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

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
s : Set 𝕜
hs : UniqueDiffOn 𝕜 s
⊢ DifferentiableOn 𝕜 f s ∧ (∞ = ω → AnalyticOn 𝕜 f s) ∧ ContDiffOn 𝕜 ∞ (derivWithin f s) s ↔
    DifferentiableOn 𝕜 f s ∧ ContDiffOn 𝕜 (∞ + 1) (derivWithin f s) s

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
s : Set 𝕜
hs : IsOpen s
⊢ DifferentiableOn 𝕜 f s ∧ (∞ = ω → AnalyticOn 𝕜 f s) ∧ ContDiffOn 𝕜 ∞ (deriv f) s ↔
    DifferentiableOn 𝕜 f s ∧ ContDiffOn 𝕜 (∞ + 1) (deriv f) s

Tactic:
simp

Goal state after:
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
hn : n ≠ ∞
h'n : n + 1 ≠ ∞
h : ContDiffWithinAt 𝕜 (n + 1) f s x
u : Set E
hu : u ∈ 𝓝[insert x s] x
p : E → FormalMultilinearSeries 𝕜 E F
Hp : HasFTaylorSeriesUpToOn (n + 1) f p u
H'p : n + 1 = ω → ∀ (i : ℕ), AnalyticOn 𝕜 (fun x ↦ p x i) u
y : E
hy : y ∈ u
⊢ n + 1 ≠ 0

Tactic:
simp

Goal state after:
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
hn : n ≠ ∞
h'n : n + 1 ≠ ∞
h : ContDiffWithinAt 𝕜 (n + 1) f s x
u : Set E
hu : u ∈ 𝓝[insert x s] x
p : E → FormalMultilinearSeries 𝕜 E F
Hp : HasFTaylorSeriesUpToOn (n + 1) f p u
H'p : n + 1 = ω → ∀ (i : ℕ), AnalyticOn 𝕜 (fun x ↦ p x i) u
⊢ x ∈ insert x s

Tactic:
simp

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

Goal state before:
pp: case mp.refine_2.refine_2
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
hn : n ≠ ∞
h'n : n + 1 ≠ ∞
h : ContDiffWithinAt 𝕜 (n + 1) f s x
u : Set E
hu : u ∈ 𝓝[insert x s] x
p : E → FormalMultilinearSeries 𝕜 E F
Hp : HasFTaylorSeriesUpToOn (n + 1) f p u
H'p : n + 1 = ω → ∀ (i : ℕ), AnalyticOn 𝕜 (fun x ↦ p x i) u
⊢ HasFTaylorSeriesUpToOn n (fun y ↦ (continuousMultilinearCurryFin1 𝕜 E F) (p y 1)) (fun y ↦ (p y).shift) u ∧
    (n = ω → ∀ (i : ℕ), AnalyticOn 𝕜 (fun x ↦ (fun y ↦ (p y).shift) x i) u)

Tactic:
hasFTaylorSeriesUpToOn_succ_iff_right

Goal state after:
case mp.refine_2.refine_2
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
hn : n ≠ ∞
h'n : n + 1 ≠ ∞
h : ContDiffWithinAt 𝕜 (n + 1) f s x
u : Set E
hu : u ∈ 𝓝[insert x s] x
p : E → FormalMultilinearSeries 𝕜 E F
Hp :
  (∀ x ∈ u, (p x 0).curry0 = f x) ∧
    (∀ x ∈ u, HasFDerivWithinAt (fun y ↦ p y 0) (p x 1).curryLeft u x) ∧
      HasFTaylorSeriesUpToOn n (fun x ↦ (continuousMultilinearCurryFin1 𝕜 E F) (p x 1)) (fun x ↦ (p x).shift) u
H'p : n + 1 = ω → ∀ (i : ℕ), AnalyticOn 𝕜 (fun x ↦ p x i) u
⊢ HasFTaylorSeriesUpToOn n (fun y ↦ (continuousMultilinearCurryFin1 𝕜 E F) (p y 1)) (fun y ↦ (p y).shift) u ∧
    (n = ω → ∀ (i : ℕ), AnalyticOn 𝕜 (fun x ↦ (fun y ↦ (p y).shift) x i) u)
Module: Mathlib.Analysis.Calculus.ContDiff.Deriv

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
⊢ Differentiable 𝕜 f ∧ (0 = ω → AnalyticOn 𝕜 f univ) ∧ ContDiff 𝕜 0 (deriv f) ↔
    Differentiable 𝕜 f ∧ Continuous (deriv f)

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
⊢ Differentiable 𝕜 f ∧ (∞ = ω → AnalyticOn 𝕜 f univ) ∧ ContDiff 𝕜 ∞ (deriv f) ↔
    Differentiable 𝕜 f ∧ ContDiff 𝕜 (∞ + 1) (deriv f)

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
m n : ℕ∞ω
f : 𝕜 → F
x : 𝕜
H : ContDiffAt 𝕜 n f x
hmn : m + 1 ≤ n
⊢ UniqueDiffOn 𝕜 univ

Tactic:
simp

Goal state after:
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
⊢ ↑k ≠ ∞

Tactic:
simp

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

Goal state before:
pp: case mpr.refine_2
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
hn : n ≠ ∞
h'n : n + 1 ≠ ∞
u : Set E
hu : u ∈ 𝓝[insert x s] x
hf : n = ω → AnalyticOn 𝕜 f u
f' : E → E →L[𝕜] F
f'_eq_deriv : ∀ x ∈ u, HasFDerivWithinAt f (f' x) u x
Hf' : ContDiffWithinAt 𝕜 n f' u x
v : Set E
hv : v ∈ 𝓝[insert x u] x
p' : E → FormalMultilinearSeries 𝕜 E (E →L[𝕜] F)
Hp' : HasFTaylorSeriesUpToOn n f' p' v
p'_an : n = ω → ∀ (i : ℕ), AnalyticOn 𝕜 (fun x ↦ p' x i) v
⊢ HasFTaylorSeriesUpToOn (n + 1) f (fun x ↦ (p' x).unshift (f x)) (v ∩ u)

Tactic:
hasFTaylorSeriesUpToOn_succ_iff_right

Goal state after:
case mpr.refine_2
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
hn : n ≠ ∞
h'n : n + 1 ≠ ∞
u : Set E
hu : u ∈ 𝓝[insert x s] x
hf : n = ω → AnalyticOn 𝕜 f u
f' : E → E →L[𝕜] F
f'_eq_deriv : ∀ x ∈ u, HasFDerivWithinAt f (f' x) u x
Hf' : ContDiffWithinAt 𝕜 n f' u x
v : Set E
hv : v ∈ 𝓝[insert x u] x
p' : E → FormalMultilinearSeries 𝕜 E (E →L[𝕜] F)
Hp' : HasFTaylorSeriesUpToOn n f' p' v
p'_an : n = ω → ∀ (i : ℕ), AnalyticOn 𝕜 (fun x ↦ p' x i) v
⊢ (∀ x ∈ v ∩ u, ((p' x).unshift (f x) 0).curry0 = f x) ∧
    (∀ x ∈ v ∩ u, HasFDerivWithinAt (fun y ↦ (p' y).unshift (f y) 0) ((p' x).unshift (f x) 1).curryLeft (v ∩ u) x) ∧
      HasFTaylorSeriesUpToOn n (fun x ↦ (continuousMultilinearCurryFin1 𝕜 E F) ((p' x).unshift (f x) 1))
        (fun x ↦ ((p' x).unshift (f x)).shift) (v ∩ u)
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∞ω
f : E → F → G
g : E → F
t : Set F
ht : ∀ᶠ (x : E) in 𝓝[insert x₀ s] x₀, UniqueDiffWithinAt 𝕜 t (g x)
hgt : t ∈ 𝓝[g '' s] g x₀
hg : ContDiffWithinAt 𝕜 ω g s x₀
this : ∀ (k : ℕ), ↑k ≤ ω → ContDiffWithinAt 𝕜 (↑k) (fun x ↦ fderivWithin 𝕜 (f x) t (g x)) s x₀
hf : ContDiffWithinAt 𝕜 ω (uncurry f) (insert x₀ s ×ˢ t) (x₀, g x₀)
hmn : ω + 1 ≤ ω
⊢ ω ≠ ∞

Tactic:
simp

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

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
g : E → F
f : F → G
h : E → G
f' : F →L[𝕜] G
h' : E →L[𝕜] G
f'symm : G →L[𝕜] F
a : E
s : Set E
t : Set F
hst : Tendsto g (𝓝[s] a) (𝓝[t] g a)
hf : HasFDerivWithinAt f f' t (g a)
hh : HasFDerivWithinAt h h' s a
hcomp : f ∘ g =ᶠ[𝓝[s] a] h
hf'symm : Function.LeftInverse ⇑f'symm ⇑f'
ha : a ∈ s
⊢ Tendsto g (pure a) (pure (g a))

Tactic:
simp

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

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
g : E → F
f : F → G
h : E → G
f' : F →L[𝕜] G
h' : E →L[𝕜] G
f'symm : G →L[𝕜] F
a : E
hgc : ContinuousAt g a
hf : HasFDerivAt f f' (g a)
hh : HasFDerivAt h h' a
hcomp : f ∘ g =ᶠ[𝓝 a] h
hf'symm : Function.LeftInverse ⇑f'symm ⇑f'
⊢ Tendsto g (pure a) (pure (g a))

Tactic:
simp

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
m : Fin n
⊢ ∃ m_1, m ∈ range fun x ↦ m_1

Tactic:
simp

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

Goal state before:
pp: case ho
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
g : E → F
f : F → G
h : E → G
g' : E →L[𝕜] F
f' : F →L[𝕜] G
lE : Filter (E × E)
lF : Filter (F × F)
hg : Tendsto (Prod.map g g) lE lF
hf : HasFDerivAtFilter f f' lF
hf' : Topology.IsEmbedding ⇑f'
hh : HasFDerivAtFilter h (f' ∘SL g') lE
hcomp : Prod.map (f ∘ g) (f ∘ g) =ᶠ[lE] Prod.map h h
⊢ ∀ (x : E × E),
    (⇑f' ∘ fun x ↦
          match x with
          | (x, y) => g x - g y - g' (x - y))
        x =
      match x with
      | (x, y) => f' (g x - g y) - (f' ∘SL g') (x - y)

Tactic:
simp

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

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
g : E → F
f : F → G
h : E → G
g' : E →L[𝕜] F
f' : F →L[𝕜] G
a : E
s : Set E
t : Set F
hg : Tendsto g (𝓝[s] a) (𝓝[t] g a)
hf : HasFDerivWithinAt f f' t (g a)
hf' : Topology.IsEmbedding ⇑f'
hh : HasFDerivWithinAt h (f' ∘SL g') s a
hcomp : f ∘ g =ᶠ[𝓝[s] a] h
ha : a ∈ s
⊢ Tendsto g (pure a) (pure (g a))

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.OpenPartialHomeomorph.IsImage

Goal state before:
pp: case h
X : Type u_1
inst✝ : TopologicalSpace X
x✝ : X
⊢ ↑(ofSet univ ⋯) x✝ = ↑(OpenPartialHomeomorph.refl X) x✝

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.OpenPartialHomeomorph.IsImage

Goal state before:
pp: case hinv
X : Type u_1
inst✝ : TopologicalSpace X
x✝ : X
⊢ ↑(ofSet univ ⋯).symm x✝ = ↑(OpenPartialHomeomorph.refl X).symm x✝

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.OpenPartialHomeomorph.IsImage

Goal state before:
pp: case hs
X : Type u_1
inst✝ : TopologicalSpace X
x✝ : X
⊢ x✝ ∈ (ofSet univ ⋯).source ↔ x✝ ∈ (OpenPartialHomeomorph.refl X).source

Tactic:
simp

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

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
g : E → F
f : F → G
h : E → G
g' : E →L[𝕜] F
f' : F →L[𝕜] G
a : E
hg : ContinuousAt g a
hf : HasFDerivAt f f' (g a)
hf' : Topology.IsEmbedding ⇑f'
hh : HasFDerivAt h (f' ∘SL g') a
hcomp : f ∘ g =ᶠ[𝓝 a] h
⊢ Tendsto g (pure a) (pure (g a))

Tactic:
simp

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 : OrderedFinpartition n
x✝ : (i : Fin c.length) × Fin (c.partSize i)
⊢ (fun p ↦ c.emb p.fst p.snd) ((fun i ↦ ⟨c.index i, c.invEmbedding i⟩) ((fun p ↦ c.emb p.fst p.snd) x✝)) =
    (fun p ↦ c.emb p.fst p.snd) x✝

Tactic:
simp

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 : OrderedFinpartition n
x✝ : Fin n
⊢ (fun p ↦ c.emb p.fst p.snd) ((fun i ↦ ⟨c.index i, c.invEmbedding i⟩) x✝) = x✝

Tactic:
simp

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

Goal state before:
pp: n : ℕ
c : OrderedFinpartition (n + 1)
hc : range (c.emb 0) ≠ {0}
⊢ c.partSize (c.index 0) = Nat.card (Fin (c.partSize (c.index 0)))

Tactic:
simp

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
⊢ 0 < cons 1 c.partSize 0

Tactic:
simp

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
⊢ Subsingleton (Fin (cons 1 c.partSize 0))

Tactic:
simp

Goal state after:
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
⊢ Subsingleton (Fin 1)
Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno

Goal state before:
pp: case succ.zero
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
j : Fin c.length
a✝ :
  0 < j.castSucc →
    (fun m ↦
          cases (motive := fun x ↦ Fin (cons 1 c.partSize x) → Fin (n + 1)) (fun x ↦ 0) (fun m ↦ succ ∘ c.emb m) m
            ⟨cons 1 c.partSize m - 1, ⋯⟩)
        0 <
      (fun m ↦
          cases (motive := fun x ↦ Fin (cons 1 c.partSize x) → Fin (n + 1)) (fun x ↦ 0) (fun m ↦ succ ∘ c.emb m) m
            ⟨cons 1 c.partSize m - 1, ⋯⟩)
        j.castSucc
hij : 0 < j.succ
⊢ (fun m ↦
        cases (motive := fun x ↦ Fin (cons 1 c.partSize x) → Fin (n + 1)) (fun x ↦ 0) (fun m ↦ succ ∘ c.emb m) m
          ⟨cons 1 c.partSize m - 1, ⋯⟩)
      0 <
    (fun m ↦
        cases (motive := fun x ↦ Fin (cons 1 c.partSize x) → Fin (n + 1)) (fun x ↦ 0) (fun m ↦ succ ∘ c.emb m) m
          ⟨cons 1 c.partSize m - 1, ⋯⟩)
      j.succ

Tactic:
simp

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

Goal state before:
pp: case refine_1
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
hn : n ≠ ∞
h : ContDiffWithinAt 𝕜 n f s x
⊢ ∃ u ∈ 𝓝[insert x s] x, ContDiffOn 𝕜 n f u

Tactic:
rcases h.contDiffOn le_rfl (by simp [hn]) with ⟨u, hu, h'u⟩

Goal state after:
case refine_1
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
hn : n ≠ ∞
h : ContDiffWithinAt 𝕜 n f s x
u : Set E
hu : u ∈ 𝓝[insert x s] x
h'u : u ⊆ insert x s ∧ ContDiffOn 𝕜 n f u
⊢ ∃ u ∈ 𝓝[insert x s] x, ContDiffOn 𝕜 n f 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
s : Set 𝕜
x : 𝕜
m : Fin n → 𝕜
⊢ (iteratedFDerivWithin 𝕜 n f s x) m = (iteratedFDerivWithin 𝕜 n f s x) fun i ↦ m i • 1

Tactic:
simp

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
f : 𝕜 → F
s : Set 𝕜
n : ℕ
hs : UniqueDiffOn 𝕜 s
⊢ ((∀ (m : ℕ), ↑m ≤ ↑n → ContinuousOn (iteratedDerivWithin m f s) s) ∧
      ∀ (m : ℕ), ↑m < ↑n → DifferentiableOn 𝕜 (iteratedDerivWithin m f s) s) ↔
    (∀ m ≤ n, ContinuousOn (iteratedDerivWithin m f s) s) ∧ ∀ m < n, DifferentiableOn 𝕜 (iteratedDerivWithin m f s) s

Tactic:
simp

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

Goal state before:
pp: case pos
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
hxs : AccPt x (𝓟 s)
⊢ ((ContinuousMultilinearMap.mkPiRing 𝕜 (Fin n) ((fderivWithin 𝕜 (iteratedDerivWithin n f s) s x) 1)) fun x ↦ 1) =
    (fderivWithin 𝕜 (iteratedDerivWithin n f s) s x) 1

Tactic:
simp

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
⊢ cases (motive := fun x ↦ Fin (cons 1 c.partSize x) → Fin (n + 1)) (fun x ↦ 0) (fun m ↦ succ ∘ c.emb m) 0 ⟨0, ⋯⟩ = 0

Tactic:
simp

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

Goal state before:
pp: case zero
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
⊢ iteratedDerivWithin 0 f s x = (fun g ↦ derivWithin g s)^[0] f x

Tactic:
simp

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
i : Fin n
⊢ c.partSize (c.index i) = cons 1 c.partSize (c.index i).succ

Tactic:
simp

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
x : 𝕜
m : Fin n → 𝕜
⊢ (iteratedFDeriv 𝕜 n f x) m = (iteratedFDeriv 𝕜 n f x) fun i ↦ m i • 1

Tactic:
simp

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
i : Fin n
⊢ cases (motive := fun x ↦ Fin (cons 1 c.partSize x) → Fin (n + 1)) (fun x ↦ 0) (fun m ↦ succ ∘ c.emb m)
      (c.index i).succ (Fin.cast ⋯ (c.invEmbedding i)) =
    i.succ

Tactic:
simp

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

Goal state before:
pp: n : ℕ
c : OrderedFinpartition n
⊢ Nonempty (Fin (cons 1 c.partSize 0))

Tactic:
simp

Goal state after:
n : ℕ
c : OrderedFinpartition n
⊢ Nonempty (Fin 1)
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
f : 𝕜 → F
n : ℕ
⊢ ((∀ (m : ℕ),
        ↑m ≤ ↑n →
          Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace,
              PseudoMetricSpace.toUniformSpace.toTopologicalSpace]
            (iteratedDeriv m f)) ∧
      ∀ (m : ℕ), ↑m < ↑n → Differentiable 𝕜 (iteratedDeriv m f)) ↔
    (∀ m ≤ n,
        Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace,
            PseudoMetricSpace.toUniformSpace.toTopologicalSpace]
          (iteratedDeriv m f)) ∧
      ∀ m < n, Differentiable 𝕜 (iteratedDeriv m f)

Tactic:
simp

Goal state after:
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✝ c : OrderedFinpartition n
m : Fin c.length
⊢ 0 < update c.partSize m (c.partSize m + 1) m

Tactic:
simp

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 m : Fin c.length
h : m = k
⊢ update c.partSize k (c.partSize k + 1) k = c.partSize k + 1

Tactic:
simp

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

Goal state before:
pp: case zero
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
x : 𝕜
c : F
⊢ iteratedDeriv 0 (fun x ↦ c) x = if 0 = 0 then c else 0

Tactic:
simp

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

Goal state before:
pp: case zero
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set 𝕜
x : 𝕜
c : F
⊢ iteratedDerivWithin 0 (fun x ↦ c) s x = if 0 = 0 then c else 0

Tactic:
simp

Goal state after: