ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Algebra/Order/Field/Basic.lean
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝ : LinearOrderedSemifield α
a b c d e : α
m n : ℤ
h : a < b
hc : 0 < c
\u22a2 a / c < b / c
-- Mathlib/Algebra/Order/Field/Basic.lean
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝ : LinearOrderedSemifield α
a b c d e : α
m n : ℤ
h : a < b
hc : 0 < c
\u22a2 a * (1 / c) < b * (1 / c)
-- Mathlib/Topology/MetricSpace/PartitionOfUnity.lean
ι : Type u_1
X : Type u_2
inst✝ : EMetricSpace X
K U : ι → Set X
hK : ∀ (i : ι), IsClosed (K i)
hU : ∀ (i : ι), IsOpen (U i)
hKU : ∀ (i : ι), K i ⊆ U i
hfin : LocallyFinite K
\u22a2 ∃ δ, (∀ (x : X), 0 < δ x) ∧ ∀ (i : ι), ∀ x ∈ K i, closedBall x (ENNReal.ofReal (δ x)) ⊆ U i
-- Mathlib/Combinatorics/SetFamily/Shatter.lean
α : Type u_1
inst✝ : DecidableEq α
𝒜 ℬ : Finset (Finset α)
s t : Finset α
a : α
n : ℕ
\u22a2 𝒜.shatterer.Shatters s ↔ 𝒜.Shatters s
-- Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean
R : Type u
inst✝³ : CommRing R
n G : Type v
inst✝² : DecidableEq n
inst✝¹ : Fintype n
α β : Type v
inst✝ : DecidableEq α
M : Matrix n n R
\u22a2 (M.charpoly - ∏ i : n, (X - C (M i i))).degree < ↑(Fintype.card n - 1)
-- Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean
R : Type u
inst✝³ : CommRing R
n G : Type v
inst✝² : DecidableEq n
inst✝¹ : Fintype n
α β : Type v
inst✝ : DecidableEq α
M : Matrix n n R
\u22a2 (∑ x ∈ univ.erase (Equiv.refl n), ↑↑(Equiv.Perm.sign x) * ∏ i : n, M.charmatrix (x i) i + ↑↑(Equiv.Perm.sign (Equiv.refl n)) * ∏ i : n, M.charmatrix ((Equiv.refl n) i) i - ∏ i : n, (X - C (M i i))).degree < ↑(Fintype.card n - 1)
-- Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean
R : Type u
inst✝³ : CommRing R
n G : Type v
inst✝² : DecidableEq n
inst✝¹ : Fintype n
α β : Type v
inst✝ : DecidableEq α
M : Matrix n n R
\u22a2 (∑ x ∈ univ.erase (Equiv.refl n), ↑↑(Equiv.Perm.sign x) * ∏ i : n, M.charmatrix (x i) i).degree < ↑(Fintype.card n - 1)
-- Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean
R : Type u
inst✝³ : CommRing R
n G : Type v
inst✝² : DecidableEq n
inst✝¹ : Fintype n
α β : Type v
inst✝ : DecidableEq α
M : Matrix n n R
\u22a2 ∑ x ∈ univ.erase (Equiv.refl n), ↑↑(Equiv.Perm.sign x) * ∏ i : n, M.charmatrix (x i) i ∈ degreeLT R (Fintype.card n - 1)
-- Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean
R : Type u
inst✝³ : CommRing R
n G : Type v
inst✝² : DecidableEq n
inst✝¹ : Fintype n
α β : Type v
inst✝ : DecidableEq α
M : Matrix n n R
\u22a2 ∀ c ∈ univ.erase (Equiv.refl n), ↑↑(Equiv.Perm.sign c) * ∏ i : n, M.charmatrix (c i) i ∈ degreeLT R (Fintype.card n - 1)
-- Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean
R : Type u
inst✝³ : CommRing R
n G : Type v
inst✝² : DecidableEq n
inst✝¹ : Fintype n
α β : Type v
inst✝ : DecidableEq α
M : Matrix n n R
c : n ≃ n
hc : c ∈ univ.erase (Equiv.refl n)
\u22a2 ↑↑(Equiv.Perm.sign c) * ∏ i : n, M.charmatrix (c i) i ∈ degreeLT R (Fintype.card n - 1)
-- Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean
R : Type u
inst✝³ : CommRing R
n G : Type v
inst✝² : DecidableEq n
inst✝¹ : Fintype n
α β : Type v
inst✝ : DecidableEq α
M : Matrix n n R
c : n ≃ n
hc : c ∈ univ.erase (Equiv.refl n)
\u22a2 ↑↑(Equiv.Perm.sign c) • ∏ i : n, M.charmatrix (c i) i ∈ degreeLT R (Fintype.card n - 1)
-- Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean
R : Type u
inst✝³ : CommRing R
n G : Type v
inst✝² : DecidableEq n
inst✝¹ : Fintype n
α β : Type v
inst✝ : DecidableEq α
M : Matrix n n R
c : n ≃ n
hc : c ∈ univ.erase (Equiv.refl n)
\u22a2 ∏ i : n, M.charmatrix (c i) i ∈ degreeLT R (Fintype.card n - 1)
-- Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean
R : Type u
inst✝³ : CommRing R
n G : Type v
inst✝² : DecidableEq n
inst✝¹ : Fintype n
α β : Type v
inst✝ : DecidableEq α
M : Matrix n n R
c : n ≃ n
hc : c ∈ univ.erase (Equiv.refl n)
\u22a2 (∏ i : n, M.charmatrix (c i) i).degree < ↑(Fintype.card n - 1)
-- Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean
R : Type u
inst✝³ : CommRing R
n G : Type v
inst✝² : DecidableEq n
inst✝¹ : Fintype n
α β : Type v
inst✝ : DecidableEq α
M : Matrix n n R
c : n ≃ n
hc : c ∈ univ.erase (Equiv.refl n)
\u22a2 ↑(∏ i : n, M.charmatrix (c i) i).natDegree < ↑(Fintype.card n - 1)
-- Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean
R : Type u
inst✝³ : CommRing R
n G : Type v
inst✝² : DecidableEq n
inst✝¹ : Fintype n
α β : Type v
inst✝ : DecidableEq α
M : Matrix n n R
c : n ≃ n
hc : c ∈ univ.erase (Equiv.refl n)
\u22a2 (∏ i : n, M.charmatrix (c i) i).natDegree < Fintype.card n - 1
-- Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean
R : Type u
inst✝³ : CommRing R
n G : Type v
inst✝² : DecidableEq n
inst✝¹ : Fintype n
α β : Type v
inst✝ : DecidableEq α
M : Matrix n n R
c : n ≃ n
hc : c ∈ univ.erase (Equiv.refl n)
\u22a2 (∏ i : n, M.charmatrix (c i) i).natDegree ≤ (filter (fun x => c x = x) univ).card
-- Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean
R : Type u
inst✝³ : CommRing R
n G : Type v
inst✝² : DecidableEq n
inst✝¹ : Fintype n
α β : Type v
inst✝ : DecidableEq α
M : Matrix n n R
c : n ≃ n
hc : c ∈ univ.erase (Equiv.refl n)
\u22a2 ∑ i : n, (M.charmatrix (c i) i).natDegree ≤ (filter (fun x => c x = x) univ).card
-- Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean
R : Type u
inst✝³ : CommRing R
n G : Type v
inst✝² : DecidableEq n
inst✝¹ : Fintype n
α β : Type v
inst✝ : DecidableEq α
M : Matrix n n R
c : n ≃ n
hc : c ∈ univ.erase (Equiv.refl n)
\u22a2 ∑ i : n, (M.charmatrix (c i) i).natDegree ≤ ∑ x ∈ filter (fun x => c x = x) univ, 1
-- Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean
R : Type u
inst✝³ : CommRing R
n G : Type v
inst✝² : DecidableEq n
inst✝¹ : Fintype n
α β : Type v
inst✝ : DecidableEq α
M : Matrix n n R
c : n ≃ n
hc : c ∈ univ.erase (Equiv.refl n)
\u22a2 ∑ i : n, (M.charmatrix (c i) i).natDegree ≤ ∑ a : n, if c a = a then 1 else 0
-- Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean
R : Type u
inst✝³ : CommRing R
n G : Type v
inst✝² : DecidableEq n
inst✝¹ : Fintype n
α β : Type v
inst✝ : DecidableEq α
M : Matrix n n R
c : n ≃ n
hc : c ∈ univ.erase (Equiv.refl n)
\u22a2 ∀ i ∈ univ, (M.charmatrix (c i) i).natDegree ≤ if c i = i then 1 else 0
-- Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean
R : Type u
inst✝³ : CommRing R
n G : Type v
inst✝² : DecidableEq n
inst✝¹ : Fintype n
α β : Type v
inst✝ : DecidableEq α
M : Matrix n n R
c : n ≃ n
hc : c ∈ univ.erase (Equiv.refl n)
i✝ : n
a✝ : i✝ ∈ univ
\u22a2 (M.charmatrix (c i✝) i✝).natDegree ≤ if c i✝ = i✝ then 1 else 0
-- Mathlib/ModelTheory/Substructures.lean
L : Language
M : Type w
N : Type u_1
P : Type u_2
inst✝² : L.Structure M
inst✝¹ : L.Structure N
inst✝ : L.Structure P
S : L.Substructure M
f : M ≃[L] N
n : N
\u22a2 n ∈ f.toHom.range ↔ n ∈ ⊤
-- Mathlib/ModelTheory/Substructures.lean
L : Language
M : Type w
N : Type u_1
P : Type u_2
inst✝² : L.Structure M
inst✝¹ : L.Structure N
inst✝ : L.Structure P
S : L.Substructure M
f : M ≃[L] N
n : N
\u22a2 ∃ y, f y = n
-- Mathlib/LinearAlgebra/Matrix/ToLin.lean
R : Type u_1
inst✝⁹ : CommSemiring R
l : Type u_2
m : Type u_3
n : Type u_4
inst✝⁸ : Fintype n
inst✝⁷ : Finite m
inst✝⁶ : DecidableEq n
M₁ : Type u_5
M₂ : Type u_6
inst✝⁵ : AddCommMonoid M₁
inst✝⁴ : AddCommMonoid M₂
inst✝³ : Module R M₁
inst✝² : Module R M₂
v₁ : Basis n R M₁
v₂ : Basis m R M₂
inst✝¹ : Fintype l
inst✝ : DecidableEq l
b : Basis l R M₁
b' : Basis l R M₂
i j : l
\u22a2 (toMatrix b' b) (↑(b'.equiv b (Equiv.refl l))) i j = 1 i j
-- Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean
p n : ℕ+
A : Type w
B : Type z
K : Type u
L : Type v
C : Type w
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra A B
inst✝⁵ : IsCyclotomicExtension {n} A B
inst✝⁴ : CommRing L
ζ : L
hζ : IsPrimitiveRoot ζ ↑n
inst✝³ : Field K
inst✝² : Algebra K L
inst✝¹ : IsDomain L
inst✝ : IsCyclotomicExtension {n} K L
hn : n ≠ 2
hirr : Irreducible (cyclotomic (↑n) K)
\u22a2 (Algebra.norm K) ζ = 1
-- Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean
p n : ℕ+
A : Type w
B : Type z
K : Type u
L : Type v
C : Type w
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra A B
inst✝⁵ : IsCyclotomicExtension {n} A B
inst✝⁴ : CommRing L
ζ : L
hζ : IsPrimitiveRoot ζ ↑n
inst✝³ : Field K
inst✝² : Algebra K L
inst✝¹ : IsDomain L
inst✝ : IsCyclotomicExtension {n} K L
hn : n ≠ 2
hirr : Irreducible (cyclotomic (↑n) K)
this : NeZero ↑↑n
\u22a2 (Algebra.norm K) ζ = 1
-- Mathlib/FieldTheory/Perfect.lean
R : Type u_1
inst✝³ : CommRing R
inst✝² : IsDomain R
p n : ℕ
inst✝¹ : ExpChar R p
f : R[X]
inst✝ : PerfectRing R p
\u22a2 ((expand R (p ^ n)) f).roots = p ^ n • Multiset.map (⇑(iterateFrobeniusEquiv R p n).symm) f.roots
-- Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : o.oangle x y = ↑(π / 2)
\u22a2 (o.oangle (x + y) y).sin * ‖x + y‖ = ‖x‖
-- Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : (-o).oangle y x = ↑(π / 2)
\u22a2 ((-o).oangle y (x + y)).sin * ‖x + y‖ = ‖x‖
-- Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : (-o).oangle y x = ↑(π / 2)
\u22a2 ((-o).oangle y (y + x)).sin * ‖y + x‖ = ‖x‖
-- Mathlib/GroupTheory/MonoidLocalization/Basic.lean
M : Type u_1
inst✝² : CommMonoid M
S : Submonoid M
N : Type u_2
inst✝¹ : CommMonoid N
P : Type u_3
inst✝ : CommMonoid P
f : S.LocalizationMap N
g : M →* P
hg : ∀ (y : ↥S), IsUnit (g ↑y)
z : N
\u22a2 (f.lift hg) z * g ↑(f.sec z).2 = g (f.sec z).1
-- Mathlib/Algebra/Ring/Prod.lean
α : Type u_1
β : Type u_2
R : Type u_3
R' : Type u_4
S : Type u_5
S' : Type u_6
T : Type u_7
T' : Type u_8
inst✝⁴ : NonAssocSemiring R
inst✝³ : NonAssocSemiring S
inst✝² : NonAssocSemiring R'
inst✝¹ : NonAssocSemiring S'
inst✝ : Subsingleton S
fst✝ : S
snd✝ : R
\u22a2 (fun x => (0, x)) (fst✝, snd✝).2 = (fst✝, snd✝)
-- Mathlib/Topology/MetricSpace/Lipschitz.lean
α : Type u
β : Type v
γ : Type w
ι : Type x
inst✝ : PseudoEMetricSpace α
f g : α → ℝ
Kf Kg : ℝ≥0
hf : LipschitzWith Kf f
a : ℝ
\u22a2 LipschitzWith Kf fun x => max a (f x)
-- Mathlib/CategoryTheory/Abelian/Basic.lean
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
\u22a2 kernel.ι f ≫ factorThruImage f = 0
-- Mathlib/CategoryTheory/Abelian/Basic.lean
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
\u22a2 coequalizer.π (kernel.ι f) 0 ≫ (coimageIsoImage' f).hom = coequalizer.π (kernel.ι f) 0 ≫ cokernel.desc (kernel.ι f) (factorThruImage f) ⋯
-- Mathlib/LinearAlgebra/Dual.lean
K : Type uK
inst✝⁴ : Field K
V₁ : Type uV₁
V₂ : Type uV₂
inst✝³ : AddCommGroup V₁
inst✝² : Module K V₁
inst✝¹ : AddCommGroup V₂
inst✝ : Module K V₂
f : V₁ →ₗ[K] V₂
\u22a2 Function.Surjective ⇑f.dualMap ↔ Function.Injective ⇑f
-- Mathlib/Topology/Order/OrderClosed.lean
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderClosedTopology α
f g : β → α
inst✝ : TopologicalSpace β
hf : Continuous f
hg : Continuous g
\u22a2 Continuous fun b => min (f b) (g b)
-- Mathlib/Topology/Order/OrderClosed.lean
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderClosedTopology α
f g : β → α
inst✝ : TopologicalSpace β
hf : Continuous f
hg : Continuous g
\u22a2 Continuous fun b => if f b ≤ g b then f b else g b
-- Mathlib/Combinatorics/Quiver/SingleObj.lean
α : Type u_1
β : Type u_2
γ : Type u_3
f : SingleObj α ⥤q SingleObj β
g : SingleObj β ⥤q SingleObj γ
\u22a2 toPrefunctor.symm (f ⋙q g) = toPrefunctor.symm g ∘ toPrefunctor.symm f
-- Mathlib/Analysis/Normed/Algebra/Spectrum.lean
𝕜 : Type u_1
A✝ : Type u_2
A : Type u_3
inst✝¹ : Ring A
inst✝ : Algebra ℝ A
a : A
t : ℝ≥0
ht : spectralRadius ℝ a ≤ ↑t
\u22a2 SpectrumRestricts a ⇑ContinuousMap.realToNNReal ↔ spectralRadius ℝ ((algebraMap ℝ A) ↑t - a) ≤ ↑t
-- Mathlib/Analysis/Normed/Algebra/Spectrum.lean
𝕜 : Type u_1
A✝ : Type u_2
A : Type u_3
inst✝¹ : Ring A
inst✝ : Algebra ℝ A
a : A
t : ℝ≥0
ht : spectralRadius ℝ a ≤ ↑t
this : spectrum ℝ a ⊆ Set.Icc (-↑t) ↑t
\u22a2 SpectrumRestricts a ⇑ContinuousMap.realToNNReal ↔ spectralRadius ℝ ((algebraMap ℝ A) ↑t - a) ≤ ↑t
-- Mathlib/Analysis/Normed/Algebra/Spectrum.lean
𝕜 : Type u_1
A✝ : Type u_2
A : Type u_3
inst✝¹ : Ring A
inst✝ : Algebra ℝ A
a : A
t : ℝ≥0
ht : spectralRadius ℝ a ≤ ↑t
this : spectrum ℝ a ⊆ Set.Icc (-↑t) ↑t
\u22a2 (∀ x ∈ spectrum ℝ a, 0 ≤ x) ↔ spectralRadius ℝ ((algebraMap ℝ A) ↑t - a) ≤ ↑t
-- Mathlib/Analysis/Normed/Algebra/Spectrum.lean
𝕜 : Type u_1
A✝ : Type u_2
A : Type u_3
inst✝¹ : Ring A
inst✝ : Algebra ℝ A
a : A
t : ℝ≥0
ht : spectralRadius ℝ a ≤ ↑t
this : spectrum ℝ a ⊆ Set.Icc (-↑t) ↑t
h : ∀ x ∈ spectrum ℝ a, 0 ≤ x
x : ℝ
hx : x ∈ spectrum ℝ ((algebraMap ℝ A) ↑t - a)
\u22a2 ↑‖x‖₊ ≤ ↑t
-- Mathlib/Analysis/Normed/Algebra/Spectrum.lean
𝕜 : Type u_1
A✝ : Type u_2
A : Type u_3
inst✝¹ : Ring A
inst✝ : Algebra ℝ A
a : A
t : ℝ≥0
ht : spectralRadius ℝ a ≤ ↑t
this : spectrum ℝ a ⊆ Set.Icc (-↑t) ↑t
h : spectralRadius ℝ ((algebraMap ℝ A) ↑t - a) ≤ ↑t
\u22a2 ∀ x ∈ spectrum ℝ a, 0 ≤ x
-- Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
a b : A
ha : a ∈ carrier f_deg q
hb : b ∈ carrier f_deg q
\u22a2 a + b ∈ carrier f_deg q
-- Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
a b : A
ha : a ∈ carrier f_deg q
hb : b ∈ carrier f_deg q
i : ℕ
\u22a2 HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) (a + b) ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } * HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) (a + b) ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } ∈ q.asIdeal
-- Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
a b : A
ha : a ∈ carrier f_deg q
hb : b ∈ carrier f_deg q
i : ℕ
\u22a2 HomogeneousLocalization.mk { deg := { deg := m * i, num := ⟨(proj 𝒜 i) (a + b) ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ }.deg + { deg := m * i, num := ⟨(proj 𝒜 i) (a + b) ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ }.deg, num := ⟨↑{ deg := m * i, num := ⟨(proj 𝒜 i) (a + b) ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ }.num * ↑{ deg := m * i, num := ⟨(proj 𝒜 i) (a + b) ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ }.num, ⋯⟩, den := ⟨↑{ deg := m * i, num := ⟨(proj 𝒜 i) (a + b) ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ }.den * ↑{ deg := m * i, num := ⟨(proj 𝒜 i) (a + b) ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ }.den, ⋯⟩, den_mem := ⋯ } ∈ q.asIdeal
-- Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
a b : A
ha : a ∈ carrier f_deg q
hb : b ∈ carrier f_deg q
i : ℕ
\u22a2 HomogeneousLocalization.mk { deg := m * i + m * i, num := ⟨(proj 𝒜 i) (a + b) ^ m * (proj 𝒜 i) (a + b) ^ m, ⋯⟩, den := ⟨f ^ i * f ^ i, ⋯⟩, den_mem := ⋯ } ∈ q.asIdeal
-- Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
a b : A
ha : a ∈ carrier f_deg q
hb : b ∈ carrier f_deg q
i : ℕ
\u22a2 HomogeneousLocalization.mk { deg := m * i + m * i, num := ⟨∑ x ∈ range (m + m + 1), (m + m).choose x • ((proj 𝒜 i) a ^ x * (proj 𝒜 i) b ^ (m + m - x)), ⋯⟩, den := ⟨f ^ (i + i), ⋯⟩, den_mem := ⋯ } ∈ q.asIdeal
-- Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
a b : A
ha : a ∈ carrier f_deg q
hb : b ∈ carrier f_deg q
i : ℕ
g : ℕ → A⁰_ f := fun j => (m + m).choose j • if h2 : m + m < j then 0 else if h1 : j ≤ m then HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ j * (proj 𝒜 i) b ^ (m - j), ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } * HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) b ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } else HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } * HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ (j - m) * (proj 𝒜 i) b ^ (m + m - j), ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ }
\u22a2 HomogeneousLocalization.mk { deg := m * i + m * i, num := ⟨∑ x ∈ range (m + m + 1), (m + m).choose x • ((proj 𝒜 i) a ^ x * (proj 𝒜 i) b ^ (m + m - x)), ⋯⟩, den := ⟨f ^ (i + i), ⋯⟩, den_mem := ⋯ } ∈ q.asIdeal
-- Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
a b : A
ha : a ∈ carrier f_deg q
hb : b ∈ carrier f_deg q
i j : ℕ
h2 : ¬m + m < j
h1 : j ≤ m
\u22a2 (proj 𝒜 i) a ^ j * (proj 𝒜 i) b ^ (m - j) ∈ 𝒜 (m * i)
-- Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
a b : A
ha : a ∈ carrier f_deg q
hb : b ∈ carrier f_deg q
i j : ℕ
h2 : ¬m + m < j
h1 : ¬j ≤ m
l : A⁰_ f := HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ }
\u22a2 (proj 𝒜 i) a ^ (j - m) * (proj 𝒜 i) b ^ (m + m - j) ∈ 𝒜 (m * i)
-- Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
a b : A
ha : a ∈ carrier f_deg q
hb : b ∈ carrier f_deg q
i : ℕ
g : ℕ → A⁰_ f := fun j => (m + m).choose j • if h2 : m + m < j then 0 else if h1 : j ≤ m then HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ j * (proj 𝒜 i) b ^ (m - j), ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } * HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) b ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } else HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } * HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ (j - m) * (proj 𝒜 i) b ^ (m + m - j), ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ }
\u22a2 HomogeneousLocalization.mk { deg := m * i + m * i, num := ⟨∑ x ∈ range (m + m + 1), (m + m).choose x • ((proj 𝒜 i) a ^ x * (proj 𝒜 i) b ^ (m + m - x)), ⋯⟩, den := ⟨f ^ (i + i), ⋯⟩, den_mem := ⋯ } = ∑ i ∈ range (m + m + 1), g i
-- Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
a b : A
ha : a ∈ carrier f_deg q
hb : b ∈ carrier f_deg q
i : ℕ
g : ℕ → A⁰_ f := fun j => (m + m).choose j • if h2 : m + m < j then 0 else if h1 : j ≤ m then HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ j * (proj 𝒜 i) b ^ (m - j), ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } * HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) b ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } else HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } * HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ (j - m) * (proj 𝒜 i) b ^ (m + m - j), ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ }
\u22a2 Localization.mk ↑{ deg := m * i + m * i, num := ⟨∑ x ∈ range (m + m + 1), (m + m).choose x • ((proj 𝒜 i) a ^ x * (proj 𝒜 i) b ^ (m + m - x)), ⋯⟩, den := ⟨f ^ (i + i), ⋯⟩, den_mem := ⋯ }.num ⟨↑{ deg := m * i + m * i, num := ⟨∑ x ∈ range (m + m + 1), (m + m).choose x • ((proj 𝒜 i) a ^ x * (proj 𝒜 i) b ^ (m + m - x)), ⋯⟩, den := ⟨f ^ (i + i), ⋯⟩, den_mem := ⋯ }.den, ⋯⟩ = HomogeneousLocalization.val (∑ i ∈ range (m + m + 1), g i)
-- Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
a b : A
ha : a ∈ carrier f_deg q
hb : b ∈ carrier f_deg q
i : ℕ
g : ℕ → A⁰_ f := fun j => (m + m).choose j • if h2 : m + m < j then 0 else if h1 : j ≤ m then HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ j * (proj 𝒜 i) b ^ (m - j), ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } * HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) b ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } else HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } * HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ (j - m) * (proj 𝒜 i) b ^ (m + m - j), ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ }
\u22a2 Localization.mk ↑{ deg := m * i + m * i, num := ⟨∑ x ∈ range (m + m + 1), (m + m).choose x • ((proj 𝒜 i) a ^ x * (proj 𝒜 i) b ^ (m + m - x)), ⋯⟩, den := ⟨f ^ (i + i), ⋯⟩, den_mem := ⋯ }.num ⟨↑{ deg := m * i + m * i, num := ⟨∑ x ∈ range (m + m + 1), (m + m).choose x • ((proj 𝒜 i) a ^ x * (proj 𝒜 i) b ^ (m + m - x)), ⋯⟩, den := ⟨f ^ (i + i), ⋯⟩, den_mem := ⋯ }.den, ⋯⟩ = (algebraMap (A⁰_ f) (Localization.Away f)) (∑ i ∈ range (m + m + 1), g i)
-- Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
a b : A
ha : a ∈ carrier f_deg q
hb : b ∈ carrier f_deg q
i : ℕ
g : ℕ → A⁰_ f := fun j => (m + m).choose j • if h2 : m + m < j then 0 else if h1 : j ≤ m then HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ j * (proj 𝒜 i) b ^ (m - j), ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } * HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) b ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } else HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } * HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ (j - m) * (proj 𝒜 i) b ^ (m + m - j), ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ }
\u22a2 Localization.mk (∑ x ∈ range (m + m + 1), (m + m).choose x • ((proj 𝒜 i) a ^ x * (proj 𝒜 i) b ^ (m + m - x))) ⟨f ^ (i + i), ⋯⟩ = (algebraMap (A⁰_ f) (Localization.Away f)) (∑ i ∈ range (m + m + 1), g i)
-- Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
a b : A
ha : a ∈ carrier f_deg q
hb : b ∈ carrier f_deg q
i : ℕ
g : ℕ → A⁰_ f := fun j => (m + m).choose j • if h2 : m + m < j then 0 else if h1 : j ≤ m then HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ j * (proj 𝒜 i) b ^ (m - j), ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } * HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) b ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } else HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } * HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ (j - m) * (proj 𝒜 i) b ^ (m + m - j), ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ }
\u22a2 ∑ i_1 ∈ range (m + m + 1), Localization.mk ((m + m).choose i_1 • ((proj 𝒜 i) a ^ i_1 * (proj 𝒜 i) b ^ (m + m - i_1))) ⟨f ^ (i + i), ⋯⟩ = ∑ x ∈ range (m + m + 1), (algebraMap (A⁰_ f) (Localization.Away f)) (g x)