mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Group.Prod

Goal state before:
pp: G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
inst✝⁴ : MeasurableInv G
μ' ν' : Measure G
inst✝³ : SigmaFinite μ'
inst✝² : SigmaFinite ν'
inst✝¹ : μ'.IsMulLeftInvariant
inst✝ : ν'.IsMulLeftInvariant
s : Set G
h2s : ν' s ≠ 0
h3s : ν' s ≠ ∞
hs : MeasurableSet s
t : Set G
ht : MeasurableSet t
h1 : ν' s * ∫⁻ (y : G), t.indicator (fun x ↦ 1) y⁻¹ / ν' ((fun x ↦ x * y⁻¹) ⁻¹' s) ∂ν' = ν' t
h2 : μ' s * ∫⁻ (y : G), t.indicator (fun x ↦ 1) y⁻¹ / ν' ((fun x ↦ x * y⁻¹) ⁻¹' s) ∂ν' = μ' t
⊢ ν' s * (μ' s * ∫⁻ (y : G), t.indicator (fun x ↦ 1) y⁻¹ / ν' ((fun x ↦ x * y⁻¹) ⁻¹' s) ∂ν') = ν' s * μ' t

Tactic:
h2

Goal state after:
G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
inst✝⁴ : MeasurableInv G
μ' ν' : Measure G
inst✝³ : SigmaFinite μ'
inst✝² : SigmaFinite ν'
inst✝¹ : μ'.IsMulLeftInvariant
inst✝ : ν'.IsMulLeftInvariant
s : Set G
h2s : ν' s ≠ 0
h3s : ν' s ≠ ∞
hs : MeasurableSet s
t : Set G
ht : MeasurableSet t
h1 : ν' s * ∫⁻ (y : G), t.indicator (fun x ↦ 1) y⁻¹ / ν' ((fun x ↦ x * y⁻¹) ⁻¹' s) ∂ν' = ν' t
h2 : μ' s * ∫⁻ (y : G), t.indicator (fun x ↦ 1) y⁻¹ / ν' ((fun x ↦ x * y⁻¹) ⁻¹' s) ∂ν' = μ' t
⊢ ν' s * μ' t = ν' s * μ' t
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0
hf : AEMeasurable f μ
g : α → ℝ≥0∞
f' : α → ℝ≥0
hf'_m : Measurable f'
hf'_ae : f =ᵐ[μ] f'
g' : α → ℝ≥0∞
g'meas : Measurable g'
hg' : (fun x ↦ ↑(f x) * g x) =ᵐ[μ] g'
a : α
hfga : ↑(f' a) * g a = g' a
hff'a : f a = f' a
h'a : ↑(f' a) ≠ 0
⊢ g a = (↑(f' a))⁻¹ * g' a

Tactic:
rw [← hfga, ← mul_assoc, ENNReal.inv_mul_cancel h'a ENNReal.coe_ne_top, one_mul]

Goal state after:
Module: Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g h : α → β
hf : AEStronglyMeasurable f μ
hh : AEStronglyMeasurable h μ
a : α
⊢ edist (f a) (g a) ≤ edist (f a) (h a) + edist (g a) (h a)

Tactic:
apply edist_triangle_right

Goal state after:
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: case iSup
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
h_mf : Measurable f
g : ℕ → α → ℝ≥0∞
h_mea_g : ∀ (n : ℕ), Measurable (g n)
h_mono_g : Monotone g
h_ind : ∀ (n : ℕ), ∫⁻ (a : α), g n a ∂μ.withDensity f = ∫⁻ (a : α), (f * g n) a ∂μ
this : Monotone fun n a ↦ f a * g n a
⊢ ∫⁻ (a : α), (fun x ↦ ⨆ n, g n x) a ∂μ.withDensity f = ∫⁻ (a : α), (f * fun x ↦ ⨆ n, g n x) a ∂μ

Tactic:
simp [lintegral_iSup, ENNReal.mul_iSup, h_mf.fun_mul (h_mea_g _), *]

Goal state after:
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: x : ℝ
hx : 0 ≤ x
y : ℝ
⊢ ↑(NNReal.sqrt (x * y).toNNReal) = ↑(NNReal.sqrt x.toNNReal * NNReal.sqrt y.toNNReal)

Tactic:
NNReal.coe_inj,

Goal state after:
x : ℝ
hx : 0 ≤ x
y : ℝ
⊢ NNReal.sqrt (x * y).toNNReal = NNReal.sqrt x.toNNReal * NNReal.sqrt y.toNNReal
Module: Mathlib.Analysis.Complex.Norm

Goal state before:
pp: z w : ℂ
h : z.im = w.im
⊢ dist z w = dist z.re w.re

Tactic:
by
  rw [dist_eq_re_im, h, sub_self, zero_pow two_ne_zero, add_zero, Real.sqrt_sq_eq_abs, Real.dist_eq]

Goal state after:
Module: Mathlib.Data.Complex.Basic

Goal state before:
pp: m : ℤ
| I ^ m

Tactic:
← Int.mul_ediv_add_emod m 4

Goal state after:
m : ℤ
| I ^ (4 * (m / 4) + m % 4)
Module: Mathlib.Data.Complex.Basic

Goal state before:
pp: case mp
a₁ a₂ b : ℝ
x : ℂ
⊢ x ∈ (fun x ↦ ↑x + ↑b * I) '' [[a₁, a₂]] → x ∈ ⇑equivRealProd ⁻¹' [[a₁, a₂]] ×ˢ {b}

Tactic:
intro hx

Goal state after:
case mp
a₁ a₂ b : ℝ
x : ℂ
hx : x ∈ (fun x ↦ ↑x + ↑b * I) '' [[a₁, a₂]]
⊢ x ∈ ⇑equivRealProd ⁻¹' [[a₁, a₂]] ×ˢ {b}
Module: Mathlib.Data.Complex.Basic

Goal state before:
pp: case mpr
a₁ a₂ b : ℝ
x : ℂ
⊢ x ∈ ⇑equivRealProd ⁻¹' [[a₁, a₂]] ×ˢ {b} → x ∈ (fun x ↦ ↑x + ↑b * I) '' [[a₁, a₂]]

Tactic:
intro hx

Goal state after:
case mpr
a₁ a₂ b : ℝ
x : ℂ
hx : x ∈ ⇑equivRealProd ⁻¹' [[a₁, a₂]] ×ˢ {b}
⊢ x ∈ (fun x ↦ ↑x + ↑b * I) '' [[a₁, a₂]]
Module: Mathlib.Data.Complex.Basic

Goal state before:
pp: case mp
a b₁ b₂ : ℝ
x : ℂ
⊢ x ∈ (fun y ↦ ↑a + ↑y * I) '' [[b₁, b₂]] → x ∈ ⇑equivRealProd ⁻¹' {a} ×ˢ [[b₁, b₂]]

Tactic:
intro hx

Goal state after:
case mp
a b₁ b₂ : ℝ
x : ℂ
hx : x ∈ (fun y ↦ ↑a + ↑y * I) '' [[b₁, b₂]]
⊢ x ∈ ⇑equivRealProd ⁻¹' {a} ×ˢ [[b₁, b₂]]
Module: Mathlib.Data.Complex.Basic

Goal state before:
pp: case mpr
a b₁ b₂ : ℝ
x : ℂ
⊢ x ∈ ⇑equivRealProd ⁻¹' {a} ×ˢ [[b₁, b₂]] → x ∈ (fun y ↦ ↑a + ↑y * I) '' [[b₁, b₂]]

Tactic:
intro hx

Goal state after:
case mpr
a b₁ b₂ : ℝ
x : ℂ
hx : x ∈ ⇑equivRealProd ⁻¹' {a} ×ˢ [[b₁, b₂]]
⊢ x ∈ (fun y ↦ ↑a + ↑y * I) '' [[b₁, b₂]]
Module: Mathlib.Analysis.CStarAlgebra.Basic

Goal state before:
pp: E : Type u_2
inst✝³ : NormedRing E
inst✝² : StarRing E
inst✝¹ : CStarRing E
inst✝ : Nontrivial E
this : 0 < ‖1‖
⊢ ‖1‖ * ‖1‖ = 1 * ‖1‖

Tactic:
← norm_star_mul_self,

Goal state after:
E : Type u_2
inst✝³ : NormedRing E
inst✝² : StarRing E
inst✝¹ : CStarRing E
inst✝ : Nontrivial E
this : 0 < ‖1‖
⊢ ‖1⋆ * 1‖ = 1 * ‖1‖
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
a : ℝ
n : ℕ
⊢ im (↑a ^ n) = 0

Tactic:
← @ofReal_pow,

Goal state after:
K : Type u_1
inst✝ : RCLike K
a : ℝ
n : ℕ
⊢ im ↑(a ^ n) = 0
Module: Mathlib.Analysis.Complex.Basic

Goal state before:
pp: 𝕜 : Type u_2
𝕜' : Type u_3
inst✝¹ : RCLike 𝕜
inst✝ : RCLike 𝕜'
h : RCLike.im RCLike.I = 1
a : 𝕜
⊢ 0 ≤ RCLike.re ((RCLike.map 𝕜 𝕜') a) ∧ RCLike.im ((RCLike.map 𝕜 𝕜') a) = 0 ↔ 0 ≤ a

Tactic:
RCLike.nonneg_iff (K := 𝕜)

Goal state after:
𝕜 : Type u_2
𝕜' : Type u_3
inst✝¹ : RCLike 𝕜
inst✝ : RCLike 𝕜'
h : RCLike.im RCLike.I = 1
a : 𝕜
⊢ 0 ≤ RCLike.re ((RCLike.map 𝕜 𝕜') a) ∧ RCLike.im ((RCLike.map 𝕜 𝕜') a) = 0 ↔ 0 ≤ RCLike.re a ∧ RCLike.im a = 0
Module: Mathlib.LinearAlgebra.Complex.Module

Goal state before:
pp: A : Type u_1
inst✝³ : AddCommGroup A
inst✝² : Module ℂ A
inst✝¹ : StarAddMonoid A
inst✝ : StarModule ℂ A
a : A
⊢ I • a ∈ skewAdjoint A ↔ IsSelfAdjoint a

Tactic:
simp [skewAdjoint.mem_iff, IsSelfAdjoint, smul_right_inj]

Goal state after:
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
z : K
⊢ √(‖z‖ ^ 2) = ‖z‖

Tactic:
Real.sqrt_sq (norm_nonneg _)

Goal state after:
K : Type u_1
inst✝ : RCLike K
z : K
⊢ ‖z‖ = ‖z‖
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
E : Type u_2
inst✝ : RCLike K
z : K
r : ℤ
x : K
⊢ ‖r • x‖ = ‖r‖ * ‖x‖

Tactic:
by
    rw [zsmul_eq_mul, norm_mul, ← ofReal_intCast, norm_ofReal, Int.norm_eq_abs]

Goal state after:
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
⊢ map K ℝ = reCLM

Tactic:
by
  ext; simp only [map_apply, I, mul_zero, add_zero]; rfl

Goal state after:
Module: Mathlib.Analysis.Asymptotics.Defs

Goal state before:
pp: α : Type u_1
E' : Type u_6
inst✝ : SeminormedAddCommGroup E'
l : Filter α
ι : Type u_18
A : ι → α → E'
C : ι → ℝ
s : Finset ι
B : ι → α → ℝ
hs : s.Nonempty
hAB : ∀ i ∈ s, ∀ᶠ (x : α) in l, ‖A i x‖ ≤ C i * ‖B i x‖
x : α
hx : ∀ i ∈ s, ‖A i x‖ ≤ C i * ‖B i x‖
⊢ ∑ i ∈ s, C i * ‖B i x‖ ≤ ∑ i ∈ s, sSup (C '' ↑s) * ‖B i x‖

Tactic:
refine Finset.sum_le_sum ?_
        intro j hj; gcongr
        rw [← s.sup'_eq_csSup_image hs, Finset.le_sup'_iff]; use j

Goal state after:
Module: Mathlib.Analysis.Asymptotics.Defs

Goal state before:
pp: α : Type u_1
E' : Type u_6
inst✝ : SeminormedAddCommGroup E'
l : Filter α
ι : Type u_18
A : ι → α → E'
B : ι → α → ℝ
C : ℝ
i : α → Finset ι
hAB : IsBigOWith C (⊤ ×ˢ l) (Function.uncurry A) (Function.uncurry B)
⊢ IsBigOWith C l (fun H ↦ ∑ j ∈ i H, A j H) fun H ↦ ∑ j ∈ i H, ‖B j H‖

Tactic:
simp only [IsBigOWith_def] at *
  obtain ⟨s₁, hs₁, s₂, hs₂, hbound⟩ := Filter.eventually_prod_iff.mp hAB
  filter_upwards [hs₂] with H hH
  calc
    ‖∑ j ∈ i H, A j H‖ ≤ ∑ j ∈ i H, ‖A j H‖ := norm_sum_le ..
    _ ≤ ∑ j ∈ i H, C * ‖B j H‖ :=
        Finset.sum_le_sum fun j _ => hbound (Filter.eventually_top.mp hs₁ j) hH
    _ = C * ∑ j ∈ i H, ‖B j H‖ := (Finset.mul_sum ..).symm
    _ = C * ‖∑ j ∈ i H, ‖B j H‖‖ := by
        congr; rw [Real.norm_of_nonneg (Finset.sum_nonneg (fun _ _ ↦ norm_nonneg _))]

Goal state after:
Module: Mathlib.Analysis.Asymptotics.Lemmas

Goal state before:
pp: α : Type u_1
E'' : Type u_9
F'' : Type u_10
inst✝¹ : NormedAddCommGroup E''
inst✝ : NormedAddCommGroup F''
g'' : α → F''
l : Filter α
c : E''
⊢ (fun _x ↦ c) =o[l] g'' ↔ c = 0 ∨ Tendsto (norm ∘ g'') l atTop

Tactic:
rcases eq_or_ne c 0 with (rfl | hc)

Goal state after:
case inl
α : Type u_1
E'' : Type u_9
F'' : Type u_10
inst✝¹ : NormedAddCommGroup E''
inst✝ : NormedAddCommGroup F''
g'' : α → F''
l : Filter α
⊢ (fun _x ↦ 0) =o[l] g'' ↔ 0 = 0 ∨ Tendsto (norm ∘ g'') l atTop

case inr
α : Type u_1
E'' : Type u_9
F'' : Type u_10
inst✝¹ : NormedAddCommGroup E''
inst✝ : NormedAddCommGroup F''
g'' : α → F''
l : Filter α
c : E''
hc : c ≠ 0
⊢ (fun _x ↦ c) =o[l] g'' ↔ c = 0 ∨ Tendsto (norm ∘ g'') l atTop
Module: Mathlib.Analysis.Complex.Trigonometric

Goal state before:
pp: x : ℂ
⊢ cos x + sin x * I = cexp (x * I)

Tactic:
← cosh_add_sinh,

Goal state after:
x : ℂ
⊢ cos x + sin x * I = cosh (x * I) + sinh (x * I)
Module: Mathlib.Analysis.Complex.Trigonometric

Goal state before:
pp: x : ℂ
⊢ sin (2 * x) = 2 * sin x * cos x

Tactic:
by
  rw [two_mul, sin_add, two_mul, add_mul, mul_comm]

Goal state after:
Module: Mathlib.Algebra.Order.CauSeq.BigOperators

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁴ : Field α
inst✝³ : LinearOrder α
inst✝² : IsStrictOrderedRing α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : ℕ → β
ha : IsCauSeq abs fun m ↦ ∑ n ∈ range m, abv (f n)
hb : IsCauSeq abv fun m ↦ ∑ n ∈ range m, g n
ε : α
ε0 : 0 < ε
P : α
hP : ∀ (i : ℕ), |∑ n ∈ range i, abv (f n)| < P
Q : α
hQ : ∀ (i : ℕ), abv (∑ n ∈ range i, g n) < Q
hP0 : 0 < P
hPε0 : 0 < ε / (2 * P)
N : ℕ
hN : ∀ j ≥ N, ∀ k ≥ N, abv (∑ n ∈ range j, g n - ∑ n ∈ range k, g n) < ε / (2 * P)
hQε0 : 0 < ε / (4 * Q)
M : ℕ
hM : ∀ j ≥ M, ∀ k ≥ M, |∑ n ∈ range j, abv (f n) - ∑ n ∈ range k, abv (f n)| < ε / (4 * Q)
K : ℕ
hK : K ≥ 2 * (max N M + 1)
h₁ : ∑ m ∈ range K, ∑ k ∈ range (m + 1), f k * g (m - k) = ∑ m ∈ range K, ∑ n ∈ range (K - m), f m * g n
h₂ : (fun i ↦ ∑ k ∈ range (K - i), f i * g k) = fun i ↦ f i * ∑ k ∈ range (K - i), g k
h₃ :
  ∑ i ∈ range K, f i * ∑ k ∈ range (K - i), g k =
    ∑ i ∈ range K, f i * (∑ k ∈ range (K - i), g k - ∑ k ∈ range K, g k) + ∑ i ∈ range K, f i * ∑ k ∈ range K, g k
two_mul_two : 4 = 2 * 2
hQ0 : Q ≠ 0
h2Q0 : 2 * Q ≠ 0
hε : ε / (2 * P) * P + ε / (4 * Q) * (2 * Q) = ε
hNMK : max N M + 1 < K
hKN : N < K
hsumlesum :
  ∑ i ∈ range (max N M + 1), abv (f i) * abv (∑ k ∈ range (K - i), g k - ∑ k ∈ range K, g k) ≤
    ∑ i ∈ range (max N M + 1), abv (f i) * (ε / (2 * P))
hsumltP : ∑ n ∈ range (max N M + 1), abv (f n) < P
⊢ abv ((∑ k ∈ range K, f k) * ∑ k ∈ range K, g k - ∑ n ∈ range K, ∑ m ∈ range (n + 1), f m * g (n - m)) < ε

Tactic:
rw [h₁, h₂, h₃, sum_mul, ← sub_sub, sub_right_comm, sub_self, zero_sub, abv_neg abv]

Goal state after:
α : Type u_1
β : Type u_2
inst✝⁴ : Field α
inst✝³ : LinearOrder α
inst✝² : IsStrictOrderedRing α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : ℕ → β
ha : IsCauSeq abs fun m ↦ ∑ n ∈ range m, abv (f n)
hb : IsCauSeq abv fun m ↦ ∑ n ∈ range m, g n
ε : α
ε0 : 0 < ε
P : α
hP : ∀ (i : ℕ), |∑ n ∈ range i, abv (f n)| < P
Q : α
hQ : ∀ (i : ℕ), abv (∑ n ∈ range i, g n) < Q
hP0 : 0 < P
hPε0 : 0 < ε / (2 * P)
N : ℕ
hN : ∀ j ≥ N, ∀ k ≥ N, abv (∑ n ∈ range j, g n - ∑ n ∈ range k, g n) < ε / (2 * P)
hQε0 : 0 < ε / (4 * Q)
M : ℕ
hM : ∀ j ≥ M, ∀ k ≥ M, |∑ n ∈ range j, abv (f n) - ∑ n ∈ range k, abv (f n)| < ε / (4 * Q)
K : ℕ
hK : K ≥ 2 * (max N M + 1)
h₁ : ∑ m ∈ range K, ∑ k ∈ range (m + 1), f k * g (m - k) = ∑ m ∈ range K, ∑ n ∈ range (K - m), f m * g n
h₂ : (fun i ↦ ∑ k ∈ range (K - i), f i * g k) = fun i ↦ f i * ∑ k ∈ range (K - i), g k
h₃ :
  ∑ i ∈ range K, f i * ∑ k ∈ range (K - i), g k =
    ∑ i ∈ range K, f i * (∑ k ∈ range (K - i), g k - ∑ k ∈ range K, g k) + ∑ i ∈ range K, f i * ∑ k ∈ range K, g k
two_mul_two : 4 = 2 * 2
hQ0 : Q ≠ 0
h2Q0 : 2 * Q ≠ 0
hε : ε / (2 * P) * P + ε / (4 * Q) * (2 * Q) = ε
hNMK : max N M + 1 < K
hKN : N < K
hsumlesum :
  ∑ i ∈ range (max N M + 1), abv (f i) * abv (∑ k ∈ range (K - i), g k - ∑ k ∈ range K, g k) ≤
    ∑ i ∈ range (max N M + 1), abv (f i) * (ε / (2 * P))
hsumltP : ∑ n ∈ range (max N M + 1), abv (f n) < P
⊢ abv (∑ i ∈ range K, f i * (∑ k ∈ range (K - i), g k - ∑ k ∈ range K, g k)) < ε
Module: Mathlib.Analysis.Asymptotics.Lemmas

Goal state before:
pp: case refine_1
α : Type u_1
𝕜 : Type u_15
inst✝ : NormedDivisionRing 𝕜
l : Filter α
f g h✝ : α → 𝕜
c : ℝ
hf : ∀ᶠ (x : α) in l, f x ≠ 0
h : ∀ᶠ (x : α) in l, ‖f x * g x‖ ≤ c * ‖h✝ x‖
⊢ ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h✝ x / f x‖

Tactic:
refine h.congr <| Eventually.mp hf <| Eventually.of_forall fun x hx ↦ ?_

Goal state after:
case refine_1
α : Type u_1
𝕜 : Type u_15
inst✝ : NormedDivisionRing 𝕜
l : Filter α
f g h✝ : α → 𝕜
c : ℝ
hf : ∀ᶠ (x : α) in l, f x ≠ 0
h : ∀ᶠ (x : α) in l, ‖f x * g x‖ ≤ c * ‖h✝ x‖
x : α
hx : f x ≠ 0
⊢ ‖f x * g x‖ ≤ c * ‖h✝ x‖ ↔ ‖g x‖ ≤ c * ‖h✝ x / f x‖
Module: Mathlib.Analysis.Asymptotics.Lemmas

Goal state before:
pp: case refine_2
α : Type u_1
𝕜 : Type u_15
inst✝ : NormedDivisionRing 𝕜
l : Filter α
f g h✝ : α → 𝕜
c : ℝ
hf : ∀ᶠ (x : α) in l, f x ≠ 0
h : ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h✝ x / f x‖
⊢ ∀ᶠ (x : α) in l, ‖f x * g x‖ ≤ c * ‖h✝ x‖

Tactic:
refine h.congr <| Eventually.mp hf <| Eventually.of_forall fun x hx ↦ ?_

Goal state after:
case refine_2
α : Type u_1
𝕜 : Type u_15
inst✝ : NormedDivisionRing 𝕜
l : Filter α
f g h✝ : α → 𝕜
c : ℝ
hf : ∀ᶠ (x : α) in l, f x ≠ 0
h : ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h✝ x / f x‖
x : α
hx : f x ≠ 0
⊢ ‖g x‖ ≤ c * ‖h✝ x / f x‖ ↔ ‖f x * g x‖ ≤ c * ‖h✝ x‖
Module: Mathlib.Analysis.Complex.Trigonometric

Goal state before:
pp: x : ℝ
⊢ sin x ^ 2 ≤ 1

Tactic:
by
  rw [← sin_sq_add_cos_sq x]; exact le_add_of_nonneg_right (sq_nonneg _)

Goal state after:
Module: Mathlib.Analysis.Complex.Trigonometric

Goal state before:
pp: x : ℝ
⊢ |sin x| = √(1 - cos x ^ 2)

Tactic:
← sin_sq,

Goal state after:
x : ℝ
⊢ |sin x| = √(sin x ^ 2)
Module: Mathlib.Analysis.Complex.Trigonometric

Goal state before:
pp: x : ℝ
⊢ ‖cexp (I * ↑x) - 1‖ = ‖cexp (-(↑(x / 2) * I)) * (1 - cexp (↑x * I)) * I‖

Tactic:
← neg_mul,

Goal state after:
x : ℝ
⊢ ‖cexp (I * ↑x) - 1‖ = ‖cexp (-↑(x / 2) * I) * (1 - cexp (↑x * I)) * I‖
Module: Mathlib.Algebra.Order.CauSeq.BigOperators

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
inst✝⁵ : Field α
inst✝⁴ : LinearOrder α
inst✝³ : IsStrictOrderedRing α
inst✝² : Ring β
abv : β → α
inst✝¹ : IsAbsoluteValue abv
inst✝ : Archimedean α
f : ℕ → β
n m : ℕ
hmn : n.succ ≤ m
hr0 : 0 ≤ 0
hr1 : 0 < 1
h : ∀ (m : ℕ), n ≤ m → abv (f m.succ) ≤ 0 * abv (f m)
har1 : |0| < 1
⊢ abv (f m) ≤ abv (f n.succ) * 0⁻¹ ^ n.succ * 0 ^ m

pp: case inr
α : Type u_1
β : Type u_2
inst✝⁵ : Field α
inst✝⁴ : LinearOrder α
inst✝³ : IsStrictOrderedRing α
inst✝² : Ring β
abv : β → α
inst✝¹ : IsAbsoluteValue abv
inst✝ : Archimedean α
f : ℕ → β
n : ℕ
r : α
hr0 : 0 ≤ r
hr1 : r < 1
h : ∀ (m : ℕ), n ≤ m → abv (f m.succ) ≤ r * abv (f m)
har1 : |r| < 1
m : ℕ
hmn : n.succ ≤ m
hr : 0 < r
⊢ abv (f m) ≤ abv (f n.succ) * r⁻¹ ^ n.succ * r ^ m

Tactic:
· have m_pos := lt_of_lt_of_le (Nat.succ_pos n) hmn
    have := h m.pred (Nat.le_of_succ_le_succ (by rwa [Nat.succ_pred_eq_of_pos m_pos]))
    simpa [Nat.sub_add_cancel m_pos, pow_succ] using this

Goal state after:
case inr
α : Type u_1
β : Type u_2
inst✝⁵ : Field α
inst✝⁴ : LinearOrder α
inst✝³ : IsStrictOrderedRing α
inst✝² : Ring β
abv : β → α
inst✝¹ : IsAbsoluteValue abv
inst✝ : Archimedean α
f : ℕ → β
n : ℕ
r : α
hr0 : 0 ≤ r
hr1 : r < 1
h : ∀ (m : ℕ), n ≤ m → abv (f m.succ) ≤ r * abv (f m)
har1 : |r| < 1
m : ℕ
hmn : n.succ ≤ m
hr : 0 < r
⊢ abv (f m) ≤ abv (f n.succ) * r⁻¹ ^ n.succ * r ^ m
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic

Goal state before:
pp: ⊢ cos π = -1

Tactic:
rw [← mul_div_cancel_left₀ π two_ne_zero, mul_div_assoc, cos_two_mul, cos_pi_div_two]

Goal state after:
⊢ 2 * 0 ^ 2 - 1 = -1
Module: Mathlib.Analysis.SpecialFunctions.Exp

Goal state before:
pp: a : ℝ
⊢ rexp '' Set.Iic a = Set.Ioc 0 (rexp a)

Tactic:
by
  rw [← coe_comp_expOrderIso, image_comp, expOrderIso.image_Iic, image_subtype_val_Ioi_Iic,
    Function.comp_apply]

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

Goal state before:
pp: n : ℕ
⊢ cos (↑n * π) = (-1) ^ n

Tactic:
by
  simpa using cos_add_nat_mul_pi 0 n

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Pointwise

Goal state before:
pp: E : Type u_1
inst✝ : SeminormedGroup E
s t : Set E
hs : IsBounded s
ht : IsBounded t
⊢ IsBounded (s * t)

Tactic:
obtain ⟨Rs, hRs⟩ : ∃ R, ∀ x ∈ s, ‖x‖ ≤ R := hs.exists_norm_le'

Goal state after:
E : Type u_1
inst✝ : SeminormedGroup E
s t : Set E
hs : IsBounded s
ht : IsBounded t
Rs : ℝ
hRs : ∀ x ∈ s, ‖x‖ ≤ Rs
⊢ IsBounded (s * t)
Module: Mathlib.Analysis.Normed.Module.RCLike.Real

Goal state before:
pp: E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
r : ℝ
hr : r ≠ 0
⊢ closedBall x r \ ball x r = sphere x r

Tactic:
closedBall_sdiff_ball

Goal state after:
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
r : ℝ
hr : r ≠ 0
⊢ sphere x r = sphere x r
Module: Mathlib.Analysis.Normed.Module.RCLike.Real

Goal state before:
pp: case inr.inr
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x y : E
hr✝ : dist y x ≠ 0
hr : 0 < dist y x
hy : y ∈ interior (closedBall x (dist y x))
f : ℝ → E := fun c ↦ c • (y - x) + x
c : ℝ
hc : c ∈ f ⁻¹' closedBall x (dist y x)
⊢ ‖c‖ * dist y x ≤ 1 * dist y x

Tactic:
simpa [f, dist_eq_norm, norm_smul] using hc

Goal state after:
Module: Mathlib.Analysis.Normed.Module.RCLike.Real

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : Nontrivial E
x : E
r : ℝ
⊢ closedBall x r \ ball x r = sphere x r

Tactic:
closedBall_sdiff_ball

Goal state after:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : Nontrivial E
x : E
r : ℝ
⊢ sphere x r = sphere x r
Module: Mathlib.Analysis.Normed.Module.Ball.Pointwise

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
s : Set E
hs : Bornology.IsBounded s
u : Set E
hu : u ∈ 𝓝 x
ε : ℝ
εpos : 0 < ε
hε : closedBall x ε ⊆ u
R : ℝ
Rpos : 0 < R
hR : s ⊆ closedBall 0 R
this : closedBall 0 (ε / R) ∈ 𝓝 0
⊢ ∀ᶠ (r : 𝕜) in 𝓝 0, {x} + r • s ⊆ u

Tactic:
filter_upwards [this] with r hr

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
s : Set E
hs : Bornology.IsBounded s
u : Set E
hu : u ∈ 𝓝 x
ε : ℝ
εpos : 0 < ε
hε : closedBall x ε ⊆ u
R : ℝ
Rpos : 0 < R
hR : s ⊆ closedBall 0 R
this : closedBall 0 (ε / R) ∈ 𝓝 0
r : 𝕜
hr : r ∈ closedBall 0 (ε / R)
⊢ {x} + r • s ⊆ u
Module: Mathlib.Analysis.Normed.Module.Ball.Pointwise

Goal state before:
pp: E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
δ ε : ℝ
hε : 0 < ε
hδ : 0 < δ
x : E
⊢ thickening ε (Metric.ball x δ) = Metric.ball x (ε + δ)

Tactic:
← thickening_singleton,

Goal state after:
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
δ ε : ℝ
hε : 0 < ε
hδ : 0 < δ
x : E
⊢ thickening ε (thickening δ {x}) = Metric.ball x (ε + δ)
Module: Mathlib.Analysis.Normed.Module.Ball.Pointwise

Goal state before:
pp: E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
δ ε : ℝ
hε : 0 ≤ ε
hδ : 0 < δ
x : E
⊢ cthickening ε (Metric.ball x δ) = Metric.closedBall x (ε + δ)

Tactic:
← thickening_singleton,

Goal state after:
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
δ ε : ℝ
hε : 0 ≤ ε
hδ : 0 < δ
x : E
⊢ cthickening ε (thickening δ {x}) = Metric.closedBall x (ε + δ)
Module: Mathlib.Analysis.Normed.Module.Ball.Pointwise

Goal state before:
pp: E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
δ ε : ℝ
hε : 0 ≤ ε
hδ : 0 ≤ δ
x : E
⊢ cthickening ε (cthickening δ {x}) = Metric.closedBall x (ε + δ)

Tactic:
cthickening_cthickening hε hδ,

Goal state after:
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
δ ε : ℝ
hε : 0 ≤ ε
hδ : 0 ≤ δ
x : E
⊢ cthickening (ε + δ) {x} = Metric.closedBall x (ε + δ)
Module: Mathlib.Analysis.Normed.Module.Ball.Pointwise

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
x : E
r : ℝ
hr : 0 ≤ r
⊢ c • closedBall x r = closedBall (c • x) (‖c‖ * r)

Tactic:
rcases eq_or_ne c 0 with (rfl | hc)

Goal state after:
case inl
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
r : ℝ
hr : 0 ≤ r
⊢ 0 • closedBall x r = closedBall (0 • x) (‖0‖ * r)

case inr
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
x : E
r : ℝ
hr : 0 ≤ r
hc : c ≠ 0
⊢ c • closedBall x r = closedBall (c • x) (‖c‖ * r)
Module: Mathlib.Analysis.Normed.Module.Ball.Pointwise

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace ℝ E
inst✝ : Nontrivial E
c : 𝕜
x : E
r : ℝ
hr : 0 ≤ r
⊢ c • sphere x r = sphere (c • x) (‖c‖ * r)

Tactic:
rcases eq_or_ne c 0 with (rfl | hc)

Goal state after:
case inl
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace ℝ E
inst✝ : Nontrivial E
x : E
r : ℝ
hr : 0 ≤ r
⊢ 0 • sphere x r = sphere (0 • x) (‖0‖ * r)

case inr
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace ℝ E
inst✝ : Nontrivial E
c : 𝕜
x : E
r : ℝ
hr : 0 ≤ r
hc : c ≠ 0
⊢ c • sphere x r = sphere (c • x) (‖c‖ * r)
Module: Mathlib.Topology.Connected.PathConnected

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
x y z : X
ι : Type u_3
F : Set X
inst✝ : PathConnectedSpace X
⊢ ConnectedSpace X

Tactic:
rw [connectedSpace_iff_connectedComponent]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
x y z : X
ι : Type u_3
F : Set X
inst✝ : PathConnectedSpace X
⊢ ∃ x, connectedComponent x = univ
Module: Mathlib.Analysis.SpecificLimits.Normed

Goal state before:
pp: α : Type u_1
inst✝¹ : NormedDivisionRing α
inst✝ : CompleteSpace α
f : ℕ → α
w z : α
h : CauchySeq fun n ↦ ∑ i ∈ Finset.range n, f i * w ^ i
hz : ‖z‖ < ‖w‖
hw : 0 < ‖w‖
C : ℝ
hC : ∀ (n : ℕ), ‖f n * w ^ n‖ ≤ C
n : ℕ
| ∀ (n : ℕ), ‖f n * w ^ n‖ ≤ C

Tactic:
enter [n]; rw [norm_mul, norm_pow, ← _root_.le_div_iff₀ (by positivity)]

Goal state after:
α : Type u_1
inst✝¹ : NormedDivisionRing α
inst✝ : CompleteSpace α
f : ℕ → α
w z : α
h : CauchySeq fun n ↦ ∑ i ∈ Finset.range n, f i * w ^ i
hz : ‖z‖ < ‖w‖
hw : 0 < ‖w‖
C : ℝ
hC : ∀ (n : ℕ), ‖f n * w ^ n‖ ≤ C
n✝ n : ℕ
| ‖f n‖ ≤ C / ‖w‖ ^ n
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse

Goal state before:
pp: case inr.inr.refine_1
x : ℝ
hx₁ : -1 ≤ x
hx₂ : x ≤ 1
⊢ sin (-arcsin x) = -x

pp: case inr.inr.refine_2
x : ℝ
hx₁ : -1 ≤ x
hx₂ : x ≤ 1
⊢ -arcsin x ∈ Icc (-(π / 2)) (π / 2)

Tactic:
· rw [sin_neg, sin_arcsin hx₁ hx₂]

Goal state after:
case inr.inr.refine_2
x : ℝ
hx₁ : -1 ≤ x
hx₂ : x ≤ 1
⊢ -arcsin x ∈ Icc (-(π / 2)) (π / 2)
Module: Mathlib.Analysis.SpecificLimits.Normed

Goal state before:
pp: x : ℝ
⊢ Summable fun n ↦ x ^ n / ↑n !

Tactic:
have A : (0 : ℝ) < ⌊‖x‖⌋₊ + 1 := zero_lt_one.trans_le (by simp)
  have B : ‖x‖ / (⌊‖x‖⌋₊ + 1) < 1 := (div_lt_one A).2 (Nat.lt_floor_add_one _)
  -- Then we apply the ratio test. The estimate works for `n ≥ ⌊‖x‖⌋₊`.
  suffices ∀ n ≥ ⌊‖x‖⌋₊, ‖x ^ (n + 1) / (n + 1)!‖ ≤ ‖x‖ / (⌊‖x‖⌋₊ + 1) * ‖x ^ n / ↑n !‖ from
    summable_of_ratio_norm_eventually_le B (eventually_atTop.2 ⟨⌊‖x‖⌋₊, this⟩)
  -- Finally, we prove the upper estimate
  intro n hn
  calc
    ‖x ^ (n + 1) / (n + 1)!‖ = ‖x‖ / (n + 1) * ‖x ^ n / (n !)‖ := by
      rw [_root_.pow_succ', Nat.factorial_succ, Nat.cast_mul, ← _root_.div_mul_div_comm, norm_mul,
        norm_div, Real.norm_natCast, Nat.cast_succ]
    _ ≤ ‖x‖ / (⌊‖x‖⌋₊ + 1) * ‖x ^ n / (n !)‖ := by gcongr

Goal state after: