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

Goal state before:
pp: 𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
x : 𝕜
s : Set 𝕜
𝕜' : Type u_1
inst✝³ : NontriviallyNormedField 𝕜'
inst✝² : NormedAlgebra 𝕜 𝕜'
inst✝¹ : NormedSpace 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
h : 𝕜 → 𝕜'
h' : 𝕜'
g₁ : 𝕜' → F
g₁' : F
y : 𝕜'
hg : HasDerivAt g₁ g₁' y
hh : HasDerivWithinAt h h' s x
hy : y = h x
⊢ HasDerivWithinAt (g₁ ∘ h) (h' • g₁') s x

Tactic:
rw [hy] at hg

Goal state after:
𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
x : 𝕜
s : Set 𝕜
𝕜' : Type u_1
inst✝³ : NontriviallyNormedField 𝕜'
inst✝² : NormedAlgebra 𝕜 𝕜'
inst✝¹ : NormedSpace 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
h : 𝕜 → 𝕜'
h' : 𝕜'
g₁ : 𝕜' → F
g₁' : F
y : 𝕜'
hg : HasDerivAt g₁ g₁' (h x)
hh : HasDerivWithinAt h h' s x
hy : y = h x
⊢ HasDerivWithinAt (g₁ ∘ h) (h' • g₁') s x
Module: Mathlib.Analysis.Calculus.Deriv.Comp

Goal state before:
pp: 𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
x : 𝕜
s : Set 𝕜
𝕜' : Type u_1
inst✝³ : NontriviallyNormedField 𝕜'
inst✝² : NormedAlgebra 𝕜 𝕜'
inst✝¹ : NormedSpace 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
t' : Set 𝕜'
h : 𝕜 → 𝕜'
g₁ : 𝕜' → F
y : 𝕜'
hg : DifferentiableWithinAt 𝕜' g₁ t' y
hh : DifferentiableWithinAt 𝕜 h s x
hs : MapsTo h s t'
hy : y = h x
⊢ derivWithin (g₁ ∘ h) s x = derivWithin h s x • derivWithin g₁ t' (h x)

Tactic:
rw [hy] at hg

Goal state after:
𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
x : 𝕜
s : Set 𝕜
𝕜' : Type u_1
inst✝³ : NontriviallyNormedField 𝕜'
inst✝² : NormedAlgebra 𝕜 𝕜'
inst✝¹ : NormedSpace 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
t' : Set 𝕜'
h : 𝕜 → 𝕜'
g₁ : 𝕜' → F
y : 𝕜'
hg : DifferentiableWithinAt 𝕜' g₁ t' (h x)
hh : DifferentiableWithinAt 𝕜 h s x
hs : MapsTo h s t'
hy : y = h x
⊢ derivWithin (g₁ ∘ h) s x = derivWithin h s x • derivWithin g₁ t' (h x)
Module: Mathlib.Analysis.Calculus.Deriv.Comp

Goal state before:
pp: 𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
x : 𝕜
𝕜' : Type u_1
inst✝³ : NontriviallyNormedField 𝕜'
inst✝² : NormedAlgebra 𝕜 𝕜'
inst✝¹ : NormedSpace 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
h : 𝕜 → 𝕜'
g₁ : 𝕜' → F
y : 𝕜'
hg : DifferentiableAt 𝕜' g₁ y
hh : DifferentiableAt 𝕜 h x
hy : y = h x
⊢ deriv (g₁ ∘ h) x = deriv h x • deriv g₁ (h x)

Tactic:
rw [hy] at hg

Goal state after:
𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
x : 𝕜
𝕜' : Type u_1
inst✝³ : NontriviallyNormedField 𝕜'
inst✝² : NormedAlgebra 𝕜 𝕜'
inst✝¹ : NormedSpace 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
h : 𝕜 → 𝕜'
g₁ : 𝕜' → F
y : 𝕜'
hg : DifferentiableAt 𝕜' g₁ (h x)
hh : DifferentiableAt 𝕜 h x
hy : y = h x
⊢ deriv (g₁ ∘ h) x = deriv h x • deriv g₁ (h x)
Module: Mathlib.Analysis.Calculus.LogDeriv

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NontriviallyNormedField 𝕜'
inst✝¹ : NormedAlgebra 𝕜 𝕜'
inst✝ : IsRCLikeNormedField 𝕜
f g : 𝕜 → 𝕜'
s : Set 𝕜
hf : DifferentiableOn 𝕜 f s
hg : DifferentiableOn 𝕜 g s
hs2 : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hsc : IsPreconnected s
hgn : ∀ x ∈ s, g x ≠ 0
hfn : ∀ x ∈ s, f x ≠ 0
⊢ EqOn (logDeriv f) (logDeriv g) s ↔ ∃ z, z ≠ 0 ∧ EqOn f (z • g) s

Tactic:
rcases s.eq_empty_or_nonempty with rfl | ⟨t, ht⟩

Goal state after:
case inl
𝕜 : Type u_1
𝕜' : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NontriviallyNormedField 𝕜'
inst✝¹ : NormedAlgebra 𝕜 𝕜'
inst✝ : IsRCLikeNormedField 𝕜
f g : 𝕜 → 𝕜'
hf : DifferentiableOn 𝕜 f ∅
hg : DifferentiableOn 𝕜 g ∅
hs2 : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ∅
hsc : IsPreconnected ∅
hgn : ∀ x ∈ ∅, g x ≠ 0
hfn : ∀ x ∈ ∅, f x ≠ 0
⊢ EqOn (logDeriv f) (logDeriv g) ∅ ↔ ∃ z, z ≠ 0 ∧ EqOn f (z • g) ∅

case inr
𝕜 : Type u_1
𝕜' : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NontriviallyNormedField 𝕜'
inst✝¹ : NormedAlgebra 𝕜 𝕜'
inst✝ : IsRCLikeNormedField 𝕜
f g : 𝕜 → 𝕜'
s : Set 𝕜
hf : DifferentiableOn 𝕜 f s
hg : DifferentiableOn 𝕜 g s
hs2 : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hsc : IsPreconnected s
hgn : ∀ x ∈ s, g x ≠ 0
hfn : ∀ x ∈ s, f x ≠ 0
t : 𝕜
ht : t ∈ s
⊢ EqOn (logDeriv f) (logDeriv g) s ↔ ∃ z, z ≠ 0 ∧ EqOn f (z • g) s
Module: Mathlib.Analysis.Calculus.ContDiff.Basic

Goal state before:
pp: case fderivWithin
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
f : E → F
n : ℕ∞ω
p : E → FormalMultilinearSeries 𝕜 E F
hf : HasFTaylorSeriesUpToOn n f p s
g : G →ᴬ[𝕜] E
A : (m : ℕ) → (E [×m]→L[𝕜] F) → G [×m]→L[𝕜] F := fun m h ↦ h.compContinuousLinearMap fun x ↦ g.contLinear
hA : ∀ (m : ℕ), IsBoundedLinearMap 𝕜 (A m)
m : ℕ
hm : ↑m < n
x : G
hx : x ∈ ⇑g ⁻¹' s
⊢ HasFDerivWithinAt (fun x ↦ (p (g x) m).compContinuousLinearMap fun x ↦ g.contLinear)
    ((p (g x) m.succ).compContinuousLinearMap fun x ↦ g.contLinear).curryLeft (⇑g ⁻¹' s) x

Tactic:
convert!
      (hA m).hasFDerivAt.comp_hasFDerivWithinAt x
        ((hf.fderivWithin m hm (g x) hx).comp x g.hasFDerivWithinAt (Subset.refl _))

Goal state after:
case e'_12
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
f : E → F
n : ℕ∞ω
p : E → FormalMultilinearSeries 𝕜 E F
hf : HasFTaylorSeriesUpToOn n f p s
g : G →ᴬ[𝕜] E
A : (m : ℕ) → (E [×m]→L[𝕜] F) → G [×m]→L[𝕜] F := ⋯
hA : ∀ (m : ℕ), IsBoundedLinearMap 𝕜 (A m)
m : ℕ
hm : ↑m < n
x : G
hx : x ∈ ⇑g ⁻¹' s
e_8✝ : ContinuousMultilinearMap.instAddCommGroup = ContinuousMultilinearMap.normedAddCommGroup'.toAddCommGroup
e_9✝ : ContinuousMultilinearMap.instModule ≍ ContinuousMultilinearMap.normedSpace'.toModule
e_10✝ : ContinuousMultilinearMap.instTopologicalSpace = PseudoMetricSpace.toUniformSpace.toTopologicalSpace
⊢ ((p (g x) m.succ).compContinuousLinearMap fun x ↦ g.contLinear).curryLeft =
    IsBoundedLinearMap.toContinuousLinearMap (A m) ⋯ ∘SL (p (g x) m.succ).curryLeft ∘SL g.contLinear
Module: Mathlib.Analysis.Calculus.ContDiff.Comp

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
x₀ : E
m n : ℕ∞ω
f : E → F → G
g : E → F
t : Set F
hf : ContDiffWithinAt 𝕜 n (uncurry f) (insert x₀ s ×ˢ t) (x₀, g x₀)
hg : ContDiffWithinAt 𝕜 m g s x₀
ht : ∀ᶠ (x : E) in 𝓝[insert x₀ s] x₀, UniqueDiffWithinAt 𝕜 t (g x)
hmn : m + 1 ≤ n
hgt : t ∈ 𝓝[g '' s] g x₀
k : ℕ
hkm : ↑k ≤ m
⊢ m + 1 ≤ n

Tactic:
hmn

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
x₀ : E
m n : ℕ∞ω
f : E → F → G
g : E → F
t : Set F
hf : ContDiffWithinAt 𝕜 n (uncurry f) (insert x₀ s ×ˢ t) (x₀, g x₀)
hg : ContDiffWithinAt 𝕜 m g s x₀
ht : ∀ᶠ (x : E) in 𝓝[insert x₀ s] x₀, UniqueDiffWithinAt 𝕜 t (g x)
hmn : m + 1 ≤ n
hgt : t ∈ 𝓝[g '' s] g x₀
k : ℕ
hkm : ↑k ≤ m
⊢ n ≤ n
Module: Mathlib.Analysis.Calculus.ContDiff.Defs

Goal state before:
pp: 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
n : ℕ∞ω
h : ContDiffOn 𝕜 n f s
hs : UniqueDiffOn 𝕜 s
m : ℕ
hm : ↑m < n
x : E
hx : x ∈ s
this : ↑(m + 1) ≤ n
u : Set E
hu : u ∈ 𝓝[s] x
p : E → FormalMultilinearSeries 𝕜 E F
Hp : HasFTaylorSeriesUpToOn (↑(m + 1)) f p u
o : Set E
o_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] o
xo : x ∈ o
ho : s ∩ o ⊆ u
⊢ p x m.succ = iteratedFDerivWithin 𝕜 m.succ f (s ∩ o) x

Tactic:
exact (Hp.mono ho).eq_iteratedFDerivWithin_of_uniqueDiffOn le_rfl (hs.inter o_open) ⟨hx, xo⟩

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
c✝ c : OrderedFinpartition n
k : Fin c.length
⊢ ∀ (x : Fin (n + 1)),
    ∃ m,
      x ∈
        range
          (if h : m = k then
            have this := ⋯;
            (fun i ↦ cases 0 (succ ∘ c.emb k) i) ∘ Fin.cast this
          else
            have this := ⋯;
            succ ∘ c.emb m ∘ Fin.cast this)

Tactic:
by
    refine Fin.cases ?_ (fun i ↦ ?_)
    · simp only [mem_range]
      exact ⟨k, ⟨0, by simp⟩, by simp⟩
    · simp only [mem_range]
      rcases eq_or_ne (c.index i) k with rfl | hi
      · have A : update c.partSize (c.index i) (c.partSize (c.index i) + 1) (c.index i) =
          c.partSize (c.index i) + 1 := by simp
        exact ⟨c.index i, (succ (c.invEmbedding i)).cast A.symm, by simp⟩
      · have A : update c.partSize k (c.partSize k + 1) (c.index i) = c.partSize (c.index i) := by
          simp [hi]
        exact ⟨c.index i, (c.invEmbedding i).cast A.symm, by simp [hi]⟩

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.Defs

Goal state before:
pp: case mp
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
n : ℕ∞ω
⊢ ContDiffOn 𝕜 ω f univ → ContDiff 𝕜 ω f

pp: case mpr
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
n : ℕ∞ω
⊢ ContDiff 𝕜 ω f → ContDiffOn 𝕜 ω f univ

Tactic:
· intro H
      use ftaylorSeriesWithin 𝕜 f univ
      rw [← hasFTaylorSeriesUpToOn_univ_iff]
      refine ⟨H.ftaylorSeriesWithin uniqueDiffOn_univ, fun i ↦ ?_⟩
      rw [← analyticOn_univ]
      exact H.analyticOn.iteratedFDerivWithin uniqueDiffOn_univ _

Goal state after:
case mpr
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
n : ℕ∞ω
⊢ ContDiff 𝕜 ω f → ContDiffOn 𝕜 ω f univ
Module: Mathlib.Analysis.Complex.RealDeriv

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
x : ℂ
f' : E
h : HasDerivAt f f' x
⊢ HasFDerivAt f (reCLM.smulRight f' + I • imCLM.smulRight f') x

Tactic:
by
  simpa only [Complex.restrictScalars_toSpanSingleton'] using h.hasFDerivAt.restrictScalars ℝ

Goal state after:
Module: Mathlib.Analysis.Normed.Algebra.Exponential

Goal state before:
pp: 𝕂 : Type u_1
𝔸 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕂
inst✝³ : NormedRing 𝔸
inst✝² : NormedAlgebra 𝕂 𝔸
inst✝¹ : CompleteSpace 𝔸
inst✝ : CharZero 𝕂
x y : 𝔸
hxy : Commute x y
hx : x ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius
hy : y ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius
⊢ exp (x + y) = exp x * exp y

Tactic:
by
  rw [exp_eq_tsum 𝕂,
    tsum_mul_tsum_eq_tsum_sum_antidiagonal_of_summable_norm
      (norm_expSeries_summable_of_mem_ball' x hx) (norm_expSeries_summable_of_mem_ball' y hy)]
  dsimp only
  conv_lhs =>
    congr
    ext
    rw [hxy.add_pow' _, Finset.smul_sum]
  refine tsum_congr fun n => Finset.sum_congr rfl fun kl hkl => ?_
  rw [← Nat.cast_smul_eq_nsmul 𝕂, smul_smul, smul_mul_smul_comm, ← Finset.mem_antidiagonal.mp hkl,
    Nat.cast_add_choose, Finset.mem_antidiagonal.mp hkl]
  field_simp [n.factorial_ne_zero]

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: case refine_1
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
G : Type u_4
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
α : Type u_5
H : Type u_6
inst✝ : NormedAddCommGroup H
l : Filter α
p₁ p₂ : α → FormalMultilinearSeries 𝕜 F G
q₁ q₂ : α → FormalMultilinearSeries 𝕜 E F
f : α → H
n : ℕ
hp_bdd : ∀ k ≤ n, IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l fun x ↦ ‖p₁ x k‖
hpf : ∀ k ≤ n, (fun a ↦ p₁ a k - p₂ a k) =O[l] f
hq₁_bdd : ∀ k ≤ n, IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l fun x ↦ ‖q₁ x k‖
hq₂_bdd : ∀ k ≤ n, IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l fun x ↦ ‖q₂ x k‖
hqf : ∀ k ≤ n, (fun a ↦ q₁ a k - q₂ a k) =O[l] f
c : OrderedFinpartition n
x✝ : c ∈ Finset.univ
H₁ : (fun x ↦ p₁ x c.length) =O[l] 1
H₂ : ∀ (m : Fin c.length), (fun x ↦ q₁ x (c.partSize m)) =O[l] 1
H₃ : ∀ (m : Fin c.length), (fun x ↦ q₂ x (c.partSize m)) =O[l] 1
H₄ : ∀ (m : Fin c.length), (fun a ↦ q₁ a (c.partSize m) - q₂ a (c.partSize m)) =O[l] f
⊢ (fun x ↦
      ‖p₁ x c.length‖ * ↑c.length * max ‖fun m ↦ q₁ x (c.partSize m)‖ ‖fun m ↦ q₂ x (c.partSize m)‖ ^ (c.length - 1) *
        ‖(fun m ↦ q₁ x (c.partSize m)) - fun m ↦ q₂ x (c.partSize m)‖) =O[l]
    f

Tactic:
rw [← Asymptotics.isBigO_pi] at H₂ H₃ H₄

Goal state after:
case refine_1
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
G : Type u_4
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
α : Type u_5
H : Type u_6
inst✝ : NormedAddCommGroup H
l : Filter α
p₁ p₂ : α → FormalMultilinearSeries 𝕜 F G
q₁ q₂ : α → FormalMultilinearSeries 𝕜 E F
f : α → H
n : ℕ
hp_bdd : ∀ k ≤ n, IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l fun x ↦ ‖p₁ x k‖
hpf : ∀ k ≤ n, (fun a ↦ p₁ a k - p₂ a k) =O[l] f
hq₁_bdd : ∀ k ≤ n, IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l fun x ↦ ‖q₁ x k‖
hq₂_bdd : ∀ k ≤ n, IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l fun x ↦ ‖q₂ x k‖
hqf : ∀ k ≤ n, (fun a ↦ q₁ a k - q₂ a k) =O[l] f
c : OrderedFinpartition n
x✝ : c ∈ Finset.univ
H₁ : (fun x ↦ p₁ x c.length) =O[l] 1
H₂ : (fun x m ↦ q₁ x (c.partSize m)) =O[l] 1
H₃ : (fun x m ↦ q₂ x (c.partSize m)) =O[l] 1
H₄ : (fun a m ↦ q₁ a (c.partSize m) - q₂ a (c.partSize m)) =O[l] f
⊢ (fun x ↦
      ‖p₁ x c.length‖ * ↑c.length * max ‖fun m ↦ q₁ x (c.partSize m)‖ ‖fun m ↦ q₂ x (c.partSize m)‖ ^ (c.length - 1) *
        ‖(fun m ↦ q₁ x (c.partSize m)) - fun m ↦ q₂ x (c.partSize m)‖) =O[l]
    f
Module: Mathlib.Analysis.Calculus.ContDiff.Operations

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
ι : Type u_3
inst✝³ : Fintype ι
F' : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (F' i)
inst✝¹ : (i : ι) → NormedSpace 𝕜 (F' i)
inst✝ : DecidableEq ι
k : ℕ∞ω
x : (i : ι) → F' i
i j : ι
h : ¬j = i
⊢ ContDiff 𝕜 k fun x_1 ↦ x j

Tactic:
· exact contDiff_const

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.Operations

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
x : E
i : ℕ
f : E → F
hu : UniqueDiffOn 𝕜 s
hx : x ∈ s
⊢ iteratedFDerivWithin 𝕜 i (-f) s x = -iteratedFDerivWithin 𝕜 i f s x

Tactic:
induction i generalizing x with ext h
  | zero => simp
  | succ i hi =>
    calc
      iteratedFDerivWithin 𝕜 (i + 1) (-f) s x h =
          fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 i (-f) s) s x (h 0) (Fin.tail h) :=
        iteratedFDerivWithin_succ_apply_left _
      _ = fderivWithin 𝕜 (-iteratedFDerivWithin 𝕜 i f s) s x (h 0) (Fin.tail h) := by
        rw [fderivWithin_congr' (@hi) hx, Pi.neg_def]
      _ = -(fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 i f s) s) x (h 0) (Fin.tail h) := by
        rw [fderivWithin_neg (hu x hx), neg_apply, neg_apply]
      _ = -(iteratedFDerivWithin 𝕜 (i + 1) f s) x h := by
        rw [iteratedFDerivWithin_succ_apply_left]

Goal state after:
Module: Mathlib.Analysis.Calculus.LocalExtr.Basic

Goal state before:
pp: f : ℝ → ℝ
a : ℝ
h : IsLocalMin f a
⊢ deriv f a = 0

Tactic:
by
  classical
  exact if hf : DifferentiableAt ℝ f a then h.hasDerivAt_eq_zero hf.hasDerivAt
  else deriv_zero_of_not_differentiableAt hf

Goal state after:
Module: Mathlib.Analysis.Calculus.LocalExtr.Basic

Goal state before:
pp: f : ℝ → ℝ
a : ℝ
h : IsLocalMax f a
⊢ deriv f a = 0

Tactic:
by
  classical
  exact if hf : DifferentiableAt ℝ f a then h.hasDerivAt_eq_zero hf.hasDerivAt
  else deriv_zero_of_not_differentiableAt hf

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.FundThmCalculus

Goal state before:
pp: X : Type u_1
E : Type u_2
ι : Type u_3
inst✝⁴ : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure X
l : Filter X
inst✝ : l.IsMeasurablyGenerated
f : X → E
b : E
h : Tendsto f (l ⊓ ae μ) (𝓝 b)
hfm✝ : StronglyMeasurableAtFilter f l μ
hμ : μ.FiniteAtFilter l
s✝ : ι → Set X
li : Filter ι
hs : Tendsto s✝ li l.smallSets
m : ι → ℝ
hsμ : (fun i ↦ μ.real (s✝ i)) =ᶠ[li] m
ε : ℝ
ε₀ : 0 < ε
this : ∀ᶠ (s : Set X) in l.smallSets, ∀ᵐ (x : X) ∂μ, x ∈ s → f x ∈ closedBall b ε
s : Set X
hμs : μ s < ⊤
h_integrable : IntegrableOn f s μ
hfm : AEStronglyMeasurable f (μ.restrict s)
h_norm : ∀ᵐ (x : X) ∂μ, x ∈ s → ‖f x - b‖ ≤ ε
⊢ ‖∫ (x : X) in s, f x ∂μ - ∫ (x : X) in s, b ∂μ‖ ≤ ε * ‖μ.real s‖

Tactic:
← integral_sub h_integrable (integrableOn_const hμs.ne),

Goal state after:
X : Type u_1
E : Type u_2
ι : Type u_3
inst✝⁴ : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure X
l : Filter X
inst✝ : l.IsMeasurablyGenerated
f : X → E
b : E
h : Tendsto f (l ⊓ ae μ) (𝓝 b)
hfm✝ : StronglyMeasurableAtFilter f l μ
hμ : μ.FiniteAtFilter l
s✝ : ι → Set X
li : Filter ι
hs : Tendsto s✝ li l.smallSets
m : ι → ℝ
hsμ : (fun i ↦ μ.real (s✝ i)) =ᶠ[li] m
ε : ℝ
ε₀ : 0 < ε
this : ∀ᶠ (s : Set X) in l.smallSets, ∀ᵐ (x : X) ∂μ, x ∈ s → f x ∈ closedBall b ε
s : Set X
hμs : μ s < ⊤
h_integrable : IntegrableOn f s μ
hfm : AEStronglyMeasurable f (μ.restrict s)
h_norm : ∀ᵐ (x : X) ∂μ, x ∈ s → ‖f x - b‖ ≤ ε
⊢ ‖∫ (a : X) in s, f a - b ∂μ‖ ≤ ε * ‖μ.real s‖
Module: Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory

Goal state before:
pp: case pos
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
c : ℝ≥0
s : Set α
hs : MeasurableSet s
ε : ℝ≥0∞
ε0 : ε ≠ 0
f : α →ₛ ℝ≥0 := piecewise s hs (const α c) (const α 0)
h : ∫⁻ (x : α), ↑(f x) ∂μ = ∞
⊢ ∃ g,
    (∀ (x : α), (piecewise s hs (const α c) (const α 0)) x ≤ g x) ∧
      LowerSemicontinuous g ∧ ∫⁻ (x : α), ↑(g x) ∂μ ≤ ∫⁻ (x : α), ↑((piecewise s hs (const α c) (const α 0)) x) ∂μ + ε

Tactic:
refine
        ⟨fun _ => c, fun x => ?_, lowerSemicontinuous_const, by
          simp only [f, _root_.top_add, le_top, h]⟩

Goal state after:
case pos
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
c : ℝ≥0
s : Set α
hs : MeasurableSet s
ε : ℝ≥0∞
ε0 : ε ≠ 0
f : α →ₛ ℝ≥0 := piecewise s hs (const α c) (const α 0)
h : ∫⁻ (x : α), ↑(f x) ∂μ = ∞
x : α
⊢ (piecewise s hs (const α c) (const α 0)) x ≤ (fun x ↦ c) x
Module: Mathlib.Analysis.Calculus.Deriv.MeanValue

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

Tactic:
rw [deriv_eqOn isOpen_interior hf' hx]

Goal state after:
D : Set ℝ
hD : Convex ℝ D
f f' : ℝ → ℝ
hf : ContinuousOn f D
hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x
hf'₀ : ∀ x ∈ interior D, 0 < f' x
x : ℝ
hx : x ∈ interior D
⊢ 0 < f' x
Module: Mathlib.Analysis.Calculus.Deriv.MeanValue

Goal state before:
pp: D : Set ℝ
hD : Convex ℝ D
f f' : ℝ → ℝ
hf : ContinuousOn f D
hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x
hf'₀ : ∀ x ∈ interior D, 0 ≤ f' x
x : ℝ
hx : x ∈ interior D
⊢ 0 ≤ deriv f x

Tactic:
rw [deriv_eqOn isOpen_interior hf' hx]

Goal state after:
D : Set ℝ
hD : Convex ℝ D
f f' : ℝ → ℝ
hf : ContinuousOn f D
hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x
hf'₀ : ∀ x ∈ interior D, 0 ≤ f' x
x : ℝ
hx : x ∈ interior D
⊢ 0 ≤ f' x
Module: Mathlib.Analysis.Calculus.Deriv.MeanValue

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

Tactic:
rw [deriv_eqOn isOpen_interior hf' hx]

Goal state after:
D : Set ℝ
hD : Convex ℝ D
f f' : ℝ → ℝ
hf : ContinuousOn f D
hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x
hf'₀ : ∀ x ∈ interior D, f' x < 0
x : ℝ
hx : x ∈ interior D
⊢ f' x < 0
Module: Mathlib.Analysis.Calculus.Deriv.MeanValue

Goal state before:
pp: D : Set ℝ
hD : Convex ℝ D
f f' : ℝ → ℝ
hf : ContinuousOn f D
hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x
hf'₀ : ∀ x ∈ interior D, f' x ≤ 0
x : ℝ
hx : x ∈ interior D
⊢ deriv f x ≤ 0

Tactic:
rw [deriv_eqOn isOpen_interior hf' hx]

Goal state after:
D : Set ℝ
hD : Convex ℝ D
f f' : ℝ → ℝ
hf : ContinuousOn f D
hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x
hf'₀ : ∀ x ∈ interior D, f' x ≤ 0
x : ℝ
hx : x ∈ interior D
⊢ f' x ≤ 0
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: case h₁.hz
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
c : 𝕜
hc : 1 < ‖c‖
r ε : ℝ
hε : 0 < ε
hr : 0 < r
x : E
L₁ L₂ : E →L[𝕜] F
h₁ : x ∈ A f L₁ r ε
h₂ : x ∈ A f L₂ r ε
y : E
ley : r ≤ 2 * ‖c‖ * ‖y‖
ylt : ‖y‖ < r / 2
⊢ x + y ∈ closedBall x (r / 2)

Tactic:
· simp only [dist_eq_norm, add_sub_cancel_left, mem_closedBall, ylt.le]

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: case h₂.hz
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
c : 𝕜
hc : 1 < ‖c‖
r ε : ℝ
hε : 0 < ε
hr : 0 < r
x : E
L₁ L₂ : E →L[𝕜] F
h₁ : x ∈ A f L₁ r ε
h₂ : x ∈ A f L₂ r ε
y : E
ley : r ≤ 2 * ‖c‖ * ‖y‖
ylt : ‖y‖ < r / 2
⊢ x + y ∈ closedBall x (r / 2)

Tactic:
· simp only [dist_eq_norm, add_sub_cancel_left, mem_closedBall, ylt.le]

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → E →L[𝕜] F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
e p q e' p' q' : ℕ
hp : n e ≤ p
hq : n e ≤ q
hp' : n e' ≤ p'
hq' : n e' ≤ q'
⊢ e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e

Tactic:
he'

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → E →L[𝕜] F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
e p q e' p' q' : ℕ
hp : n e ≤ p
hq : n e ≤ q
hp' : n e' ≤ p'
hq' : n e' ≤ q'
he' : e ≤ e'
⊢ ‖L e p q - L e' p' q'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e