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

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NontriviallyNormedField 𝕜'
inst✝⁸ : NormedAlgebra 𝕜 𝕜'
E : Type u_3
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedSpace 𝕜' E
inst✝⁴ : IsScalarTower 𝕜 𝕜' E
F : Type u_4
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedSpace 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
x : E
f : E → F
n✝ : ℕ
s : Set E
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
hn :
  ContDiffWithinAt 𝕜' (↑n) f s x →
    restrictScalars 𝕜 ∘ iteratedFDerivWithin 𝕜' n f s =ᶠ[𝓝[s] x] iteratedFDerivWithin 𝕜 n f s
h : ContDiffWithinAt 𝕜' (↑(n + 1)) f s x
t₀ : ContDiffWithinAt 𝕜' (↑n) f s x
t₁ : ∀ᶠ (y : E) in 𝓝[s] x, ContDiffWithinAt 𝕜' (↑(n + 1)) f s y
a : E
h₁a : restrictScalars 𝕜 ∘ iteratedFDerivWithin 𝕜' n f s =ᶠ[𝓝[s] a] iteratedFDerivWithin 𝕜 n f s
h₃a : ContDiffWithinAt 𝕜' (↑(n + 1)) f s a
h₄a : a ∈ s
m : Fin (n + 1) → E
⊢ ((restrictScalars 𝕜 ∘ iteratedFDerivWithin 𝕜' (n + 1) f s) a) m = (iteratedFDerivWithin 𝕜 (n + 1) f s a) m

Tactic:
simp only [Function.comp_apply, coe_restrictScalars, iteratedFDerivWithin_succ_apply_left]

Goal state after:
𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NontriviallyNormedField 𝕜'
inst✝⁸ : NormedAlgebra 𝕜 𝕜'
E : Type u_3
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedSpace 𝕜' E
inst✝⁴ : IsScalarTower 𝕜 𝕜' E
F : Type u_4
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedSpace 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
x : E
f : E → F
n✝ : ℕ
s : Set E
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
hn :
  ContDiffWithinAt 𝕜' (↑n) f s x →
    restrictScalars 𝕜 ∘ iteratedFDerivWithin 𝕜' n f s =ᶠ[𝓝[s] x] iteratedFDerivWithin 𝕜 n f s
h : ContDiffWithinAt 𝕜' (↑(n + 1)) f s x
t₀ : ContDiffWithinAt 𝕜' (↑n) f s x
t₁ : ∀ᶠ (y : E) in 𝓝[s] x, ContDiffWithinAt 𝕜' (↑(n + 1)) f s y
a : E
h₁a : restrictScalars 𝕜 ∘ iteratedFDerivWithin 𝕜' n f s =ᶠ[𝓝[s] a] iteratedFDerivWithin 𝕜 n f s
h₃a : ContDiffWithinAt 𝕜' (↑(n + 1)) f s a
h₄a : a ∈ s
m : Fin (n + 1) → E
⊢ ((fderivWithin 𝕜' (iteratedFDerivWithin 𝕜' n f s) s a) (m 0)) (Fin.tail m) =
    ((fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n f s) s a) (m 0)) (Fin.tail m)
Module: Mathlib.Topology.MetricSpace.Holder

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : PseudoEMetricSpace X
inst✝ : PseudoEMetricSpace Y
r : ℝ≥0
f : X → Y
C₁ C₂ : ℝ≥0
A : Set X
hf : HolderOnWith C₁ r f A
hC : C₁ ≤ C₂
x : X
hx : x ∈ A
y : X
hy : y ∈ A
⊢ edist (f x) (f y) ≤ ↑C₁ * edist x y ^ ↑r

Tactic:
exact hf x hx y hy

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiffHolder.Pointwise

Goal state before:
pp: E : Type u_1
F : Type u_2
G : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
k : ℕ
α : ↑I
f : E → F
a : E
g : F ≃L[ℝ] G
h : ContDiffPointwiseHolderAt k α (⇑g ∘ f) a
⊢ ContDiffPointwiseHolderAt k α f a

Tactic:
simpa [Function.comp_def] using h.continuousLinearMap_comp (g.symm : G →L[ℝ] F)

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

Goal state before:
pp: case inr
𝕜 : Type u_1
ι : Type u_2
inst✝³ : DecidableEq ι
inst✝² : NontriviallyNormedField 𝕜
E : ι → Type u_3
inst✝¹ : (i : ι) → NormedAddCommGroup (E i)
inst✝ : (i : ι) → NormedSpace 𝕜 (E i)
x : (i : ι) → E i
i : ι
y : E i
j : ι
hij : j ≠ i
⊢ HasFDerivAt (fun x_1 ↦ Function.update x i x_1 j) (Pi.single i (ContinuousLinearMap.id 𝕜 (E i)) j) y

Tactic:
· simpa [hij] using! hasFDerivAt_const _ _

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

Goal state before:
pp: f : ℝ → ℝ
x₀ : ℝ
hf : deriv f x₀ > 0
hx : f x₀ = 0
⊢ ∀ᶠ (x : ℝ) in 𝓝 x₀, sign (f x) = sign (x - x₀)

Tactic:
rw [← nhdsNE_sup_pure x₀, eventually_sup]

Goal state after:
f : ℝ → ℝ
x₀ : ℝ
hf : deriv f x₀ > 0
hx : f x₀ = 0
⊢ (∀ᶠ (x : ℝ) in 𝓝[≠] x₀, sign (f x) = sign (x - x₀)) ∧ ∀ᶠ (x : ℝ) in pure x₀, sign (f x) = sign (x - x₀)
Module: Mathlib.Analysis.Calculus.FDeriv.Symmetric

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s t : Set E
f : E → F
x : E
h : IsSymmSndFDerivWithinAt 𝕜 f t x
hst : t ∈ 𝓝[s] x
hf : ContDiffWithinAt 𝕜 2 f t x
hs : UniqueDiffOn 𝕜 s
ht : UniqueDiffOn 𝕜 t
hx : x ∈ s
⊢ IsSymmSndFDerivWithinAt 𝕜 f s x

Tactic:
intro v

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s t : Set E
f : E → F
x : E
h : IsSymmSndFDerivWithinAt 𝕜 f t x
hst : t ∈ 𝓝[s] x
hf : ContDiffWithinAt 𝕜 2 f t x
hs : UniqueDiffOn 𝕜 s
ht : UniqueDiffOn 𝕜 t
hx : x ∈ s
v : E
⊢ ∀ (w : E), ((fderivWithin 𝕜 (fderivWithin 𝕜 f s) s x) v) w = ((fderivWithin 𝕜 (fderivWithin 𝕜 f s) s x) w) v
Module: Mathlib.Analysis.Calculus.FDeriv.Symmetric

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s t : Set E
f : E → F
x : E
h : IsSymmSndFDerivWithinAt 𝕜 f s x
hst : s =ᶠ[𝓝 x] t
⊢ IsSymmSndFDerivWithinAt 𝕜 f t x

Tactic:
intro v

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s t : Set E
f : E → F
x : E
h : IsSymmSndFDerivWithinAt 𝕜 f s x
hst : s =ᶠ[𝓝 x] t
v : E
⊢ ∀ (w : E), ((fderivWithin 𝕜 (fderivWithin 𝕜 f t) t x) v) w = ((fderivWithin 𝕜 (fderivWithin 𝕜 f t) t x) w) v
Module: Mathlib.Analysis.Calculus.FDeriv.ContinuousMultilinearMap

