ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Algebra/Category/ModuleCat/Adjunctions.lean
R : Type u
inst✝ : CommRing R
X : Type u

⊢ ∀ (a : X), (TensorProduct.mk R ↑((free R).obj X) ↑(𝟙_ (ModuleCat R))).compr₂ (ρ_ ((free R).obj X)).hom ∘ₗ Finsupp.lsingle a = (TensorProduct.mk R ↑((free R).obj X) ↑(𝟙_ (ModuleCat R))).compr₂ ((𝟙 ((free R).obj X) ⊗ ε R) ≫ (μ R X (𝟙_ (Type u))).hom ≫ map (free R).obj (ρ_ X).hom) ∘ₗ Finsupp.lsingle a
-- Mathlib/Algebra/Category/ModuleCat/Adjunctions.lean
R : Type u
inst✝ : CommRing R
X : Type u
x : X

⊢ (TensorProduct.mk R ↑((free R).obj X) ↑(𝟙_ (ModuleCat R))).compr₂ (ρ_ ((free R).obj X)).hom ∘ₗ Finsupp.lsingle x = (TensorProduct.mk R ↑((free R).obj X) ↑(𝟙_ (ModuleCat R))).compr₂ ((𝟙 ((free R).obj X) ⊗ ε R) ≫ (μ R X (𝟙_ (Type u))).hom ≫ map (free R).obj (ρ_ X).hom) ∘ₗ Finsupp.lsingle x
-- Mathlib/Algebra/Category/ModuleCat/Adjunctions.lean
R : Type u
inst✝ : CommRing R
X : Type u
x : X

⊢ ((TensorProduct.mk R ↑((free R).obj X) ↑(𝟙_ (ModuleCat R))).compr₂ (ρ_ ((free R).obj X)).hom ∘ₗ Finsupp.lsingle x) 1 = ((TensorProduct.mk R ↑((free R).obj X) ↑(𝟙_ (ModuleCat R))).compr₂ ((𝟙 ((free R).obj X) ⊗ ε R) ≫ (μ R X (𝟙_ (Type u))).hom ≫ map (free R).obj (ρ_ X).hom) ∘ₗ Finsupp.lsingle x) 1
-- Mathlib/Algebra/Category/ModuleCat/Adjunctions.lean
R : Type u
inst✝ : CommRing R
X : Type u
x : X

⊢ (((TensorProduct.mk R ↑((free R).obj X) ↑(𝟙_ (ModuleCat R))).compr₂ (ρ_ ((free R).obj X)).hom ∘ₗ Finsupp.lsingle x) 1) 1 = (((TensorProduct.mk R ↑((free R).obj X) ↑(𝟙_ (ModuleCat R))).compr₂ ((𝟙 ((free R).obj X) ⊗ ε R) ≫ (μ R X (𝟙_ (Type u))).hom ≫ map (free R).obj (ρ_ X).hom) ∘ₗ Finsupp.lsingle x) 1) 1
-- Mathlib/Algebra/Category/ModuleCat/Adjunctions.lean
R : Type u
inst✝ : CommRing R
X : Type u
x : X

⊢ ∀ (a : X), ((((TensorProduct.mk R ↑((free R).obj X) ↑(𝟙_ (ModuleCat R))).compr₂ (ρ_ ((free R).obj X)).hom ∘ₗ Finsupp.lsingle x) 1) 1) a = ((((TensorProduct.mk R ↑((free R).obj X) ↑(𝟙_ (ModuleCat R))).compr₂ ((𝟙 ((free R).obj X) ⊗ ε R) ≫ (μ R X (𝟙_ (Type u))).hom ≫ map (free R).obj (ρ_ X).hom) ∘ₗ Finsupp.lsingle x) 1) 1) a
-- Mathlib/Algebra/Category/ModuleCat/Adjunctions.lean
R : Type u
inst✝ : CommRing R
X : Type u
x x' : X
q : X →₀ R := (ρ_ (of R (X →₀ R))).hom (Finsupp.single x 1 ⊗ₜ[R] 1)

⊢ q x' = (Finsupp.mapDomain (ρ_ X).hom ((finsuppTensorFinsupp' R X (𝟙_ (Type u))) (Finsupp.single x 1 ⊗ₜ[R] Finsupp.single PUnit.unit 1))) x'
-- Mathlib/RingTheory/MvPolynomial/Homogeneous.lean
σ : Type u_1
τ : Type u_2
R : Type u_3
S : Type u_4
inst✝² inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
φ ψ : MvPolynomial σ R
m n : ℕ
hφ : φ.IsHomogeneous n
f : R →+* S

⊢ ((MvPolynomial.map f) φ).IsHomogeneous n
-- Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α
m' : MeasurableSpace β
inst✝ : BooleanAlgebra β
h : Measurable compl
f : α → β

⊢ MeasurableSpace.comap (fun a => (f a)ᶜ) inferInstance = MeasurableSpace.comap f inferInstance
-- Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α
m' : MeasurableSpace β
inst✝ : BooleanAlgebra β
h : Measurable compl
f : α → β

⊢ MeasurableSpace.comap f (MeasurableSpace.comap compl inferInstance) = MeasurableSpace.comap f inferInstance
-- Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α
m' : MeasurableSpace β
inst✝ : BooleanAlgebra β
h : Measurable compl
f : α → β

⊢ MeasurableSpace.comap compl inferInstance = inferInstance
-- Mathlib/SetTheory/Cardinal/Basic.lean
α β : Type u
s : Set α

⊢ #↑(ULift.down ⁻¹' s) = lift.{v, u} #↑s
-- Mathlib/SetTheory/Cardinal/Basic.lean
α β : Type u
s : Set α

⊢ Nonempty (↑(ULift.down ⁻¹' s) ≃ ULift.{v, u} ↑s)
-- Mathlib/SetTheory/Cardinal/Basic.lean
α β : Type u
s : Set α
f : ↑(ULift.down ⁻¹' s) → ULift.{?u.6919, u} ↑s := fun x => { down := s.restrictPreimage ULift.down x }

⊢ ↑(ULift.down ⁻¹' s) ≃ ULift.{v, u} ↑s
-- Mathlib/SetTheory/Cardinal/Basic.lean
α β : Type u
s : Set α
f : ↑(ULift.down ⁻¹' s) → ULift.{?u.6966, u} ↑s := fun x => { down := s.restrictPreimage ULift.down x }
this : Bijective f

⊢ ↑(ULift.down ⁻¹' s) ≃ ULift.{v, u} ↑s
-- Mathlib/Combinatorics/Enumerative/DoubleCounting.lean
α : Type u_1
β : Type u_2
r : α → β → Prop
s : Finset α
t : Finset β
a a' : α
b b' : β
inst✝¹ : DecidablePred (r a)
inst✝ : (a : α) → Decidable (r a b)
m n : ℕ
hs : ∀ a ∈ s, ∃ b ∈ t, r a b
ht : ∀ b ∈ t, {a | a ∈ s ∧ r a b}.Subsingleton

⊢ s.card ≤ t.card
-- Mathlib/Data/Set/MemPartition.lean
α : Type u_1
f : ℕ → Set α
n : ℕ
a : α
s : Set α
hs : s ∈ memPartition f n

⊢ memPartitionSet f n a = s ↔ a ∈ s
-- Mathlib/Data/Set/MemPartition.lean
α : Type u_1
f : ℕ → Set α
n : ℕ
a : α
s : Set α
hs : s ∈ memPartition f n
h : a ∈ s

⊢ memPartitionSet f n a = s
-- Mathlib/Data/Set/MemPartition.lean
α : Type u_1
f : ℕ → Set α
n : ℕ
a : α
s : Set α
hs : s ∈ memPartition f n
h : a ∈ s
h_ne : ¬memPartitionSet f n a = s

⊢ False
-- Mathlib/Data/Set/MemPartition.lean
α : Type u_1
f : ℕ → Set α
n : ℕ
a : α
s : Set α
hs : s ∈ memPartition f n
h : a ∈ s
h_ne : ¬memPartitionSet f n a = s
h_disj : Disjoint s (memPartitionSet f n a)

⊢ False
-- Mathlib/Data/Set/MemPartition.lean
α : Type u_1
f : ℕ → Set α
n : ℕ
a : α
s : Set α
hs : s ∈ memPartition f n
h : a ∈ s
h_ne : ¬memPartitionSet f n a = s
h_disj : Disjoint s (memPartitionSet f n a)

⊢ ¬Disjoint s (memPartitionSet f n a)
-- Mathlib/Data/Set/MemPartition.lean
α : Type u_1
f : ℕ → Set α
n : ℕ
a : α
s : Set α
hs : s ∈ memPartition f n
h : a ∈ s
h_ne : ¬memPartitionSet f n a = s
h_disj : Disjoint s (memPartitionSet f n a)

⊢ (s ∩ memPartitionSet f n a).Nonempty
-- Mathlib/Topology/Constructions.lean
X : Type u
Y : Type v
Z : Type u_1
W : Type u_2
ε : Type u_3
ζ : Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : TopologicalSpace Z
inst✝² : TopologicalSpace W
inst✝¹ : TopologicalSpace ε
inst✝ : TopologicalSpace ζ
f : X → Y
g : Z → W
hf : _root_.Embedding f
hg : _root_.Embedding g
x✝¹ x✝ : X × Z
x₁ : X
z₁ : Z
x₂ : X
z₂ : Z

⊢ Prod.map f g (x₁, z₁) = Prod.map f g (x₂, z₂) → (x₁, z₁) = (x₂, z₂)
-- Mathlib/Logic/IsEmpty.lean
α✝ : Sort u_1
β✝ : Sort u_2
γ : Sort u_3
α : Type u_4
β : Type u_5

⊢ IsEmpty (α × β) ↔ IsEmpty α ∨ IsEmpty β
-- Mathlib/Logic/Relation.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ε : Type u_5
ζ : Type u_6
r : α → α → Prop
a b c d : α

⊢ ReflTransGen r a b ↔ a = b ∨ ∃ c, r a c ∧ ReflTransGen r c b
-- Mathlib/Logic/Relation.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ε : Type u_5
ζ : Type u_6
r : α → α → Prop
a b c d : α

⊢ (a = b ∨ ∃ c, r a c ∧ ReflTransGen r c b) → ReflTransGen r a b
-- Mathlib/Algebra/Order/Ring/WithTop.lean
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : MulZeroClass α
a b : WithTop α
hb : b ≠ 0

⊢ ⊤ * b = ⊤
-- Mathlib/Analysis/Convex/Combination.lean
R : Type u_1
R' : Type u_2
E : Type u_3
F : Type u_4
ι : Type u_5
ι' : Type u_6
α : Type u_7
inst✝⁸ : LinearOrderedField R
inst✝⁷ : LinearOrderedField R'
inst✝⁶ : AddCommGroup E
inst✝⁵ : AddCommGroup F
inst✝⁴ : LinearOrderedAddCommGroup α
inst✝³ : Module R E
inst✝² : Module R F
inst✝¹ : Module R α
inst✝ : OrderedSMul R α
s : Set E
i j : ι
c : R
t : Finset ι
w : ι → R
z : ι → E
hw : w i ≠ 0

⊢ {i}.centerMass w z = z i
-- Mathlib/Data/List/Infix.lean
α : Type u_1
β : Type u_2
l l₁ l₂ l₃ : List α
a b : α
m n✝ : ℕ
x y : List α
h : x <+: y
n : ℕ
hn : n < x.length

⊢ x.get ⟨n, hn⟩ = y.get ⟨n, ⋯⟩
-- Mathlib/Data/Fin/Basic.lean
n m : ℕ
p : Fin (n + 1)
i j : Fin n
inst✝ : NeZero n
a : Fin (n + 1)
b : Fin n
ha : a ≠ 0

⊢ a.succAbove b = 0 ↔ b = 0
-- Mathlib/Algebra/GroupWithZero/Semiconj.lean
α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝ : GroupWithZero G₀
a x y x' y' : G₀
h : SemiconjBy a x y
h' : SemiconjBy a x' y'

⊢ SemiconjBy a (x / x') (y / y')
-- Mathlib/Algebra/GroupWithZero/Semiconj.lean
α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝ : GroupWithZero G₀
a x y x' y' : G₀
h : SemiconjBy a x y
h' : SemiconjBy a x' y'

⊢ SemiconjBy a (x * x'⁻¹) (y * y'⁻¹)
-- Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean
k G✝ : Type u
inst✝⁴ : CommRing k
inst✝³ : Group G✝
A✝ : Rep k G✝
G : Type u_1
A : Type u_2
inst✝² : Group G
inst✝¹ : AddCommGroup A
inst✝ : MulAction G A
f : G → A
hf : IsOneCocycle f
g : G

⊢ g • f g⁻¹ = -f g
-- Mathlib/Algebra/Polynomial/Degree/Lemmas.lean
R : Type u
S : Type v
ι : Type w
a b : R
m n : ℕ
inst✝ : Ring R
p q : R[X]
df : p.natDegree < n

⊢ (p - q).coeff n = -q.coeff n
-- Mathlib/MeasureTheory/Integral/Lebesgue.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f

⊢ ∫⁻ (a : α), ⨆ b, f b a ∂μ = ⨆ b, ∫⁻ (a : α), f b a ∂μ
-- Mathlib/MeasureTheory/Integral/Lebesgue.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f

⊢ ∫⁻ (a : α), (⨆ i, f i) a ∂μ = ⨆ b, ∫⁻ (a : α), f b a ∂μ
-- Mathlib/MeasureTheory/Integral/Lebesgue.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'

⊢ ∫⁻ (a : α), (⨆ i, f i) a ∂μ = ⨆ b, ∫⁻ (a : α), f b a ∂μ
-- Mathlib/MeasureTheory/Integral/Lebesgue.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x

⊢ ∫⁻ (a : α), (⨆ i, f i) a ∂μ = ⨆ b, ∫⁻ (a : α), f b a ∂μ
-- Mathlib/MeasureTheory/Integral/Lebesgue.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
h_ae_seq_directed : Directed LE.le (aeSeq hf p)

⊢ ∫⁻ (a : α), (⨆ i, f i) a ∂μ = ⨆ b, ∫⁻ (a : α), f b a ∂μ
-- Mathlib/MeasureTheory/Integral/Lebesgue.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
h_ae_seq_directed : Directed LE.le (aeSeq hf p)

⊢ ∫⁻ (a : α), (⨆ i, f i) a ∂μ = ∫⁻ (a : α), ⨆ b, aeSeq hf p b a ∂?m.329415
-- Mathlib/MeasureTheory/Integral/Lebesgue.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
h_ae_seq_directed : Directed LE.le (aeSeq hf p)

⊢ ⨆ b, ∫⁻ (a : α), f b a ∂μ = ⨆ b, ∫⁻ (a : α), aeSeq hf p b a ∂?m.329415
-- Mathlib/MeasureTheory/Integral/Lebesgue.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x x_1 => x ≤ x_1) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
h_ae_seq_directed : Directed LE.le (aeSeq hf p)

⊢ Measure α
-- Mathlib/Algebra/Polynomial/Module/Basic.lean
R : Type u_1
M : Type u_2
inst✝¹² : CommRing R
inst✝¹¹ : AddCommGroup M
inst✝¹⁰ : Module R M
I : Ideal R
S : Type u_3
inst✝⁹ : CommSemiring S
inst✝⁸ : Algebra S R
inst✝⁷ : Module S M
inst✝⁶ : IsScalarTower S R M
R' : Type u_4
M' : Type u_5
inst✝⁵ : CommRing R'
inst✝⁴ : AddCommGroup M'
inst✝³ : Module R' M'
inst✝² : Algebra R R'
inst✝¹ : Module R M'
inst✝ : IsScalarTower R R' M'
p p' : R[X]
q : PolynomialModule R M

⊢ (comp p) (p' • q) = p'.comp p • (comp p) q
-- Mathlib/Algebra/Polynomial/Module/Basic.lean
R : Type u_1
M : Type u_2
inst✝¹² : CommRing R
inst✝¹¹ : AddCommGroup M
inst✝¹⁰ : Module R M
I : Ideal R
S : Type u_3
inst✝⁹ : CommSemiring S
inst✝⁸ : Algebra S R
inst✝⁷ : Module S M
inst✝⁶ : IsScalarTower S R M
R' : Type u_4
M' : Type u_5
inst✝⁵ : CommRing R'
inst✝⁴ : AddCommGroup M'
inst✝³ : Module R' M'
inst✝² : Algebra R R'
inst✝¹ : Module R M'
inst✝ : IsScalarTower R R' M'
p p' : R[X]
q : PolynomialModule R M

⊢ eval₂ (algebraMap R R[X]) p p' • (eval p) ((map R[X] (lsingle R 0)) q) = eval₂ C p p' • (eval p) ((map R[X] (lsingle R 0)) q)
-- Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
f : SimpleFunc α ℝ
g : α → ℝ
hg : Integrable g μ
this : ∀ (s : Set α) (c : ℝ) (f : α → ℝ), s.indicator (Function.const α c) * f = s.indicator (c • f)

⊢ μ[↑f * g|m] =ᶠ[ae μ] ↑f * μ[g|m]
-- Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
f : SimpleFunc α ℝ
g : α → ℝ
hg : Integrable g μ
this : ∀ (s : Set α) (c : ℝ) (f : α → ℝ), s.indicator (Function.const α c) * f = s.indicator (c • f)
c : ℝ
s : Set α
hs : MeasurableSet s

⊢ μ[↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0)) * g|m] =ᶠ[ae μ] ↑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0)) * μ[g|m]
-- Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
f : SimpleFunc α ℝ
g : α → ℝ
hg : Integrable g μ
this : ∀ (s : Set α) (c : ℝ) (f : α → ℝ), s.indicator (Function.const α c) * f = s.indicator (c • f)
g₁ g₂ : SimpleFunc α ℝ
x✝ : Disjoint (Function.support ↑g₁) (Function.support ↑g₂)
h_eq₁ : (fun f => μ[↑f * g|m] =ᶠ[ae μ] ↑f * μ[g|m]) g₁
h_eq₂ : (fun f => μ[↑f * g|m] =ᶠ[ae μ] ↑f * μ[g|m]) g₂

⊢ μ[↑(g₁ + g₂) * g|m] =ᶠ[ae μ] ↑(g₁ + g₂) * μ[g|m]
-- Mathlib/MeasureTheory/Decomposition/Jordan.lean
α : Type u_1
β : Type u_2
inst✝ : MeasurableSpace α
s : SignedMeasure α

⊢ (-s).totalVariation = s.totalVariation
-- Mathlib/MeasureTheory/Measure/VectorMeasure.lean
α : Type u_1
β : Type u_2
m : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
inst✝⁵ : AddCommMonoid L
inst✝⁴ : TopologicalSpace L
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
n : MeasurableSpace α
v : VectorMeasure α M
hle : m ≤ n
i : Set α
hi : MeasurableSet i

⊢ ↑(trim 0 hle) i = ↑0 i
-- Mathlib/Topology/Category/Profinite/Product.lean
ι : Type u
X : ι → Type
inst✝ : (i : ι) → TopologicalSpace (X i)
C : Set ((i : ι) → X i)
J K : ι → Prop
x : ↑(obj C J)

⊢ ∃ a, (π_app C J) a = x
-- Mathlib/Topology/Category/Profinite/Product.lean
ι : Type u
X : ι → Type
inst✝ : (i : ι) → TopologicalSpace (X i)
C : Set ((i : ι) → X i)
J K : ι → Prop
x : ↑(obj C J)
y : (i : ι) → X i
hy : y ∈ C ∧ (precomp Subtype.val) y = ↑x

⊢ ∃ a, (π_app C J) a = x
-- Mathlib/NumberTheory/Zsqrtd/Basic.lean
d x y z w : ℕ
xy : { re := ↑x, im := -↑y }.Nonneg
zw : { re := -↑z, im := ↑w }.Nonneg

⊢ ({ re := ↑x, im := -↑y } + { re := -↑z, im := ↑w }).Nonneg
-- Mathlib/NumberTheory/Zsqrtd/Basic.lean
d x y z w : ℕ
xy : { re := ↑x, im := -↑y }.Nonneg
zw : { re := -↑z, im := ↑w }.Nonneg
this : { re := subNatNat x z, im := subNatNat w y }.Nonneg

⊢ ({ re := ↑x, im := -↑y } + { re := -↑z, im := ↑w }).Nonneg
-- Mathlib/NumberTheory/Zsqrtd/Basic.lean
d x y z w : ℕ
xy : { re := ↑x, im := -↑y }.Nonneg
zw : { re := -↑z, im := ↑w }.Nonneg
this : { re := subNatNat x z, im := subNatNat w y }.Nonneg

⊢ { re := ↑x + -↑z, im := ↑w - ↑y }.Nonneg
-- Mathlib/Topology/Category/Profinite/Nobeling.lean
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
l : Products I
o₁ o₂ : Ordinal.{u}
h : o₁ ≤ o₂
hl : ∀ i ∈ ↑l, ord I i < o₁
f : LocallyConstant ↑(π C fun x => ord I x < o₂) ℤ

⊢ f ∈ eval (π C fun x => ord I x < o₂) '' {m | m < l} ↔ f ∈ ⇑(πs' C h) '' (eval (π C fun x => ord I x < o₁) '' {m | m < l})
-- Mathlib/Topology/Category/Profinite/Nobeling.lean
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
l : Products I
o₁ o₂ : Ordinal.{u}
h : o₁ ≤ o₂
hl : ∀ i ∈ ↑l, ord I i < o₁
f : LocallyConstant ↑(π C fun x => ord I x < o₂) ℤ

⊢ (∃ x < l, eval (π C fun x => ord I x < o₂) x = f) ↔ ∃ a < l, (πs' C h) (eval (π C fun x => ord I x < o₁) a) = f
-- Mathlib/Topology/Category/Profinite/Nobeling.lean
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
l : Products I
o₁ o₂ : Ordinal.{u}
h : o₁ ≤ o₂
hl : ∀ i ∈ ↑l, ord I i < o₁
f : LocallyConstant ↑(π C fun x => ord I x < o₂) ℤ

⊢ ∀ (a : Products I), a < l ∧ eval (π C fun x => ord I x < o₂) a = f ↔ a < l ∧ (πs' C h) (eval (π C fun x => ord I x < o₁) a) = f
-- Mathlib/Topology/Category/Profinite/Nobeling.lean
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
l : Products I
o₁ o₂ : Ordinal.{u}
h : o₁ ≤ o₂
hl : ∀ i ∈ ↑l, ord I i < o₁
f : LocallyConstant ↑(π C fun x => ord I x < o₂) ℤ
m : Products I

⊢ m < l ∧ eval (π C fun x => ord I x < o₂) m = f ↔ m < l ∧ (πs' C h) (eval (π C fun x => ord I x < o₁) m) = f
-- Mathlib/Topology/Category/Profinite/Nobeling.lean
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
l : Products I
o₁ o₂ : Ordinal.{u}
h : o₁ ≤ o₂
hl : ∀ i ∈ ↑l, ord I i < o₁
f : LocallyConstant ↑(π C fun x => ord I x < o₂) ℤ
m : Products I

⊢ m < l → (eval (π C fun x => ord I x < o₂) m = f ↔ (πs' C h) (eval (π C fun x => ord I x < o₁) m) = f)
-- Mathlib/Topology/Category/Profinite/Nobeling.lean
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
l : Products I
o₁ o₂ : Ordinal.{u}
h : o₁ ≤ o₂
hl : ∀ i ∈ ↑l, ord I i < o₁
f : LocallyConstant ↑(π C fun x => ord I x < o₂) ℤ
m : Products I
hm : m < l

⊢ eval (π C fun x => ord I x < o₂) m = f ↔ (πs' C h) (eval (π C fun x => ord I x < o₁) m) = f
-- Mathlib/LinearAlgebra/Dimension/Basic.lean
R✝ : Type u
R'✝ : Type u'
M M₁ : Type v
M' : Type v'
inst✝¹⁸ : Ring R✝
inst✝¹⁷ : Ring R'✝
inst✝¹⁶ : AddCommGroup M
inst✝¹⁵ : AddCommGroup M'
inst✝¹⁴ : AddCommGroup M₁
inst✝¹³ : Module R✝ M
inst✝¹² : Module R✝ M'
inst✝¹¹ : Module R✝ M₁
inst✝¹⁰ : Module R'✝ M'
inst✝⁹ : Module R'✝ M₁
R : Type w
S : Type v
inst✝⁸ : CommRing R
inst✝⁷ : Ring S
inst✝⁶ : Algebra R S
R' : Type w'
S'✝ : Type v'
inst✝⁵ : CommRing R'
inst✝⁴ : Ring S'✝
inst✝³ : Algebra R' S'✝
S' : Type v
inst✝² : CommRing R'
inst✝¹ : Ring S'
inst✝ : Algebra R' S'
i : R →+* R'
j : S →+* S'
hi : Surjective ⇑i
hj : Injective ⇑j
hc : (algebraMap R' S').comp i = j.comp (algebraMap R S)

⊢ Module.rank R S ≤ Module.rank R' S'
-- Mathlib/Data/Matrix/Basic.lean
l : Type u_1
m : Type u_2
n : Type u_3
o : Type u_4
m' : o → Type u_5
n' : o → Type u_6
R : Type u_7
S : Type u_8
α : Type v
β : Type w
γ : Type u_9
inst✝¹ : NonUnitalCommSemiring α
inst✝ : Fintype m
A : Matrix m n α
x : m → α
x✝ : n

⊢ (Aᵀ *ᵥ x) x✝ = (x ᵥ* A) x✝
-- Mathlib/MeasureTheory/Integral/SetIntegral.lean
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝³ : MeasurableSpace X
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f g : X → E
s t : Set X
μ ν : Measure X
l l' : Filter X
inst✝ : DecidablePred fun x => x ∈ s
hs : MeasurableSet s
hf : IntegrableOn f s μ
hg : IntegrableOn g sᶜ μ

⊢ ∫ (x : X), s.piecewise f g x ∂μ = ∫ (x : X) in s, f x ∂μ + ∫ (x : X) in sᶜ, g x ∂μ
-- Mathlib/Combinatorics/Enumerative/Composition.lean
n✝ : ℕ
c✝ : Composition n✝
n : ℕ
hn : 0 < n
c : Composition n

⊢ c ≠ single n hn ↔ ∀ (i : Fin c.length), c.blocksFun i < n
-- Mathlib/Combinatorics/Enumerative/Composition.lean
n✝ : ℕ
c✝ : Composition n✝
n : ℕ
hn : 0 < n
c : Composition n

⊢ ¬c ≠ single n hn ↔ ¬∀ (i : Fin c.length), c.blocksFun i < n
-- Mathlib/Analysis/Calculus/LagrangeMultipliers.lean
E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f✝ : E → F
φ : E → ℝ
x₀ : E
f'✝ : E →L[ℝ] F
φ' : E →L[ℝ] ℝ
f : E → ℝ
f' : E →L[ℝ] ℝ
hextr : IsLocalExtrOn φ {x | f x = f x₀} x₀
hf' : HasStrictFDerivAt f f' x₀
hφ' : HasStrictFDerivAt φ φ' x₀

⊢ ∃ a b, (a, b) ≠ 0 ∧ a • f' + b • φ' = 0
-- Mathlib/Analysis/Calculus/LagrangeMultipliers.lean
E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f✝ : E → F
φ : E → ℝ
x₀ : E
f'✝ : E →L[ℝ] F
φ' : E →L[ℝ] ℝ
f : E → ℝ
f' : E →L[ℝ] ℝ
hextr : IsLocalExtrOn φ {x | f x = f x₀} x₀
hf' : HasStrictFDerivAt f f' x₀
hφ' : HasStrictFDerivAt φ φ' x₀
Λ : Module.Dual ℝ ℝ
Λ₀ : ℝ
hΛ : (Λ, Λ₀) ≠ 0
hfΛ : ∀ (x : E), Λ (f' x) + Λ₀ • φ' x = 0

⊢ ∃ a b, (a, b) ≠ 0 ∧ a • f' + b • φ' = 0
-- Mathlib/Topology/Separation.lean
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X

⊢ T2Space X ↔ ∀ {x y : X}, (𝓝 x ⊓ 𝓝 y).NeBot → x = y
-- Mathlib/AlgebraicGeometry/Scheme.lean
R : CommRingCat
f : ↑R
x : ↑↑(Spec R).toPresheafedSpace

⊢ x ∈ ↑((Spec R).basicOpen ((Scheme.ΓSpecIso R).inv f)) ↔ x ∈ ↑(PrimeSpectrum.basicOpen f)
-- Mathlib/AlgebraicGeometry/Scheme.lean
R : CommRingCat
f : ↑R
x : ↑↑(Spec R).toPresheafedSpace

⊢ IsUnit (((Spec R).presheaf.germ ⟨x, _root_.trivial⟩) ((Scheme.ΓSpecIso R).inv f)) ↔ x ∈ PrimeSpectrum.basicOpen f
-- Mathlib/AlgebraicGeometry/Scheme.lean
R : CommRingCat
f : ↑R
x : ↑↑(Spec R).toPresheafedSpace

⊢ IsUnit ((StructureSheaf.toStalk (↑R) x) f) ↔ f ∉ x.asIdeal
-- Mathlib/AlgebraicGeometry/Scheme.lean
R : CommRingCat
f : ↑R
x : ↑↑(Spec R).toPresheafedSpace

⊢ IsUnit ((algebraMap (↑R) (Localization.AtPrime x.asIdeal)) f) ↔ f ∉ x.asIdeal
-- Mathlib/Order/Interval/Set/OrdConnectedComponent.lean
α : Type u_1
inst✝ : LinearOrder α
s t : Set α

⊢ Disjoint (s.ordT5Nhd t) (t.ordT5Nhd s)
-- Mathlib/Order/Interval/Set/OrdConnectedComponent.lean
α : Type u_1
inst✝ : LinearOrder α
s t : Set α

⊢ s.ordT5Nhd t ⊓ t.ordT5Nhd s ≤ ⊥
-- Mathlib/Order/Interval/Set/OrdConnectedComponent.lean
α : Type u_1
inst✝ : LinearOrder α
s t : Set α
x : α
hx₁ : x ∈ s.ordT5Nhd t
hx₂ : x ∈ t.ordT5Nhd s

⊢ x ∈ ⊥
-- Mathlib/Order/Interval/Set/OrdConnectedComponent.lean
α : Type u_1
inst✝ : LinearOrder α
s t : Set α
x : α
hx₁ : x ∈ s.ordT5Nhd t
hx₂ : x ∈ t.ordT5Nhd s
a : α
has : a ∈ s
ha : x ∈ (tᶜ ∩ (s.ordSeparatingSet t).ordConnectedSectionᶜ).ordConnectedComponent a

⊢ x ∈ ⊥
-- Mathlib/Order/Interval/Set/OrdConnectedComponent.lean
α : Type u_1
inst✝ : LinearOrder α
s t : Set α
x : α
hx₂ : x ∈ t.ordT5Nhd s
a : α
has : a ∈ s
ha : x ∈ (tᶜ ∩ (s.ordSeparatingSet t).ordConnectedSectionᶜ).ordConnectedComponent a

⊢ x ∈ ⊥
-- Mathlib/Order/Interval/Set/OrdConnectedComponent.lean
α : Type u_1
inst✝ : LinearOrder α
s t : Set α
x : α
hx₂ : x ∈ t.ordT5Nhd s
a : α
has : a ∈ s
ha : x ∈ (tᶜ ∩ (s.ordSeparatingSet t).ordConnectedSectionᶜ).ordConnectedComponent a
b : α
hbt : b ∈ t
hb : x ∈ (sᶜ ∩ (t.ordSeparatingSet s).ordConnectedSectionᶜ).ordConnectedComponent b

⊢ x ∈ ⊥
-- Mathlib/Order/Interval/Set/OrdConnectedComponent.lean
α : Type u_1
inst✝ : LinearOrder α
s t : Set α
x a : α
has : a ∈ s
ha : x ∈ (tᶜ ∩ (s.ordSeparatingSet t).ordConnectedSectionᶜ).ordConnectedComponent a
b : α
hbt : b ∈ t
hb : x ∈ (sᶜ ∩ (t.ordSeparatingSet s).ordConnectedSectionᶜ).ordConnectedComponent b

⊢ x ∈ ⊥
-- Mathlib/Order/Interval/Set/OrdConnectedComponent.lean
α : Type u_1
inst✝ : LinearOrder α
s t : Set α
x a : α
has : a ∈ s
ha : [[a, x]] ⊆ tᶜ ∧ [[a, x]] ⊆ (s.ordSeparatingSet t).ordConnectedSectionᶜ
b : α
hbt : b ∈ t
hb : [[b, x]] ⊆ sᶜ ∧ [[b, x]] ⊆ (t.ordSeparatingSet s).ordConnectedSectionᶜ

⊢ x ∈ ⊥