mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn

Goal state before:
pp: case succ
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E → F
inst✝ : CompleteSpace E
s : Set E
c : ℝ≥0
f' : E →L[𝕜] F
hf : ApproximatesLinearOn f f' s c
f'symm : f'.NonlinearRightInverse
ε : ℝ
b : E
ε0 : 0 ≤ ε
hε : closedBall b ε ⊆ s
y : F
hy : y ∈ closedBall (f b) (((↑f'symm.nnnorm)⁻¹ - ↑c) * ε)
hc : ↑c < (↑f'symm.nnnorm)⁻¹
If' : 0 < ↑f'symm.nnnorm
Icf' : ↑c * ↑f'symm.nnnorm < 1
Jcf' : 1 - ↑c * ↑f'symm.nnnorm ≠ 0
g : E → E := fun x ↦ x + f'symm.toFun (y - f x)
hg : g = fun x ↦ x + f'symm.toFun (y - f x)
u : ℕ → E := fun n ↦ g^[n] b
hu : u = fun n ↦ g^[n] b
usucc : ∀ (n : ℕ), u (n + 1) = g (u n)
A : ∀ (z : E), dist (g z) z ≤ ↑f'symm.nnnorm * dist (f z) y
B : ∀ z ∈ closedBall b ε, g z ∈ closedBall b ε → dist (f (g z)) y ≤ ↑c * ↑f'symm.nnnorm * dist (f z) y
C :
  ∀ (n : ℕ) (w : E),
    dist w b ≤ ↑f'symm.nnnorm * (1 - (↑c * ↑f'symm.nnnorm) ^ n) / (1 - ↑c * ↑f'symm.nnnorm) * dist (f b) y →
      w ∈ closedBall b ε
n : ℕ
IH :
  dist (f (u n)) y ≤ (↑c * ↑f'symm.nnnorm) ^ n * dist (f b) y ∧
    dist (u n) b ≤ ↑f'symm.nnnorm * (1 - (↑c * ↑f'symm.nnnorm) ^ n) / (1 - ↑c * ↑f'symm.nnnorm) * dist (f b) y
⊢ dist (f (u (n + 1))) y ≤ (↑c * ↑f'symm.nnnorm) ^ (n + 1) * dist (f b) y ∧
    dist (u (n + 1)) b ≤
      ↑f'symm.nnnorm * (1 - (↑c * ↑f'symm.nnnorm) ^ (n + 1)) / (1 - ↑c * ↑f'symm.nnnorm) * dist (f b) y

Tactic:
| succ n IH =>

Goal state after:
case succ
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E → F
inst✝ : CompleteSpace E
s : Set E
c : ℝ≥0
f' : E →L[𝕜] F
hf : ApproximatesLinearOn f f' s c
f'symm : f'.NonlinearRightInverse
ε : ℝ
b : E
ε0 : 0 ≤ ε
hε : closedBall b ε ⊆ s
y : F
hy : y ∈ closedBall (f b) (((↑f'symm.nnnorm)⁻¹ - ↑c) * ε)
hc : ↑c < (↑f'symm.nnnorm)⁻¹
If' : 0 < ↑f'symm.nnnorm
Icf' : ↑c * ↑f'symm.nnnorm < 1
Jcf' : 1 - ↑c * ↑f'symm.nnnorm ≠ 0
g : E → E := fun x ↦ x + f'symm.toFun (y - f x)
hg : g = fun x ↦ x + f'symm.toFun (y - f x)
u : ℕ → E := fun n ↦ g^[n] b
hu : u = fun n ↦ g^[n] b
usucc : ∀ (n : ℕ), u (n + 1) = g (u n)
A : ∀ (z : E), dist (g z) z ≤ ↑f'symm.nnnorm * dist (f z) y
B : ∀ z ∈ closedBall b ε, g z ∈ closedBall b ε → dist (f (g z)) y ≤ ↑c * ↑f'symm.nnnorm * dist (f z) y
C :
  ∀ (n : ℕ) (w : E),
    dist w b ≤ ↑f'symm.nnnorm * (1 - (↑c * ↑f'symm.nnnorm) ^ n) / (1 - ↑c * ↑f'symm.nnnorm) * dist (f b) y →
      w ∈ closedBall b ε
n : ℕ
IH :
  dist (f (u n)) y ≤ (↑c * ↑f'symm.nnnorm) ^ n * dist (f b) y ∧
    dist (u n) b ≤ ↑f'symm.nnnorm * (1 - (↑c * ↑f'symm.nnnorm) ^ n) / (1 - ↑c * ↑f'symm.nnnorm) * dist (f b) y
⊢ dist (f (u (n + 1))) y ≤ (↑c * ↑f'symm.nnnorm) ^ (n + 1) * dist (f b) y ∧
    dist (u (n + 1)) b ≤
      ↑f'symm.nnnorm * (1 - (↑c * ↑f'symm.nnnorm) ^ (n + 1)) / (1 - ↑c * ↑f'symm.nnnorm) * dist (f b) y
Module: Mathlib.AlgebraicTopology.SimplicialSet.NerveAdjunction

Goal state before:
pp: X : Truncated 2
C D : Type u
inst✝¹ : SmallCategory C
inst✝ : SmallCategory D
F : X.HomotopyCategory ⥤ C
y : ((truncation 2).obj (nerve C)).obj (op { obj := ⦋2⦌, property := _proof_14 })
h' :
  ∀ {a b : X.obj (op { obj := ⦋0⦌, property := ⋯ })} (e : Edge a b),
    ComposableArrows.mk₁ (F.map (homMk (Edge.mk' e.edge))) = ComposableArrows.mk₁ (F.map (homMk e))
x₀ x₁ x₂ : X.obj (op { obj := ⦋0⦌, property := Edge._proof_1 })
e₀₁ : Edge x₀ x₁
e₁₂ : Edge x₁ x₂
e₀₂ : Edge x₀ x₂
h : e₀₁.CompStruct e₁₂ e₀₂
h₂ :
  ComposableArrows.mk₁ (F.map (homMk e₀₁)) =
    (ConcreteCategory.hom (((truncation 2).obj (nerve C)).map (δ₂ 2 _proof_12 _proof_15).op)) y
h₀ :
  ComposableArrows.mk₁
      (F.map (homMk (Edge.mk' ((ConcreteCategory.hom (X.map (δ₂ 0 _proof_12 _proof_15).op)) h.simplex)))) =
    (ConcreteCategory.hom (((truncation 2).obj (nerve C)).map (δ₂ 0 _proof_12 _proof_15).op)) y
⊢ ComposableArrows.mk₂ (F.map (homMk e₀₁)) (F.map (homMk e₁₂)) = y

Tactic:
rw [h.d₀, h'] at h₀

Goal state after:
X : Truncated 2
C D : Type u
inst✝¹ : SmallCategory C
inst✝ : SmallCategory D
F : X.HomotopyCategory ⥤ C
y : ((truncation 2).obj (nerve C)).obj (op { obj := ⦋2⦌, property := _proof_14 })
h' :
  ∀ {a b : X.obj (op { obj := ⦋0⦌, property := ⋯ })} (e : Edge a b),
    ComposableArrows.mk₁ (F.map (homMk (Edge.mk' e.edge))) = ComposableArrows.mk₁ (F.map (homMk e))
x₀ x₁ x₂ : X.obj (op { obj := ⦋0⦌, property := Edge._proof_1 })
e₀₁ : Edge x₀ x₁
e₁₂ : Edge x₁ x₂
e₀₂ : Edge x₀ x₂
h : e₀₁.CompStruct e₁₂ e₀₂
h₂ :
  ComposableArrows.mk₁ (F.map (homMk e₀₁)) =
    (ConcreteCategory.hom (((truncation 2).obj (nerve C)).map (δ₂ 2 _proof_12 _proof_15).op)) y
h₀ :
  ComposableArrows.mk₁ (F.map (homMk e₁₂)) =
    (ConcreteCategory.hom (((truncation 2).obj (nerve C)).map (δ₂ 0 _proof_12 _proof_15).op)) y
⊢ ComposableArrows.mk₂ (F.map (homMk e₀₁)) (F.map (homMk e₁₂)) = y
Module: Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E → F
inst✝ : CompleteSpace E
s : Set E
c : ℝ≥0
f' : E →L[𝕜] F
hf : ApproximatesLinearOn f f' s c
f'symm : f'.NonlinearRightInverse
x : E
hs : s ∈ 𝓝 x
hc : Subsingleton F ∨ c < f'symm.nnnorm⁻¹
⊢ map f (𝓝 x) = 𝓝 (f x)

Tactic:
by
  refine
    le_antisymm ((hf.continuousOn x (mem_of_mem_nhds hs)).continuousAt hs) (le_map fun t ht => ?_)
  have : f '' (s ∩ t) ∈ 𝓝 (f x) :=
    (hf.mono_set inter_subset_left).image_mem_nhds f'symm (inter_mem hs ht) hc
  exact mem_of_superset this (image_mono inter_subset_right)

Goal state after:
Module: Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
f' : E ≃L[𝕜] F
s : Set E
c : ℝ≥0
hf : ApproximatesLinearOn f (↑f') s c
hc : Subsingleton E ∨ c < ‖↑f'.symm‖₊⁻¹
⊢ Continuous[instTopologicalSpaceSubtype, PseudoMetricSpace.toUniformSpace.toTopologicalSpace]
    ((f '' s).domRestrict ↑(hf.toPartialEquiv hc).symm)

Tactic:
refine ((hf.antilipschitz hc).to_rightInvOn' ?_ (hf.toPartialEquiv hc).right_inv').continuous

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
f' : E ≃L[𝕜] F
s : Set E
c : ℝ≥0
hf : ApproximatesLinearOn f (↑f') s c
hc : Subsingleton E ∨ c < ‖↑f'.symm‖₊⁻¹
⊢ MapsTo (hf.toPartialEquiv hc).invFun (hf.toPartialEquiv hc).target s
Module: Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
f' : E ≃L[𝕜] F
s : Set E
c : ℝ≥0
hf : ApproximatesLinearOn f (↑f') s c
hc : Subsingleton E ∨ c < ‖↑f'.symm‖₊⁻¹
x : F
hx : x ∈ f '' s
y : F
hy : y ∈ f '' s
⊢ ‖↑(hf.toPartialEquiv hc).symm x - ↑(hf.toPartialEquiv hc).symm y - ↑f'.symm (x - y)‖ ≤
    ↑(‖↑f'.symm‖₊ * (‖↑f'.symm‖₊⁻¹ - c)⁻¹ * c) * ‖x - y‖

Tactic:
by
  set A := hf.toPartialEquiv hc
  have Af : ∀ z, A z = f z := fun z => rfl
  rcases (mem_image _ _ _).1 hx with ⟨x', x's, rfl⟩
  rcases (mem_image _ _ _).1 hy with ⟨y', y's, rfl⟩
  rw [← Af x', ← Af y', A.left_inv x's, A.left_inv y's]
  calc
    ‖x' - y' - f'.symm (A x' - A y')‖ ≤ N * ‖f' (x' - y' - f'.symm (A x' - A y'))‖ :=
      (f' : E →L[𝕜] F).bound_of_antilipschitz f'.antilipschitz _
    _ = N * ‖A y' - A x' - f' (y' - x')‖ := by
      congr 2
      simp only [ContinuousLinearEquiv.apply_symm_apply, map_sub]
      abel
    _ ≤ N * (c * ‖y' - x'‖) := by gcongr; exact hf _ y's _ x's
    _ ≤ N * (c * (((N⁻¹ - c)⁻¹ : ℝ≥0) * ‖A y' - A x'‖)) := by
      gcongr
      rw [← dist_eq_norm, ← dist_eq_norm]
      exact (hf.antilipschitz hc).le_mul_dist ⟨y', y's⟩ ⟨x', x's⟩
    _ = (N * (N⁻¹ - c)⁻¹ * c : ℝ≥0) * ‖A x' - A y'‖ := by
      simp only [norm_sub_rev, NNReal.coe_mul]; ring

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

Goal state before:
pp: case succ
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
z₀ : 𝕜
inst✝¹ : CharZero 𝕜
inst✝ : CompleteSpace E
n : ℕ
IH : ∀ {f : 𝕜 → E}, AnalyticAt 𝕜 f z₀ → (↑n ≤ analyticOrderAt f z₀ ↔ ∀ i < n, iteratedDeriv i f z₀ = 0)
f : 𝕜 → E
hf : AnalyticAt 𝕜 f z₀
⊢ ↑(n + 1) ≤ analyticOrderAt f z₀ ↔ ∀ i < n + 1, iteratedDeriv i f z₀ = 0

Tactic:
| succ n IH =>

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

Goal state before:
pp: case zero
𝕜 : Type u_3
E : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : CompleteSpace E
f : 𝕜 → E
z₀ : 𝕜
hf : AnalyticAt 𝕜 f z₀
inst✝ : CharZero 𝕜
n : ℕ
⊢ ↑n = analyticOrderAt f z₀ → n ≠ 0 → 0 ≤ n → analyticOrderAt (deriv^[0] f) z₀ = ↑(n - 0)

Tactic:
exact fun Hn Hpos Hk ↦ Hn.symm

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

Goal state before:
pp: case succ
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : CharZero 𝕜
inst✝ : CompleteSpace E
z₀ : 𝕜
n : ℕ
IH :
  ∀ {f : 𝕜 → E},
    AnalyticAt 𝕜 f z₀ → (analyticOrderAt f z₀ = ↑n ↔ (∀ k < n, iteratedDeriv k f z₀ = 0) ∧ iteratedDeriv n f z₀ ≠ 0)
f : 𝕜 → E
hf : AnalyticAt 𝕜 f z₀
⊢ analyticOrderAt f z₀ = ↑(n + 1) ↔ (∀ k < n + 1, iteratedDeriv k f z₀ = 0) ∧ iteratedDeriv (n + 1) f z₀ ≠ 0

Tactic:
| succ n IH =>

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f : 𝕜 → E
hf : AnalyticOnNhd 𝕜 f U
hU : IsConnected U
⊢ (∃ u, analyticOrderAt f ↑u ≠ ⊤) ↔ ∀ (u : ↑U), analyticOrderAt f ↑u ≠ ⊤

Tactic:
by
  have : ConnectedSpace U := Subtype.connectedSpace hU
  obtain ⟨v⟩ : Nonempty U := inferInstance
  suffices (∀ (u : U), analyticOrderAt f u ≠ ⊤) ∨ ∀ (u : U), analyticOrderAt f u = ⊤ by tauto
  simpa [Set.eq_empty_iff_forall_notMem, Set.eq_univ_iff_forall] using
      isClopen_iff.1 hf.isClopen_setOfPred_analyticOrderAt_eq_top

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

Goal state before:
pp: x✝ : ℂ
⊢ 1 = 1 x✝

Tactic:
Pi.one_apply

Goal state after:
x✝ : ℂ
⊢ 1 = 1
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv

Goal state before:
pp: x✝ : ℝ
⊢ 1 = 1 x✝

Tactic:
Pi.one_apply

Goal state after:
x✝ : ℝ
⊢ 1 = 1
Module: Mathlib.Analysis.Meromorphic.Basic

Goal state before:
pp: case succ
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace E
f : 𝕜 → E
x : 𝕜
h : MeromorphicAt f x
n : ℕ
IH : MeromorphicAt (deriv^[n] f) x
⊢ MeromorphicAt (deriv^[n + 1] f) x

Tactic:
| succ n IH =>

Goal state after:
Module: Mathlib.RingTheory.Binomial

Goal state before:
pp: case succ
R : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : Pow R ℕ
inst✝ : NatPowAssoc R
n k : ℕ
ih : (descPochhammer ℤ k).smeval ↑n = ↑(n.descFactorial k)
⊢ ↑(n.descFactorial k) * (↑n - ↑k) = ↑(n.descFactorial k) * ↑(n - k)

Tactic:
by_cases! h : n < k

Goal state after:
case pos
R : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : Pow R ℕ
inst✝ : NatPowAssoc R
n k : ℕ
ih : (descPochhammer ℤ k).smeval ↑n = ↑(n.descFactorial k)
h : n < k
⊢ ↑(n.descFactorial k) * (↑n - ↑k) = ↑(n.descFactorial k) * ↑(n - k)

case neg
R : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : Pow R ℕ
inst✝ : NatPowAssoc R
n k : ℕ
ih : (descPochhammer ℤ k).smeval ↑n = ↑(n.descFactorial k)
h : k ≤ n
⊢ ↑(n.descFactorial k) * (↑n - ↑k) = ↑(n.descFactorial k) * ↑(n - k)
Module: Mathlib.RingTheory.Binomial

Goal state before:
pp: case neg
R : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : Pow R ℕ
inst✝ : NatPowAssoc R
n k : ℕ
ih : (descPochhammer ℤ k).smeval ↑n = ↑(n.descFactorial k)
h : k ≤ n
⊢ ↑(n.descFactorial k) * (↑n - ↑k) = ↑(n.descFactorial k) * ↑(n - k)

Tactic:
rw [Nat.cast_sub h]

Goal state after:
Module: Mathlib.RingTheory.Binomial

Goal state before:
pp: R : Type u_1
inst✝² : NonAssocRing R
inst✝¹ : Pow R ℕ
inst✝ : NatPowAssoc R
r : R
k : ℕ
ih : (ascPochhammer ℕ k).smeval (-r) = (↑k).negOnePow • (descPochhammer ℤ k).smeval r
⊢ (X + ↑k).smeval (-r) = -(X + -↑k).smeval r

Tactic:
simp [smeval_natCast, add_comm]

Goal state after:
Module: Mathlib.Analysis.Meromorphic.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f g : 𝕜 → E
U : Set 𝕜
hf : MeromorphicOn f U
h₁ : f =ᶠ[codiscreteWithin U] g
h₂ : EqOn f g Uᶜ
x : 𝕜
hx : x ∈ U
⊢ f =ᶠ[𝓝[≠] x] g

Tactic:
simp_rw [EventuallyEq, Filter.Eventually, mem_codiscreteWithin,
    disjoint_principal_right] at h₁

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f g : 𝕜 → E
U : Set 𝕜
hf : MeromorphicOn f U
h₂ : EqOn f g Uᶜ
x : 𝕜
hx : x ∈ U
h₁ : ∀ x ∈ U, (U \ {x | f x = g x})ᶜ ∈ 𝓝[≠] x
⊢ f =ᶠ[𝓝[≠] x] g
Module: Mathlib.Analysis.Meromorphic.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f g : 𝕜 → E
U : Set 𝕜
hf : MeromorphicOn f U
h₁ : f =ᶠ[codiscreteWithin U] g
h₂ : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
x : 𝕜
hx : x ∈ U
⊢ f =ᶠ[𝓝[≠] x] g

Tactic:
simp_rw [EventuallyEq, Filter.Eventually, mem_codiscreteWithin,
    disjoint_principal_right] at h₁

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f g : 𝕜 → E
U : Set 𝕜
hf : MeromorphicOn f U
h₂ : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
x : 𝕜
hx : x ∈ U
h₁ : ∀ x ∈ U, (U \ {x | f x = g x})ᶜ ∈ 𝓝[≠] x
⊢ f =ᶠ[𝓝[≠] x] g
Module: Mathlib.RingTheory.Binomial

Goal state before:
pp: R : Type u_1
inst✝³ : NonAssocRing R
inst✝² : Pow R ℕ
inst✝¹ : BinomialRing R
inst✝ : NatPowAssoc R
r : R
⊢ choose r 0 = 1

Tactic:
by
  rw [choose_zero_right', npow_zero]

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

Goal state before:
pp: x : ℝ
hx : x ≠ 0
r : ℂ
hr : ¬r + 1 = 0
⊢ HasDerivAt (fun y ↦ ↑y ^ (r + 1) / (r + 1)) (↑x ^ r) x

Tactic:
← Ne

Goal state after:
x : ℝ
hx : x ≠ 0
r : ℂ
hr : r + 1 ≠ 0
⊢ HasDerivAt (fun y ↦ ↑y ^ (r + 1) / (r + 1)) (↑x ^ r) x
Module: Mathlib.Topology.MetricSpace.Cover

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : PseudoEMetricSpace X
inst✝ : PseudoEMetricSpace Y
ε : ℝ≥0
s : Set X
f : X → Y
hf : Isometry f
C : Set X
h : IsCover ε s C
⊢ IsCover ε (f '' s) (f '' C)

Tactic:
simpa using h.image_lipschitz hf.lipschitz

Goal state after:
Module: Mathlib.Analysis.Asymptotics.ExpGrowth

Goal state before:
pp: u : ℕ → ℝ≥0∞
a : EReal
h : ∃ᶠ (n : ℕ) in atTop, u n ≤ (a * ↑n).exp
⊢ expGrowthInf u ≤ a

Tactic:
apply expGrowthInf_le_iff.2 fun c c_u ↦ h.mono fun n hn ↦ hn.trans ?_

Goal state after:
u : ℕ → ℝ≥0∞
a : EReal
h : ∃ᶠ (n : ℕ) in atTop, u n ≤ (a * ↑n).exp
c : EReal
c_u : c > a
n : ℕ
hn : u n ≤ (a * ↑n).exp
⊢ (a * ↑n).exp ≤ (c * ↑n).exp
Module: Mathlib.Analysis.Asymptotics.ExpGrowth

Goal state before:
pp: u v : ℕ → ℝ≥0∞
h : expGrowthSup u ≠ ⊥ ∨ expGrowthInf v ≠ ⊤
h' : expGrowthSup u ≠ ⊤ ∨ expGrowthInf v ≠ ⊥
⊢ expGrowthInf (u * v) ≤ expGrowthSup u + expGrowthInf v

Tactic:
by
  refine (liminf_add_le h h').trans_eq' (liminf_congr (Eventually.of_forall fun n ↦ ?_))
  rw [Pi.add_apply, Pi.mul_apply, ← add_div_of_nonneg_right n.cast_nonneg', log_mul_add]

Goal state after:
Module: Mathlib.MeasureTheory.Covering.VitaliFamily

Goal state before:
pp: X : Type u_1
inst✝ : PseudoMetricSpace X
m0 : MeasurableSpace X
μ : Measure X
v : VitaliFamily μ
x : X
s : Set (Set X)
⊢ s ∈ v.filterAt x ↔ ∃ ε > 0, ∀ t ∈ v.setsAt x, t ⊆ closedBall x ε → t ∈ s

Tactic:
by
  simp only [(v.filterAt_basis_closedBall x).mem_iff, ← and_imp, subset_def, mem_ofPred]

Goal state after:
Module: Mathlib.Analysis.Asymptotics.LinearGrowth

Goal state before:
pp: case pos
u : ℕ → EReal
v : ℕ → ℕ
h : Monotone u
hv₀ : (linearGrowthSup fun n ↦ ↑(v n)) ≠ 0
hv₁ : (linearGrowthSup fun n ↦ ↑(v n)) ≠ ⊤
u_0 : u = ⊥
⊢ linearGrowthInf (u ∘ v) ≤ (linearGrowthSup fun n ↦ ↑(v n)) * linearGrowthInf u

Tactic:
rw [u_0, Pi.bot_comp, linearGrowthInf_bot]

Goal state after:
case pos
u : ℕ → EReal
v : ℕ → ℕ
h : Monotone u
hv₀ : (linearGrowthSup fun n ↦ ↑(v n)) ≠ 0
hv₁ : (linearGrowthSup fun n ↦ ↑(v n)) ≠ ⊤
u_0 : u = ⊥
⊢ ⊥ ≤ (linearGrowthSup fun n ↦ ↑(v n)) * ⊥
Module: Mathlib.MeasureTheory.Function.AEMeasurableOrder

Goal state before:
pp: case pos
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_2
inst✝⁶ : CompleteLinearOrder β
inst✝⁵ : DenselyOrdered β
inst✝⁴ : TopologicalSpace β
inst✝³ : OrderTopology β
inst✝² : SecondCountableTopology β
inst✝¹ : MeasurableSpace β
inst✝ : BorelSpace β
s : Set β
s_count : s.Countable
s_dense : Dense s
f : α → β
h :
  ∀ p ∈ s,
    ∀ q ∈ s, p < q → ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧ {x | f x < p} ⊆ u ∧ {x | q < f x} ⊆ v ∧ μ (u ∩ v) = 0
this : Encodable ↑s
p q : β
H : p ∈ s ∧ q ∈ s ∧ p < q
u v : Set α
hu : MeasurableSet u
hv : MeasurableSet v
h'u : {x | f x < p} ⊆ u
h'v : {x | q < f x} ⊆ v
hμ : μ (u ∩ v) = 0
⊢ ∃ u v,
    MeasurableSet u ∧ MeasurableSet v ∧ {x | f x < p} ⊆ u ∧ {x | q < f x} ⊆ v ∧ (p ∈ s → q ∈ s → p < q → μ (u ∩ v) = 0)

Tactic:
exact ⟨u, v, hu, hv, h'u, h'v, fun _ _ _ => hμ⟩

Goal state after:
Module: Mathlib.Analysis.Asymptotics.ExpGrowth

Goal state before:
pp: case inr
u v : ℕ → ℝ≥0∞
b : ℝ≥0∞
hb : b ≠ ∞
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ b * v n
b_pos : 0 < b
⊢ 0 + expGrowthSup v = expGrowthSup v

pp: u v : ℕ → ℝ≥0∞
b : ℝ≥0∞
hb : b ≠ ∞
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ b * v n
b_pos : 0 < b
⊢ 0 ≠ ⊥ ∨ expGrowthSup v ≠ ⊤

pp: u v : ℕ → ℝ≥0∞
b : ℝ≥0∞
hb : b ≠ ∞
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ b * v n
b_pos : 0 < b
⊢ 0 ≠ ⊤ ∨ expGrowthSup v ≠ ⊥

Tactic:
· exact zero_add (expGrowthSup v)

Goal state after:
u v : ℕ → ℝ≥0∞
b : ℝ≥0∞
hb : b ≠ ∞
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ b * v n
b_pos : 0 < b
⊢ 0 ≠ ⊥ ∨ expGrowthSup v ≠ ⊤

u v : ℕ → ℝ≥0∞
b : ℝ≥0∞
hb : b ≠ ∞
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ b * v n
b_pos : 0 < b
⊢ 0 ≠ ⊤ ∨ expGrowthSup v ≠ ⊥
Module: Mathlib.Analysis.Asymptotics.ExpGrowth

Goal state before:
pp: u : ℕ → ℝ≥0∞
v : ℕ → ℕ
hu : 1 ≤ᶠ[atTop] u
hv : Tendsto v atTop atTop
n : ℕ
h : 1 n ≤ u n
⊢ 0 n ≤ (u n).log

Tactic:
Pi.one_apply

Goal state after:
u : ℕ → ℝ≥0∞
v : ℕ → ℕ
hu : 1 ≤ᶠ[atTop] u
hv : Tendsto v atTop atTop
n : ℕ
h : 1 ≤ u n
⊢ 0 n ≤ (u n).log
Module: Mathlib.MeasureTheory.Covering.Vitali

Goal state before:
pp: case refine_3
α : Type u_1
ι : Type u_2
B : ι → Set α
t : Set ι
δ : ι → ℝ
τ : ℝ
hτ : 1 < τ
δnonneg : ∀ a ∈ t, 0 ≤ δ a
R : ℝ
δle : ∀ a ∈ t, δ a ≤ R
hne : ∀ a ∈ t, (B a).Nonempty
T : Set (Set ι) :=
  {u |
    u ⊆ t ∧
      u.PairwiseDisjoint B ∧ ∀ a ∈ t, ∀ b ∈ u, (B a ∩ B b).Nonempty → ∃ c ∈ u, (B a ∩ B c).Nonempty ∧ δ a ≤ τ * δ c}
u : Set ι
hu : Maximal (fun x ↦ x ∈ T) u
A : Set ι := {a' | a' ∈ t ∧ ∀ c ∈ u, Disjoint (B a') (B c)}
Anonempty : A.Nonempty
m : ℝ := sSup (δ '' A)
bddA : BddAbove (δ '' A)
a' : ι
a'A : a' ∈ A
ha' : m / τ ≤ δ a'
a'_ne_u : a' ∉ u
c : ι
ct : c ∈ t
b : ι
ba'u : b ∈ insert a' u
hcb : (B c ∩ B b).Nonempty
⊢ ∃ c_1 ∈ insert a' u, (B c ∩ B c_1).Nonempty ∧ δ c ≤ τ * δ c_1

Tactic:
by_cases! H : ∃ d ∈ u, (B c ∩ B d).Nonempty

Goal state after:
case pos
α : Type u_1
ι : Type u_2
B : ι → Set α
t : Set ι
δ : ι → ℝ
τ : ℝ
hτ : 1 < τ
δnonneg : ∀ a ∈ t, 0 ≤ δ a
R : ℝ
δle : ∀ a ∈ t, δ a ≤ R
hne : ∀ a ∈ t, (B a).Nonempty
T : Set (Set ι) :=
  {u |
    u ⊆ t ∧
      u.PairwiseDisjoint B ∧ ∀ a ∈ t, ∀ b ∈ u, (B a ∩ B b).Nonempty → ∃ c ∈ u, (B a ∩ B c).Nonempty ∧ δ a ≤ τ * δ c}
u : Set ι
hu : Maximal (fun x ↦ x ∈ T) u
A : Set ι := {a' | a' ∈ t ∧ ∀ c ∈ u, Disjoint (B a') (B c)}
Anonempty : A.Nonempty
m : ℝ := sSup (δ '' A)
bddA : BddAbove (δ '' A)
a' : ι
a'A : a' ∈ A
ha' : m / τ ≤ δ a'
a'_ne_u : a' ∉ u
c : ι
ct : c ∈ t
b : ι
ba'u : b ∈ insert a' u
hcb : (B c ∩ B b).Nonempty
H : ∃ d ∈ u, (B c ∩ B d).Nonempty
⊢ ∃ c_1 ∈ insert a' u, (B c ∩ B c_1).Nonempty ∧ δ c ≤ τ * δ c_1

case neg
α : Type u_1
ι : Type u_2
B : ι → Set α
t : Set ι
δ : ι → ℝ
τ : ℝ
hτ : 1 < τ
δnonneg : ∀ a ∈ t, 0 ≤ δ a
R : ℝ
δle : ∀ a ∈ t, δ a ≤ R
hne : ∀ a ∈ t, (B a).Nonempty
T : Set (Set ι) :=
  {u |
    u ⊆ t ∧
      u.PairwiseDisjoint B ∧ ∀ a ∈ t, ∀ b ∈ u, (B a ∩ B b).Nonempty → ∃ c ∈ u, (B a ∩ B c).Nonempty ∧ δ a ≤ τ * δ c}
u : Set ι
hu : Maximal (fun x ↦ x ∈ T) u
A : Set ι := {a' | a' ∈ t ∧ ∀ c ∈ u, Disjoint (B a') (B c)}
Anonempty : A.Nonempty
m : ℝ := sSup (δ '' A)
bddA : BddAbove (δ '' A)
a' : ι
a'A : a' ∈ A
ha' : m / τ ≤ δ a'
a'_ne_u : a' ∉ u
c : ι
ct : c ∈ t
b : ι
ba'u : b ∈ insert a' u
hcb : (B c ∩ B b).Nonempty
H : ∀ d ∈ u, B c ∩ B d = ∅
⊢ ∃ c_1 ∈ insert a' u, (B c ∩ B c_1).Nonempty ∧ δ c ≤ τ * δ c_1
Module: Mathlib.MeasureTheory.Function.AEMeasurableOrder

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_2
inst✝⁶ : CompleteLinearOrder β
inst✝⁵ : DenselyOrdered β
inst✝⁴ : TopologicalSpace β
inst✝³ : OrderTopology β
inst✝² : SecondCountableTopology β
inst✝¹ : MeasurableSpace β
inst✝ : BorelSpace β
s : Set β
s_count : s.Countable
s_dense : Dense s
f : α → β
h :
  ∀ p ∈ s,
    ∀ q ∈ s, p < q → ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧ {x | f x < p} ⊆ u ∧ {x | q < f x} ⊆ v ∧ μ (u ∩ v) = 0
this : Encodable ↑s
u v : β → β → Set α
huv :
  ∀ (p q : β),
    MeasurableSet (u p q) ∧
      MeasurableSet (v p q) ∧
        {x | f x < p} ⊆ u p q ∧ {x | q < f x} ⊆ v p q ∧ (p ∈ s → q ∈ s → p < q → μ (u p q ∩ v p q) = 0)
u' : β → Set α := fun p ↦ ⋂ q ∈ s ∩ Ioi p, u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β := fun x ↦ ⨅ i, (u' ↑i).piecewise (fun x ↦ ↑i) (fun x ↦ ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ p, ⋃ q, u' ↑p ∩ v ↑p ↑q
μt : μ t ≤ 0
ff' : ∀ᵐ (x : α) ∂μ, f x = f' x
⊢ AEMeasurable f μ

Tactic:
exact ⟨f', f'_meas, ff'⟩

Goal state after:
Module: Mathlib.MeasureTheory.Function.AEEqOfLIntegral

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : TopologicalSpace β
inst✝¹ : OrderTopology β
inst✝ : FirstCountableTopology β
f : α → β
c : β
⊢ (∀ᵐ (x : α) ∂μ, c ≤ f x) ↔ ∀ b < c, μ {x | f x ≤ b} = 0

Tactic:
eventually_const_le_iff_forall_lt_eventually_const_lt,

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
β : Type u_2
inst✝³ : LinearOrder β
inst✝² : TopologicalSpace β
inst✝¹ : OrderTopology β
inst✝ : FirstCountableTopology β
f : α → β
c : β
⊢ (∀ b < c, ∀ᵐ (x : α) ∂μ, b < f x) ↔ ∀ b < c, μ {x | f x ≤ b} = 0
Module: Mathlib.MeasureTheory.Function.AEEqOfLIntegral

Goal state before:
pp: case refine_3
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set (Set α)
f g : α → ℝ≥0∞
h_eq : m0 = MeasurableSpace.generateFrom s
h_inter : IsPiSystem s
basic : ∀ t ∈ s, ∫⁻ (x : α) in t, f x ∂μ = ∫⁻ (x : α) in t, g x ∂μ
h_univ : ∫⁻ (x : α), f x ∂μ = ∫⁻ (x : α), g x ∂μ
hf_int : ∫⁻ (x : α), f x ∂μ ≠ ∞
⊢ ∀ (f_1 : ℕ → Set α),
    Pairwise (Function.onFun Disjoint f_1) →
      (∀ (i : ℕ), MeasurableSet (f_1 i)) →
        (∀ (i : ℕ), ∫⁻ (x : α) in f_1 i, f x ∂μ = ∫⁻ (x : α) in f_1 i, g x ∂μ) →
          ∫⁻ (x : α) in ⋃ i, f_1 i, f x ∂μ = ∫⁻ (x : α) in ⋃ i, f_1 i, g x ∂μ

Tactic:
· intro t htd htm h
    simp_rw [lintegral_iUnion htm htd, h]

Goal state after:
Module: Mathlib.MeasureTheory.Function.AEEqOfLIntegral

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hfi : ∫⁻ (x : α), f x ∂μ ≠ ∞
hfg : μ.withDensity f = μ.withDensity g
⊢ f =ᵐ[μ] g

Tactic:
refine AEMeasurable.ae_eq_of_forall_setLIntegral_eq hf hg hfi ?_ fun s hs _ ↦ ?_
    · rwa [← setLIntegral_univ, ← withDensity_apply g MeasurableSet.univ, ← hfg,
        withDensity_apply f MeasurableSet.univ, setLIntegral_univ]
    · rw [← withDensity_apply f hs, ← withDensity_apply g hs, ← hfg]

Goal state after: