mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: case e'_3
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
der1 :
  ∀ x ∈ uIcc 0 (π / 2),
    HasDerivAt (fun y ↦ ↑(sin y) * ↑(cos y) ^ (n - 1)) (↑(cos x) ^ n - (↑n - 1) * ↑(sin x) ^ 2 * ↑(cos x) ^ (n - 2)) x
⊢ (↑n / (2 * z) * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n) -
      (↑n - 1) / (2 * z) * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (n - 2) =
    ↑(sin (π / 2)) * ↑(cos (π / 2)) ^ (n - 1) * (-Complex.cos (2 * z * ↑(π / 2)) / (2 * z)) -
        ↑(sin 0) * ↑(cos 0) ^ (n - 1) * (-Complex.cos (2 * z * ↑0) / (2 * z)) -
      ∫ (x : ℝ) in 0..π / 2,
        (↑(cos x) ^ n - (↑n - 1) * ↑(sin x) ^ 2 * ↑(cos x) ^ (n - 2)) * (-Complex.cos (2 * z * ↑x) / (2 * z))

Tactic:
rw [sin_zero, cos_pi_div_two, Complex.ofReal_zero, zero_pow, zero_mul,
      mul_zero, zero_mul, zero_mul, sub_zero, zero_sub, ←
      integral_neg, ← integral_const_mul, ← integral_const_mul, ← integral_sub]
    rotate_left
    · apply Continuous.intervalIntegrable <| by fun_prop
    · apply Continuous.intervalIntegrable <| by fun_prop
    · exact Nat.sub_ne_zero_of_lt hn
    refine integral_congr fun x _ => ?_
    -- get rid of real trig functions and divisions by 2 * z:
    rw [Complex.ofReal_cos, Complex.ofReal_sin, Complex.sin_sq, ← mul_div_right_comm, ←
      mul_div_right_comm, ← sub_div, mul_div, ← neg_div]
    congr 1
    have : Complex.cos x ^ n = Complex.cos x ^ (n - 2) * Complex.cos x ^ 2 := by
      conv_lhs => rw [← Nat.sub_add_cancel hn, pow_add]
    rw [this]
    ring

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: s t : ℂ
a : ℝ
ha : 0 < a
⊢ ∫ (x : ℝ) in 0..a, ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) = ↑a ^ (s + t - 1) * s.betaIntegral t

Tactic:
have ha' : (a : ℂ) ≠ 0 := ofReal_ne_zero.mpr ha.ne'
  rw [betaIntegral]
  have A : (a : ℂ) ^ (s + t - 1) = a * ((a : ℂ) ^ (s - 1) * (a : ℂ) ^ (t - 1)) := by
    rw [(by abel : s + t - 1 = 1 + (s - 1) + (t - 1)), cpow_add _ _ ha', cpow_add 1 _ ha', cpow_one,
      mul_assoc]
  rw [A, mul_assoc, ← intervalIntegral.integral_const_mul, ← real_smul, ← zero_div a, ←
    div_self ha.ne', ← intervalIntegral.integral_comp_div _ ha.ne', zero_div]
  simp_rw [intervalIntegral.integral_of_le ha.le]
  refine setIntegral_congr_fun measurableSet_Ioc fun x hx => ?_
  rw [mul_mul_mul_comm]
  congr 1
  · rw [← mul_cpow_ofReal_nonneg ha.le (div_pos hx.1 ha).le, ofReal_div, mul_div_cancel₀ _ ha']
  · rw [(by norm_cast : (1 : ℂ) - ↑(x / a) = ↑(1 - x / a)), ←
      mul_cpow_ofReal_nonneg ha.le (sub_nonneg.mpr <| (div_le_one ha).mpr hx.2)]
    push_cast
    rw [mul_sub, mul_one, mul_div_cancel₀ _ ha']

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
F : ℝ → ℂ := fun x ↦ ↑x ^ u * (1 - ↑x) ^ v
hu' : 0 < (u + 1).re
⊢ u * u.betaIntegral (v + 1) = v * (u + 1).betaIntegral v

Tactic:
have hv' : 0 < re (v + 1) := by rw [add_re, one_re]; positivity

Goal state after:
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
F : ℝ → ℂ := fun x ↦ ↑x ^ u * (1 - ↑x) ^ v
hu' : 0 < (u + 1).re
hv' : 0 < (v + 1).re
⊢ u * u.betaIntegral (v + 1) = v * (u + 1).betaIntegral v
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: ⊢ Γ 2 = 1

Tactic:
by simp [Nat.factorial_one]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
𝕜' : Type u_2
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
U : Set 𝕜
ι : Type u_3
F : ι → 𝕜 → 𝕜'
hF : (mulSupport F).Finite
h : ∀ (i : ι), MeromorphicOn (F i) U
h' : ∀ (i : ι), ∀ x ∈ U, meromorphicOrderAt (F i) x ≠ ⊤
hsub : (support fun i ↦ logDeriv (F i)) ⊆ ↑hF.toFinset
⊢ logDeriv (∏ᶠ (i : ι), F i) =ᶠ[codiscreteWithin U] ∑ᶠ (i : ι), logDeriv (F i)

Tactic:
rw [finprod_eq_prod_of_mulSupport_subset F (s := hF.toFinset) (by simp),
    finsum_eq_sum_of_support_subset _ hsub]

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
𝕜' : Type u_2
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
U : Set 𝕜
ι : Type u_3
F : ι → 𝕜 → 𝕜'
hF : (mulSupport F).Finite
h : ∀ (i : ι), MeromorphicOn (F i) U
h' : ∀ (i : ι), ∀ x ∈ U, meromorphicOrderAt (F i) x ≠ ⊤
hsub : (support fun i ↦ logDeriv (F i)) ⊆ ↑hF.toFinset
⊢ logDeriv (∏ i ∈ hF.toFinset, F i) =ᶠ[codiscreteWithin U] ∑ i ∈ hF.toFinset, logDeriv (F i)
Module: Mathlib.Analysis.Normed.Affine.AsymptoticCone

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : NormedSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
inst✝ : FiniteDimensional ℝ V
⊢ cobounded P = ⨆ v ∈ Metric.sphere 0 1, asymptoticNhds ℝ P v

Tactic:
by
  refine le_antisymm ?_ <| iSup₂_le fun _ h => asymptoticNhds_le_cobounded <|
    Metric.ne_of_mem_sphere h one_ne_zero
  intro s hs
  have ⟨p⟩ : Nonempty P := inferInstance
  simp_rw [mem_iSup, asymptoticNhds_eq_smul_vadd _ p, vadd_pure] at hs
  choose! t ht u hu smul_subset_s using hs
  have ⟨cover, h₁, h₂⟩ := (isCompact_sphere 0 1).elim_nhds_subcover u hu
  rw [← Metric.comap_dist_left_atTop p]
  refine ⟨Set.Ioi 0 ∩ ⋂ x ∈ cover, t x, inter_mem (Ioi_mem_atTop 0)
    (cover.iInter_mem_sets.mpr fun x hx => ht x (h₁ x hx)), fun x hx => ?_⟩
  rw [Set.mem_preimage, dist_eq_norm_vsub'] at hx
  let x' := ‖x -ᵥ p‖⁻¹ • (x -ᵥ p)
  have x'_mem : x' ∈ Metric.sphere 0 1 := by
    rw [mem_sphere_zero_iff_norm, norm_smul, norm_inv, norm_norm, inv_mul_cancel₀ hx.1.ne']
  have ⟨y, y_mem, hy⟩ := Set.mem_iUnion₂.mp (h₂ x'_mem)
  rw [← vsub_vadd x p, ← show ‖x -ᵥ p‖ • x' = x -ᵥ p from smul_inv_smul₀ hx.1.ne' (x -ᵥ p)]
  exact smul_subset_s y (h₁ y y_mem) <| Set.smul_mem_smul (Set.biInter_subset_of_mem y_mem hx.2) hy

