mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.LinearAlgebra.AffineSpace.Ceva

Goal state before:
pp: 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
⊢ w' (i + 1) = 0

Tactic:
simpa [hi] using (hc1 i).symm

Goal state after:
Module: Mathlib.Analysis.Normed.Algebra.TrivSqZeroExt

Goal state before:
pp: 𝕜 : Type u_1
R : Type u_3
M : Type u_4
inst✝¹⁶ : Field 𝕜
inst✝¹⁵ : CharZero 𝕜
inst✝¹⁴ : Ring R
inst✝¹³ : AddCommGroup M
inst✝¹² : Algebra 𝕜 R
inst✝¹¹ : Module 𝕜 M
inst✝¹⁰ : Module R M
inst✝⁹ : Module Rᵐᵒᵖ M
inst✝⁸ : SMulCommClass R Rᵐᵒᵖ M
inst✝⁷ : IsScalarTower 𝕜 R M
inst✝⁶ : IsScalarTower 𝕜 Rᵐᵒᵖ M
inst✝⁵ : TopologicalSpace R
inst✝⁴ : TopologicalSpace M
inst✝³ : IsTopologicalRing R
inst✝² : IsTopologicalAddGroup M
inst✝¹ : ContinuousSMul R M
inst✝ : ContinuousSMul Rᵐᵒᵖ M
x : tsze R M
hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd
e : R
h : HasSum (fun n ↦ (expSeries 𝕜 R n) fun x_1 ↦ x.fst) e
⊢ HasSum (fun n ↦ (expSeries 𝕜 (tsze R M) n) fun x_1 ↦ x) (inl e + inr (e • x.snd))

Tactic:
by
  have : HasSum (fun n => fst (expSeries 𝕜 (tsze R M) n fun _ => x)) e := by
    simpa [fst_expSeries] using h
  simpa only [inl_fst_add_inr_snd_eq] using
    (hasSum_inl _ <| this).add (hasSum_inr _ <| hasSum_snd_expSeries_of_smul_comm 𝕜 x hx h)

Goal state after:
Module: Mathlib.Analysis.Normed.Algebra.TrivSqZeroExt

Goal state before:
pp: R : Type u_3
M : Type u_4
inst✝¹⁵ : CommRing R
inst✝¹⁴ : AddCommGroup M
inst✝¹³ : Algebra ℚ R
inst✝¹² : Module ℚ M
inst✝¹¹ : Module R M
inst✝¹⁰ : Module Rᵐᵒᵖ M
inst✝⁹ : IsCentralScalar R M
inst✝⁸ : TopologicalSpace R
inst✝⁷ : TopologicalSpace M
inst✝⁶ : IsTopologicalRing R
inst✝⁵ : IsTopologicalAddGroup M
inst✝⁴ : ContinuousSMul R M
inst✝³ : ContinuousSMul Rᵐᵒᵖ M
inst✝² : T2Space R
inst✝¹ : T2Space M
x : tsze R M
inst✝ : Invertible x.fst
⊢ x = x.fst • exp (⅟x.fst • inr x.snd)

Tactic:
← inr_smul,

Goal state after:
R : Type u_3
M : Type u_4
inst✝¹⁵ : CommRing R
inst✝¹⁴ : AddCommGroup M
inst✝¹³ : Algebra ℚ R
inst✝¹² : Module ℚ M
inst✝¹¹ : Module R M
inst✝¹⁰ : Module Rᵐᵒᵖ M
inst✝⁹ : IsCentralScalar R M
inst✝⁸ : TopologicalSpace R
inst✝⁷ : TopologicalSpace M
inst✝⁶ : IsTopologicalRing R
inst✝⁵ : IsTopologicalAddGroup M
inst✝⁴ : ContinuousSMul R M
inst✝³ : ContinuousSMul Rᵐᵒᵖ M
inst✝² : T2Space R
inst✝¹ : T2Space M
x : tsze R M
inst✝ : Invertible x.fst
⊢ x = x.fst • exp (inr (⅟x.fst • x.snd))
Module: Mathlib.Analysis.Normed.Algebra.TrivSqZeroExt

Goal state before:
pp: R : Type u_3
M : Type u_4
inst✝¹⁵ : CommRing R
inst✝¹⁴ : AddCommGroup M
inst✝¹³ : Algebra ℚ R
inst✝¹² : Module ℚ M
inst✝¹¹ : Module R M
inst✝¹⁰ : Module Rᵐᵒᵖ M
inst✝⁹ : IsCentralScalar R M
inst✝⁸ : TopologicalSpace R
inst✝⁷ : TopologicalSpace M
inst✝⁶ : IsTopologicalRing R
inst✝⁵ : IsTopologicalAddGroup M
inst✝⁴ : ContinuousSMul R M
inst✝³ : ContinuousSMul Rᵐᵒᵖ M
inst✝² : T2Space R
inst✝¹ : T2Space M
x : tsze R M
inst✝ : Invertible x.fst
⊢ x = inl (x.fst • 1) + x.fst • inr (⅟x.fst • x.snd)

Tactic:
← inr_smul,

Goal state after:
R : Type u_3
M : Type u_4
inst✝¹⁵ : CommRing R
inst✝¹⁴ : AddCommGroup M
inst✝¹³ : Algebra ℚ R
inst✝¹² : Module ℚ M
inst✝¹¹ : Module R M
inst✝¹⁰ : Module Rᵐᵒᵖ M
inst✝⁹ : IsCentralScalar R M
inst✝⁸ : TopologicalSpace R
inst✝⁷ : TopologicalSpace M
inst✝⁶ : IsTopologicalRing R
inst✝⁵ : IsTopologicalAddGroup M
inst✝⁴ : ContinuousSMul R M
inst✝³ : ContinuousSMul Rᵐᵒᵖ M
inst✝² : T2Space R
inst✝¹ : T2Space M
x : tsze R M
inst✝ : Invertible x.fst
⊢ x = inl (x.fst • 1) + inr (x.fst • ⅟x.fst • x.snd)
Module: Mathlib.LinearAlgebra.AffineSpace.Ceva

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Field k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
t : Triangle k P
r : Fin 3 → k
hr0 : ∀ (i : Fin 3), r i ≠ 0
p' : P
hp' : ∀ (i : Fin 3), p' ∈ line[k, t.points i, (AffineMap.lineMap (t.points (i + 1)) (t.points (i + 2))) (r i)]
⊢ ∏ i, r i / (1 - r i) = 1

Tactic:
by
  rw [Finset.prod_div_distrib, ← prod_eq_prod_one_sub_of_mem_line_point_lineMap hp', div_self]
  exact Finset.prod_ne_zero_iff.2 fun _ _ ↦ hr0 _

Goal state after:
Module: Mathlib.Analysis.Normed.Algebra.GelfandMazur

Goal state before:
pp: F : Type u_1
inst✝⁴ : NormedRing F
inst✝³ : NormOneClass F
inst✝² : NormMulClass F
inst✝¹ : NormedAlgebra ℂ F
inst✝ : Nontrivial F
x : F
z : ℂ
hz : ‖x - (algebraMap ℂ F) z‖ = 0
⊢ ∃ a, (ofId ℂ F) a = x

Tactic:
refine ⟨z, ?_⟩

Goal state after:
F : Type u_1
inst✝⁴ : NormedRing F
inst✝³ : NormOneClass F
inst✝² : NormMulClass F
inst✝¹ : NormedAlgebra ℂ F
inst✝ : Nontrivial F
x : F
z : ℂ
hz : ‖x - (algebraMap ℂ F) z‖ = 0
⊢ (ofId ℂ F) z = x
Module: Mathlib.Analysis.Normed.Algebra.GelfandMazur

Goal state before:
pp: F : Type u_1
inst✝³ : NormedRing F
inst✝² : NormedAlgebra ℝ F
inst✝¹ : NormOneClass F
inst✝ : NormMulClass F
x : F
z : ℝ × ℝ
h : IsMinOn (fun x_1 ↦ ‖φ x x_1‖) Set.univ z
w✝ : ℝ × ℝ
M : ℝ := ‖φ x z‖
H : M ≠ 0
hM : M = ‖φ x z‖
hM₀ : 0 < M
w u : ℝ × ℝ
hw : ‖φ x w‖ = M
n : ℕ
hn : n > 0
q : ℝ × ℝ → ℝ[X] := fun y ↦ X ^ 2 - C y.1 * X + C y.2
hq : ∀ (y : ℝ × ℝ), (q y).IsMonicOfDegree 2
hsub : q w - q u = (C u.1 - C w.1) * X + C w.2 - C u.2
⊢ q u ∣ q w ^ n - (q w - q u) ^ n

Tactic:
← sub_sub_self (q w) (q u)

Goal state after:
F : Type u_1
inst✝³ : NormedRing F
inst✝² : NormedAlgebra ℝ F
inst✝¹ : NormOneClass F
inst✝ : NormMulClass F
x : F
z : ℝ × ℝ
h : IsMinOn (fun x_1 ↦ ‖φ x x_1‖) Set.univ z
w✝ : ℝ × ℝ
M : ℝ := ‖φ x z‖
H : M ≠ 0
hM : M = ‖φ x z‖
hM₀ : 0 < M
w u : ℝ × ℝ
hw : ‖φ x w‖ = M
n : ℕ
hn : n > 0
q : ℝ × ℝ → ℝ[X] := fun y ↦ X ^ 2 - C y.1 * X + C y.2
hq : ∀ (y : ℝ × ℝ), (q y).IsMonicOfDegree 2
hsub : q w - q u = (C u.1 - C w.1) * X + C w.2 - C u.2
⊢ q w - (q w - q u) ∣ q w ^ n - (q w - q u) ^ n
Module: Mathlib.Analysis.Normed.Algebra.GelfandMazur

Goal state before:
pp: case refine_2
F : Type u_1
inst✝² : NormedRing F
inst✝¹ : NormedAlgebra ℝ F
inst✝ : NormOneClass F
x : F
c : ℝ
hc₀ : 0 < c
hbd : ∀ (r : ℝ), c ≤ ‖x - (algebraMap ℝ F) r‖
⊢ Tendsto (fun x_1 ↦ ‖x_1.1 • x - (algebraMap ℝ F) x_1.2‖) (cobounded ℝ ×ˢ ⊤) atTop

Tactic:
· suffices Tendsto (fun y : ℝ × ℝ ↦ ‖y.1‖ * c) (cobounded ℝ ×ˢ ⊤) atTop by
      refine tendsto_atTop_mono' _ ?_ this
      filter_upwards [prod_mem_prod (isBounded_singleton (x := 0)) univ_mem] with y hy
      calc ‖y.1‖ * c
        _ ≤ ‖y.1‖ * ‖x - algebraMap ℝ F (y.1⁻¹ * y.2)‖ := by gcongr; exact hbd _
        _ = ‖y.1 • x - algebraMap ℝ F y.2‖ := by
          simp only [← norm_smul, smul_sub, smul_smul, Algebra.algebraMap_eq_smul_one]
          simp_all
    rw [tendsto_mul_const_atTop_of_pos hc₀, tendsto_norm_atTop_iff_cobounded]
    exact tendsto_fst

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

Goal state before:
pp: case inr
R : Type u_1
inst✝³ : NonAssocRing R
inst✝² : DecidableEq R
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
x y : R
hx : x ≠ 0
⊢ AddGroupSeminorm.toFun 1 (x * y) = AddGroupSeminorm.toFun 1 x * AddGroupSeminorm.toFun 1 y

Tactic:
obtain rfl | hy := eq_or_ne y 0

Goal state after:
case inr.inl
R : Type u_1
inst✝³ : NonAssocRing R
inst✝² : DecidableEq R
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
x : R
hx : x ≠ 0
⊢ AddGroupSeminorm.toFun 1 (x * 0) = AddGroupSeminorm.toFun 1 x * AddGroupSeminorm.toFun 1 0

case inr.inr
R : Type u_1
inst✝³ : NonAssocRing R
inst✝² : DecidableEq R
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
x y : R
hx : x ≠ 0
hy : y ≠ 0
⊢ AddGroupSeminorm.toFun 1 (x * y) = AddGroupSeminorm.toFun 1 x * AddGroupSeminorm.toFun 1 y
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

Goal state before:
pp: case a
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_ne_zero : f ≠ 0
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
⊢ 1 ≤ ⨆ y, f y / f y

Tactic:
· rw [← div_self (map_one_ne_zero f_ne_zero f_nonneg f_mul)]
    have h_bdd : BddAbove (Set.range fun y ↦ f y / f y) := by
      use (1 : ℝ)
      rintro r ⟨y, rfl⟩
      by_cases hy : f y = 0
      · simp only [hy, div_zero, zero_le_one]
      · simp only [div_self hy, le_refl]
    exact le_ciSup h_bdd (1 : R)

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
f_add : ∀ (a b : R), f (a + b) ≤ f a + f b
x y : R
⊢ seminormFromBounded' f (x + y) ≤ seminormFromBounded' f x + seminormFromBounded' f y

Tactic:
refine ciSup_le (fun z ↦ ?_)
  suffices hf : f ((x + y) * z) / f z ≤ f (x * z) / f z + f (y * z) / f z by
    exact le_trans hf (add_le_add
      (le_ciSup_of_le (seminormFromBounded_bddAbove_range f_nonneg f_mul x) z (le_refl _))
      (le_ciSup_of_le (seminormFromBounded_bddAbove_range f_nonneg f_mul y) z (le_refl _)))
  by_cases hz : f z = 0
  · simp only [hz, div_zero, zero_add, le_refl]
  · rw [← add_div, div_le_div_iff_of_pos_right (lt_of_le_of_ne' (f_nonneg _) hz), add_mul]
    exact f_add _ _

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

Goal state before:
pp: K : Type u_2
inst✝² : NormedField K
L : Type u_3
inst✝¹ : Field L
inst✝ : Algebra K L
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
hf_na : IsNonarchimedean ⇑f
p : K[X]
hp : p.Monic
x : L
hx : (aeval x) p = 0
hx0 : ¬f x = 0
h_ge : ¬f x ≤ iSup (spectralValueTerms p)
n : ℕ
hn : n < p.natDegree
hexp : (‖p.coeff n‖ ^ (1 / (↑p.natDegree - ↑n))) ^ (p.natDegree - n) = ‖p.coeff n‖
⊢ ‖p.coeff n‖ ^ (1 / (↑p.natDegree - ↑n)) < f x

Tactic:
iSup,

Goal state after:
K : Type u_2
inst✝² : NormedField K
L : Type u_3
inst✝¹ : Field L
inst✝ : Algebra K L
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
hf_na : IsNonarchimedean ⇑f
p : K[X]
hp : p.Monic
x : L
hx : (aeval x) p = 0
hx0 : ¬f x = 0
h_ge : ¬f x ≤ sSup (Set.range (spectralValueTerms p))
n : ℕ
hn : n < p.natDegree
hexp : (‖p.coeff n‖ ^ (1 / (↑p.natDegree - ↑n))) ^ (p.natDegree - n) = ‖p.coeff n‖
⊢ ‖p.coeff n‖ ^ (1 / (↑p.natDegree - ↑n)) < f x
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: case pos
K : Type u_2
inst✝² : NormedField K
L : Type u_3
inst✝¹ : Field L
inst✝ : Algebra K L
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
hf_na : IsNonarchimedean ⇑f
p : K[X]
hp : p.Monic
x : L
hx : (aeval x) p = 0
hx0 : ¬f x = 0
h_ge : ¬f x ≤ spectralValue p
hn_lt : ∀ n < p.natDegree, ‖p.coeff n‖ < f x ^ (p.natDegree - n)
h_deg : 0 < p.natDegree
n : ℕ
hn : n < p.natDegree
hn0 : n = 0
⊢ ‖p.coeff 0‖ * f 1 < f (x ^ p.natDegree)

Tactic:
hf_pm _ (succ_le_iff.mpr h_deg),

Goal state after:
case pos
K : Type u_2
inst✝² : NormedField K
L : Type u_3
inst✝¹ : Field L
inst✝ : Algebra K L
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
hf_na : IsNonarchimedean ⇑f
p : K[X]
hp : p.Monic
x : L
hx : (aeval x) p = 0
hx0 : ¬f x = 0
h_ge : ¬f x ≤ spectralValue p
hn_lt : ∀ n < p.natDegree, ‖p.coeff n‖ < f x ^ (p.natDegree - n)
h_deg : 0 < p.natDegree
n : ℕ
hn : n < p.natDegree
hn0 : n = 0
⊢ ‖p.coeff 0‖ * f 1 < f x ^ p.natDegree
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: case neg
K : Type u_2
inst✝² : NormedField K
L : Type u_3
inst✝¹ : Field L
inst✝ : Algebra K L
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
hf_na : IsNonarchimedean ⇑f
p : K[X]
hp : p.Monic
x : L
hx : (aeval x) p = 0
hx0 : ¬f x = 0
h_ge : ¬f x ≤ spectralValue p
hn_lt : ∀ n < p.natDegree, ‖p.coeff n‖ < f x ^ (p.natDegree - n)
h_deg : 0 < p.natDegree
n : ℕ
hn : n < p.natDegree
hn0 : ¬n = 0
this : p.natDegree = p.natDegree - n + n
⊢ ‖p.coeff n‖ * f x ^ n < f (x ^ p.natDegree)

Tactic:
hf_pm _ (succ_le_iff.mpr h_deg),

Goal state after:
case neg
K : Type u_2
inst✝² : NormedField K
L : Type u_3
inst✝¹ : Field L
inst✝ : Algebra K L
f : AlgebraNorm K L
hf_pm : IsPowMul ⇑f
hf_na : IsNonarchimedean ⇑f
p : K[X]
hp : p.Monic
x : L
hx : (aeval x) p = 0
hx0 : ¬f x = 0
h_ge : ¬f x ≤ spectralValue p
hn_lt : ∀ n < p.natDegree, ‖p.coeff n‖ < f x ^ (p.natDegree - n)
h_deg : 0 < p.natDegree
n : ℕ
hn : n < p.natDegree
hn0 : ¬n = 0
this : p.natDegree = p.natDegree - n + n
⊢ ‖p.coeff n‖ * f x ^ n < f x ^ p.natDegree
Module: Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
x y : R
hn : ∀ (n : ℕ), ∃ m < n + 1, μ ((x + y) ^ n) ^ (1 / ↑n) ≤ (μ (x ^ m) * μ (y ^ (n - m))) ^ (1 / ↑n)
n : ℕ
⊢ mu μ hn n ≤ n

Tactic:
by
  simpa [mu] using (Classical.choose_spec (hn n)).1

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
s : ℕ → ℕ
hs_le : ∀ (n : ℕ), s n ≤ n
x : R
a : ℝ
a_in : a ∈ Set.Icc 0 1
ψ : ℕ → ℕ
hψ_mono : StrictMono ψ
hψ_lim : Tendsto ((fun n ↦ ↑(s n) / ↑n) ∘ ψ) atTop (𝓝 a)
ha : a = 0
⊢ limsup (fun n ↦ μ (x ^ s (ψ n)) ^ (1 / ↑(ψ n))) atTop ≤ smoothingFun μ x ^ a

Tactic:
rw [ha] at hψ_lim

Goal state after:
case pos
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
s : ℕ → ℕ
hs_le : ∀ (n : ℕ), s n ≤ n
x : R
a : ℝ
a_in : a ∈ Set.Icc 0 1
ψ : ℕ → ℕ
hψ_mono : StrictMono ψ
hψ_lim : Tendsto ((fun n ↦ ↑(s n) / ↑n) ∘ ψ) atTop (𝓝 0)
ha : a = 0
⊢ limsup (fun n ↦ μ (x ^ s (ψ n)) ^ (1 / ↑(ψ n))) atTop ≤ smoothingFun μ x ^ a
Module: Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm

Goal state before:
pp: case neg
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
s : ℕ → ℕ
hs_le : ∀ (n : ℕ), s n ≤ n
x : R
a : ℝ
a_in : a ∈ Set.Icc 0 1
ψ : ℕ → ℕ
hψ_mono : StrictMono ψ
hψ_lim : Tendsto ((fun n ↦ ↑(s n) / ↑n) ∘ ψ) atTop (𝓝 a)
ha : ¬a = 0
⊢ limsup (fun n ↦ μ (x ^ s (ψ n)) ^ (1 / ↑(ψ n))) atTop ≤ smoothingFun μ x ^ a

Tactic:
have ha_pos : 0 < a := lt_of_le_of_ne a_in.1 (Ne.symm ha)
    have h_eq : (fun n : ℕ =>
        (μ (x ^ s (ψ n)) ^ (1 / (s (ψ n) : ℝ))) ^ ((s (ψ n) : ℝ) / (ψ n : ℝ))) =ᶠ[atTop]
        fun n : ℕ => μ (x ^ s (ψ n)) ^ (1 / (ψ n : ℝ)) := by
      have h : (fun n : ℕ => (1 : ℝ) / (s (ψ n) : ℝ) * (s (ψ n) : ℝ)) =ᶠ[atTop] 1 := by
        apply Filter.EventuallyEq.div_mul_cancel_atTop
        exact (tendsto_natCast_atTop_atTop.comp hψ_mono.tendsto_atTop).num ha_pos hψ_lim
      simp_rw [← rpow_mul (apply_nonneg μ _), mul_div]
      exact EventuallyEq.comp₂ EventuallyEq.rfl HPow.hPow (h.div EventuallyEq.rfl)
    exact ((tendsto_smoothingFun_of_map_one_le_one μ hμ1 x |>.comp <|
      tendsto_natCast_atTop_iff.mp <| (tendsto_natCast_atTop_atTop.comp
        hψ_mono.tendsto_atTop).num ha_pos hψ_lim).rpow
          hψ_lim <| .inr ha_pos).congr' h_eq |>.limsup_eq.le

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

Goal state before:
pp: R : Type u_1
K : Type u
inst✝⁴ : NontriviallyNormedField K
L : Type v
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra.IsAlgebraic K L
hu : IsUltrametricDist K
inst✝ : CompleteSpace K
x y : L
⊢ spectralNorm K L (x - y) = spectralNorm K L (-x + y)

Tactic:
rw [← spectralNorm_neg, sub_eq_add_neg, neg_add, neg_neg]

Goal state after:
R : Type u_1
K : Type u
inst✝⁴ : NontriviallyNormedField K
L : Type v
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra.IsAlgebraic K L
hu : IsUltrametricDist K
inst✝ : CompleteSpace K
x y : L
⊢ IsAlgebraic K (x - y)
Module: Mathlib.Analysis.Normed.Module.ContinuousInverse

Goal state before:
pp: R : Type u_1
inst✝¹² : Semiring R
E : Type u_2
E' : Type u_3
F : Type u_4
F' : Type u_5
inst✝¹¹ : TopologicalSpace E
inst✝¹⁰ : AddCommMonoid E
inst✝⁹ : Module R E
inst✝⁸ : TopologicalSpace E'
inst✝⁷ : AddCommMonoid E'
inst✝⁶ : Module R E'
inst✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommMonoid F
inst✝³ : Module R F
inst✝² : TopologicalSpace F'
inst✝¹ : AddCommMonoid F'
inst✝ : Module R F'
f : E →L[R] F
g : E' →L[R] F'
hg : g.HasLeftInverse
finv : F →L[R] E
hfinv : LeftInverse ⇑finv ⇑f
⊢ (f.prodMap g).HasLeftInverse

Tactic:
obtain ⟨ginv, hginv⟩ := hg

Goal state after:
R : Type u_1
inst✝¹² : Semiring R
E : Type u_2
E' : Type u_3
F : Type u_4
F' : Type u_5
inst✝¹¹ : TopologicalSpace E
inst✝¹⁰ : AddCommMonoid E
inst✝⁹ : Module R E
inst✝⁸ : TopologicalSpace E'
inst✝⁷ : AddCommMonoid E'
inst✝⁶ : Module R E'
inst✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommMonoid F
inst✝³ : Module R F
inst✝² : TopologicalSpace F'
inst✝¹ : AddCommMonoid F'
inst✝ : Module R F'
f : E →L[R] F
g : E' →L[R] F'
finv : F →L[R] E
hfinv : LeftInverse ⇑finv ⇑f
ginv : F' →L[R] E'
hginv : LeftInverse ⇑ginv ⇑g
⊢ (f.prodMap g).HasLeftInverse
Module: Mathlib.Analysis.Normed.Module.ContinuousInverse

Goal state before:
pp: R : Type u_1
inst✝⁹ : Semiring R
E : Type u_2
F : Type u_4
G : Type u_6
inst✝⁸ : TopologicalSpace E
inst✝⁷ : AddCommMonoid E
inst✝⁶ : Module R E
inst✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommMonoid F
inst✝³ : Module R F
f : E →L[R] F
inst✝² : TopologicalSpace G
inst✝¹ : AddCommMonoid G
inst✝ : Module R G
g : F →L[R] G
hg : g.HasLeftInverse
finv : F →L[R] E
hfinv : LeftInverse ⇑finv ⇑f
⊢ (g ∘SL f).HasLeftInverse

Tactic:
obtain ⟨ginv, hginv⟩ := hg

Goal state after:
R : Type u_1
inst✝⁹ : Semiring R
E : Type u_2
F : Type u_4
G : Type u_6
inst✝⁸ : TopologicalSpace E
inst✝⁷ : AddCommMonoid E
inst✝⁶ : Module R E
inst✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommMonoid F
inst✝³ : Module R F
f : E →L[R] F
inst✝² : TopologicalSpace G
inst✝¹ : AddCommMonoid G
inst✝ : Module R G
g : F →L[R] G
finv : F →L[R] E
hfinv : LeftInverse ⇑finv ⇑f
ginv : G →L[R] F
hginv : LeftInverse ⇑ginv ⇑g
⊢ (g ∘SL f).HasLeftInverse
Module: Mathlib.Analysis.Normed.Module.ContinuousInverse

Goal state before:
pp: R : Type u_1
inst✝⁹ : Semiring R
E : Type u_2
F : Type u_4
G : Type u_6
inst✝⁸ : TopologicalSpace E
inst✝⁷ : AddCommMonoid E
inst✝⁶ : Module R E
inst✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommMonoid F
inst✝³ : Module R F
f : E →L[R] F
inst✝² : TopologicalSpace G
inst✝¹ : AddCommMonoid G
inst✝ : Module R G
g : F →L[R] G
hg : g.HasLeftInverse
hf : f.HasLeftInverse
⊢ (g ∘SL f).HasLeftInverse

Tactic:
by
  obtain ⟨finv, hfinv⟩ := hf
  obtain ⟨ginv, hginv⟩ := hg
  refine ⟨finv.comp ginv, fun x ↦ ?_⟩
  simp only [comp_apply]
  rw [hginv, hfinv]

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

Goal state before:
pp: R : Type u_1
inst✝¹² : Semiring R
E : Type u_2
E' : Type u_3
F : Type u_4
F' : Type u_5
inst✝¹¹ : TopologicalSpace E
inst✝¹⁰ : AddCommMonoid E
inst✝⁹ : Module R E
inst✝⁸ : TopologicalSpace E'
inst✝⁷ : AddCommMonoid E'
inst✝⁶ : Module R E'
inst✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommMonoid F
inst✝³ : Module R F
inst✝² : TopologicalSpace F'
inst✝¹ : AddCommMonoid F'
inst✝ : Module R F'
f : E →L[R] F
g : E' →L[R] F'
hg : g.HasRightInverse
finv : F →L[R] E
hfinv : RightInverse ⇑finv ⇑f
⊢ (f.prodMap g).HasRightInverse

Tactic:
obtain ⟨ginv, hginv⟩ := hg

Goal state after:
R : Type u_1
inst✝¹² : Semiring R
E : Type u_2
E' : Type u_3
F : Type u_4
F' : Type u_5
inst✝¹¹ : TopologicalSpace E
inst✝¹⁰ : AddCommMonoid E
inst✝⁹ : Module R E
inst✝⁸ : TopologicalSpace E'
inst✝⁷ : AddCommMonoid E'
inst✝⁶ : Module R E'
inst✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommMonoid F
inst✝³ : Module R F
inst✝² : TopologicalSpace F'
inst✝¹ : AddCommMonoid F'
inst✝ : Module R F'
f : E →L[R] F
g : E' →L[R] F'
finv : F →L[R] E
hfinv : RightInverse ⇑finv ⇑f
ginv : F' →L[R] E'
hginv : RightInverse ⇑ginv ⇑g
⊢ (f.prodMap g).HasRightInverse
Module: Mathlib.Analysis.Normed.Module.ContinuousInverse

Goal state before:
pp: R : Type u_1
inst✝⁹ : Semiring R
E : Type u_2
F : Type u_4
G : Type u_6
inst✝⁸ : TopologicalSpace E
inst✝⁷ : AddCommMonoid E
inst✝⁶ : Module R E
inst✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommMonoid F
inst✝³ : Module R F
f : E →L[R] F
inst✝² : TopologicalSpace G
inst✝¹ : AddCommMonoid G
inst✝ : Module R G
g : F →L[R] G
hg : g.HasRightInverse
finv : F →L[R] E
hfinv : RightInverse ⇑finv ⇑f
⊢ (g ∘SL f).HasRightInverse

Tactic:
obtain ⟨ginv, hginv⟩ := hg

Goal state after:
R : Type u_1
inst✝⁹ : Semiring R
E : Type u_2
F : Type u_4
G : Type u_6
inst✝⁸ : TopologicalSpace E
inst✝⁷ : AddCommMonoid E
inst✝⁶ : Module R E
inst✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommMonoid F
inst✝³ : Module R F
f : E →L[R] F
inst✝² : TopologicalSpace G
inst✝¹ : AddCommMonoid G
inst✝ : Module R G
g : F →L[R] G
finv : F →L[R] E
hfinv : RightInverse ⇑finv ⇑f
ginv : G →L[R] F
hginv : RightInverse ⇑ginv ⇑g
⊢ (g ∘SL f).HasRightInverse
Module: Mathlib.Analysis.Normed.Module.ContinuousInverse

Goal state before:
pp: R : Type u_1
inst✝⁶ : Semiring R
F : Type u_4
G : Type u_6
inst✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommMonoid F
inst✝³ : Module R F
inst✝² : TopologicalSpace G
inst✝¹ : AddCommMonoid G
inst✝ : Module R G
⊢ (fst R F G).HasRightInverse

Tactic:
use (ContinuousLinearMap.id _ _).prod 0

Goal state after:
case h
R : Type u_1
inst✝⁶ : Semiring R
F : Type u_4
G : Type u_6
inst✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommMonoid F
inst✝³ : Module R F
inst✝² : TopologicalSpace G
inst✝¹ : AddCommMonoid G
inst✝ : Module R G
⊢ RightInverse ⇑((ContinuousLinearMap.id R F).prod 0) ⇑(fst R F G)
Module: Mathlib.Analysis.Normed.Module.DoubleDual

Goal state before:
pp: 𝕜 : Type v
inst✝² : RCLike 𝕜
E : Type u
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
g : StrongDual 𝕜 E
hg : ‖g‖ ≤ 1 ∧ g x = ↑‖x‖
⊢ ‖x‖ ≤ ‖((inclusionInDoubleDual 𝕜 E) x) g‖

Tactic:
simp [hg.right]

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpHolder

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
E : ι → Type u_3
F : ι → Type u_4
G : ι → Type u_5
inst✝⁶ : RCLike 𝕜
inst✝⁵ : (i : ι) → NormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝³ : (i : ι) → NormedAddCommGroup (F i)
inst✝² : (i : ι) → NormedSpace 𝕜 (F i)
inst✝¹ : (i : ι) → NormedAddCommGroup (G i)
inst✝ : (i : ι) → NormedSpace 𝕜 (G i)
p q r : ℝ≥0∞
hpqr : p.HolderTriple q r
e : (i : ι) → E i
f : (i : ι) → F i
he : Memℓp e p
hf : Memℓp f q
B : (i : ι) → E i →L[𝕜] F i →L[𝕜] G i
K : ℝ
hBK : ∀ (i : ι), ‖B i‖ ≤ K
⊢ Memℓp (fun i ↦ ((B i) (e i)) (f i)) r

Tactic:
obtain (h | h) := isEmpty_or_nonempty ι
  · exact all _
  have hK : 0 ≤ K := norm_nonneg (B (Classical.arbitrary ι)) |>.trans <| hBK _
  have hpqr' := hpqr.inv_eq
  obtain (rfl | rfl | hp) := p.trichotomy
  · simp_all only [ENNReal.inv_zero, top_add, inv_eq_top]
    exact he.bilin_of_zero_left B
  · simp_all only [inv_top, zero_add, inv_inj]
    exact he.bilin_of_top_left B hBK hf
  obtain (rfl | rfl | hq) := q.trichotomy
  · simp_all only [ENNReal.inv_zero, add_top, inv_eq_top]
    exact hf.bilin_of_zero_right B
  · simp_all only [inv_top, add_zero, inv_inj]
    exact he.bilin_of_top_right B hBK hf
  obtain ⟨C, hC, hCe⟩ := memℓp_gen_iff'' hp |>.mp he
  obtain ⟨D, hD, hDf⟩ := memℓp_gen_iff'' hq |>.mp hf
  exact memℓp_gen' <| holder_gen_bound r hp hq B hBK hK hC hCe hDf

Goal state after: