ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean
μ : Measure ℝ
inst✝ : NoAtoms μ
s : Set ℝ
p : ℝ → Prop
h : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ.restrict (s ∩ Ioo a b), p x
T : ↑s × ↑s → Set ℝ := fun p => Ioo ↑p.1 ↑p.2
u : Set ℝ := ⋃ i, T i
hfinite : (s \ u).Finite
A : Set (↑s × ↑s)
A_count : A.Countable
hA : ⋃ i ∈ A, T i = ⋃ i, T i
this : s ⊆ s \ u ∪ ⋃ p ∈ A, s ∩ T p

⊢ (∀ᵐ (x : ℝ) ∂μ.restrict (s \ u), p x) ∧ ∀ i ∈ A, ∀ᵐ (x : ℝ) ∂μ.restrict (s ∩ T i), p x
-- Mathlib/Data/Finset/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝ : DecidableEq α
s✝ t✝ u v : Finset α
a✝ b a : α
s t : Finset α

⊢ s ⊆ insert a t ↔ s.erase a ⊆ t
-- Mathlib/Data/Finset/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝ : DecidableEq α
s✝ t✝ u v : Finset α
a✝ b a : α
s t : Finset α

⊢ (∀ ⦃x : α⦄, x ∈ s → ¬x = a → x ∈ t) ↔ ∀ ⦃x : α⦄, x ≠ a → x ∈ s → x ∈ t
-- Mathlib/Data/Set/Pairwise/Lattice.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Type u_5
κ : Sort u_6
r✝ p q : α → α → Prop
f g : ι → α
s✝ t u : Set α
a b : α
r : α → α → Prop
s : Set (Set α)
h : DirectedOn (fun x x_1 => x ⊆ x_1) s

⊢ (⋃₀ s).Pairwise r ↔ ∀ a ∈ s, a.Pairwise r
-- Mathlib/Data/Multiset/Dedup.lean
α : Type u_1
s t : Multiset α
n : ℕ
h : s.Nodup
hn : n ≠ 0

⊢ s ≤ n • t ↔ s ≤ t
-- Mathlib/Algebra/Order/BigOperators/Group/Multiset.lean
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝ : OrderedCommMonoid α
s✝ t : Multiset α
a : α
s : Multiset α
n : α
h : ∀ x ∈ s, x ≤ n

⊢ s.prod ≤ n ^ card s
-- Mathlib/Algebra/Order/BigOperators/Group/Multiset.lean
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝ : OrderedCommMonoid α
s t : Multiset α
a n : α
a✝ : List α
h : ∀ x ∈ ⟦a✝⟧, x ≤ n

⊢ prod ⟦a✝⟧ ≤ n ^ card ⟦a✝⟧
-- Mathlib/Algebra/BigOperators/Group/Finset.lean
ι : Type u_1
κ : Type u_2
α : Type u_3
β : Type u_4
γ : Type u_5
s s₁ s₂ : Finset α
a : α
f✝ g✝ : α → β
inst✝¹ : CommMonoid β
inst✝ : DecidableEq α
m : Multiset α
f : { x // x ∈ m } → β
g : α → β
hfg : ∀ (x : { x // x ∈ m }), f x = g ↑x

⊢ ∏ x : { x // x ∈ m }, f x = ∏ x ∈ m.toFinset, g x
-- Mathlib/Data/List/MinMax.lean
α : Type u_1
β : Type u_2
r : α → α → Prop
inst✝ : DecidableRel r
l : List α
o✝ : Option α
a✝ m✝ : α
hr₀ : Irreflexive r
hr₁ : Transitive r
tl : List α
a : α
ih : ∀ {a m : α} {o : Option α}, a ∈ tl → m ∈ foldl (argAux r) o tl → ¬r a m
b m : α
o : Option α
hb : b ∈ tl ++ [a]
ho : m ∈ foldl (argAux r) o (tl ++ [a])

⊢ ¬r b m
-- Mathlib/Data/List/MinMax.lean
α : Type u_1
β : Type u_2
r : α → α → Prop
inst✝ : DecidableRel r
l : List α
o✝ : Option α
a✝ m✝ : α
hr₀ : Irreflexive r
hr₁ : Transitive r
tl : List α
a : α
ih : ∀ {a m : α} {o : Option α}, a ∈ tl → m ∈ foldl (argAux r) o tl → ¬r a m
b m : α
o : Option α
hb : b ∈ tl ++ [a]
ho : m ∈ Option.casesOn (foldl (argAux r) o tl) (some a) fun c => if r a c then some a else some c

⊢ ¬r b m
-- Mathlib/Data/List/MinMax.lean
α : Type u_1
β : Type u_2
r : α → α → Prop
inst✝ : DecidableRel r
l : List α
o✝ : Option α
a✝ m✝ : α
hr₀ : Irreflexive r
hr₁ : Transitive r
tl : List α
a : α
ih : ∀ {a m : α} {o : Option α}, a ∈ tl → m ∈ foldl (argAux r) o tl → ¬r a m
b m : α
o : Option α
hb : b ∈ tl ++ [a]
ho : m ∈ Option.casesOn (foldl (argAux r) o tl) (some a) fun c => if r a c then some a else some c
c : α
hf : foldl (argAux r) o tl = some c

⊢ ¬r b m
-- Mathlib/Data/List/MinMax.lean
α : Type u_1
β : Type u_2
r : α → α → Prop
inst✝ : DecidableRel r
l : List α
o✝ : Option α
a✝ m✝ : α
hr₀ : Irreflexive r
hr₁ : Transitive r
tl : List α
a : α
ih : ∀ {a m : α} {o : Option α}, a ∈ tl → m ∈ foldl (argAux r) o tl → ¬r a m
b m : α
o : Option α
hb : b ∈ tl ++ [a]
c : α
ho : (Option.casesOn (some c) (some a) fun c => if r a c then some a else some c) = some m
hf : foldl (argAux r) o tl = some c

⊢ ¬r b m
-- Mathlib/Data/List/MinMax.lean
α : Type u_1
β : Type u_2
r : α → α → Prop
inst✝ : DecidableRel r
l : List α
o✝ : Option α
a✝ m✝ : α
hr₀ : Irreflexive r
hr₁ : Transitive r
tl : List α
a : α
ih : ∀ {a m : α} {o : Option α}, a ∈ tl → m ∈ foldl (argAux r) o tl → ¬r a m
b m : α
o : Option α
hb : b ∈ tl ++ [a]
c : α
ho : (if r a c then some a else some c) = some m
hf : foldl (argAux r) o tl = some c

⊢ ¬r b m
-- Mathlib/Data/List/MinMax.lean
α : Type u_1
β : Type u_2
r : α → α → Prop
inst✝ : DecidableRel r
l : List α
o✝ : Option α
a✝ m : α
hr₀ : Irreflexive r
hr₁ : Transitive r
tl : List α
a : α
ih : ∀ {a m : α} {o : Option α}, a ∈ tl → m ∈ foldl (argAux r) o tl → ¬r a m
b : α
o : Option α
hb : b ∈ tl ++ [a]
c : α
hf : foldl (argAux r) o tl = some c
hac : r a c
h : b ∈ tl

⊢ ¬r b a
-- Mathlib/Data/List/MinMax.lean
α : Type u_1
β : Type u_2
r : α → α → Prop
inst✝ : DecidableRel r
l : List α
o✝ : Option α
a✝ m : α
hr₀ : Irreflexive r
hr₁ : Transitive r
tl : List α
a : α
ih : ∀ {a m : α} {o : Option α}, a ∈ tl → m ∈ foldl (argAux r) o tl → ¬r a m
b : α
o : Option α
hb : b ∈ tl ++ [a]
c : α
hf : foldl (argAux r) o tl = some c
hac : r a c
h : b ∈ [a]

⊢ ¬r b a
-- Mathlib/Data/List/MinMax.lean
α : Type u_1
β : Type u_2
r : α → α → Prop
inst✝ : DecidableRel r
l : List α
o✝ : Option α
a✝ m : α
hr₀ : Irreflexive r
hr₁ : Transitive r
tl : List α
a : α
ih : ∀ {a m : α} {o : Option α}, a ∈ tl → m ∈ foldl (argAux r) o tl → ¬r a m
b : α
o : Option α
hb : b ∈ tl ++ [a]
c : α
hf : foldl (argAux r) o tl = some c
hac : ¬r a c
h : b ∈ tl

⊢ ¬r b c
-- Mathlib/Data/List/MinMax.lean
α : Type u_1
β : Type u_2
r : α → α → Prop
inst✝ : DecidableRel r
l : List α
o✝ : Option α
a✝ m : α
hr₀ : Irreflexive r
hr₁ : Transitive r
tl : List α
a : α
ih : ∀ {a m : α} {o : Option α}, a ∈ tl → m ∈ foldl (argAux r) o tl → ¬r a m
b : α
o : Option α
hb : b ∈ tl ++ [a]
c : α
hf : foldl (argAux r) o tl = some c
hac : ¬r a c
h : b ∈ [a]

⊢ ¬r b c
-- Mathlib/Topology/MetricSpace/Infsep.lean
α : Type u_1
β : Type u_2
inst✝² : PseudoMetricSpace α
x y z : α
s t : Set α
inst✝¹ : DecidableEq α
inst✝ : Fintype ↑s
hs : s.Nontrivial

⊢ s.infsep = s.offDiag.toFinset.inf' ⋯ (uncurry dist)
-- Mathlib/Topology/Semicontinuous.lean
α : Type u_1
inst✝³ : TopologicalSpace α
β : Type u_2
inst✝² : Preorder β
f✝ g : α → β
x : α
s t : Set α
y z : β
γ : Type u_3
inst✝¹ : CompleteLinearOrder γ
inst✝ : DenselyOrdered γ
f : α → γ

⊢ f x ≤ liminf f (𝓝[s] x) → LowerSemicontinuousWithinAt f s x
-- Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean
f g : CircleDeg1Lift
x : ℝ
n : ℕ
m : ℤ
h : (f ^ n) x = x + ↑m
hn : 0 < n

⊢ τ f = ↑m / ↑n
-- Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean
f g : CircleDeg1Lift
x : ℝ
n : ℕ
m : ℤ
h : (f ^ n) x = x + ↑m
hn : 0 < n
this : τ (f ^ n) = ↑m

⊢ τ f = ↑m / ↑n
-- Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean
f g : CircleDeg1Lift
x : ℝ
n : ℕ
m : ℤ
h : (f ^ n) x = x + ↑m
hn : 0 < n
this : τ f * ↑n = ↑m

⊢ ↑n ≠ 0
-- Mathlib/Algebra/Category/ModuleCat/Adjunctions.lean
R : Type u
inst✝ : CommRing R
X Y Z : Type u

⊢ ((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫ (μ R (X ⊗ Y) Z).hom ≫ map (free R).obj (α_ X Y Z).hom = (α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫ (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom
-- Mathlib/Algebra/Category/ModuleCat/Adjunctions.lean
R : Type u
inst✝ : CommRing R
X Y Z : Type u

⊢ (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z)).compr₂ (((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫ (μ R (X ⊗ Y) Z).hom ≫ map (free R).obj (α_ X Y Z).hom) = (TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z)).compr₂ ((α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫ (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom)
-- Mathlib/Algebra/Category/ModuleCat/Adjunctions.lean
R : Type u
inst✝ : CommRing R
X Y Z : Type u

⊢ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y)).compr₂ ((TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z)).compr₂ (((μ R X Y).hom ⊗ 𝟙 ((free R).obj Z)) ≫ (μ R (X ⊗ Y) Z).hom ≫ map (free R).obj (α_ X Y Z).hom)) = (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj Y)).compr₂ ((TensorProduct.mk R ↑((free R).obj X ⊗ (free R).obj Y) ↑((free R).obj Z)).compr₂ ((α_ ((free R).obj X) ((free R).obj Y) ((free R).obj Z)).hom ≫ (𝟙 ((free R).obj X) ⊗ (μ R Y Z).hom) ≫ (μ R X (Y ⊗ Z)).hom))
-- Mathlib/Algebra/Category/ModuleCat/Adjunctions.lean
R : Type u
inst✝ : CommRing R
X Y Z : Type u

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

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

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

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

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

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

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

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

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

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

⊢ (Finsupp.mapDomain (α_ X Y Z).hom ((finsuppTensorFinsupp' R (X ⊗ Y) Z) ((finsuppTensorFinsupp' R X Y) (Finsupp.single x 1 ⊗ₜ[R] Finsupp.single y 1) ⊗ₜ[R] Finsupp.single z 1))) a = ((finsuppTensorFinsupp' R X (Y ⊗ Z)) (Finsupp.single x 1 ⊗ₜ[R] (finsuppTensorFinsupp' R Y Z) (Finsupp.single y 1 ⊗ₜ[R] Finsupp.single z 1))) a
-- Mathlib/Algebra/Category/ModuleCat/Adjunctions.lean
R : Type u
inst✝ : CommRing R
X Y Z : Type u
x : X
y : Y
z : Z
a : X ⊗ Y ⊗ Z

⊢ (Finsupp.single (x, y, z) 1) a = (Finsupp.single (x, y, z) 1) a
-- Mathlib/Data/Nat/Factorization/Basic.lean
a b m✝ n✝ p : ℕ
β : Type u_1
inst✝ : CommMonoid β
m n : ℕ
f : ℕ → β

⊢ (m.gcd n).primeFactors.prod f * (m * n).primeFactors.prod f = m.primeFactors.prod f * n.primeFactors.prod f
-- Mathlib/Data/Nat/Factorization/Basic.lean
a b m n✝ p : ℕ
β : Type u_1
inst✝ : CommMonoid β
n : ℕ
f : ℕ → β

⊢ (gcd 0 n).primeFactors.prod f * (0 * n).primeFactors.prod f = (primeFactors 0).prod f * n.primeFactors.prod f
-- Mathlib/Data/Nat/Factorization/Basic.lean
a b m✝ n✝ p : ℕ
β : Type u_1
inst✝ : CommMonoid β
m n : ℕ
f : ℕ → β
hm₀ : m ≠ 0

⊢ (m.gcd n).primeFactors.prod f * (m * n).primeFactors.prod f = m.primeFactors.prod f * n.primeFactors.prod f
-- Mathlib/Data/Nat/Factorization/Basic.lean
a b m✝ n p : ℕ
β : Type u_1
inst✝ : CommMonoid β
m : ℕ
f : ℕ → β
hm₀ : m ≠ 0

⊢ (m.gcd 0).primeFactors.prod f * (m * 0).primeFactors.prod f = m.primeFactors.prod f * (primeFactors 0).prod f
-- Mathlib/Data/Nat/Factorization/Basic.lean
a b m✝ n✝ p : ℕ
β : Type u_1
inst✝ : CommMonoid β
m n : ℕ
f : ℕ → β
hm₀ : m ≠ 0
hn₀ : n ≠ 0

⊢ (m.gcd n).primeFactors.prod f * (m * n).primeFactors.prod f = m.primeFactors.prod f * n.primeFactors.prod f
-- Mathlib/MeasureTheory/Measure/Haar/Basic.lean
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ ↑K₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₂ - f K₁

⊢ chaar K₀ K₁ ≤ chaar K₀ K₂
-- Mathlib/MeasureTheory/Measure/Haar/Basic.lean
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ ↑K₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₂ - f K₁
this : Continuous eval

⊢ chaar K₀ K₁ ≤ chaar K₀ K₂
-- Mathlib/MeasureTheory/Measure/Haar/Basic.lean
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ ↑K₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₂ - f K₁
this : Continuous eval

⊢ 0 ≤ chaar K₀ K₂ - chaar K₀ K₁
-- Mathlib/MeasureTheory/Measure/Haar/Basic.lean
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ ↑K₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₂ - f K₁
this : Continuous eval

⊢ chaar K₀ ∈ eval ⁻¹' Ici 0
-- Mathlib/MeasureTheory/Measure/Haar/Basic.lean
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ ↑K₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₂ - f K₁
this : Continuous eval

⊢ clPrehaar ↑K₀ ⊤ ⊆ eval ⁻¹' Ici 0
-- Mathlib/MeasureTheory/Measure/Haar/Basic.lean
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ ↑K₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₂ - f K₁
this : Continuous eval

⊢ closure (prehaar ↑K₀ '' {U | U ⊆ ↑⊤.toOpens ∧ IsOpen U ∧ 1 ∈ U}) ⊆ eval ⁻¹' Ici 0
-- Mathlib/MeasureTheory/Measure/Haar/Basic.lean
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ ↑K₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₂ - f K₁
this : Continuous eval

⊢ prehaar ↑K₀ '' {U | U ⊆ ↑⊤.toOpens ∧ IsOpen U ∧ 1 ∈ U} ⊆ eval ⁻¹' Ici 0
-- Mathlib/MeasureTheory/Measure/Haar/Basic.lean
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ ↑K₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₂ - f K₁
this : Continuous eval

⊢ IsClosed (eval ⁻¹' Ici 0)
-- Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean
a : ℤ
R : Type u_1
inst✝ : CommSemiring R
χ : R →* ℤ
hp : ∀ (p : ℕ) (pp : Nat.Prime p), p ≠ 2 → legendreSym p a = χ ↑p
b : ℕ
hb : Odd b

⊢ J(a | b) = χ ↑b
-- Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean
a : ℤ
R : Type u_1
inst✝ : CommSemiring R
χ : R →* ℤ
hp : ∀ (p : ℕ) (pp : Nat.Prime p), p ≠ 2 → legendreSym p a = χ ↑p
b : ℕ
hb : Odd b

⊢ J(a | b) = (List.map (⇑χ) (List.map Nat.cast b.primeFactorsList)).prod
-- Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean
a : ℤ
R : Type u_1
inst✝ : CommSemiring R
χ : R →* ℤ
hp : ∀ (p : ℕ) (pp : Nat.Prime p), p ≠ 2 → legendreSym p a = χ ↑p
b : ℕ
hb : Odd b

⊢ (List.pmap (fun p pp => legendreSym p a) b.primeFactorsList ⋯).prod = (List.pmap (fun a x => (⇑χ ∘ Nat.cast) a) b.primeFactorsList ⋯).prod
-- Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean
a : ℤ
R : Type u_1
inst✝ : CommSemiring R
χ : R →* ℤ
hp : ∀ (p : ℕ) (pp : Nat.Prime p), p ≠ 2 → legendreSym p a = χ ↑p
b : ℕ
hb : Odd b

⊢ List.pmap (fun p pp => legendreSym p a) b.primeFactorsList ⋯ = List.pmap (fun a x => (⇑χ ∘ Nat.cast) a) b.primeFactorsList ⋯
-- Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean
a : ℤ
R : Type u_1
inst✝ : CommSemiring R
χ : R →* ℤ
hp : ∀ (p : ℕ) (pp : Nat.Prime p), p ≠ 2 → legendreSym p a = χ ↑p
b : ℕ
hb : Odd b

⊢ ∀ a_1 ∈ b.primeFactorsList, ∀ (h₁ : Nat.Prime a_1), Nat.Prime a_1 → legendreSym a_1 a = (⇑χ ∘ Nat.cast) a_1
-- Mathlib/Topology/AlexandrovDiscrete.lean
ι : Sort u_1
κ : ι → Sort u_2
α : Type u_3
β : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : AlexandrovDiscrete α
S✝ : Set (Set α)
f : ι → Set α
S : Set (Set α)

⊢ interior (⋂₀ S) = ⋂ s ∈ S, interior s
-- Mathlib/Topology/UniformSpace/Pi.lean
ι : Type u_1
ι' : Type u_2
β : Type u_3
α : ι → Type u
U : (i : ι) → UniformSpace (α i)
inst✝ : UniformSpace β
φ : ι' → ι

⊢ UniformSpace.comap (fun g i' => g (φ i')) (uniformSpace fun i' => α (φ i')) = ⨅ i', UniformSpace.comap (eval (φ i')) (U (φ i'))
-- Mathlib/Probability/Kernel/Disintegration/Integral.lean
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
inst✝¹ : CountableOrCountablyGenerated α β
κ : Kernel α (β × Ω)
inst✝ : IsFiniteKernel κ
f : β × Ω → ℝ≥0∞
hf : Measurable f
a : α
s : Set β
hs : MeasurableSet s
t : Set Ω
ht : MeasurableSet t

⊢ ∫⁻ (b : β) in s, ∫⁻ (ω : Ω) in t, f (b, ω) ∂κ.condKernel (a, b) ∂κ.fst a = ∫⁻ (x : β × Ω) in s ×ˢ t, f x ∂κ a
-- Mathlib/Probability/Kernel/Disintegration/Integral.lean
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
inst✝¹ : CountableOrCountablyGenerated α β
κ : Kernel α (β × Ω)
inst✝ : IsFiniteKernel κ
f : β × Ω → ℝ≥0∞
hf : Measurable f
a : α
s : Set β
hs : MeasurableSet s
t : Set Ω
ht : MeasurableSet t

⊢ ∫⁻ (b : β) in s, ∫⁻ (ω : Ω) in t, f (b, ω) ∂κ.condKernel (a, b) ∂κ.fst a = ∫⁻ (x : β × Ω) in s ×ˢ t, f x ∂(κ.fst ⊗ₖ κ.condKernel) a
-- Mathlib/Order/Filter/AtTopBot.lean
ι : Type u_1
ι' : Type u_2
α : Type u_3
β : Type u_4
γ : Type u_5
inst✝ : SemilatticeInf α
a : α
f : β → ↑(Iic a)
l : Filter β

⊢ Tendsto f l atBot ↔ Tendsto (fun x => ↑(f x)) l atBot
-- Mathlib/Algebra/Ring/Int.lean
m n : ℤ

⊢ Even (m + n) ↔ (Odd m ↔ Odd n)
-- Mathlib/GroupTheory/Coprod/Basic.lean
G : Type u_1
H : Type u_2
inst✝¹ : Group G
inst✝ : Group H
x : FreeMonoid (G ⊕ H)

⊢ (coprodCon G H) (ofList (List.map (Sum.map Inv.inv Inv.inv) (toList x)).reverse * x) 1
-- Mathlib/GroupTheory/Coprod/Basic.lean
G : Type u_1
H : Type u_2
inst✝¹ : Group G
inst✝ : Group H
x : FreeMonoid (G ⊕ H)

⊢ mk (ofList (List.map (Sum.map Inv.inv Inv.inv) (toList x)).reverse) * mk x = 1
-- Mathlib/Order/Interval/Set/Basic.lean
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
a a₁ a₂ b b₁ b₂ c d : α
h₁ : min a b < max c d
h₂ : min c d < max a b

⊢ Icc a b ∪ Icc c d = Icc (min a c) (max b d)
-- Mathlib/Order/Interval/Set/Basic.lean
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
a a₁ a₂ b b₁ b₂ c d : α
hab : a ≤ b
hcd : c ≤ d
h₁ : a < d
h₂ : c < b

⊢ Icc a b ∪ Icc c d = Icc (min a c) (max b d)
-- Mathlib/Order/Interval/Set/Basic.lean
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
a a₁ a₂ b b₁ b₂ c d : α
hab : a ≤ b
hcd : d < c
h₁ : a < c
h₂ : d < b

⊢ Icc a b ∪ Icc c d = Icc (min a c) (max b d)
-- Mathlib/Order/Interval/Set/Basic.lean
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
a a₁ a₂ b b₁ b₂ c d : α
hab : b < a
hcd : c ≤ d
h₁ : b < d
h₂ : c < a

⊢ Icc a b ∪ Icc c d = Icc (min a c) (max b d)