ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Algebra/Module/LocalizedModule.lean
R : Type u
inst✝⁵ : CommSemiring R
S : Submonoid R
M : Type v
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
T : Type u_1
inst✝² : CommSemiring T
inst✝¹ : Algebra R T
inst✝ : IsLocalization S T
s : ↥S
r : R
x : LocalizedModule S M
x✝¹ : M
x✝ : ↥S

⊢ { toFun := fun p => p.liftOn (fun p => mk p.1 (p.2 * s)) ⋯, map_add' := ⋯ }.toFun (r • mk x✝¹ x✝) = (RingHom.id R) r • { toFun := fun p => p.liftOn (fun p => mk p.1 (p.2 * s)) ⋯, map_add' := ⋯ }.toFun (mk x✝¹ x✝)
-- Mathlib/Algebra/Module/LocalizedModule.lean
R : Type u
inst✝⁵ : CommSemiring R
S : Submonoid R
M : Type v
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
T : Type u_1
inst✝² : CommSemiring T
inst✝¹ : Algebra R T
inst✝ : IsLocalization S T
s : ↥S
r : R
x : LocalizedModule S M
x✝¹ : M
x✝ : ↥S

⊢ (mk ((IsLocalization.sec S (↑(algebraMap R (Localization S)).toMonoidWithZeroHom r)).1 • (x✝¹, x✝).1) ((IsLocalization.sec S (↑(algebraMap R (Localization S)).toMonoidWithZeroHom r)).2 * (x✝¹, x✝).2)).liftOn (fun p => mk p.1 (p.2 * s)) ⋯ = r • (mk x✝¹ x✝).liftOn (fun p => mk p.1 (p.2 * s)) ⋯
-- Mathlib/Algebra/Module/LocalizedModule.lean
R : Type u
inst✝⁵ : CommSemiring R
S : Submonoid R
M : Type v
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
T : Type u_1
inst✝² : CommSemiring T
inst✝¹ : Algebra R T
inst✝ : IsLocalization S T
s : ↥S
r : R
x : LocalizedModule S M
x✝¹ : M
x✝ : ↥S

⊢ { toFun := (↑↑(algebraMap R (Localization S))).toFun, map_zero' := ⋯ } r • mk x✝¹ (x✝ * s) = r • mk x✝¹ (x✝ * s)
-- Mathlib/AlgebraicGeometry/Pullbacks.lean
C : Type u
inst✝¹ : Category.{v, u} C
X Y Z : Scheme
𝒰 : X.OpenCover
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (i : 𝒰.J), HasPullback (𝒰.map i ≫ f) g
i j k : 𝒰.J

⊢ (fun i j k => t' 𝒰 f g i j k) i j k ≫ pullback.snd ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j k) ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j i) = pullback.fst ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i j) ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i k) ≫ (fun i j => t 𝒰 f g i j) i j
-- Mathlib/AlgebraicGeometry/Pullbacks.lean
C : Type u
inst✝¹ : Category.{v, u} C
X Y Z : Scheme
𝒰 : X.OpenCover
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (i : 𝒰.J), HasPullback (𝒰.map i ≫ f) g
i j k : 𝒰.J

⊢ ((fun i j k => t' 𝒰 f g i j k) i j k ≫ pullback.snd ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j k) ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j i)) ≫ pullback.fst (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i) = (pullback.fst ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i j) ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i k) ≫ (fun i j => t 𝒰 f g i j) i j) ≫ pullback.fst (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i)
-- Mathlib/AlgebraicGeometry/Pullbacks.lean
C : Type u
inst✝¹ : Category.{v, u} C
X Y Z : Scheme
𝒰 : X.OpenCover
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (i : 𝒰.J), HasPullback (𝒰.map i ≫ f) g
i j k : 𝒰.J

⊢ ((fun i j k => t' 𝒰 f g i j k) i j k ≫ pullback.snd ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j k) ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j i)) ≫ pullback.snd (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i) = (pullback.fst ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i j) ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i k) ≫ (fun i j => t 𝒰 f g i j) i j) ≫ pullback.snd (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i)
-- Mathlib/AlgebraicGeometry/Pullbacks.lean
C : Type u
inst✝¹ : Category.{v, u} C
X Y Z : Scheme
𝒰 : X.OpenCover
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (i : 𝒰.J), HasPullback (𝒰.map i ≫ f) g
i j k : 𝒰.J

⊢ (((fun i j k => t' 𝒰 f g i j k) i j k ≫ pullback.snd ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j k) ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j i)) ≫ pullback.fst (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i)) ≫ pullback.fst (𝒰.map j ≫ f) g = ((pullback.fst ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i j) ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i k) ≫ (fun i j => t 𝒰 f g i j) i j) ≫ pullback.fst (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i)) ≫ pullback.fst (𝒰.map j ≫ f) g
-- Mathlib/AlgebraicGeometry/Pullbacks.lean
C : Type u
inst✝¹ : Category.{v, u} C
X Y Z : Scheme
𝒰 : X.OpenCover
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (i : 𝒰.J), HasPullback (𝒰.map i ≫ f) g
i j k : 𝒰.J

⊢ (((fun i j k => t' 𝒰 f g i j k) i j k ≫ pullback.snd ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j k) ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) j i)) ≫ pullback.fst (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i)) ≫ pullback.snd (𝒰.map j ≫ f) g = ((pullback.fst ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i j) ((fun i j => pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)) i k) ≫ (fun i j => t 𝒰 f g i j) i j) ≫ pullback.fst (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i)) ≫ pullback.snd (𝒰.map j ≫ f) g
-- Mathlib/AlgebraicTopology/SimplicialSet.lean
n : ℕ
i a b : Fin (n + 1)
hab : a ≤ b
H : {i, a, b}.card ≤ n

⊢ Set.range ⇑(asOrderHom (standardSimplex.edge n a b hab)) ∪ {i} ≠ Set.univ
-- Mathlib/GroupTheory/NoncommPiCoprod.lean
G : Type u_1
inst✝¹ : Group G
ι : Type u_2
hdec : DecidableEq ι
hfin : Fintype ι
H : ι → Type u_3
inst✝ : (i : ι) → Group (H i)
ϕ : (i : ι) → H i →* G
f g : (i : ι) → H i
hcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)
hind : CompleteLattice.Independent fun i => (ϕ i).range
hinj : ∀ (i : ι), Function.Injective ⇑(ϕ i)

⊢ Function.Injective ⇑(noncommPiCoprod ϕ hcomm)
-- Mathlib/Analysis/Calculus/Deriv/Slope.lean
𝕜 : Type u
inst✝⁵ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
E : Type w
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
f✝ f₀ f₁ g : 𝕜 → F
f' f₀' f₁' g' : F
x : 𝕜
s✝ t : Set 𝕜
L L₁ L₂ : Filter 𝕜
inst✝ : SeparableSpace 𝕜
f : 𝕜 → F
s : Set 𝕜

⊢ IsSeparable (range (derivWithin f s))
-- Mathlib/Analysis/Calculus/Deriv/Slope.lean
𝕜 : Type u
inst✝⁵ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
E : Type w
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
f✝ f₀ f₁ g : 𝕜 → F
f' f₀' f₁' g' : F
x : 𝕜
s✝ t✝ : Set 𝕜
L L₁ L₂ : Filter 𝕜
inst✝ : SeparableSpace 𝕜
f : 𝕜 → F
s t : Set 𝕜
ts : t ⊆ s
t_count : t.Countable
ht : s ⊆ closure t

⊢ IsSeparable (range (derivWithin f s))
-- Mathlib/Analysis/Calculus/Deriv/Slope.lean
𝕜 : Type u
inst✝⁵ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
E : Type w
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
f✝ f₀ f₁ g : 𝕜 → F
f' f₀' f₁' g' : F
x : 𝕜
s✝ t✝ : Set 𝕜
L L₁ L₂ : Filter 𝕜
inst✝ : SeparableSpace 𝕜
f : 𝕜 → F
s t : Set 𝕜
ts : t ⊆ s
t_count : t.Countable
ht : s ⊆ closure t
this : s ⊆ closure (s ∩ t)

⊢ IsSeparable (range (derivWithin f s))
-- Mathlib/Analysis/Calculus/Deriv/Slope.lean
𝕜 : Type u
inst✝⁵ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
E : Type w
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
f✝ f₀ f₁ g : 𝕜 → F
f' f₀' f₁' g' : F
x : 𝕜
s✝ t✝ : Set 𝕜
L L₁ L₂ : Filter 𝕜
inst✝ : SeparableSpace 𝕜
f : 𝕜 → F
s t : Set 𝕜
ts : t ⊆ s
t_count : t.Countable
ht : s ⊆ closure t
this : s ⊆ closure (s ∩ t)

⊢ IsSeparable (closure ↑(Submodule.span 𝕜 (f '' t)))
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean

⊢ 2 • ↑π = 0
-- Mathlib/Probability/Kernel/Condexp.lean
Ω : Type u_1
F : Type u_2
m mΩ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
μ : Measure Ω
inst✝¹ inst✝ : IsFiniteMeasure μ
hm : m ≤ mΩ
s : Set Ω
hs : MeasurableSet s

⊢ (fun ω => (((condexpKernel μ m) ω) s).toReal) =ᶠ[ae (μ.trim hm)] μ[s.indicator fun ω => 1|m]
-- Mathlib/Probability/Kernel/Condexp.lean
Ω : Type u_1
F : Type u_2
m mΩ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
μ : Measure Ω
inst✝¹ inst✝ : IsFiniteMeasure μ
hm : m ≤ mΩ
s : Set Ω
hs : MeasurableSet s

⊢ (fun ω => (((condexpKernel μ m) ω) s).toReal) =ᶠ[ae μ] μ[s.indicator fun ω => 1|m]
-- Mathlib/Probability/Kernel/Condexp.lean
Ω : Type u_1
F : Type u_2
m mΩ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
μ : Measure Ω
inst✝¹ inst✝ : IsFiniteMeasure μ
hm : m ≤ mΩ
s : Set Ω
hs : MeasurableSet s

⊢ StronglyMeasurable fun ω => (((condexpKernel μ m) ω) s).toReal
-- Mathlib/Data/Option/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
p : α → Prop
f✝ : (a : α) → p a → β
x✝ : Option α
f : β → γ
x : Option α
g : (a : α) → a ∈ x → Option β

⊢ Option.map f (x.pbind g) = x.pbind fun a H => Option.map f (g a H)
-- Mathlib/NumberTheory/Cyclotomic/Rat.lean
p : ℕ+
k : ℕ
K : Type u
inst✝² : Field K
inst✝¹ : CharZero K
ζ : K
hp : Fact (Nat.Prime ↑p)
inst✝ : IsCyclotomicExtension {p ^ k} ℚ K
hζ : IsPrimitiveRoot ζ ↑(p ^ k)

⊢ ∃ u n, Algebra.discr ℚ ⇑(IsPrimitiveRoot.subOnePowerBasis ℚ hζ).basis = ↑↑u * ↑↑p ^ n
-- Mathlib/NumberTheory/Cyclotomic/Rat.lean
p : ℕ+
k : ℕ
K : Type u
inst✝² : Field K
inst✝¹ : CharZero K
ζ : K
hp : Fact (Nat.Prime ↑p)
inst✝ : IsCyclotomicExtension {p ^ k} ℚ K
hζ : IsPrimitiveRoot ζ ↑(p ^ k)

⊢ ∃ u n, Algebra.discr ℚ ⇑(IsPrimitiveRoot.powerBasis ℚ hζ).basis = ↑↑u * ↑↑p ^ n
-- Mathlib/MeasureTheory/OuterMeasure/Operations.lean
α : Type u_1
β : Type u_2
m : OuterMeasure α
f : α → β
s : Set β

⊢ ((map f) ⊤) s = ((restrict (range f)) ⊤) s
-- Mathlib/MeasureTheory/Measure/LogLikelihoodRatio.lean
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
f : α → ℝ
inst✝¹ : IsFiniteMeasure μ
inst✝ : μ.HaveLebesgueDecomposition ν
hμν : μ ≪ ν
c : ℝ≥0∞
hc : c ≠ 0
hc_ne_top : c ≠ ⊤

⊢ llr (c • μ) ν =ᶠ[ae μ] fun x => llr μ ν x + log c.toReal
-- Mathlib/MeasureTheory/Measure/LogLikelihoodRatio.lean
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
f : α → ℝ
inst✝¹ : IsFiniteMeasure μ
inst✝ : μ.HaveLebesgueDecomposition ν
hμν : μ ≪ ν
c : ℝ≥0∞
hc : c ≠ 0
hc_ne_top : c ≠ ⊤

⊢ (fun x => log ((c • μ).rnDeriv ν x).toReal) =ᶠ[ae μ] fun x => log (μ.rnDeriv ν x).toReal + log c.toReal
-- Mathlib/MeasureTheory/Measure/LogLikelihoodRatio.lean
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
f : α → ℝ
inst✝¹ : IsFiniteMeasure μ
inst✝ : μ.HaveLebesgueDecomposition ν
hμν : μ ≪ ν
c : ℝ≥0∞
hc : c ≠ 0
hc_ne_top : c ≠ ⊤
h : (c • μ).rnDeriv ν =ᶠ[ae ν] c • μ.rnDeriv ν

⊢ (fun x => log ((c • μ).rnDeriv ν x).toReal) =ᶠ[ae μ] fun x => log (μ.rnDeriv ν x).toReal + log c.toReal
-- Mathlib/MeasureTheory/Measure/LogLikelihoodRatio.lean
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
f : α → ℝ
inst✝¹ : IsFiniteMeasure μ
inst✝ : μ.HaveLebesgueDecomposition ν
hμν : μ ≪ ν
c : ℝ≥0∞
hc : c ≠ 0
hc_ne_top : c ≠ ⊤
h : (c • μ).rnDeriv ν =ᶠ[ae ν] c • μ.rnDeriv ν
x : α
hx_eq : (c • μ).rnDeriv ν x = (c • μ.rnDeriv ν) x
hx_pos : 0 < μ.rnDeriv ν x
hx_ne_top : μ.rnDeriv ν x < ⊤

⊢ log ((c • μ).rnDeriv ν x).toReal = log (μ.rnDeriv ν x).toReal + log c.toReal
-- Mathlib/MeasureTheory/Measure/LogLikelihoodRatio.lean
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
f : α → ℝ
inst✝¹ : IsFiniteMeasure μ
inst✝ : μ.HaveLebesgueDecomposition ν
hμν : μ ≪ ν
c : ℝ≥0∞
hc : c ≠ 0
hc_ne_top : c ≠ ⊤
h : (c • μ).rnDeriv ν =ᶠ[ae ν] c • μ.rnDeriv ν
x : α
hx_eq : (c • μ).rnDeriv ν x = (c • μ.rnDeriv ν) x
hx_pos : 0 < μ.rnDeriv ν x
hx_ne_top : μ.rnDeriv ν x < ⊤

⊢ log ((c • μ.rnDeriv ν) x).toReal = log (μ.rnDeriv ν x).toReal + log c.toReal
-- Mathlib/MeasureTheory/Measure/LogLikelihoodRatio.lean
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
f : α → ℝ
inst✝¹ : IsFiniteMeasure μ
inst✝ : μ.HaveLebesgueDecomposition ν
hμν : μ ≪ ν
c : ℝ≥0∞
hc : c ≠ 0
hc_ne_top : c ≠ ⊤
h : (c • μ).rnDeriv ν =ᶠ[ae ν] c • μ.rnDeriv ν
x : α
hx_eq : (c • μ).rnDeriv ν x = (c • μ.rnDeriv ν) x
hx_pos : 0 < μ.rnDeriv ν x
hx_ne_top : μ.rnDeriv ν x < ⊤

⊢ log (c.toReal * (μ.rnDeriv ν x).toReal) = log (μ.rnDeriv ν x).toReal + log c.toReal
-- Mathlib/MeasureTheory/Measure/LogLikelihoodRatio.lean
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
f : α → ℝ
inst✝¹ : IsFiniteMeasure μ
inst✝ : μ.HaveLebesgueDecomposition ν
hμν : μ ≪ ν
c : ℝ≥0∞
hc : c ≠ 0
hc_ne_top : c ≠ ⊤
h : (c • μ).rnDeriv ν =ᶠ[ae ν] c • μ.rnDeriv ν
x : α
hx_eq : (c • μ).rnDeriv ν x = (c • μ.rnDeriv ν) x
hx_pos : 0 < μ.rnDeriv ν x
hx_ne_top : μ.rnDeriv ν x < ⊤

⊢ log c.toReal + log (μ.rnDeriv ν x).toReal = log (μ.rnDeriv ν x).toReal + log c.toReal
-- Mathlib/MeasureTheory/Measure/LogLikelihoodRatio.lean
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
f : α → ℝ
inst✝¹ : IsFiniteMeasure μ
inst✝ : μ.HaveLebesgueDecomposition ν
hμν : μ ≪ ν
c : ℝ≥0∞
hc : c ≠ 0
hc_ne_top : c ≠ ⊤
h : (c • μ).rnDeriv ν =ᶠ[ae ν] c • μ.rnDeriv ν
x : α
hx_eq : (c • μ).rnDeriv ν x = (c • μ.rnDeriv ν) x
hx_pos : 0 < μ.rnDeriv ν x
hx_ne_top : μ.rnDeriv ν x < ⊤

⊢ c.toReal ≠ 0
-- Mathlib/MeasureTheory/Measure/LogLikelihoodRatio.lean
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
f : α → ℝ
inst✝¹ : IsFiniteMeasure μ
inst✝ : μ.HaveLebesgueDecomposition ν
hμν : μ ≪ ν
c : ℝ≥0∞
hc : c ≠ 0
hc_ne_top : c ≠ ⊤
h : (c • μ).rnDeriv ν =ᶠ[ae ν] c • μ.rnDeriv ν
x : α
hx_eq : (c • μ).rnDeriv ν x = (c • μ.rnDeriv ν) x
hx_pos : 0 < μ.rnDeriv ν x
hx_ne_top : μ.rnDeriv ν x < ⊤

⊢ (μ.rnDeriv ν x).toReal ≠ 0
-- Mathlib/Algebra/Algebra/Subalgebra/Rank.lean
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
A B : Subalgebra R S
inst✝³ : Module.Free R ↥A
inst✝² : Module.Free R ↥B
inst✝¹ : Module.Free ↥A ↥(Algebra.adjoin ↥A ↑B)
inst✝ : Module.Free ↥B ↥(Algebra.adjoin ↥B ↑A)

⊢ finrank R ↥(A ⊔ B) = finrank R ↥B * finrank ↥B ↥(Algebra.adjoin ↥B ↑A)
-- Mathlib/Data/Multiset/Basic.lean
α : Type u_1
β : Type v
γ : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
hf : Injective f
x : α
s : Multiset α

⊢ map f (s.erase x) = (map f s).erase (f x)
-- Mathlib/Data/Multiset/Basic.lean
α : Type u_1
β : Type v
γ : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
hf : Injective f
x : α

⊢ map f (erase 0 x) = (map f 0).erase (f x)
-- Mathlib/Data/Multiset/Basic.lean
α : Type u_1
β : Type v
γ : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
hf : Injective f
x y : α
s : Multiset α
ih : map f (s.erase x) = (map f s).erase (f x)

⊢ map f ((y ::ₘ s).erase x) = (map f (y ::ₘ s)).erase (f x)
-- Mathlib/Order/SupIndep.lean
α✝ : Type u_1
β : Type u_2
ι : Type u_3
ι' : Type u_4
inst✝¹ : CompleteLattice α✝
s✝ : Set α✝
hs : SetIndependent s✝
α : Type u_5
inst✝ : CompleteLattice α
s : Set α

⊢ SetIndependent s ↔ Independent Subtype.val
-- Mathlib/Order/SupIndep.lean
α✝ : Type u_1
β : Type u_2
ι : Type u_3
ι' : Type u_4
inst✝¹ : CompleteLattice α✝
s✝ : Set α✝
hs : SetIndependent s✝
α : Type u_5
inst✝ : CompleteLattice α
s : Set α

⊢ (∀ ⦃a : α⦄, a ∈ s → Disjoint a (⨆ a_2 ∈ s \ {a}, a_2)) ↔ ∀ (x : α) (h : x ∈ s), Disjoint x (⨆ j, ⨆ (_ : j ≠ ⟨x, h⟩), ↑j)
-- Mathlib/Order/SupIndep.lean
α✝ : Type u_1
β : Type u_2
ι : Type u_3
ι' : Type u_4
inst✝¹ : CompleteLattice α✝
s✝ : Set α✝
hs : SetIndependent s✝
α : Type u_5
inst✝ : CompleteLattice α
s : Set α
a : α
ha : a ∈ s

⊢ Disjoint a (⨆ a_1 ∈ s \ {a}, a_1) ↔ Disjoint a (⨆ j, ⨆ (_ : j ≠ ⟨a, ha⟩), ↑j)
-- Mathlib/Topology/Basic.lean
X : Type u
Y : Type v
ι : Sort w
α : Type u_1
β : Type u_2
x : X
s s₁ s₂ t : Set X
p p₁ p₂ : X → Prop
inst✝ : TopologicalSpace X
h : IsClosed t

⊢ interior (s ∪ t) ⊆ interior s ∪ t
-- Mathlib/Algebra/Polynomial/Expand.lean
R : Type u
inst✝¹ : CommSemiring R
S : Type v
inst✝ : CommSemiring S
p✝ q p : ℕ
f : R[X]
hp : 0 < p

⊢ ((expand R p) f).Monic ↔ f.Monic
-- Mathlib/GroupTheory/OrderOfElement.lean
G : Type u_1
H : Type u_2
A : Type u_3
α : Type u_4
β : Type u_5
inst✝ : Monoid G
a b x y : G
n m p : ℕ
hp : Fact (Nat.Prime p)
hnot : ¬x ^ p ^ n = 1
hfin : x ^ p ^ (n + 1) = 1

⊢ orderOf x = p ^ (n + 1)
-- Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
inst✝ : ConditionallyCompleteLinearOrderBot α
f : ι → α

⊢ ⨅ x, ↑(f x) = ⊤ ↔ IsEmpty ι
-- Mathlib/MeasureTheory/Measure/Portmanteau.lean
Ω : Type u_1
inst✝² : MeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
μs : ι → Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
E : Set Ω
E_mble : MeasurableSet E
h : μ E ≤ liminf (fun i => (μs i) E) L

⊢ limsup (fun i => (μs i) Eᶜ) L ≤ μ Eᶜ
-- Mathlib/MeasureTheory/Measure/Portmanteau.lean
Ω : Type u_1
inst✝² : MeasurableSpace Ω
ι : Type u_2
μ : Measure Ω
μs : ι → Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
E : Set Ω
E_mble : MeasurableSet E
h : μ E ≤ liminf (fun i => (μs i) E) ⊥

⊢ limsup (fun i => (μs i) Eᶜ) ⊥ ≤ μ Eᶜ
-- Mathlib/MeasureTheory/Measure/Portmanteau.lean
Ω : Type u_1
inst✝² : MeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
μs : ι → Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
E : Set Ω
E_mble : MeasurableSet E
h : μ E ≤ liminf (fun i => (μs i) E) L
hne : L.NeBot

⊢ limsup (fun i => (μs i) Eᶜ) L ≤ μ Eᶜ
-- Mathlib/MeasureTheory/Measure/Portmanteau.lean
Ω : Type u_1
inst✝² : MeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
μs : ι → Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
E : Set Ω
E_mble : MeasurableSet E
h : μ E ≤ liminf (fun i => (μs i) E) L
hne : L.NeBot
meas_Ec : μ Eᶜ = 1 - μ E

⊢ limsup (fun i => (μs i) Eᶜ) L ≤ μ Eᶜ
-- Mathlib/MeasureTheory/Measure/Portmanteau.lean
Ω : Type u_1
inst✝² : MeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
μs : ι → Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
E : Set Ω
E_mble : MeasurableSet E
h : μ E ≤ liminf (fun i => (μs i) E) L
hne : L.NeBot
meas_Ec : μ Eᶜ = 1 - μ E
meas_i_Ec : ∀ (i : ι), (μs i) Eᶜ = 1 - (μs i) E

⊢ limsup (fun i => (μs i) Eᶜ) L ≤ μ Eᶜ
-- Mathlib/MeasureTheory/Measure/Portmanteau.lean
Ω : Type u_1
inst✝² : MeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
μs : ι → Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
E : Set Ω
E_mble : MeasurableSet E
h : μ E ≤ liminf (fun i => (μs i) E) L
hne : L.NeBot
meas_Ec : μ Eᶜ = 1 - μ E
meas_i_Ec : ∀ (i : ι), (μs i) Eᶜ = 1 - (μs i) E
obs : limsup (fun i => 1 - (μs i) E) L = limsup ((fun x => 1 - x) ∘ fun i => (μs i) E) L

⊢ limsup ((fun x => 1 - x) ∘ fun i => (μs i) E) L ≤ 1 - μ E
-- Mathlib/MeasureTheory/Measure/Portmanteau.lean
Ω : Type u_1
inst✝² : MeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
μs : ι → Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
E : Set Ω
E_mble : MeasurableSet E
h : μ E ≤ liminf (fun i => (μs i) E) L
hne : L.NeBot
meas_Ec : μ Eᶜ = 1 - μ E
meas_i_Ec : ∀ (i : ι), (μs i) Eᶜ = 1 - (μs i) E
obs : limsup (fun i => 1 - (μs i) E) L = limsup ((fun x => 1 - x) ∘ fun i => (μs i) E) L
this : 1 - liminf (fun i => (μs i) E) L = limsup ((fun x => 1 - x) ∘ fun i => (μs i) E) L

⊢ limsup ((fun x => 1 - x) ∘ fun i => (μs i) E) L ≤ 1 - μ E
-- Mathlib/MeasureTheory/Measure/Portmanteau.lean
Ω : Type u_1
inst✝² : MeasurableSpace Ω
ι : Type u_2
L : Filter ι
μ : Measure Ω
μs : ι → Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)
E : Set Ω
E_mble : MeasurableSet E
h : μ E ≤ liminf (fun i => (μs i) E) L
hne : L.NeBot
meas_Ec : μ Eᶜ = 1 - μ E
meas_i_Ec : ∀ (i : ι), (μs i) Eᶜ = 1 - (μs i) E
obs : limsup (fun i => 1 - (μs i) E) L = limsup ((fun x => 1 - x) ∘ fun i => (μs i) E) L
this : 1 - liminf (fun i => (μs i) E) L = limsup ((fun x => 1 - x) ∘ fun i => (μs i) E) L

⊢ 1 - liminf (fun i => (μs i) E) L ≤ 1 - μ E
-- Mathlib/LinearAlgebra/Dual.lean
R : Type uR
M : Type uM
K : Type uK
V : Type uV
ι : Type uι
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : DecidableEq ι
b : Basis ι R M
inst✝ : _root_.Finite ι
i j : ι

⊢ (b.dualBasis i) (b j) = if j = i then 1 else 0
-- Mathlib/LinearAlgebra/Dual.lean
R : Type uR
M : Type uM
K : Type uK
V : Type uV
ι : Type uι
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : DecidableEq ι
b : Basis ι R M
inst✝ : _root_.Finite ι
i j : ι

⊢ j = i ↔ i = j
-- Mathlib/Topology/MetricSpace/Gluing.lean
X : ℕ → Type u
inst✝ : (n : ℕ) → MetricSpace (X n)
f : (n : ℕ) → X n → X (n + 1)
I : ∀ (n : ℕ), Isometry (f n)
n : ℕ
x y : X n

⊢ inductiveLimitDist f ⟨n, x⟩ ⟨n, y⟩ = dist x y
-- Mathlib/Analysis/Calculus/MeanValue.lean
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → E
a b : ℝ
f' g : ℝ → E
derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x
derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x
fcont : ContinuousOn f (Icc a b)
gcont : ContinuousOn g (Icc a b)
hi : f a = g a

⊢ ∀ y ∈ Icc a b, f y = g y
-- Mathlib/Analysis/Calculus/MeanValue.lean
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → E
a b : ℝ
f' g : ℝ → E
derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x
derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x
fcont : ContinuousOn f (Icc a b)
gcont : ContinuousOn g (Icc a b)
hi : f a - g a = 0

⊢ ∀ y ∈ Icc a b, f y - g y = 0
-- Mathlib/RingTheory/PowerSeries/Basic.lean
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
φ ψ : R[X]
a : R

⊢ ↑(C a) = (PowerSeries.C R) a
-- Mathlib/RingTheory/PowerSeries/Basic.lean
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
φ ψ : R[X]
a : R
this : ↑((monomial 0) a) = (PowerSeries.monomial R 0) a

⊢ ↑(C a) = (PowerSeries.C R) a
-- Mathlib/MeasureTheory/Measure/Stieltjes.lean
f : StieltjesFunction
a b : ℝ

⊢ f.length (Ioc a b) = ofReal (↑f b - ↑f a)
-- Mathlib/MeasureTheory/Measure/Stieltjes.lean
f : StieltjesFunction
a b a' b' : ℝ
h : Ioc a b ⊆ Ioc a' b'

⊢ (↑f b - ↑f a).toNNReal ≤ (↑f b' - ↑f a').toNNReal
-- Mathlib/MeasureTheory/Measure/Stieltjes.lean
f : StieltjesFunction
a b a' b' : ℝ
h : Ioc a b ⊆ Ioc a' b'
ab : b ≤ a

⊢ (↑f b - ↑f a).toNNReal ≤ (↑f b' - ↑f a').toNNReal
-- Mathlib/MeasureTheory/Measure/Stieltjes.lean
f : StieltjesFunction
a b a' b' : ℝ
h : Ioc a b ⊆ Ioc a' b'
ab : a < b

⊢ (↑f b - ↑f a).toNNReal ≤ (↑f b' - ↑f a').toNNReal
-- Mathlib/MeasureTheory/Measure/Stieltjes.lean
f : StieltjesFunction
a b a' b' : ℝ
h : Ioc a b ⊆ Ioc a' b'
ab : a < b
h₁ : b ≤ b'
h₂ : a' ≤ a

⊢ (↑f b - ↑f a).toNNReal ≤ (↑f b' - ↑f a').toNNReal
-- Mathlib/RingTheory/OreLocalization/Basic.lean
R : Type u_1
inst✝¹ : CommMonoid R
S : Submonoid R
inst✝ : OreSet S
r₁ r₂ : R
s₁ s₂ : ↥S

⊢ r₁ /ₒ s₁ * (r₂ /ₒ s₂) = r₁ * r₂ /ₒ (s₁ * s₂)
-- Mathlib/SetTheory/Cardinal/ENat.lean

⊢ StrictMonoOn (⇑toENat) (Iic ℵ₀)
-- Mathlib/SetTheory/Cardinal/ENat.lean

⊢ ∀ (i i_1 : ℕ∞), i < i_1 → i < i_1
-- Mathlib/MeasureTheory/MeasurableSpace/Basic.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 α
mβ : MeasurableSpace β
s : Set α

⊢ MeasurableSet (Sum.inl '' s) ↔ MeasurableSet s
-- Mathlib/Topology/Category/TopCat/Limits/Products.lean
J : Type v
inst✝ : SmallCategory J
ι : Type v
α : ι → TopCat
i : ι
x : ↑(∏ᶜ α)

⊢ (piIsoPi α).hom x i = (Pi.π α i) x
-- Mathlib/Topology/Category/TopCat/Limits/Products.lean
J : Type v
inst✝ : SmallCategory J
ι : Type v
α : ι → TopCat
i : ι
x : ↑(∏ᶜ α)
this : (piIsoPi α).inv ≫ Pi.π α i = piπ α i

⊢ (piIsoPi α).hom x i = (Pi.π α i) x
-- Mathlib/Topology/Category/TopCat/Limits/Products.lean
J : Type v
inst✝ : SmallCategory J
ι : Type v
α : ι → TopCat
i : ι
x : ↑(∏ᶜ α)
this : Pi.π α i = (piIsoPi α).hom ≫ piπ α i

⊢ (piIsoPi α).hom x i = (Pi.π α i) x
-- Mathlib/Data/Set/Card.lean
α : Type u_1
β : Type u_2
s t : Set α
k : ℕ∞
hst : s ⊆ t
hsk : s.encard ≤ k
hkt : k ≤ t.encard

⊢ ∃ r, s ⊆ r ∧ r ⊆ t ∧ r.encard = k
-- Mathlib/Data/Set/Card.lean
α : Type u_1
β : Type u_2
s t : Set α
k : ℕ∞
hst : s ⊆ t
hsk : s.encard ≤ k
hkt : k ≤ t.encard
hs : s.encard = ⊤

⊢ ∃ r, s ⊆ r ∧ r ⊆ t ∧ r.encard = k
-- Mathlib/Data/Set/Card.lean
α : Type u_1
β : Type u_2
s t : Set α
k : ℕ∞
hst : s ⊆ t
hsk : s.encard ≤ k
hkt : k ≤ t.encard
hs : s.encard ≠ ⊤

⊢ ∃ r, s ⊆ r ∧ r ⊆ t ∧ r.encard = k