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

Goal state before:
pp: case e'_9
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
m : ℤ
x : 𝕜
h : x ≠ 0 ∨ 0 ≤ m
this✝ : ∀ (m : ℤ), 0 < m → HasStrictDerivAt (fun x ↦ x ^ m) (↑m * x ^ (m - 1)) x
hm : m < 0
hx : x ≠ 0
this : HasStrictDerivAt (fun x ↦ x ^ m) (↑(-m) * x ^ (-m - 1) * -((x ^ m)⁻¹ ^ 2)⁻¹) x
⊢ ↑m * x ^ (m - 1) = -↑m * x ^ (-m - 1) * -(x ^ m * x ^ m)

Tactic:
neg_mul,

Goal state after:
case e'_9
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
m : ℤ
x : 𝕜
h : x ≠ 0 ∨ 0 ≤ m
this✝ : ∀ (m : ℤ), 0 < m → HasStrictDerivAt (fun x ↦ x ^ m) (↑m * x ^ (m - 1)) x
hm : m < 0
hx : x ≠ 0
this : HasStrictDerivAt (fun x ↦ x ^ m) (↑(-m) * x ^ (-m - 1) * -((x ^ m)⁻¹ ^ 2)⁻¹) x
⊢ ↑m * x ^ (m - 1) = -(↑m * x ^ (-m - 1)) * -(x ^ m * x ^ m)
Module: Mathlib.Analysis.Calculus.Deriv.ZPow

Goal state before:
pp: case neg
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
m : ℤ
hm : m < 0
⊢ 0 = ↑m * 0 ^ (m - 1)

Tactic:
zero_zpow _ ((sub_one_lt _).trans hm).ne,

Goal state after:
case neg
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
m : ℤ
hm : m < 0
⊢ 0 = ↑m * 0
Module: Mathlib.Analysis.Calculus.Deriv.ZPow

Goal state before:
pp: 𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
k : ℕ
x : 𝕜
⊢ deriv^[k] Inv.inv x = deriv^[k] (fun x ↦ x ^ (-1)) x

Tactic:
by simp

Goal state after:
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
C : ℝ
s : Set E
x y : E
f' : E → E →L[𝕜] G
φ : E →L[𝕜] G
hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C
hs : Convex ℝ s
xs : x ∈ s
ys : y ∈ s
g : E → G := fun y ↦ f y - φ y
hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x
⊢ ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖

Tactic:
by simp

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

Goal state before:

Tactic:
‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp
    _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1; abel
    _ = ‖g y - g x‖ := by simp [g]
    _ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hg bound hs xs ys

Goal state after:
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
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ∅
hs' : IsPreconnected ∅
hf : DifferentiableOn 𝕜 f ∅
hf' : EqOn (fderiv 𝕜 f) 0 ∅
⊢ ∀ x ∈ ∅, f x = 0

Tactic:
by simp

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

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
c : F
⊢ n + 1 ≠ 0

Tactic:
by simp

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

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
f : E → F
x : E
n : ℕ∞ω
⊢ ∀ y ∈ {x}, f y = f x

Tactic:
by simp

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

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
x : E
n : ℕ∞ω
f : E → F
g : F →L[𝕜] G
hf : ContDiffWithinAt 𝕜 n f s x
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
i : ℕ
hi : ↑i ≤ n
⊢ ↑i = ∞ → n = ω

Tactic:
by simp

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

Goal state before:
pp: 𝕜 : Type u_3
G : Type u_4
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f f' : 𝕜 → G
s : Set 𝕜
x✝ y : 𝕜
C : ℝ
hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
bound : ∀ x ∈ s, ‖f' x‖ ≤ C
hs : Convex ℝ s
xs : x✝ ∈ s
ys : y ∈ s
x : 𝕜
hx : x ∈ s
⊢ ‖toSpanSingleton 𝕜 (f' x)‖ ≤ ‖f' x‖

Tactic:
by simp

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

Goal state before:
pp: 𝕜 : Type u_3
G : Type u_4
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f f' : 𝕜 → G
s : Set 𝕜
C : ℝ≥0
hs : Convex ℝ s
hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C
x : 𝕜
hx : x ∈ s
⊢ ‖toSpanSingleton 𝕜 (f' x)‖₊ ≤ ‖f' x‖₊

Tactic:
by simp

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

Goal state before:
pp: 𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
H : Type u_5
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
f : G → H
f' : G → G →L[𝕜] H
x : G
hder : ∀ᶠ (y : G) in 𝓝 x, HasFDerivAt f (f' y) y
hcont : ContinuousAt f' x
c : ℝ
hc : 0 < c
ε : ℝ
ε0 : ε > 0
hε : ball x ε ⊆ {x | (fun y ↦ HasFDerivAt f (f' y) y) x} ∩ f' ⁻¹' ball (f' x) c
⊢ ∃ ε > 0, ∀ y ∈ ball (x, x) ε, ‖f y.1 - f y.2 - (f' x) (y.1 - y.2)‖ ≤ c * ‖y.1 - y.2‖

