ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Analysis/Convex/Between.lean
R : Type u_1
V : Type u_2
V' : Type u_3
P : Type u_4
P' : Type u_5
inst✝⁴ : LinearOrderedRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AddTorsor V P
inst✝ : NoZeroSMulDivisors R V
t : Affine.Triangle R P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
this : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h3 : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
hu : Finset.univ = {i₁, i₂, i₃}
r₁ : R
hr₁0 : 0 < r₁
hr₁1 : r₁ < 1
h₁ : Sbtw R (t.points i₂) ((lineMap (t.points i₂) (t.points i₃)) r₁) (t.points i₃)
r₂ : R
hr₂0 : 0 < r₂
hr₂1 : r₂ < 1
h₂ : Sbtw R (t.points i₁) ((lineMap (t.points i₁) (t.points i₃)) r₂) (t.points i₃)
w : Fin (2 + 1) → R
hw : ∑ i : Fin (2 + 1), w i = 1
hp : (Finset.affineCombination R Finset.univ t.points) w ∈ affineSpan R (Set.range t.points)
h₁' : (Finset.affineCombination R Finset.univ t.points) w ∈ affineSpan R {(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationSingleWeights R i₁), (Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationLineMapWeights i₂ i₃ r₁)}
h₂' : (Finset.affineCombination R Finset.univ t.points) w ∈ affineSpan R {t.points i₂, (lineMap (t.points i₁) (t.points i₃)) r₂}
h₁s : SignType.sign (w i₂) = SignType.sign (w i₃)
h₂s : SignType.sign (w i₁) = SignType.sign (w i₃)
hs : ∀ (i : Fin 3), SignType.sign (w i) = SignType.sign (w i₃)
hss : SignType.sign (∑ i : Fin (2 + 1), w i) = 1
\u22a2 w i₁ ∈ Set.Ioo 0 1
-- Mathlib/Analysis/Convex/Between.lean
R : Type u_1
V : Type u_2
V' : Type u_3
P : Type u_4
P' : Type u_5
inst✝⁴ : LinearOrderedRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AddTorsor V P
inst✝ : NoZeroSMulDivisors R V
t : Affine.Triangle R P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
this : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h3 : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
hu : Finset.univ = {i₁, i₂, i₃}
r₁ : R
hr₁0 : 0 < r₁
hr₁1 : r₁ < 1
h₁ : Sbtw R (t.points i₂) ((lineMap (t.points i₂) (t.points i₃)) r₁) (t.points i₃)
r₂ : R
hr₂0 : 0 < r₂
hr₂1 : r₂ < 1
h₂ : Sbtw R (t.points i₁) ((lineMap (t.points i₁) (t.points i₃)) r₂) (t.points i₃)
w : Fin (2 + 1) → R
hw : ∑ i : Fin (2 + 1), w i = 1
hp : (Finset.affineCombination R Finset.univ t.points) w ∈ affineSpan R (Set.range t.points)
h₁' : (Finset.affineCombination R Finset.univ t.points) w ∈ affineSpan R {(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationSingleWeights R i₁), (Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationLineMapWeights i₂ i₃ r₁)}
h₂' : (Finset.affineCombination R Finset.univ t.points) w ∈ affineSpan R {t.points i₂, (lineMap (t.points i₁) (t.points i₃)) r₂}
h₁s : SignType.sign (w i₂) = SignType.sign (w i₃)
h₂s : SignType.sign (w i₁) = SignType.sign (w i₃)
hs : ∀ (i : Fin 3), SignType.sign (w i) = SignType.sign (w i₃)
hss : SignType.sign (∑ i : Fin (2 + 1), w i) = 1
hs' : SignType.sign (∑ i : Fin 3, w i) = SignType.sign (w i₃)
\u22a2 w i₁ ∈ Set.Ioo 0 1
-- Mathlib/Analysis/Convex/Between.lean
R : Type u_1
V : Type u_2
V' : Type u_3
P : Type u_4
P' : Type u_5
inst✝⁴ : LinearOrderedRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AddTorsor V P
inst✝ : NoZeroSMulDivisors R V
t : Affine.Triangle R P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
this : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h3 : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
hu : Finset.univ = {i₁, i₂, i₃}
r₁ : R
hr₁0 : 0 < r₁
hr₁1 : r₁ < 1
h₁ : Sbtw R (t.points i₂) ((lineMap (t.points i₂) (t.points i₃)) r₁) (t.points i₃)
r₂ : R
hr₂0 : 0 < r₂
hr₂1 : r₂ < 1
h₂ : Sbtw R (t.points i₁) ((lineMap (t.points i₁) (t.points i₃)) r₂) (t.points i₃)
w : Fin (2 + 1) → R
hw : ∑ i : Fin (2 + 1), w i = 1
hp : (Finset.affineCombination R Finset.univ t.points) w ∈ affineSpan R (Set.range t.points)
h₁' : (Finset.affineCombination R Finset.univ t.points) w ∈ affineSpan R {(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationSingleWeights R i₁), (Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationLineMapWeights i₂ i₃ r₁)}
h₂' : (Finset.affineCombination R Finset.univ t.points) w ∈ affineSpan R {t.points i₂, (lineMap (t.points i₁) (t.points i₃)) r₂}
h₁s : SignType.sign (w i₂) = SignType.sign (w i₃)
h₂s : SignType.sign (w i₁) = SignType.sign (w i₃)
hs : ∀ (i : Fin 3), SignType.sign (w i) = SignType.sign (w i₃)
hss : SignType.sign (w i₃) = 1
hs' : SignType.sign (∑ i : Fin 3, w i) = SignType.sign (w i₃)
\u22a2 w i₁ ∈ Set.Ioo 0 1
-- Mathlib/Analysis/Convex/Between.lean
R : Type u_1
V : Type u_2
V' : Type u_3
P : Type u_4
P' : Type u_5
inst✝⁴ : LinearOrderedRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AddTorsor V P
inst✝ : NoZeroSMulDivisors R V
t : Affine.Triangle R P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
this : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h3 : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
hu : Finset.univ = {i₁, i₂, i₃}
r₁ : R
hr₁0 : 0 < r₁
hr₁1 : r₁ < 1
h₁ : Sbtw R (t.points i₂) ((lineMap (t.points i₂) (t.points i₃)) r₁) (t.points i₃)
r₂ : R
hr₂0 : 0 < r₂
hr₂1 : r₂ < 1
h₂ : Sbtw R (t.points i₁) ((lineMap (t.points i₁) (t.points i₃)) r₂) (t.points i₃)
w : Fin (2 + 1) → R
hw : ∑ i : Fin (2 + 1), w i = 1
hp : (Finset.affineCombination R Finset.univ t.points) w ∈ affineSpan R (Set.range t.points)
h₁' : (Finset.affineCombination R Finset.univ t.points) w ∈ affineSpan R {(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationSingleWeights R i₁), (Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationLineMapWeights i₂ i₃ r₁)}
h₂' : (Finset.affineCombination R Finset.univ t.points) w ∈ affineSpan R {t.points i₂, (lineMap (t.points i₁) (t.points i₃)) r₂}
h₁s : SignType.sign (w i₂) = SignType.sign (w i₃)
h₂s : SignType.sign (w i₁) = SignType.sign (w i₃)
hss : SignType.sign (w i₃) = 1
hs' : SignType.sign (∑ i : Fin 3, w i) = SignType.sign (w i₃)
hs : ∀ (i : Fin 3), 0 < w i
\u22a2 w i₁ ∈ Set.Ioo 0 1
-- Mathlib/Analysis/Convex/Between.lean
R : Type u_1
V : Type u_2
V' : Type u_3
P : Type u_4
P' : Type u_5
inst✝⁴ : LinearOrderedRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AddTorsor V P
inst✝ : NoZeroSMulDivisors R V
t : Affine.Triangle R P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
this : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h3 : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
hu : Finset.univ = {i₁, i₂, i₃}
r₁ : R
hr₁0 : 0 < r₁
hr₁1 : r₁ < 1
h₁ : Sbtw R (t.points i₂) ((lineMap (t.points i₂) (t.points i₃)) r₁) (t.points i₃)
r₂ : R
hr₂0 : 0 < r₂
hr₂1 : r₂ < 1
h₂ : Sbtw R (t.points i₁) ((lineMap (t.points i₁) (t.points i₃)) r₂) (t.points i₃)
w : Fin (2 + 1) → R
hw : ∑ i ∈ {i₁, i₂, i₃}, w i = 1
hp : (Finset.affineCombination R Finset.univ t.points) w ∈ affineSpan R (Set.range t.points)
h₁' : (Finset.affineCombination R Finset.univ t.points) w ∈ affineSpan R {(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationSingleWeights R i₁), (Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationLineMapWeights i₂ i₃ r₁)}
h₂' : (Finset.affineCombination R Finset.univ t.points) w ∈ affineSpan R {t.points i₂, (lineMap (t.points i₁) (t.points i₃)) r₂}
h₁s : SignType.sign (w i₂) = SignType.sign (w i₃)
h₂s : SignType.sign (w i₁) = SignType.sign (w i₃)
hss : SignType.sign (w i₃) = 1
hs' : SignType.sign (∑ i : Fin 3, w i) = SignType.sign (w i₃)
hs : ∀ (i : Fin 3), 0 < w i
\u22a2 w i₁ < 1
-- Mathlib/Analysis/Convex/Between.lean
R : Type u_1
V : Type u_2
V' : Type u_3
P : Type u_4
P' : Type u_5
inst✝⁴ : LinearOrderedRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AddTorsor V P
inst✝ : NoZeroSMulDivisors R V
t : Affine.Triangle R P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
this : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h3 : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
hu : Finset.univ = {i₁, i₂, i₃}
r₁ : R
hr₁0 : 0 < r₁
hr₁1 : r₁ < 1
h₁ : Sbtw R (t.points i₂) ((lineMap (t.points i₂) (t.points i₃)) r₁) (t.points i₃)
r₂ : R
hr₂0 : 0 < r₂
hr₂1 : r₂ < 1
h₂ : Sbtw R (t.points i₁) ((lineMap (t.points i₁) (t.points i₃)) r₂) (t.points i₃)
w : Fin (2 + 1) → R
hw : w i₁ + (w i₂ + w i₃) = 1
hp : (Finset.affineCombination R Finset.univ t.points) w ∈ affineSpan R (Set.range t.points)
h₁' : (Finset.affineCombination R Finset.univ t.points) w ∈ affineSpan R {(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationSingleWeights R i₁), (Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationLineMapWeights i₂ i₃ r₁)}
h₂' : (Finset.affineCombination R Finset.univ t.points) w ∈ affineSpan R {t.points i₂, (lineMap (t.points i₁) (t.points i₃)) r₂}
h₁s : SignType.sign (w i₂) = SignType.sign (w i₃)
h₂s : SignType.sign (w i₁) = SignType.sign (w i₃)
hss : SignType.sign (w i₃) = 1
hs' : SignType.sign (∑ i : Fin 3, w i) = SignType.sign (w i₃)
hs : ∀ (i : Fin 3), 0 < w i
\u22a2 w i₁ < 1
-- Mathlib/Analysis/Convex/Between.lean
R : Type u_1
V : Type u_2
V' : Type u_3
P : Type u_4
P' : Type u_5
inst✝⁴ : LinearOrderedRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AddTorsor V P
inst✝ : NoZeroSMulDivisors R V
t : Affine.Triangle R P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
this : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h3 : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
hu : Finset.univ = {i₁, i₂, i₃}
r₁ : R
hr₁0 : 0 < r₁
hr₁1 : r₁ < 1
h₁ : Sbtw R (t.points i₂) ((lineMap (t.points i₂) (t.points i₃)) r₁) (t.points i₃)
r₂ : R
hr₂0 : 0 < r₂
hr₂1 : r₂ < 1
h₂ : Sbtw R (t.points i₁) ((lineMap (t.points i₁) (t.points i₃)) r₂) (t.points i₃)
w : Fin (2 + 1) → R
hw : w i₁ + ∑ x ∈ {i₂, i₃}, w x = 1
hp : (Finset.affineCombination R Finset.univ t.points) w ∈ affineSpan R (Set.range t.points)
h₁' : (Finset.affineCombination R Finset.univ t.points) w ∈ affineSpan R {(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationSingleWeights R i₁), (Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationLineMapWeights i₂ i₃ r₁)}
h₂' : (Finset.affineCombination R Finset.univ t.points) w ∈ affineSpan R {t.points i₂, (lineMap (t.points i₁) (t.points i₃)) r₂}
h₁s : SignType.sign (w i₂) = SignType.sign (w i₃)
h₂s : SignType.sign (w i₁) = SignType.sign (w i₃)
hss : SignType.sign (w i₃) = 1
hs' : SignType.sign (∑ i : Fin 3, w i) = SignType.sign (w i₃)
hs : ∀ (i : Fin 3), 0 < w i
\u22a2 i₂ ∉ {i₃}
-- Mathlib/Analysis/Convex/Between.lean
R : Type u_1
V : Type u_2
V' : Type u_3
P : Type u_4
P' : Type u_5
inst✝⁴ : LinearOrderedRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AddTorsor V P
inst✝ : NoZeroSMulDivisors R V
t : Affine.Triangle R P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
this : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h3 : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
hu : Finset.univ = {i₁, i₂, i₃}
r₁ : R
hr₁0 : 0 < r₁
hr₁1 : r₁ < 1
h₁ : Sbtw R (t.points i₂) ((lineMap (t.points i₂) (t.points i₃)) r₁) (t.points i₃)
r₂ : R
hr₂0 : 0 < r₂
hr₂1 : r₂ < 1
h₂ : Sbtw R (t.points i₁) ((lineMap (t.points i₁) (t.points i₃)) r₂) (t.points i₃)
w : Fin (2 + 1) → R
hw : ∑ i ∈ {i₁, i₂, i₃}, w i = 1
hp : (Finset.affineCombination R Finset.univ t.points) w ∈ affineSpan R (Set.range t.points)
h₁' : (Finset.affineCombination R Finset.univ t.points) w ∈ affineSpan R {(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationSingleWeights R i₁), (Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationLineMapWeights i₂ i₃ r₁)}
h₂' : (Finset.affineCombination R Finset.univ t.points) w ∈ affineSpan R {t.points i₂, (lineMap (t.points i₁) (t.points i₃)) r₂}
h₁s : SignType.sign (w i₂) = SignType.sign (w i₃)
h₂s : SignType.sign (w i₁) = SignType.sign (w i₃)
hss : SignType.sign (w i₃) = 1
hs' : SignType.sign (∑ i : Fin 3, w i) = SignType.sign (w i₃)
hs : ∀ (i : Fin 3), 0 < w i
\u22a2 i₁ ∉ {i₂, i₃}
-- Mathlib/Analysis/Complex/TaylorSeries.lean
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝¹ f✝ : ℂ → E
hf✝ : Differentiable ℂ f✝
c z : ℂ
f : ℂ → ℂ
hf : Differentiable ℂ f
\u22a2 ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f c * (z - c) ^ n = f z
-- Mathlib/Analysis/Complex/TaylorSeries.lean
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝¹ f✝ : ℂ → E
hf✝ : Differentiable ℂ f✝
c z : ℂ
f : ℂ → ℂ
hf : Differentiable ℂ f
n : ℕ
\u22a2 (↑n !)⁻¹ * iteratedDeriv n f c * (z - c) ^ n = (↑n !)⁻¹ • (z - c) ^ n • iteratedDeriv n f c
-- Mathlib/FieldTheory/Normal.lean
F : Type u_1
K : Type u_2
inst✝⁶ : Field F
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
E : Type u_3
inst✝³ : Field E
inst✝² : Algebra F E
inst✝¹ : Algebra K E
inst✝ : IsScalarTower F K E
h : Normal F E
x : E
\u22a2 IsIntegral K x ∧ Splits (algebraMap K E) (minpoly K x)
-- Mathlib/FieldTheory/Normal.lean
F : Type u_1
K : Type u_2
inst✝⁶ : Field F
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
E : Type u_3
inst✝³ : Field E
inst✝² : Algebra F E
inst✝¹ : Algebra K E
inst✝ : IsScalarTower F K E
h : Normal F E
x : E
hx : IsIntegral F x
hhx : Splits (algebraMap F E) (minpoly F x)
\u22a2 IsIntegral K x ∧ Splits (algebraMap K E) (minpoly K x)
-- Mathlib/FieldTheory/Normal.lean
F : Type u_1
K : Type u_2
inst✝⁶ : Field F
inst✝⁵ : Field K
inst✝⁴ : Algebra F K
E : Type u_3
inst✝³ : Field E
inst✝² : Algebra F E
inst✝¹ : Algebra K E
inst✝ : IsScalarTower F K E
h : Normal F E
x : E
hx : IsIntegral F x
hhx : Splits ((algebraMap K E).comp (algebraMap F K)) (minpoly F x)
\u22a2 IsIntegral K x ∧ Splits (algebraMap K E) (minpoly K x)
-- Mathlib/MeasureTheory/Function/L2Space.lean
α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
E : Type u_2
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
f✝ : α → E
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace ℝ E
f : α → E
hf : Integrable f μ
hf_int : ∫ (x : α), ⟪∫ (x : α), f x ∂μ, f x⟫_𝕜 ∂μ = 0
\u22a2 ∫ (x : α), f x ∂μ = 0
-- Mathlib/Order/Atoms.lean
ι : Sort u_1
α : Type u_2
β : Type u_3
p : Prop
\u22a2 IsCoatom p ↔ ¬p
-- Mathlib/Order/Atoms.lean
ι : Sort u_1
α : Type u_2
β : Type u_3
p : Prop
\u22a2 ¬p → ∀ (b : Prop), (p → b) → b → ¬p → b
-- Mathlib/Algebra/Polynomial/Div.lean
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝¹ : Ring R
p q : R[X]
inst✝ : Nontrivial R
hq : q.Monic
h : p.degree < q.degree
\u22a2 p %ₘ q = p
-- Mathlib/Data/List/Zip.lean
α : Type u
β : Type u_1
γ : Type u_2
δ : Type u_3
ε : Type u_4
l : List α
l' : List β
lp : List (α × β)
hl : map Prod.fst lp = l
hr : map Prod.snd lp = l'
\u22a2 lp = l.zip l'
-- Mathlib/Analysis/Normed/Lp/PiLp.lean
p✝ : ℝ≥0∞
𝕜 : Type u_1
ι : Type u_2
α : ι → Type u_3
β✝ : ι → Type u_4
inst✝³ : Fact (1 ≤ p✝)
inst✝² : Fintype ι
p : ℝ≥0∞
inst✝¹ : Fact (1 ≤ p)
β : ι → Type u_5
inst✝ : (i : ι) → SeminormedAddCommGroup (β i)
n : ℕ
h : p = ↑n
f : PiLp p β
\u22a2 ‖f‖ = (∑ i : ι, ‖f i‖ ^ n) ^ (1 / ↑n)
-- Mathlib/Analysis/Normed/Lp/PiLp.lean
p✝ : ℝ≥0∞
𝕜 : Type u_1
ι : Type u_2
α : ι → Type u_3
β✝ : ι → Type u_4
inst✝³ : Fact (1 ≤ p✝)
inst✝² : Fintype ι
p : ℝ≥0∞
inst✝¹ : Fact (1 ≤ p)
β : ι → Type u_5
inst✝ : (i : ι) → SeminormedAddCommGroup (β i)
n : ℕ
h : p = ↑n
f : PiLp p β
this : 0 < p.toReal
\u22a2 ‖f‖ = (∑ i : ι, ‖f i‖ ^ n) ^ (1 / ↑n)
-- Mathlib/RingTheory/GradedAlgebra/Basic.lean
ι : Type u_1
R : Type u_2
A : Type u_3
σ : Type u_4
inst✝⁹ : Semiring A
inst✝⁸ : DecidableEq ι
inst✝⁷ : CanonicallyOrderedAddCommMonoid ι
inst✝⁶ : SetLike σ A
inst✝⁵ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝⁴ : GradedRing 𝒜
a : A
n✝ i : ι
inst✝³ : Sub ι
inst✝² : OrderedSub ι
inst✝¹ : ContravariantClass ι ι (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
n : ι
inst✝ : Decidable (i ≤ n)
b : ↥(𝒜 i)
\u22a2 ↑(((decompose 𝒜) (a * ↑b)) n) = if i ≤ n then ↑(((decompose 𝒜) a) (n - i)) * ↑b else 0
-- Mathlib/Data/Seq/Computation.lean
α : Type u
β : Type v
γ : Type w
R : α → β → Prop
ca : Computation α
cb : Computation β
l : ∀ {a : α}, a ∈ ca → ∃ b, b ∈ cb ∧ R a b
right✝ : ∀ {b : β}, b ∈ cb → ∃ a, a ∈ ca ∧ R a b
a : α
b : β
ma : a ∈ ca
mb : b ∈ cb
b' : β
mb' : b' ∈ cb
ab' : R a b'
\u22a2 R a b
-- Mathlib/Data/Seq/Computation.lean
α : Type u
β : Type v
γ : Type w
R : α → β → Prop
ca : Computation α
cb : Computation β
l : ∀ {a : α}, a ∈ ca → ∃ b, b ∈ cb ∧ R a b
right✝ : ∀ {b : β}, b ∈ cb → ∃ a, a ∈ ca ∧ R a b
a : α
b : β
ma : a ∈ ca
mb : b ∈ cb
b' : β
mb' : b' ∈ cb
ab' : R a b'
\u22a2 R a b'
-- Mathlib/Data/Finmap.lean
α : Type u
β : α → Type v
inst✝ : DecidableEq α
f : { f // ∀ (i : α), (f.2 i).isSome = true ↔ i ∈ f.1 }
\u22a2 ((↑f).1.sigma fun i => ((↑f).2 i).toFinset).val.NodupKeys
-- Mathlib/Data/Finmap.lean
α : Type u
β : α → Type v
inst✝ : DecidableEq α
f : { f // ∀ (i : α), (f.2 i).isSome = true ↔ i ∈ f.1 }
\u22a2 ∀ x ∈ ((↑f).1.sigma fun i => ((↑f).2 i).toFinset).val, ∀ y ∈ ((↑f).1.sigma fun i => ((↑f).2 i).toFinset).val, x.fst = y.fst → x = y
-- Mathlib/Data/Finmap.lean
α : Type u
β : α → Type v
inst✝ : DecidableEq α
f : { f // ∀ (i : α), (f.2 i).isSome = true ↔ i ∈ f.1 }
\u22a2 ∀ (x : (a : α) × β a), x.fst ∈ (↑f).1 ∧ (↑f).2 x.fst = some x.snd → ∀ (y : (a : α) × β a), y.fst ∈ (↑f).1 ∧ (↑f).2 y.fst = some y.snd → x.fst = y.fst → x = y
-- Mathlib/Data/Finmap.lean
α : Type u
β : α → Type v
inst✝ : DecidableEq α
f : { f // ∀ (i : α), (f.2 i).isSome = true ↔ i ∈ f.1 }
i : α
x : β i
left✝¹ : ⟨i, x⟩.fst ∈ (↑f).1
hx : (↑f).2 ⟨i, x⟩.fst = some ⟨i, x⟩.snd
y : β i
left✝ : ⟨i, y⟩.fst ∈ (↑f).1
hy : (↑f).2 ⟨i, y⟩.fst = some ⟨i, y⟩.snd
\u22a2 ⟨i, x⟩ = ⟨i, y⟩
-- Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean
f g : CircleDeg1Lift
x : ℝ
\u22a2 Tendsto (fun n => ((f ^ n) x - x) / ↑n) atTop (𝓝 (τ f))
-- Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean
f g : CircleDeg1Lift
x : ℝ
\u22a2 Tendsto (fun n => ((f ^ n) x - x) / ↑n) atTop (𝓝 (τ (↑(translate (Multiplicative.ofAdd x))⁻¹ * f * ↑(translate (Multiplicative.ofAdd x)))))
-- Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean
f g : CircleDeg1Lift
x : ℝ
n : ℕ
\u22a2 ((↑(translate (Multiplicative.ofAdd x))⁻¹ * f * ↑(translate (Multiplicative.ofAdd x))) ^ n) 0 / ↑n = ((f ^ n) x - x) / ↑n
-- Mathlib/Analysis/Normed/Group/HomCompletion.lean
G : Type u_1
inst✝² : SeminormedAddCommGroup G
H : Type u_2
inst✝¹ : SeminormedAddCommGroup H
K : Type u_3
inst✝ : SeminormedAddCommGroup K
f : NormedAddGroupHom G H
g : NormedAddGroupHom H K
x : Completion G
\u22a2 (g.completion.comp f.completion) x = (g.comp f).completion x
-- Mathlib/Analysis/Normed/Group/HomCompletion.lean
G : Type u_1
inst✝² : SeminormedAddCommGroup G
H : Type u_2
inst✝¹ : SeminormedAddCommGroup H
K : Type u_3
inst✝ : SeminormedAddCommGroup K
f : NormedAddGroupHom G H
g : NormedAddGroupHom H K
x : Completion G
\u22a2 Completion.map (⇑g ∘ ⇑f) x = Completion.map (⇑(g.comp f)) x
-- Mathlib/Combinatorics/Colex.lean
α : Type u_1
β : Type u_2
inst✝² : LinearOrder α
inst✝¹ : LinearOrder β
f : α → β
𝒜 𝒜₁ 𝒜₂ : Finset (Finset α)
s t u : Finset α
a b : α
r : ℕ
inst✝ : Fintype α
\u22a2 IsInitSeg (initSeg s) s.card
-- Mathlib/Combinatorics/Colex.lean
α : Type u_1
β : Type u_2
inst✝² : LinearOrder α
inst✝¹ : LinearOrder β
f : α → β
𝒜 𝒜₁ 𝒜₂ : Finset (Finset α)
s t u : Finset α
a b : α
r : ℕ
inst✝ : Fintype α
t₁ t₂ : Finset α
ht₁ : t₁ ∈ initSeg s
ht₂ : { ofColex := t₂ } < { ofColex := t₁ } ∧ t₂.card = s.card
\u22a2 { ofColex := t₂ } ≤ { ofColex := s }
-- Mathlib/Combinatorics/Colex.lean
α : Type u_1
β : Type u_2
inst✝² : LinearOrder α
inst✝¹ : LinearOrder β
f : α → β
𝒜 𝒜₁ 𝒜₂ : Finset (Finset α)
s t u : Finset α
a b : α
r : ℕ
inst✝ : Fintype α
t₁ t₂ : Finset α
ht₁ : s.card = t₁.card ∧ { ofColex := t₁ } ≤ { ofColex := s }
ht₂ : { ofColex := t₂ } < { ofColex := t₁ } ∧ t₂.card = s.card
\u22a2 { ofColex := t₂ } ≤ { ofColex := s }
-- Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean
\u22a2 StrictConcaveOn ℝ (Ioi 1) fun x => √x * log x
-- Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean
\u22a2 ContinuousOn (fun x => √x * log x) (Ioi 1)
-- Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean
x : ℝ
hx : x ∈ Ioi 1
\u22a2 deriv^[2] (fun x => √x * log x) x < 0
-- Mathlib/MeasureTheory/Measure/MeasureSpace.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t : Set α
f✝¹ : α → β
μa μa' : Measure α
μb μb' : Measure β
μc : Measure γ
f✝ : α → β
s : Set α
f : α → α
hf : QuasiMeasurePreserving f μ μ
hs : f ⁻¹' s =ᵐ[μ] s
k : ℕ
ih : f^[k] ⁻¹' s =ᵐ[μ] s
\u22a2 f^[k + 1] ⁻¹' s =ᵐ[μ] s
-- Mathlib/MeasureTheory/Measure/MeasureSpace.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t : Set α
f✝¹ : α → β
μa μa' : Measure α
μb μb' : Measure β
μc : Measure γ
f✝ : α → β
s : Set α
f : α → α
hf : QuasiMeasurePreserving f μ μ
hs : f ⁻¹' s =ᵐ[μ] s
k : ℕ
ih : f^[k] ⁻¹' s =ᵐ[μ] s
\u22a2 f ⁻¹' (f^[k] ⁻¹' s) =ᵐ[μ] s
-- Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Type u_4
s s' : Set α
x : α
p : Filter ι✝
g : ι✝ → α
inst✝² : UniformSpace β
𝔖 : Set (Set α)
ι : Type u_5
inst✝¹ : Preorder ι
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
t : ι → Set α
V : ι → Set (β × β)
ht : ∀ (n : ι), t n ∈ 𝔖
hmono : Monotone t
hex : ∀ s ∈ 𝔖, ∃ n, s ⊆ t n
hb : (𝓤 β).HasAntitoneBasis V
\u22a2 (𝓤 (α →ᵤ[𝔖] β)).HasAntitoneBasis fun n => UniformOnFun.gen 𝔖 (t n) (V n)
-- Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Type u_4
s s' : Set α
x : α
p : Filter ι✝
g : ι✝ → α
inst✝² : UniformSpace β
𝔖 : Set (Set α)
ι : Type u_5
inst✝¹ : Preorder ι
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
t : ι → Set α
V : ι → Set (β × β)
ht : ∀ (n : ι), t n ∈ 𝔖
hmono : Monotone t
hex : ∀ s ∈ 𝔖, ∃ n, s ⊆ t n
hb : (𝓤 β).HasAntitoneBasis V
this : Nonempty ι
\u22a2 (𝓤 (α →ᵤ[𝔖] β)).HasAntitoneBasis fun n => UniformOnFun.gen 𝔖 (t n) (V n)
-- Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Type u_4
s s' : Set α
x : α
p : Filter ι✝
g : ι✝ → α
inst✝² : UniformSpace β
𝔖 : Set (Set α)
ι : Type u_5
inst✝¹ : Preorder ι
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
t : ι → Set α
V : ι → Set (β × β)
ht : ∀ (n : ι), t n ∈ 𝔖
hmono : Monotone t
hex : ∀ s ∈ 𝔖, ∃ n, s ⊆ t n
hb : (𝓤 β).HasAntitoneBasis V
this : Nonempty ι
\u22a2 ∀ (i : ι × ι), True → ∃ i', True ∧ UniformOnFun.gen 𝔖 (t i') (V i') ⊆ UniformOnFun.gen 𝔖 (t i.1) (V i.2)
-- Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Type u_4
s s' : Set α
x : α
p : Filter ι✝
g : ι✝ → α
inst✝² : UniformSpace β
𝔖 : Set (Set α)
ι : Type u_5
inst✝¹ : Preorder ι
inst✝ : IsDirected ι fun x x_1 => x ≤ x_1
t : ι → Set α
V : ι → Set (β × β)
ht : ∀ (n : ι), t n ∈ 𝔖
hmono : Monotone t
hex : ∀ s ∈ 𝔖, ∃ n, s ⊆ t n
hb : (𝓤 β).HasAntitoneBasis V
this : Nonempty ι
\u22a2 Antitone fun n => UniformOnFun.gen 𝔖 (t n) (V n)
-- Mathlib/Algebra/MvPolynomial/Rename.lean
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
hf : Injective f
\u22a2 Injective ⇑(rename f)
-- Mathlib/Algebra/MvPolynomial/Rename.lean
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
hf : Injective f
this : ⇑(rename f) = Finsupp.mapDomain (Finsupp.mapDomain f)
\u22a2 Injective ⇑(rename f)
-- Mathlib/Algebra/MvPolynomial/Rename.lean
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
hf : Injective f
this : ⇑(rename f) = Finsupp.mapDomain (Finsupp.mapDomain f)
\u22a2 Injective (Finsupp.mapDomain (Finsupp.mapDomain f))
-- Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean
X : Type u_1
x : X
a : FreeAbelianGroup X
\u22a2 x ∉ a.support ↔ (coeff x) a = 0
-- Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean
X : Type u_1
x : X
a : FreeAbelianGroup X
\u22a2 (toFinsupp a) x = 0 ↔ (coeff x) a = 0
-- Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean
a : UnitAddCircle
x : ℝ
hx : x ≤ 0
\u22a2 cosKernel a x = 0
-- Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean
x : ℝ
hx : x ≤ 0
a' : ℝ
\u22a2 cosKernel (↑a') x = 0