.meromorphicTrailingCoeffAt_of_ne_zero (by fun_prop)
    simp_all [sub_ne_zero]

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

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
x : 𝕜
f₁ : 𝕜 → 𝕜
f₂ : 𝕜 → E
hf₁ : MeromorphicAt f₁ x
hf₂ : MeromorphicAt f₂ x
h₁f₁ : ¬meromorphicOrderAt f₁ x = ⊤
h₁f₂ : ¬meromorphicOrderAt f₂ x = ⊤
g₁ : 𝕜 → 𝕜
h₁g₁ : AnalyticAt 𝕜 g₁ x
h₂g₁ : g₁ x ≠ 0
h₃g₁ : f₁ =ᶠ[𝓝[≠] x] fun z ↦ (z - x) ^ (meromorphicOrderAt f₁ x).untop₀ • g₁ z
g₂ : 𝕜 → E
h₁g₂ : AnalyticAt 𝕜 g₂ x
h₂g₂ : g₂ x ≠ 0
h₃g₂ : f₂ =ᶠ[𝓝[≠] x] fun z ↦ (z - x) ^ (meromorphicOrderAt f₂ x).untop₀ • g₂ z
this : f₁ • f₂ =ᶠ[𝓝[≠] x] fun z ↦ (z - x) ^ (meromorphicOrderAt (f₁ • f₂) x).untop₀ • (g₁ • g₂) z
⊢ meromorphicTrailingCoeffAt (f₁ • f₂) x = g₁ x • meromorphicTrailingCoeffAt f₂ x

Tactic:
h₁g₂.meromorphicTrailingCoeffAt_of_ne_zero_of_eq_nhdsNE h₂g₂ h₃g₂,

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
x : 𝕜
f₁ : 𝕜 → 𝕜
f₂ : 𝕜 → E
hf₁ : MeromorphicAt f₁ x
hf₂ : MeromorphicAt f₂ x
h₁f₁ : ¬meromorphicOrderAt f₁ x = ⊤
h₁f₂ : ¬meromorphicOrderAt f₂ x = ⊤
g₁ : 𝕜 → 𝕜
h₁g₁ : AnalyticAt 𝕜 g₁ x
h₂g₁ : g₁ x ≠ 0
h₃g₁ : f₁ =ᶠ[𝓝[≠] x] fun z ↦ (z - x) ^ (meromorphicOrderAt f₁ x).untop₀ • g₁ z
g₂ : 𝕜 → E
h₁g₂ : AnalyticAt 𝕜 g₂ x
h₂g₂ : g₂ x ≠ 0
h₃g₂ : f₂ =ᶠ[𝓝[≠] x] fun z ↦ (z - x) ^ (meromorphicOrderAt f₂ x).untop₀ • g₂ z
this : f₁ • f₂ =ᶠ[𝓝[≠] x] fun z ↦ (z - x) ^ (meromorphicOrderAt (f₁ • f₂) x).untop₀ • (g₁ • g₂) z
⊢ meromorphicTrailingCoeffAt (f₁ • f₂) x = g₁ x • g₂ x
Module: Mathlib.Analysis.Meromorphic.Order

Goal state before:
pp: case top
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 : AnalyticAt 𝕜 f x
hn : analyticOrderAt f x = ⊤
⊢ meromorphicOrderAt f x = ENat.map Nat.cast ⊤

Tactic:
rw [ENat.map_top, meromorphicOrderAt_eq_top_iff]
    exact (analyticOrderAt_eq_top.mp hn).filter_mono nhdsWithin_le_nhds

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

Goal state before:
pp: case coe
𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
R : Type u_3
inst✝⁶ : NormedRing R
inst✝⁵ : NoZeroDivisors R
inst✝⁴ : Module R E
inst✝³ : IsBoundedSMul R E
inst✝² : Module.IsTorsionFree R E
x : 𝕜
inst✝¹ : NormedAlgebra 𝕜 R
inst✝ : IsScalarTower 𝕜 R E
f : 𝕜 → R
g : 𝕜 → E
hf : MeromorphicAt f x
hg : MeromorphicAt g x
m : ℤ
h₂f : meromorphicOrderAt f x = ↑m
⊢ meromorphicOrderAt (f • g) x = ↑m + meromorphicOrderAt g x

Tactic:
| coe m =>

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

Goal state before:
pp: case left
R : ℝ
F : locallyFinsuppWithin (ball 0 R) ℤ
z : ℂ
hz : z ∈ ball 0 R
a : ℂ
ha : a ∈ {σ | (canonicalFactor R σ ^ F σ) z = 0}
b : ℂ
hb : b ∈ {σ | (canonicalFactor R σ ^ F σ) z = 0}
⊢ a ∈ ball 0 R

Tactic:
by_contra

Goal state after:
case left
R : ℝ
F : locallyFinsuppWithin (ball 0 R) ℤ
z : ℂ
hz : z ∈ ball 0 R
a : ℂ
ha : a ∈ {σ | (canonicalFactor R σ ^ F σ) z = 0}
b : ℂ
hb : b ∈ {σ | (canonicalFactor R σ ^ F σ) z = 0}
this : a ∉ ball 0 R
⊢ False
Module: Mathlib.Analysis.Complex.CanonicalDecomposition

Goal state before:
pp: case right
R : ℝ
F : locallyFinsuppWithin (ball 0 R) ℤ
z : ℂ
hz : z ∈ ball 0 R
a : ℂ
ha : a ∈ {σ | (canonicalFactor R σ ^ F σ) z = 0}
b : ℂ
hb : b ∈ {σ | (canonicalFactor R σ ^ F σ) z = 0}
⊢ b ∈ ball 0 R

Tactic:
by_contra

Goal state after:
case right
R : ℝ
F : locallyFinsuppWithin (ball 0 R) ℤ
z : ℂ
hz : z ∈ ball 0 R
a : ℂ
ha : a ∈ {σ | (canonicalFactor R σ ^ F σ) z = 0}
b : ℂ
hb : b ∈ {σ | (canonicalFactor R σ ^ F σ) z = 0}
this : b ∉ ball 0 R
⊢ False