ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Algebra/Polynomial/DenomsClearable.lean
K : Type u_1
inst✝ : LinearOrderedField K
f : ℤ[X]
a b : ℤ
b0 : 0 < b
fab : eval (↑a / ↑b) (Polynomial.map (algebraMap ℤ K) f) ≠ 0
ev : ℤ
bi : K
bu : bi * (algebraMap ℤ K) b = 1
hF : (algebraMap ℤ K) b ^ f.natDegree = 0 ∨ eval ((algebraMap ℤ K) a * ((algebraMap ℤ K) b)⁻¹) (Polynomial.map (algebraMap ℤ K) f) = 0
Fa : |↑ev| = ↑b ^ f.natDegree * |eval (↑a * (↑b)⁻¹) (Polynomial.map (algebraMap ℤ K) f)|
F0 : ev = 0
\u22a2 eval (↑a / ↑b) (Polynomial.map (algebraMap ℤ K) f) = 0
-- Mathlib/Data/List/Sort.lean
α : Type u
r : α → α → Prop
inst✝² : DecidableRel r
inst✝¹ : DecidableEq α
inst✝ : IsRefl α r
x : α
xs : List α
\u22a2 (orderedInsert r x xs).erase x = xs
-- Mathlib/Data/List/Sort.lean
α : Type u
r : α → α → Prop
inst✝² : DecidableRel r
inst✝¹ : DecidableEq α
inst✝ : IsRefl α r
x : α
xs : List α
\u22a2 x ∉ takeWhile (fun b => decide ¬r x b) xs
-- Mathlib/Data/List/Sort.lean
α : Type u
r : α → α → Prop
inst✝² : DecidableRel r
inst✝¹ : DecidableEq α
inst✝ : IsRefl α r
x : α
xs : List α
h : x ∈ takeWhile (fun b => decide ¬r x b) xs
\u22a2 False
-- Mathlib/Data/List/Sort.lean
α : Type u
r : α → α → Prop
inst✝² : DecidableRel r
inst✝¹ : DecidableEq α
inst✝ : IsRefl α r
x : α
xs : List α
h : (decide ¬r x x) = true
\u22a2 False
-- Mathlib/NumberTheory/NumberField/House.lean
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
α : K
\u22a2 house α = ↑(univ.sup' ⋯ fun φ => ‖φ α‖₊)
-- Mathlib/Algebra/Regular/Pow.lean
R : Type u_1
a b : R
inst✝ : Monoid R
n : ℕ
n0 : 0 < n
\u22a2 IsRightRegular (a ^ n) ↔ IsRightRegular a
-- Mathlib/Algebra/Regular/Pow.lean
R : Type u_1
a b : R
inst✝ : Monoid R
n : ℕ
n0 : 0 < n
\u22a2 IsRightRegular (a ^ n) → IsRightRegular a
-- Mathlib/Algebra/Regular/Pow.lean
R : Type u_1
a b : R
inst✝ : Monoid R
n : ℕ
n0 : 0 < n
\u22a2 IsRightRegular (a * a ^ n.pred) → IsRightRegular a
-- Mathlib/Order/Interval/Finset/Basic.lean
ι : Type u_1
α : Type u_2
inst✝¹ : Lattice α
inst✝ : LocallyFiniteOrder α
a a₁ a₂ b b₁ b₂ c x : α
h : a ≤ b
\u22a2 [[a, b]] = Icc a b
-- Mathlib/Algebra/Order/Sub/Canonical.lean
α : Type u_1
inst✝⁵ : AddCommSemigroup α
inst✝⁴ : PartialOrder α
inst✝³ : ExistsAddOfLE α
inst✝² : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b c d : α
hab : b ≤ a
hcb : c ≤ b
\u22a2 a - b + (b - c) = a - c
-- Mathlib/Algebra/Order/Sub/Canonical.lean
α : Type u_1
inst✝⁵ : AddCommSemigroup α
inst✝⁴ : PartialOrder α
inst✝³ : ExistsAddOfLE α
inst✝² : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b c d : α
hab : b ≤ a
hcb : c ≤ b
\u22a2 a - b = a - c - (b - c)
-- Mathlib/LinearAlgebra/CliffordAlgebra/EvenEquiv.lean
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m : M
\u22a2 e0 Q * (v Q) m * e0 Q = (v Q) m
-- Mathlib/Algebra/Homology/Localization.lean
C : Type u_1
inst✝³ : Category.{u_3, u_1} C
ι : Type u_2
c : ComplexShape ι
inst✝² : Preadditive C
inst✝¹ : CategoryWithHomology C
inst✝ : (HomologicalComplex.quasiIso C c).HasLocalization
K L : HomologicalComplex C c
f : K ⟶ L
\u22a2 quasiIso C c ((quotient C c).map f) ↔ HomologicalComplex.quasiIso C c f
-- Mathlib/Algebra/Homology/Localization.lean
C : Type u_1
inst✝³ : Category.{u_3, u_1} C
ι : Type u_2
c : ComplexShape ι
inst✝² : Preadditive C
inst✝¹ : CategoryWithHomology C
inst✝ : (HomologicalComplex.quasiIso C c).HasLocalization
K L : HomologicalComplex C c
f : K ⟶ L
eq : ∀ (i : ι), IsIso ((quotient C c ⋙ homologyFunctor C c i).map f) ↔ IsIso ((HomologicalComplex.homologyFunctor C c i).map f)
\u22a2 quasiIso C c ((quotient C c).map f) ↔ HomologicalComplex.quasiIso C c f
-- Mathlib/Algebra/Homology/Localization.lean
C : Type u_1
inst✝³ : Category.{u_3, u_1} C
ι : Type u_2
c : ComplexShape ι
inst✝² : Preadditive C
inst✝¹ : CategoryWithHomology C
inst✝ : (HomologicalComplex.quasiIso C c).HasLocalization
K L : HomologicalComplex C c
f : K ⟶ L
eq : ∀ (i : ι), IsIso ((homologyFunctor C c i).map ((quotient C c).map f)) ↔ IsIso (HomologicalComplex.homologyMap f i)
\u22a2 quasiIso C c ((quotient C c).map f) ↔ HomologicalComplex.quasiIso C c f
-- Mathlib/NumberTheory/EllipticDivisibilitySequence.lean
R : Type u
S : Type v
inst✝¹ : CommRing R
inst✝ : CommRing S
W : ℤ → R
f : R →+* S
b c d : R
m : ℕ
\u22a2 preNormEDS' b c d (2 * (m + 2) + 1) = (preNormEDS' b c d (m + 4) * preNormEDS' b c d (m + 2) ^ 3 * if Even m then b else 1) - preNormEDS' b c d (m + 1) * preNormEDS' b c d (m + 3) ^ 3 * if Even m then 1 else b
-- Mathlib/NumberTheory/EllipticDivisibilitySequence.lean
R : Type u
S : Type v
inst✝¹ : CommRing R
inst✝ : CommRing S
W : ℤ → R
f : R →+* S
b c d : R
m : ℕ
\u22a2 ((preNormEDS' b c d (2 * m / 2 + 4) * preNormEDS' b c d (2 * m / 2 + 2) ^ 3 * if Even (2 * m / 2) then b else 1) - preNormEDS' b c d (2 * m / 2 + 1) * preNormEDS' b c d (2 * m / 2 + 3) ^ 3 * if Even (2 * m / 2) then 1 else b) = (preNormEDS' b c d (m + 4) * preNormEDS' b c d (m + 2) ^ 3 * if Even m then b else 1) - preNormEDS' b c d (m + 1) * preNormEDS' b c d (m + 3) ^ 3 * if Even m then 1 else b
-- 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)
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x x_1 => x < x_1
l : ↑(MaxProducts C ho)
q : Products I
hq : q < (↑l).Tail
this : Inhabited I
\u22a2 List.Chain' (fun x x_1 => x > x_1) (term I ho :: ↑q)
-- 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)
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x x_1 => x < x_1
l : ↑(MaxProducts C ho)
q : Products I
hq : q < (↑l).Tail
this : Inhabited I
\u22a2 List.Pairwise (fun x x_1 => x > x_1) (term I ho :: ↑q)
-- 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)
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x x_1 => x < x_1
l : ↑(MaxProducts C ho)
q : Products I
hq : q < (↑l).Tail
this : Inhabited I
\u22a2 (∀ a' ∈ ↑q, a' < term I ho) ∧ List.Pairwise (fun x x_1 => x_1 < x) ↑q
-- 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)
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x x_1 => x < x_1
l : ↑(MaxProducts C ho)
q : Products I
hq : q < (↑l).Tail
this : Inhabited I
a : I
ha : a ∈ ↑q
\u22a2 (↑q).head! < term I ho
-- 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)
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x x_1 => x < x_1
l : ↑(MaxProducts C ho)
q : Products I
hq : q < (↑l).Tail
this : Inhabited I
a : I
ha : a ∈ ↑q
\u22a2 (↑(↑l).Tail).head! < term I ho
-- Mathlib/Topology/Algebra/Module/Basic.lean
R : Type u_1
inst✝⁶ : Ring R
M : Type u_2
inst✝⁵ : TopologicalSpace M
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
M₂ : Type u_3
inst✝² : TopologicalSpace M₂
inst✝¹ : AddCommGroup M₂
inst✝ : Module R M₂
x : ↥⊥
\u22a2 0 ↑x = x
-- Mathlib/NumberTheory/Multiplicity.lean
R : Type u_1
n : ℕ
inst✝² : CommRing R
a b x y : R
p : ℕ
inst✝¹ : IsDomain R
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
hp : Prime ↑p
hp1 : Odd p
hxy : ↑p ∣ x - y
hx : ¬↑p ∣ x
\u22a2 multiplicity (↑p) (∑ i ∈ range p, x ^ i * y ^ (p - 1 - i)) = 1
-- Mathlib/NumberTheory/Multiplicity.lean
R : Type u_1
n : ℕ
inst✝² : CommRing R
a b x y : R
p : ℕ
inst✝¹ : IsDomain R
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
hp : Prime ↑p
hp1 : Odd p
hxy : ↑p ∣ x - y
hx : ¬↑p ∣ x
\u22a2 multiplicity (↑p) (∑ i ∈ range p, x ^ i * y ^ (p - ↑1 - i)) = 1
-- Mathlib/NumberTheory/Multiplicity.lean
R : Type u_1
n : ℕ
inst✝² : CommRing R
a b x y : R
p : ℕ
inst✝¹ : IsDomain R
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
hp : Prime ↑p
hp1 : Odd p
hxy : ↑p ∣ x - y
hx : ¬↑p ∣ x
\u22a2 ↑p ^ 1 ∣ ∑ i ∈ range p, x ^ i * y ^ (p - ↑1 - i)
-- Mathlib/NumberTheory/Multiplicity.lean
R : Type u_1
n : ℕ
inst✝² : CommRing R
a b x y : R
p : ℕ
inst✝¹ : IsDomain R
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
hp : Prime ↑p
hp1 : Odd p
hxy : ↑p ∣ x - y
hx : ¬↑p ∣ x
\u22a2 ¬↑p ^ (1 + 1) ∣ ∑ i ∈ range p, x ^ i * y ^ (p - ↑1 - i)
-- Mathlib/NumberTheory/Multiplicity.lean
R : Type u_1
n : ℕ
inst✝² : CommRing R
a b x y : R
p : ℕ
inst✝¹ : IsDomain R
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
hp : Prime ↑p
hp1 : Odd p
hxy : ↑p ∣ x - y
hx : ¬↑p ∣ y
\u22a2 ¬↑p ^ (1 + 1) ∣ ∑ i ∈ range p, x ^ i * y ^ (p - ↑1 - i)
-- Mathlib/NumberTheory/Multiplicity.lean
R : Type u_1
n : ℕ
inst✝² : CommRing R
a b x y : R
p : ℕ
inst✝¹ : IsDomain R
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
hp : Prime ↑p
hp1 : Odd p
hx : ¬↑p ∣ y
k : R
hk : x - y = ↑p * k
\u22a2 ¬↑p ^ (1 + 1) ∣ ∑ i ∈ range p, x ^ i * y ^ (p - ↑1 - i)
-- Mathlib/NumberTheory/Multiplicity.lean
R : Type u_1
n : ℕ
inst✝² : CommRing R
a b x y : R
p : ℕ
inst✝¹ : IsDomain R
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
hp : Prime ↑p
hp1 : Odd p
hx : ¬↑p ∣ y
k : R
hk : x - y = ↑p * k
\u22a2 ¬↑p ^ 2 ∣ ∑ i ∈ range p, (y + ↑p * k) ^ i * y ^ (p - ↑1 - i)
-- Mathlib/NumberTheory/Multiplicity.lean
R : Type u_1
n : ℕ
inst✝² : CommRing R
a b x y : R
p : ℕ
inst✝¹ : IsDomain R
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
hp : Prime ↑p
hp1 : Odd p
hx : ¬↑p ∣ y
k : R
hk : x - y = ↑p * k
\u22a2 ¬↑p ^ 2 ∣ ↑p * y ^ (p - 1)
-- Mathlib/NumberTheory/Multiplicity.lean
R : Type u_1
n : ℕ
inst✝² : CommRing R
a b x y : R
p : ℕ
inst✝¹ : IsDomain R
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
hp : Prime ↑p
hp1 : Odd p
hx : ¬↑p ∣ y
k : R
hk : x - y = ↑p * k
\u22a2 ¬↑p ∣ y ^ (p - 1)
-- Mathlib/Order/Filter/Archimedean.lean
α : Type u_1
R : Type u_2
inst✝¹ : LinearOrderedField R
inst✝ : Archimedean R
f : α → ℚ
l : Filter α
\u22a2 Tendsto (fun n => ↑(f n)) l atBot ↔ Tendsto f l atBot
-- Mathlib/Order/Filter/Archimedean.lean
α : Type u_1
R : Type u_2
inst✝¹ : LinearOrderedField R
inst✝ : Archimedean R
f : α → ℚ
l : Filter α
\u22a2 Tendsto (fun n => ↑(f n)) l atBot ↔ Tendsto (Rat.cast ∘ f) l atBot
-- Mathlib/Data/Finset/Interval.lean
α : Type u_1
β : Type u_2
inst✝ : DecidableEq α
s t : Finset α
h : s ⊆ t
u : Finset α
\u22a2 u ∈ Icc s t ↔ u ∈ image (fun x => s ∪ x) (t \ s).powerset
-- Mathlib/Data/Finset/Interval.lean
α : Type u_1
β : Type u_2
inst✝ : DecidableEq α
s t : Finset α
h : s ⊆ t
u : Finset α
\u22a2 s ≤ u ∧ u ≤ t ↔ ∃ a ⊆ t \ s, s ∪ a = u
-- Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w₁✝ : InfinitePlace K
B : ℕ
hB : minkowskiBound K 1 < ↑(convexBodyLTFactor K) * ↑B
w₁ : InfinitePlace K
\u22a2 ∃ u, ∀ (w : InfinitePlace K), w ≠ w₁ → Real.log (w ((algebraMap (𝓞 K) K) ↑u)) < 0
-- Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w₁✝ : InfinitePlace K
B✝ : ℕ
hB✝ : minkowskiBound K 1 < ↑(convexBodyLTFactor K) * ↑B✝
w₁ : InfinitePlace K
B : ℕ
hB : minkowskiBound K 1 < ↑(convexBodyLTFactor K) * ↑B
\u22a2 ∃ u, ∀ (w : InfinitePlace K), w ≠ w₁ → Real.log (w ((algebraMap (𝓞 K) K) ↑u)) < 0
-- Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w₁✝ : InfinitePlace K
B✝ : ℕ
hB✝ : minkowskiBound K 1 < ↑(convexBodyLTFactor K) * ↑B✝
w₁ : InfinitePlace K
B : ℕ
hB : minkowskiBound K 1 < ↑(convexBodyLTFactor K) * ↑B
\u22a2 ∃ n m, n < m ∧ Ideal.span {↑(seq K w₁ hB n)} = Ideal.span {↑(seq K w₁ hB m)}
-- Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w₁✝ : InfinitePlace K
B✝ : ℕ
hB✝ : minkowskiBound K 1 < ↑(convexBodyLTFactor K) * ↑B✝
w₁ : InfinitePlace K
B : ℕ
hB : minkowskiBound K 1 < ↑(convexBodyLTFactor K) * ↑B
n : ℕ
\u22a2 Ideal.span {↑(seq K w₁ hB n)} ∈ {I | 1 ≤ Ideal.absNorm I ∧ Ideal.absNorm I ≤ B}
-- Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w₁✝ : InfinitePlace K
B✝ : ℕ
hB✝ : minkowskiBound K 1 < ↑(convexBodyLTFactor K) * ↑B✝
w₁ : InfinitePlace K
B : ℕ
hB : minkowskiBound K 1 < ↑(convexBodyLTFactor K) * ↑B
\u22a2 {I | 1 ≤ Ideal.absNorm I ∧ Ideal.absNorm I ≤ B}.Finite
-- Mathlib/CategoryTheory/Closed/Ideal.lean
C : Type u₁
D : Type u₂
inst✝⁶ : Category.{v₁, u₁} C
inst✝⁵ : Category.{v₁, u₂} D
i : D ⥤ C
inst✝⁴ : HasFiniteProducts C
inst✝³ : Reflective i
inst✝² : CartesianClosed C
inst✝¹ : HasFiniteProducts D
inst✝ : ExponentialIdeal i
A B : C
\u22a2 ((reflectorAdjunction i).homEquiv (A ⨯ B) ((reflector i).obj A ⨯ (reflector i).obj B)).symm (prod.lift prod.snd prod.fst ≫ ((exp.adjunction B).homEquiv A (i.obj ((reflector i).obj A ⨯ (reflector i).obj B))).symm ((unitCompPartialBijective A ⋯).symm (((exp.adjunction B).homEquiv (i.obj ((reflector i).obj A)) (i.obj ((reflector i).obj A ⨯ (reflector i).obj B))) (prod.lift prod.snd prod.fst ≫ ((exp.adjunction (i.obj ((reflector i).obj A))).homEquiv B (i.obj ((reflector i).obj A ⨯ (reflector i).obj B))).symm ((unitCompPartialBijective B ⋯).symm (((exp.adjunction (i.obj ((reflector i).obj A))).homEquiv (i.obj ((reflector i).obj B)) (i.obj ((reflector i).obj A ⨯ (reflector i).obj B))) ((PreservesLimitPair.iso i ((reflector i).obj A) ((reflector i).obj B)).inv ≫ i.map (𝟙 ((reflector i).obj A ⨯ (reflector i).obj B)) ≫ 𝟙 (i.obj ((reflector i).obj A ⨯ (reflector i).obj B))))) ≫ 𝟙 (i.obj ((reflector i).obj A ⨯ (reflector i).obj B))))) ≫ 𝟙 (i.obj ((reflector i).obj A ⨯ (reflector i).obj B))) = prodComparison (reflector i) A B
-- Mathlib/CategoryTheory/Closed/Ideal.lean
C : Type u₁
D : Type u₂
inst✝⁶ : Category.{v₁, u₁} C
inst✝⁵ : Category.{v₁, u₂} D
i : D ⥤ C
inst✝⁴ : HasFiniteProducts C
inst✝³ : Reflective i
inst✝² : CartesianClosed C
inst✝¹ : HasFiniteProducts D
inst✝ : ExponentialIdeal i
A B : C
\u22a2 ((reflectorAdjunction i).homEquiv (A ⨯ B) ((reflector i).obj A ⨯ (reflector i).obj B)).symm (prod.lift prod.snd prod.fst ≫ uncurry ((unitCompPartialBijective A ⋯).symm (curry (prod.lift prod.snd prod.fst ≫ uncurry ((unitCompPartialBijective B ⋯).symm (curry ((PreservesLimitPair.iso i ((reflector i).obj A) ((reflector i).obj B)).inv ≫ i.map (𝟙 ((reflector i).obj A ⨯ (reflector i).obj B)) ≫ 𝟙 (i.obj ((reflector i).obj A ⨯ (reflector i).obj B))))) ≫ 𝟙 (i.obj ((reflector i).obj A ⨯ (reflector i).obj B))))) ≫ 𝟙 (i.obj ((reflector i).obj A ⨯ (reflector i).obj B))) = prodComparison (reflector i) A B
-- Mathlib/CategoryTheory/Closed/Ideal.lean
C : Type u₁
D : Type u₂
inst✝⁶ : Category.{v₁, u₁} C
inst✝⁵ : Category.{v₁, u₂} D
i : D ⥤ C
inst✝⁴ : HasFiniteProducts C
inst✝³ : Reflective i
inst✝² : CartesianClosed C
inst✝¹ : HasFiniteProducts D
inst✝ : ExponentialIdeal i
A B : C
\u22a2 ((reflectorAdjunction i).homEquiv (A ⨯ B) ((reflector i).obj A ⨯ (reflector i).obj B)).symm (prod.lift prod.snd (prod.fst ≫ (reflectorAdjunction i).unit.app A) ≫ prod.lift prod.snd (prod.fst ≫ (reflectorAdjunction i).unit.app B) ≫ (PreservesLimitPair.iso i ((reflector i).obj A) ((reflector i).obj B)).inv) = prodComparison (reflector i) A B
-- Mathlib/CategoryTheory/Closed/Ideal.lean
C : Type u₁
D : Type u₂
inst✝⁶ : Category.{v₁, u₁} C
inst✝⁵ : Category.{v₁, u₂} D
i : D ⥤ C
inst✝⁴ : HasFiniteProducts C
inst✝³ : Reflective i
inst✝² : CartesianClosed C
inst✝¹ : HasFiniteProducts D
inst✝ : ExponentialIdeal i
A B : C
\u22a2 prod.map ((reflectorAdjunction i).unit.app A) ((reflectorAdjunction i).unit.app B) = (reflectorAdjunction i).unit.app (A ⨯ B) ≫ i.map (prodComparison (reflector i) A B) ≫ (PreservesLimitPair.iso i ((reflector i).obj A) ((reflector i).obj B)).hom
-- Mathlib/CategoryTheory/Closed/Ideal.lean
C : Type u₁
D : Type u₂
inst✝⁶ : Category.{v₁, u₁} C
inst✝⁵ : Category.{v₁, u₂} D
i : D ⥤ C
inst✝⁴ : HasFiniteProducts C
inst✝³ : Reflective i
inst✝² : CartesianClosed C
inst✝¹ : HasFiniteProducts D
inst✝ : ExponentialIdeal i
A B : C
\u22a2 prod.map ((reflectorAdjunction i).unit.app A) ((reflectorAdjunction i).unit.app B) = (reflectorAdjunction i).unit.app (A ⨯ B) ≫ i.map (prodComparison (reflector i) A B) ≫ prodComparison i ((reflector i).obj A) ((reflector i).obj B)
-- Mathlib/CategoryTheory/Closed/Ideal.lean
C : Type u₁
D : Type u₂
inst✝⁶ : Category.{v₁, u₁} C
inst✝⁵ : Category.{v₁, u₂} D
i : D ⥤ C
inst✝⁴ : HasFiniteProducts C
inst✝³ : Reflective i
inst✝² : CartesianClosed C
inst✝¹ : HasFiniteProducts D
inst✝ : ExponentialIdeal i
A B : C
\u22a2 prod.map ((reflectorAdjunction i).unit.app A) ((reflectorAdjunction i).unit.app B) ≫ prod.fst = ((reflectorAdjunction i).unit.app (A ⨯ B) ≫ i.map (prodComparison (reflector i) A B) ≫ prodComparison i ((reflector i).obj A) ((reflector i).obj B)) ≫ prod.fst
-- Mathlib/CategoryTheory/Closed/Ideal.lean
C : Type u₁
D : Type u₂
inst✝⁶ : Category.{v₁, u₁} C
inst✝⁵ : Category.{v₁, u₂} D
i : D ⥤ C
inst✝⁴ : HasFiniteProducts C
inst✝³ : Reflective i
inst✝² : CartesianClosed C
inst✝¹ : HasFiniteProducts D
inst✝ : ExponentialIdeal i
A B : C
\u22a2 prod.map ((reflectorAdjunction i).unit.app A) ((reflectorAdjunction i).unit.app B) ≫ prod.snd = ((reflectorAdjunction i).unit.app (A ⨯ B) ≫ i.map (prodComparison (reflector i) A B) ≫ prodComparison i ((reflector i).obj A) ((reflector i).obj B)) ≫ prod.snd
-- Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean
R : Type u1
inst✝⁸ : CommRing R
M : Type u2
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
A : Type u_1
inst✝⁵ : Semiring A
inst✝⁴ : Algebra R A
N : Type u4
N' : Type u5
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : AddCommGroup N'
inst✝ : Module R N'
f : M →ₗ[R] N
g : N →ₗ[R] M
\u22a2 LeftInverse ⇑(map g) ⇑(map f) ↔ LeftInverse ⇑g ⇑f
-- Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean
R : Type u1
inst✝⁸ : CommRing R
M : Type u2
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
A : Type u_1
inst✝⁵ : Semiring A
inst✝⁴ : Algebra R A
N : Type u4
N' : Type u5
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : AddCommGroup N'
inst✝ : Module R N'
f : M →ₗ[R] N
g : N →ₗ[R] M
h : LeftInverse ⇑(map g) ⇑(map f)
x : M
\u22a2 g (f x) = x
-- Mathlib/RingTheory/Regular/RegularSequence.lean
R✝ : Type u_1
S : Type u_2
M✝ : Type u_3
M₂ : Type u_4
M₃ : Type u_5
M₄ : Type u_6
inst✝⁴ : Semiring R✝
inst✝³ : Semiring S
R : Type u_7
inst✝² : CommSemiring R
r : R
rs : List R
M : Type u_8
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
N : Submodule R M
\u22a2 ofList (r :: rs) • N = r • N ⊔ ofList rs • N
-- Mathlib/Analysis/LocallyConvex/WithSeminorms.lean
𝕜 : Type u_1
𝕜₂ : Type u_2
𝕝 : Type u_3
𝕝₂ : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
ι : Type u_8
ι' : Type u_9
inst✝⁸ : NormedField 𝕜
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : NormedField 𝕜₂
inst✝⁴ : AddCommGroup F
inst✝³ : Module 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝² : RingHomIsometric σ₁₂
inst✝¹ : TopologicalSpace F
hι : Nonempty ι
q : SeminormFamily 𝕜₂ F ι
hq : WithSeminorms q
inst✝ : TopologicalSpace E
f : E →ₛₗ[σ₁₂] F
hf : Inducing ⇑f
\u22a2 WithSeminorms (q.comp f)
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
\u22a2 (↑π).sign = 0
-- .lake/packages/batteries/Batteries/Data/List/Basic.lean
\u22a2 @takeD = @takeDTR
-- .lake/packages/batteries/Batteries/Data/List/Basic.lean
α : Type u_1
f : Nat
n : List α
l : α
\u22a2 takeD f n l = takeDTR f n l
-- Mathlib/Data/PNat/Prime.lean
k m n : ℕ+
\u22a2 m.Coprime k → n.Coprime k → (m * n).Coprime k
-- Mathlib/Data/PNat/Prime.lean
k m n : ℕ+
\u22a2 (↑m).Coprime ↑k → (↑n).Coprime ↑k → (↑(m * n)).Coprime ↑k
-- Mathlib/Data/PNat/Prime.lean
k m n : ℕ+
\u22a2 (↑m).Coprime ↑k → (↑n).Coprime ↑k → (↑m * ↑n).Coprime ↑k
-- 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 : α
s t : Finset α
x : α
hxs : x ∈ s
hxt : x ∉ t
\u22a2 insert x (s \ insert x t) = s \ t
-- Mathlib/RingTheory/Congruence/Basic.lean
α : Type u_1
R : Type u_2
inst✝¹ : Add R
inst✝ : Mul R
ι : Sort u_3
f : ι → RingCon R
\u22a2 ⇑(iInf f) = ⨅ i, ⇑(f i)
-- Mathlib/Algebra/Group/Subgroup/Basic.lean
G : Type u_1
G' : Type u_2
G'' : Type u_3
inst✝³ : Group G
inst✝² : Group G'
inst✝¹ : Group G''
A : Type u_4
inst✝ : AddGroup A
s : Set G
x c : G
H : x ∈ conjugatesOfSet s
\u22a2 c * x * c⁻¹ ∈ conjugatesOfSet s
-- Mathlib/Algebra/Group/Subgroup/Basic.lean
G : Type u_1
G' : Type u_2
G'' : Type u_3
inst✝³ : Group G
inst✝² : Group G'
inst✝¹ : Group G''
A : Type u_4
inst✝ : AddGroup A
s : Set G
x c : G
H : x ∈ conjugatesOfSet s
a : G
h₁ : a ∈ s
h₂ : IsConj a x
\u22a2 c * x * c⁻¹ ∈ conjugatesOfSet s
-- Mathlib/SetTheory/Cardinal/Finite.lean
α✝ : Type u_1
β✝ : Type u_2
α : Type u_3
β : Type u_4
\u22a2 card (α ⊕ β) = card α + card β
-- Mathlib/ModelTheory/DirectLimit.lean
L : Language
ι✝ : Type v
inst✝¹³ : Preorder ι✝
G✝ : ι✝ → Type w
inst✝¹² : (i : ι✝) → L.Structure (G✝ i)
f✝ : (i j : ι✝) → i ≤ j → G✝ i ↪[L] G✝ j
inst✝¹¹ : IsDirected ι✝ fun x x_1 => x ≤ x_1
inst✝¹⁰ : DirectedSystem G✝ fun i j h => ⇑(f✝ i j h)
inst✝⁹ : Nonempty ι✝
P : Type u₁
inst✝⁸ : L.Structure P
g✝ : (i : ι✝) → G✝ i ↪[L] P
Hg : ∀ (i j : ι✝) (hij : i ≤ j) (x : G✝ i), (g✝ j) ((f✝ i j hij) x) = (g✝ i) x
G' : ι✝ → Type w'
inst✝⁷ : (i : ι✝) → L.Structure (G' i)
f' : (i j : ι✝) → i ≤ j → G' i ↪[L] G' j
g : (i : ι✝) → G✝ i ≃[L] G' i
H_commuting : ∀ (i j : ι✝) (hij : i ≤ j) (x : G✝ i), (g j) ((f✝ i j hij) x) = (f' i j hij) ((g i) x)
inst✝⁶ : DirectedSystem G' fun i j h => ⇑(f' i j h)
ι : Type u_1
inst✝⁵ : Countable ι
inst✝⁴ : Preorder ι
inst✝³ : IsDirected ι fun x x_1 => x ≤ x_1
inst✝² : Nonempty ι
G : ι → Type w
inst✝¹ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
h : ∀ (i : ι), CG L (G i)
inst✝ : DirectedSystem G fun i j h => ⇑(f i j h)
\u22a2 CG L (DirectLimit G f)
-- Mathlib/ModelTheory/DirectLimit.lean
L : Language
ι✝ : Type v
inst✝¹³ : Preorder ι✝
G✝ : ι✝ → Type w
inst✝¹² : (i : ι✝) → L.Structure (G✝ i)
f✝ : (i j : ι✝) → i ≤ j → G✝ i ↪[L] G✝ j
inst✝¹¹ : IsDirected ι✝ fun x x_1 => x ≤ x_1
inst✝¹⁰ : DirectedSystem G✝ fun i j h => ⇑(f✝ i j h)
inst✝⁹ : Nonempty ι✝
P : Type u₁
inst✝⁸ : L.Structure P
g✝ : (i : ι✝) → G✝ i ↪[L] P
Hg : ∀ (i j : ι✝) (hij : i ≤ j) (x : G✝ i), (g✝ j) ((f✝ i j hij) x) = (g✝ i) x
G' : ι✝ → Type w'
inst✝⁷ : (i : ι✝) → L.Structure (G' i)
f' : (i j : ι✝) → i ≤ j → G' i ↪[L] G' j
g : (i : ι✝) → G✝ i ≃[L] G' i
H_commuting : ∀ (i j : ι✝) (hij : i ≤ j) (x : G✝ i), (g j) ((f✝ i j hij) x) = (f' i j hij) ((g i) x)
inst✝⁶ : DirectedSystem G' fun i j h => ⇑(f' i j h)
ι : Type u_1
inst✝⁵ : Countable ι
inst✝⁴ : Preorder ι
inst✝³ : IsDirected ι fun x x_1 => x ≤ x_1
inst✝² : Nonempty ι
G : ι → Type w
inst✝¹ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
h : ∀ (i : ι), CG L (G i)
inst✝ : DirectedSystem G fun i j h => ⇑(f i j h)
\u22a2 (⋃ i, ⇑(of L ι G f i) '' Classical.choose ⋯).Countable
-- Mathlib/ModelTheory/DirectLimit.lean
L : Language
ι✝ : Type v
inst✝¹³ : Preorder ι✝
G✝ : ι✝ → Type w
inst✝¹² : (i : ι✝) → L.Structure (G✝ i)
f✝ : (i j : ι✝) → i ≤ j → G✝ i ↪[L] G✝ j
inst✝¹¹ : IsDirected ι✝ fun x x_1 => x ≤ x_1
inst✝¹⁰ : DirectedSystem G✝ fun i j h => ⇑(f✝ i j h)
inst✝⁹ : Nonempty ι✝
P : Type u₁
inst✝⁸ : L.Structure P
g✝ : (i : ι✝) → G✝ i ↪[L] P
Hg : ∀ (i j : ι✝) (hij : i ≤ j) (x : G✝ i), (g✝ j) ((f✝ i j hij) x) = (g✝ i) x
G' : ι✝ → Type w'
inst✝⁷ : (i : ι✝) → L.Structure (G' i)
f' : (i j : ι✝) → i ≤ j → G' i ↪[L] G' j
g : (i : ι✝) → G✝ i ≃[L] G' i
H_commuting : ∀ (i j : ι✝) (hij : i ≤ j) (x : G✝ i), (g j) ((f✝ i j hij) x) = (f' i j hij) ((g i) x)
inst✝⁶ : DirectedSystem G' fun i j h => ⇑(f' i j h)
ι : Type u_1
inst✝⁵ : Countable ι
inst✝⁴ : Preorder ι
inst✝³ : IsDirected ι fun x x_1 => x ≤ x_1
inst✝² : Nonempty ι
G : ι → Type w
inst✝¹ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
h : ∀ (i : ι), CG L (G i)
inst✝ : DirectedSystem G fun i j h => ⇑(f i j h)
\u22a2 (Substructure.closure L).toFun (⋃ i, ⇑(of L ι G f i) '' Classical.choose ⋯) = ⊤
-- .lake/packages/batteries/Batteries/WF.lean
\u22a2 @ndrec = @Acc.ndrecC
-- .lake/packages/batteries/Batteries/WF.lean
α : Sort u_1
r : α → α → Prop
motive : α → Sort u_2
intro : (x : α) → (∀ (y : α), r y x → Acc r y) → ((y : α) → r y x → motive y) → motive x
a : α
t : Acc r a
\u22a2 ndrec intro t = Acc.ndrecC intro t
-- Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
x✝ y✝ z x : ℝ
hx : 0 ≤ x
y : ℝ
\u22a2 0 ≤ x ^ y