Goal state before:
pp: case e'_12.e'_6.hl
𝕜 : Type u_1
ι : Type u_2
F : ι → Type u_4
G : ι → Type u_5
H : Type u_6
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : (i : ι) → NormedAddCommGroup (F i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (F i)
inst✝⁵ : (i : ι) → NormedAddCommGroup (G i)
inst✝⁴ : (i : ι) → NormedSpace 𝕜 (G i)
inst✝³ : NormedAddCommGroup H
inst✝² : NormedSpace 𝕜 H
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
fg : ContinuousMultilinearMap 𝕜 G H × ((i : ι) → F i →L[𝕜] G i)
this :
  HasStrictFDerivAt (⇑(compContinuousLinearMapContinuousMultilinear 𝕜 F G H))
    ((compContinuousLinearMapContinuousMultilinear 𝕜 F G H).linearDeriv fg.2) fg.2
e_4✝ : Prod.instAddCommGroup = Prod.normedAddCommGroup.toAddCommGroup
e_5✝ : Prod.instModule ≍ Prod.normedSpace.toModule
e_6✝ : instTopologicalSpaceProd = PseudoMetricSpace.toUniformSpace.toTopologicalSpace
e_8✝ : instAddCommGroup = normedAddCommGroup.toAddCommGroup
e_9✝ : instModule ≍ normedSpace.toModule
e_10✝ : instTopologicalSpace = PseudoMetricSpace.toUniformSpace.toTopologicalSpace
x✝¹ : ContinuousMultilinearMap 𝕜 G H
x✝ : (i : ι) → F i
⊢ (((fg.1.fderivCompContinuousLinearMap fg.2 ∘SL
            ContinuousLinearMap.snd 𝕜 (ContinuousMultilinearMap 𝕜 G H) ((i : ι) → F i →L[𝕜] G i)) ∘SL
          ContinuousLinearMap.inl 𝕜 (ContinuousMultilinearMap 𝕜 G H) ((i : ι) → F i →L[𝕜] G i))
        x✝¹)
      x✝ =
    ((((compContinuousLinearMapContinuousMultilinear 𝕜 F G H).linearDeriv fg.2 ∘SL
                ContinuousLinearMap.snd 𝕜 (ContinuousMultilinearMap 𝕜 G H) ((i : ι) → F i →L[𝕜] G i)).flip
            fg.1 ∘SL
          ContinuousLinearMap.inl 𝕜 (ContinuousMultilinearMap 𝕜 G H) ((i : ι) → F i →L[𝕜] G i))
        x✝¹)
      x✝

Tactic:
simp [fderivCompContinuousLinearMap]

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

Goal state before:
pp: case hr
𝕜 : Type u_1
ι : Type u_2
F : ι → Type u_4
G : ι → Type u_5
H : Type u_6
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : (i : ι) → NormedAddCommGroup (F i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (F i)
inst✝⁵ : (i : ι) → NormedAddCommGroup (G i)
inst✝⁴ : (i : ι) → NormedSpace 𝕜 (G i)
inst✝³ : NormedAddCommGroup H
inst✝² : NormedSpace 𝕜 H
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
fg : ContinuousMultilinearMap 𝕜 G H × ((i : ι) → F i →L[𝕜] G i)
this :
  HasStrictFDerivAt (⇑(compContinuousLinearMapContinuousMultilinear 𝕜 F G H))
    ((compContinuousLinearMapContinuousMultilinear 𝕜 F G H).linearDeriv fg.2) fg.2
e_4✝ : Prod.instAddCommGroup = Prod.normedAddCommGroup.toAddCommGroup
e_5✝ : Prod.instModule ≍ Prod.normedSpace.toModule
e_6✝ : instTopologicalSpaceProd = PseudoMetricSpace.toUniformSpace.toTopologicalSpace
e_8✝ : instAddCommGroup = normedAddCommGroup.toAddCommGroup
e_9✝ : instModule ≍ normedSpace.toModule
e_10✝ : instTopologicalSpace = PseudoMetricSpace.toUniformSpace.toTopologicalSpace
x✝¹ : (i : ι) → F i →L[𝕜] G i
x✝ : (i : ι) → F i
⊢ (((fg.1.fderivCompContinuousLinearMap fg.2 ∘SL
            ContinuousLinearMap.snd 𝕜 (ContinuousMultilinearMap 𝕜 G H) ((i : ι) → F i →L[𝕜] G i)) ∘SL
          ContinuousLinearMap.inr 𝕜 (ContinuousMultilinearMap 𝕜 G H) ((i : ι) → F i →L[𝕜] G i))
        x✝¹)
      x✝ =
    ((((compContinuousLinearMapContinuousMultilinear 𝕜 F G H).linearDeriv fg.2 ∘SL
                ContinuousLinearMap.snd 𝕜 (ContinuousMultilinearMap 𝕜 G H) ((i : ι) → F i →L[𝕜] G i)).flip
            fg.1 ∘SL
          ContinuousLinearMap.inr 𝕜 (ContinuousMultilinearMap 𝕜 G H) ((i : ι) → F i →L[𝕜] G i))
        x✝¹)
      x✝

Tactic:
simp [fderivCompContinuousLinearMap]

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

Goal state before:
pp: case hx
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
g : F → G
f : E → F
n : ℕ
s : Set E
t : Set F
x : E
N : ℕ∞ω
hg : ContDiffOn 𝕜 N g t
hf : ContDiffOn 𝕜 N f s
hn : ↑n ≤ N
ht : UniqueDiffOn 𝕜 t
hs : UniqueDiffOn 𝕜 s
hst : MapsTo f s t
hx : x ∈ s
C D : ℝ
hC : ∀ i ≤ n, ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C
Fu : Type (max uF uG) := ULift.{uG, uF} F
Gu : Type (max uF uG) := ULift.{uF, uG} G
isoF : Fu ≃ₗᵢ[𝕜] F
isoG : Gu ≃ₗᵢ[𝕜] G
fu : E → Fu := ⇑isoF.symm ∘ f
gu : Fu → Gu := ⇑isoG.symm ∘ g ∘ ⇑isoF
tu : Set Fu := ⇑isoF ⁻¹' t
htu : UniqueDiffOn 𝕜 tu
hstu : MapsTo fu s tu
Ffu : isoF (fu x) = f x
hfu : ContDiffOn 𝕜 (↑n) fu s
hgu : ContDiffOn 𝕜 (↑n) gu tu
Nfu : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 i fu s x‖ = ‖iteratedFDerivWithin 𝕜 i f s x‖
hD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i fu s x‖ ≤ D ^ i
i : ℕ
⊢ f x ∈ t

Tactic:
exact hst hx

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

Goal state before:
pp: case intro
𝕜 : Type u_1
ι : Type u_2
E : Type u_3
F : ι → Type u_4
G : ι → Type u_5
H : Type u_6
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : (i : ι) → NormedAddCommGroup (F i)
inst✝⁵ : (i : ι) → NormedSpace 𝕜 (F i)
inst✝⁴ : (i : ι) → NormedAddCommGroup (G i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (G i)
inst✝² : NormedAddCommGroup H
inst✝¹ : NormedSpace 𝕜 H
f : E → ContinuousMultilinearMap 𝕜 G H
g : (i : ι) → E → F i →L[𝕜] G i
s : Set E
x : E
inst✝ : Finite ι
hf : DifferentiableWithinAt 𝕜 f s x
hg : ∀ (i : ι), DifferentiableWithinAt 𝕜 (g i) s x
val✝ : Fintype ι
⊢ DifferentiableWithinAt 𝕜 (fun x ↦ (f x).compContinuousLinearMap fun x_1 ↦ g x_1 x) s x

Tactic:
classical
  exact hf.hasFDerivWithinAt.continuousMultilinearMapCompContinuousLinearMap
    (fun i ↦ (hg i).hasFDerivWithinAt) |>.differentiableWithinAt

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

Goal state before:
pp: case hf
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
V W : E → E
s : Set E
x : E
m n : ℕ∞ω
hV : ContDiffWithinAt 𝕜 n V s x
hW : ContDiffWithinAt 𝕜 n W s x
hs : UniqueDiffOn 𝕜 s
hmn : m + 1 ≤ n
hx : x ∈ s
⊢ ContDiffWithinAt 𝕜 m (fun x ↦ (fderivWithin 𝕜 W s x) (V x)) s x

pp: case hg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
V W : E → E
s : Set E
x : E
m n : ℕ∞ω
hV : ContDiffWithinAt 𝕜 n V s x
hW : ContDiffWithinAt 𝕜 n W s x
hs : UniqueDiffOn 𝕜 s
hmn : m + 1 ≤ n
hx : x ∈ s
⊢ ContDiffWithinAt 𝕜 m (fun x ↦ (fderivWithin 𝕜 V s x) (W x)) s x

Tactic:
· exact ContDiffWithinAt.clm_apply (hW.fderivWithin_right hs hmn hx)
      (hV.of_le (le_trans le_self_add hmn))

Goal state after:
case hg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
V W : E → E
s : Set E
x : E
m n : ℕ∞ω
hV : ContDiffWithinAt 𝕜 n V s x
hW : ContDiffWithinAt 𝕜 n W s x
hs : UniqueDiffOn 𝕜 s
hmn : m + 1 ≤ n
hx : x ∈ s
⊢ ContDiffWithinAt 𝕜 m (fun x ↦ (fderivWithin 𝕜 V s x) (W x)) s x
Module: Mathlib.Analysis.Calculus.LineDeriv.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
F✝ : Type u_2
inst✝³ : NormedAddCommGroup F✝
inst✝² : NormedSpace 𝕜 F✝
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : E → F✝
f' : F✝
s t : Set E
x v : E
h : s =ᶠ[𝓝 x] t
F : 𝕜 → E := fun t ↦ x + t • v
⊢ ContinuousAt F 0

Tactic:
apply Continuous.continuousAt

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
F✝ : Type u_2
inst✝³ : NormedAddCommGroup F✝
inst✝² : NormedSpace 𝕜 F✝
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : E → F✝
f' : F✝
s t : Set E
x v : E
h : s =ᶠ[𝓝 x] t
F : 𝕜 → E := ⋯
⊢ Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] F
Module: Mathlib.Analysis.Calculus.LineDeriv.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
F✝ : Type u_2
inst✝³ : NormedAddCommGroup F✝
inst✝² : NormedSpace 𝕜 F✝
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : E → F✝
s t : Set E
x v : E
h : s =ᶠ[𝓝 x] t
F : 𝕜 → E := fun t ↦ x + t • v
⊢ ContinuousAt F 0

Tactic:
apply Continuous.continuousAt

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
F✝ : Type u_2
inst✝³ : NormedAddCommGroup F✝
inst✝² : NormedSpace 𝕜 F✝
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : E → F✝
s t : Set E
x v : E
h : s =ᶠ[𝓝 x] t
F : 𝕜 → E := ⋯
⊢ Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] F
Module: Mathlib.Analysis.Calculus.LineDeriv.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
F✝ : Type u_2
inst✝³ : NormedAddCommGroup F✝
inst✝² : NormedSpace 𝕜 F✝
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f₀ f₁ : E → F✝
f' : F✝
x v : E
h : f₀ =ᶠ[𝓝 x] f₁
F : 𝕜 → E := fun t ↦ x + t • v
⊢ ContinuousAt F 0

Tactic:
apply Continuous.continuousAt

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
F✝ : Type u_2
inst✝³ : NormedAddCommGroup F✝
inst✝² : NormedSpace 𝕜 F✝
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f₀ f₁ : E → F✝
f' : F✝
x v : E
h : f₀ =ᶠ[𝓝 x] f₁
F : 𝕜 → E := ⋯
⊢ Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] F
Module: Mathlib.Analysis.Calculus.LineDeriv.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type u_2
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type u_3
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : E → F
s : Set E
x v : E
f' : F
h : HasLineDerivWithinAt 𝕜 f f' s x v
c : 𝕜
⊢ HasLineDerivWithinAt 𝕜 f (c • f') s x (c • v)

Tactic:
simp only [HasLineDerivWithinAt] at h ⊢

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type u_2
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type u_3
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f : E → F
s : Set E
x v : E
f' : F
h : HasDerivWithinAt (fun t ↦ f (x + t • v)) f' ((fun t ↦ x + t • v) ⁻¹' s) 0
c : 𝕜
⊢ HasDerivWithinAt (fun t ↦ f (x + t • c • v)) (c • f') ((fun t ↦ x + t • c • v) ⁻¹' s) 0
Module: Mathlib.Analysis.Calculus.FDeriv.Symmetric

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
x : E
n : ℕ∞ω
hf : ContDiffAt 𝕜 n f x
hn : minSmoothness 𝕜 2 ≤ n
h : IsRCLikeNormedField 𝕜
⊢ IsSymmSndFDerivAt 𝕜 f x

Tactic:
intro v

Goal state after:
case pos
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
x : E
n : ℕ∞ω
hf : ContDiffAt 𝕜 n f x
hn : minSmoothness 𝕜 2 ≤ n
h : IsRCLikeNormedField 𝕜
v : E
⊢ ∀ (w : E), ((fderiv 𝕜 (fderiv 𝕜 f) x) v) w = ((fderiv 𝕜 (fderiv 𝕜 f) x) w) v
Module: Mathlib.Analysis.Calculus.FDeriv.Symmetric

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
x : E
n : ℕ∞ω
hf : ContDiffWithinAt 𝕜 n f s x
hn : minSmoothness 𝕜 2 ≤ n
hs : UniqueDiffOn 𝕜 s
hx : x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (interior s)
h'x : x ∈ s
m : ℕ∞ω
hm : minSmoothness 𝕜 ↑2 ≤ m
hmn : m ≤ n
m_ne : m ≠ ∞
u : Set E
u_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u
xu : x ∈ u
hu : ContDiffOn 𝕜 m f (s ∩ u)
h'u : UniqueDiffOn 𝕜 (s ∩ u)
y : ℕ → E
hy : ∀ (n : ℕ), y n ∈ interior s
y_lim : Tendsto y atTop (𝓝 x)
L : ∀ᶠ (k : ℕ) in atTop, y k ∈ u
k : ℕ
hk : y k ∈ u
s_mem : s ∈ 𝓝 (y k)
this : IsSymmSndFDerivAt 𝕜 f (y k)
⊢ IsSymmSndFDerivWithinAt 𝕜 f s (y k)

Tactic:
intro v

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
x : E
n : ℕ∞ω
hf : ContDiffWithinAt 𝕜 n f s x
hn : minSmoothness 𝕜 2 ≤ n
hs : UniqueDiffOn 𝕜 s
hx : x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (interior s)
h'x : x ∈ s
m : ℕ∞ω
hm : minSmoothness 𝕜 ↑2 ≤ m
hmn : m ≤ n
m_ne : m ≠ ∞
u : Set E
u_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u
xu : x ∈ u
hu : ContDiffOn 𝕜 m f (s ∩ u)
h'u : UniqueDiffOn 𝕜 (s ∩ u)
y : ℕ → E
hy : ∀ (n : ℕ), y n ∈ interior s
y_lim : Tendsto y atTop (𝓝 x)
L : ∀ᶠ (k : ℕ) in atTop, y k ∈ u
k : ℕ
hk : y k ∈ u
s_mem : s ∈ 𝓝 (y k)
this : IsSymmSndFDerivAt 𝕜 f (y k)
v : E
⊢ ∀ (w : E), ((fderivWithin 𝕜 (fderivWithin 𝕜 f s) s (y k)) v) w = ((fderivWithin 𝕜 (fderivWithin 𝕜 f s) s (y k)) w) v
Module: Mathlib.Analysis.Calculus.FDeriv.Norm

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
h : DifferentiableAt ℝ (fun x ↦ ‖x‖) x
this : ∀ (t : ℝ), ‖x + t • x‖ = |1 + t| * ‖x‖
⊢ deriv (fun t ↦ |1 + t| * ‖x‖) 0 = ‖x‖

Tactic:
deriv_mul_const

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
h : DifferentiableAt ℝ (fun x ↦ ‖x‖) x
this : ∀ (t : ℝ), ‖x + t • x‖ = |1 + t| * ‖x‖
⊢ deriv (fun t ↦ |1 + t|) 0 * ‖x‖ = ‖x‖

case hc
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
h : DifferentiableAt ℝ (fun x ↦ ‖x‖) x
this : ∀ (t : ℝ), ‖x + t • x‖ = |1 + t| * ‖x‖
⊢ DifferentiableAt ℝ (fun t ↦ |1 + t|) 0
Module: Mathlib.Analysis.Calculus.Gradient.Basic

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
inst✝ : CompleteSpace F
f : F → 𝕜
x : F
⊢ (toDual 𝕜 F) (∇ f x) = fderiv 𝕜 f x

Tactic:
rw [gradient, (toDual 𝕜 F).apply_symm_apply]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
x : E
n : ℕ∞ω
hf : ContDiffWithinAt 𝕜 n f s x
hn : minSmoothness 𝕜 2 ≤ n
hs : UniqueDiffOn 𝕜 s
hx : x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (interior s)
h'x : x ∈ s
m : ℕ∞ω
hm : minSmoothness 𝕜 ↑2 ≤ m
hmn : m ≤ n
m_ne : m ≠ ∞
u : Set E
u_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u
xu : x ∈ u
hu : ContDiffOn 𝕜 m f (s ∩ u)
h'u : UniqueDiffOn 𝕜 (s ∩ u)
y : ℕ → E
hy : ∀ (n : ℕ), y n ∈ interior s
y_lim : Tendsto y atTop (𝓝 x)
L : ∀ᶠ (k : ℕ) in atTop, y k ∈ u
I : ∀ᶠ (k : ℕ) in atTop, IsSymmSndFDerivWithinAt 𝕜 f s (y k)
A : ContinuousOn (fderivWithin 𝕜 (fderivWithin 𝕜 f s) s) (s ∩ u)
B : Tendsto (fun k ↦ fderivWithin 𝕜 (fderivWithin 𝕜 f s) s (y k)) atTop (𝓝 (fderivWithin 𝕜 (fderivWithin 𝕜 f s) s x))
C :
  ∀ (v w : E),
    Tendsto (fun k ↦ ((fderivWithin 𝕜 (fderivWithin 𝕜 f s) s (y k)) v) w) atTop
      (𝓝 (((fderivWithin 𝕜 (fderivWithin 𝕜 f s) s x) v) w))
C' :
  ∀ (v w : E),
    Tendsto (fun k ↦ ((fderivWithin 𝕜 (fderivWithin 𝕜 f s) s (y k)) w) v) atTop
      (𝓝 (((fderivWithin 𝕜 (fderivWithin 𝕜 f s) s x) v) w))
⊢ IsSymmSndFDerivWithinAt 𝕜 f s x

Tactic:
intro v

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
x : E
n : ℕ∞ω
hf : ContDiffWithinAt 𝕜 n f s x
hn : minSmoothness 𝕜 2 ≤ n
hs : UniqueDiffOn 𝕜 s
hx : x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (interior s)
h'x : x ∈ s
m : ℕ∞ω
hm : minSmoothness 𝕜 ↑2 ≤ m
hmn : m ≤ n
m_ne : m ≠ ∞
u : Set E
u_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u
xu : x ∈ u
hu : ContDiffOn 𝕜 m f (s ∩ u)
h'u : UniqueDiffOn 𝕜 (s ∩ u)
y : ℕ → E
hy : ∀ (n : ℕ), y n ∈ interior s
y_lim : Tendsto y atTop (𝓝 x)
L : ∀ᶠ (k : ℕ) in atTop, y k ∈ u
I : ∀ᶠ (k : ℕ) in atTop, IsSymmSndFDerivWithinAt 𝕜 f s (y k)
A : ContinuousOn (fderivWithin 𝕜 (fderivWithin 𝕜 f s) s) (s ∩ u)
B : Tendsto (fun k ↦ fderivWithin 𝕜 (fderivWithin 𝕜 f s) s (y k)) atTop (𝓝 (fderivWithin 𝕜 (fderivWithin 𝕜 f s) s x))
C :
  ∀ (v w : E),
    Tendsto (fun k ↦ ((fderivWithin 𝕜 (fderivWithin 𝕜 f s) s (y k)) v) w) atTop
      (𝓝 (((fderivWithin 𝕜 (fderivWithin 𝕜 f s) s x) v) w))
C' :
  ∀ (v w : E),
    Tendsto (fun k ↦ ((fderivWithin 𝕜 (fderivWithin 𝕜 f s) s (y k)) w) v) atTop
      (𝓝 (((fderivWithin 𝕜 (fderivWithin 𝕜 f s) s x) v) w))
v : E
⊢ ∀ (w : E), ((fderivWithin 𝕜 (fderivWithin 𝕜 f s) s x) v) w = ((fderivWithin 𝕜 (fderivWithin 𝕜 f s) s x) w) v
Module: Mathlib.Analysis.Calculus.ImplicitFunction.ProdDomain

Goal state before:
pp: case codisjoint
𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E₁ : Type u_2
inst✝⁸ : NormedAddCommGroup E₁
inst✝⁷ : NormedSpace 𝕜 E₁
inst✝⁶ : CompleteSpace E₁
E₂ : Type u_3
inst✝⁵ : NormedAddCommGroup E₂
inst✝⁴ : NormedSpace 𝕜 E₂
inst✝³ : CompleteSpace E₂
F : Type u_4
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
u : E₁ × E₂
f : E₁ × E₂ → F
f'u : E₁ × E₂ →L[𝕜] F
dfu : HasStrictFDerivAt f f'u u
if₂u : (f'u ∘SL ContinuousLinearMap.inr 𝕜 E₁ E₂).IsInvertible
⊢ ∀ (z : E₁ × E₂), ∃ x y, x ∈ (↑f'u).ker ∧ y ∈ (↑(ContinuousLinearMap.fst 𝕜 E₁ E₂)).ker ∧ x + y = z

Tactic:
intro v

Goal state after:
case codisjoint
𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E₁ : Type u_2
inst✝⁸ : NormedAddCommGroup E₁
inst✝⁷ : NormedSpace 𝕜 E₁
inst✝⁶ : CompleteSpace E₁
E₂ : Type u_3
inst✝⁵ : NormedAddCommGroup E₂
inst✝⁴ : NormedSpace 𝕜 E₂
inst✝³ : CompleteSpace E₂
F : Type u_4
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
u : E₁ × E₂
f : E₁ × E₂ → F
f'u : E₁ × E₂ →L[𝕜] F
dfu : HasStrictFDerivAt f f'u u
if₂u : (f'u ∘SL ContinuousLinearMap.inr 𝕜 E₁ E₂).IsInvertible
v : E₁ × E₂
⊢ ∃ x y, x ∈ (↑f'u).ker ∧ y ∈ (↑(ContinuousLinearMap.fst 𝕜 E₁ E₂)).ker ∧ x + y = v