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

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
𝕜' : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
c d : 𝕜 → 𝕜'
c' d' : 𝕜'
hc : HasStrictDerivAt c c' x
hd : HasStrictDerivAt d d' x
hx : d x ≠ 0
⊢ HasStrictDerivAt (fun y ↦ c y / d y) ((c' * d x - c x * d') / d x ^ 2) x

Tactic:
convert hc.fun_mul ((hasStrictDerivAt_inv hx).comp x hd)
  · simp only [div_eq_mul_inv, (· ∘ ·)]
  · simp [field]
    ring

Goal state after:
Module: Mathlib.Analysis.Analytic.IsolatedZeros

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
U : Set 𝕜
hf : AnalyticOnNhd 𝕜 f U
hU : IsPreconnected U
⊢ EqOn f 0 U ∨ ∀ᶠ (x : 𝕜) in codiscreteWithin U, f x ≠ 0

Tactic:
simp only [or_iff_not_imp_right, ne_eq, eventually_iff, mem_codiscreteWithin,
    disjoint_principal_right, not_forall]

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
U : Set 𝕜
hf : AnalyticOnNhd 𝕜 f U
hU : IsPreconnected U
⊢ (∃ x, ∃ (_ : x ∈ U), (U \ {x | ¬f x = 0})ᶜ ∉ 𝓝[≠] x) → EqOn f 0 U
Module: Mathlib.Analysis.Calculus.MeanValue

Goal state before:
pp: E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
𝕜 : Type u_3
G : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : E → G
s : Set E
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hs' : IsPreconnected s
hf : DifferentiableOn 𝕜 f s
hf' : EqOn (fderiv 𝕜 f) 0 s
x y : E
hx : x ∈ s
hy : y ∈ s
a : G
ha : ∀ x ∈ s, f x = a
⊢ a = f y

Tactic:
ha y hy

Goal state after:
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
𝕜 : Type u_3
G : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : E → G
s : Set E
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hs' : IsPreconnected s
hf : DifferentiableOn 𝕜 f s
hf' : EqOn (fderiv 𝕜 f) 0 s
x y : E
hx : x ∈ s
hy : y ∈ s
a : G
ha : ∀ x ∈ s, f x = a
⊢ a = a
Module: Mathlib.Analysis.Calculus.ContDiff.Basic

Goal state before:
pp: case succ
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
g : G ≃L[𝕜] E
f : E → F
hs : UniqueDiffOn 𝕜 s
i : ℕ
IH :
  ∀ {x : G},
    g x ∈ s →
      iteratedFDerivWithin 𝕜 i (f ∘ ⇑g) (⇑g ⁻¹' s) x =
        (iteratedFDerivWithin 𝕜 i f s (g x)).compContinuousLinearMap fun x ↦ ↑g
x : G
hx : g x ∈ s
m : Fin (i + 1) → G
⊢ ((fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 i (f ∘ ⇑g) (⇑g ⁻¹' s)) (⇑g ⁻¹' s) x) (m 0)) (tail m) =
    ((fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 i f s) s (g x)) (g (m 0))) (tail fun i ↦ g (m i))

Tactic:
have : fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 i (f ∘ g) (g ⁻¹' s)) (g ⁻¹' s) x =
        fderivWithin 𝕜
          (ContinuousLinearEquiv.continuousMultilinearMapCongrLeft _ (fun _x : Fin i => g) ∘
            (iteratedFDerivWithin 𝕜 i f s ∘ g)) (g ⁻¹' s) x :=
      fderivWithin_congr' (@IH) hx

Goal state after:
case succ
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
g : G ≃L[𝕜] E
f : E → F
hs : UniqueDiffOn 𝕜 s
i : ℕ
IH :
  ∀ {x : G},
    g x ∈ s →
      iteratedFDerivWithin 𝕜 i (f ∘ ⇑g) (⇑g ⁻¹' s) x =
        (iteratedFDerivWithin 𝕜 i f s (g x)).compContinuousLinearMap fun x ↦ ↑g
x : G
hx : g x ∈ s
m : Fin (i + 1) → G
this :
  fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 i (f ∘ ⇑g) (⇑g ⁻¹' s)) (⇑g ⁻¹' s) x =
    fderivWithin 𝕜 (⇑(continuousMultilinearMapCongrLeft F fun _x ↦ g) ∘ iteratedFDerivWithin 𝕜 i f s ∘ ⇑g) (⇑g ⁻¹' s) x
⊢ ((fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 i (f ∘ ⇑g) (⇑g ⁻¹' s)) (⇑g ⁻¹' s) x) (m 0)) (tail m) =
    ((fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 i f s) s (g x)) (g (m 0))) (tail fun i ↦ g (m i))
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
n : ℕ∞ω
f : E → F → G
g : E → F
t : Set F
hn : n ≠ ∞
x₀ : E
hf : ContDiffWithinAt 𝕜 (n + 1) (uncurry f) (insert x₀ s ×ˢ t) (x₀, g x₀)
hg : ContDiffWithinAt 𝕜 n g s x₀
hgt : t ∈ 𝓝[g '' s] g x₀
⊢ insert x₀ s ×ˢ t ∈ 𝓝[(fun x ↦ (x, g x)) '' s] (x₀, g x₀)

Tactic:
refine nhdsWithin_mono _ ?_ (nhdsWithin_prod self_mem_nhdsWithin hgt)

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
n : ℕ∞ω
f : E → F → G
g : E → F
t : Set F
hn : n ≠ ∞
x₀ : E
hf : ContDiffWithinAt 𝕜 (n + 1) (uncurry f) (insert x₀ s ×ˢ t) (x₀, g x₀)
hg : ContDiffWithinAt 𝕜 n g s x₀
hgt : t ∈ 𝓝[g '' s] g x₀
⊢ (fun x ↦ (x, g x)) '' s ⊆ insert x₀ s ×ˢ (g '' s)
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
x : E
n : ℕ∞ω
y : E
⊢ ContDiffWithinAt 𝕜 n f (s \ {y}) x ↔ ContDiffWithinAt 𝕜 n f s x

Tactic:
rw [← contDiffWithinAt_insert, insert_sdiff_singleton, contDiffWithinAt_insert]

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

Goal state before:
pp: case succ
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
s : Set E
f : E → F
x₀ : E
n : ℕ∞ω
hf : ContDiffWithinAt 𝕜 n f s x₀
hs : UniqueDiffOn 𝕜 s
hx₀s : x₀ ∈ s
i : ℕ
hi : ∀ {m : ℕ∞ω}, m + ↑i ≤ n → ContDiffWithinAt 𝕜 m (iteratedFDerivWithin 𝕜 i f s) s x₀
m : ℕ∞ω
hmn : m + (↑i + 1) ≤ n
⊢ ContDiffWithinAt 𝕜 m (iteratedFDerivWithin 𝕜 (i + 1) f s) s x₀

Tactic:
add_comm _ 1,

Goal state after:
case succ
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
s : Set E
f : E → F
x₀ : E
n : ℕ∞ω
hf : ContDiffWithinAt 𝕜 n f s x₀
hs : UniqueDiffOn 𝕜 s
hx₀s : x₀ ∈ s
i : ℕ
hi : ∀ {m : ℕ∞ω}, m + ↑i ≤ n → ContDiffWithinAt 𝕜 m (iteratedFDerivWithin 𝕜 i f s) s x₀
m : ℕ∞ω
hmn : m + (1 + ↑i) ≤ n
⊢ ContDiffWithinAt 𝕜 m (iteratedFDerivWithin 𝕜 (i + 1) f s) s x₀
Module: Mathlib.Topology.OpenPartialHomeomorph.IsImage

Goal state before:
pp: X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e : OpenPartialHomeomorph X Y
s : Set X
t : Set Y
h : e.IsImage s t
x : X
hx : x ∈ e.source
⊢ ↑e x ∈ closure[inst✝] t ↔ x ∈ closure[inst✝¹] s

Tactic:
simp only [mem_closure_iff_nhdsWithin_neBot, ← h.map_nhdsWithin_eq hx, map_neBot_iff]

