ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/MeasureTheory/Integral/SetToL1.lean
α : Type u_1
E : Type u_2
F : Type u_3
F' : Type u_4
G : Type u_5
𝕜 : Type u_6
p : ℝ≥0∞
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace ℝ F
inst✝⁸ : NormedAddCommGroup F'
inst✝⁷ : NormedSpace ℝ F'
inst✝⁶ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
inst✝⁵ : NormedField 𝕜
inst✝⁴ : NormedSpace 𝕜 E
G'' : Type u_7
G' : Type u_8
inst✝³ : NormedLatticeAddCommGroup G'
inst✝² : NormedSpace ℝ G'
inst✝¹ : NormedLatticeAddCommGroup G''
inst✝ : NormedSpace ℝ G''
T : Set α → G'' →L[ℝ] G'
h_zero : ∀ (s : Set α), MeasurableSet s → μ s = 0 → T s = 0
h_add : FinMeasAdditive μ T
hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∀ (x : G''), 0 ≤ x → 0 ≤ (T s) x
f g : ↥(simpleFunc G'' 1 μ)
hfg : 0 ≤ g - f
\u22a2 0 ≤ setToL1S T (g - f)
-- Mathlib/AlgebraicGeometry/StructureSheaf.lean
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
\u22a2 ∃ a' h' iDh', U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h' i) ∧ (∀ i ∈ t, ∀ j ∈ t, a' i * h' j = h' i * a' j) ∧ ∀ i ∈ t, ((structureSheaf R).val.map (iDh' i).op) s = const R (a' i) (h' i) (PrimeSpectrum.basicOpen (h' i)) ⋯
-- Mathlib/AlgebraicGeometry/StructureSheaf.lean
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq : ∀ (i j : ι), IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ = IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
\u22a2 ∃ a' h' iDh', U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h' i) ∧ (∀ i ∈ t, ∀ j ∈ t, a' i * h' j = h' i * a' j) ∧ ∀ i ∈ t, ((structureSheaf R).val.map (iDh' i).op) s = const R (a' i) (h' i) (PrimeSpectrum.basicOpen (h' i)) ⋯
-- Mathlib/AlgebraicGeometry/StructureSheaf.lean
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq : ∀ (i j : ι), IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ = IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
\u22a2 ∃ a' h' iDh', U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h' i) ∧ (∀ i ∈ t, ∀ j ∈ t, a' i * h' j = h' i * a' j) ∧ ∀ i ∈ t, ((structureSheaf R).val.map (iDh' i).op) s = const R (a' i) (h' i) (PrimeSpectrum.basicOpen (h' i)) ⋯
-- Mathlib/AlgebraicGeometry/StructureSheaf.lean
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq : ∀ (i j : ι), IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ = IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
n : ι × ι → ℕ := fun p => ⋯.choose
\u22a2 ∃ a' h' iDh', U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h' i) ∧ (∀ i ∈ t, ∀ j ∈ t, a' i * h' j = h' i * a' j) ∧ ∀ i ∈ t, ((structureSheaf R).val.map (iDh' i).op) s = const R (a' i) (h' i) (PrimeSpectrum.basicOpen (h' i)) ⋯
-- Mathlib/AlgebraicGeometry/StructureSheaf.lean
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq : ∀ (i j : ι), IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ = IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
n : ι × ι → ℕ := fun p => ⋯.choose
n_spec : ∀ (p : ι × ι), a p.1 * h p.2 * (h p.1 * h p.2) ^ ⋯.choose = h p.1 * a p.2 * (h p.1 * h p.2) ^ ⋯.choose
\u22a2 ∃ a' h' iDh', U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h' i) ∧ (∀ i ∈ t, ∀ j ∈ t, a' i * h' j = h' i * a' j) ∧ ∀ i ∈ t, ((structureSheaf R).val.map (iDh' i).op) s = const R (a' i) (h' i) (PrimeSpectrum.basicOpen (h' i)) ⋯
-- Mathlib/AlgebraicGeometry/StructureSheaf.lean
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq : ∀ (i j : ι), IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ = IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
n : ι × ι → ℕ := fun p => ⋯.choose
n_spec : ∀ (p : ι × ι), a p.1 * h p.2 * (h p.1 * h p.2) ^ ⋯.choose = h p.1 * a p.2 * (h p.1 * h p.2) ^ ⋯.choose
N : ℕ := (t ×ˢ t).sup n
\u22a2 ∃ a' h' iDh', U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h' i) ∧ (∀ i ∈ t, ∀ j ∈ t, a' i * h' j = h' i * a' j) ∧ ∀ i ∈ t, ((structureSheaf R).val.map (iDh' i).op) s = const R (a' i) (h' i) (PrimeSpectrum.basicOpen (h' i)) ⋯
-- Mathlib/AlgebraicGeometry/StructureSheaf.lean
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq : ∀ (i j : ι), IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ = IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
n : ι × ι → ℕ := fun p => ⋯.choose
n_spec : ∀ (p : ι × ι), a p.1 * h p.2 * (h p.1 * h p.2) ^ ⋯.choose = h p.1 * a p.2 * (h p.1 * h p.2) ^ ⋯.choose
N : ℕ := (t ×ˢ t).sup n
basic_opens_eq : ∀ (i : ι), PrimeSpectrum.basicOpen (h i ^ (N + 1)) = PrimeSpectrum.basicOpen (h i)
\u22a2 ∃ a' h' iDh', U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h' i) ∧ (∀ i ∈ t, ∀ j ∈ t, a' i * h' j = h' i * a' j) ∧ ∀ i ∈ t, ((structureSheaf R).val.map (iDh' i).op) s = const R (a' i) (h' i) (PrimeSpectrum.basicOpen (h' i)) ⋯
-- Mathlib/AlgebraicGeometry/StructureSheaf.lean
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq : ∀ (i j : ι), IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ = IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
n : ι × ι → ℕ := fun p => ⋯.choose
n_spec : ∀ (p : ι × ι), a p.1 * h p.2 * (h p.1 * h p.2) ^ ⋯.choose = h p.1 * a p.2 * (h p.1 * h p.2) ^ ⋯.choose
N : ℕ := (t ×ˢ t).sup n
basic_opens_eq : ∀ (i : ι), PrimeSpectrum.basicOpen (h i ^ (N + 1)) = PrimeSpectrum.basicOpen (h i)
i : ι
a✝ : i ∈ t
\u22a2 ((structureSheaf R).val.map ((fun i => eqToHom ⋯ ≫ iDh i) i).op) s = const R ((fun i => a i * h i ^ N) i) ((fun i => h i ^ (N + 1)) i) (PrimeSpectrum.basicOpen ((fun i => h i ^ (N + 1)) i)) ⋯
-- Mathlib/AlgebraicGeometry/StructureSheaf.lean
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq : ∀ (i j : ι), IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ = IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
n : ι × ι → ℕ := fun p => ⋯.choose
n_spec : ∀ (p : ι × ι), a p.1 * h p.2 * (h p.1 * h p.2) ^ ⋯.choose = h p.1 * a p.2 * (h p.1 * h p.2) ^ ⋯.choose
N : ℕ := (t ×ˢ t).sup n
basic_opens_eq : ∀ (i : ι), PrimeSpectrum.basicOpen (h i ^ (N + 1)) = PrimeSpectrum.basicOpen (h i)
i : ι
a✝ : i ∈ t
\u22a2 ((structureSheaf R).val.map (iDh i).op ≫ (structureSheaf R).val.map (eqToHom ⋯).op) s = const R ((fun i => a i * h i ^ N) i) ((fun i => h i ^ (N + 1)) i) (PrimeSpectrum.basicOpen ((fun i => h i ^ (N + 1)) i)) ⋯
-- Mathlib/AlgebraicGeometry/StructureSheaf.lean
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq : ∀ (i j : ι), IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ = IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
n : ι × ι → ℕ := fun p => ⋯.choose
n_spec : ∀ (p : ι × ι), a p.1 * h p.2 * (h p.1 * h p.2) ^ ⋯.choose = h p.1 * a p.2 * (h p.1 * h p.2) ^ ⋯.choose
N : ℕ := (t ×ˢ t).sup n
basic_opens_eq : ∀ (i : ι), PrimeSpectrum.basicOpen (h i ^ (N + 1)) = PrimeSpectrum.basicOpen (h i)
i : ι
a✝ : i ∈ t
\u22a2 ((structureSheaf R).val.map (eqToHom ⋯).op) (((structureSheaf R).val.map (iDh i).op) s) = const R ((fun i => a i * h i ^ N) i) ((fun i => h i ^ (N + 1)) i) (PrimeSpectrum.basicOpen ((fun i => h i ^ (N + 1)) i)) ⋯
-- Mathlib/AlgebraicGeometry/StructureSheaf.lean
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq : ∀ (i j : ι), IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ = IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
n : ι × ι → ℕ := fun p => ⋯.choose
n_spec : ∀ (p : ι × ι), a p.1 * h p.2 * (h p.1 * h p.2) ^ ⋯.choose = h p.1 * a p.2 * (h p.1 * h p.2) ^ ⋯.choose
N : ℕ := (t ×ˢ t).sup n
basic_opens_eq : ∀ (i : ι), PrimeSpectrum.basicOpen (h i ^ (N + 1)) = PrimeSpectrum.basicOpen (h i)
i : ι
a✝ : i ∈ t
\u22a2 const R (a i) (h i) (PrimeSpectrum.basicOpen (h i ^ (N + 1))) ?refine_3.hv = const R ((fun i => a i * h i ^ N) i) ((fun i => h i ^ (N + 1)) i) (PrimeSpectrum.basicOpen ((fun i => h i ^ (N + 1)) i)) ⋯
-- Mathlib/AlgebraicGeometry/StructureSheaf.lean
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq : ∀ (i j : ι), IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ = IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
n : ι × ι → ℕ := fun p => ⋯.choose
n_spec : ∀ (p : ι × ι), a p.1 * h p.2 * (h p.1 * h p.2) ^ ⋯.choose = h p.1 * a p.2 * (h p.1 * h p.2) ^ ⋯.choose
N : ℕ := (t ×ˢ t).sup n
basic_opens_eq : ∀ (i : ι), PrimeSpectrum.basicOpen (h i ^ (N + 1)) = PrimeSpectrum.basicOpen (h i)
i : ι
a✝ : i ∈ t
\u22a2 ∀ x ∈ PrimeSpectrum.basicOpen (h i ^ (N + 1)), h i ∈ x.asIdeal.primeCompl
-- Mathlib/AlgebraicGeometry/StructureSheaf.lean
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq : ∀ (i j : ι), IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ = IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
n : ι × ι → ℕ := fun p => ⋯.choose
n_spec : ∀ (p : ι × ι), a p.1 * h p.2 * (h p.1 * h p.2) ^ ⋯.choose = h p.1 * a p.2 * (h p.1 * h p.2) ^ ⋯.choose
N : ℕ := (t ×ˢ t).sup n
basic_opens_eq : ∀ (i : ι), PrimeSpectrum.basicOpen (h i ^ (N + 1)) = PrimeSpectrum.basicOpen (h i)
i : ι
a✝ : i ∈ t
\u22a2 const R (a i) (h i) (PrimeSpectrum.basicOpen (h i ^ (N + 1))) ⋯ = const R ((fun i => a i * h i ^ N) i) ((fun i => h i ^ (N + 1)) i) (PrimeSpectrum.basicOpen ((fun i => h i ^ (N + 1)) i)) ⋯
-- Mathlib/AlgebraicGeometry/StructureSheaf.lean
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq : ∀ (i j : ι), IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ = IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
n : ι × ι → ℕ := fun p => ⋯.choose
n_spec : ∀ (p : ι × ι), a p.1 * h p.2 * (h p.1 * h p.2) ^ ⋯.choose = h p.1 * a p.2 * (h p.1 * h p.2) ^ ⋯.choose
N : ℕ := (t ×ˢ t).sup n
basic_opens_eq : ∀ (i : ι), PrimeSpectrum.basicOpen (h i ^ (N + 1)) = PrimeSpectrum.basicOpen (h i)
i : ι
a✝ : i ∈ t
\u22a2 a i * (fun i => h i ^ (N + 1)) i = (fun i => a i * h i ^ N) i * h i
-- Mathlib/AlgebraicGeometry/StructureSheaf.lean
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq : ∀ (i j : ι), IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ = IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
n : ι × ι → ℕ := fun p => ⋯.choose
n_spec : ∀ (p : ι × ι), a p.1 * h p.2 * (h p.1 * h p.2) ^ ⋯.choose = h p.1 * a p.2 * (h p.1 * h p.2) ^ ⋯.choose
N : ℕ := (t ×ˢ t).sup n
basic_opens_eq : ∀ (i : ι), PrimeSpectrum.basicOpen (h i ^ (N + 1)) = PrimeSpectrum.basicOpen (h i)
i : ι
a✝ : i ∈ t
\u22a2 a i * h i ^ (N + 1) = a i * h i ^ N * h i
-- Mathlib/AlgebraicGeometry/StructureSheaf.lean
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq : ∀ (i j : ι), IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ = IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
n : ι × ι → ℕ := fun p => ⋯.choose
n_spec : ∀ (p : ι × ι), a p.1 * h p.2 * (h p.1 * h p.2) ^ ⋯.choose = h p.1 * a p.2 * (h p.1 * h p.2) ^ ⋯.choose
N : ℕ := (t ×ˢ t).sup n
basic_opens_eq : ∀ (i : ι), PrimeSpectrum.basicOpen (h i ^ (N + 1)) = PrimeSpectrum.basicOpen (h i)
i : ι
a✝ : i ∈ t
\u22a2 a i * (h i ^ N * h i) = a i * h i ^ N * h i
-- Mathlib/Data/Set/Pointwise/Basic.lean
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝ : InvolutiveInv α
s t : Set α
a : α
ι : Sort u_5
f : ι → α
\u22a2 (range f)⁻¹ = range fun i => (f i)⁻¹
-- Mathlib/Data/Set/Pointwise/Basic.lean
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝ : InvolutiveInv α
s t : Set α
a : α
ι : Sort u_5
f : ι → α
\u22a2 Inv.inv '' range f = range fun i => (f i)⁻¹
-- Mathlib/RingTheory/Polynomial/Pochhammer.lean
R : Type u
inst✝¹ : Ring R
S : Type u_1
inst✝ : Ring S
n : ℕ
k : S
\u22a2 eval k (descPochhammer S (n + 1)) = eval k (descPochhammer S n) * (k - ↑n)
-- Mathlib/RepresentationTheory/GroupCohomology/Hilbert90.lean
K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : FiniteDimensional K L
f : (L ≃ₐ[K] L) → Lˣ
x y : L ≃ₐ[K] L
h : (fun f => ↑f) x = (fun f => ↑f) y
a✝ : L
\u22a2 x a✝ = y a✝
-- Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
x y : V
\u22a2 (o.oangle (x - y) x).sign = (o.oangle x y).sign
-- Mathlib/CategoryTheory/Limits/Presheaf.lean
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
ℰ : Type u₂
inst✝³ : Category.{v₁, u₂} ℰ
A✝ A : C ⥤ ℰ
inst✝² : yoneda.HasPointwiseLeftKanExtension A
D : Type u₂
inst✝¹ : Category.{v₁, u₂} D
F : C ⥤ D
inst✝ : ∀ (P : Cᵒᵖ ⥤ Type v₁), F.op.HasLeftKanExtension P
X : C
G : (Cᵒᵖ ⥤ Type v₁) ⥤ Dᵒᵖ ⥤ Type v₁
φ : F ⋙ yoneda ⟶ yoneda ⋙ G
P : Cᵒᵖ ⥤ Type v₁
x y : P.Elements
f : x ⟶ y
\u22a2 yoneda.map (↑f).unop ≫ coconeApp φ x = coconeApp φ y
-- Mathlib/CategoryTheory/Limits/Presheaf.lean
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
ℰ : Type u₂
inst✝³ : Category.{v₁, u₂} ℰ
A✝ A : C ⥤ ℰ
inst✝² : yoneda.HasPointwiseLeftKanExtension A
D : Type u₂
inst✝¹ : Category.{v₁, u₂} D
F : C ⥤ D
inst✝ : ∀ (P : Cᵒᵖ ⥤ Type v₁), F.op.HasLeftKanExtension P
X : C
G : (Cᵒᵖ ⥤ Type v₁) ⥤ Dᵒᵖ ⥤ Type v₁
φ : F ⋙ yoneda ⟶ yoneda ⋙ G
P : Cᵒᵖ ⥤ Type v₁
x y : P.Elements
f : x ⟶ y
eq₁ : yoneda.map (↑f).unop ≫ yonedaEquiv.symm x.snd = yonedaEquiv.symm y.snd
\u22a2 yoneda.map (↑f).unop ≫ coconeApp φ x = coconeApp φ y
-- Mathlib/CategoryTheory/Limits/Presheaf.lean
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
ℰ : Type u₂
inst✝³ : Category.{v₁, u₂} ℰ
A✝ A : C ⥤ ℰ
inst✝² : yoneda.HasPointwiseLeftKanExtension A
D : Type u₂
inst✝¹ : Category.{v₁, u₂} D
F : C ⥤ D
inst✝ : ∀ (P : Cᵒᵖ ⥤ Type v₁), F.op.HasLeftKanExtension P
X : C
G : (Cᵒᵖ ⥤ Type v₁) ⥤ Dᵒᵖ ⥤ Type v₁
φ : F ⋙ yoneda ⟶ yoneda ⋙ G
P : Cᵒᵖ ⥤ Type v₁
x y : P.Elements
f : x ⟶ y
eq₁ : yoneda.map (↑f).unop ≫ yonedaEquiv.symm x.snd = yonedaEquiv.symm y.snd
eq₂ : ((G.obj (yoneda.obj (Opposite.unop x.1))).map (F.map (↑f).unop).op ≫ (G.map (yonedaEquiv.symm x.snd)).app (Opposite.op (F.obj (Opposite.unop y.fst)))) ((φ.app (Opposite.unop x.fst)).app (Opposite.op (F.obj (Opposite.unop x.fst))) (𝟙 (Opposite.unop (Opposite.op (F.obj (Opposite.unop x.fst)))))) = ((G.map (yonedaEquiv.symm x.snd)).app (Opposite.op (F.obj (Opposite.unop x.fst))) ≫ (G.obj P).map (F.map (↑f).unop).op) ((φ.app (Opposite.unop x.fst)).app (Opposite.op (F.obj (Opposite.unop x.fst))) (𝟙 (Opposite.unop (Opposite.op (F.obj (Opposite.unop x.fst))))))
\u22a2 yoneda.map (↑f).unop ≫ coconeApp φ x = coconeApp φ y
-- Mathlib/CategoryTheory/Limits/Presheaf.lean
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
ℰ : Type u₂
inst✝³ : Category.{v₁, u₂} ℰ
A✝ A : C ⥤ ℰ
inst✝² : yoneda.HasPointwiseLeftKanExtension A
D : Type u₂
inst✝¹ : Category.{v₁, u₂} D
F : C ⥤ D
inst✝ : ∀ (P : Cᵒᵖ ⥤ Type v₁), F.op.HasLeftKanExtension P
X : C
G : (Cᵒᵖ ⥤ Type v₁) ⥤ Dᵒᵖ ⥤ Type v₁
φ : F ⋙ yoneda ⟶ yoneda ⋙ G
P : Cᵒᵖ ⥤ Type v₁
x y : P.Elements
f : x ⟶ y
eq₁ : yoneda.map (↑f).unop ≫ yonedaEquiv.symm x.snd = yonedaEquiv.symm y.snd
eq₂ : ((G.obj (yoneda.obj (Opposite.unop x.1))).map (F.map (↑f).unop).op ≫ (G.map (yonedaEquiv.symm x.snd)).app (Opposite.op (F.obj (Opposite.unop y.fst)))) ((φ.app (Opposite.unop x.fst)).app (Opposite.op (F.obj (Opposite.unop x.fst))) (𝟙 (Opposite.unop (Opposite.op (F.obj (Opposite.unop x.fst)))))) = ((G.map (yonedaEquiv.symm x.snd)).app (Opposite.op (F.obj (Opposite.unop x.fst))) ≫ (G.obj P).map (F.map (↑f).unop).op) ((φ.app (Opposite.unop x.fst)).app (Opposite.op (F.obj (Opposite.unop x.fst))) (𝟙 (Opposite.unop (Opposite.op (F.obj (Opposite.unop x.fst))))))
eq₃ : ((F ⋙ yoneda).map (↑f).unop ≫ φ.app (Opposite.unop x.fst)).app (Opposite.op (F.toPrefunctor.1 (Opposite.unop y.fst))) (𝟙 (Opposite.unop (Opposite.op (F.toPrefunctor.1 (Opposite.unop y.fst))))) = (φ.app (Opposite.unop y.fst) ≫ (yoneda ⋙ G).map (↑f).unop).app (Opposite.op (F.toPrefunctor.1 (Opposite.unop y.fst))) (𝟙 (Opposite.unop (Opposite.op (F.toPrefunctor.1 (Opposite.unop y.fst)))))
\u22a2 yoneda.map (↑f).unop ≫ coconeApp φ x = coconeApp φ y
-- Mathlib/CategoryTheory/Limits/Presheaf.lean
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
ℰ : Type u₂
inst✝³ : Category.{v₁, u₂} ℰ
A✝ A : C ⥤ ℰ
inst✝² : yoneda.HasPointwiseLeftKanExtension A
D : Type u₂
inst✝¹ : Category.{v₁, u₂} D
F : C ⥤ D
inst✝ : ∀ (P : Cᵒᵖ ⥤ Type v₁), F.op.HasLeftKanExtension P
X : C
G : (Cᵒᵖ ⥤ Type v₁) ⥤ Dᵒᵖ ⥤ Type v₁
φ : F ⋙ yoneda ⟶ yoneda ⋙ G
P : Cᵒᵖ ⥤ Type v₁
x y : P.Elements
f : x ⟶ y
eq₁ : yoneda.map (↑f).unop ≫ yonedaEquiv.symm x.snd = yonedaEquiv.symm y.snd
eq₂ : ((G.obj (yoneda.obj (Opposite.unop x.1))).map (F.map (↑f).unop).op ≫ (G.map (yonedaEquiv.symm x.snd)).app (Opposite.op (F.obj (Opposite.unop y.fst)))) ((φ.app (Opposite.unop x.fst)).app (Opposite.op (F.obj (Opposite.unop x.fst))) (𝟙 (Opposite.unop (Opposite.op (F.obj (Opposite.unop x.fst)))))) = ((G.map (yonedaEquiv.symm x.snd)).app (Opposite.op (F.obj (Opposite.unop x.fst))) ≫ (G.obj P).map (F.map (↑f).unop).op) ((φ.app (Opposite.unop x.fst)).app (Opposite.op (F.obj (Opposite.unop x.fst))) (𝟙 (Opposite.unop (Opposite.op (F.obj (Opposite.unop x.fst))))))
eq₃ : ((F ⋙ yoneda).map (↑f).unop ≫ φ.app (Opposite.unop x.fst)).app (Opposite.op (F.toPrefunctor.1 (Opposite.unop y.fst))) (𝟙 (Opposite.unop (Opposite.op (F.toPrefunctor.1 (Opposite.unop y.fst))))) = (φ.app (Opposite.unop y.fst) ≫ (yoneda ⋙ G).map (↑f).unop).app (Opposite.op (F.toPrefunctor.1 (Opposite.unop y.fst))) (𝟙 (Opposite.unop (Opposite.op (F.toPrefunctor.1 (Opposite.unop y.fst)))))
eq₄ : ∀ (a : ((F ⋙ yoneda).obj (Opposite.unop x.fst)).obj (Opposite.op (F.obj (Opposite.unop x.fst)))), (((F ⋙ yoneda).obj (Opposite.unop x.fst)).map (F.map (↑f).unop).op ≫ (φ.app (Opposite.unop x.fst)).app (Opposite.op (F.obj (Opposite.unop y.fst)))) a = ((φ.app (Opposite.unop x.fst)).app (Opposite.op (F.obj (Opposite.unop x.fst))) ≫ ((yoneda ⋙ G).obj (Opposite.unop x.fst)).map (F.map (↑f).unop).op) a
\u22a2 yoneda.map (↑f).unop ≫ coconeApp φ x = coconeApp φ y