mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Covering.Besicovitch

Goal state before:
pp: case ht
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : HasBesicovitchCovering α
μ : Measure α
inst✝ : IsFiniteMeasure μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
P : Finset (α × ℝ) → Prop :=
  fun t ↦ ((↑t).PairwiseDisjoint fun p ↦ closedBall p.1 p.2) ∧ (∀ p ∈ t, p.1 ∈ s) ∧ ∀ p ∈ t, p.2 ∈ f p.1
F : Finset (α × ℝ) → Finset (α × ℝ)
hF :
  ∀ (t : Finset (α × ℝ)),
    P t →
      t ⊆ F t ∧ P (F t) ∧ μ (s \ ⋃ p ∈ F t, closedBall p.1 p.2) ≤ ↑N / (↑N + 1) * μ (s \ ⋃ p ∈ t, closedBall p.1 p.2)
u : ℕ → Finset (α × ℝ) := fun n ↦ F^[n] ∅
u_succ : ∀ (n : ℕ), u n.succ = F (u n)
Pu : ∀ (n : ℕ), P (u n)
A : ∀ (n : ℕ), μ (s \ ⋃ p ∈ ⋃ n, ↑(u n), closedBall p.1 p.2) ≤ μ (s \ ⋃ p ∈ u n, closedBall p.1 p.2)
B : ∀ (n : ℕ), μ (s \ ⋃ p ∈ u n, closedBall p.1 p.2) ≤ (↑N / (↑N + 1)) ^ n * μ s
⊢ ↑N + 1 ≠ ∞ ∨ ↑N ≠ ∞

Tactic:
·

Goal state after:
case ht
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : HasBesicovitchCovering α
μ : Measure α
inst✝ : IsFiniteMeasure μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
P : Finset (α × ℝ) → Prop :=
  fun t ↦ ((↑t).PairwiseDisjoint fun p ↦ closedBall p.1 p.2) ∧ (∀ p ∈ t, p.1 ∈ s) ∧ ∀ p ∈ t, p.2 ∈ f p.1
F : Finset (α × ℝ) → Finset (α × ℝ)
hF :
  ∀ (t : Finset (α × ℝ)),
    P t →
      t ⊆ F t ∧ P (F t) ∧ μ (s \ ⋃ p ∈ F t, closedBall p.1 p.2) ≤ ↑N / (↑N + 1) * μ (s \ ⋃ p ∈ t, closedBall p.1 p.2)
u : ℕ → Finset (α × ℝ) := fun n ↦ F^[n] ∅
u_succ : ∀ (n : ℕ), u n.succ = F (u n)
Pu : ∀ (n : ℕ), P (u n)
A : ∀ (n : ℕ), μ (s \ ⋃ p ∈ ⋃ n, ↑(u n), closedBall p.1 p.2) ≤ μ (s \ ⋃ p ∈ u n, closedBall p.1 p.2)
B : ∀ (n : ℕ), μ (s \ ⋃ p ∈ u n, closedBall p.1 p.2) ≤ (↑N / (↑N + 1)) ^ n * μ s
⊢ ↑N + 1 ≠ ∞ ∨ ↑N ≠ ∞
Module: Mathlib.Analysis.Calculus.DerivativeTest

Goal state before:
pp: case refine_1
f : ℝ → ℝ
a b : ℝ
ha : ContinuousAt f a
hb : ContinuousAt f b
hd₀ : DifferentiableOn ℝ f (Ioo a b)
hd₁ : DifferentiableOn ℝ f (Ioi b)
h₀ : ∀ x ∈ Ioo a b, deriv f x ≤ 0
h₁ : ∀ x ∈ Ioi b, 0 ≤ deriv f x
⊢ AntitoneOn f (Icc a b)

pp: case refine_2
f : ℝ → ℝ
a b : ℝ
ha : ContinuousAt f a
hb : ContinuousAt f b
hd₀ : DifferentiableOn ℝ f (Ioo a b)
hd₁ : DifferentiableOn ℝ f (Ioi b)
h₀ : ∀ x ∈ Ioo a b, deriv f x ≤ 0
h₁ : ∀ x ∈ Ioi b, 0 ≤ deriv f x
⊢ MonotoneOn f (Ici b)

Tactic:
·

Goal state after:
case refine_1
f : ℝ → ℝ
a b : ℝ
ha : ContinuousAt f a
hb : ContinuousAt f b
hd₀ : DifferentiableOn ℝ f (Ioo a b)
hd₁ : DifferentiableOn ℝ f (Ioi b)
h₀ : ∀ x ∈ Ioo a b, deriv f x ≤ 0
h₁ : ∀ x ∈ Ioi b, 0 ≤ deriv f x
⊢ AntitoneOn f (Icc a b)
Module: Mathlib.Analysis.Calculus.DerivativeTest

Goal state before:
pp: case refine_2
f : ℝ → ℝ
a b : ℝ
ha : ContinuousAt f a
hb : ContinuousAt f b
hd₀ : DifferentiableOn ℝ f (Ioo a b)
hd₁ : DifferentiableOn ℝ f (Ioi b)
h₀ : ∀ x ∈ Ioo a b, deriv f x ≤ 0
h₁ : ∀ x ∈ Ioi b, 0 ≤ deriv f x
⊢ MonotoneOn f (Ici b)

Tactic:
·

Goal state after:
case refine_2
f : ℝ → ℝ
a b : ℝ
ha : ContinuousAt f a
hb : ContinuousAt f b
hd₀ : DifferentiableOn ℝ f (Ioo a b)
hd₁ : DifferentiableOn ℝ f (Ioi b)
h₀ : ∀ x ∈ Ioo a b, deriv f x ≤ 0
h₁ : ∀ x ∈ Ioi b, 0 ≤ deriv f x
⊢ MonotoneOn f (Ici b)
Module: Mathlib.Analysis.Calculus.DerivativeTest

Goal state before:
pp: case refine_1
f : ℝ → ℝ
b c : ℝ
hb : ContinuousAt f b
hd₀ : DifferentiableOn ℝ f (Iio b)
hd₁ : DifferentiableOn ℝ f (Ioo b c)
h₀ : ∀ x ∈ Iio b, deriv f x ≤ 0
h₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x
⊢ AntitoneOn f (Iic b)

pp: case refine_2
f : ℝ → ℝ
b c : ℝ
hb : ContinuousAt f b
hd₀ : DifferentiableOn ℝ f (Iio b)
hd₁ : DifferentiableOn ℝ f (Ioo b c)
h₀ : ∀ x ∈ Iio b, deriv f x ≤ 0
h₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x
⊢ MonotoneOn f (Ico b c)

Tactic:
·

Goal state after:
case refine_1
f : ℝ → ℝ
b c : ℝ
hb : ContinuousAt f b
hd₀ : DifferentiableOn ℝ f (Iio b)
hd₁ : DifferentiableOn ℝ f (Ioo b c)
h₀ : ∀ x ∈ Iio b, deriv f x ≤ 0
h₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x
⊢ AntitoneOn f (Iic b)
Module: Mathlib.Analysis.Calculus.DerivativeTest

Goal state before:
pp: case refine_2
f : ℝ → ℝ
b c : ℝ
hb : ContinuousAt f b
hd₀ : DifferentiableOn ℝ f (Iio b)
hd₁ : DifferentiableOn ℝ f (Ioo b c)
h₀ : ∀ x ∈ Iio b, deriv f x ≤ 0
h₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x
⊢ MonotoneOn f (Ico b c)

Tactic:
·

Goal state after:
case refine_2
f : ℝ → ℝ
b c : ℝ
hb : ContinuousAt f b
hd₀ : DifferentiableOn ℝ f (Iio b)
hd₁ : DifferentiableOn ℝ f (Ioo b c)
h₀ : ∀ x ∈ Iio b, deriv f x ≤ 0
h₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x
⊢ MonotoneOn f (Ico b c)
Module: Mathlib.Analysis.Calculus.DerivativeTest

Goal state before:
pp: case refine_1
f : ℝ → ℝ
b c : ℝ
hb : ContinuousAt f b
hc : ContinuousAt f c
hd₀ : DifferentiableOn ℝ f (Iio b)
hd₁ : DifferentiableOn ℝ f (Ioo b c)
h₀ : ∀ x ∈ Iio b, deriv f x ≤ 0
h₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x
⊢ AntitoneOn f (Iic b)

pp: case refine_2
f : ℝ → ℝ
b c : ℝ
hb : ContinuousAt f b
hc : ContinuousAt f c
hd₀ : DifferentiableOn ℝ f (Iio b)
hd₁ : DifferentiableOn ℝ f (Ioo b c)
h₀ : ∀ x ∈ Iio b, deriv f x ≤ 0
h₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x
⊢ MonotoneOn f (Icc b c)

Tactic:
·

Goal state after:
case refine_1
f : ℝ → ℝ
b c : ℝ
hb : ContinuousAt f b
hc : ContinuousAt f c
hd₀ : DifferentiableOn ℝ f (Iio b)
hd₁ : DifferentiableOn ℝ f (Ioo b c)
h₀ : ∀ x ∈ Iio b, deriv f x ≤ 0
h₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x
⊢ AntitoneOn f (Iic b)
Module: Mathlib.Analysis.Normed.Module.Alternating.Uncurry.Fin

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
f : E →L[𝕜] E →L[𝕜] E [⋀^Fin n]→L[𝕜] F
hf : ∀ (x y : E), (f x) y = (f y) x
⊢ alternatizeUncurryFin (alternatizeUncurryFinCLM 𝕜 E F ∘SL f) = 0

Tactic:
by
  ext v
  simp [alternatizeUncurryFin_alternatizeUncurryFinCLM_comp_apply, hf]

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

Goal state before:
pp: case refine_2
f : ℝ → ℝ
b c : ℝ
hb : ContinuousAt f b
hc : ContinuousAt f c
hd₀ : DifferentiableOn ℝ f (Iio b)
hd₁ : DifferentiableOn ℝ f (Ioo b c)
h₀ : ∀ x ∈ Iio b, deriv f x ≤ 0
h₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x
⊢ MonotoneOn f (Icc b c)

Tactic:
·

Goal state after:
case refine_2
f : ℝ → ℝ
b c : ℝ
hb : ContinuousAt f b
hc : ContinuousAt f c
hd₀ : DifferentiableOn ℝ f (Iio b)
hd₁ : DifferentiableOn ℝ f (Ioo b c)
h₀ : ∀ x ∈ Iio b, deriv f x ≤ 0
h₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x
⊢ MonotoneOn f (Icc b c)
Module: Mathlib.Analysis.Calculus.DerivativeTest

Goal state before:
pp: case refine_1
f : ℝ → ℝ
b : ℝ
hb : ContinuousAt f b
hd₀ : DifferentiableOn ℝ f (Iio b)
hd₁ : DifferentiableOn ℝ f (Ioi b)
h₀ : ∀ x ∈ Iio b, deriv f x ≤ 0
h₁ : ∀ x ∈ Ioi b, 0 ≤ deriv f x
⊢ AntitoneOn f (Iic b)

pp: case refine_2
f : ℝ → ℝ
b : ℝ
hb : ContinuousAt f b
hd₀ : DifferentiableOn ℝ f (Iio b)
hd₁ : DifferentiableOn ℝ f (Ioi b)
h₀ : ∀ x ∈ Iio b, deriv f x ≤ 0
h₁ : ∀ x ∈ Ioi b, 0 ≤ deriv f x
⊢ MonotoneOn f (Ici b)

Tactic:
·

Goal state after:
case refine_1
f : ℝ → ℝ
b : ℝ
hb : ContinuousAt f b
hd₀ : DifferentiableOn ℝ f (Iio b)
hd₁ : DifferentiableOn ℝ f (Ioi b)
h₀ : ∀ x ∈ Iio b, deriv f x ≤ 0
h₁ : ∀ x ∈ Ioi b, 0 ≤ deriv f x
⊢ AntitoneOn f (Iic b)
Module: Mathlib.Analysis.Calculus.DerivativeTest

Goal state before:
pp: case refine_2
f : ℝ → ℝ
b : ℝ
hb : ContinuousAt f b
hd₀ : DifferentiableOn ℝ f (Iio b)
hd₁ : DifferentiableOn ℝ f (Ioi b)
h₀ : ∀ x ∈ Iio b, deriv f x ≤ 0
h₁ : ∀ x ∈ Ioi b, 0 ≤ deriv f x
⊢ MonotoneOn f (Ici b)

Tactic:
·

Goal state after:
case refine_2
f : ℝ → ℝ
b : ℝ
hb : ContinuousAt f b
hd₀ : DifferentiableOn ℝ f (Iio b)
hd₁ : DifferentiableOn ℝ f (Ioi b)
h₀ : ∀ x ∈ Iio b, deriv f x ≤ 0
h₁ : ∀ x ∈ Ioi b, 0 ≤ deriv f x
⊢ MonotoneOn f (Ici b)
Module: Mathlib.Analysis.Calculus.ContDiff.Bounds

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
Fu : Type u
inst✝³ : NormedAddCommGroup Fu
inst✝² : NormedSpace 𝕜 Fu
f : E → Fu
s : Set E
t : Set Fu
x : E
ht : UniqueDiffOn 𝕜 t
hs : UniqueDiffOn 𝕜 s
hst : MapsTo f s t
hx : x ∈ s
C D : ℝ
n : ℕ
IH :
  ∀ m ≤ n,
    ∀ {Gu : Type u} [inst : NormedAddCommGroup Gu] [inst_1 : NormedSpace 𝕜 Gu] {g : Fu → Gu},
      ContDiffOn 𝕜 (↑m) g t →
        ContDiffOn 𝕜 (↑m) f s →
          (∀ i ≤ m, ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C) →
            (∀ (i : ℕ), 1 ≤ i → i ≤ m → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i) →
              ‖iteratedFDerivWithin 𝕜 m (g ∘ f) s x‖ ≤ ↑m ! * C * D ^ m
Gu : Type u
inst✝¹ : NormedAddCommGroup Gu
inst✝ : NormedSpace 𝕜 Gu
g : Fu → Gu
hg : ContDiffOn 𝕜 (↑(n + 1)) g t
hf : ContDiffOn 𝕜 (↑(n + 1)) f s
hC : ∀ i ≤ n + 1, ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C
hD : ∀ (i : ℕ), 1 ≤ i → i ≤ n + 1 → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i
M : ↑n < ↑n.succ
Cnonneg : 0 ≤ C
Dnonneg : 0 ≤ D
I : ∀ i ∈ Finset.range (n + 1), ‖iteratedFDerivWithin 𝕜 i (fderivWithin 𝕜 g t ∘ f) s x‖ ≤ ↑i ! * C * D ^ i
J : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 (n - i) (fderivWithin 𝕜 f s) s x‖ ≤ D ^ (n - i + 1)
y : E
hy : y ∈ s
⊢ DifferentiableWithinAt 𝕜 f s y

Tactic:
·

Goal state after:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
Fu : Type u
inst✝³ : NormedAddCommGroup Fu
inst✝² : NormedSpace 𝕜 Fu
f : E → Fu
s : Set E
t : Set Fu
x : E
ht : UniqueDiffOn 𝕜 t
hs : UniqueDiffOn 𝕜 s
hst : MapsTo f s t
hx : x ∈ s
C D : ℝ
n : ℕ
IH :
  ∀ m ≤ n,
    ∀ {Gu : Type u} [inst : NormedAddCommGroup Gu] [inst_1 : NormedSpace 𝕜 Gu] {g : Fu → Gu},
      ContDiffOn 𝕜 (↑m) g t →
        ContDiffOn 𝕜 (↑m) f s →
          (∀ i ≤ m, ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C) →
            (∀ (i : ℕ), 1 ≤ i → i ≤ m → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i) →
              ‖iteratedFDerivWithin 𝕜 m (g ∘ f) s x‖ ≤ ↑m ! * C * D ^ m
Gu : Type u
inst✝¹ : NormedAddCommGroup Gu
inst✝ : NormedSpace 𝕜 Gu
g : Fu → Gu
hg : ContDiffOn 𝕜 (↑(n + 1)) g t
hf : ContDiffOn 𝕜 (↑(n + 1)) f s
hC : ∀ i ≤ n + 1, ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C
hD : ∀ (i : ℕ), 1 ≤ i → i ≤ n + 1 → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i
M : ↑n < ↑n.succ
Cnonneg : 0 ≤ C
Dnonneg : 0 ≤ D
I : ∀ i ∈ Finset.range (n + 1), ‖iteratedFDerivWithin 𝕜 i (fderivWithin 𝕜 g t ∘ f) s x‖ ≤ ↑i ! * C * D ^ i
J : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 (n - i) (fderivWithin 𝕜 f s) s x‖ ≤ D ^ (n - i + 1)
y : E
hy : y ∈ s
⊢ DifferentiableWithinAt 𝕜 f s y
Module: Mathlib.MeasureTheory.Covering.Besicovitch

Goal state before:
pp: case refine_4
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : HasBesicovitchCovering α
μ : Measure α
inst✝ : IsFiniteMeasure μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
P : Finset (α × ℝ) → Prop :=
  fun t ↦ ((↑t).PairwiseDisjoint fun p ↦ closedBall p.1 p.2) ∧ (∀ p ∈ t, p.1 ∈ s) ∧ ∀ p ∈ t, p.2 ∈ f p.1
F : Finset (α × ℝ) → Finset (α × ℝ)
hF :
  ∀ (t : Finset (α × ℝ)),
    P t →
      t ⊆ F t ∧ P (F t) ∧ μ (s \ ⋃ p ∈ F t, closedBall p.1 p.2) ≤ ↑N / (↑N + 1) * μ (s \ ⋃ p ∈ t, closedBall p.1 p.2)
u : ℕ → Finset (α × ℝ) := fun n ↦ F^[n] ∅
u_succ : ∀ (n : ℕ), u n.succ = F (u n)
Pu : ∀ (n : ℕ), P (u n)
⊢ (⋃ n, ↑(u n)).PairwiseDisjoint fun p ↦ closedBall p.1 p.2

Tactic:
·

Goal state after:
case refine_4
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : HasBesicovitchCovering α
μ : Measure α
inst✝ : IsFiniteMeasure μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
P : Finset (α × ℝ) → Prop :=
  fun t ↦ ((↑t).PairwiseDisjoint fun p ↦ closedBall p.1 p.2) ∧ (∀ p ∈ t, p.1 ∈ s) ∧ ∀ p ∈ t, p.2 ∈ f p.1
F : Finset (α × ℝ) → Finset (α × ℝ)
hF :
  ∀ (t : Finset (α × ℝ)),
    P t →
      t ⊆ F t ∧ P (F t) ∧ μ (s \ ⋃ p ∈ F t, closedBall p.1 p.2) ≤ ↑N / (↑N + 1) * μ (s \ ⋃ p ∈ t, closedBall p.1 p.2)
u : ℕ → Finset (α × ℝ) := fun n ↦ F^[n] ∅
u_succ : ∀ (n : ℕ), u n.succ = F (u n)
Pu : ∀ (n : ℕ), P (u n)
⊢ (⋃ n, ↑(u n)).PairwiseDisjoint fun p ↦ closedBall p.1 p.2
Module: Mathlib.Analysis.Calculus.DerivativeTest

Goal state before:
pp: case inl
f : ℝ → ℝ
x₀ : ℝ
hf : deriv f x₀ > 0
hx : f x₀ = 0
h_tendsto : Tendsto (slope f x₀) (𝓝[≠] x₀) (𝓝 (deriv f x₀))
x : ℝ
hx₀ : 0 < slope f x₀ x
hx₁ : x ≠ x₀
hx' : x < x₀
⊢ sign (f x) = sign (x - x₀)

pp: case inr
f : ℝ → ℝ
x₀ : ℝ
hf : deriv f x₀ > 0
hx : f x₀ = 0
h_tendsto : Tendsto (slope f x₀) (𝓝[≠] x₀) (𝓝 (deriv f x₀))
x : ℝ
hx₀ : 0 < slope f x₀ x
hx₁ : x ≠ x₀
hx' : x₀ < x
⊢ sign (f x) = sign (x - x₀)

Tactic:
·

Goal state after:
case inl
f : ℝ → ℝ
x₀ : ℝ
hf : deriv f x₀ > 0
hx : f x₀ = 0
h_tendsto : Tendsto (slope f x₀) (𝓝[≠] x₀) (𝓝 (deriv f x₀))
x : ℝ
hx₀ : 0 < slope f x₀ x
hx₁ : x ≠ x₀
hx' : x < x₀
⊢ sign (f x) = sign (x - x₀)
Module: Mathlib.Analysis.Calculus.DerivativeTest

Goal state before:
pp: case inr
f : ℝ → ℝ
x₀ : ℝ
hf : deriv f x₀ > 0
hx : f x₀ = 0
h_tendsto : Tendsto (slope f x₀) (𝓝[≠] x₀) (𝓝 (deriv f x₀))
x : ℝ
hx₀ : 0 < slope f x₀ x
hx₁ : x ≠ x₀
hx' : x₀ < x
⊢ sign (f x) = sign (x - x₀)

Tactic:
·

Goal state after:
case inr
f : ℝ → ℝ
x₀ : ℝ
hf : deriv f x₀ > 0
hx : f x₀ = 0
h_tendsto : Tendsto (slope f x₀) (𝓝[≠] x₀) (𝓝 (deriv f x₀))
x : ℝ
hx₀ : 0 < slope f x₀ x
hx₁ : x ≠ x₀
hx' : x₀ < x
⊢ sign (f x) = sign (x - x₀)
Module: Mathlib.Analysis.Calculus.ContDiff.Bounds

Goal state before:
pp: case h₂
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
Fu : Type u
inst✝³ : NormedAddCommGroup Fu
inst✝² : NormedSpace 𝕜 Fu
f : E → Fu
s : Set E
t : Set Fu
x : E
ht : UniqueDiffOn 𝕜 t
hs : UniqueDiffOn 𝕜 s
hst : MapsTo f s t
hx : x ∈ s
C D : ℝ
n : ℕ
IH :
  ∀ m ≤ n,
    ∀ {Gu : Type u} [inst : NormedAddCommGroup Gu] [inst_1 : NormedSpace 𝕜 Gu] {g : Fu → Gu},
      ContDiffOn 𝕜 (↑m) g t →
        ContDiffOn 𝕜 (↑m) f s →
          (∀ i ≤ m, ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C) →
            (∀ (i : ℕ), 1 ≤ i → i ≤ m → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i) →
              ‖iteratedFDerivWithin 𝕜 m (g ∘ f) s x‖ ≤ ↑m ! * C * D ^ m
Gu : Type u
inst✝¹ : NormedAddCommGroup Gu
inst✝ : NormedSpace 𝕜 Gu
g : Fu → Gu
hg : ContDiffOn 𝕜 (↑(n + 1)) g t
hf : ContDiffOn 𝕜 (↑(n + 1)) f s
hC : ∀ i ≤ n + 1, ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C
hD : ∀ (i : ℕ), 1 ≤ i → i ≤ n + 1 → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i
M : ↑n < ↑n.succ
Cnonneg : 0 ≤ C
Dnonneg : 0 ≤ D
I : ∀ i ∈ Finset.range (n + 1), ‖iteratedFDerivWithin 𝕜 i (fderivWithin 𝕜 g t ∘ f) s x‖ ≤ ↑i ! * C * D ^ i
J : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 (n - i) (fderivWithin 𝕜 f s) s x‖ ≤ D ^ (n - i + 1)
i : ℕ
hi : i ∈ Finset.range (n + 1)
⊢ ‖iteratedFDerivWithin 𝕜 (n - i) (fderivWithin 𝕜 f s) s x‖ ≤ D ^ (n - i + 1)

Tactic:
·

Goal state after:
case h₂
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
Fu : Type u
inst✝³ : NormedAddCommGroup Fu
inst✝² : NormedSpace 𝕜 Fu
f : E → Fu
s : Set E
t : Set Fu
x : E
ht : UniqueDiffOn 𝕜 t
hs : UniqueDiffOn 𝕜 s
hst : MapsTo f s t
hx : x ∈ s
C D : ℝ
n : ℕ
IH :
  ∀ m ≤ n,
    ∀ {Gu : Type u} [inst : NormedAddCommGroup Gu] [inst_1 : NormedSpace 𝕜 Gu] {g : Fu → Gu},
      ContDiffOn 𝕜 (↑m) g t →
        ContDiffOn 𝕜 (↑m) f s →
          (∀ i ≤ m, ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C) →
            (∀ (i : ℕ), 1 ≤ i → i ≤ m → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i) →
              ‖iteratedFDerivWithin 𝕜 m (g ∘ f) s x‖ ≤ ↑m ! * C * D ^ m
Gu : Type u
inst✝¹ : NormedAddCommGroup Gu
inst✝ : NormedSpace 𝕜 Gu
g : Fu → Gu
hg : ContDiffOn 𝕜 (↑(n + 1)) g t
hf : ContDiffOn 𝕜 (↑(n + 1)) f s
hC : ∀ i ≤ n + 1, ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C
hD : ∀ (i : ℕ), 1 ≤ i → i ≤ n + 1 → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i
M : ↑n < ↑n.succ
Cnonneg : 0 ≤ C
Dnonneg : 0 ≤ D
I : ∀ i ∈ Finset.range (n + 1), ‖iteratedFDerivWithin 𝕜 i (fderivWithin 𝕜 g t ∘ f) s x‖ ≤ ↑i ! * C * D ^ i
J : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 (n - i) (fderivWithin 𝕜 f s) s x‖ ≤ D ^ (n - i + 1)
i : ℕ
hi : i ∈ Finset.range (n + 1)
⊢ ‖iteratedFDerivWithin 𝕜 (n - i) (fderivWithin 𝕜 f s) s x‖ ≤ D ^ (n - i + 1)
Module: Mathlib.Analysis.Calculus.ContDiff.Bounds

Goal state before:
pp: case a.e'_6
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
Fu : Type u
inst✝³ : NormedAddCommGroup Fu
inst✝² : NormedSpace 𝕜 Fu
f : E → Fu
s : Set E
t : Set Fu
x : E
ht : UniqueDiffOn 𝕜 t
hs : UniqueDiffOn 𝕜 s
hst : MapsTo f s t
hx : x ∈ s
C D : ℝ
n : ℕ
IH :
  ∀ m ≤ n,
    ∀ {Gu : Type u} [inst : NormedAddCommGroup Gu] [inst_1 : NormedSpace 𝕜 Gu] {g : Fu → Gu},
      ContDiffOn 𝕜 (↑m) g t →
        ContDiffOn 𝕜 (↑m) f s →
          (∀ i ≤ m, ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C) →
            (∀ (i : ℕ), 1 ≤ i → i ≤ m → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i) →
              ‖iteratedFDerivWithin 𝕜 m (g ∘ f) s x‖ ≤ ↑m ! * C * D ^ m
Gu : Type u
inst✝¹ : NormedAddCommGroup Gu
inst✝ : NormedSpace 𝕜 Gu
g : Fu → Gu
hg : ContDiffOn 𝕜 (↑(n + 1)) g t
hf : ContDiffOn 𝕜 (↑(n + 1)) f s
hC : ∀ i ≤ n + 1, ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C
hD : ∀ (i : ℕ), 1 ≤ i → i ≤ n + 1 → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i
M : ↑n < ↑n.succ
Cnonneg : 0 ≤ C
Dnonneg : 0 ≤ D
I : ∀ i ∈ Finset.range (n + 1), ‖iteratedFDerivWithin 𝕜 i (fderivWithin 𝕜 g t ∘ f) s x‖ ≤ ↑i ! * C * D ^ i
J : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 (n - i) (fderivWithin 𝕜 f s) s x‖ ≤ D ^ (n - i + 1)
i : ℕ
hi : i ∈ Finset.range (n + 1)
⊢ D ^ i * D ^ (n - i + 1) = D ^ (n + 1)

Tactic:
·

Goal state after:
case a.e'_6
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
Fu : Type u
inst✝³ : NormedAddCommGroup Fu
inst✝² : NormedSpace 𝕜 Fu
f : E → Fu
s : Set E
t : Set Fu
x : E
ht : UniqueDiffOn 𝕜 t
hs : UniqueDiffOn 𝕜 s
hst : MapsTo f s t
hx : x ∈ s
C D : ℝ
n : ℕ
IH :
  ∀ m ≤ n,
    ∀ {Gu : Type u} [inst : NormedAddCommGroup Gu] [inst_1 : NormedSpace 𝕜 Gu] {g : Fu → Gu},
      ContDiffOn 𝕜 (↑m) g t →
        ContDiffOn 𝕜 (↑m) f s →
          (∀ i ≤ m, ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C) →
            (∀ (i : ℕ), 1 ≤ i → i ≤ m → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i) →
              ‖iteratedFDerivWithin 𝕜 m (g ∘ f) s x‖ ≤ ↑m ! * C * D ^ m
Gu : Type u
inst✝¹ : NormedAddCommGroup Gu
inst✝ : NormedSpace 𝕜 Gu
g : Fu → Gu
hg : ContDiffOn 𝕜 (↑(n + 1)) g t
hf : ContDiffOn 𝕜 (↑(n + 1)) f s
hC : ∀ i ≤ n + 1, ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C
hD : ∀ (i : ℕ), 1 ≤ i → i ≤ n + 1 → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i
M : ↑n < ↑n.succ
Cnonneg : 0 ≤ C
Dnonneg : 0 ≤ D
I : ∀ i ∈ Finset.range (n + 1), ‖iteratedFDerivWithin 𝕜 i (fderivWithin 𝕜 g t ∘ f) s x‖ ≤ ↑i ! * C * D ^ i
J : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 (n - i) (fderivWithin 𝕜 f s) s x‖ ≤ D ^ (n - i + 1)
i : ℕ
hi : i ∈ Finset.range (n + 1)
⊢ D ^ i * D ^ (n - i + 1) = D ^ (n + 1)
Module: Mathlib.Analysis.Calculus.FDeriv.Symmetric

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
x : E
s : Set E
h : s ∈ 𝓝 x
⊢ fderivWithin 𝕜 (fderivWithin 𝕜 f s) s x = fderiv 𝕜 (fderiv 𝕜 f) x

Tactic:
simp only [← fderivWithin_univ]

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
x : E
s : Set E
h : s ∈ 𝓝 x
⊢ fderivWithin 𝕜 (fderivWithin 𝕜 f s) s x = fderivWithin 𝕜 (fderivWithin 𝕜 f univ) univ x
Module: Mathlib.Analysis.Calculus.ContDiff.Bounds

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
g : F → G
f : E → F
n : ℕ
N : ℕ∞ω
hg : ContDiff 𝕜 N g
hf : ContDiff 𝕜 N f
hn : ↑n ≤ N
x : E
C D : ℝ
hC : ∀ i ≤ n, ‖iteratedFDeriv 𝕜 i g (f x)‖ ≤ C
hD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDeriv 𝕜 i f x‖ ≤ D ^ i
⊢ ‖iteratedFDeriv 𝕜 n (g ∘ f) x‖ ≤ ↑n ! * C * D ^ n

Tactic:
simp_rw [← iteratedFDerivWithin_univ] at hC

Goal state after:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
g : F → G
f : E → F
n : ℕ
N : ℕ∞ω
hg : ContDiff 𝕜 N g
hf : ContDiff 𝕜 N f
hn : ↑n ≤ N
x : E
C D : ℝ
hD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDeriv 𝕜 i f x‖ ≤ D ^ i
hC : ∀ i ≤ n, ‖iteratedFDerivWithin 𝕜 i g univ (f x)‖ ≤ C
⊢ ‖iteratedFDeriv 𝕜 n (g ∘ f) x‖ ≤ ↑n ! * C * D ^ n
Module: Mathlib.Analysis.Calculus.FDeriv.Symmetric

Goal state before:
pp: case hg
E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set E
s_conv : Convex ℝ s
f : E → F
f' : E → E →L[ℝ] F
f'' : E →L[ℝ] E →L[ℝ] F
hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x
x : E
xs : x ∈ s
hx : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x_1 : E) in 𝓝[interior s] x, ‖f' x_1 - f' x - f'' (x_1 - x)‖ ≤ c * ‖x_1 - x‖
v w : E
hv : x + v ∈ interior s
hw : x + v + w ∈ interior s
ε : ℝ
εpos : 0 < ε
δ : ℝ
δpos : δ > 0
sδ : Metric.ball x δ ∩ interior s ⊆ {x_1 | (fun x_2 ↦ ‖f' x_2 - f' x - f'' (x_2 - x)‖ ≤ ε * ‖x_2 - x‖) x_1}
E1 : ∀ᶠ (h : ℝ) in 𝓝[>] 0, h * (‖v‖ + ‖w‖) < δ
E2 : ∀ᶠ (h : ℝ) in 𝓝[>] 0, h < 1
h : ℝ
hδ : h * (‖v‖ + ‖w‖) < δ
h_lt_1 : h < 1
hpos : 0 < h
xt_mem : ∀ t ∈ Icc 0 1, x + h • v + (t * h) • w ∈ interior s
g : ℝ → F := ⋯
g' : ℝ → F := ⋯
hg' : g' = fun t ↦ (f' (x + h • v + (t * h) • w)) (h • w) - h • (f' x) w - h ^ 2 • (f'' v) w - (t * h ^ 2) • (f'' w) w
t : ℝ
ht : t ∈ Icc 0 1
⊢ HasDerivWithinAt (fun t ↦ ((t * h) ^ 2 / 2) • (f'' w) w) ((t * h ^ 2) • (f'' w) w) (Icc 0 1) t

Tactic:
·

Goal state after:
case hg
E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set E
s_conv : Convex ℝ s
f : E → F
f' : E → E →L[ℝ] F
f'' : E →L[ℝ] E →L[ℝ] F
hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x
x : E
xs : x ∈ s
hx : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x_1 : E) in 𝓝[interior s] x, ‖f' x_1 - f' x - f'' (x_1 - x)‖ ≤ c * ‖x_1 - x‖
v w : E
hv : x + v ∈ interior s
hw : x + v + w ∈ interior s
ε : ℝ
εpos : 0 < ε
δ : ℝ
δpos : δ > 0
sδ : Metric.ball x δ ∩ interior s ⊆ {x_1 | (fun x_2 ↦ ‖f' x_2 - f' x - f'' (x_2 - x)‖ ≤ ε * ‖x_2 - x‖) x_1}
E1 : ∀ᶠ (h : ℝ) in 𝓝[>] 0, h * (‖v‖ + ‖w‖) < δ
E2 : ∀ᶠ (h : ℝ) in 𝓝[>] 0, h < 1
h : ℝ
hδ : h * (‖v‖ + ‖w‖) < δ
h_lt_1 : h < 1
hpos : 0 < h
xt_mem : ∀ t ∈ Icc 0 1, x + h • v + (t * h) • w ∈ interior s
g : ℝ → F := ⋯
g' : ℝ → F := ⋯
hg' : g' = fun t ↦ (f' (x + h • v + (t * h) • w)) (h • w) - h • (f' x) w - h ^ 2 • (f'' v) w - (t * h ^ 2) • (f'' w) w
t : ℝ
ht : t ∈ Icc 0 1
⊢ HasDerivWithinAt (fun t ↦ ((t * h) ^ 2 / 2) • (f'' w) w) ((t * h ^ 2) • (f'' w) w) (Icc 0 1) t