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

Goal state before:
pp: case inl
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
x : E
m : ℕ∞ω
hm : m ≤ ω
h' : m = ∞ → ω = ω
h : ContDiffWithinAt 𝕜 ω f s x
⊢ ∃ u, IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u ∧ x ∈ u ∧ ContDiffOn 𝕜 m f (insert x s ∩ u)

Tactic:
obtain ⟨t, ht, p, hp, h'p⟩ := h

Goal state after:
case inl
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
x : E
m : ℕ∞ω
hm : m ≤ ω
h' : m = ∞ → ω = ω
t : Set E
ht : t ∈ 𝓝[insert x s] x
p : E → FormalMultilinearSeries 𝕜 E F
hp : HasFTaylorSeriesUpToOn ω f p t
h'p : ∀ (i : ℕ), AnalyticOn 𝕜 (fun x ↦ p x i) t
⊢ ∃ u, IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u ∧ x ∈ u ∧ ContDiffOn 𝕜 m f (insert x s ∩ u)
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
f : 𝕜 → F
x : 𝕜
⊢ iteratedDerivWithin n f univ x = iteratedDeriv n f x

Tactic:
iteratedDerivWithin,

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
f : 𝕜 → F
x : 𝕜
⊢ ((iteratedFDerivWithin 𝕜 n f univ x) fun x ↦ 1) = iteratedDeriv n f x
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
f : 𝕜 → F
s : Set 𝕜
x : 𝕜
hs : UniqueDiffOn 𝕜 s
h : ContDiffAt 𝕜 (↑n) f x
hx : x ∈ s
⊢ iteratedDerivWithin n f s x = iteratedDeriv n f x

Tactic:
iteratedDerivWithin,

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
f : 𝕜 → F
s : Set 𝕜
x : 𝕜
hs : UniqueDiffOn 𝕜 s
h : ContDiffAt 𝕜 (↑n) f x
hx : x ∈ s
⊢ ((iteratedFDerivWithin 𝕜 n f s x) fun x ↦ 1) = iteratedDeriv n f x
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 : ∀ x ∈ s, ∃ u, IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u ∧ x ∈ u ∧ ContDiffOn 𝕜 n f (s ∩ u)
⊢ ContDiffOn 𝕜 n f s

Tactic:
intro x xs
  rcases h x xs with ⟨u, u_open, xu, hu⟩
  apply (contDiffWithinAt_inter _).1 (hu x ⟨xs, xu⟩)
  exact IsOpen.mem_nhds u_open xu

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
f : 𝕜 → F
s : Set 𝕜
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
x : 𝕜
hx : x ∈ s
⊢ iteratedDerivWithin n f s x = iteratedDeriv n f x

Tactic:
iteratedDerivWithin,

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
f : 𝕜 → F
s : Set 𝕜
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
x : 𝕜
hx : x ∈ s
⊢ ((iteratedFDerivWithin 𝕜 n f s x) fun x ↦ 1) = iteratedDeriv n f x
Module: Mathlib.Analysis.Calculus.ContDiff.Defs

Goal state before:
pp: case h
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
H : ContDiffOn 𝕜 ω f univ
i : ℕ
⊢ AnalyticOn 𝕜 (fun x ↦ ftaylorSeriesWithin 𝕜 f univ x i) univ

Tactic:
exact H.analyticOn.iteratedFDerivWithin uniqueDiffOn_univ _

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
n : ℕ∞
⊢ ContDiffOn 𝕜 (↑n) f univ → ContDiff 𝕜 (↑n) 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
n : ℕ∞
⊢ ContDiff 𝕜 (↑n) f → ContDiffOn 𝕜 (↑n) f univ

Tactic:
· intro H
      use ftaylorSeriesWithin 𝕜 f univ
      rw [← hasFTaylorSeriesUpToOn_univ_iff]
      exact H.ftaylorSeriesWithin 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
n : ℕ∞
⊢ ContDiff 𝕜 (↑n) f → ContDiffOn 𝕜 (↑n) f univ
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: case inl
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✝ : OrderedFinpartition n
c : OrderedFinpartition (n + 1)
hc : range (c.emb 0) ≠ {0}
x : Fin n
j : Fin (c.partSize (c.index 0))
hij : c.emb (c.index 0) j = x.succ
⊢ ∃ m y, (if h : m = c.index 0 then (c.emb m (Fin.cast ⋯ y.succ)).pred ⋯ else (c.emb m (Fin.cast ⋯ y)).pred ⋯) = x

Tactic:
refine ⟨c.index 0, ?_⟩

Goal state after:
case inl
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✝ : OrderedFinpartition n
c : OrderedFinpartition (n + 1)
hc : range (c.emb 0) ≠ {0}
x : Fin n
j : Fin (c.partSize (c.index 0))
hij : c.emb (c.index 0) j = x.succ
⊢ ∃ y,
    (if h : c.index 0 = c.index 0 then (c.emb (c.index 0) (Fin.cast ⋯ y.succ)).pred ⋯
      else (c.emb (c.index 0) (Fin.cast ⋯ y)).pred ⋯) =
      x
Module: Mathlib.Analysis.Normed.Algebra.Exponential

Goal state before:
pp: 𝔸 : Type u_1
inst✝² : NormedRing 𝔸
inst✝¹ : NormedAlgebra ℚ 𝔸
inst✝ : CompleteSpace 𝔸
x a b : 𝔸
h : SemiconjBy x a b
this : Invertible (exp b) := invertibleExp b
⊢ exp (-b) * x * exp a = x

Tactic:
simpa [← invOf_exp, mul_assoc, invOf_mul_eq_iff_eq_mul_left] using! h.exp_right

Goal state after:
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
x : 𝕜
s : Set 𝕜
hx : x ∈ s
h : UniqueDiffOn 𝕜 s
f g : 𝕜 → F
hf : ContDiffWithinAt 𝕜 (↑n) f s x
hg : ContDiffWithinAt 𝕜 (↑n) g s x
⊢ iteratedDerivWithin n (f + g) s x = iteratedDerivWithin n f s x + iteratedDerivWithin n g s x

Tactic:
iteratedDerivWithin,

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
x : 𝕜
s : Set 𝕜
hx : x ∈ s
h : UniqueDiffOn 𝕜 s
f g : 𝕜 → F
hf : ContDiffWithinAt 𝕜 (↑n) f s x
hg : ContDiffWithinAt 𝕜 (↑n) g s x
⊢ ((iteratedFDerivWithin 𝕜 n (f + g) s x) fun x ↦ 1) =
    ((iteratedFDerivWithin 𝕜 n f s x) fun x ↦ 1) + (iteratedFDerivWithin 𝕜 n g s x) fun x ↦ 1
Module: Mathlib.Analysis.Calculus.ContDiff.Operations

Goal state before:
pp: case refine_3
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
s : Set E
ι : Type u_3
inst✝² : Fintype ι
F' : ι → Type u_5
inst✝¹ : (i : ι) → NormedAddCommGroup (F' i)
inst✝ : (i : ι) → NormedSpace 𝕜 (F' i)
φ : (i : ι) → E → F' i
p' : (i : ι) → E → FormalMultilinearSeries 𝕜 E (F' i)
n : ℕ∞ω
pr : (i : ι) → ((i : ι) → F' i) →L[𝕜] F' i := ContinuousLinearMap.proj
L : (m : ℕ) → ((i : ι) → E [×m]→L[𝕜] F' i) ≃ₗᵢ[𝕜] E [×m]→L[𝕜] (i : ι) → F' i :=
  fun m ↦ ContinuousMultilinearMap.piₗᵢ 𝕜 fun _i ↦ E
h : ∀ (i : ι), HasFTaylorSeriesUpToOn n (φ i) (p' i) s
⊢ ∀ (m : ℕ),
    ↑m < n →
      ∀ x ∈ s,
        HasFDerivWithinAt (fun x ↦ ContinuousMultilinearMap.pi fun i ↦ p' i x m)
          (ContinuousMultilinearMap.pi fun i ↦ p' i x m.succ).curryLeft s x

Tactic:
intro m hm x hx
    exact (L m).hasFDerivAt.comp_hasFDerivWithinAt x <|
      hasFDerivWithinAt_pi.2 fun i => (h i).fderivWithin m hm x hx

Goal state after:
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: case succ.hf
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
F : Type u_2
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
n✝ : ℕ
x : 𝕜
s : Set 𝕜
hx : x ∈ s
h : UniqueDiffOn 𝕜 s
𝔸 : Type u_5
inst✝⁴ : NormedRing 𝔸
inst✝³ : NormedAlgebra 𝕜 𝔸
inst✝² : Module 𝔸 F
inst✝¹ : IsBoundedSMul 𝔸 F
inst✝ : IsScalarTower 𝕜 𝔸 F
n : ℕ
IH :
  ∀ {f : 𝕜 → 𝔸} {g : 𝕜 → F},
    ContDiffWithinAt 𝕜 (↑n) f s x →
      ContDiffWithinAt 𝕜 (↑n) g s x →
        iteratedDerivWithin n (f • g) s x =
          ∑ i ∈ Finset.range (n + 1), n.choose i • iteratedDerivWithin i f s x • iteratedDerivWithin (n - i) g s x
f : 𝕜 → 𝔸
g : 𝕜 → F
hf : ContDiffWithinAt 𝕜 (↑(n + 1)) f s x
hg : ContDiffWithinAt 𝕜 (↑(n + 1)) g s x
U : Set 𝕜
hU : U ∈ 𝓝[insert x s] x
H : ∀ y ∈ U, ContDiffWithinAt 𝕜 (↑(n + 1)) f s y ∧ ContDiffWithinAt 𝕜 (↑(n + 1)) g s y
⊢ ContDiffWithinAt 𝕜 (↑n) (derivWithin f s) s x

Tactic:
clear IH H U hU; fun_prop (disch := simp_all)

Goal state after:
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: case succ.hg
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
F : Type u_2
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
n✝ : ℕ
x : 𝕜
s : Set 𝕜
hx : x ∈ s
h : UniqueDiffOn 𝕜 s
𝔸 : Type u_5
inst✝⁴ : NormedRing 𝔸
inst✝³ : NormedAlgebra 𝕜 𝔸
inst✝² : Module 𝔸 F
inst✝¹ : IsBoundedSMul 𝔸 F
inst✝ : IsScalarTower 𝕜 𝔸 F
n : ℕ
IH :
  ∀ {f : 𝕜 → 𝔸} {g : 𝕜 → F},
    ContDiffWithinAt 𝕜 (↑n) f s x →
      ContDiffWithinAt 𝕜 (↑n) g s x →
        iteratedDerivWithin n (f • g) s x =
          ∑ i ∈ Finset.range (n + 1), n.choose i • iteratedDerivWithin i f s x • iteratedDerivWithin (n - i) g s x
f : 𝕜 → 𝔸
g : 𝕜 → F
hf : ContDiffWithinAt 𝕜 (↑(n + 1)) f s x
hg : ContDiffWithinAt 𝕜 (↑(n + 1)) g s x
U : Set 𝕜
hU : U ∈ 𝓝[insert x s] x
H : ∀ y ∈ U, ContDiffWithinAt 𝕜 (↑(n + 1)) f s y ∧ ContDiffWithinAt 𝕜 (↑(n + 1)) g s y
⊢ ContDiffWithinAt 𝕜 (↑n) g s x

Tactic:
clear IH H U hU; fun_prop (disch := simp_all)

Goal state after:
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: case succ.hf
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
F : Type u_2
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
n✝ : ℕ
x : 𝕜
s : Set 𝕜
hx : x ∈ s
h : UniqueDiffOn 𝕜 s
𝔸 : Type u_5
inst✝⁴ : NormedRing 𝔸
inst✝³ : NormedAlgebra 𝕜 𝔸
inst✝² : Module 𝔸 F
inst✝¹ : IsBoundedSMul 𝔸 F
inst✝ : IsScalarTower 𝕜 𝔸 F
n : ℕ
IH :
  ∀ {f : 𝕜 → 𝔸} {g : 𝕜 → F},
    ContDiffWithinAt 𝕜 (↑n) f s x →
      ContDiffWithinAt 𝕜 (↑n) g s x →
        iteratedDerivWithin n (f • g) s x =
          ∑ i ∈ Finset.range (n + 1), n.choose i • iteratedDerivWithin i f s x • iteratedDerivWithin (n - i) g s x
f : 𝕜 → 𝔸
g : 𝕜 → F
hf : ContDiffWithinAt 𝕜 (↑(n + 1)) f s x
hg : ContDiffWithinAt 𝕜 (↑(n + 1)) g s x
U : Set 𝕜
hU : U ∈ 𝓝[insert x s] x
H : ∀ y ∈ U, ContDiffWithinAt 𝕜 (↑(n + 1)) f s y ∧ ContDiffWithinAt 𝕜 (↑(n + 1)) g s y
⊢ ContDiffWithinAt 𝕜 (↑n) f s x

Tactic:
clear IH H U hU; fun_prop (disch := simp_all)

Goal state after:
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: case succ.hg
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
F : Type u_2
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
n✝ : ℕ
x : 𝕜
s : Set 𝕜
hx : x ∈ s
h : UniqueDiffOn 𝕜 s
𝔸 : Type u_5
inst✝⁴ : NormedRing 𝔸
inst✝³ : NormedAlgebra 𝕜 𝔸
inst✝² : Module 𝔸 F
inst✝¹ : IsBoundedSMul 𝔸 F
inst✝ : IsScalarTower 𝕜 𝔸 F
n : ℕ
IH :
  ∀ {f : 𝕜 → 𝔸} {g : 𝕜 → F},
    ContDiffWithinAt 𝕜 (↑n) f s x →
      ContDiffWithinAt 𝕜 (↑n) g s x →
        iteratedDerivWithin n (f • g) s x =
          ∑ i ∈ Finset.range (n + 1), n.choose i • iteratedDerivWithin i f s x • iteratedDerivWithin (n - i) g s x
f : 𝕜 → 𝔸
g : 𝕜 → F
hf : ContDiffWithinAt 𝕜 (↑(n + 1)) f s x
hg : ContDiffWithinAt 𝕜 (↑(n + 1)) g s x
U : Set 𝕜
hU : U ∈ 𝓝[insert x s] x
H : ∀ y ∈ U, ContDiffWithinAt 𝕜 (↑(n + 1)) f s y ∧ ContDiffWithinAt 𝕜 (↑(n + 1)) g s y
⊢ ContDiffWithinAt 𝕜 (↑n) (derivWithin g s) s x

Tactic:
clear IH H U hU; fun_prop (disch := simp_all)

Goal state after:
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: case succ.hf
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
F : Type u_2
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
n✝ : ℕ
x : 𝕜
s : Set 𝕜
hx : x ∈ s
h : UniqueDiffOn 𝕜 s
𝔸 : Type u_5
inst✝⁴ : NormedRing 𝔸
inst✝³ : NormedAlgebra 𝕜 𝔸
inst✝² : Module 𝔸 F
inst✝¹ : IsBoundedSMul 𝔸 F
inst✝ : IsScalarTower 𝕜 𝔸 F
n : ℕ
IH :
  ∀ {f : 𝕜 → 𝔸} {g : 𝕜 → F},
    ContDiffWithinAt 𝕜 (↑n) f s x →
      ContDiffWithinAt 𝕜 (↑n) g s x →
        iteratedDerivWithin n (f • g) s x =
          ∑ i ∈ Finset.range (n + 1), n.choose i • iteratedDerivWithin i f s x • iteratedDerivWithin (n - i) g s x
f : 𝕜 → 𝔸
g : 𝕜 → F
hf : ContDiffWithinAt 𝕜 (↑(n + 1)) f s x
hg : ContDiffWithinAt 𝕜 (↑(n + 1)) g s x
U : Set 𝕜
hU : U ∈ 𝓝[insert x s] x
H : ∀ y ∈ U, ContDiffWithinAt 𝕜 (↑(n + 1)) f s y ∧ ContDiffWithinAt 𝕜 (↑(n + 1)) g s y
⊢ ContDiffWithinAt 𝕜 (↑n) (f • derivWithin g s) s x

Tactic:
clear IH H U hU; fun_prop (disch := simp_all)

Goal state after:
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: case succ.hg
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
F : Type u_2
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
n✝ : ℕ
x : 𝕜
s : Set 𝕜
hx : x ∈ s
h : UniqueDiffOn 𝕜 s
𝔸 : Type u_5
inst✝⁴ : NormedRing 𝔸
inst✝³ : NormedAlgebra 𝕜 𝔸
inst✝² : Module 𝔸 F
inst✝¹ : IsBoundedSMul 𝔸 F
inst✝ : IsScalarTower 𝕜 𝔸 F
n : ℕ
IH :
  ∀ {f : 𝕜 → 𝔸} {g : 𝕜 → F},
    ContDiffWithinAt 𝕜 (↑n) f s x →
      ContDiffWithinAt 𝕜 (↑n) g s x →
        iteratedDerivWithin n (f • g) s x =
          ∑ i ∈ Finset.range (n + 1), n.choose i • iteratedDerivWithin i f s x • iteratedDerivWithin (n - i) g s x
f : 𝕜 → 𝔸
g : 𝕜 → F
hf : ContDiffWithinAt 𝕜 (↑(n + 1)) f s x
hg : ContDiffWithinAt 𝕜 (↑(n + 1)) g s x
U : Set 𝕜
hU : U ∈ 𝓝[insert x s] x
H : ∀ y ∈ U, ContDiffWithinAt 𝕜 (↑(n + 1)) f s y ∧ ContDiffWithinAt 𝕜 (↑(n + 1)) g s y
⊢ ContDiffWithinAt 𝕜 (↑n) (derivWithin f s • g) s x

Tactic:
clear IH H U hU; fun_prop (disch := simp_all)

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 : WithTop ℕ∞
g : F → G
f : E → F
hg : HasFTaylorSeriesUpToOn n g q t
hf : HasFTaylorSeriesUpToOn n f p s
h : MapsTo f s t
m : ℕ
hm : ↑m < n
x : E
hx : x ∈ s
A :
  ∀ (c : OrderedFinpartition m),
    HasFDerivWithinAt (fun x ↦ (q (f x)).compAlongOrderedFinpartition (p x) c)
      (∑ i, ((q (f x)).compAlongOrderedFinpartition (p x) (c.extend i)).curryLeft) s x
B :
  HasFDerivWithinAt (fun x ↦ (q (f x)).taylorComp (p x) m)
    (∑ c, ∑ i, ((q (f x)).compAlongOrderedFinpartition (p x) (c.extend i)).curryLeft) s x
this : ∑ c, ∑ i, (q (f x)).compAlongOrderedFinpartition (p x) (c.extend i) = (q (f x)).taylorComp (p x) (m + 1)
⊢ HasFDerivWithinAt (fun x ↦ (q (f x)).taylorComp (p x) m) ((q (f x)).taylorComp (p x) m.succ).curryLeft s x

Tactic:
rw [← this]
      convert! B
      ext v
      simp only [Nat.succ_eq_add_one, Fintype.sum_option, ContinuousMultilinearMap.curryLeft_apply,
        FormalMultilinearSeries.compAlongOrderedFinpartition_apply, sum_apply, add_apply]

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

Goal state before:

Tactic:
(∫⁻ x : α, g x ∂μ) ≤ (∫⁻ x : α, f x + w x ∂μ) + ε / 2 := gint
      _ = ((∫⁻ x : α, f x ∂μ) + ∫⁻ x : α, w x ∂μ) + ε / 2 := by
        rw [lintegral_add_right _ wmeas.coe_nnreal_ennreal]
      _ ≤ (∫⁻ x : α, f x ∂μ) + ε / 2 + ε / 2 := by grw [wint]
      _ = (∫⁻ x : α, f x ∂μ) + ε := by rw [add_assoc, ENNReal.add_halves]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Log.Deriv

Goal state before:
pp: case hf
n : ℕ∞ω
x : ℝ
hx : x ≠ 0
y : ℝ
hy : 0 < y
⊢ ContDiffAt ℝ n (↑expPartialHomeomorph) (↑expPartialHomeomorph.symm y)

Tactic:
exact analyticAt_rexp.contDiffAt

Goal state after: