ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/RingTheory/LocalProperties.lean
R✝ S✝ : Type u
inst✝⁷ : CommRing R✝
inst✝⁶ : CommRing S✝
M : Submonoid R✝
N : Submonoid S✝
R' S' : Type u
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
f✝ : R✝ →+* S✝
inst✝³ : Algebra R✝ R'
inst✝² : Algebra S✝ S'
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset R
hs : Ideal.span ↑s = ⊤
H : ∀ (r : { x // x ∈ s }), (Localization.awayMap f ↑r).Finite
this✝ : Algebra R S := f.toAlgebra
this : (r : { x // x ∈ s }) → Algebra (Localization.Away ↑r) (Localization.Away (f ↑r)) := fun r => (Localization.awayMap f ↑r).toAlgebra

⊢ f.Finite
-- Mathlib/RingTheory/LocalProperties.lean
R✝ S✝ : Type u
inst✝⁷ : CommRing R✝
inst✝⁶ : CommRing S✝
M : Submonoid R✝
N : Submonoid S✝
R' S' : Type u
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
f✝ : R✝ →+* S✝
inst✝³ : Algebra R✝ R'
inst✝² : Algebra S✝ S'
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset R
hs : Ideal.span ↑s = ⊤
H : ∀ (r : { x // x ∈ s }), (Localization.awayMap f ↑r).Finite
this✝¹ : Algebra R S := f.toAlgebra
this✝ : (r : { x // x ∈ s }) → Algebra (Localization.Away ↑r) (Localization.Away (f ↑r)) := fun r => (Localization.awayMap f ↑r).toAlgebra
this : ∀ (r : { x // x ∈ s }), IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers ↑r)) (Localization.Away (f ↑r))

⊢ f.Finite
-- Mathlib/RingTheory/LocalProperties.lean
R✝ S✝ : Type u
inst✝⁷ : CommRing R✝
inst✝⁶ : CommRing S✝
M : Submonoid R✝
N : Submonoid S✝
R' S' : Type u
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
f✝ : R✝ →+* S✝
inst✝³ : Algebra R✝ R'
inst✝² : Algebra S✝ S'
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset R
hs : Ideal.span ↑s = ⊤
H : ∀ (r : { x // x ∈ s }), (Localization.awayMap f ↑r).Finite
this✝² : Algebra R S := f.toAlgebra
this✝¹ : (r : { x // x ∈ s }) → Algebra (Localization.Away ↑r) (Localization.Away (f ↑r)) := fun r => (Localization.awayMap f ↑r).toAlgebra
this✝ : ∀ (r : { x // x ∈ s }), IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers ↑r)) (Localization.Away (f ↑r))
this : ∀ (r : { x // x ∈ s }), IsScalarTower R (Localization.Away ↑r) (Localization.Away (f ↑r))

⊢ f.Finite
-- Mathlib/RingTheory/LocalProperties.lean
R✝ S✝ : Type u
inst✝⁷ : CommRing R✝
inst✝⁶ : CommRing S✝
M : Submonoid R✝
N : Submonoid S✝
R' S' : Type u
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
f✝ : R✝ →+* S✝
inst✝³ : Algebra R✝ R'
inst✝² : Algebra S✝ S'
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset R
hs : Ideal.span ↑s = ⊤
H : ∀ (r : { x // x ∈ s }), (Localization.awayMap f ↑r).Finite
this✝² : Algebra R S := f.toAlgebra
this✝¹ : (r : { x // x ∈ s }) → Algebra (Localization.Away ↑r) (Localization.Away (f ↑r)) := fun r => (Localization.awayMap f ↑r).toAlgebra
this✝ : ∀ (r : { x // x ∈ s }), IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers ↑r)) (Localization.Away (f ↑r))
this : ∀ (r : { x // x ∈ s }), IsScalarTower R (Localization.Away ↑r) (Localization.Away (f ↑r))

⊢ ⊤.FG
-- Mathlib/RingTheory/LocalProperties.lean
R✝ S✝ : Type u
inst✝⁷ : CommRing R✝
inst✝⁶ : CommRing S✝
M : Submonoid R✝
N : Submonoid S✝
R' S' : Type u
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
f✝ : R✝ →+* S✝
inst✝³ : Algebra R✝ R'
inst✝² : Algebra S✝ S'
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset R
hs : Ideal.span ↑s = ⊤
this✝² : Algebra R S := f.toAlgebra
this✝¹ : (r : { x // x ∈ s }) → Algebra (Localization.Away ↑r) (Localization.Away (f ↑r)) := fun r => (Localization.awayMap f ↑r).toAlgebra
this✝ : ∀ (r : { x // x ∈ s }), IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers ↑r)) (Localization.Away (f ↑r))
this : ∀ (r : { x // x ∈ s }), IsScalarTower R (Localization.Away ↑r) (Localization.Away (f ↑r))
H : ∀ (r : { x // x ∈ s }), ⊤.FG

⊢ ⊤.FG
-- Mathlib/RingTheory/LocalProperties.lean
R✝ S✝ : Type u
inst✝⁷ : CommRing R✝
inst✝⁶ : CommRing S✝
M : Submonoid R✝
N : Submonoid S✝
R' S' : Type u
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
f✝ : R✝ →+* S✝
inst✝³ : Algebra R✝ R'
inst✝² : Algebra S✝ S'
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset R
hs : Ideal.span ↑s = ⊤
this✝² : Algebra R S := f.toAlgebra
this✝¹ : (r : { x // x ∈ s }) → Algebra (Localization.Away ↑r) (Localization.Away (f ↑r)) := fun r => (Localization.awayMap f ↑r).toAlgebra
this✝ : ∀ (r : { x // x ∈ s }), IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers ↑r)) (Localization.Away (f ↑r))
this : ∀ (r : { x // x ∈ s }), IsScalarTower R (Localization.Away ↑r) (Localization.Away (f ↑r))
s₁ : (r : { x // x ∈ s }) → Finset (Localization.Away (f ↑r))
s₂ : ∀ (r : { x // x ∈ s }), Submodule.span (Localization.Away ↑r) ↑(s₁ r) = ⊤
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (f ↑x)) (s₁ x)

⊢ ⊤.FG
-- Mathlib/RingTheory/LocalProperties.lean
R✝ S✝ : Type u
inst✝⁷ : CommRing R✝
inst✝⁶ : CommRing S✝
M : Submonoid R✝
N : Submonoid S✝
R' S' : Type u
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
f✝ : R✝ →+* S✝
inst✝³ : Algebra R✝ R'
inst✝² : Algebra S✝ S'
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset R
hs : Ideal.span ↑s = ⊤
this✝² : Algebra R S := f.toAlgebra
this✝¹ : (r : { x // x ∈ s }) → Algebra (Localization.Away ↑r) (Localization.Away (f ↑r)) := fun r => (Localization.awayMap f ↑r).toAlgebra
this✝ : ∀ (r : { x // x ∈ s }), IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers ↑r)) (Localization.Away (f ↑r))
this : ∀ (r : { x // x ∈ s }), IsScalarTower R (Localization.Away ↑r) (Localization.Away (f ↑r))
s₁ : (r : { x // x ∈ s }) → Finset (Localization.Away (f ↑r))
s₂ : ∀ (r : { x // x ∈ s }), Submodule.span (Localization.Away ↑r) ↑(s₁ r) = ⊤
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (f ↑x)) (s₁ x)

⊢ Submodule.span R ↑(s.attach.biUnion sf) = ⊤
-- Mathlib/RingTheory/LocalProperties.lean
R✝ S✝ : Type u
inst✝⁷ : CommRing R✝
inst✝⁶ : CommRing S✝
M : Submonoid R✝
N : Submonoid S✝
R' S' : Type u
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
f✝ : R✝ →+* S✝
inst✝³ : Algebra R✝ R'
inst✝² : Algebra S✝ S'
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset R
hs : Ideal.span ↑s = ⊤
this✝² : Algebra R S := f.toAlgebra
this✝¹ : (r : { x // x ∈ s }) → Algebra (Localization.Away ↑r) (Localization.Away (f ↑r)) := fun r => (Localization.awayMap f ↑r).toAlgebra
this✝ : ∀ (r : { x // x ∈ s }), IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers ↑r)) (Localization.Away (f ↑r))
this : ∀ (r : { x // x ∈ s }), IsScalarTower R (Localization.Away ↑r) (Localization.Away (f ↑r))
s₁ : (r : { x // x ∈ s }) → Finset (Localization.Away (f ↑r))
s₂ : ∀ (r : { x // x ∈ s }), Submodule.span (Localization.Away ↑r) ↑(s₁ r) = ⊤
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (f ↑x)) (s₁ x)

⊢ ⊤ ≤ ⨆ x, Submodule.span R ↑(sf x)
-- Mathlib/RingTheory/LocalProperties.lean
R✝ S✝ : Type u
inst✝⁷ : CommRing R✝
inst✝⁶ : CommRing S✝
M : Submonoid R✝
N : Submonoid S✝
R' S' : Type u
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
f✝ : R✝ →+* S✝
inst✝³ : Algebra R✝ R'
inst✝² : Algebra S✝ S'
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset R
hs : Ideal.span ↑s = ⊤
this✝² : Algebra R S := f.toAlgebra
this✝¹ : (r : { x // x ∈ s }) → Algebra (Localization.Away ↑r) (Localization.Away (f ↑r)) := fun r => (Localization.awayMap f ↑r).toAlgebra
this✝ : ∀ (r : { x // x ∈ s }), IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers ↑r)) (Localization.Away (f ↑r))
this : ∀ (r : { x // x ∈ s }), IsScalarTower R (Localization.Away ↑r) (Localization.Away (f ↑r))
s₁ : (r : { x // x ∈ s }) → Finset (Localization.Away (f ↑r))
s₂ : ∀ (r : { x // x ∈ s }), Submodule.span (Localization.Away ↑r) ↑(s₁ r) = ⊤
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (f ↑x)) (s₁ x)
x : S

⊢ x ∈ ⨆ x, Submodule.span R ↑(sf x)
-- Mathlib/RingTheory/LocalProperties.lean
R✝ S✝ : Type u
inst✝⁷ : CommRing R✝
inst✝⁶ : CommRing S✝
M : Submonoid R✝
N : Submonoid S✝
R' S' : Type u
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
f✝ : R✝ →+* S✝
inst✝³ : Algebra R✝ R'
inst✝² : Algebra S✝ S'
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset R
hs : Ideal.span ↑s = ⊤
this✝² : Algebra R S := f.toAlgebra
this✝¹ : (r : { x // x ∈ s }) → Algebra (Localization.Away ↑r) (Localization.Away (f ↑r)) := fun r => (Localization.awayMap f ↑r).toAlgebra
this✝ : ∀ (r : { x // x ∈ s }), IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers ↑r)) (Localization.Away (f ↑r))
this : ∀ (r : { x // x ∈ s }), IsScalarTower R (Localization.Away ↑r) (Localization.Away (f ↑r))
s₁ : (r : { x // x ∈ s }) → Finset (Localization.Away (f ↑r))
s₂ : ∀ (r : { x // x ∈ s }), Submodule.span (Localization.Away ↑r) ↑(s₁ r) = ⊤
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (f ↑x)) (s₁ x)
x : S

⊢ ∀ (r : ↑↑s), ∃ n, ↑r ^ n • x ∈ ⨆ x, Submodule.span R ↑(sf x)
-- Mathlib/RingTheory/LocalProperties.lean
R✝ S✝ : Type u
inst✝⁷ : CommRing R✝
inst✝⁶ : CommRing S✝
M : Submonoid R✝
N : Submonoid S✝
R' S' : Type u
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
f✝ : R✝ →+* S✝
inst✝³ : Algebra R✝ R'
inst✝² : Algebra S✝ S'
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset R
hs : Ideal.span ↑s = ⊤
this✝² : Algebra R S := f.toAlgebra
this✝¹ : (r : { x // x ∈ s }) → Algebra (Localization.Away ↑r) (Localization.Away (f ↑r)) := fun r => (Localization.awayMap f ↑r).toAlgebra
this✝ : ∀ (r : { x // x ∈ s }), IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers ↑r)) (Localization.Away (f ↑r))
this : ∀ (r : { x // x ∈ s }), IsScalarTower R (Localization.Away ↑r) (Localization.Away (f ↑r))
s₁ : (r : { x // x ∈ s }) → Finset (Localization.Away (f ↑r))
s₂ : ∀ (r : { x // x ∈ s }), Submodule.span (Localization.Away ↑r) ↑(s₁ r) = ⊤
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (f ↑x)) (s₁ x)
x : S
r : ↑↑s

⊢ ∃ n, ↑r ^ n • x ∈ ⨆ x, Submodule.span R ↑(sf x)
-- Mathlib/RingTheory/LocalProperties.lean
R✝ S✝ : Type u
inst✝⁷ : CommRing R✝
inst✝⁶ : CommRing S✝
M : Submonoid R✝
N : Submonoid S✝
R' S' : Type u
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
f✝ : R✝ →+* S✝
inst✝³ : Algebra R✝ R'
inst✝² : Algebra S✝ S'
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset R
hs : Ideal.span ↑s = ⊤
this✝² : Algebra R S := f.toAlgebra
this✝¹ : (r : { x // x ∈ s }) → Algebra (Localization.Away ↑r) (Localization.Away (f ↑r)) := fun r => (Localization.awayMap f ↑r).toAlgebra
this✝ : ∀ (r : { x // x ∈ s }), IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers ↑r)) (Localization.Away (f ↑r))
this : ∀ (r : { x // x ∈ s }), IsScalarTower R (Localization.Away ↑r) (Localization.Away (f ↑r))
s₁ : (r : { x // x ∈ s }) → Finset (Localization.Away (f ↑r))
s₂ : ∀ (r : { x // x ∈ s }), Submodule.span (Localization.Away ↑r) ↑(s₁ r) = ⊤
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (f ↑x)) (s₁ x)
x : S
r : ↑↑s
n₁ : ℕ
hn₁ : ⟨(fun x => ↑r ^ x) n₁, ⋯⟩ • (algebraMap S (Localization.Away (f ↑r))) x ∈ Submodule.span R ↑(s₁ r)

⊢ ∃ n, ↑r ^ n • x ∈ ⨆ x, Submodule.span R ↑(sf x)
-- Mathlib/RingTheory/LocalProperties.lean
R✝ S✝ : Type u
inst✝⁷ : CommRing R✝
inst✝⁶ : CommRing S✝
M : Submonoid R✝
N : Submonoid S✝
R' S' : Type u
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
f✝ : R✝ →+* S✝
inst✝³ : Algebra R✝ R'
inst✝² : Algebra S✝ S'
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset R
hs : Ideal.span ↑s = ⊤
this✝² : Algebra R S := f.toAlgebra
this✝¹ : (r : { x // x ∈ s }) → Algebra (Localization.Away ↑r) (Localization.Away (f ↑r)) := fun r => (Localization.awayMap f ↑r).toAlgebra
this✝ : ∀ (r : { x // x ∈ s }), IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers ↑r)) (Localization.Away (f ↑r))
this : ∀ (r : { x // x ∈ s }), IsScalarTower R (Localization.Away ↑r) (Localization.Away (f ↑r))
s₁ : (r : { x // x ∈ s }) → Finset (Localization.Away (f ↑r))
s₂ : ∀ (r : { x // x ∈ s }), Submodule.span (Localization.Away ↑r) ↑(s₁ r) = ⊤
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (f ↑x)) (s₁ x)
x : S
r : ↑↑s
n₁ : ℕ
hn₁ : ⟨↑r ^ n₁, ⋯⟩ • (algebraMap S (Localization.Away (f ↑r))) x ∈ Submodule.span R ↑(s₁ r)

⊢ ∃ n, ↑r ^ n • x ∈ ⨆ x, Submodule.span R ↑(sf x)
-- Mathlib/RingTheory/LocalProperties.lean
R✝ S✝ : Type u
inst✝⁷ : CommRing R✝
inst✝⁶ : CommRing S✝
M : Submonoid R✝
N : Submonoid S✝
R' S' : Type u
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
f✝ : R✝ →+* S✝
inst✝³ : Algebra R✝ R'
inst✝² : Algebra S✝ S'
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset R
hs : Ideal.span ↑s = ⊤
this✝² : Algebra R S := f.toAlgebra
this✝¹ : (r : { x // x ∈ s }) → Algebra (Localization.Away ↑r) (Localization.Away (f ↑r)) := fun r => (Localization.awayMap f ↑r).toAlgebra
this✝ : ∀ (r : { x // x ∈ s }), IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers ↑r)) (Localization.Away (f ↑r))
this : ∀ (r : { x // x ∈ s }), IsScalarTower R (Localization.Away ↑r) (Localization.Away (f ↑r))
s₁ : (r : { x // x ∈ s }) → Finset (Localization.Away (f ↑r))
s₂ : ∀ (r : { x // x ∈ s }), Submodule.span (Localization.Away ↑r) ↑(s₁ r) = ⊤
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (f ↑x)) (s₁ x)
x : S
r : ↑↑s
n₁ : ℕ
hn₁ : (algebraMap S (Localization.Away (f ↑r))) ((algebraMap R S) ↑⟨↑r ^ n₁, ⋯⟩ * x) ∈ Submodule.span R ↑(s₁ r)

⊢ ∃ n, ↑r ^ n • x ∈ ⨆ x, Submodule.span R ↑(sf x)
-- Mathlib/RingTheory/LocalProperties.lean
R✝ S✝ : Type u
inst✝⁷ : CommRing R✝
inst✝⁶ : CommRing S✝
M : Submonoid R✝
N : Submonoid S✝
R' S' : Type u
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
f✝ : R✝ →+* S✝
inst✝³ : Algebra R✝ R'
inst✝² : Algebra S✝ S'
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset R
hs : Ideal.span ↑s = ⊤
this✝² : Algebra R S := f.toAlgebra
this✝¹ : (r : { x // x ∈ s }) → Algebra (Localization.Away ↑r) (Localization.Away (f ↑r)) := fun r => (Localization.awayMap f ↑r).toAlgebra
this✝ : ∀ (r : { x // x ∈ s }), IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers ↑r)) (Localization.Away (f ↑r))
this : ∀ (r : { x // x ∈ s }), IsScalarTower R (Localization.Away ↑r) (Localization.Away (f ↑r))
s₁ : (r : { x // x ∈ s }) → Finset (Localization.Away (f ↑r))
s₂ : ∀ (r : { x // x ∈ s }), Submodule.span (Localization.Away ↑r) ↑(s₁ r) = ⊤
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (f ↑x)) (s₁ x)
x : S
r : ↑↑s
n₁ : ℕ
hn₁ : (algebraMap S (Localization.Away (f ↑r))) ((algebraMap R S) ↑⟨↑r ^ n₁, ⋯⟩ * x) ∈ Submodule.span R ↑(s₁ r)
n₂ : ℕ
hn₂ : ⟨(fun x => ↑r ^ x) n₂, ⋯⟩ • ((algebraMap R S) ↑⟨↑r ^ n₁, ⋯⟩ * x) ∈ Submodule.span R ↑(IsLocalization.finsetIntegerMultiple (Submonoid.map (algebraMap R S) (Submonoid.powers ↑r)) (s₁ r))

⊢ ∃ n, ↑r ^ n • x ∈ ⨆ x, Submodule.span R ↑(sf x)
-- Mathlib/RingTheory/LocalProperties.lean
R✝ S✝ : Type u
inst✝⁷ : CommRing R✝
inst✝⁶ : CommRing S✝
M : Submonoid R✝
N : Submonoid S✝
R' S' : Type u
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
f✝ : R✝ →+* S✝
inst✝³ : Algebra R✝ R'
inst✝² : Algebra S✝ S'
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset R
hs : Ideal.span ↑s = ⊤
this✝² : Algebra R S := f.toAlgebra
this✝¹ : (r : { x // x ∈ s }) → Algebra (Localization.Away ↑r) (Localization.Away (f ↑r)) := fun r => (Localization.awayMap f ↑r).toAlgebra
this✝ : ∀ (r : { x // x ∈ s }), IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers ↑r)) (Localization.Away (f ↑r))
this : ∀ (r : { x // x ∈ s }), IsScalarTower R (Localization.Away ↑r) (Localization.Away (f ↑r))
s₁ : (r : { x // x ∈ s }) → Finset (Localization.Away (f ↑r))
s₂ : ∀ (r : { x // x ∈ s }), Submodule.span (Localization.Away ↑r) ↑(s₁ r) = ⊤
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (f ↑x)) (s₁ x)
x : S
r : ↑↑s
n₁ : ℕ
hn₁ : (algebraMap S (Localization.Away (f ↑r))) ((algebraMap R S) ↑⟨↑r ^ n₁, ⋯⟩ * x) ∈ Submodule.span R ↑(s₁ r)
n₂ : ℕ
hn₂ : ↑r ^ (n₂ + n₁) • x ∈ Submodule.span R ↑(IsLocalization.finsetIntegerMultiple (Submonoid.map (algebraMap R S) (Submonoid.powers ↑r)) (s₁ r))

⊢ ∃ n, ↑r ^ n • x ∈ ⨆ x, Submodule.span R ↑(sf x)
-- Mathlib/RingTheory/LocalProperties.lean
R✝ S✝ : Type u
inst✝⁷ : CommRing R✝
inst✝⁶ : CommRing S✝
M : Submonoid R✝
N : Submonoid S✝
R' S' : Type u
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
f✝ : R✝ →+* S✝
inst✝³ : Algebra R✝ R'
inst✝² : Algebra S✝ S'
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset R
hs : Ideal.span ↑s = ⊤
this✝² : Algebra R S := f.toAlgebra
this✝¹ : (r : { x // x ∈ s }) → Algebra (Localization.Away ↑r) (Localization.Away (f ↑r)) := fun r => (Localization.awayMap f ↑r).toAlgebra
this✝ : ∀ (r : { x // x ∈ s }), IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers ↑r)) (Localization.Away (f ↑r))
this : ∀ (r : { x // x ∈ s }), IsScalarTower R (Localization.Away ↑r) (Localization.Away (f ↑r))
s₁ : (r : { x // x ∈ s }) → Finset (Localization.Away (f ↑r))
s₂ : ∀ (r : { x // x ∈ s }), Submodule.span (Localization.Away ↑r) ↑(s₁ r) = ⊤
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (f ↑x)) (s₁ x)
x : S
r : ↑↑s
n₁ : ℕ
hn₁ : (algebraMap S (Localization.Away (f ↑r))) ((algebraMap R S) ↑⟨↑r ^ n₁, ⋯⟩ * x) ∈ Submodule.span R ↑(s₁ r)
n₂ : ℕ
hn₂ : ↑r ^ (n₂ + n₁) • x ∈ Submodule.span R ↑(IsLocalization.finsetIntegerMultiple (Submonoid.powers ((algebraMap R S) ↑r)) (s₁ r))

⊢ ∃ n, ↑r ^ n • x ∈ ⨆ x, Submodule.span R ↑(sf x)
-- Mathlib/RingTheory/LocalProperties.lean
R✝ S✝ : Type u
inst✝⁷ : CommRing R✝
inst✝⁶ : CommRing S✝
M : Submonoid R✝
N : Submonoid S✝
R' S' : Type u
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
f✝ : R✝ →+* S✝
inst✝³ : Algebra R✝ R'
inst✝² : Algebra S✝ S'
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset R
hs : Ideal.span ↑s = ⊤
this✝² : Algebra R S := f.toAlgebra
this✝¹ : (r : { x // x ∈ s }) → Algebra (Localization.Away ↑r) (Localization.Away (f ↑r)) := fun r => (Localization.awayMap f ↑r).toAlgebra
this✝ : ∀ (r : { x // x ∈ s }), IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers ↑r)) (Localization.Away (f ↑r))
this : ∀ (r : { x // x ∈ s }), IsScalarTower R (Localization.Away ↑r) (Localization.Away (f ↑r))
s₁ : (r : { x // x ∈ s }) → Finset (Localization.Away (f ↑r))
s₂ : ∀ (r : { x // x ∈ s }), Submodule.span (Localization.Away ↑r) ↑(s₁ r) = ⊤
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers (f ↑x)) (s₁ x)
x : S
r : ↑↑s
n₁ : ℕ
hn₁ : (algebraMap S (Localization.Away (f ↑r))) ((algebraMap R S) ↑⟨↑r ^ n₁, ⋯⟩ * x) ∈ Submodule.span R ↑(s₁ r)
n₂ : ℕ
hn₂ : ↑r ^ (n₂ + n₁) • x ∈ Submodule.span R ↑(IsLocalization.finsetIntegerMultiple (Submonoid.powers ((algebraMap R S) ↑r)) (s₁ r))

⊢ ↑r ^ (n₂ + n₁) • x ∈ ⨆ x, Submodule.span R ↑(sf x)
-- Mathlib/Data/ZMod/Basic.lean
m n✝ n : ℕ
a : ZMod n

⊢ -a = a ↔ a = 0 ∨ 2 * a.val = n
-- Mathlib/Data/ZMod/Basic.lean
m n✝ n : ℕ
a : ZMod n

⊢ 2 * a = 0 ↔ a = 0 ∨ 2 * a.val = n
-- Mathlib/Data/ZMod/Basic.lean
m n n✝ : ℕ
a : ZMod (n✝ + 1)

⊢ 2 * a = 0 ↔ a = 0 ∨ 2 * a.val = n✝ + 1
-- Mathlib/Data/ZMod/Basic.lean
m n n✝ : ℕ
a : ZMod (n✝ + 1)

⊢ n✝ + 1 ∣ 2 * a.val ↔ a = 0 ∨ 2 * a.val = n✝ + 1
-- Mathlib/Geometry/RingedSpace/LocallyRingedSpace.lean
X✝ : LocallyRingedSpace
U : TopCat
X : LocallyRingedSpace
f : U ⟶ X.toTopCat
h : OpenEmbedding ⇑f
x : ↑↑(X.restrict h).toPresheafedSpace

⊢ LocalRing ↑((X.restrict h).presheaf.stalk x)
-- Mathlib/Geometry/RingedSpace/LocallyRingedSpace.lean
X✝ : LocallyRingedSpace
U : TopCat
X : LocallyRingedSpace
f : U ⟶ X.toTopCat
h : OpenEmbedding ⇑f
x : ↑↑(X.restrict h).toPresheafedSpace

⊢ ↑(X.presheaf.stalk (f x)) ≃+* ↑((X.restrict h).presheaf.stalk x)
-- Mathlib/Analysis/Convex/Jensen.lean
𝕜 : Type u_1
E : Type u_2
F : Type u_3
β : Type u_4
ι : Type u_5
inst✝⁵ : LinearOrderedField 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : OrderedAddCommGroup β
inst✝² : Module 𝕜 E
inst✝¹ : Module 𝕜 β
inst✝ : OrderedSMul 𝕜 β
s : Set E
f : E → β
t : Finset ι
w : ι → 𝕜
p : ι → E
v : 𝕜
q : E
hf : StrictConcaveOn 𝕜 s f
h₀ : ∀ i ∈ t, 0 < w i
h₁ : ∑ i ∈ t, w i = 1
hmem : ∀ i ∈ t, p i ∈ s
h_eq : f (∑ i ∈ t, w i • p i) ≤ ∑ i ∈ t, w i • f (p i)

⊢ ∀ ⦃j : ι⦄, j ∈ t → ∀ ⦃k : ι⦄, k ∈ t → p j = p k
-- Mathlib/Analysis/Convex/Jensen.lean
𝕜 : Type u_1
E : Type u_2
F : Type u_3
β : Type u_4
ι : Type u_5
inst✝⁵ : LinearOrderedField 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : OrderedAddCommGroup β
inst✝² : Module 𝕜 E
inst✝¹ : Module 𝕜 β
inst✝ : OrderedSMul 𝕜 β
s : Set E
f : E → β
t : Finset ι
w : ι → 𝕜
p : ι → E
v : 𝕜
q : E
hf : StrictConcaveOn 𝕜 s f
h₀ : ∀ i ∈ t, 0 < w i
h₁ : ∑ i ∈ t, w i = 1
hmem : ∀ i ∈ t, p i ∈ s
h_eq : f (∑ i ∈ t, w i • p i) ≤ ∑ i ∈ t, w i • f (p i)
this : ∃ j ∈ t, ∃ k ∈ t, p j ≠ p k

⊢ False
-- Mathlib/Data/Matrix/Kronecker.lean
R : Type u_1
α : Type u_2
α' : Type u_3
β : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
l : Type u_8
m : Type u_9
n : Type u_10
p : Type u_11
q : Type u_12
r : Type u_13
l' : Type u_14
m' : Type u_15
n' : Type u_16
p' : Type u_17
inst✝⁸ : CommRing R
inst✝⁷ : CommRing α
inst✝⁶ : CommRing β
inst✝⁵ : Algebra R α
inst✝⁴ : Algebra R β
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : DecidableEq m
inst✝ : DecidableEq n
A : Matrix m m α
B : Matrix n n β

⊢ (kroneckerMap (tmul R) A B).det = (A.det ^ Fintype.card n) ⊗ₜ[R] (B.det ^ Fintype.card m)
-- Mathlib/Data/Matrix/Kronecker.lean
R : Type u_1
α : Type u_2
α' : Type u_3
β : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
l : Type u_8
m : Type u_9
n : Type u_10
p : Type u_11
q : Type u_12
r : Type u_13
l' : Type u_14
m' : Type u_15
n' : Type u_16
p' : Type u_17
inst✝⁸ : CommRing R
inst✝⁷ : CommRing α
inst✝⁶ : CommRing β
inst✝⁵ : Algebra R α
inst✝⁴ : Algebra R β
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : DecidableEq m
inst✝ : DecidableEq n
A : Matrix m m α
B : Matrix n n β

⊢ (A.map fun a => ((mk R α β) a) 1).det ^ Fintype.card n * (B.map fun b => ((mk R α β) 1) b).det ^ Fintype.card m = (A.det ^ Fintype.card n) ⊗ₜ[R] (B.det ^ Fintype.card m)
-- Mathlib/Data/Matrix/Kronecker.lean
R : Type u_1
α : Type u_2
α' : Type u_3
β : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
l : Type u_8
m : Type u_9
n : Type u_10
p : Type u_11
q : Type u_12
r : Type u_13
l' : Type u_14
m' : Type u_15
n' : Type u_16
p' : Type u_17
inst✝⁸ : CommRing R
inst✝⁷ : CommRing α
inst✝⁶ : CommRing β
inst✝⁵ : Algebra R α
inst✝⁴ : Algebra R β
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : DecidableEq m
inst✝ : DecidableEq n
A : Matrix m m α
B : Matrix n n β

⊢ (A.map fun a => includeLeft a).det ^ Fintype.card n * (B.map fun b => includeRight b).det ^ Fintype.card m = (A.det ^ Fintype.card n) ⊗ₜ[R] (B.det ^ Fintype.card m)
-- Mathlib/Data/Matrix/Kronecker.lean
R : Type u_1
α : Type u_2
α' : Type u_3
β : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
l : Type u_8
m : Type u_9
n : Type u_10
p : Type u_11
q : Type u_12
r : Type u_13
l' : Type u_14
m' : Type u_15
n' : Type u_16
p' : Type u_17
inst✝⁸ : CommRing R
inst✝⁷ : CommRing α
inst✝⁶ : CommRing β
inst✝⁵ : Algebra R α
inst✝⁴ : Algebra R β
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : DecidableEq m
inst✝ : DecidableEq n
A : Matrix m m α
B : Matrix n n β

⊢ includeLeft A.det ^ Fintype.card n * includeRight B.det ^ Fintype.card m = (A.det ^ Fintype.card n) ⊗ₜ[R] (B.det ^ Fintype.card m)
-- Mathlib/Topology/Instances/EReal.lean
α : Type u_1
inst✝ : TopologicalSpace α
a : ℝ

⊢ ContinuousAt (fun p => p.1 + p.2) (⊥, ↑a)
-- Mathlib/Topology/Instances/EReal.lean
α : Type u_1
inst✝ : TopologicalSpace α
a : ℝ

⊢ ∀ (x : ℝ), ∀ᶠ (a : EReal × EReal) in 𝓝 (⊥, ↑a), a.1 + a.2 < ↑x
-- Mathlib/Topology/Instances/EReal.lean
α : Type u_1
inst✝ : TopologicalSpace α
a r : ℝ
x✝ : EReal × EReal
h : x✝.1 < ↑(r - (a + 1)) ∧ x✝.2 < ↑(a + 1)

⊢ x✝.1 + x✝.2 < ↑r
-- Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean
C : Type u_1
D : Type u_2
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Category.{?u.100731, u_2} D
inst✝² : Preadditive C
inst✝¹ : Preadditive D
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
n : ℤ

⊢ (mappingCone φ).d (n - 1) n ≫ (snd φ).v n n ⋯ = (↑(fst φ)).v (n - 1) n ⋯ ≫ φ.f n + (snd φ).v (n - 1) (n - 1) ⋯ ≫ G.d (n - 1) n