mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.LocallyConvex.WeakOperatorTopology

Goal state before:
pp: 𝕜₁ : Type u_5
𝕜₂ : Type u_6
𝕜₃ : Type u_7
E : Type u_9
F : Type u_10
G : Type u_11
inst✝¹⁸ : NormedField 𝕜₁
inst✝¹⁷ : NormedField 𝕜₂
inst✝¹⁶ : NormedField 𝕜₃
σ₁₂ : 𝕜₁ →+* 𝕜₂
σ₁₃ : 𝕜₁ →+* 𝕜₃
σ₂₃ : 𝕜₂ →+* 𝕜₃
inst✝¹⁵ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝¹⁴ : AddCommGroup E
inst✝¹³ : TopologicalSpace E
inst✝¹² : Module 𝕜₁ E
inst✝¹¹ : AddCommGroup F
inst✝¹⁰ : TopologicalSpace F
inst✝⁹ : Module 𝕜₂ F
inst✝⁸ : AddCommGroup G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : Module 𝕜₃ G
inst✝⁵ : IsTopologicalAddGroup F
inst✝⁴ : ContinuousConstSMul 𝕜₂ F
inst✝³ : IsTopologicalAddGroup G
inst✝² : ContinuousConstSMul 𝕜₃ G
inst✝¹ : RingHomSurjective σ₂₃
inst✝ : RingHomIsometric σ₂₃
g : F →SWOT[σ₂₃] G
x : E
z : StrongDual 𝕜₃ G
σ_bij : Function.Bijective ⇑σ₂₃
σ_equiv : 𝕜₂ ≃+* 𝕜₃ := RingEquiv.ofBijective σ₂₃ σ_bij
invPair : RingHomInvPair σ₂₃ ↑σ_equiv.symm := RingHomInvPair.of_ringEquiv σ_equiv
invPair_symm : RingHomInvPair (↑σ_equiv.symm) σ₂₃ := RingHomInvPair.symm σ₂₃ ↑σ_equiv.symm
σ_li : 𝕜₂ ≃ₛₗᵢ[σ₂₃] 𝕜₃ :=
  let __LinearEquiv := LinearEquiv.ofBijective σ₂₃.toSemilinearMap σ_bij;
  { toLinearEquiv := __LinearEquiv, norm_map' := ⋯ }
| Continuous[instTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun a ↦ z ((g.comp a) x)

Tactic:
enter [1, a]; rw [← σ_li.apply_symm_apply (z _), comp_apply, ← toCLM_apply]

Goal state after:
𝕜₁ : Type u_5
𝕜₂ : Type u_6
𝕜₃ : Type u_7
E : Type u_9
F : Type u_10
G : Type u_11
inst✝¹⁸ : NormedField 𝕜₁
inst✝¹⁷ : NormedField 𝕜₂
inst✝¹⁶ : NormedField 𝕜₃
σ₁₂ : 𝕜₁ →+* 𝕜₂
σ₁₃ : 𝕜₁ →+* 𝕜₃
σ₂₃ : 𝕜₂ →+* 𝕜₃
inst✝¹⁵ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝¹⁴ : AddCommGroup E
inst✝¹³ : TopologicalSpace E
inst✝¹² : Module 𝕜₁ E
inst✝¹¹ : AddCommGroup F
inst✝¹⁰ : TopologicalSpace F
inst✝⁹ : Module 𝕜₂ F
inst✝⁸ : AddCommGroup G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : Module 𝕜₃ G
inst✝⁵ : IsTopologicalAddGroup F
inst✝⁴ : ContinuousConstSMul 𝕜₂ F
inst✝³ : IsTopologicalAddGroup G
inst✝² : ContinuousConstSMul 𝕜₃ G
inst✝¹ : RingHomSurjective σ₂₃
inst✝ : RingHomIsometric σ₂₃
g : F →SWOT[σ₂₃] G
x : E
z : StrongDual 𝕜₃ G
σ_bij : Function.Bijective ⇑σ₂₃
σ_equiv : 𝕜₂ ≃+* 𝕜₃ := RingEquiv.ofBijective σ₂₃ σ_bij
invPair : RingHomInvPair σ₂₃ ↑σ_equiv.symm := RingHomInvPair.of_ringEquiv σ_equiv
invPair_symm : RingHomInvPair (↑σ_equiv.symm) σ₂₃ := RingHomInvPair.symm σ₂₃ ↑σ_equiv.symm
σ_li : 𝕜₂ ≃ₛₗᵢ[σ₂₃] 𝕜₃ :=
  let __LinearEquiv := LinearEquiv.ofBijective σ₂₃.toSemilinearMap σ_bij;
  { toLinearEquiv := __LinearEquiv, norm_map' := ⋯ }
a : E →SWOT[σ₁₂] F
| σ_li (σ_li.symm (z (g.toCLM (a x))))
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Deriv

Goal state before:
pp: s : ℂ
hs : ∀ (m : ℕ), s ≠ -↑m
⊢ DifferentiableAt ℂ Gamma s

Tactic:
by
  -- We will show, by induction on `n`, that `Gamma` is differentiable on `-n < Re s`.
  suffices ∀ (n : ℕ) (s : ℂ) (hsre : -n < s.re) (hs : ∀ m : ℕ, s ≠ -m), DifferentiableAt ℂ _ s from
    this (⌊-s.re⌋₊ + 1) s (by grind [Nat.lt_floor_add_one (-s.re)]) hs
  intro n s hsre hs
  induction n generalizing s with
  | zero =>
    -- Case `n = 0`: use relation to `gammaIntegral`
    replace hsre : 0 < s.re := by simpa using hsre
    have : IsOpen {s : ℂ | 0 < s.re} := continuous_re.isOpen_preimage _ isOpen_Ioi
    apply (hasDerivAt_GammaIntegral (by simpa using hsre)).differentiableAt.congr_of_eventuallyEq
    filter_upwards [this.mem_nhds hsre] with a using Gamma_eq_integral
  | succ n IH =>
    -- Induction step: use recurrence relation
    have hsne : s ≠ 0 := by grind [hs 0]
    specialize IH (s + 1) (by grind [add_re, one_re]) (fun m ↦ by grind [hs (m + 1)])
    have := IH.comp s (show DifferentiableAt ℂ (fun s ↦ s + 1) s by fun_prop)
    apply (this.fun_div differentiableAt_id hsne).congr_of_eventuallyEq
    filter_upwards [isOpen_ne.mem_nhds hsne] using by grind

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

Goal state before:
pp: s t : ℂ
hs : 0 < s.re
ht : 0 < t.re
conv_int :
  ∫ (x : ℝ) in Ioi 0,
      ∫ (t_1 : ℝ) in 0..x, ↑(rexp (-t_1)) * ↑t_1 ^ (s - 1) * (↑(rexp (-(x - t_1))) * ↑(x - t_1) ^ (t - 1)) ∂volume =
    (∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (s - 1) ∂volume) * ∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (t - 1) ∂volume
hst : 0 < (s + t).re
⊢ s.GammaIntegral * t.GammaIntegral = (s + t).GammaIntegral * s.betaIntegral t

Tactic:
GammaIntegral,

Goal state after:
s t : ℂ
hs : 0 < s.re
ht : 0 < t.re
conv_int :
  ∫ (x : ℝ) in Ioi 0,
      ∫ (t_1 : ℝ) in 0..x, ↑(rexp (-t_1)) * ↑t_1 ^ (s - 1) * (↑(rexp (-(x - t_1))) * ↑(x - t_1) ^ (t - 1)) ∂volume =
    (∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (s - 1) ∂volume) * ∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (t - 1) ∂volume
hst : 0 < (s + t).re
⊢ (∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (s - 1)) * t.GammaIntegral = (s + t).GammaIntegral * s.betaIntegral t
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: s t : ℂ
hs : 0 < s.re
ht : 0 < t.re
conv_int :
  ∫ (x : ℝ) in Ioi 0,
      ∫ (t_1 : ℝ) in 0..x, ↑(rexp (-t_1)) * ↑t_1 ^ (s - 1) * (↑(rexp (-(x - t_1))) * ↑(x - t_1) ^ (t - 1)) ∂volume =
    (∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (s - 1) ∂volume) * ∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (t - 1) ∂volume
hst : 0 < (s + t).re
⊢ (∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (s - 1)) * t.GammaIntegral = (s + t).GammaIntegral * s.betaIntegral t

Tactic:
GammaIntegral,

Goal state after:
s t : ℂ
hs : 0 < s.re
ht : 0 < t.re
conv_int :
  ∫ (x : ℝ) in Ioi 0,
      ∫ (t_1 : ℝ) in 0..x, ↑(rexp (-t_1)) * ↑t_1 ^ (s - 1) * (↑(rexp (-(x - t_1))) * ↑(x - t_1) ^ (t - 1)) ∂volume =
    (∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (s - 1) ∂volume) * ∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (t - 1) ∂volume
hst : 0 < (s + t).re
⊢ (∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (s - 1)) * ∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (t - 1) =
    (s + t).GammaIntegral * s.betaIntegral t
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: s t : ℂ
hs : 0 < s.re
ht : 0 < t.re
conv_int :
  ∫ (x : ℝ) in Ioi 0,
      ∫ (t_1 : ℝ) in 0..x, ↑(rexp (-t_1)) * ↑t_1 ^ (s - 1) * (↑(rexp (-(x - t_1))) * ↑(x - t_1) ^ (t - 1)) ∂volume =
    (∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (s - 1) ∂volume) * ∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (t - 1) ∂volume
hst : 0 < (s + t).re
⊢ (∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (s - 1)) * ∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (t - 1) =
    (s + t).GammaIntegral * s.betaIntegral t

Tactic:
GammaIntegral,

Goal state after:
s t : ℂ
hs : 0 < s.re
ht : 0 < t.re
conv_int :
  ∫ (x : ℝ) in Ioi 0,
      ∫ (t_1 : ℝ) in 0..x, ↑(rexp (-t_1)) * ↑t_1 ^ (s - 1) * (↑(rexp (-(x - t_1))) * ↑(x - t_1) ^ (t - 1)) ∂volume =
    (∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (s - 1) ∂volume) * ∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (t - 1) ∂volume
hst : 0 < (s + t).re
⊢ (∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (s - 1)) * ∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (t - 1) =
    (∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (s + t - 1)) * s.betaIntegral t
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:
Complex.ofReal_mul,

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✝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)
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: s : ℂ
hs : 0 < s.re
aux1 : -1 < (s - 1).re
⊢ HasMellin ((Ioc 0 1).indicator fun x ↦ 1) s (1 / s)

Tactic:
have aux2 : s ≠ 0 := by contrapose! hs; rw [hs, zero_re]

Goal state after:
s : ℂ
hs : 0 < s.re
aux1 : -1 < (s - 1).re
aux2 : s ≠ 0
⊢ HasMellin ((Ioc 0 1).indicator fun x ↦ 1) s (1 / s)
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
s : Finset ι
F : ι → 𝕜 → 𝕜'
h : ∀ i ∈ s, MeromorphicOn (F i) U
h' : ∀ i ∈ s, ∀ x ∈ U, meromorphicOrderAt (F i) x ≠ ⊤
hA : ∀ᶠ (y : 𝕜) in codiscreteWithin U, ∀ i ∈ s, AnalyticAt 𝕜 (F i) y
hN : ∀ᶠ (y : 𝕜) in codiscreteWithin U, ∀ i ∈ s, F i y ≠ 0
y : 𝕜
h₁y : ∀ i ∈ s, AnalyticAt 𝕜 (F i) y
h₂y : ∀ i ∈ s, F i y ≠ 0
⊢ logDeriv (fun a ↦ ∏ c ∈ s, F c a) y = ∑ c ∈ s, logDeriv (F c) y

Tactic:
exact logDeriv_prod h₂y fun i hi ↦ (h₁y i hi).differentiableAt

Goal state after:
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
α : Type u_4
l : Filter α
f : α → V
v : V
p : P
hf : Tendsto f l (asymptoticNhds k V v)
⊢ Tendsto (fun x ↦ f x +ᵥ p) l (map (fun x ↦ x +ᵥ ?m.44) (asymptoticNhds k V v))

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
α : Type u_4
l : Filter α
f : α → V
v : V
p : P
hf : Tendsto f l (asymptoticNhds k V v)
⊢ P

Tactic:
exact tendsto_map.comp hf

Goal state after:
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
α : Type u_4
l : Filter α
f : α → P
v u : V
hf : Tendsto f l (asymptoticNhds k P v)
⊢ Tendsto (fun x ↦ u +ᵥ f x) l (map (fun b ↦ u +ᵥ b) (asymptoticNhds k P v))

Tactic:
exact tendsto_map.comp hf

Goal state after:
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
s t : Set P
⊢ asymptoticCone k (s ∪ t) = asymptoticCone k s ∪ asymptoticCone k t

Tactic:
ext
  simp only [Set.mem_union, mem_asymptoticCone_iff, Filter.frequently_or_distrib]

Goal state after:
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 : Set P
c : k
v : V
hc : 0 < c
⊢ (∃ᶠ (p : P) in asymptoticNhds k P (c • v), p ∈ s) ↔ ∃ᶠ (p : P) in asymptoticNhds k P v, p ∈ s

Tactic:
asymptoticNhds_smul v hc

Goal state after:
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 : Set P
c : k
v : V
hc : 0 < c
⊢ c • v ∈ asymptoticCone k s ↔ v ∈ asymptoticCone k s

Tactic:
simp_rw [mem_asymptoticCone_iff, asymptoticNhds_smul v hc]

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

Goal state before:
pp: case h
s : ℂ
hs : 0 < s.re
f : ℕ → ℝ → ℂ := fun n ↦ (Ioc 0 ↑n).indicator fun x ↦ ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)
f_ible : ∀ (n : ℕ), Integrable (f n) (volume.restrict (Ioi 0))
x : ℝ
hx : x ∈ Ioi 0
n : ℕ
hn : x ≤ ↑n
⊢ x ∈ Ioc 0 ↑n

Tactic:
exact ⟨hx, hn⟩

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

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_3
W : Type u_4
W₂ : Type u_5
inst✝⁶ : SeminormedAddCommGroup V
inst✝⁵ : SeminormedAddCommGroup W
inst✝⁴ : SeminormedAddCommGroup W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : V →ᴬ[𝕜] W
g : W₂ →ᴬ[𝕜] V
⊢ ‖f.contLinear (g 0)‖ + ‖f 0‖ ≤ ‖f.contLinear‖ * ‖g 0‖ + ‖f 0‖

Tactic:
f.contLinear.le_opNorm

Goal state after:
𝕜 : Type u_1
V : Type u_3
W : Type u_4
W₂ : Type u_5
inst✝⁶ : SeminormedAddCommGroup V
inst✝⁵ : SeminormedAddCommGroup W
inst✝⁴ : SeminormedAddCommGroup W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : V →ᴬ[𝕜] W
g : W₂ →ᴬ[𝕜] V
⊢ ‖f.contLinear‖ * ‖g 0‖ + ‖f 0‖ ≤ ‖f.contLinear‖ * ‖g 0‖ + ‖f 0‖
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: z : ℂ
n : ℕ
hn : n ≠ 0
aux : ∀ (a b c d : ℂ), a * b * (c * d) = a * c * (b * d)
⊢ ↑n ^ z * ↑n ! * (↑n ^ (1 - z) * ↑n !) /
      ((∏ j ∈ Finset.range (n + 1), (z + ↑j)) * ∏ j ∈ Finset.range (n + 1), (1 - z + ↑j)) =
    ↑n / (↑n + 1 - z) * (1 / (z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)))

Tactic:
aux,

Goal state after:
z : ℂ
n : ℕ
hn : n ≠ 0
aux : ∀ (a b c d : ℂ), a * b * (c * d) = a * c * (b * d)
⊢ ↑n ^ z * ↑n ^ (1 - z) * (↑n ! * ↑n !) /
      ((∏ j ∈ Finset.range (n + 1), (z + ↑j)) * ∏ j ∈ Finset.range (n + 1), (1 - z + ↑j)) =
    ↑n / (↑n + 1 - z) * (1 / (z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)))
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

Goal state before:
pp: z : ℂ
n : ℕ
hn : n ≠ 0
aux : ∀ (a b c d : ℂ), a * b * (c * d) = a * c * (b * d)
this : ↑n ^ z * ↑n ^ (1 - z) = ↑n
⊢ ↑n * ↑n ! ^ 2 /
      ((∏ k ∈ Finset.range n, (z + ↑(k + 1))) * (z + ↑0) * ((∏ x ∈ Finset.range n, (1 - z + ↑x)) * (1 - z + ↑n))) =
    ↑n / (↑n + 1 - z) * (1 / (z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)))

Tactic:
aux,

Goal state after:
z : ℂ
n : ℕ
hn : n ≠ 0
aux : ∀ (a b c d : ℂ), a * b * (c * d) = a * c * (b * d)
this : ↑n ^ z * ↑n ^ (1 - z) = ↑n
⊢ ↑n * ↑n ! ^ 2 /
      (((∏ k ∈ Finset.range n, (z + ↑(k + 1))) * ∏ x ∈ Finset.range n, (1 - z + ↑x)) * ((z + ↑0) * (1 - z + ↑n))) =
    ↑n / (↑n + 1 - z) * (1 / (z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)))
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
⊢ ∀ (e : PE ≃ᵢ PE), e x = x → e y = y → e z = z

Tactic:
set s := { e : PE ≃ᵢ PE | e x = x ∧ e y = y }

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}
⊢ ∀ (e : PE ≃ᵢ PE), e x = x → e y = y → e z = z
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Beta

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

Tactic:
Complex.ofReal_mul,

Goal state after:
s : ℝ
⊢ ↑(Gamma s) * ↑(Gamma (1 - s)) = ↑π / Complex.sin (↑π * ↑s)
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) w‖ * ‖(aeval (x - (algebraMap ℂ F) y)) p‖ ≤ M ^ n + ‖w - y‖ ^ n

Tactic:
hrel

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

Goal state before:
pp: case calc_2
q : ℍ
hq : q.re = 0
n : ℕ
hq2 : q ^ 2 = -↑(normSq q)
k : ℝ := ↑(2 * n)!
⊢ k⁻¹ • ↑((-1) ^ n * ‖q‖ ^ (2 * n)) = ↑((-1) ^ n * ‖q‖ ^ (2 * n) / k)

Tactic:
← coe_mul_eq_smul,

Goal state after:
case calc_2
q : ℍ
hq : q.re = 0
n : ℕ
hq2 : q ^ 2 = -↑(normSq q)
k : ℝ := ↑(2 * n)!
⊢ ↑k⁻¹ * ↑((-1) ^ n * ‖q‖ ^ (2 * n)) = ↑((-1) ^ n * ‖q‖ ^ (2 * n) / k)
Module: Mathlib.Analysis.Normed.Algebra.QuaternionExponential

Goal state before:
pp: q : ℍ
this : NormedAlgebra ℚ ℍ
⊢ exp q = exp q.re • exp q.im

Tactic:
← coe_mul_eq_smul,

Goal state after:
q : ℍ
this : NormedAlgebra ℚ ℍ
⊢ exp q = ↑(exp q.re) * exp q.im
Module: Mathlib.Analysis.Normed.Algebra.QuaternionExponential

Goal state before:
pp: q : ℍ
this : NormedAlgebra ℚ ℍ
⊢ exp q = ↑(exp q.re) * exp q.im

Tactic:
← exp_coe,

Goal state after:
q : ℍ
this : NormedAlgebra ℚ ℍ
⊢ exp q = exp ↑q.re * exp q.im
Module: Mathlib.Analysis.Normed.Field.Ultra

Goal state before:
pp: 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)

Tactic:
replace h (x : R) (n : ℕ) : ‖n • x‖ ≤ ‖x‖ := by
    rw [nsmul_eq_mul, norm_mul]
    rcases (norm_nonneg x).eq_or_lt with hx | hx
    · simp only [← hx, mul_zero, le_refl]
    · simpa only [mul_le_iff_le_one_left hx] using h _

Goal state after:
R : Type u_1
inst✝ : NormedDivisionRing R
x : R
m : ℕ
h : ∀ (x : R) (n : ℕ), ‖n • x‖ ≤ ‖x‖
⊢ ‖x + 1‖ ^ m ≤ (m + 1) • max 1 (‖x‖ ^ m)
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
x y : R
⊢ ∀ (x_1 : R), f (x * y * x_1) / f x_1 ≤ seminormFromBounded' f x * seminormFromBounded' f y

Tactic:
by_cases hy : seminormFromBounded' f y = 0

Goal state after:
case pos
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
⊢ ∀ (x_1 : R), f (x * y * x_1) / f x_1 ≤ seminormFromBounded' f x * seminormFromBounded' f y

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
⊢ ∀ (x_1 : R), f (x * y * x_1) / f x_1 ≤ seminormFromBounded' f x * seminormFromBounded' f y
Module: Mathlib.Analysis.Normed.Field.Approximation

Goal state before:
pp: case h
K : Type u_1
L : Type u_2
inst✝² : Field K
inst✝¹ : NormedField L
inst✝ : Algebra K L
hd : DenseRange ⇑(algebraMap K L)
f : L[X]
hf : f.Monic
ε : ℝ
hε : ε > 0
h : f.natDegree = 0
⊢ Monic 1 ∧ f.natDegree = natDegree 1 ∧ ∀ (n : ℕ), ‖(map (algebraMap K L) 1).coeff n - f.coeff n‖ < ε

Tactic:
hf.natDegree_eq_zero.mp

Goal state after:
case h
K : Type u_1
L : Type u_2
inst✝² : Field K
inst✝¹ : NormedField L
inst✝ : Algebra K L
hd : DenseRange ⇑(algebraMap K L)
f : L[X]
hf : f.Monic
ε : ℝ
hε : ε > 0
h : f.natDegree = 0
⊢ Monic 1 ∧ natDegree 1 = natDegree 1 ∧ ∀ (n : ℕ), ‖(map (algebraMap K L) 1).coeff n - coeff 1 n‖ < ε

case h
K : Type u_1
L : Type u_2
inst✝² : Field K
inst✝¹ : NormedField L
inst✝ : Algebra K L
hd : DenseRange ⇑(algebraMap K L)
f : L[X]
hf : f.Monic
ε : ℝ
hε : ε > 0
h : f.natDegree = 0
⊢ f.natDegree = 0
Module: Mathlib.Analysis.Normed.Field.Approximation

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝² : Field K
inst✝¹ : NormedField L
inst✝ : Algebra K L
hd : DenseRange ⇑(algebraMap K L)
f : L[X]
hf : f.Monic
ε : ℝ
hε : ε > 0
h : ¬f.natDegree = 0
c : ℕ → K
hc : ∀ (i : ℕ), dist (f.coeff i) ((algebraMap K L) (c i)) < ε
⊢ ∀ i ∈ Finset.Iio f.natDegree, (C (c i) * X ^ i).natDegree ≤ f.natDegree - 1

Tactic:
refine fun i hi ↦ (Polynomial.natDegree_C_mul_X_pow_le _ _).trans ?_

Goal state after:
K : Type u_1
L : Type u_2
inst✝² : Field K
inst✝¹ : NormedField L
inst✝ : Algebra K L
hd : DenseRange ⇑(algebraMap K L)
f : L[X]
hf : f.Monic
ε : ℝ
hε : ε > 0
h : ¬f.natDegree = 0
c : ℕ → K
hc : ∀ (i : ℕ), dist (f.coeff i) ((algebraMap K L) (c i)) < ε
i : ℕ
hi : i ∈ Finset.Iio f.natDegree
⊢ i ≤ f.natDegree - 1
Module: Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded

Goal state before:
pp: case pos
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
f_nonneg : 0 ≤ f
x : R
hx : ∀ (y : R), f (x * y) ≤ f x * f y
h_one : f 1 ≤ 1
y : R
hy : f y = 0
⊢ f (x * y) / f y ≤ f x

Tactic:
rw [hy, div_zero]

Goal state after:
case pos
R : Type u_1
inst✝ : CommRing R
f : R → ℝ
f_nonneg : 0 ≤ f
x : R
hx : ∀ (y : R), f (x * y) ≤ f x * f y
h_one : f 1 ≤ 1
y : R
hy : f y = 0
⊢ 0 ≤ f x
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
x y : L
⊢ (Classical.choose ⋯) (σ (x + y)) ≤ (Classical.choose ⋯) (σ x) + (Classical.choose ⋯) (σ y)

Tactic:
simp [map_add σ, map_add_le_add]

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

Goal state before:
pp: case h
F : Type u_1
α : Type u_2
inst✝³ : Ring α
inst✝² : FunLike F α ℝ
inst✝¹ : RingSeminormClass F α ℝ
β : Type u_3
inst✝ : Ring β
nα : F
nβ : β → ℝ
hβ : IsPowMul nβ
f : α →+* β
x : α
C : ℝ
hC0 : 0 < C
hC : ∀ (x : α), nβ (f x) ≤ C * nα x
hlim : Tendsto (fun n ↦ C ^ (1 / ↑n) * nα x) atTop (𝓝 (nα x))
n : ℕ
hn : 1 ≤ n
h : (C ^ (1 / ↑n)) ^ n = C
⊢ C * nα (x ^ n) ≤ C * nα x ^ n

Tactic:
rw [mul_le_mul_iff_right₀ hC0]

Goal state after:
case h
F : Type u_1
α : Type u_2
inst✝³ : Ring α
inst✝² : FunLike F α ℝ
inst✝¹ : RingSeminormClass F α ℝ
β : Type u_3
inst✝ : Ring β
nα : F
nβ : β → ℝ
hβ : IsPowMul nβ
f : α →+* β
x : α
C : ℝ
hC0 : 0 < C
hC : ∀ (x : α), nβ (f x) ≤ C * nα x
hlim : Tendsto (fun n ↦ C ^ (1 / ↑n) * nα x) atTop (𝓝 (nα x))
n : ℕ
hn : 1 ≤ n
h : (C ^ (1 / ↑n)) ^ n = C
⊢ nα (x ^ n) ≤ nα x ^ n