mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.InnerProductSpace.NormDet

Goal state before:
pp: U : Type u_5
V : Type u_6
inst✝⁸ : NormedAddCommGroup U
inst✝⁷ : InnerProductSpace ℝ U
inst✝⁶ : FiniteDimensional ℝ U
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : MeasurableSpace U
inst✝² : BorelSpace U
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
f : U →ₗ[ℝ] V
s : Set U
h : f.ker = ⊥
hrank : finrank ℝ ↥f.range = finrank ℝ U
bv : OrthonormalBasis (Fin (finrank ℝ U)) ℝ ↥f.range
g : U ≃ₗᵢ[ℝ] ↥f.range := (stdOrthonormalBasis ℝ U).equiv bv (Equiv.refl (Fin (finrank ℝ U)))
this :
  μH[↑(finrank ℝ U)]
      (⇑(f.range.subtypeₗᵢ.comp g.toLinearIsometry) '' ⇑(g.symm.toLinearIsometry.toLinearMap ∘ₗ f.rangeRestrict) '' s) =
    ENNReal.ofReal f.normDet * μH[↑(finrank ℝ U)] s
⊢ μH[↑(finrank ℝ U)] (⇑f '' s) = ENNReal.ofReal f.normDet * μH[↑(finrank ℝ U)] s

Tactic:
by
      simpa [Set.image_image]

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Reproducing

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : RCLike 𝕜
X : Type u_2
V : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : InnerProductSpace 𝕜 V
H : Type u_4
inst✝⁴ : NormedAddCommGroup H
inst✝³ : InnerProductSpace 𝕜 H
inst✝² : RKHS 𝕜 H X V
inst✝¹ : CompleteSpace H
inst✝ : CompleteSpace V
C : ℝ
s : Set X
hC : ∀ x ∈ s, ‖kerFun H x‖ ≤ C
ι : Type u_5
l : Filter ι
F : ι → H
f : H
h : Tendsto F l (𝓝 f)
ε : ℝ
hε : ε > 0
hnorm : Tendsto (fun x ↦ ‖F x - f‖ * C) l (𝓝 0)
n : ι
hn : ‖F n - f‖ * C < ε
x : X
hx : x ∈ s
⊢ ‖F n - f‖ * √‖kernel H x x‖ < ε

Tactic:
← norm_kerFun_eq_sqrt_norm_kernel,

Goal state after:
𝕜 : Type u_1
inst✝⁷ : RCLike 𝕜
X : Type u_2
V : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : InnerProductSpace 𝕜 V
H : Type u_4
inst✝⁴ : NormedAddCommGroup H
inst✝³ : InnerProductSpace 𝕜 H
inst✝² : RKHS 𝕜 H X V
inst✝¹ : CompleteSpace H
inst✝ : CompleteSpace V
C : ℝ
s : Set X
hC : ∀ x ∈ s, ‖kerFun H x‖ ≤ C
ι : Type u_5
l : Filter ι
F : ι → H
f : H
h : Tendsto F l (𝓝 f)
ε : ℝ
hε : ε > 0
hnorm : Tendsto (fun x ↦ ‖F x - f‖ * C) l (𝓝 0)
n : ι
hn : ‖F n - f‖ * C < ε
x : X
hx : x ∈ s
⊢ ‖F n - f‖ * ‖kerFun H x‖ < ε
Module: Mathlib.Analysis.InnerProductSpace.Reproducing

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁸ : RCLike 𝕜
X : Type u_2
V : Type u_3
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace 𝕜 V
H : Type u_4
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : InnerProductSpace 𝕜 H
inst✝³ : RKHS 𝕜 H X V
inst✝² : CompleteSpace H
inst✝¹ : CompleteSpace V
K : Matrix X X (V →L[𝕜] V)
inst✝ : Fact K.PosSemidef
x : X
x✝¹ x✝ : V
⊢ ↑(Finsupp.single (x, x✝¹ + x✝) 1) = ↑(Finsupp.single (x, x✝¹) 1) + ↑(Finsupp.single (x, x✝) 1)

Tactic:
refine UniformSpace.Completion.denseRange_coe.eq_of_inner_left 𝕜 fun f ↦ ?_
    simp [inner_add_left, inner_H₀_def, ← Finsupp.sum_add, ← mul_add]

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.WeakOperatorTopology

Goal state before:
pp: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝⁷ : NormedField 𝕜₁
inst✝⁶ : NormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
E : Type u_3
F : Type u_4
inst✝⁵ : AddCommGroup E
inst✝⁴ : TopologicalSpace E
inst✝³ : Module 𝕜₁ E
inst✝² : AddCommGroup F
inst✝¹ : TopologicalSpace F
inst✝ : Module 𝕜₂ F
f : E →SWOT[σ] F
x y : E
⊢ f (x + y) = f x + f y

Tactic:
simp [← coe_toCLM]

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.WeakOperatorTopology

Goal state before:
pp: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝⁷ : NormedField 𝕜₁
inst✝⁶ : NormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
E : Type u_3
F : Type u_4
inst✝⁵ : AddCommGroup E
inst✝⁴ : TopologicalSpace E
inst✝³ : Module 𝕜₁ E
inst✝² : AddCommGroup F
inst✝¹ : TopologicalSpace F
inst✝ : Module 𝕜₂ F
f : E →SWOT[σ] F
r : 𝕜₁
x : E
⊢ f (r • x) = σ r • f x

Tactic:
simp [← coe_toCLM]

Goal state after:
Module: Mathlib.Analysis.Matrix.LDL

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : WellFoundedLT n
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
⊢ (lowerInv hS)⁻¹ * (diag hS * (lowerInv hS)ᴴ⁻¹) = S

Tactic:
Matrix.inv_mul_eq_iff_eq_mul_of_invertible (LDL.lowerInv hS),

Goal state after:
𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : WellFoundedLT n
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
⊢ diag hS * (lowerInv hS)ᴴ⁻¹ = lowerInv hS * S
Module: Mathlib.Analysis.MellinInversion

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
u : ℝ
| cexp (-s * ↑u) • f (rexp (-u)) = cexp (-↑s.im * ↑u * I) • rexp (-s.re * u) • f (rexp (-u))

Tactic:
lhs; rw [← re_add_im s]

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
u : ℝ
| cexp (-(↑s.re + ↑s.im * I) * ↑u) • f (rexp (-u))
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
hf : MellinConvergent f s
R : Type u_2
inst✝³ : NormedRing R
inst✝² : Module R E
inst✝¹ : IsBoundedSMul R E
inst✝ : SMulCommClass ℂ R E
c : R
⊢ mellin (fun t ↦ c • f t) s = c • mellin f s

Tactic:
simp [mellin, smul_comm, hf.integral_smul]

Goal state after:
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 ↦ ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x
⊢ ↑n / (2 * z) * ∫ (x : ℝ) in 0..π / 2, Complex.sin (2 * z * ↑x) * ↑(sin x) * ↑(cos x) ^ (n - 1) =
    ↑(cos (π / 2)) ^ n * (Complex.sin (2 * z * ↑(π / 2)) / (2 * z)) -
        ↑(cos 0) ^ n * (Complex.sin (2 * z * ↑0) / (2 * z)) -
      ∫ (x : ℝ) in 0..π / 2, -↑n * ↑(sin x) * ↑(cos x) ^ (n - 1) * (Complex.sin (2 * z * ↑x) / (2 * z))

Tactic:
rw [Complex.ofReal_zero, mul_zero, Complex.sin_zero, zero_div, mul_zero, sub_zero,
      cos_pi_div_two, Complex.ofReal_zero, zero_pow (by positivity : n ≠ 0), zero_mul, zero_sub,
      ← integral_neg, ← integral_const_mul]
    refine integral_congr fun x _ => ?_
    ring

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

Goal state before:
pp: g : ℝ → ℂ
t : ℂ
hg : Tendsto (fun x ↦ ↑x * g x) atTop (𝓝 t)
hg0 : Tendsto g atTop (𝓝 0)
⊢ (fun x ↦ ↑x * (log (1 + g x) - g x)) =O[atTop] fun x ↦ ↑x * g x ^ 2

Tactic:
exact (isBigO_refl _ _).mul (log_sub_self_isBigO.comp_tendsto hg0)

Goal state after:
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b : ℝ
f : ℝ → E
s : ℂ
hfc : LocallyIntegrableOn f (Ioi 0) volume
hf_top : f =O[atTop] fun x ↦ x ^ (-a)
hs_top : s.re < a
hf_bot : f =O[𝓝[>] 0] fun x ↦ x ^ (-b)
hs_bot : b < s.re
F : ℂ → ℝ → E := fun z t ↦ ↑t ^ (z - 1) • f t
F' : ℂ → ℝ → E := fun z t ↦ (↑t ^ (z - 1) * ↑(log t)) • f t
v : ℝ
hv0 : 0 < v
hv1 : v < s.re - b
hv2 : v < a - s.re
bound : ℝ → ℝ := fun t ↦ (t ^ (s.re + v - 1) + t ^ (s.re - v - 1)) * |log t| * ‖f t‖
h1 : ∀ᶠ (z : ℂ) in 𝓝 s, AEStronglyMeasurable (F z) (volume.restrict (Ioi 0))
h2 : IntegrableOn (F s) (Ioi 0) volume
h3 : AEStronglyMeasurable (F' s) (volume.restrict (Ioi 0))
t : ℝ
ht : t ∈ Ioi 0
z : ℂ
hz : z ∈ Metric.ball s v
⊢ ‖F' z t‖ ≤ bound t

Tactic:
F',

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b : ℝ
f : ℝ → E
s : ℂ
hfc : LocallyIntegrableOn f (Ioi 0) volume
hf_top : f =O[atTop] fun x ↦ x ^ (-a)
hs_top : s.re < a
hf_bot : f =O[𝓝[>] 0] fun x ↦ x ^ (-b)
hs_bot : b < s.re
F : ℂ → ℝ → E := fun z t ↦ ↑t ^ (z - 1) • f t
F' : ℂ → ℝ → E := fun z t ↦ (↑t ^ (z - 1) * ↑(log t)) • f t
v : ℝ
hv0 : 0 < v
hv1 : v < s.re - b
hv2 : v < a - s.re
bound : ℝ → ℝ := fun t ↦ (t ^ (s.re + v - 1) + t ^ (s.re - v - 1)) * |log t| * ‖f t‖
h1 : ∀ᶠ (z : ℂ) in 𝓝 s, AEStronglyMeasurable (F z) (volume.restrict (Ioi 0))
h2 : IntegrableOn (F s) (Ioi 0) volume
h3 : AEStronglyMeasurable (F' s) (volume.restrict (Ioi 0))
t : ℝ
ht : t ∈ Ioi 0
z : ℂ
hz : z ∈ Metric.ball s v
⊢ ‖(↑t ^ (z - 1) * ↑(log t)) • f t‖ ≤ bound t
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: s t : ℂ
a : ℝ
ha : 0 < a
ha' : ↑a ≠ 0
A : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))
x : ℝ
hx : x ∈ Ioc 0 a
⊢ ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) = ↑a ^ (s - 1) * ↑a ^ (t - 1) * (↑(x / a) ^ (s - 1) * (1 - ↑(x / a)) ^ (t - 1))

Tactic:
rw [mul_mul_mul_comm]

Goal state after:
s t : ℂ
a : ℝ
ha : 0 < a
ha' : ↑a ≠ 0
A : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))
x : ℝ
hx : x ∈ Ioc 0 a
⊢ ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) = ↑a ^ (s - 1) * ↑(x / a) ^ (s - 1) * (↑a ^ (t - 1) * (1 - ↑(x / a)) ^ (t - 1))
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: case inr.zero
z : ℂ
hz : z ≠ 0
⊢ Complex.sin (↑π * z) =
    ((↑π * z * ∏ j ∈ Finset.range 0, (1 - z ^ 2 / (↑j + 1) ^ 2)) *
        ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ 0) /
      ↑(∫ (x : ℝ) in 0..π / 2, cos x ^ 0)

Tactic:
pow_zero,

Goal state after:
case inr.zero
z : ℂ
hz : z ≠ 0
⊢ Complex.sin (↑π * z) =
    ((↑π * z * ∏ j ∈ Finset.range 0, (1 - z ^ 2 / (↑j + 1) ^ 2)) *
        ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * 1) /
      ↑(∫ (x : ℝ) in 0..π / 2, 1)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: f : ℝ → ℝ
x : ℝ
n : ℕ
hf_conv : ConvexOn ℝ (Ioi 0) f
hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y
hn : n ≠ 0
hx : 0 < x
hx' : x ≤ 1
hn' : 0 < ↑n
this : f ↑n + x * log ↑n = (1 - x) * f ↑n + x * f (↑n + 1)
⊢ f ((1 - x) * ↑n + x * (↑n + 1)) ≤ (1 - x) * f ↑n + x * f (↑n + 1)

Tactic:
simpa only [smul_eq_mul] using
    hf_conv.2 hn' (by linarith : 0 < (n + 1 : ℝ)) (by linarith : 0 ≤ 1 - x) hx.le (by linarith)

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

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

Tactic:
Finset.prod_range_succ

Goal state after:
case inr.succ
z : ℂ
hz : z ≠ 0
n : ℕ
hn :
  Complex.sin (↑π * z) =
    ((↑π * z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) *
        ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) /
      ↑(∫ (x : ℝ) in 0..π / 2, cos x ^ (2 * n))
⊢ ((↑π * z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) *
        ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) /
      ↑(∫ (x : ℝ) in 0..π / 2, cos x ^ (2 * n)) =
    (↑π * z * ((∏ x ∈ Finset.range n, (1 - z ^ 2 / (↑x + 1) ^ 2)) * (1 - z ^ 2 / (↑n + 1) ^ 2)) *
        ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * (n + 1))) /
      ↑(∫ (x : ℝ) in 0..π / 2, cos x ^ (2 * (n + 1)))
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
⊢ 2 * n.succ = 2 * n + 2

Tactic:
rw [Nat.succ_eq_add_one, mul_add, mul_one]

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

Goal state before:
pp: case a
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
v u : V
⊢ map₂ (fun a b ↦ a • b + u) atTop (𝓝 v) ≤ map₂ (fun x1 x2 ↦ x1 • x2) atTop (𝓝 v)

Tactic:
← map_prod_eq_map₂

Goal state after:
case a
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
v u : V
⊢ map (fun p ↦ p.1 • p.2 + u) (atTop ×ˢ 𝓝 v) ≤ map (fun p ↦ p.1 • p.2) (atTop ×ˢ 𝓝 v)
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
v : V
this : (fun x ↦ v) =ᶠ[atTop] (Function.uncurry fun x1 x2 ↦ x1 • x2) ∘ fun c ↦ (c, c⁻¹ • v)
⊢ map (Function.uncurry fun x1 x2 ↦ x1 • x2) (map (fun c ↦ (c, c⁻¹ • v)) atTop) ≤ map₂ (fun x1 x2 ↦ x1 • x2) atTop (𝓝 0)

Tactic:
← map_prod_eq_map₂

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
v : V
this : (fun x ↦ v) =ᶠ[atTop] (Function.uncurry fun x1 x2 ↦ x1 • x2) ∘ fun c ↦ (c, c⁻¹ • v)
⊢ map (Function.uncurry fun x1 x2 ↦ x1 • x2) (map (fun c ↦ (c, c⁻¹ • v)) atTop) ≤ map (fun p ↦ p.1 • p.2) (atTop ×ˢ 𝓝 0)
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
α : Type u_4
l : Filter α
inst✝⁴ : TopologicalSpace k
inst✝³ : OrderTopology k
inst✝² : IsStrictOrderedRing k
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul k V
f : α → k
g : α → V
v : V
hf : Tendsto f l atTop
hg : Tendsto g l (𝓝 v)
⊢ Tendsto (fun x ↦ f x • g x) l (map₂ (fun x1 x2 ↦ x1 • x2) atTop (𝓝 v))

Tactic:
← map_prod_eq_map₂

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
α : Type u_4
l : Filter α
inst✝⁴ : TopologicalSpace k
inst✝³ : OrderTopology k
inst✝² : IsStrictOrderedRing k
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul k V
f : α → k
g : α → V
v : V
hf : Tendsto f l atTop
hg : Tendsto g l (𝓝 v)
⊢ Tendsto (fun x ↦ f x • g x) l (map (fun p ↦ p.1 • p.2) (atTop ×ˢ 𝓝 v))
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 ↦ (↑n.factorial)⁻¹ • x.fst ^ n) e
⊢ HasSum (fun n ↦ ((↑n.factorial)⁻¹ • x.fst ^ n) • x.snd) (e • x.snd - (1⁻¹ • x ^ 0).snd)

Tactic:
pow_zero,

Goal state after:
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 ↦ (↑n.factorial)⁻¹ • x.fst ^ n) e
⊢ HasSum (fun n ↦ ((↑n.factorial)⁻¹ • x.fst ^ n) • x.snd) (e • x.snd - (1⁻¹ • 1).snd)
Module: Mathlib.Analysis.Normed.Algebra.GelfandMazur

Goal state before:
pp: F : Type u_1
inst✝³ : NormedRing F
inst✝² : NormOneClass F
inst✝¹ : NormMulClass F
inst✝ : NormedAlgebra ℂ F
x : F
z : ℂ
hz : IsMinOn (fun x_1 ↦ ‖x - (algebraMap ℂ F) x_1‖) univ z
H : ∀ (z' : ℂ), ‖x - (algebraMap ℂ F) z'‖ ≠ 0
c : ℂ
M : ℝ := ‖x - (algebraMap ℂ F) z‖
hMdef : M = ‖x - (algebraMap ℂ F) z‖
hM₀ : 0 < M
y w : ℂ
hy : ‖x - (algebraMap ℂ F) y‖ = M
n : ℕ
hn : n > 0
p : ℂ[X]
hp : p.IsMonicOfDegree (n - 1)
hrel :
  ‖x - (algebraMap ℂ F) w‖ * ‖(aeval (x - (algebraMap ℂ F) y)) p‖ =
    ‖(x - (algebraMap ℂ F) y) ^ n - ((algebraMap ℂ F) w - (algebraMap ℂ F) y) ^ n‖
⊢ ‖(x - (algebraMap ℂ F) y) ^ n‖ + ‖((algebraMap ℂ F) w - (algebraMap ℂ F) y) ^ n‖ ≤ M ^ n + ‖w - y‖ ^ n

Tactic:
by simp [hy, ← map_sub]

Goal state after:
Module: Mathlib.Analysis.Quaternion

Goal state before:
pp: x : ℍ
⊢ √((equivTuple ℝ) x 0 * (equivTuple ℝ) x 0 + (equivTuple ℝ) x 1 * (equivTuple ℝ) x 1 +
          (equivTuple ℝ) x 2 * (equivTuple ℝ) x 2 +
        (equivTuple ℝ) x 3 * (equivTuple ℝ) x 3) =
    √(x.re ^ 2 + x.imI ^ 2 + x.imJ ^ 2 + x.imK ^ 2)

Tactic:
← 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.Algebra.QuaternionExponential

Goal state before:
pp: q : ℍ
hq : q.re = 0
n : ℕ
hq2 : q ^ 2 = -↑(normSq q)
⊢ (↑(2 * n)!)⁻¹ • q ^ (2 * n) = ↑((-1) ^ n * ‖q‖ ^ (2 * n) / ↑(2 * n)!)

Tactic:
let k : ℝ := ↑(2 * n)!

Goal state after:
q : ℍ
hq : q.re = 0
n : ℕ
hq2 : q ^ 2 = -↑(normSq q)
k : ℝ := ↑(2 * n)!
⊢ (↑(2 * n)!)⁻¹ • q ^ (2 * n) = ↑((-1) ^ n * ‖q‖ ^ (2 * n) / ↑(2 * n)!)
Module: Mathlib.Analysis.Normed.Algebra.GelfandMazur

Goal state before:
pp: case hbc
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‖
this : Tendsto (fun y ↦ ‖y.1‖ * c) (cobounded ℝ ×ˢ ⊤) atTop
y : ℝ × ℝ
hy : y ∈ {0}ᶜ ×ˢ Set.univ
⊢ c ≤ ‖x - (algebraMap ℝ F) (y.1⁻¹ * y.2)‖

Tactic:
exact hbd _

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

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : SeminormedCommRing R
inst✝¹ : Ring S
inst✝ : Algebra R S
f✝ : AlgebraNorm R S
f f' : MulAlgebraNorm R S
h : (fun f ↦ f.toFun) f = (fun f ↦ f.toFun) f'
⊢ f = f'

Tactic:
by
    simp only [AddGroupSeminorm.toFun_eq_coe, MulRingSeminorm.toFun_eq_coe, DFunLike.coe_fn_eq] at h
    obtain ⟨⟨_, _⟩, _⟩ := f; obtain ⟨⟨_, _⟩, _⟩ := f'; congr

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

Goal state before:
pp: 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

Tactic:
simp [hx, hy]

Goal state after:
Module: Mathlib.Analysis.Normed.Field.Approximation

Goal state before:
pp: K : Type u_1
inst✝ : NormedField K
f g : K[X]
ε : ℝ
hε : 0 < ε
a : K
ha : eval a f = 0
hfm : f.Monic
hgm : g.Monic
hdeg : g.natDegree = f.natDegree
hcoeff : ∀ (i : ℕ), ‖g.coeff i - f.coeff i‖ < ε
hg : g.Splits
⊢ ‖eval a (∑ i ∈ Finset.range (g.natDegree + 1), C (g.coeff i - f.coeff i) * X ^ i)‖ ≤
    ∑ i ∈ Finset.range (g.natDegree + 1), ‖(g.coeff i - f.coeff i) * a ^ i‖

Tactic:
have := norm_sum_le (Finset.range (g.natDegree + 1))
        (fun i ↦ (C (g.coeff i - f.coeff i) * X ^ i).eval a)
    simpa [eval_mul, eval_finsetSum] using this

Goal state after:
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

Goal state after:
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
x : L
n : ℕ
hn : 1 ≤ n
⊢ (algNormOfAlgEquiv σ) (x ^ n) = (algNormOfAlgEquiv σ) x ^ n
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
⊢ IsPowMul ⇑(invariantExtension K L)

Tactic:
intro x n hn

Goal state after:
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
x : L
n : ℕ
hn : 1 ≤ n
⊢ (invariantExtension K L) (x ^ n) = (invariantExtension K L) x ^ n
Module: Mathlib.FieldTheory.Minpoly.IsConjRoot

Goal state before:
pp: K : Type u_2
S : Type u_4
inst✝³ : CommRing S
inst✝² : Field K
inst✝¹ : Algebra K S
inst✝ : IsDomain S
x y : S
h : IsIntegral K x
⊢ IsConjRoot K x y ↔ minpoly K x ≠ 0 ∧ (Polynomial.aeval y) (minpoly K x) = 0

Tactic:
isConjRoot_iff_aeval_eq_zero h

Goal state after:
K : Type u_2
S : Type u_4
inst✝³ : CommRing S
inst✝² : Field K
inst✝¹ : Algebra K S
inst✝ : IsDomain S
x y : S
h : IsIntegral K x
⊢ (Polynomial.aeval y) (minpoly K x) = 0 ↔ minpoly K x ≠ 0 ∧ (Polynomial.aeval y) (minpoly K x) = 0
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: case pos
R : Type u_1
inst✝¹ : SeminormedRing R
inst✝ : Nontrivial R
r : R
this : (⨆ n, if n = 0 then ‖r‖ else 0) = ‖r‖
n : ℕ
hn : n = 0
⊢ ‖X.coeff 0 - (C r).coeff 0‖ ^ (1 - ↑0)⁻¹ = ‖r‖

Tactic:
cast_zero,

Goal state after:
case pos
R : Type u_1
inst✝¹ : SeminormedRing R
inst✝ : Nontrivial R
r : R
this : (⨆ n, if n = 0 then ‖r‖ else 0) = ‖r‖
n : ℕ
hn : n = 0
⊢ ‖X.coeff 0 - (C r).coeff 0‖ ^ (1 - 0)⁻¹ = ‖r‖
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: R : Type u_1
inst✝¹ : NormedRing R
inst✝ : Nontrivial R
p : R[X]
hp : p.Monic
⊢ spectralValue p = 0 ↔ p = X ^ p.natDegree

Tactic:
refine ⟨fun h ↦ ?_, fun h ↦ h ▸ spectralValue_X_pow p.natDegree⟩
  refine hp.eq_X_pow_iff_natDegree_le_natTrailingDegree.mpr <|
    le_natTrailingDegree hp.ne_zero fun n hn ↦ ?_
  have h0 : spectralValueTerms p n = 0 := by
    apply le_antisymm ((le_ciSup (spectralValueTerms_bddAbove p) n).trans h.le)
    exact spectralValueTerms_nonneg _ _
  rw [spectralValueTerms_of_lt_natDegree _ hn,
    Real.rpow_eq_zero_iff_of_nonneg (norm_nonneg _)] at h0
  exact norm_eq_zero.mp h0.1

Goal state after:
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
⊢ f (p.coeff 0 • x ^ 0) < f (x ^ p.natDegree)

Tactic:
pow_zero,

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
⊢ f (p.coeff 0 • 1) < f (x ^ p.natDegree)
Module: Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm

Goal state before:
pp: case pos
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
hn0 : n = 0
⊢ ↑(mu μ hn 0) / ↑0 ≤ 1

Tactic:
cast_zero,

Goal state after:
case pos
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
hn0 : n = 0
⊢ ↑(mu μ hn 0) / 0 ≤ 1