mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.AlgebraicTopology.SimplicialSet.Homotopy

Goal state before:
pp: X Y : SSet
f g : X ⟶ Y
H : Homotopy f g
n : ℕ
x : X _⦋n⦌
⊢ (ConcreteCategory.hom
          ((↾fun x ↦
              (ConcreteCategory.hom ((yonedaEquiv.symm x ▷ Δ[1] ≫ H.h).app (op ⦋n + 1⦌)))
                ↑(prodStdSimplex.nonDegenerateEquiv₁ 0)) ≫
            δ Y 0)).toFun
      x =
    (ConcreteCategory.hom (g.app (op ⦋n⦌))).toFun x

Tactic:
simp only [TypeCat.Fun.toFun_apply, types_comp_apply, TypeCat.hom_ofHom, TypeCat.Fun.coe_mk,
      ← SSet.δ_naturality_apply, ← H.h₁]

Goal state after:
X Y : SSet
f g : X ⟶ Y
H : Homotopy f g
n : ℕ
x : X _⦋n⦌
⊢ (ConcreteCategory.hom ((yonedaEquiv.symm x ▷ Δ[1] ≫ H.h).app (op ⦋n⦌)))
      ((ConcreteCategory.hom (δ (Δ[n] ⊗ Δ[1]) 0)) ↑(prodStdSimplex.nonDegenerateEquiv₁ 0)) =
    (ConcreteCategory.hom ((ι₁ ≫ H.h).app (op ⦋n⦌))) x
Module: Mathlib.AlgebraicTopology.SimplicialSet.KanComplex.MulStruct

Goal state before:
pp: X : SSet
n : ℕ
x : X _⦋0⦌
f : X.PtSimplex n x
j : Fin (n + 2)
i : Fin n
hj : j < i.succ.castSucc
⊢ stdSimplex.δ j ≫ stdSimplex.σ i.succ ≫ f.map = const x

Tactic:
obtain ⟨j, rfl⟩ := j.eq_castSucc_of_ne_last (by grind)

Goal state after:
X : SSet
n : ℕ
x : X _⦋0⦌
f : X.PtSimplex n x
i : Fin n
j : Fin (n + 1)
hj : j.castSucc < i.succ.castSucc
⊢ stdSimplex.δ j.castSucc ≫ stdSimplex.σ i.succ ≫ f.map = const x
Module: Mathlib.SetTheory.Cardinal.CountableCover

Goal state before:
pp: case a
α : Type u
ι : Type v
a : Cardinal.{u}
inst✝¹ : Countable ι
f : ι → Set α
l : Filter ι
inst✝ : l.NeBot
ht : ∀ (x : α), ∀ᶠ (i : ι) in l, x ∈ f i
h'f : ∀ (i : ι), #↑(f i) = a
⊢ #α ≤ a

pp: case a
α : Type u
ι : Type v
a : Cardinal.{u}
inst✝¹ : Countable ι
f : ι → Set α
l : Filter ι
inst✝ : l.NeBot
ht : ∀ (x : α), ∀ᶠ (i : ι) in l, x ∈ f i
h'f : ∀ (i : ι), #↑(f i) = a
⊢ a ≤ #α

Tactic:
· apply mk_le_of_countable_eventually_mem ht (fun i ↦ (h'f i).le)

Goal state after:
case a
α : Type u
ι : Type v
a : Cardinal.{u}
inst✝¹ : Countable ι
f : ι → Set α
l : Filter ι
inst✝ : l.NeBot
ht : ∀ (x : α), ∀ᶠ (i : ι) in l, x ∈ f i
h'f : ∀ (i : ι), #↑(f i) = a
⊢ a ≤ #α
Module: Mathlib.Analysis.Normed.Module.Connected

Goal state before:
pp: E : Type u_1
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
inst✝ : ContinuousSMul ℝ E
h : 1 < Module.rank ℝ E
s : Set E
hs : s.Countable
this : Nontrivial E
a : E
ha : a ∈ sᶜ
b : E
hb : b ∈ sᶜ
hab : a ≠ b
c : E := 2⁻¹ • (a + b)
x : E := 2⁻¹ • (b - a)
Ia : c - x = a
Ib : c + x = b
x_ne_zero : x ≠ 0
y : E
hy : LinearIndependent ℝ ![x, y]
A : {t | ([c + x -[ℝ] c + t • y] ∩ s).Nonempty}.Countable
B : {t | ([c - x -[ℝ] c + t • y] ∩ s).Nonempty}.Countable
t : ℝ
z : E := c + t • y
ht : [c + x -[ℝ] c + t • y] ∩ s = ∅ ∧ [c - x -[ℝ] c + t • y] ∩ s = ∅
JA : JoinedIn sᶜ a z
⊢ b = c + x

Tactic:
exact Ib.symm

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.NerveAdjunction

Goal state before:
pp: case refine_1.zero.«0»
X Y : Truncated 2
f₀ : X.obj (op { obj := ⦋0⦌, property := _proof_11 }) → Y.obj (op { obj := ⦋0⦌, property := _proof_11 })
f₁ : X.obj (op { obj := ⦋1⦌, property := _proof_12 }) → Y.obj (op { obj := ⦋1⦌, property := _proof_12 })
hδ₁ :
  ∀ (x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })),
    f₀ ((ConcreteCategory.hom (X.map (δ₂ 1 _proof_11 _proof_13).op)) x) =
      (ConcreteCategory.hom (Y.map (δ₂ 1 _proof_11 _proof_13).op)) (f₁ x)
hδ₀ :
  ∀ (x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })),
    f₀ ((ConcreteCategory.hom (X.map (δ₂ 0 _proof_11 _proof_13).op)) x) =
      (ConcreteCategory.hom (Y.map (δ₂ 0 _proof_11 _proof_13).op)) (f₁ x)
H :
  ∀ (x : X.obj (op { obj := ⦋2⦌, property := _proof_14 })) (y : Y.obj (op { obj := ⦋2⦌, property := _proof_14 })),
    f₁ ((ConcreteCategory.hom (X.map (δ₂ 2 _proof_12 _proof_15).op)) x) =
        (ConcreteCategory.hom (Y.map (δ₂ 2 _proof_12 _proof_15).op)) y →
      f₁ ((ConcreteCategory.hom (X.map (δ₂ 0 _proof_12 _proof_15).op)) x) =
          (ConcreteCategory.hom (Y.map (δ₂ 0 _proof_12 _proof_15).op)) y →
        f₁ ((ConcreteCategory.hom (X.map (δ₂ 1 _proof_12 _proof_15).op)) x) =
          (ConcreteCategory.hom (Y.map (δ₂ 1 _proof_12 _proof_15).op)) y
hσ :
  ∀ (x : X.obj (op { obj := ⦋0⦌, property := _proof_11 })),
    f₁ ((ConcreteCategory.hom (X.map (σ₂ 0 _proof_13 _proof_11).op)) x) =
      (ConcreteCategory.hom (Y.map (σ₂ 0 _proof_13 _proof_11).op)) (f₀ x)
hY : Y.StrictSegal
hn : 0 < 2
⊢ naturalityProperty f₀ f₁ hδ₁ hδ₀ hY (δ 2 ((fun i ↦ i) ⟨0, ⋯⟩) ⋯ ⋯)

Tactic:
ext; apply hδ₀

Goal state after:
Module: Mathlib.Analysis.AbsoluteValue.Equivalence

Goal state before:
pp: F : Type u_1
inst✝ : Field F
v w : AbsoluteValue F ℝ
h : v.IsEquiv w
U : Set (WithAbs v)
hU : U ∈ 𝓝 0
⊢ ⇑(WithAbs.congr v w (RingEquiv.refl F)) ⁻¹' ⇑(WithAbs.congr v w (RingEquiv.refl F)) '' U ⊆ U

Tactic:
#adaptation_note /-- Before https://github.com/leanprover/lean4/pull/13166
        (replacing grind's canonicalizer with a type-directed normalizer), `grind` closed this
        goal. It is not yet clear whether this is due to defeq abuse in Mathlib or a problem in
        the new canonicalizer; a minimization would help. The original proof was:
        `grind [RingEquiv.image_eq_preimage_symm, Set.preimage_preimage]` -/

Goal state after:
F : Type u_1
inst✝ : Field F
v w : AbsoluteValue F ℝ
h : v.IsEquiv w
U : Set (WithAbs v)
hU : U ∈ 𝓝 0
⊢ ⇑(WithAbs.congr v w (RingEquiv.refl F)) ⁻¹' ⇑(WithAbs.congr v w (RingEquiv.refl F)) '' U ⊆ U
Module: Mathlib.Analysis.AbsoluteValue.Equivalence

Goal state before:
pp: F : Type u_1
inst✝¹ : Field F
ι : Type u_2
inst✝ : Finite ι
v : ι → AbsoluteValue F ℝ
h : ∀ (i : ι), (v i).IsNontrivial
hv : Pairwise fun i j ↦ ¬(v i).IsEquiv (v j)
this : Fintype ι
z : (i : ι) → WithAbs (v i)
r : ℝ
hr : r > 0
a : ι → F
hx : ∀ (i : ι), 1 < (v i) (a i) ∧ ∀ (j : ι), j ≠ i → (v j) (a i) < 1
y : ℕ → F := fun n ↦ ∑ i, 1 / (1 + (a i)⁻¹ ^ n) * (WithAbs.equiv (v i)) (z i)
⊢ ∃ y, dist z ((algebraMap F ((i : ι) → WithAbs (v i))) y) < r

Tactic:
have htend : atTop.Tendsto (fun n i ↦ (WithAbs.equiv (v i)).symm (y n)) (𝓝 z) := by
    refine tendsto_pi_nhds.mpr fun u ↦ ?_
    simp_rw [← Fintype.sum_pi_single u z, y, map_sum, map_mul]
    refine tendsto_finsetSum _ fun w _ ↦ ?_
    by_cases hw : u = w
    · rw [← hw, Pi.single_eq_same]
      have hlt : (v u) (a u)⁻¹ < 1 := by
        simpa [← inv_pow, inv_lt_one_iff₀] using Or.inr (hx u).1
      simpa using (WithAbs.tendsto_one_div_one_add_pow_nhds_one hlt).mul_const (z u)
    · rw [Pi.single_eq_of_ne (M := fun i ↦ WithAbs (v i)) hw (z w)]
      have hgt : 1 < (v u) (a w)⁻¹ := by
        rw [map_inv₀]
        refine one_lt_inv_iff₀.mpr ⟨(v u).pos_iff.mpr fun ha ↦ ?_, (hx w).2 u hw⟩
        linarith [map_zero (v w) ▸ ha ▸ (hx w).1]
      have := (v u).tendsto_div_one_add_pow_nhds_zero hgt
      simp_rw [← WithAbs.norm_toAbs_eq] at this
      simpa using (tendsto_zero_iff_norm_tendsto_zero.2 this).mul_const
        ((WithAbs.equiv (v u)).symm _)

Goal state after:
F : Type u_1
inst✝¹ : Field F
ι : Type u_2
inst✝ : Finite ι
v : ι → AbsoluteValue F ℝ
h : ∀ (i : ι), (v i).IsNontrivial
hv : Pairwise fun i j ↦ ¬(v i).IsEquiv (v j)
this : Fintype ι
z : (i : ι) → WithAbs (v i)
r : ℝ
hr : r > 0
a : ι → F
hx : ∀ (i : ι), 1 < (v i) (a i) ∧ ∀ (j : ι), j ≠ i → (v j) (a i) < 1
y : ℕ → F := fun n ↦ ∑ i, 1 / (1 + (a i)⁻¹ ^ n) * (WithAbs.equiv (v i)) (z i)
htend : Tendsto (fun n i ↦ (WithAbs.equiv (v i)).symm (y n)) atTop (𝓝 z)
⊢ ∃ y, dist z ((algebraMap F ((i : ι) → WithAbs (v i))) y) < r
Module: Mathlib.Analysis.Analytic.Order

Goal state before:
pp: case neg.refine_2
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
g : 𝕜 → E
z₀ : 𝕜
f : 𝕜 → 𝕜
hf : AnalyticAt 𝕜 f z₀
hg : AnalyticAt 𝕜 g z₀
hf' : ¬analyticOrderAt f z₀ = ⊤
hg' : ¬analyticOrderAt g z₀ = ⊤
f' : 𝕜 → 𝕜
h₁f' : AnalyticAt 𝕜 f' z₀
h₂f' : f' z₀ ≠ 0
h₃f' : f =ᶠ[𝓝 z₀] fun z ↦ (z - z₀) ^ analyticOrderNatAt f z₀ • f' z
g' : 𝕜 → E
h₁g' : AnalyticAt 𝕜 g' z₀
h₂g' : g' z₀ ≠ 0
h₃g' : g =ᶠ[𝓝 z₀] fun z ↦ (z - z₀) ^ analyticOrderNatAt g z₀ • g' z
⊢ ∀ᶠ (z : 𝕜) in 𝓝 z₀, (f • g) z = (z - z₀) ^ (analyticOrderNatAt f z₀ + analyticOrderNatAt g z₀) • (f' • g') z

Tactic:
obtain ⟨t, h₁t, h₂t, h₃t⟩ := eventually_nhds_iff.1 h₃f'
    obtain ⟨s, h₁s, h₂s, h₃s⟩ := eventually_nhds_iff.1 h₃g'
    exact eventually_nhds_iff.2
      ⟨t ∩ s, fun y hy ↦ (by simp [h₁t y hy.1, h₁s y hy.2]; module), h₂t.inter h₂s, h₃t, h₃s⟩

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.OrdinaryHypergeometric

Goal state before:
pp: 𝕂 : Type u_1
𝔸 : Type u_2
inst✝⁴ : Field 𝕂
inst✝³ : Ring 𝔸
inst✝² : Algebra 𝕂 𝔸
inst✝¹ : TopologicalSpace 𝔸
inst✝ : IsTopologicalRing 𝔸
a b c : 𝕂
⊢ ordinaryHypergeometricSeries 𝔸 a b c = ordinaryHypergeometricSeries 𝔸 b a c

Tactic:
by
  unfold ordinaryHypergeometricSeries ordinaryHypergeometricCoefficient
  simp [mul_assoc, mul_left_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 : 𝕜 → E
z₀ : 𝕜
n : ℕ
h : AnalyticAt 𝕜 (fun z ↦ (z - z₀) ^ n • f z) z₀
h₁ : ∀ᶠ (z : 𝕜) in 𝓝 z₀, (z - z₀) ^ n • f z = 0
y : 𝕜
h₁y : (y - z₀) ^ n • f y = 0
⊢ y ∈ {z₀}ᶜ → f y = 0

Tactic:
h₂y

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
n : ℕ
h : AnalyticAt 𝕜 (fun z ↦ (z - z₀) ^ n • f z) z₀
h₁ : ∀ᶠ (z : 𝕜) in 𝓝 z₀, (z - z₀) ^ n • f z = 0
y : 𝕜
h₁y : (y - z₀) ^ n • f y = 0
h₂y : y ∈ {z₀}ᶜ
⊢ f y = 0
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 : 𝕜 → E
z₀ : 𝕜
n : ℕ
h : AnalyticAt 𝕜 (fun z ↦ (z - z₀) ^ n • f z) z₀
h₂ : ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, (z - z₀) ^ n • f z ≠ 0
y : 𝕜
h₁y : (y - z₀) ^ n • f y ≠ 0
⊢ y ∈ {z₀}ᶜ → f y ≠ 0

Tactic:
h₂y

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
n : ℕ
h : AnalyticAt 𝕜 (fun z ↦ (z - z₀) ^ n • f z) z₀
h₂ : ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, (z - z₀) ^ n • f z ≠ 0
y : 𝕜
h₁y : (y - z₀) ^ n • f y ≠ 0
h₂y : y ∈ {z₀}ᶜ
⊢ f y ≠ 0
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
x : 𝕜
f g : 𝕜 → E
m : ℕ
hf : AnalyticAt 𝕜 (fun z ↦ (z - x) ^ m • f z) x
n : ℕ
hg : AnalyticAt 𝕜 (fun z ↦ (z - x) ^ n • g z) x
⊢ (fun z ↦ (z - x) ^ max m n • (f + g) z) = fun z ↦
    (z - x) ^ (max m n - m) • (z - x) ^ m • f z + (z - x) ^ (max m n - n) • (z - x) ^ n • g z

Tactic:
simp_rw [← mul_smul, ← pow_add, Nat.sub_add_cancel (Nat.le_max_left _ _),
      Nat.sub_add_cancel (Nat.le_max_right _ _), Pi.add_apply, smul_add]

Goal state after:
Module: Mathlib.RingTheory.Binomial

Goal state before:
pp: R : Type u_2
inst✝³ : NonAssocSemiring R
inst✝² : Pow R ℕ
inst✝¹ : NatPowAssoc R
inst✝ : BinomialRing R
r : R
k : ℕ
⊢ ↑k • (ascPochhammer ℕ k).smeval (r + 1) +
      (1 * (ascPochhammer ℕ k).smeval (r + 1) + r * (ascPochhammer ℕ k).smeval (r + 1)) =
    k.succ • (ascPochhammer ℕ k).smeval (r + 1) + r * (ascPochhammer ℕ k).smeval (r + 1)

Tactic:
@succ_nsmul,

Goal state after:
R : Type u_2
inst✝³ : NonAssocSemiring R
inst✝² : Pow R ℕ
inst✝¹ : NatPowAssoc R
inst✝ : BinomialRing R
r : R
k : ℕ
⊢ ↑k • (ascPochhammer ℕ k).smeval (r + 1) +
      (1 * (ascPochhammer ℕ k).smeval (r + 1) + r * (ascPochhammer ℕ k).smeval (r + 1)) =
    k • (ascPochhammer ℕ k).smeval (r + 1) + (ascPochhammer ℕ k).smeval (r + 1) + r * (ascPochhammer ℕ k).smeval (r + 1)
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
n : ℕ
⊢ multichoose (-↑n) (n + 1) = 0

Tactic:
← nsmul_right_inj (Nat.factorial_ne_zero (n + 1)),

Goal state after:
R : Type u_1
inst✝³ : NonAssocRing R
inst✝² : Pow R ℕ
inst✝¹ : BinomialRing R
inst✝ : NatPowAssoc R
n : ℕ
⊢ (n + 1).factorial • multichoose (-↑n) (n + 1) = (n + 1).factorial • 0
Module: Mathlib.Analysis.Analytic.Binomial

Goal state before:
pp: case convert_8
a : ℂ
this : binomialSeries ℂ a = FormalMultilinearSeries.ofScalars ℂ fun n ↦ iteratedDeriv n (fun x ↦ (1 + x) ^ a) 0 / ↑n !
⊢ 1 ≤ (binomialSeries ℂ a).radius

Tactic:
exact binomialSeries_radius_ge_one

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

Goal state before:
pp: case succ
a : ℂ
B : Set ℂ := Metric.ball 0 1
n : ℕ
ih :
  Set.EqOn (iteratedDerivWithin n (fun x ↦ (1 + x) ^ a) B) (fun x ↦ (descPochhammer ℤ n).smeval a * (1 + x) ^ (a - ↑n))
    B
this✝ :
  iteratedDerivWithin (n + 1) (fun x ↦ (1 + x) ^ a) B = derivWithin (iteratedDerivWithin n (fun x ↦ (1 + x) ^ a) B) B
this :
  Set.EqOn (derivWithin (iteratedDerivWithin n (fun x ↦ (1 + x) ^ a) B) B)
    (derivWithin (fun x ↦ (descPochhammer ℤ n).smeval a * (1 + x) ^ (a - ↑n)) B) B
z : ℂ
hz : z ∈ B
⊢ (descPochhammer ℤ n).smeval a * (a - ↑n) * (1 + z) ^ (a - ↑n - 1) =
    (descPochhammer ℤ (n + 1)).smeval a * (1 + z) ^ (a - ↑n - 1)

Tactic:
congr
      simp [descPochhammer_succ_right, Polynomial.smeval_mul, Polynomial.smeval_natCast]

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

Goal state before:
pp: case succ
n : ℕ
ihn : ∀ {p : ℝ}, ↑n ≤ p → ContDiff ℝ ↑n fun x ↦ x ^ p
p : ℝ
h : ↑n ≤ p - 1
h1 : 1 ≤ p
⊢ (Differentiable ℝ fun x ↦ x ^ p) ∧
    (↑n = ⊤ → AnalyticOn ℝ (fun x ↦ x ^ p) Set.univ) ∧ ContDiff ℝ (↑n) (deriv fun x ↦ x ^ p)

Tactic:
deriv_rpow_const' p

Goal state after:
case succ
n : ℕ
ihn : ∀ {p : ℝ}, ↑n ≤ p → ContDiff ℝ ↑n fun x ↦ x ^ p
p : ℝ
h : ↑n ≤ p - 1
h1 : 1 ≤ p
⊢ (Differentiable ℝ fun x ↦ x ^ p) ∧
    (↑n = ⊤ → AnalyticOn ℝ (fun x ↦ x ^ p) Set.univ) ∧ ContDiff ℝ ↑n fun x ↦ p * x ^ (p - 1)
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:

Tactic:
deriv (fun (x : ℝ) ↦ x ^ p) = fun x => p * x ^ (p - 1) := Real.deriv_rpow_const' p
    _ =Θ[atTop] fun x ↦ x ^ (p - 1) :=
      Asymptotics.IsTheta.const_mul_left hp Asymptotics.isTheta_rfl

Goal state after:
Module: Mathlib.Analysis.AperiodicOrder.Delone.Basic

Goal state before:
pp: X : Type u_1
inst✝ : MetricSpace X
D : DeloneSet X
r : ℝ≥0
hr : r ≤ D.packingRadius / 2
x y z : X
hx : x ∈ D
hy : y ∈ D
hxz : x ∈ ball z ↑r
hyz : y ∈ ball z ↑r
hne : ¬x = y
⊢ dist x z + dist y z < ↑r + ↑r

Tactic:
by gcongr <;> simpa

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

Goal state before:
pp: α : Type u_1
β : Type u_2
l : Filter α
k f : α → β
inst✝⁴ : TopologicalSpace β
inst✝³ : CommRing β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
inst✝ : OrderTopology β
⊢ (∀ (n : ℕ), Tendsto (fun a ↦ |k a ^ n * f a|) l (𝓝 0)) ↔ ∀ (n : ℕ), Tendsto (fun a ↦ |k a| ^ n * |f a|) l (𝓝 0)

Tactic:
abs_mul,

Goal state after:
α : Type u_1
β : Type u_2
l : Filter α
k f : α → β
inst✝⁴ : TopologicalSpace β
inst✝³ : CommRing β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
inst✝ : OrderTopology β
⊢ (∀ (n : ℕ), Tendsto (fun a ↦ |k a ^ n| * |f a|) l (𝓝 0)) ↔ ∀ (n : ℕ), Tendsto (fun a ↦ |k a| ^ n * |f a|) l (𝓝 0)
Module: Mathlib.Analysis.Asymptotics.LinearGrowth

Goal state before:
pp: case inl
u : ℕ → EReal
h : Monotone u
m : ℕ
hm : ¬u m = ⊥ m
m_n : ∀ᶠ (n : ℕ) in atTop, u m ≤ u n
hm' : u m = ⊤
⊢ 0 ≤ linearGrowthInf u

Tactic:
hm'

Goal state after:
case inl
u : ℕ → EReal
h : Monotone u
m : ℕ
hm : ¬u m = ⊥ m
m_n : ∀ᶠ (n : ℕ) in atTop, ⊤ ≤ u n
hm' : u m = ⊤
⊢ 0 ≤ linearGrowthInf u
Module: Mathlib.Analysis.Asymptotics.ExpGrowth

Goal state before:
pp: u : ℕ → ℝ≥0∞
h : Monotone u
h' : u ≠ 0
⊢ 0 ≤ expGrowthInf u

Tactic:
apply (log_monotone.comp h).linearGrowthInf_nonneg

Goal state after:
u : ℕ → ℝ≥0∞
h : Monotone u
h' : u ≠ 0
⊢ log ∘ u ≠ ⊥
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : IsProbabilityMeasure μ
f : α → ℝ≥0∞
⊢ ⨍⁻ (x : α), f x ∂μ = ∫⁻ (x : α), f x ∂μ

Tactic:
rw [laverage, measure_univ, inv_one, one_smul]

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

Goal state before:
pp: case pos
u : ℕ → EReal
v : ℕ → ℕ
a : EReal
h : Monotone u
hv : Tendsto (fun n ↦ ↑(v n) / ↑n) atTop (𝓝 a)
ha : a ≠ 0
ha' : a ≠ ⊤
hv₁ : 0 < liminf (fun n ↦ ↑(v n) / ↑n) atTop
v_top : Tendsto v atTop atTop
u_0 : ¬u = ⊥
u_1 : ∀ᶠ (n : ℕ) in atTop, u n ≤ 0
⊢ linearGrowthSup (u ∘ v) = a * linearGrowthSup u

Tactic:
have u_0' : linearGrowthSup u = 0 := by
      apply le_antisymm _ (h.linearGrowthSup_nonneg u_0)
      apply (linearGrowthSup_eventually_monotone u_1).trans_eq
      exact (linearGrowthSup_const zero_ne_bot zero_ne_top)
    rw [u_0', mul_zero]
    apply le_antisymm _ (linearGrowthSup_comp_nonneg h u_0 v_top)
    apply (linearGrowthSup_eventually_monotone (v_top.eventually u_1)).trans_eq
    exact linearGrowthSup_const zero_ne_bot zero_ne_top

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

Goal state before:
pp: u : ℕ → EReal
m : ℕ
h : Monotone u
hm : m ≠ 0
x : ℕ
hx : 0 < x
⊢ ↑x * (↑m / ↑x) = ↑m

Tactic:
exact mul_div_cancel (natCast_ne_bot x) (natCast_ne_top x) (Nat.cast_ne_zero.2 hx.ne.symm)

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: case neg
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ≥0∞
hμ : ¬IsFiniteMeasure (μ.restrict s)
⊢ ⨍⁻ (x : α) in s, f x ∂μ ≤ essSup f μ

Tactic:
simp [laverage, not_isFiniteMeasure_iff.mp hμ]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Decomposition.Hahn

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s ↦ ↑(μ s).toNNReal - ↑(ν s).toNNReal
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), μ s ≠ ∞
hν : ∀ (s : Set α), ν s ≠ ∞
to_nnreal_μ : ∀ (s : Set α), ↑(μ s).toNNReal = μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ν s).toNNReal = ν s
d_split : ∀ (s t : Set α), MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n ↦ d (s n)) atTop (𝓝 (d (⋃ n, s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n ↦ d (s n)) atTop (𝓝 (d (⋂ n, s n)))
bdd_c : BddAbove c
c_nonempty : c.Nonempty
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m ↦ (Finset.Ico n (m + 1)).inf e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
n✝ m : ℕ
h : m ≤ n✝
n : ℕ
hmn : m ≤ n
ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
⊢ γ + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n / 2) = γ + (γ - 2 * (1 / 2) ^ m + ((1 / 2) ^ n - (1 / 2) ^ n / 2))

Tactic:
_root_.sub_half

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s ↦ ↑(μ s).toNNReal - ↑(ν s).toNNReal
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), μ s ≠ ∞
hν : ∀ (s : Set α), ν s ≠ ∞
to_nnreal_μ : ∀ (s : Set α), ↑(μ s).toNNReal = μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ν s).toNNReal = ν s
d_split : ∀ (s t : Set α), MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n ↦ d (s n)) atTop (𝓝 (d (⋃ n, s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n ↦ d (s n)) atTop (𝓝 (d (⋂ n, s n)))
bdd_c : BddAbove c
c_nonempty : c.Nonempty
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m ↦ (Finset.Ico n (m + 1)).inf e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
n✝ m : ℕ
h : m ≤ n✝
n : ℕ
hmn : m ≤ n
ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
⊢ γ + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n / 2) = γ + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n / 2)
Module: Mathlib.MeasureTheory.Covering.Vitali

Goal state before:
pp: α : Type u_1
ι : Type u_2
inst✝⁴ : PseudoMetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)
μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)
ht : ∀ a ∈ t, (interior (B a)).Nonempty
h't : ∀ a ∈ t, IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (B a)
hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), μ (closedBall x (20 * R x)) < ∞
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : u.PairwiseDisjoint B
hu : ∀ a ∈ t', ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : u.Countable
x : α
x✝ : x ∈ s \ ⋃ a ∈ u, B a
v : Set ι := {a | a ∈ u ∧ (B a ∩ ball x (R x)).Nonempty}
vu : v ⊆ u
K : ℝ
μK : μ (closedBall x K) < ∞
hK : ∀ a ∈ u, (B a ∩ ball x (R x)).Nonempty → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : ∑' (a : ↑v), μ (B ↑a) < ∞
w : Finset ↑v
hw : ∑' (a : { a // a ∉ w }), μ (B ↑↑a) < ε / ↑C
z : α
hz : z ∈ (s \ ⋃ a ∈ u, B a) ∩ ball x (R x)
⊢ z ∈ ⋃ a, closedBall (c ↑↑a) (3 * r ↑↑a)

Tactic:
set k := ⋃ (a : v) (_ : a ∈ w), B a

Goal state after:
α : Type u_1
ι : Type u_2
inst✝⁴ : PseudoMetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)
μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)
ht : ∀ a ∈ t, (interior (B a)).Nonempty
h't : ∀ a ∈ t, IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (B a)
hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), μ (closedBall x (20 * R x)) < ∞
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : u.PairwiseDisjoint B
hu : ∀ a ∈ t', ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : u.Countable
x : α
x✝ : x ∈ s \ ⋃ a ∈ u, B a
v : Set ι := {a | a ∈ u ∧ (B a ∩ ball x (R x)).Nonempty}
vu : v ⊆ u
K : ℝ
μK : μ (closedBall x K) < ∞
hK : ∀ a ∈ u, (B a ∩ ball x (R x)).Nonempty → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : ∑' (a : ↑v), μ (B ↑a) < ∞
w : Finset ↑v
hw : ∑' (a : { a // a ∉ w }), μ (B ↑↑a) < ε / ↑C
z : α
hz : z ∈ (s \ ⋃ a ∈ u, B a) ∩ ball x (R x)
k : Set α := ⋃ a ∈ w, B ↑a
⊢ z ∈ ⋃ a, closedBall (c ↑↑a) (3 * r ↑↑a)
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝ : SigmaFinite μ
⊢ ∀ᵐ (x : α) ∂ν, μ.rnDeriv ν x < ∞

Tactic:
suffices ∀ n, ∀ᵐ x ∂ν, x ∈ spanningSets μ n → μ.rnDeriv ν x < ∞ by
    filter_upwards [ae_all_iff.2 this] with _ hx using hx _ (mem_spanningSetsIndex _ _)
  intro n
  rw [← ae_restrict_iff' (measurableSet_spanningSets _ _)]
  apply ae_lt_top (measurable_rnDeriv _ _)
  refine (lintegral_rnDeriv_lt_top_of_measure_ne_top _ ?_).ne
  exact (measure_spanningSets_lt_top _ _).ne

Goal state after: