mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Unbundled.FiniteExtension

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝² : NormedField K
inst✝¹ : Field L
inst✝ : Algebra K L
hfd : FiniteDimensional K L
hna : IsNonarchimedean norm
h1 : LinearIndepOn K id {1}
ι : Type u_2 := { x // x ∈ h1.extend ⋯ }
B : Basis ι K L := Basis.extend h1
hfin : Fintype ι := FiniteDimensional.fintypeBasisIndex B
hem : Nonempty ι
h1L : 1 ∈ h1.extend ⋯
hB1 : B ⟨1, h1L⟩ = 1
g : L → ℝ := B.norm
hg0 : g 0 = 0
hg_nonneg : ∀ (x : L), 0 ≤ g x
hg_ext : ∀ (x : K), g ((algebraMap K L) x) = ‖x‖
hg_na : IsNonarchimedean g
hg_add : ∀ (a b : L), g (a + b) ≤ g a + g b
hg_neg : ∀ (a : L), g (-a) = g a
w✝¹ : ℝ
w✝ : 0 < w✝¹
hg_bdd : ∀ (x y : L), B.norm (x * y) ≤ w✝¹ * B.norm x * B.norm y
hg_mul : ∀ (k : K) (y : L), g ((algebraMap K L) k * y) = g ((algebraMap K L) k) * g y
f : RingSeminorm L := seminormFromBounded hg0 hg_nonneg hg_bdd hg_add hg_neg
hf_na : IsNonarchimedean ⇑f
hf_1 : f 1 ≤ 1
hf_ext : ∀ (x : K), f ((algebraMap K L) x) = ‖x‖
F' : RingSeminorm L := smoothingSeminorm f hf_1 hf_na
hF' : F' = smoothingSeminorm f hf_1 hf_na
hF'_ext : ∀ (k : K), F' ((algebraMap K L) k) = ‖k‖
hF'_1 : F' 1 = 1
hF'_0 : F' ≠ 0
k : K
y : L
hk : ∀ (y : L), f ((algebraMap K L) k * y) = f ((algebraMap K L) k) * f y
hfk : ‖k‖ = (smoothingSeminorm f hf_1 hf_na) ((algebraMap K L) k)
⊢ ((smoothingSeminorm f hf_1 hf_na).toRingNorm ⋯).toFun (k • y) =
    (smoothingSeminorm f hf_1 hf_na) ((algebraMap K L) k) * ((smoothingSeminorm f hf_1 hf_na).toRingNorm ⋯).toFun y

Tactic:
erw [← smoothingSeminorm_of_mul f hf_1 hf_na hk y]

Goal state after:
K : Type u_1
L : Type u_2
inst✝² : NormedField K
inst✝¹ : Field L
inst✝ : Algebra K L
hfd : FiniteDimensional K L
hna : IsNonarchimedean norm
h1 : LinearIndepOn K id {1}
ι : Type u_2 := { x // x ∈ h1.extend ⋯ }
B : Basis ι K L := Basis.extend h1
hfin : Fintype ι := FiniteDimensional.fintypeBasisIndex B
hem : Nonempty ι
h1L : 1 ∈ h1.extend ⋯
hB1 : B ⟨1, h1L⟩ = 1
g : L → ℝ := B.norm
hg0 : g 0 = 0
hg_nonneg : ∀ (x : L), 0 ≤ g x
hg_ext : ∀ (x : K), g ((algebraMap K L) x) = ‖x‖
hg_na : IsNonarchimedean g
hg_add : ∀ (a b : L), g (a + b) ≤ g a + g b
hg_neg : ∀ (a : L), g (-a) = g a
w✝¹ : ℝ
w✝ : 0 < w✝¹
hg_bdd : ∀ (x y : L), B.norm (x * y) ≤ w✝¹ * B.norm x * B.norm y
hg_mul : ∀ (k : K) (y : L), g ((algebraMap K L) k * y) = g ((algebraMap K L) k) * g y
f : RingSeminorm L := seminormFromBounded hg0 hg_nonneg hg_bdd hg_add hg_neg
hf_na : IsNonarchimedean ⇑f
hf_1 : f 1 ≤ 1
hf_ext : ∀ (x : K), f ((algebraMap K L) x) = ‖x‖
F' : RingSeminorm L := smoothingSeminorm f hf_1 hf_na
hF' : F' = smoothingSeminorm f hf_1 hf_na
hF'_ext : ∀ (k : K), F' ((algebraMap K L) k) = ‖k‖
hF'_1 : F' 1 = 1
hF'_0 : F' ≠ 0
k : K
y : L
hk : ∀ (y : L), f ((algebraMap K L) k * y) = f ((algebraMap K L) k) * f y
hfk : ‖k‖ = (smoothingSeminorm f hf_1 hf_na) ((algebraMap K L) k)
⊢ ((smoothingSeminorm f hf_1 hf_na).toRingNorm ⋯).toFun (k • y) =
    (smoothingSeminorm f hf_1 hf_na) ((algebraMap K L) k * y)
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: K : Type u_2
inst✝³ : NormedField K
L : Type u_3
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : DecidableEq L
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
hf_na : IsNonarchimedean ⇑f
hf1 : f 1 = 1
p : K[X]
s : Multiset L
hp : (mapAlg K L) p = (Multiset.map (fun a ↦ X - C a) s).prod
h_le : 0 ≤ ⨆ x, if x ∈ s then f x else 0
m : ℕ
hm : m < p.natDegree
h✝ : 0 < ↑p.natDegree - ↑m
hps : s.card = p.natDegree
hc : ‖p.coeff m‖ = f (((mapAlg K L) p).coeff m)
h : f ((-1) ^ (s.card - m) * s.esymm (s.card - m)) = f (s.esymm (s.card - m))
t : Multiset L
ht_card : t.card = s.card - m
hts : ∀ x ∈ t, x ∈ s
ht_ge : f (Multiset.map prod (powersetCard (s.card - m) s)).sum ≤ f t.prod
h_pr : f t.prod ≤ (Multiset.map (⇑f) t).prod
hs_ne : s ≠ 0
y : L
hyx : y ∈ s
hy_max : ∀ z ∈ s, f z ≤ f y
g : L → NNReal := fun x ↦ ⟨f x, ⋯⟩
⊢ p.natDegree - m = s.card - m

Tactic:
← hps

Goal state after:
K : Type u_2
inst✝³ : NormedField K
L : Type u_3
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : DecidableEq L
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
hf_na : IsNonarchimedean ⇑f
hf1 : f 1 = 1
p : K[X]
s : Multiset L
hp : (mapAlg K L) p = (Multiset.map (fun a ↦ X - C a) s).prod
h_le : 0 ≤ ⨆ x, if x ∈ s then f x else 0
m : ℕ
hm : m < p.natDegree
h✝ : 0 < ↑p.natDegree - ↑m
hps : s.card = p.natDegree
hc : ‖p.coeff m‖ = f (((mapAlg K L) p).coeff m)
h : f ((-1) ^ (s.card - m) * s.esymm (s.card - m)) = f (s.esymm (s.card - m))
t : Multiset L
ht_card : t.card = s.card - m
hts : ∀ x ∈ t, x ∈ s
ht_ge : f (Multiset.map prod (powersetCard (s.card - m) s)).sum ≤ f t.prod
h_pr : f t.prod ≤ (Multiset.map (⇑f) t).prod
hs_ne : s ≠ 0
y : L
hyx : y ∈ s
hy_max : ∀ z ∈ s, f z ≤ f y
g : L → NNReal := fun x ↦ ⟨f x, ⋯⟩
⊢ s.card - m = s.card - m
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: R : Type u_1
K : Type u_2
inst✝³ : NormedField K
L : Type u_3
inst✝² : Field L
inst✝¹ : Algebra K L
E : IntermediateField K L
y : L
h_fin : FiniteDimensional K L
hn : Normal K L
inst✝ : IsUltrametricDist K
x : L
⊢ spectralNorm K L x = 0 → x = 0

Tactic:
simp only [spectralNorm_eq_invariantExtension]
    exact eq_zero_of_map_eq_zero _

Goal state after:
Module: Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
⊢ IsPowMul (smoothingFun μ)

Tactic:
intro x m hm
  have hlim : Tendsto (fun n => smoothingSeminormSeq μ x (m * n)) atTop
      (𝓝 (smoothingFun μ x)) :=
    Tendsto.comp (tendsto_smoothingFun_of_map_one_le_one μ hμ1 x) (tendsto_atTop_atTop_of_monotone
      (fun n k hnk ↦ mul_le_mul_right hnk m) (fun n ↦ ⟨n, le_mul_of_one_le_left' hm⟩))
  apply tendsto_nhds_unique _ (Tendsto.pow hlim m)
  have h_eq (n : ℕ) : smoothingSeminormSeq μ x (m * n) ^ m = smoothingSeminormSeq μ (x ^ m) n := by
    have hm' : (m : ℝ) ≠ 0 := cast_ne_zero.mpr (_root_.ne_of_gt (lt_of_lt_of_le zero_lt_one hm))
    simp only [smoothingSeminormSeq]
    rw [pow_mul, ← rpow_natCast, ← rpow_mul (apply_nonneg μ _), cast_mul, ← one_div_mul_one_div,
      mul_comm (1 / (m : ℝ)), mul_assoc, one_div_mul_cancel hm', mul_one]
  simpa [h_eq] using tendsto_smoothingFun_of_map_one_le_one μ hμ1 _

Goal state after:
Module: Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm

Goal state before:
pp: case pos
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
x : R
hx : ∀ (y : R), μ (x * y) = μ x * μ y
n : ℕ
hn : 1 ≤ n
hx0 : μ x = 0
hxn : μ (x ^ n) = 0
⊢ 0 ^ (1 / ↑n) = 0

Tactic:
zero_rpow (one_div_cast_ne_zero (one_le_iff_ne_zero.mp hn))

Goal state after:
case pos
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
x : R
hx : ∀ (y : R), μ (x * y) = μ x * μ y
n : ℕ
hn : 1 ≤ n
hx0 : μ x = 0
hxn : μ (x ^ n) = 0
⊢ 0 = 0
Module: Mathlib.Analysis.SpecialFunctions.Log.Summable

Goal state before:
pp: ι : Type u_1
R : Type u_2
inst✝¹ : NormedCommRing R
inst✝ : NormOneClass R
f : ι → R
x : ι
t : Finset ι
hx : x ∉ t
A : ℝ
h : Real.exp (∑ i ∈ t, ‖f i‖) = A
IH : ‖∏ i ∈ t, (1 + f i) - 1‖ ≤ A - 1
⊢ A + ‖f x‖ * A ≤ Real.exp ‖f x‖ * A

Tactic:
← one_add_mul,

Goal state after:
ι : Type u_1
R : Type u_2
inst✝¹ : NormedCommRing R
inst✝ : NormOneClass R
f : ι → R
x : ι
t : Finset ι
hx : x ∉ t
A : ℝ
h : Real.exp (∑ i ∈ t, ‖f i‖) = A
IH : ‖∏ i ∈ t, (1 + f i) - 1‖ ≤ A - 1
⊢ (1 + ‖f x‖) * A ≤ Real.exp ‖f x‖ * A
Module: Mathlib.Analysis.Normed.Module.MultipliableUniformlyOn

Goal state before:
pp: α : Type u_1
ι : Type u_2
K : Set α
u : ι → ℝ
f : ι → α → ℂ
hu : Summable u
h : ∀ᶠ (i : ι) in cofinite, ∀ x ∈ K, ‖f i x‖ ≤ u i
⊢ HasSumUniformlyOn (fun i x ↦ log (1 + f i x)) (fun x ↦ ∑' (i : ι), log (1 + f i x)) K

Tactic:
simp only [hasSumUniformlyOn_iff_tendstoUniformlyOn]
  apply tendstoUniformlyOn_tsum_of_cofinite_eventually <| hu.mul_left (3 / 2)
  filter_upwards [h, hu.tendsto_cofinite_zero.eventually_le_const one_half_pos] with i hi hi' x hx
    using (norm_log_one_add_half_le_self <| (hi x hx).trans hi').trans (by simpa using hi x hx)

Goal state after:
Module: Mathlib.Topology.Maps.Strict.Group

Goal state before:
pp: G : Type u_1
H : Type u_2
G' : Type u_3
H' : Type u_4
inst✝⁹ : Group G'
inst✝⁸ : Group H'
inst✝⁷ : Group G
inst✝⁶ : Group H
f : G →* H
g : G' →* H'
inst✝⁵ : TopologicalSpace G
inst✝⁴ : TopologicalSpace H
inst✝³ : IsTopologicalGroup G
inst✝² : TopologicalSpace G'
inst✝¹ : IsTopologicalGroup G'
inst✝ : TopologicalSpace H'
⊢ ↑(f.prodMap g).range = ↑f.range ×ˢ ↑g.range

Tactic:
by simp [Subgroup.coe_prod]

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
W : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace 𝕜 V
inst✝³ : CompleteSpace V
inst✝² : NormedAddCommGroup W
inst✝¹ : InnerProductSpace 𝕜 W
inst✝ : CompleteSpace W
f : (V →L[𝕜] V) ≃⋆ₐ[𝕜] W →L[𝕜] W
hf : Continuous ⇑f
h✝ : Nontrivial V
y : V ≃L[𝕜] W
hy : ⇑{ toAlgEquiv := f.toAlgEquiv, continuous_toFun := hf, continuous_invFun := ⋯ } = ⇑y.conjContinuousAlgEquiv
this✝¹ : ∀ (x : V →L[𝕜] V), Commute x (adjoint ↑y ∘SL ↑y)
α : 𝕜
hα : α • 1 = adjoint ↑y ∘SL ↑y
this✝ : IsUnit (adjoint ↑y ∘SL ↑y)
this : (starRingEnd 𝕜) α • 1 = α • 1
⊢ α = ↑(RCLike.re α)

Tactic:
(smul_left_injective 𝕜 one_ne_zero).eq_iff,

Goal state after:
𝕜 : Type u_1
V : Type u_2
W : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace 𝕜 V
inst✝³ : CompleteSpace V
inst✝² : NormedAddCommGroup W
inst✝¹ : InnerProductSpace 𝕜 W
inst✝ : CompleteSpace W
f : (V →L[𝕜] V) ≃⋆ₐ[𝕜] W →L[𝕜] W
hf : Continuous ⇑f
h✝ : Nontrivial V
y : V ≃L[𝕜] W
hy : ⇑{ toAlgEquiv := f.toAlgEquiv, continuous_toFun := hf, continuous_invFun := ⋯ } = ⇑y.conjContinuousAlgEquiv
this✝¹ : ∀ (x : V →L[𝕜] V), Commute x (adjoint ↑y ∘SL ↑y)
α : 𝕜
hα : α • 1 = adjoint ↑y ∘SL ↑y
this✝ : IsUnit (adjoint ↑y ∘SL ↑y)
this : (starRingEnd 𝕜) α = α
⊢ α = ↑(RCLike.re α)
Module: Mathlib.Analysis.ODE.Transform

Goal state before:
pp: case e'_6
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
γ : ℝ → E
v : ℝ → E → E
s : Set ℝ
a : ℝ
ha : a ≠ 0
heq : a⁻¹ • s = {t | t * a ∈ s}
hγ : IsIntegralCurveOn (γ ∘ fun x ↦ x * a) (a • v ∘ fun x ↦ x * a) (a⁻¹ • s)
⊢ s = {t | t * a⁻¹ ∈ a⁻¹ • s}

Tactic:
· simp only [mul_comm _ a⁻¹, ← smul_eq_mul, mem_inv_smul_set_iff₀ ha, smul_inv_smul₀ ha,
      ofPred_mem_eq]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hdeg : P.degree < Q.degree
⊢ Tendsto (fun x ↦ eval x P / eval x Q) atBot (𝓝 0)

Tactic:
rw [← P.degree_comp_neg_X, ← Q.degree_comp_neg_X] at hdeg
  convert! (div_tendsto_atTop_zero_of_degree_lt _ _ hdeg).comp tendsto_neg_atBot_atTop using 2
  simp

Goal state after:
Module: Mathlib.Analysis.ODE.PicardLindelof

Goal state before:
pp: case e'_5
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : ℝ → E → E
tmin tmax : ℝ
t₀ : ↑(Icc tmin tmax)
x₀ : E
a r L K : ℝ≥0
inst✝ : CompleteSpace E
hf : IsPicardLindelof f t₀ x₀ a r L K
m : ℕ
C : ℝ≥0
h : ∀ (x : E) (hx : x ∈ closedBall x₀ ↑r), ContractingWith C (next hf hx)^[m]
L' : ℝ := (∑ i ∈ Finset.range m, (↑K * max (tmax - ↑t₀) (↑t₀ - tmin)) ^ i / ↑i !) * ↑(1 - C)⁻¹
hL' : 0 ≤ L'
x y : E
hx : x ∈ closedBall x₀ ↑r
hy : y ∈ closedBall x₀ ↑r
α β : FunSpace t₀ x₀ r L
hα : IsFixedPt (next hf hx) α
hβ : IsFixedPt (next hf hy) β
⊢ ↑(1 - C)⁻¹ = (1 - ↑C)⁻¹

Tactic:
simp [NNReal.coe_sub <| le_of_lt (h y hy).1, NNReal.coe_one]

Goal state after:
Module: Mathlib.Analysis.Rat.NatSqrt.Real

Goal state before:
pp: x prec : ℕ
h : 0 < prec
⊢ √↑x ∈ Set.Ico (↑(x.ratSqrt prec)) (↑(x.ratSqrt prec) + 1 / ↑prec)

Tactic:
grind [ratSqrt_le_realSqrt, realSqrt_lt_ratSqrt_add_inv_prec]

Goal state after:
Module: Mathlib.Analysis.Real.Pi.Bounds

Goal state before:
pp: n : ℕ
⊢ (4 / 2 ^ (n + 2)) ^ 3 / 4 = 1 / (2 ^ n) ^ 3 / 4

Tactic:
simp [add_comm n, pow_add, div_mul_eq_div_div]; norm_num

Goal state after:
Module: Mathlib.Analysis.Real.Hyperreal

Goal state before:
pp: x : ℝ*
r : ℝ
hxr : x.IsSt r
⊢ (x - ↑r).Infinitesimal

Tactic:
simpa only [sub_self] using! hxr.sub (isSt_refl_real r)

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.Arctan

Goal state before:
pp: z : ℂ
h₁ : z ≠ I
h₂ : z ≠ -I
z₁ : 1 + z * I ≠ 0
z₂ : 1 - z * I ≠ 0
key : cexp (2 * (z.arctan * I)) = (1 + z * I) / (1 - z * I)
⊢ (1 - cexp (2 * (z.arctan * I))) / (cexp (2 * (z.arctan * I)) + 1) * I = z

Tactic:
key,

Goal state after:
z : ℂ
h₁ : z ≠ I
h₂ : z ≠ -I
z₁ : 1 + z * I ≠ 0
z₂ : 1 - z * I ≠ 0
key : cexp (2 * (z.arctan * I)) = (1 + z * I) / (1 - z * I)
⊢ (1 - (1 + z * I) / (1 - z * I)) / ((1 + z * I) / (1 - z * I) + 1) * I = z
Module: Mathlib.Analysis.SpecialFunctions.Complex.Arctan

Goal state before:
pp: case e'_6
z : ℂ
hz : ‖z‖ < 1
this :
  HasSum (fun x ↦ -I / 2 * (((-1) ^ (2 * x.1 + ↑x.2 + 1) + 1) * (z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)))
    (-I / 2 * log ((1 + z * I) / (1 - z * I)))
k : ℕ
⊢ (-1) ^ k * z ^ (2 * k + 1) / ↑(2 * k + 1) =
    -I / 2 * (((-1) ^ (2 * k + ↑0 + 1) + 1) * (z * I) ^ (2 * k + ↑0) / ↑(2 * k + ↑0)) +
      -I / 2 * (((-1) ^ (2 * k + ↑1 + 1) + 1) * (z * I) ^ (2 * k + ↑1) / ↑(2 * k + ↑1))

Tactic:
Fin.val_zero,

Goal state after:
case e'_6
z : ℂ
hz : ‖z‖ < 1
this :
  HasSum (fun x ↦ -I / 2 * (((-1) ^ (2 * x.1 + ↑x.2 + 1) + 1) * (z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)))
    (-I / 2 * log ((1 + z * I) / (1 - z * I)))
k : ℕ
⊢ (-1) ^ k * z ^ (2 * k + 1) / ↑(2 * k + 1) =
    -I / 2 * (((-1) ^ (2 * k + 0 + 1) + 1) * (z * I) ^ (2 * k + 0) / ↑(2 * k + 0)) +
      -I / 2 * (((-1) ^ (2 * k + ↑1 + 1) + 1) * (z * I) ^ (2 * k + ↑1) / ↑(2 * k + ↑1))
Module: Mathlib.Analysis.Real.Pi.Irrational

Goal state before:
pp: n : ℕ
⊢ I n (π / 2) ≤ 2

Tactic:
← norm_of_nonneg I_pos.le

Goal state after:
n : ℕ
⊢ ‖I n (π / 2)‖ ≤ 2
Module: Mathlib.Analysis.Real.Pi.Irrational

Goal state before:
pp: ⊢ Irrational π

Tactic:
apply Irrational.of_div_natCast 2

Goal state after:
⊢ Irrational (π / ↑2)
Module: Mathlib.Analysis.SpecialFunctions.FrullaniIntegral

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
c ε r : ℝ
hc : c ≠ 0
u : ℝ → E := fun x ↦ x⁻¹ • f x
key : (fun x ↦ x⁻¹ • f (c * x)) = fun x ↦ c • u (c * x)
⊢ ∫ (x : ℝ) in ε..r, x⁻¹ • f (c * x) = ∫ (x : ℝ) in c * ε..c * r, x⁻¹ • f x

Tactic:
key,

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
c ε r : ℝ
hc : c ≠ 0
u : ℝ → E := fun x ↦ x⁻¹ • f x
key : (fun x ↦ x⁻¹ • f (c * x)) = fun x ↦ c • u (c * x)
⊢ ∫ (x : ℝ) in ε..r, c • u (c * x) = ∫ (x : ℝ) in c * ε..c * r, x⁻¹ • f x
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation

Goal state before:
pp: case refine_1
p : ℝ
hp : p ∈ Ioo 0 1
⊢ IntegrableOn (fun t ↦ 1 / 2 * t ^ (p - 2)) (Ioi 1) volume

Tactic:
refine Integrable.const_mul ?_ _

Goal state after:
case refine_1
p : ℝ
hp : p ∈ Ioo 0 1
⊢ Integrable (fun t ↦ t ^ (p - 2)) (volume.restrict (Ioi 1))
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation

Goal state before:
pp: p x : ℝ
hp : p ∈ Ioo 1 2
hx : 0 ≤ x
hmain : (fun x_1 ↦ p.rpowIntegrand₁₂ x_1 x) =ᵐ[volume.restrict (Ioi 0)] fun x_1 ↦ x * (p - 1).rpowIntegrand₀₁ x_1 x
⊢ IntegrableOn (fun x_1 ↦ x * (p - 1).rpowIntegrand₀₁ x_1 x) (Ioi 0) volume

Tactic:
refine Integrable.const_mul ?_ _

Goal state after:
p x : ℝ
hp : p ∈ Ioo 1 2
hx : 0 ≤ x
hmain : (fun x_1 ↦ p.rpowIntegrand₁₂ x_1 x) =ᵐ[volume.restrict (Ioi 0)] fun x_1 ↦ x * (p - 1).rpowIntegrand₀₁ x_1 x
⊢ Integrable (fun x_1 ↦ (p - 1).rpowIntegrand₀₁ x_1 x) (volume.restrict (Ioi 0))
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation

Goal state before:
pp: A : Type u_1
inst✝⁹ : NonUnitalNormedRing A
inst✝⁸ : StarRing A
inst✝⁷ : NormedSpace ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : CompleteSpace A
p : ℝ≥0
hp : p ∈ Ioo 0 1
μ : Measure ℝ
hμ :
  ∀ x ∈ Ici 0,
    IntegrableOn (fun t ↦ (↑p).rpowIntegrand₀₁ t x) (Ioi 0) μ ∧ x ^ ↑p = ∫ (t : ℝ) in Ioi 0, (↑p).rpowIntegrand₀₁ t x ∂μ
a : A
ha : 0 ≤ a
a✝ : Nontrivial A
p_pos : 0 < ↑p
f : ℝ → ℝ → ℝ := fun t ↦ (↑p).rpowIntegrand₀₁ t
maxr : ℝ := sSup (quasispectrum ℝ a)
maxr_nonneg : 0 ≤ maxr
⊢ IntegrableOn (fun t ↦ cfcₙ ((↑p).rpowIntegrand₀₁ t) a) (Ioi 0) μ ∧
    a ^ p = ∫ (t : ℝ) in Ioi 0, cfcₙ ((↑p).rpowIntegrand₀₁ t) a ∂μ

Tactic:
let bound (t : ℝ) := ‖f t maxr‖

Goal state after:
A : Type u_1
inst✝⁹ : NonUnitalNormedRing A
inst✝⁸ : StarRing A
inst✝⁷ : NormedSpace ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : CompleteSpace A
p : ℝ≥0
hp : p ∈ Ioo 0 1
μ : Measure ℝ
hμ :
  ∀ x ∈ Ici 0,
    IntegrableOn (fun t ↦ (↑p).rpowIntegrand₀₁ t x) (Ioi 0) μ ∧ x ^ ↑p = ∫ (t : ℝ) in Ioi 0, (↑p).rpowIntegrand₀₁ t x ∂μ
a : A
ha : 0 ≤ a
a✝ : Nontrivial A
p_pos : 0 < ↑p
f : ℝ → ℝ → ℝ := fun t ↦ (↑p).rpowIntegrand₀₁ t
maxr : ℝ := sSup (quasispectrum ℝ a)
maxr_nonneg : 0 ≤ maxr
bound : ℝ → ℝ := fun t ↦ ‖f t maxr‖
⊢ IntegrableOn (fun t ↦ cfcₙ ((↑p).rpowIntegrand₀₁ t) a) (Ioi 0) μ ∧
    a ^ p = ∫ (t : ℝ) in Ioi 0, cfcₙ ((↑p).rpowIntegrand₀₁ t) a ∂μ
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation

Goal state before:
pp: A : Type u_1
inst✝⁹ : NonUnitalNormedRing A
inst✝⁸ : StarRing A
inst✝⁷ : NormedSpace ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : CompleteSpace A
p : ℝ≥0
hp : p ∈ Ioo 1 2
μ : Measure ℝ
hμ :
  ∀ x ∈ Ici 0,
    IntegrableOn (fun t ↦ (↑p).rpowIntegrand₁₂ t x) (Ioi 0) μ ∧ x ^ ↑p = ∫ (t : ℝ) in Ioi 0, (↑p).rpowIntegrand₁₂ t x ∂μ
a : A
ha : 0 ≤ a
hpcoe : ↑p ∈ Ioo 1 2
f : ℝ → ℝ → ℝ := fun t ↦ (↑p).rpowIntegrand₁₂ t
maxr : ℝ := sSup (quasispectrum ℝ a)
maxr_nonneg : 0 ≤ maxr
⊢ IntegrableOn (fun t ↦ cfcₙ ((↑p).rpowIntegrand₁₂ t) a) (Ioi 0) μ ∧
    a ^ p = ∫ (t : ℝ) in Ioi 0, cfcₙ ((↑p).rpowIntegrand₁₂ t) a ∂μ

Tactic:
let bound (t : ℝ) := ‖f t maxr‖

Goal state after:
A : Type u_1
inst✝⁹ : NonUnitalNormedRing A
inst✝⁸ : StarRing A
inst✝⁷ : NormedSpace ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : CompleteSpace A
p : ℝ≥0
hp : p ∈ Ioo 1 2
μ : Measure ℝ
hμ :
  ∀ x ∈ Ici 0,
    IntegrableOn (fun t ↦ (↑p).rpowIntegrand₁₂ t x) (Ioi 0) μ ∧ x ^ ↑p = ∫ (t : ℝ) in Ioi 0, (↑p).rpowIntegrand₁₂ t x ∂μ
a : A
ha : 0 ≤ a
hpcoe : ↑p ∈ Ioo 1 2
f : ℝ → ℝ → ℝ := fun t ↦ (↑p).rpowIntegrand₁₂ t
maxr : ℝ := sSup (quasispectrum ℝ a)
maxr_nonneg : 0 ≤ maxr
bound : ℝ → ℝ := fun t ↦ ‖f t maxr‖
⊢ IntegrableOn (fun t ↦ cfcₙ ((↑p).rpowIntegrand₁₂ t) a) (Ioi 0) μ ∧
    a ^ p = ∫ (t : ℝ) in Ioi 0, cfcₙ ((↑p).rpowIntegrand₁₂ t) a ∂μ
Module: Mathlib.Data.Int.Log

Goal state before:
pp: R : Type u_1
inst✝³ : Semifield R
inst✝² : LinearOrder R
inst✝¹ : IsStrictOrderedRing R
inst✝ : FloorSemiring R
b : ℕ
hb : 1 < b
r : R
⊢ r < ↑b ^ (log b r + 1)

Tactic:
rcases le_or_gt r 0 with hr | hr
  · rw [log_of_right_le_zero _ hr, zero_add, zpow_one]
    exact hr.trans_lt (zero_lt_one.trans_le <| mod_cast hb.le)
  rcases le_or_gt 1 r with hr1 | hr1
  · rw [log_of_one_le_right _ hr1, Int.ofNat_add_one_out]
    exact_mod_cast Nat.lt_of_floor_lt <| Nat.lt_pow_succ_log_self hb _
  · rw [log_of_right_le_one _ hr1.le]
    have hcri : 1 < r⁻¹ := (one_lt_inv₀ hr).2 hr1
    have : 1 ≤ Nat.clog b ⌈r⁻¹⌉₊ :=
      Nat.succ_le_of_lt (Nat.clog_pos hb <| Nat.one_lt_cast.1 <| hcri.trans_le (Nat.le_ceil _))
    rw [neg_add_eq_sub, ← neg_sub, ← Int.ofNat_one, ← Int.ofNat_sub this, zpow_neg, zpow_natCast,
      lt_inv_comm₀ hr (pow_pos (Nat.cast_pos.mpr <| zero_lt_one.trans hb) _), ← Nat.cast_pow]
    refine Nat.lt_ceil.1 ?_
    exact Nat.pow_pred_clog_lt_self hb <| Nat.one_lt_cast.1 <| hcri.trans_le <| Nat.le_ceil _

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Log.Base

Goal state before:
pp: a b c : ℝ
h₁ : b ≠ 0
h₂ : b ≠ 1
h₃ : b ≠ -1
⊢ log b / log a * (log c / log b) = log c / log a

Tactic:
rw [mul_comm, div_mul_div_cancel₀ (log_ne_zero.mpr ⟨h₁, h₂, h₃⟩)]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p t : ℝ
hp : p ∈ Ioo 0 1
ht : 0 < t
a : A
ha : 0 ≤ a
b : A
hb : 0 ≤ b
hab : a ≤ b
⊢ t ^ (p - 1) • cfcₙ (p.rpowIntegrand₀₁ 1) (t⁻¹ • b) = cfcₙ (p.rpowIntegrand₀₁ t) b

Tactic:
cfcₙ_rpowIntegrand₀₁_eq_cfcₙ_rpowIntegrand₀₁_one hp ht b hb

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p t : ℝ
hp : p ∈ Ioo 0 1
ht : 0 < t
a : A
ha : 0 ≤ a
b : A
hb : 0 ≤ b
hab : a ≤ b
⊢ t ^ (p - 1) • cfcₙ (p.rpowIntegrand₀₁ 1) (t⁻¹ • b) = t ^ (p - 1) • cfcₙ (p.rpowIntegrand₀₁ 1) (t⁻¹ • b)
Module: Mathlib.Analysis.SpecialFunctions.MulExpNegMulSqIntegral

Goal state before:
pp: case h_bound
E : Type u_1
inst✝³ : TopologicalSpace E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
P : Measure E
inst✝ : IsFiniteMeasure P
ε : ℝ
g : E →ᵇ ℝ
hε : 0 < ε
⊢ ∃ C, ∀ᶠ (n : ℕ) in atTop, ∀ᵐ (ω : E) ∂P, ‖(g * (1 + (↑n)⁻¹ • -(ε • g * g)) ^ n) ω‖ ≤ C

Tactic:
obtain ⟨N, hgN⟩ := exists_nat_gt (ε * (norm g * norm g))
    use norm g
    rw [eventually_atTop]
    use N
    intro n hn
    have hnpos : 0 < (n : ℝ) := by
      apply lt_of_lt_of_le (lt_of_le_of_lt _ hgN) (Nat.cast_le.mpr hn)
      exact (mul_nonneg (le_of_lt hε) (mul_self_nonneg (norm g)))
    apply Eventually.of_forall
    intro x
    simp only [smul_neg, BoundedContinuousFunction.coe_mul, Pi.mul_apply, pow_apply,
      BoundedContinuousFunction.coe_add, BoundedContinuousFunction.coe_one, coe_neg,
      BoundedContinuousFunction.coe_smul, smul_eq_mul, Pi.add_apply, Pi.one_apply, Pi.neg_apply,
      norm_mul, norm_eq_abs, norm_pow]
    refine (mul_le_mul_of_nonneg_right (norm_coe_le_norm g x) (pow_nonneg (abs_nonneg _) n)).trans
      <| mul_le_of_le_one_right (norm_nonneg _) ?_
    apply pow_le_one₀ (abs_nonneg _)
    rw [mul_assoc, inv_mul_eq_div, abs_le]
    refine ⟨?_, (add_le_iff_nonpos_right 1).mpr (Left.neg_nonpos_iff.mpr
      (div_nonneg (mul_nonneg (le_of_lt hε) (mul_self_nonneg (g x))) (le_of_lt hnpos)))⟩
    apply le_trans (by linarith) (sub_nonneg_of_le ((div_le_one hnpos).mpr _))
    apply le_trans (le_trans _ (le_of_lt hgN)) (Nat.cast_le.mpr hn)
    apply mul_le_mul (le_refl ε) _ (mul_self_nonneg (g x)) (le_of_lt hε)
    rw [← abs_le_iff_mul_self_le, abs_norm]
    exact norm_coe_le_norm g x

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.HaarToSphere

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
μ : Measure E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
a✝ : Nontrivial E
s : Set ↑(sphere 0 1)
hs : s ∈ {s | MeasurableSet s}
r : ↑(Ioi 0)
this : Ioo 0 ↑r = ↑r • Ioo 0 1
hpos : 0 < dim E
⊢ ENNReal.ofReal (↑r ^ dim E) * μ (Ioo 0 1 • Subtype.val '' s) =
    ↑(dim E) * ENNReal.ofReal (↑r ^ dim E / ↑(dim E)) * μ (Ioo 0 1 • Subtype.val '' s)

Tactic:
← ENNReal.ofReal_natCast,

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
μ : Measure E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
a✝ : Nontrivial E
s : Set ↑(sphere 0 1)
hs : s ∈ {s | MeasurableSet s}
r : ↑(Ioi 0)
this : Ioo 0 ↑r = ↑r • Ioo 0 1
hpos : 0 < dim E
⊢ ENNReal.ofReal (↑r ^ dim E) * μ (Ioo 0 1 • Subtype.val '' s) =
    ENNReal.ofReal ↑(dim E) * ENNReal.ofReal (↑r ^ dim E / ↑(dim E)) * μ (Ioo 0 1 • Subtype.val '' s)