Tactic:
refine ⟨ε, ε0, ?_⟩

Goal state after:
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
H : Type u_5
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
f : G → H
f' : G → G →L[𝕜] H
x : G
hder : ∀ᶠ (y : G) in 𝓝 x, HasFDerivAt f (f' y) y
hcont : ContinuousAt f' x
c : ℝ
hc : 0 < c
ε : ℝ
ε0 : ε > 0
hε : ball x ε ⊆ {x | (fun y ↦ HasFDerivAt f (f' y) y) x} ∩ f' ⁻¹' ball (f' x) c
⊢ ∀ y ∈ ball (x, x) ε, ‖f y.1 - f y.2 - (f' x) (y.1 - y.2)‖ ≤ c * ‖y.1 - y.2‖
Module: Mathlib.Analysis.Calculus.MeanValue

Goal state before:
pp: 𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
H : Type u_5
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
f : G → H
f' : G → G →L[𝕜] H
x : G
hder : ∀ᶠ (y : G) in 𝓝 x, HasFDerivAt f (f' y) y
hcont : ContinuousAt f' x
c : ℝ
hc : 0 < c
ε : ℝ
ε0 : ε > 0
hε : ball x ε ⊆ {x | (fun y ↦ HasFDerivAt f (f' y) y) x} ∩ f' ⁻¹' ball (f' x) c
a b : G
h : (a, b) ∈ ball (x, x) ε
⊢ ‖f (a, b).1 - f (a, b).2 - (f' x) ((a, b).1 - (a, b).2)‖ ≤ c * ‖(a, b).1 - (a, b).2‖

Tactic:
rw [← ball_prod_same, prodMk_mem_set_prod_eq] at h

Goal state after:
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
H : Type u_5
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
f : G → H
f' : G → G →L[𝕜] H
x : G
hder : ∀ᶠ (y : G) in 𝓝 x, HasFDerivAt f (f' y) y
hcont : ContinuousAt f' x
c : ℝ
hc : 0 < c
ε : ℝ
ε0 : ε > 0
hε : ball x ε ⊆ {x | (fun y ↦ HasFDerivAt f (f' y) y) x} ∩ f' ⁻¹' ball (f' x) c
a b : G
h : a ∈ ball x ε ∧ b ∈ ball x ε
⊢ ‖f (a, b).1 - f (a, b).2 - (f' x) ((a, b).1 - (a, b).2)‖ ≤ c * ‖(a, b).1 - (a, b).2‖
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
n : ℕ∞ω
s : Set E
t : Set F
g : F → G
f : E → F
x : E
hf : ContDiffWithinAt 𝕜 n f s x
hs : f ⁻¹' t ∈ 𝓝[s] x
hg : ContDiffWithinAt 𝕜 n g t (f x)
⊢ ContDiffWithinAt 𝕜 n (g ∘ f) s x

Tactic:
exact hg.comp_of_preimage_mem_nhdsWithin x hf hs

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
h : ContDiffWithinAt 𝕜 ω f s x
u : Set E
hu : u ∈ 𝓝[insert x s] x
hf : AnalyticOn 𝕜 f u
⊢ x ∈ insert x s

Tactic:
by 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:
by 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:
by simp

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

Goal state before:
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
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
⊢ ContDiffWithinAt 𝕜 (n + 1) f s x

Tactic:
rw [contDiffWithinAt_iff_of_ne_infty h'n]

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
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
⊢ ∃ u ∈ 𝓝[insert x s] x,
    ∃ p, HasFTaylorSeriesUpToOn (n + 1) f p u ∧ (n + 1 = ω → ∀ (i : ℕ), AnalyticOn 𝕜 (fun x ↦ p x i) 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 : 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:
by 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∞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:
by 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:
by simp

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

Goal state before:
pp: case e'_10
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 y : E
⊢ ((p' x 0) (init (snoc 0 y))) y = ((p' x 0) 0) y

Tactic:
rw [init_snoc]

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
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
m n : ℕ∞ω
f : E → F
hf : ContDiffOn 𝕜 n f univ
hmn : m + 1 ≤ n
⊢ ContDiffOn 𝕜 m (fun p ↦ (fderiv 𝕜 f p.1) p.2) univ

Tactic:
← fderivWithin_univ,

Goal state after:
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
m n : ℕ∞ω
f : E → F
hf : ContDiffOn 𝕜 n f univ
hmn : m + 1 ≤ n
⊢ ContDiffOn 𝕜 m (fun p ↦ (fderivWithin 𝕜 f univ p.1) p.2) univ
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:
by 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:
by 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:
by simp

Goal state after:
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
x : X
h : e.IsImage s t
hx : x ∈ e.source
⊢ 𝓝[↑e '' (e.source ∩ s)] ↑e x = 𝓝[t] ↑e x

Tactic:
h.image_eq,

Goal state after:
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
x : X
h : e.IsImage s t
hx : x ∈ e.source
⊢ 𝓝[e.target ∩ t] ↑e x = 𝓝[t] ↑e x
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:
by 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:
by 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:
by 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:
by simp

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

Goal state before:
pp: case succ
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
n : ℕ
IH : ∀ {x : 𝕜}, iteratedDerivWithin n f s x = (fun g ↦ derivWithin g s)^[n] f x
x : 𝕜
⊢ iteratedDerivWithin (n + 1) f s x = (fun g ↦ derivWithin g s)^[n + 1] f x

Tactic:
rw [iteratedDerivWithin_succ, Function.iterate_succ']
    exact derivWithin_congr (fun y hy => IH) IH

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:
by 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:
by simp

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

Goal state before:
pp: 𝕜 : Type u_3
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : CompleteSpace 𝕜
inst✝ : CharZero 𝕜
f : 𝕜 → 𝕜
x : 𝕜
p : FormalMultilinearSeries 𝕜 𝕜 𝕜
r : ENNReal
hpr : HasFPowerSeriesOnBall f p x r
n : ℕ
h_fact_smul : ∀ (n : ℕ), ↑n.factorial * p.coeff n = (iteratedFDeriv 𝕜 n f x) fun x ↦ 1
⊢ iteratedDeriv n f x = ↑n.factorial * p.coeff n

pp: 𝕜 : Type u_3
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : CompleteSpace 𝕜
inst✝ : CharZero 𝕜
f : 𝕜 → 𝕜
x : 𝕜
p : FormalMultilinearSeries 𝕜 𝕜 𝕜
r : ENNReal
hpr : HasFPowerSeriesOnBall f p x r
n : ℕ
h_fact_smul : ∀ (n : ℕ), ↑n.factorial * p.coeff n = (iteratedFDeriv 𝕜 n f x) fun x ↦ 1
⊢ ↑n.factorial ≠ 0

Tactic:
h_fact_smul,

Goal state after:
𝕜 : Type u_3
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : CompleteSpace 𝕜
inst✝ : CharZero 𝕜
f : 𝕜 → 𝕜
x : 𝕜
p : FormalMultilinearSeries 𝕜 𝕜 𝕜
r : ENNReal
hpr : HasFPowerSeriesOnBall f p x r
n : ℕ
h_fact_smul : ∀ (n : ℕ), ↑n.factorial * p.coeff n = (iteratedFDeriv 𝕜 n f x) fun x ↦ 1
⊢ iteratedDeriv n f x = (iteratedFDeriv 𝕜 n f x) fun x ↦ 1

𝕜 : Type u_3
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : CompleteSpace 𝕜
inst✝ : CharZero 𝕜
f : 𝕜 → 𝕜
x : 𝕜
p : FormalMultilinearSeries 𝕜 𝕜 𝕜
r : ENNReal
hpr : HasFPowerSeriesOnBall f p x r
n : ℕ
h_fact_smul : ∀ (n : ℕ), ↑n.factorial * p.coeff n = (iteratedFDeriv 𝕜 n f x) fun x ↦ 1
⊢ ↑n.factorial ≠ 0
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
⊢ 0 < update c.partSize k (c.partSize k + 1) k

Tactic:
by 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 : Fin c.length
⊢ (if h : k = k then (fun i ↦ cases 0 (succ ∘ c.emb k) i) ∘ Fin.cast ⋯ else succ ∘ c.emb k ∘ Fin.cast ⋯) ⟨0, ⋯⟩ = 0

Tactic:
by 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
A : update c.partSize (c.index i) (c.partSize (c.index i) + 1) (c.index i) = c.partSize (c.index i) + 1
⊢ (if h : c.index i = c.index i then (fun i_1 ↦ cases 0 (succ ∘ c.emb (c.index i)) i_1) ∘ Fin.cast ⋯
      else succ ∘ c.emb (c.index i) ∘ Fin.cast ⋯)
      (Fin.cast ⋯ (c.invEmbedding i).succ) =
    i.succ

Tactic:
by 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
n : ℕ∞ω
f : E → F
s : Set E
a : E
h : ContDiffWithinAt 𝕜 n f s a
hs : UniqueDiffOn 𝕜 s
ha : a ∈ s
m : ℕ
hm : ↑m ≤ n
⊢ ↑m = ∞ → n = ω

Tactic:
by simp

Goal state after: