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

Goal state before:
pp: z : ℂ
hz : z ≠ 0
n : ℕ
A : ℂ := ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)
B : ℂ := ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)
C : ℝ := ∫ (x : ℝ) in 0..π / 2, cos x ^ (2 * n)
hn : Complex.sin (↑π * z) = ↑π * z * A * B / ↑C
aux' : 2 * n.succ = 2 * n + 2
this✝¹ : ∫ (x : ℝ) in 0..π / 2, cos x ^ (2 * n.succ) = (2 * ↑n + 1) / (2 * ↑n + 2) * C
this✝ :
  ↑π * z * (A * (1 - z ^ 2 / (↑n + 1) ^ 2)) *
      ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n.succ) =
    ↑π * z * A * ((1 - z ^ 2 / ↑n.succ ^ 2) * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n.succ))
this :
  (1 - z ^ 2 / ↑n.succ ^ 2) * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n.succ) =
    (2 * ↑n + 1) / (2 * ↑n + 2) * B
⊢ ↑π * z * A * B / ↑C = ↑π * z * A * ((2 * ↑n + 1) / (2 * ↑n + 2) * B) / (↑(2 * ↑n + 1) / ↑(2 * ↑n + 2) * ↑C)

Tactic:
have : (C : ℂ) ≠ 0 := Complex.ofReal_ne_zero.mpr (integral_cos_pow_pos _).ne'

Goal state after:
z : ℂ
hz : z ≠ 0
n : ℕ
A : ℂ := ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)
B : ℂ := ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)
C : ℝ := ∫ (x : ℝ) in 0..π / 2, cos x ^ (2 * n)
hn : Complex.sin (↑π * z) = ↑π * z * A * B / ↑C
aux' : 2 * n.succ = 2 * n + 2
this✝² : ∫ (x : ℝ) in 0..π / 2, cos x ^ (2 * n.succ) = (2 * ↑n + 1) / (2 * ↑n + 2) * C
this✝¹ :
  ↑π * z * (A * (1 - z ^ 2 / (↑n + 1) ^ 2)) *
      ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n.succ) =
    ↑π * z * A * ((1 - z ^ 2 / ↑n.succ ^ 2) * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n.succ))
this✝ :
  (1 - z ^ 2 / ↑n.succ ^ 2) * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n.succ) =
    (2 * ↑n + 1) / (2 * ↑n + 2) * B
this : ↑C ≠ 0
⊢ ↑π * z * A * B / ↑C = ↑π * z * A * ((2 * ↑n + 1) / (2 * ↑n + 2) * B) / (↑(2 * ↑n + 1) / ↑(2 * ↑n + 2) * ↑C)
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
hv' : 0 < (v + 1).re
hc : ContinuousOn F (Icc 0 1)
hder : ∀ x ∈ Ioo 0 1, HasDerivAt F (u * (↑x ^ (u - 1) * (1 - ↑x) ^ v) - v * (↑x ^ u * (1 - ↑x) ^ (v - 1))) x
h_int : IntervalIntegrable (fun x ↦ u * (↑x ^ (u - 1) * (1 - ↑x) ^ v) - v * (↑x ^ u * (1 - ↑x) ^ (v - 1))) volume 0 1
int_ev : ∫ (y : ℝ) in 0..1, u * (↑y ^ (u - 1) * (1 - ↑y) ^ v) - v * (↑y ^ u * (1 - ↑y) ^ (v - 1)) = F 1 - F 0
⊢ u * u.betaIntegral (v + 1) = v * (u + 1).betaIntegral v

Tactic:
have hF0 : F 0 = 0 := by
    simp only [F, mul_eq_zero, ofReal_zero, cpow_eq_zero_iff, Ne,
      true_and, sub_zero, one_cpow, one_ne_zero, or_false]
    contrapose! hu; rw [hu, zero_re]

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
hc : ContinuousOn F (Icc 0 1)
hder : ∀ x ∈ Ioo 0 1, HasDerivAt F (u * (↑x ^ (u - 1) * (1 - ↑x) ^ v) - v * (↑x ^ u * (1 - ↑x) ^ (v - 1))) x
h_int : IntervalIntegrable (fun x ↦ u * (↑x ^ (u - 1) * (1 - ↑x) ^ v) - v * (↑x ^ u * (1 - ↑x) ^ (v - 1))) volume 0 1
int_ev : ∫ (y : ℝ) in 0..1, u * (↑y ^ (u - 1) * (1 - ↑y) ^ v) - v * (↑y ^ u * (1 - ↑y) ^ (v - 1)) = F 1 - F 0
hF0 : F 0 = 0
⊢ u * u.betaIntegral (v + 1) = v * (u + 1).betaIntegral v
Module: Mathlib.Topology.Algebra.AsymptoticCone

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : Field k
inst✝⁴ : LinearOrder k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AddTorsor V P
inst✝ : TopologicalSpace V
v : V
p : P
⊢ (⨆ p, atTop • 𝓝 v +ᵥ pure p) +ᵥ pure p = ⨆ p, atTop • 𝓝 v +ᵥ pure p

Tactic:
vadd_pure,

Goal state after:
k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : Field k
inst✝⁴ : LinearOrder k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AddTorsor V P
inst✝ : TopologicalSpace V
v : V
p : P
⊢ map (fun x ↦ x +ᵥ p) (⨆ p, map (fun x ↦ x +ᵥ p) (atTop • 𝓝 v)) = ⨆ p, map (fun x ↦ x +ᵥ p) (atTop • 𝓝 v)
Module: Mathlib.Topology.Algebra.AsymptoticCone

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : Field k
inst✝⁴ : LinearOrder k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AddTorsor V P
inst✝ : TopologicalSpace V
u v : V
p : P
⊢ u +ᵥ asymptoticNhds k V v +ᵥ pure p = asymptoticNhds k V v +ᵥ pure (u +ᵥ p)

Tactic:
vadd_pure,

Goal state after:
k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : Field k
inst✝⁴ : LinearOrder k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AddTorsor V P
inst✝ : TopologicalSpace V
u v : V
p : P
⊢ u +ᵥ map (fun x ↦ x +ᵥ p) (asymptoticNhds k V v) = map (fun x ↦ x +ᵥ u +ᵥ p) (asymptoticNhds k V v)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: s : ℝ
hs : s ≠ 0
⊢ Γ (s / 2 + 1 / 2) * Γ (s / 2 + (1 / 2 + 1 / 2)) * 2 ^ (s - 1 + 1) / √π =
    s * (Γ (s / 2) * Γ (s / 2 + 1 / 2) * 2 ^ (s - 1) / √π)

Tactic:
add_halves (1 : ℝ),

Goal state after:
s : ℝ
hs : s ≠ 0
⊢ Γ (s / 2 + 1 / 2) * Γ (s / 2 + 1) * 2 ^ (s - 1 + 1) / √π = s * (Γ (s / 2) * Γ (s / 2 + 1 / 2) * 2 ^ (s - 1) / √π)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: ⊢ √π * Γ (1 / 2 + 1 / 2) * 2 ^ (1 - 1) / √π = 1

Tactic:
add_halves (1 : ℝ),

Goal state after:
⊢ √π * Γ 1 * 2 ^ (1 - 1) / √π = 1
Module: Mathlib.Analysis.Normed.Affine.AsymptoticCone

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
v : V
hv : v ≠ 0
p : P
⊢ Tendsto (fun x ↦ dist (id x) p) (atTop • 𝓝 v +ᵥ pure p) atTop

Tactic:
vadd_pure,

Goal state after:
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
v : V
hv : v ≠ 0
p : P
⊢ Tendsto (fun x ↦ dist (id x) p) (map (fun x ↦ x +ᵥ p) (atTop • 𝓝 v)) atTop
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
v : V
⊢ v ∈ asymptoticCone k s ↔ v ∈ closure[inst✝⁵] s

Tactic:
mem_closure_iff_frequently,

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 ↔ ∃ᶠ (x : V) in 𝓝 v, x ∈ 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
v : V
⊢ (∃ᶠ (p : P) in asymptoticNhds k V v +ᵥ pure p, p ∈ ↑s) ↔ ∃ᶠ (p : V) in asymptoticNhds k V v, p ∈ ↑s.direction

Tactic:
vadd_pure,

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
⊢ (∃ᶠ (p : P) in map (fun x ↦ x +ᵥ p) (asymptoticNhds k V v), p ∈ ↑s) ↔
    ∃ᶠ (p : V) in asymptoticNhds k V v, p ∈ ↑s.direction
Module: Mathlib.Topology.Algebra.AsymptoticCone

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝¹² : Field k
inst✝¹¹ : LinearOrder k
inst✝¹⁰ : AddCommGroup V
inst✝⁹ : Module k V
inst✝⁸ : AddTorsor V P
inst✝⁷ : TopologicalSpace V
inst✝⁶ : TopologicalSpace k
inst✝⁵ : OrderTopology k
inst✝⁴ : IsStrictOrderedRing k
inst✝³ : IsTopologicalAddGroup V
inst✝² : ContinuousSMul k V
inst✝¹ : TopologicalSpace P
inst✝ : IsTopologicalAddTorsor P
s : Set P
x✝ : V
⊢ (∃ᶠ (p : P) in asymptoticNhds k P x✝, p ∈ closure[inst✝¹] s) ↔ ∃ᶠ (p : P) in asymptoticNhds k P x✝, p ∈ s

Tactic:
mem_closure_iff_frequently,

Goal state after:
k : Type u_1
V : Type u_2
P : Type u_3
inst✝¹² : Field k
inst✝¹¹ : LinearOrder k
inst✝¹⁰ : AddCommGroup V
inst✝⁹ : Module k V
inst✝⁸ : AddTorsor V P
inst✝⁷ : TopologicalSpace V
inst✝⁶ : TopologicalSpace k
inst✝⁵ : OrderTopology k
inst✝⁴ : IsStrictOrderedRing k
inst✝³ : IsTopologicalAddGroup V
inst✝² : ContinuousSMul k V
inst✝¹ : TopologicalSpace P
inst✝ : IsTopologicalAddTorsor P
s : Set P
x✝ : V
⊢ (∃ᶠ (p : P) in asymptoticNhds k P x✝, ∃ᶠ (x : P) in 𝓝 p, x ∈ s) ↔ ∃ᶠ (x : P) in asymptoticNhds k P x✝, x ∈ 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 : IsStrictOrderedRing k
inst✝⁶ : TopologicalSpace k
inst✝⁵ : OrderTopology k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : TopologicalSpace V
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul k V
s : Set V
c : k
v p : V
hs₁ : StarConvex k p s
hs₂ : IsClosed[inst✝²] s
hc : 0 ≤ c
hv : ∃ᶠ (p : V) in atTop • 𝓝 v +ᵥ pure p, p ∈ s
⊢ ∃ᶠ (x : k × V) in atTop ×ˢ 𝓝 v, c • x.2 +ᵥ p ∈ s

Tactic:
vadd_pure,

Goal state after:
k : Type u_1
V : Type u_2
inst✝⁹ : Field k
inst✝⁸ : LinearOrder k
inst✝⁷ : IsStrictOrderedRing k
inst✝⁶ : TopologicalSpace k
inst✝⁵ : OrderTopology k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : TopologicalSpace V
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul k V
s : Set V
c : k
v p : V
hs₁ : StarConvex k p s
hs₂ : IsClosed[inst✝²] s
hc : 0 ≤ c
hv : ∃ᶠ (p : V) in map (fun x ↦ x +ᵥ p) (atTop • 𝓝 v), p ∈ s
⊢ ∃ᶠ (x : k × V) in atTop ×ˢ 𝓝 v, c • x.2 +ᵥ p ∈ 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 : IsStrictOrderedRing k
inst✝⁶ : TopologicalSpace k
inst✝⁵ : OrderTopology k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : TopologicalSpace V
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul k V
s : Set V
c : k
v p : V
hs : Convex k s
hc : 0 ≤ c
hp : s ∈ 𝓝 p
hv : ∃ᶠ (p : V) in atTop • 𝓝 v +ᵥ pure (c • v +ᵥ p), p ∈ s
⊢ c • v +ᵥ p ∈ s

Tactic:
vadd_pure,

Goal state after:
k : Type u_1
V : Type u_2
inst✝⁹ : Field k
inst✝⁸ : LinearOrder k
inst✝⁷ : IsStrictOrderedRing k
inst✝⁶ : TopologicalSpace k
inst✝⁵ : OrderTopology k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : TopologicalSpace V
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul k V
s : Set V
c : k
v p : V
hs : Convex k s
hc : 0 ≤ c
hp : s ∈ 𝓝 p
hv : ∃ᶠ (p : V) in map (fun x ↦ x +ᵥ c • v +ᵥ p) (atTop • 𝓝 v), p ∈ s
⊢ c • v +ᵥ p ∈ s
Module: Mathlib.Analysis.Normed.Affine.MazurUlam

Goal state before:
pp: E : Type u_1
PE : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : MetricSpace PE
inst✝ : NormedAddTorsor E PE
x y : PE
z : PE := midpoint ℝ x y
s : Set (PE ≃ᵢ PE) := {e | e x = x ∧ e y = y}
this : Nonempty ↑s
h_bdd : BddAbove (range fun e ↦ dist (↑e z) z)
R : PE ≃ᵢ PE := (pointReflection ℝ z).toIsometryEquiv
f : PE ≃ᵢ PE → PE ≃ᵢ PE := fun e ↦ ((e.trans R).trans e.symm).trans R
e : PE ≃ᵢ PE
⊢ dist ((pointReflection ℝ z) (e z)) (e z) = 2 * dist (e z) z

Tactic:
dist_pointReflection_self_real,

Goal state after:
E : Type u_1
PE : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : MetricSpace PE
inst✝ : NormedAddTorsor E PE
x y : PE
z : PE := midpoint ℝ x y
s : Set (PE ≃ᵢ PE) := {e | e x = x ∧ e y = y}
this : Nonempty ↑s
h_bdd : BddAbove (range fun e ↦ dist (↑e z) z)
R : PE ≃ᵢ PE := (pointReflection ℝ z).toIsometryEquiv
f : PE ≃ᵢ PE → PE ≃ᵢ PE := fun e ↦ ((e.trans R).trans e.symm).trans R
e : PE ≃ᵢ PE
⊢ 2 * dist z (e z) = 2 * dist (e z) z
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: s : ℝ
⊢ Gamma s * Gamma (1 - s) = π / sin (π * s)

Tactic:
simp_rw [← Complex.ofReal_inj, Complex.ofReal_div, Complex.ofReal_sin, Complex.ofReal_mul, ←
    Complex.Gamma_ofReal, Complex.ofReal_sub, Complex.ofReal_one]
  exact Complex.Gamma_mul_Gamma_one_sub s

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

Goal state before:
pp: u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
⊢ u.betaIntegral v = Gamma u * Gamma v / Gamma (u + v)

Tactic:
rw [Gamma_mul_Gamma_eq_betaIntegral hu hv,
      mul_div_cancel_left₀ _ (Gamma_ne_zero_of_re_pos (add_pos hu hv))]

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.Simplex

Goal state before:
pp: case refine_2
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
t : Triangle R P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h :
  dist (t.points i₁) (t.points i₂) = dist (t.points i₁) (t.points i₃) ∧
    dist (t.points i₁) (t.points i₂) = dist (t.points i₂) (t.points i₃)
i j : Fin (2 + 1)
⊢ i ≠ j → dist (t.points i) (t.points j) = dist (t.points i₁) (t.points i₂)

Tactic:
hij

Goal state after:
case refine_2
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
t : Triangle R P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h :
  dist (t.points i₁) (t.points i₂) = dist (t.points i₁) (t.points i₃) ∧
    dist (t.points i₁) (t.points i₂) = dist (t.points i₂) (t.points i₃)
i j : Fin (2 + 1)
hij : i ≠ j
⊢ dist (t.points i) (t.points j) = dist (t.points i₁) (t.points i₂)
Module: Mathlib.Analysis.Quaternion

Goal state before:
pp: x : ℍ
⊢ √(⟪(WithLp.toLp 2 ((equivTuple ℝ) x)).ofLp 0, (WithLp.toLp 2 ((equivTuple ℝ) x)).ofLp 0⟫ +
            ⟪(WithLp.toLp 2 ((equivTuple ℝ) x)).ofLp 1, (WithLp.toLp 2 ((equivTuple ℝ) x)).ofLp 1⟫ +
          ⟪(WithLp.toLp 2 ((equivTuple ℝ) x)).ofLp 2, (WithLp.toLp 2 ((equivTuple ℝ) x)).ofLp 2⟫ +
        ⟪(WithLp.toLp 2 ((equivTuple ℝ) x)).ofLp 3, (WithLp.toLp 2 ((equivTuple ℝ) x)).ofLp 3⟫) =
    √(x.re ^ 2 + x.imI ^ 2 + x.imJ ^ 2 + x.imK ^ 2)

Tactic:
simp_rw [RCLike.inner_apply, starRingEnd_apply, star_trivial, ← sq]

Goal state after:
x : ℍ
⊢ √((equivTuple ℝ) x 0 ^ 2 + (equivTuple ℝ) x 1 ^ 2 + (equivTuple ℝ) x 2 ^ 2 + (equivTuple ℝ) x 3 ^ 2) =
    √(x.re ^ 2 + x.imI ^ 2 + x.imJ ^ 2 + x.imK ^ 2)
Module: Mathlib.Analysis.Normed.Field.Ultra

Goal state before:
pp: R : Type u_1
inst✝ : NormedDivisionRing R
h : ∀ (n : ℕ), ‖↑n‖ ≤ 1
⊢ IsUltrametricDist R

Tactic:
refine isUltrametricDist_of_forall_pow_norm_le_nsmul_pow_max_one_norm (fun x m ↦ ?_)

Goal state after:
R : Type u_1
inst✝ : NormedDivisionRing R
h : ∀ (n : ℕ), ‖↑n‖ ≤ 1
x : R
m : ℕ
⊢ ‖x + 1‖ ^ m ≤ (m + 1) • max 1 (‖x‖ ^ m)
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

Goal state before:
pp: case neg
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
x y : R
hy : ¬seminormFromBounded' f y = 0
z : R
hz : ¬f z = 0
hxz : ¬f (x * z) = 0
⊢ f (x * y * z) ≤ seminormFromBounded' f y * f (x * z)

Tactic:
← div_le_iff₀ (lt_of_le_of_ne' (f_nonneg _) hxz)

Goal state after:
case neg
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
x y : R
hy : ¬seminormFromBounded' f y = 0
z : R
hz : ¬f z = 0
hxz : ¬f (x * z) = 0
⊢ f (x * y * z) / f (x * z) ≤ seminormFromBounded' f y
Module: Mathlib.Analysis.Normed.Unbundled.InvariantExtension

Goal state before:
pp: K : Type u_1
inst✝² : NormedField K
L : Type u_2
inst✝¹ : Field L
inst✝ : Algebra K L
h_fin : FiniteDimensional K L
hu : IsUltrametricDist K
σ : L ≃ₐ[K] L
⊢ IsPowMul ⇑(algNormOfAlgEquiv σ)

Tactic:
intro x n hn
  simp only [algNormOfAlgEquiv_apply, map_pow σ x n]
  exact (Classical.choose_spec (exists_nonarchimedean_pow_mul_seminorm_of_finiteDimensional
    h_fin hu.isNonarchimedean_norm)).1 _ hn

Goal state after:
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
⊢ F' 1 ≠ 0 1

Tactic:
hF'_1

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
⊢ 1 ≠ 0 1
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.mp (pos_iff_ne_zero.mpr hn0)),

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: case h
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
s : ℕ → ℕ
hs_le : ∀ (n : ℕ), s n ≤ n
x : R
ψ : ℕ → ℕ
hμx : ¬μ x < 1
⊢ ∀ (b : ℕ), 0 ≤ b → μ x ^ (↑(s (ψ b)) * (1 / ↑(ψ b))) ≤ μ x

Tactic:
intro b _

Goal state after:
case h
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
s : ℕ → ℕ
hs_le : ∀ (n : ℕ), s n ≤ n
x : R
ψ : ℕ → ℕ
hμx : ¬μ x < 1
b : ℕ
a✝ : 0 ≤ b
⊢ μ x ^ (↑(s (ψ b)) * (1 / ↑(ψ b))) ≤ μ x
Module: Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm

Goal state before:
pp: case pos
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
s : ℕ → ℕ
hs_le : ∀ (n : ℕ), s n ≤ n
x : R
ψ : ℕ → ℕ
hψ_lim : Tendsto ((fun n ↦ ↑(s n) / ↑n) ∘ ψ) atTop (𝓝 0)
c : ℝ
hc_bd : ∀ (x_1 : ℝ) (x_2 : ℕ), (∀ (b : ℕ), x_2 ≤ b → μ x ^ (↑(s (ψ b)) * (1 / ↑(ψ b))) ≤ x_1) → c ≤ x_1
hμx : μ x < 1
⊢ ∀ (b : ℕ), 0 ≤ b → μ x ^ (↑(s (ψ b)) * (1 / ↑(ψ b))) ≤ 1

Tactic:
intro b _

Goal state after:
case pos
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
s : ℕ → ℕ
hs_le : ∀ (n : ℕ), s n ≤ n
x : R
ψ : ℕ → ℕ
hψ_lim : Tendsto ((fun n ↦ ↑(s n) / ↑n) ∘ ψ) atTop (𝓝 0)
c : ℝ
hc_bd : ∀ (x_1 : ℝ) (x_2 : ℕ), (∀ (b : ℕ), x_2 ≤ b → μ x ^ (↑(s (ψ b)) * (1 / ↑(ψ b))) ≤ x_1) → c ≤ x_1
hμx : μ x < 1
b : ℕ
a✝ : 0 ≤ b
⊢ μ x ^ (↑(s (ψ b)) * (1 / ↑(ψ b))) ≤ 1
Module: Mathlib.Analysis.Normed.Module.ContinuousInverse

Goal state before:
pp: R : Type u_1
inst✝⁶ : Semiring R
E : Type u_2
F : Type u_4
inst✝⁵ : TopologicalSpace E
inst✝⁴ : AddCommMonoid E
inst✝³ : Module R E
inst✝² : TopologicalSpace F
inst✝¹ : AddCommMonoid F
inst✝ : Module R F
e : E ≃L[R] F
⊢ (↑e).HasLeftInverse

Tactic:
exact e.hasLeftInverse

Goal state after:
Module: Mathlib.Analysis.Normed.Module.PiTensorProduct.InjectiveSeminorm

Goal state before:
pp: case h
ι : Type uι
inst✝³ : Fintype ι
𝕜 : Type u𝕜
inst✝² : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝¹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝ : (i : ι) → NormedSpace 𝕜 (E i)
⊢ ∀ (x : Seminorm 𝕜 (⨂[𝕜] (i : ι), E i)) (x_1 : Type (max uι u𝕜 uE)) (x_2 : SeminormedAddCommGroup x_1)
    (x_3 : NormedSpace 𝕜 x_1),
    x = (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E x_1 →L[𝕜] x_1)).comp (toDualContinuousMultilinearMap x_1) →
      x ≤ projectiveSeminorm

Tactic:
intro p G _ _ hp x

Goal state after:
case h
ι : Type uι
inst✝³ : Fintype ι
𝕜 : Type u𝕜
inst✝² : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝¹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝ : (i : ι) → NormedSpace 𝕜 (E i)
p : Seminorm 𝕜 (⨂[𝕜] (i : ι), E i)
G : Type (max uι u𝕜 uE)
x✝¹ : SeminormedAddCommGroup G
x✝ : NormedSpace 𝕜 G
hp : p = (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp (toDualContinuousMultilinearMap G)
x : ⨂[𝕜] (i : ι), E i
⊢ p x ≤ projectiveSeminorm x
Module: Mathlib.Analysis.Normed.Module.PiTensorProduct.InjectiveSeminorm

Goal state before:
pp: ι : Type uι
inst✝⁵ : Fintype ι
𝕜 : Type u𝕜
inst✝⁴ : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
F : Type u_1
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : ContinuousMultilinearMap 𝕜 E F
x : ⨂[𝕜] (i : ι), E i
G : Type (max (max uE u𝕜) uι) := (⨂[𝕜] (i : ι), E i) ⧸ (lift f.toMultilinearMap).ker
G' : Submodule 𝕜 F := (lift f.toMultilinearMap).range
e : ((⨂[𝕜] (i : ι), E i) ⧸ (lift f.toMultilinearMap).ker) ≃ₗ[𝕜] ↥(lift f.toMultilinearMap).range :=
  (lift f.toMultilinearMap).quotKerEquivRange
this✝ : SeminormedAddCommGroup G := SeminormedAddCommGroup.induced G (↥G') e
this : NormedSpace 𝕜 G := NormedSpace.induced 𝕜 G (↥G') e
f'₀ : MultilinearMap 𝕜 E ((⨂[𝕜] (i : ι), E i) ⧸ (lift f.toMultilinearMap).ker) :=
  lift.symm (↑e.symm ∘ₗ (lift f.toMultilinearMap).rangeRestrict)
hf'₀ : ∀ (x : (i : ι) → E i), ‖f'₀ x‖ ≤ ‖f‖ * ∏ i, ‖x i‖
f' : ContinuousMultilinearMap 𝕜 E ((⨂[𝕜] (i : ι), E i) ⧸ (lift f.toMultilinearMap).ker) := f'₀.mkContinuous ‖f‖ hf'₀
hnorm : ‖f'‖ ≤ ‖f‖
heq : ↑(e ((lift f'.toMultilinearMap) x)) = (lift f.toMultilinearMap) x
h : ‖(lift f'.toMultilinearMap) x‖ ≤ ‖f'‖ * injectiveSeminorm x
⊢ ‖(lift f.toMultilinearMap) x‖ ≤ ‖f‖ * injectiveSeminorm x

Tactic:
change ‖(e (lift f'.toMultilinearMap x)).1‖ ≤ _ at h
    rw [heq] at h
    exact h.trans (by gcongr)

Goal state after:
Module: Mathlib.Analysis.Normed.Order.UpperLower

Goal state before:
pp: ι : Type u_2
inst✝ : Finite ι
s : Set (ι → ℝ)
x y : ι → ℝ
hs : IsUpperSet s
hx : x ∈ closure s
h : ∀ (i : ι), x i < y i
val✝ : Fintype ι
ε : ℝ
hε : 0 < ε
hxy : ∀ (i : ι), x i + ε < y i
z : ι → ℝ
hz : z ∈ s
hxz : ∀ (b : ι), dist (x b) (z b) < ε
i : ι
⊢ dist (x i) (z i) ≤ ε

Tactic:
exact (hxz _).le

Goal state after: