mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Calculus.TangentCone.ProperSpace

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : ProperSpace E
s : Set E
x : E
hx : AccPt x (𝓟 s)
u : ℕ → ℝ
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
v : ℕ → E
hvx : ∀ (n : ℕ), v n ≠ x
hvu : ∀ (n : ℕ), v n ∈ closedBall x (u n)
hvs : ∀ (n : ℕ), v n ∈ s
d : ℕ → E := fun n ↦ v n - x
M : ∀ (n : ℕ), x + d n ∈ s \ {x}
r : 𝕜
hr : 1 < ‖r‖
⊢ (tangentConeAt 𝕜 s x ∩ {0}ᶜ).Nonempty

Tactic:
have W n := rescale_to_shell hr zero_lt_one (x := d n) (by simpa using (M n).2)

Goal state after:
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : ProperSpace E
s : Set E
x : E
hx : AccPt x (𝓟 s)
u : ℕ → ℝ
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
v : ℕ → E
hvx : ∀ (n : ℕ), v n ≠ x
hvu : ∀ (n : ℕ), v n ∈ closedBall x (u n)
hvs : ∀ (n : ℕ), v n ∈ s
d : ℕ → E := fun n ↦ v n - x
M : ∀ (n : ℕ), x + d n ∈ s \ {x}
r : 𝕜
hr : 1 < ‖r‖
W : ∀ (n : ℕ), ∃ d_1, d_1 ≠ 0 ∧ ‖d_1 • d n‖ < 1 ∧ 1 / ‖r‖ ≤ ‖d_1 • d n‖ ∧ ‖d_1‖⁻¹ ≤ 1⁻¹ * ‖r‖ * ‖d n‖
⊢ (tangentConeAt 𝕜 s x ∩ {0}ᶜ).Nonempty
Module: Mathlib.Order.SuccPred.IntervalSucc

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : SuccOrder α
inst✝¹ : IsSuccArchimedean α
inst✝ : LinearOrder β
f : α → β
hf : Monotone f
m n : α
hmn : m ≤ n
k : α
hmk : m ≤ k
ihk : ⋃ i ∈ Ico m k, Ioc (f i) (f (succ i)) = Ioc (f m) (f k)
hk : IsMax k
⊢ ⋃ i ∈ Ico m (succ k), Ioc (f i) (f (succ i)) = Ioc (f k) (f (succ k)) ∪ ⋃ i ∈ Ico m k, Ioc (f i) (f (succ i))

Tactic:
hk.succ_eq,

Goal state after:
case pos
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : SuccOrder α
inst✝¹ : IsSuccArchimedean α
inst✝ : LinearOrder β
f : α → β
hf : Monotone f
m n : α
hmn : m ≤ n
k : α
hmk : m ≤ k
ihk : ⋃ i ∈ Ico m k, Ioc (f i) (f (succ i)) = Ioc (f m) (f k)
hk : IsMax k
⊢ ⋃ i ∈ Ico m k, Ioc (f i) (f (succ i)) = Ioc (f k) (f k) ∪ ⋃ i ∈ Ico m k, Ioc (f i) (f (succ i))
Module: Mathlib.Analysis.Calculus.Rademacher

Goal state before:
pp: case refine_1
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
C : ℝ≥0
f : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
ι : Type u_3
s : Finset ι
a : ι → ℝ
v : ι → E
g : E → ℝ
g_smooth : ContDiff ℝ (↑⊤) g
g_comp : HasCompactSupport g
A : ∀ i ∈ s, Integrable (fun x ↦ g x • (a i • fun x ↦ lineDeriv ℝ f x (v i)) x) μ
i : ι
_hi : i ∈ s
L : StrongDual ℝ E → ℝ := fun f ↦ f (v i)
⊢ Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] fun x ↦ (L ∘ fderiv ℝ g) x * f x

pp: case refine_2
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
C : ℝ≥0
f : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
ι : Type u_3
s : Finset ι
a : ι → ℝ
v : ι → E
g : E → ℝ
g_smooth : ContDiff ℝ (↑⊤) g
g_comp : HasCompactSupport g
A : ∀ i ∈ s, Integrable (fun x ↦ g x • (a i • fun x ↦ lineDeriv ℝ f x (v i)) x) μ
i : ι
_hi : i ∈ s
L : StrongDual ℝ E → ℝ := fun f ↦ f (v i)
⊢ HasCompactSupport fun x ↦ (L ∘ fderiv ℝ g) x * f x

Tactic:
· exact ((g_smooth.continuous_fderiv (by simp)).clm_apply continuous_const).mul
      hf.continuous

Goal state after:
case refine_2
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
C : ℝ≥0
f : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
ι : Type u_3
s : Finset ι
a : ι → ℝ
v : ι → E
g : E → ℝ
g_smooth : ContDiff ℝ (↑⊤) g
g_comp : HasCompactSupport g
A : ∀ i ∈ s, Integrable (fun x ↦ g x • (a i • fun x ↦ lineDeriv ℝ f x (v i)) x) μ
i : ι
_hi : i ∈ s
L : StrongDual ℝ E → ℝ := fun f ↦ f (v i)
⊢ HasCompactSupport fun x ↦ (L ∘ fderiv ℝ g) x * f x
Module: Mathlib.MeasureTheory.Function.AbsolutelyContinuous

Goal state before:
pp: F : Type u_2
inst✝ : SeminormedAddCommGroup F
a b : ℝ
f g : ℝ → F
hf : AbsolutelyContinuousOnInterval f a b
hg : AbsolutelyContinuousOnInterval g a b
t : ℕ × (ℕ → ℝ × ℝ)
⊢ 0 ≤ ∑ i ∈ Finset.range t.1, dist ((f + g) (t.2 i).1) ((f + g) (t.2 i).2)

Tactic:
exact Finset.sum_nonneg (fun i hi ↦ by positivity)

Goal state after:
Module: Mathlib.MeasureTheory.Function.AbsolutelyContinuous

Goal state before:
pp: F : Type u_2
inst✝ : SeminormedAddCommGroup F
a b : ℝ
f : ℝ → F
hf : AbsolutelyContinuousOnInterval f a b
t : ℕ × (ℕ → ℝ × ℝ)
⊢ 0 ≤ ∑ i ∈ Finset.range t.1, dist ((-f) (t.2 i).1) ((-f) (t.2 i).2)

Tactic:
exact Finset.sum_nonneg (fun i hi ↦ by positivity)

Goal state after:
Module: Mathlib.MeasureTheory.Function.AbsolutelyContinuous

Goal state before:
pp: F : Type u_2
inst✝³ : SeminormedAddCommGroup F
a b : ℝ
f : ℝ → F
M : Type u_3
inst✝² : SeminormedRing M
inst✝¹ : Module M F
inst✝ : NormSMulClass M F
α : M
hf : AbsolutelyContinuousOnInterval f a b
t : ℕ × (ℕ → ℝ × ℝ)
⊢ 0 ≤ ∑ i ∈ Finset.range t.1, dist ((fun x ↦ α • f x) (t.2 i).1) ((fun x ↦ α • f x) (t.2 i).2)

Tactic:
exact Finset.sum_nonneg (fun i hi ↦ by positivity)

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.DerivIntegrable

Goal state before:
pp: f : ℝ → ℝ
a b : ℝ
hf : MonotoneOn f (Icc a b)
hab : a ≤ b
G : ℕ → ℝ → ℝ
hGf : ∀ᵐ (x : ℝ), x ∈ uIcc a b → Tendsto (fun n ↦ G n x) atTop (𝓝 (deriv f x))
hG : ∀ (n : ℕ), AEStronglyMeasurable (G n) (volume.restrict (uIcc a b))
hG' : liminf (fun n ↦ ∫⁻ (x : ℝ) in uIcc a b, ‖G n x‖ₑ) atTop ≤ ENNReal.ofReal (f b - f a)
hG'₀ : liminf (fun n ↦ ∫⁻ (x : ℝ) in Icc a b, ‖G n x‖ₑ) atTop ≠ ⊤
integrable_f_deriv : Integrable (deriv f) (volume.restrict (Icc a b))
this : f a ≤ f b
f_deriv_nonneg : ∀ {x : ℝ}, x ∈ Ioo a b → 0 ≤ deriv f x
⊢ MeasurableSet (uIcc a b)

Tactic:
measurability

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.AbsolutelyContinuousFun

Goal state before:
pp: X : Type u_1
inst✝ : PseudoMetricSpace X
f : ℝ → X
d b y : ℝ
hdb : d ≤ b
hf : AbsolutelyContinuousOnInterval f d b
u : Set (ℝ × ℝ)
hu₃ : HasSum (fun z ↦ (↑z).2 - (↑z).1) (b - d)
hu₄ : HasSum (fun z ↦ dist (f (↑z).1) (f (↑z).2)) y
u_coe : Finset ↑u → Finset (ℝ × ℝ) := fun s ↦ Finset.image Subtype.val s
hu₁ : ∀ (s : Finset ↑u) ⦃z : ℝ × ℝ⦄, z ∈ u_coe s → d ≤ z.1 ∧ z.1 ≤ z.2 ∧ z.2 ≤ b
hu₂ : ∀ (s : Finset ↑u), (↑(u_coe s)).PairwiseDisjoint fun z ↦ Icc z.1 z.2
T : Finset ↑u → ℕ × (ℕ → ℝ × ℝ) := fun s ↦ ((u_coe s).card + 1, fun i ↦ (u_coe s).intervalGapsWithin ⋯ d b ↑i)
hT₁ : ∀ (s : Finset ↑u) (i : ℕ), d ≤ ((u_coe s).intervalGapsWithin ⋯ d b ↑i).1
hT₂ : ∀ (s : Finset ↑u) (i : ℕ), ((u_coe s).intervalGapsWithin ⋯ d b ↑i).1 ≤ ((u_coe s).intervalGapsWithin ⋯ d b ↑i).2
hT₃ : ∀ (s : Finset ↑u) (i : ℕ), ((u_coe s).intervalGapsWithin ⋯ d b ↑i).2 ≤ b
hT₄ :
  ∀ (s : Finset ↑u),
    (Iio ((u_coe s).card + 1)).PairwiseDisjoint fun j ↦
      Ioc ((u_coe s).intervalGapsWithin ⋯ d b ↑j).1 ((u_coe s).intervalGapsWithin ⋯ d b ↑j).2
⊢ dist (f d) (f b) ≤ y

Tactic:
have hT : univ.MapsTo T (disjWithin d b) := by
    intro s _
    #adaptation_note /-- Before https://github.com/leanprover/lean4/pull/13166
    (replacing grind's canonicalizer with a type-directed normalizer), `grind` closed this goal
    without the `simp`. It is not yet clear whether this is due to defeq abuse in Mathlib or a
    problem in the new canonicalizer; a minimization would help. The original proof was:
    `grind [disjWithin, uIcc_of_le]` -/
    simp [disjWithin]
    grind [uIcc_of_le]

Goal state after:
X : Type u_1
inst✝ : PseudoMetricSpace X
f : ℝ → X
d b y : ℝ
hdb : d ≤ b
hf : AbsolutelyContinuousOnInterval f d b
u : Set (ℝ × ℝ)
hu₃ : HasSum (fun z ↦ (↑z).2 - (↑z).1) (b - d)
hu₄ : HasSum (fun z ↦ dist (f (↑z).1) (f (↑z).2)) y
u_coe : Finset ↑u → Finset (ℝ × ℝ) := fun s ↦ Finset.image Subtype.val s
hu₁ : ∀ (s : Finset ↑u) ⦃z : ℝ × ℝ⦄, z ∈ u_coe s → d ≤ z.1 ∧ z.1 ≤ z.2 ∧ z.2 ≤ b
hu₂ : ∀ (s : Finset ↑u), (↑(u_coe s)).PairwiseDisjoint fun z ↦ Icc z.1 z.2
T : Finset ↑u → ℕ × (ℕ → ℝ × ℝ) := fun s ↦ ((u_coe s).card + 1, fun i ↦ (u_coe s).intervalGapsWithin ⋯ d b ↑i)
hT₁ : ∀ (s : Finset ↑u) (i : ℕ), d ≤ ((u_coe s).intervalGapsWithin ⋯ d b ↑i).1
hT₂ : ∀ (s : Finset ↑u) (i : ℕ), ((u_coe s).intervalGapsWithin ⋯ d b ↑i).1 ≤ ((u_coe s).intervalGapsWithin ⋯ d b ↑i).2
hT₃ : ∀ (s : Finset ↑u) (i : ℕ), ((u_coe s).intervalGapsWithin ⋯ d b ↑i).2 ≤ b
hT₄ :
  ∀ (s : Finset ↑u),
    (Iio ((u_coe s).card + 1)).PairwiseDisjoint fun j ↦
      Ioc ((u_coe s).intervalGapsWithin ⋯ d b ↑j).1 ((u_coe s).intervalGapsWithin ⋯ d b ↑j).2
hT : MapsTo T univ (disjWithin d b)
⊢ dist (f d) (f b) ≤ y
Module: Mathlib.Analysis.Calculus.Rademacher

Goal state before:
pp: E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : BorelSpace E
F : Type u_2
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
C : ℝ≥0
s : Set E
μ : Measure E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : FiniteDimensional ℝ F
inst✝ : μ.IsAddHaarMeasure
f : E → F
hf : LipschitzOnWith C f s
hs : MeasurableSet s
⊢ ∀ᵐ (x : E) ∂μ, x ∈ s → DifferentiableWithinAt ℝ f s x

Tactic:
exact hf.ae_differentiableWithinAt_of_mem

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

Goal state before:
pp: case h.h.h.h
s : ℝ
hs : 0 < s
M ε : ℝ
hM : 0 < M
hε : 0 < ε
H : ∀ (x y : ℝ), 0 < x → x < ε → |y| < s * x → √(x ^ 2 + y ^ 2) < M * (1 - √((1 - x) ^ 2 + y ^ 2))
z : ℂ
hzl : z ∈ {z | 1 - ε < z.re}
hzr : z ∈ stolzCone s
⊢ z ∈ stolzSet M

Tactic:
Set.mem_ofPred_eq,

Goal state after:
case h.h.h.h
s : ℝ
hs : 0 < s
M ε : ℝ
hM : 0 < M
hε : 0 < ε
H : ∀ (x y : ℝ), 0 < x → x < ε → |y| < s * x → √(x ^ 2 + y ^ 2) < M * (1 - √((1 - x) ^ 2 + y ^ 2))
z : ℂ
hzl : 1 - ε < z.re
hzr : z ∈ stolzCone s
⊢ z ∈ stolzSet M
Module: Mathlib.Analysis.Complex.AbelLimit

Goal state before:
pp: case h.h.h.h
s : ℝ
hs : 0 < s
M ε : ℝ
hM : 0 < M
hε : 0 < ε
H : ∀ (x y : ℝ), 0 < x → x < ε → |y| < s * x → √(x ^ 2 + y ^ 2) < M * (1 - √((1 - x) ^ 2 + y ^ 2))
z : ℂ
hzl : (1 - z).re < ε
hzr : z ∈ {z | |z.im| < s * (1 - z.re)}
⊢ z ∈ stolzSet M

Tactic:
Set.mem_ofPred_eq,

Goal state after:
case h.h.h.h
s : ℝ
hs : 0 < s
M ε : ℝ
hM : 0 < M
hε : 0 < ε
H : ∀ (x y : ℝ), 0 < x → x < ε → |y| < s * x → √(x ^ 2 + y ^ 2) < M * (1 - √((1 - x) ^ 2 + y ^ 2))
z : ℂ
hzl : (1 - z).re < ε
hzr : |z.im| < s * (1 - z.re)
⊢ z ∈ stolzSet M
Module: Mathlib.Analysis.Calculus.Taylor

Goal state before:
pp: F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : ℝ → F
x x₀ : ℝ
n : ℕ
hf : ContDiffOn ℝ (↑(n + 1)) f [[x₀, x]]
this✝ : x₀ ≠ x
this : UniqueDiffOn ℝ [[x₀, x]]
k : ℕ
hk : k ≤ n
⊢ MeasurableSet (Ι x₀ x)

Tactic:
measurability

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

Goal state before:
pp: n : ℤ
⊢ ∃ k, ↑k * ↑π / 2 = ↑n * ↑π / 2

Tactic:
by use n

Goal state after:
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f w = 0
hxw : x < w
hwy : w < y
⊢ ∃ a ∈ Ioo x w, (f w - f x) / (w - x) < deriv f a

Tactic:
apply StrictMonoOn.exists_slope_lt_deriv_aux _ hxw _ _

Goal state after:
x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f w = 0
hxw : x < w
hwy : w < y
⊢ ContinuousOn f (Icc x w)

x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f w = 0
hxw : x < w
hwy : w < y
⊢ StrictMonoOn (deriv f) (Ioo x w)

x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f w = 0
hxw : x < w
hwy : w < y
⊢ ∀ w_1 ∈ Ioo x w, deriv f w_1 ≠ 0
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex

Goal state before:
pp: x : ℂ
hw : 1 * (0 * 0) + -2 * x * 0 + 1 = 0
⊢ False

Tactic:
simp only [zero_add, one_ne_zero, mul_zero] at hw

Goal state after:
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
⊢ ∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x)

Tactic:
by_cases! h : ∀ w ∈ Ioo x y, deriv f w ≠ 0
  · apply StrictMonoOn.exists_deriv_lt_slope_aux hf hxy hf'_mono h
  · rcases h with ⟨w, ⟨hxw, hwy⟩, hw⟩
    obtain ⟨a, ⟨hxa, haw⟩, ha⟩ : ∃ a ∈ Ioo x w, deriv f a < (f w - f x) / (w - x) := by
      apply StrictMonoOn.exists_deriv_lt_slope_aux _ hxw _ _
      · exact hf.mono (Icc_subset_Icc le_rfl hwy.le)
      · exact hf'_mono.mono (Ioo_subset_Ioo le_rfl hwy.le)
      · intro z hz
        rw [← hw]
        apply ne_of_lt
        exact hf'_mono ⟨hz.1, hz.2.trans hwy⟩ ⟨hxw, hwy⟩ hz.2
    obtain ⟨b, ⟨hwb, hby⟩, hb⟩ : ∃ b ∈ Ioo w y, deriv f b < (f y - f w) / (y - w) := by
      apply StrictMonoOn.exists_deriv_lt_slope_aux _ hwy _ _
      · refine hf.mono (Icc_subset_Icc hxw.le le_rfl)
      · exact hf'_mono.mono (Ioo_subset_Ioo hxw.le le_rfl)
      · intro z hz
        rw [← hw]
        apply ne_of_gt
        exact hf'_mono ⟨hxw, hwy⟩ ⟨hxw.trans hz.1, hz.2⟩ hz.1
    refine ⟨a, ⟨hxa, haw.trans hwy⟩, ?_⟩
    simp only [lt_div_iff₀, hxy, hxw, hwy, sub_pos] at ha hb ⊢
    have : deriv f a * (y - w) < deriv f b * (y - w) := by
      apply mul_lt_mul _ le_rfl (sub_pos.2 hwy) _
      · exact hf'_mono ⟨hxa, haw.trans hwy⟩ ⟨hxw.trans hwb, hby⟩ (haw.trans hwb)
      · rw [← hw]
        exact (hf'_mono ⟨hxw, hwy⟩ ⟨hxw.trans hwb, hby⟩ hwb).le
    linarith

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

Goal state before:
pp: case e'_9
x : ℂ
h : ¬cos x = 0
⊢ 1 / cos x ^ 2 = (cos x * cos x - sin x * -sin x) / cos x ^ 2

Tactic:
← sin_sq_add_cos_sq x

Goal state after:
case e'_9
x : ℂ
h : ¬cos x = 0
⊢ (sin x ^ 2 + cos x ^ 2) / cos x ^ 2 = (cos x * cos x - sin x * -sin x) / cos x ^ 2
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan

Goal state before:
pp: n : ℤ
⊢ ∃ k, ↑k * π / 2 = ↑n * π / 2

Tactic:
by use n

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

Goal state before:
pp: case inr
x : ℝ
hx_gt : ¬x ≤ -(π / 2)
hx_lt : x < π / 2
r : ℤ
hxr_eq : x = (2 * ↑r + 1) * π / 2
h : r < 0
hr_le : r ≤ -1
⊢ 0 < 2

Tactic:
exact zero_lt_two

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

Goal state before:
pp: x : ℝ
h1 : 0 < x
h2 : x < π / 2
U : Set ℝ := Ico 0 (π / 2)
intU : interior U = Ioo 0 (π / 2)
half_pi_pos : 0 < π / 2
cos_pos : ∀ {y : ℝ}, y ∈ U → 0 < cos y
y : ℝ
hy : y ∈ interior U
⊢ 0 < sin y

Tactic:
rw [intU] at hy

Goal state after:
x : ℝ
h1 : 0 < x
h2 : x < π / 2
U : Set ℝ := Ico 0 (π / 2)
intU : interior U = Ioo 0 (π / 2)
half_pi_pos : 0 < π / 2
cos_pos : ∀ {y : ℝ}, y ∈ U → 0 < cos y
y : ℝ
hy : y ∈ Ioo 0 (π / 2)
⊢ 0 < sin y
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
x : 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
hfc : ConvexOn 𝕜 s f
hxs : x ∈ interior s
y : 𝕜
hyx : y < x
hys : y ∈ s
y' : 𝕜
x✝ : y' ∈ slope f x '' {y | y ∈ s ∧ x < y}
z : 𝕜
hz : z ∈ {y | y ∈ s ∧ x < y}
hz' : slope f x z = y'
⊢ y' ∈ Ici (slope f x y)

Tactic:
simp_rw [mem_Ici, ← hz']

Goal state after:
𝕜 : Type u_1
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
x : 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
hfc : ConvexOn 𝕜 s f
hxs : x ∈ interior s
y : 𝕜
hyx : y < x
hys : y ∈ s
y' : 𝕜
x✝ : y' ∈ slope f x '' {y | y ∈ s ∧ x < y}
z : 𝕜
hz : z ∈ {y | y ∈ s ∧ x < y}
hz' : slope f x z = y'
⊢ slope f x y ≤ slope f x z
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: case refine_2
𝕜 : Type u_1
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
x : 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
hfc : ConvexOn 𝕜 s f
hxs : x ∈ interior s
y : 𝕜
hyx : y < x
hys : y ∈ s
y' : 𝕜
x✝ : y' ∈ slope f x '' {y | y ∈ s ∧ x < y}
z : 𝕜
hz : z ∈ {y | y ∈ s ∧ x < y}
hz' : slope f x z = y'
⊢ z ∈ s \ {x}

Tactic:
· simp [hz.2.ne', hz.1]

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

Goal state before:
pp: x y : ℝ
h : 1 < x * y
hx : 0 < x
hy : 0 < y
k : arctan x⁻¹ + arctan y⁻¹ = arctan ((x⁻¹ + y⁻¹) / (1 - x⁻¹ * y⁻¹))
⊢ arctan x + arctan y = arctan ((x + y) / (1 - x * y)) + π

Tactic:
arctan_inv_of_pos hx,

Goal state after:
x y : ℝ
h : 1 < x * y
hx : 0 < x
hy : 0 < y
k : π / 2 - arctan x + arctan y⁻¹ = arctan ((x⁻¹ + y⁻¹) / (1 - x⁻¹ * y⁻¹))
⊢ arctan x + arctan y = arctan ((x + y) / (1 - x * y)) + π
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds

Goal state before:
pp: x : ℝ
hx1 : -(3 * π / 2) ≤ x
hx2 : x ≤ 3 * π / 2
hx3 : x ≠ 0
y : ℝ
hy1 : 0 < y
⊢ y ≤ 3 * π / 2 → cos y < 1 / √(y ^ 2 + 1)

Tactic:
hy2

Goal state after:
x : ℝ
hx1 : -(3 * π / 2) ≤ x
hx2 : x ≤ 3 * π / 2
hx3 : x ≠ 0
y : ℝ
hy1 : 0 < y
hy2 : y ≤ 3 * π / 2
⊢ cos y < 1 / √(y ^ 2 + 1)
Module: Mathlib.Analysis.Convex.Deriv

Goal state before:
pp: S : Set ℝ
f : ℝ → ℝ
x y f' : ℝ
hfc : StrictConvexOn ℝ S f
hx : x ∈ S
hy : y ∈ S
hxy : x < y
hf' : HasDerivWithinAt f f' (Iio y) y
⊢ slope f x y < f'

Tactic:
obtain ⟨u, hxu, huy⟩ := exists_between hxy
  have hu : u ∈ S := hfc.1.ordConnected.out hx hy ⟨hxu.le, huy.le⟩
  have := hfc.secant_strict_mono hy hx hu hxy.ne huy.ne hxu
  simp_rw [← slope_def_field, slope_comm _ y] at this
  exact this.trans_le <| hfc.convexOn.slope_le_of_hasDerivWithinAt_Iio hu hy huy hf'

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
⊢ angle (NormedSpace.normalize x) y = angle x y

Tactic:
by_cases hx : x = 0
  · simp [hx]
  · rw [NormedSpace.normalize, angle_smul_left_of_pos _ _ (by positivity)]

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
⊢ angle x (NormedSpace.normalize y) = angle x y

Tactic:
by
  rw [angle_comm, angle_normalize_left, angle_comm]

Goal state after:
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: E : Type u_1
X : Type u_2
A : Type u_3
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
p : E → X
f : C(↑I × A, X)
g : ↑I × A → E
g_lifts : p ∘ g = ⇑f
cont_0 : Continuous[inst✝, inst✝²] fun x ↦ g (0, x)
a : A
cont_a : Continuous[_, inst✝²] fun x ↦ g (x, a)
q : E → OpenPartialHomeomorph E X
mem_source : ∀ (x : E), x ∈ (q x).source
hpq : ∀ (x : E), p = ↑(q x)
t : ℕ → ↑I
t_0 : t 0 = 0
t_mono : Monotone t
t_sub : ∀ (n : ℕ), ∃ i, Set.Icc (t n) (t (n + 1)) ⊆ (fun x ↦ g (x, a)) ⁻¹' (q i).source
n_max : ℕ
h_max : ∀ m ≥ n_max, t m = 1
n : ℕ
x✝ :
  ∃ N,
    a ∈ N ∧
      IsOpen[inst✝] N ∧
        ∃ g',
          ContinuousOn g' (Set.Icc 0 (t n) ×ˢ N) ∧
            p ∘ g' = ⇑f ∧ (∀ (a : A), g' (0, a) = g (0, a)) ∧ ∀ t' ≤ t n, g' (t', a) = g (t', a)
N : Set A
haN : a ∈ N
N_open : IsOpen[inst✝] N
g' : ↑I × A → E
cont_g' : ContinuousOn g' (Set.Icc 0 (t n) ×ˢ N)
g'_lifts : p ∘ g' = ⇑f
g'_0 : ∀ (a : A), g' (0, a) = g (0, a)
g'_a : ∀ t' ≤ t n, g' (t', a) = g (t', a)
e : E
h_sub : Set.Icc (t n) (t (n + 1)) ⊆ (fun x ↦ g (x, a)) ⁻¹' (q e).source
t0 : ↑I
a' : A
ht : (t0, a').1 ∈ Set.Icc (t n) (t (n + 1))
ha : a' = a
⊢ (t0, a) ∈ ↑(q e) ∘ g ⁻¹' (q e).target

Tactic:
exact (q e).map_source (h_sub ht)

Goal state after:
Module: Mathlib.Analysis.Meromorphic.Divisor

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
U : Set 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
V : Set 𝕜
hf : MeromorphicOn f U
hU : IsCompact U
hV : V ⊆ U
b : 𝕜
hb : b ∈ (divisor f V).support
⊢ b ∈ (divisor f U).support

Tactic:
rw [Function.mem_support, ne_eq, divisor_apply hf (hV ((divisor f V).supportWithinDomain hb))]

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
U : Set 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
V : Set 𝕜
hf : MeromorphicOn f U
hU : IsCompact U
hV : V ⊆ U
b : 𝕜
hb : b ∈ (divisor f V).support
⊢ ¬(meromorphicOrderAt f b).untop₀ = 0
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: E : Type u_1
X : Type u_2
A : Type u_3
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
p : E → X
f : C(↑I × A, X)
g : ↑I × A → E
g_lifts : p ∘ g = ⇑f
cont_0 : Continuous[inst✝, inst✝²] fun x ↦ g (0, x)
a : A
cont_a : Continuous[_, inst✝²] fun x ↦ g (x, a)
q : E → OpenPartialHomeomorph E X
mem_source : ∀ (x : E), x ∈ (q x).source
hpq : ∀ (x : E), p = ↑(q x)
t : ℕ → ↑I
t_0 : t 0 = 0
t_mono : Monotone t
t_sub : ∀ (n : ℕ), ∃ i, Set.Icc (t n) (t (n + 1)) ⊆ (fun x ↦ g (x, a)) ⁻¹' (q i).source
n_max : ℕ
h_max : ∀ m ≥ n_max, t m = 1
n : ℕ
x✝¹ :
  ∃ N,
    a ∈ N ∧
      IsOpen[inst✝] N ∧
        ∃ g',
          ContinuousOn g' (Set.Icc 0 (t n) ×ˢ N) ∧
            p ∘ g' = ⇑f ∧ (∀ (a : A), g' (0, a) = g (0, a)) ∧ ∀ t' ≤ t n, g' (t', a) = g (t', a)
N : Set A
haN : a ∈ N
N_open : IsOpen[inst✝] N
g' : ↑I × A → E
cont_g' : ContinuousOn g' (Set.Icc 0 (t n) ×ˢ N)
g'_lifts : p ∘ g' = ⇑f
g'_0 : ∀ (a : A), g' (0, a) = g (0, a)
g'_a : ∀ t' ≤ t n, g' (t', a) = g (t', a)
e : E
h_sub : Set.Icc (t n) (t (n + 1)) ⊆ (fun x ↦ g (x, a)) ⁻¹' (q e).source
this : Set.Icc (t n) (t (n + 1)) ×ˢ {a} ⊆ ⇑f ⁻¹' (q e).target
u : Set ↑I
v : Set A
v_open : IsOpen[inst✝] v
hu : Set.Icc (t n) (t (n + 1)) ⊆ u
hav✝ : {a} ⊆ v
huv : u ×ˢ v ⊆ ⇑f ⁻¹' (q e).target
ta : ↑I × A
x✝ : ta ∈ Set.Icc 0 (t n.succ) ×ˢ (v ∩ (N ∩ (g' ∘ fun y ↦ (t n, y)) ⁻¹' (q e).source)) ∩ frontier {a | a.1 ≤ t n}
left✝¹ : ta.1 ∈ Set.Icc 0 (t n.succ)
hav : ta.2 ∈ v
left✝ : ta.2 ∈ N
ha : ta.2 ∈ (g' ∘ fun y ↦ (t n, y)) ⁻¹' (q e).source
hfr : ta ∈ frontier {a | a.1 ≤ t n}
ht : ta.1 = t n
⊢ ta.1 ∈ Set.Icc (t n) (t (n + 1))

Tactic:
by rw [ht]; exact ⟨le_rfl, t_mono n.le_succ⟩

Goal state after:
Module: Mathlib.Analysis.Meromorphic.TrailingCoefficient

Goal state before:
pp: case pos.hl
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
h : MeromorphicAt f x
h₂ : meromorphicOrderAt f x = ⊤
⊢ 0 =ᶠ[𝓝[≠] x] f

pp: case pos.h
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
h : MeromorphicAt f x
h₂ : meromorphicOrderAt f x = ⊤
⊢ Tendsto 0 (𝓝[≠] x) (𝓝 0)

Tactic:
· filter_upwards [meromorphicOrderAt_eq_top_iff.1 h₂] with y hy
      simp_all

Goal state after:
case pos.h
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
h : MeromorphicAt f x
h₂ : meromorphicOrderAt f x = ⊤
⊢ Tendsto 0 (𝓝[≠] x) (𝓝 0)