Goal state after:
Module: Mathlib.Topology.Algebra.AsymptoticCone

Goal state before:
pp: case refine_1
k : Type u_1
V : Type u_2
P : Type u_3
inst✝¹⁰ : Field k
inst✝⁹ : LinearOrder k
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module k V
inst✝⁶ : AddTorsor V P
inst✝⁵ : TopologicalSpace V
inst✝⁴ : TopologicalSpace k
inst✝³ : OrderTopology k
inst✝² : IsStrictOrderedRing k
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul k V
s : Set P
⊢ ¬s.Nonempty → 0 ∉ asymptoticCone k s

pp: case refine_2
k : Type u_1
V : Type u_2
P : Type u_3
inst✝¹⁰ : Field k
inst✝⁹ : LinearOrder k
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module k V
inst✝⁶ : AddTorsor V P
inst✝⁵ : TopologicalSpace V
inst✝⁴ : TopologicalSpace k
inst✝³ : OrderTopology k
inst✝² : IsStrictOrderedRing k
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul k V
s : Set P
x✝ : s.Nonempty
⊢ 0 ∈ asymptoticCone k s

Tactic:
· simp +contextual [Set.not_nonempty_iff_eq_empty]

Goal state after:
case refine_2
k : Type u_1
V : Type u_2
P : Type u_3
inst✝¹⁰ : Field k
inst✝⁹ : LinearOrder k
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module k V
inst✝⁶ : AddTorsor V P
inst✝⁵ : TopologicalSpace V
inst✝⁴ : TopologicalSpace k
inst✝³ : OrderTopology k
inst✝² : IsStrictOrderedRing k
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul k V
s : Set P
x✝ : s.Nonempty
⊢ 0 ∈ asymptoticCone k s
Module: Mathlib.Topology.Algebra.AsymptoticCone

Goal state before:
pp: k : Type u_1
V : Type u_2
inst✝⁹ : Field k
inst✝⁸ : LinearOrder k
inst✝⁷ : AddCommGroup V
inst✝⁶ : Module k V
inst✝⁵ : TopologicalSpace V
inst✝⁴ : TopologicalSpace k
inst✝³ : OrderTopology k
inst✝² : IsStrictOrderedRing k
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul k V
s : Set V
hs : ∀ (c : k), 0 < c → ∀ x ∈ s, c • x ∈ s
⊢ asymptoticCone k s = closure[inst✝⁵] s

Tactic:
ext v

Goal state after:
k : Type u_1
V : Type u_2
inst✝⁹ : Field k
inst✝⁸ : LinearOrder k
inst✝⁷ : AddCommGroup V
inst✝⁶ : Module k V
inst✝⁵ : TopologicalSpace V
inst✝⁴ : TopologicalSpace k
inst✝³ : OrderTopology k
inst✝² : IsStrictOrderedRing k
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul k V
s : Set V
hs : ∀ (c : k), 0 < c → ∀ x ∈ s, c • x ∈ s
v : V
⊢ v ∈ asymptoticCone k s ↔ v ∈ closure[inst✝⁵] s
Module: Mathlib.Topology.Algebra.AsymptoticCone

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝¹⁰ : Field k
inst✝⁹ : LinearOrder k
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module k V
inst✝⁶ : AddTorsor V P
inst✝⁵ : TopologicalSpace V
inst✝⁴ : TopologicalSpace k
inst✝³ : OrderTopology k
inst✝² : IsStrictOrderedRing k
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul k V
s : AffineSubspace k P
hs : (↑s).Nonempty
p : P
hp : p ∈ ↑s
⊢ asymptoticCone k ↑s = closure[inst✝⁵] ↑s.direction

Tactic:
ext v

Goal state after:
k : Type u_1
V : Type u_2
P : Type u_3
inst✝¹⁰ : Field k
inst✝⁹ : LinearOrder k
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module k V
inst✝⁶ : AddTorsor V P
inst✝⁵ : TopologicalSpace V
inst✝⁴ : TopologicalSpace k
inst✝³ : OrderTopology k
inst✝² : IsStrictOrderedRing k
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul k V
s : AffineSubspace k P
hs : (↑s).Nonempty
p : P
hp : p ∈ ↑s
v : V
⊢ v ∈ asymptoticCone k ↑s ↔ v ∈ closure[inst✝⁵] ↑s.direction
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: z : ℂ
pi_ne : ↑π ≠ 0
hs : ¬sin (↑π * z) = 0
this✝ : ↑π / sin (↑π * z) = 1 * (↑π / sin (↑π * z))
this : ↑π / sin (↑π * z) = 1 / (sin (↑π * z) / ↑π)
n : ℕ
⊢ ↑π * (z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) =
    ↑π * z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)

Tactic:
← mul_assoc

Goal state after:
z : ℂ
pi_ne : ↑π ≠ 0
hs : ¬sin (↑π * z) = 0
this✝ : ↑π / sin (↑π * z) = 1 * (↑π / sin (↑π * z))
this : ↑π / sin (↑π * z) = 1 / (sin (↑π * z) / ↑π)
n : ℕ
⊢ ↑π * z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2) = ↑π * z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)
Module: Mathlib.Analysis.Normed.Affine.Simplex

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : Ring R
inst✝³ : SeminormedAddCommGroup V
inst✝² : PseudoMetricSpace P
inst✝¹ : Module R V
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex R P n
hr : s.Regular
i j : Fin (n + 1)
hij : i ≠ j
hn : n ≠ 0
⊢ dist (s.points i) (s.points j) = dist (s.points 0) (s.points 1)

Tactic:
by_cases hi : i = 1

Goal state after:
case pos
R : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : Ring R
inst✝³ : SeminormedAddCommGroup V
inst✝² : PseudoMetricSpace P
inst✝¹ : Module R V
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex R P n
hr : s.Regular
i j : Fin (n + 1)
hij : i ≠ j
hn : n ≠ 0
hi : i = 1
⊢ dist (s.points i) (s.points j) = dist (s.points 0) (s.points 1)

case neg
R : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : Ring R
inst✝³ : SeminormedAddCommGroup V
inst✝² : PseudoMetricSpace P
inst✝¹ : Module R V
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex R P n
hr : s.Regular
i j : Fin (n + 1)
hij : i ≠ j
hn : n ≠ 0
hi : ¬i = 1
⊢ dist (s.points i) (s.points j) = dist (s.points 0) (s.points 1)
Module: Mathlib.LinearAlgebra.AffineSpace.Ceva

Goal state before:
pp: case pos
k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : CommRing k
inst✝³ : NoZeroDivisors k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
t : Triangle k P
r : Fin 3 → k
h✝ : Nontrivial k
w : ↑Set.univ → Fin 3 → k := fun i ↦ Finset.affineCombinationLineMapWeights (↑i + 1) (↑i + 2) (r ↑i)
hw : ∀ (i : ↑Set.univ), ∑ j, w i j = 1
w' : Fin (2 + 1) → k
hw' : ∑ j, w' j = 1
h : ∀ (i : ↑Set.univ), ∃ r, ∀ (j : Fin (2 + 1)), r * {↑i}ᶜ.indicator (w i) j = {↑i}ᶜ.indicator w' j
hp' :
  ∀ (i : Fin 3),
    (Finset.affineCombination k Finset.univ t.points) w' ∈
      line[k, t.points i, (AffineMap.lineMap (t.points (i + 1)) (t.points (i + 2))) (r i)]
hp'w :
  ∀ (i : ↑Set.univ),
    (Finset.affineCombination k Finset.univ t.points) w' ∈
      line[k, t.points ↑i, (Finset.affineCombination k Finset.univ t.points) (w i)]
h' :
  ∀ (i : Fin 3), ∃ c, ∀ (j : Fin 3), j ≠ i → c * Finset.affineCombinationLineMapWeights (i + 1) (i + 2) (r i) j = w' j
c : Fin 3 → k := fun i ↦ ⋯.choose
hc✝ : ∀ (i j : Fin 3), j ≠ i → c i * Finset.affineCombinationLineMapWeights (i + 1) (i + 2) (r i) j = w' j
hc1 : ∀ (i : Fin 3), c i * (1 - r i) = w' (i + 1)
hc2 : ∀ (i : Fin 3), c i * r i = w' (i + 2)
hcr : (∏ i, c i) * ∏ i, r i = (∏ i, c i) * ∏ i, (1 - r i)
hc : ∃ a ∈ Finset.univ, c a = 0
⊢ ∏ i, r i = ∏ i, (1 - r i)

Tactic:
obtain ⟨i, -, hi⟩ := hc

Goal state after:
case pos
k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : CommRing k
inst✝³ : NoZeroDivisors k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
t : Triangle k P
r : Fin 3 → k
h✝ : Nontrivial k
w : ↑Set.univ → Fin 3 → k := fun i ↦ Finset.affineCombinationLineMapWeights (↑i + 1) (↑i + 2) (r ↑i)
hw : ∀ (i : ↑Set.univ), ∑ j, w i j = 1
w' : Fin (2 + 1) → k
hw' : ∑ j, w' j = 1
h : ∀ (i : ↑Set.univ), ∃ r, ∀ (j : Fin (2 + 1)), r * {↑i}ᶜ.indicator (w i) j = {↑i}ᶜ.indicator w' j
hp' :
  ∀ (i : Fin 3),
    (Finset.affineCombination k Finset.univ t.points) w' ∈
      line[k, t.points i, (AffineMap.lineMap (t.points (i + 1)) (t.points (i + 2))) (r i)]
hp'w :
  ∀ (i : ↑Set.univ),
    (Finset.affineCombination k Finset.univ t.points) w' ∈
      line[k, t.points ↑i, (Finset.affineCombination k Finset.univ t.points) (w i)]
h' :
  ∀ (i : Fin 3), ∃ c, ∀ (j : Fin 3), j ≠ i → c * Finset.affineCombinationLineMapWeights (i + 1) (i + 2) (r i) j = w' j
c : Fin 3 → k := fun i ↦ ⋯.choose
hc : ∀ (i j : Fin 3), j ≠ i → c i * Finset.affineCombinationLineMapWeights (i + 1) (i + 2) (r i) j = w' j
hc1 : ∀ (i : Fin 3), c i * (1 - r i) = w' (i + 1)
hc2 : ∀ (i : Fin 3), c i * r i = w' (i + 2)
hcr : (∏ i, c i) * ∏ i, r i = (∏ i, c i) * ∏ i, (1 - r i)
i : Fin 3
hi : c i = 0
⊢ ∏ i, r i = ∏ i, (1 - r i)
Module: Mathlib.Analysis.Polynomial.Factorization

Goal state before:
pp: case inl.inl
f : ℝ[X]
n : ℕ
g₁ g₂ : ℝ[X]
hd₁ : g₁.IsMonicOfDegree 1
hg₂ : g₂.IsMonicOfDegree (n + 1)
p₁ p₂ : ℝ[X]
hf : (g₁ * (p₁ * p₂)).IsMonicOfDegree (2 + n)
h : f = g₁ * (p₁ * p₂)
h' : g₂ = p₁ * p₂
hp₁ : p₁.IsMonicOfDegree 1
⊢ ∃ f₁ f₂, f₁.IsMonicOfDegree 2 ∧ f₂.IsMonicOfDegree n ∧ f = f₁ * f₂

Tactic:
← mul_assoc

Goal state after:
case inl.inl
f : ℝ[X]
n : ℕ
g₁ g₂ : ℝ[X]
hd₁ : g₁.IsMonicOfDegree 1
hg₂ : g₂.IsMonicOfDegree (n + 1)
p₁ p₂ : ℝ[X]
hf : (g₁ * p₁ * p₂).IsMonicOfDegree (2 + n)
h : f = g₁ * p₁ * p₂
h' : g₂ = p₁ * p₂
hp₁ : p₁.IsMonicOfDegree 1
⊢ ∃ f₁ f₂, f₁.IsMonicOfDegree 2 ∧ f₂.IsMonicOfDegree n ∧ f = f₁ * f₂
Module: Mathlib.Topology.Algebra.Order.LiminfLimsup

Goal state before:
pp: case inr
ι : Type u_1
R : Type u_4
inst✝⁶ : ConditionallyCompleteLinearOrder R
inst✝⁵ : TopologicalSpace R
inst✝⁴ : OrderTopology R
F : Filter ι
inst✝³ : AddCommSemigroup R
inst✝² : Sub R
inst✝¹ : ContinuousSub R
inst✝ : OrderedSub R
f : ι → R
c : R
bdd_above : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) F f
cobdd : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) F f
h✝ : F.NeBot
⊢ limsup (fun i ↦ f i - c) F = limsup f F - c

Tactic:
apply (Monotone.map_limsSup_of_continuousAt (F := F.map f) (f := fun (x : R) ↦ x - c) _ _).symm

Goal state after:
ι : Type u_1
R : Type u_4
inst✝⁶ : ConditionallyCompleteLinearOrder R
inst✝⁵ : TopologicalSpace R
inst✝⁴ : OrderTopology R
F : Filter ι
inst✝³ : AddCommSemigroup R
inst✝² : Sub R
inst✝¹ : ContinuousSub R
inst✝ : OrderedSub R
f : ι → R
c : R
bdd_above : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) F f
cobdd : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) F f
h✝ : F.NeBot
⊢ Monotone fun x ↦ x - c

ι : Type u_1
R : Type u_4
inst✝⁶ : ConditionallyCompleteLinearOrder R
inst✝⁵ : TopologicalSpace R
inst✝⁴ : OrderTopology R
F : Filter ι
inst✝³ : AddCommSemigroup R
inst✝² : Sub R
inst✝¹ : ContinuousSub R
inst✝ : OrderedSub R
f : ι → R
c : R
bdd_above : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) F f
cobdd : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) F f
h✝ : F.NeBot
⊢ ContinuousAt (fun x ↦ x - c) (map f F).limsSup
Module: Mathlib.Analysis.Normed.Unbundled.FiniteExtension

Goal state before:
pp: case h.h
K : Type u_1
L : Type u_2
inst✝⁴ : NormedField K
inst✝³ : Ring L
inst✝² : Algebra K L
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
B : Basis ι K L
i : ι
hBi : B i = 1
hna : IsNonarchimedean Norm.norm
c : ι × ι
left✝¹ : c ∈ univ
hc : (univ.sup' ⋯ fun i ↦ B.norm (B i.1 * B i.2)) = B.norm (B c.1 * B c.2)
x y : L
ixy : ι
left✝ : ixy ∈ univ
hixy_def : (univ.sup' ⋯ fun i ↦ ‖(B.repr (x * y)) i‖) = ‖(B.repr (x * y)) ixy‖
hna' : IsNonarchimedean ⇑(NormedField.toMulRingNorm K)
k : ι
hk :
  ‖∑ x_1, (B.repr x) x_1 * ∑ x, (B.repr y) x * (B.repr (B x_1 * B x)) ixy‖ ≤
    ‖(B.repr x) k‖ * ‖∑ x, (B.repr y) x * (B.repr (B k * B x)) ixy‖
k' : ι
hk' : ‖∑ j, (B.repr y) j • (B.repr (B k * B j)) ixy‖ ≤ ‖(B.repr y) k' • (B.repr (B k * B k')) ixy‖
⊢ ‖(B.repr x) k‖ * (‖(B.repr y) k'‖ * ‖(B.repr (B k * B k')) ixy‖) ≤ B.norm x * B.norm y * B.norm (B c.1 * B c.2)

Tactic:
← mul_assoc

Goal state after:
case h.h
K : Type u_1
L : Type u_2
inst✝⁴ : NormedField K
inst✝³ : Ring L
inst✝² : Algebra K L
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
B : Basis ι K L
i : ι
hBi : B i = 1
hna : IsNonarchimedean Norm.norm
c : ι × ι
left✝¹ : c ∈ univ
hc : (univ.sup' ⋯ fun i ↦ B.norm (B i.1 * B i.2)) = B.norm (B c.1 * B c.2)
x y : L
ixy : ι
left✝ : ixy ∈ univ
hixy_def : (univ.sup' ⋯ fun i ↦ ‖(B.repr (x * y)) i‖) = ‖(B.repr (x * y)) ixy‖
hna' : IsNonarchimedean ⇑(NormedField.toMulRingNorm K)
k : ι
hk :
  ‖∑ x_1, (B.repr x) x_1 * ∑ x, (B.repr y) x * (B.repr (B x_1 * B x)) ixy‖ ≤
    ‖(B.repr x) k‖ * ‖∑ x, (B.repr y) x * (B.repr (B k * B x)) ixy‖
k' : ι
hk' : ‖∑ j, (B.repr y) j • (B.repr (B k * B j)) ixy‖ ≤ ‖(B.repr y) k' • (B.repr (B k * B k')) ixy‖
⊢ ‖(B.repr x) k‖ * ‖(B.repr y) k'‖ * ‖(B.repr (B k * B k')) ixy‖ ≤ B.norm x * B.norm y * B.norm (B c.1 * B c.2)
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromConst

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
c : R
f : RingSeminorm R
hf1 : f 1 ≤ 1
hc : f c ≠ 0
hpm : IsPowMul ⇑f
x : R
m n : ℕ
hmn : m ≤ n
⊢ f (x * (c ^ m * c ^ (n - m))) / f c ^ n ≤ f (x * c ^ m) / f c ^ m

Tactic:
← mul_assoc

Goal state after:
R : Type u_1
inst✝ : CommRing R
c : R
f : RingSeminorm R
hf1 : f 1 ≤ 1
hc : f c ≠ 0
hpm : IsPowMul ⇑f
x : R
m n : ℕ
hmn : m ≤ n
⊢ f (x * c ^ m * c ^ (n - m)) / f c ^ n ≤ f (x * c ^ m) / f c ^ m
Module: Mathlib.Analysis.Normed.Group.HomCompletion

Goal state before:
pp: G : Type u_1
inst✝³ : SeminormedAddCommGroup G
H : Type u_2
inst✝² : SeminormedAddCommGroup H
inst✝¹ : T0Space H
inst✝ : CompleteSpace H
f : NormedAddGroupHom G H
g : NormedAddGroupHom (Completion G) H
hg : ∀ (v : G), f v = g ↑v
⊢ f.extension = g

Tactic:
ext v

Goal state after:
G : Type u_1
inst✝³ : SeminormedAddCommGroup G
H : Type u_2
inst✝² : SeminormedAddCommGroup H
inst✝¹ : T0Space H
inst✝ : CompleteSpace H
f : NormedAddGroupHom G H
g : NormedAddGroupHom (Completion G) H
hg : ∀ (v : G), f v = g ↑v
v : Completion G
⊢ f.extension v = g v
Module: Mathlib.Analysis.Normed.Group.SemiNormedGrp.Kernels

Goal state before:
pp: V W : SemiNormedGrp
f g : V ⟶ W
⊢ ofHom (NormedAddGroupHom.incl (Hom.hom (f - g)).ker) ≫ f = ofHom (NormedAddGroupHom.incl (Hom.hom (f - g)).ker) ≫ g

Tactic:
ext v

Goal state after:
V W : SemiNormedGrp
f g : V ⟶ W
v : { carrier := ↥(Hom.hom (f - g)).ker, str := AddSubgroup.seminormedAddCommGroup }.carrier
⊢ (Hom.hom (ofHom (NormedAddGroupHom.incl (Hom.hom (f - g)).ker) ≫ f)) v =
    (Hom.hom (ofHom (NormedAddGroupHom.incl (Hom.hom (f - g)).ker) ≫ g)) v
Module: Mathlib.Analysis.Normed.Module.Ball.RadialEquiv

Goal state before:
pp: E✝ : Type u_1
inst✝³ : NormedAddCommGroup E✝
inst✝² : NormedSpace ℝ E✝
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
r : ℝ
hr : 0 < r
x : ↑{0}ᶜ
⊢ 0 < ‖↑x‖

Tactic:
simpa [-Subtype.coe_prop] using x.2

Goal state after:
Module: Mathlib.Analysis.Normed.Module.MStructure

Goal state before:
pp: X : Type u_1
inst✝³ : NormedAddCommGroup X
M : Type u_2
inst✝² : Ring M
inst✝¹ : Module M X
inst✝ : FaithfulSMul M X
P Q R : { P // IsLprojection X P }
⊢ ↑P * ↑P + ↑P * (↑Q * ↑R * ↑Pᶜ) + (↑Pᶜ * ↑R * ↑P + ↑Pᶜ * (↑R * (↑Q * ↑R) * ↑Pᶜ)) = ↑P + ↑Q * ↑R * ↑Pᶜ

Tactic:
← coe_inf Q,

Goal state after:
X : Type u_1
inst✝³ : NormedAddCommGroup X
M : Type u_2
inst✝² : Ring M
inst✝¹ : Module M X
inst✝ : FaithfulSMul M X
P Q R : { P // IsLprojection X P }
⊢ ↑P * ↑P + ↑P * (↑(Q ⊓ R) * ↑Pᶜ) + (↑Pᶜ * ↑R * ↑P + ↑Pᶜ * (↑R * ↑(Q ⊓ R) * ↑Pᶜ)) = ↑P + ↑(Q ⊓ R) * ↑Pᶜ
Module: Mathlib.Analysis.Normed.Module.MStructure

Goal state before:
pp: X : Type u_1
inst✝³ : NormedAddCommGroup X
M : Type u_2
inst✝² : Ring M
inst✝¹ : Module M X
inst✝ : FaithfulSMul M X
P Q R : { P // IsLprojection X P }
⊢ ↑P + ↑Pᶜ * (↑Q * ↑R) * ↑Pᶜ = ↑P + ↑Pᶜ * (↑Q * ↑R)

Tactic:
← coe_inf Q,

Goal state after:
X : Type u_1
inst✝³ : NormedAddCommGroup X
M : Type u_2
inst✝² : Ring M
inst✝¹ : Module M X
inst✝ : FaithfulSMul M X
P Q R : { P // IsLprojection X P }
⊢ ↑P + ↑Pᶜ * ↑(Q ⊓ R) * ↑Pᶜ = ↑P + ↑Pᶜ * ↑(Q ⊓ R)
Module: Mathlib.Analysis.Normed.Module.MStructure

Goal state before:
pp: X : Type u_1
inst✝³ : NormedAddCommGroup X
M : Type u_2
inst✝² : Ring M
inst✝¹ : Module M X
inst✝ : FaithfulSMul M X
P Q : { P // IsLprojection X P }
⊢ P ≤ P ⊔ Q

Tactic:
rw [le_def, coe_inf, coe_sup, ← add_sub, mul_add, mul_sub, ← mul_assoc, P.prop.proj.eq,
      sub_self, add_zero]

Goal state after:
Module: Mathlib.Analysis.Normed.Module.MStructure

Goal state before:
pp: X : Type u_1
inst✝³ : NormedAddCommGroup X
M : Type u_2
inst✝² : Ring M
inst✝¹ : Module M X
inst✝ : FaithfulSMul M X
P Q R : { P // IsLprojection X P }
⊢ ↑P = ↑P * ↑Q → ↑P = ↑P * ↑R → ↑P = ↑P * (↑Q * ↑R)

Tactic:
← mul_assoc

Goal state after:
X : Type u_1
inst✝³ : NormedAddCommGroup X
M : Type u_2
inst✝² : Ring M
inst✝¹ : Module M X
inst✝ : FaithfulSMul M X
P Q R : { P // IsLprojection X P }
⊢ ↑P = ↑P * ↑Q → ↑P = ↑P * ↑R → ↑P = ↑P * ↑Q * ↑R
Module: Mathlib.Analysis.Normed.Module.MStructure

Goal state before:
pp: X : Type u_1
inst✝³ : NormedAddCommGroup X
M : Type u_2
inst✝² : Ring M
inst✝¹ : Module M X
inst✝ : FaithfulSMul M X
P Q R : { P // IsLprojection X P }
⊢ ↑P + ↑Q * (↑R * ↑Pᶜ) = ↑P + ↑Q * ↑R * ↑Pᶜ

Tactic:
← mul_assoc

Goal state after:
X : Type u_1
inst✝³ : NormedAddCommGroup X
M : Type u_2
inst✝² : Ring M
inst✝¹ : Module M X
inst✝ : FaithfulSMul M X
P Q R : { P // IsLprojection X P }
⊢ ↑P + ↑Q * ↑R * ↑Pᶜ = ↑P + ↑Q * ↑R * ↑Pᶜ
Module: Mathlib.Analysis.Normed.Module.Ball.RadialEquiv

Goal state before:
pp: case ha
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
r : ℝ
hr : r ≠ 0
V : Set ↑(sphere 0 r)
hV : IsOpen[instTopologicalSpaceSubtype] V
y : { x // x ∈ sphere 0 r }
hyV : y ∈ V
U : Set ℝ
hU : IsOpen U
hU₀ : 0 ∉ U
hr₀ : 0 < r
x : { x // x ∈ Ioi 0 }
hxU : ↑x ∈ U
this :
  Subtype.val ∘ ⇑(homeomorphSphereProd E r hr₀).symm '' V ×ˢ Ioi 0 ↓∩ U ∈
    𝓝[{0}ᶜ] ↑((homeomorphSphereProd E r hr₀).symm (y, x))
⊢ ↑((homeomorphSphereProd E r hr₀).symm (y, x)) ∈ {0}ᶜ

Tactic:
· simp [x.2.out.ne', ne_zero_of_mem_sphere, hr₀.ne']

Goal state after: