ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/GroupTheory/CoprodI.lean
ι : Type u_1
M : ι → Type u_2
inst✝⁴ : (i : ι) → Monoid (M i)
N : Type u_3
inst✝³ : Monoid N
inst✝² : Nontrivial ι
G : Type u_1
inst✝¹ : Group G
a : ι → G
α : Type u_4
inst✝ : MulAction G α
X Y : ι → Set α
hXnonempty : ∀ (i : ι), (X i).Nonempty
hXdisj : Pairwise fun i j => Disjoint (X i) (X j)
hYdisj : Pairwise fun i j => Disjoint (Y i) (Y j)
hXYdisj : ∀ (i j : ι), Disjoint (X i) (Y j)
hX : ∀ (i : ι), a i • (Y i)ᶜ ⊆ X i
hY : ∀ (i : ι), a⁻¹ i • (X i)ᶜ ⊆ Y i
H : ι → Type := fun _i => FreeGroup Unit
f : (i : ι) → H i →* G := fun i => FreeGroup.lift fun x => a i
X' : ι → Set α := fun i => X i ∪ Y i

⊢ 3 ≤ #ι ∨ ∃ i, 3 ≤ #(H i)
-- Mathlib/GroupTheory/CoprodI.lean
ι : Type u_1
M : ι → Type u_2
inst✝⁴ : (i : ι) → Monoid (M i)
N : Type u_3
inst✝³ : Monoid N
inst✝² : Nontrivial ι
G : Type u_1
inst✝¹ : Group G
a : ι → G
α : Type u_4
inst✝ : MulAction G α
X Y : ι → Set α
hXnonempty : ∀ (i : ι), (X i).Nonempty
hXdisj : Pairwise fun i j => Disjoint (X i) (X j)
hYdisj : Pairwise fun i j => Disjoint (Y i) (Y j)
hXYdisj : ∀ (i j : ι), Disjoint (X i) (Y j)
hX : ∀ (i : ι), a i • (Y i)ᶜ ⊆ X i
hY : ∀ (i : ι), a⁻¹ i • (X i)ᶜ ⊆ Y i
H : ι → Type := fun _i => FreeGroup Unit
f : (i : ι) → H i →* G := fun i => FreeGroup.lift fun x => a i
X' : ι → Set α := fun i => X i ∪ Y i
inhabited_h : Inhabited ι

⊢ ∃ i, 3 ≤ #(H i)
-- Mathlib/GroupTheory/CoprodI.lean
ι : Type u_1
M : ι → Type u_2
inst✝⁴ : (i : ι) → Monoid (M i)
N : Type u_3
inst✝³ : Monoid N
inst✝² : Nontrivial ι
G : Type u_1
inst✝¹ : Group G
a : ι → G
α : Type u_4
inst✝ : MulAction G α
X Y : ι → Set α
hXnonempty : ∀ (i : ι), (X i).Nonempty
hXdisj : Pairwise fun i j => Disjoint (X i) (X j)
hYdisj : Pairwise fun i j => Disjoint (Y i) (Y j)
hXYdisj : ∀ (i j : ι), Disjoint (X i) (Y j)
hX : ∀ (i : ι), a i • (Y i)ᶜ ⊆ X i
hY : ∀ (i : ι), a⁻¹ i • (X i)ᶜ ⊆ Y i
H : ι → Type := fun _i => FreeGroup Unit
f : (i : ι) → H i →* G := fun i => FreeGroup.lift fun x => a i
X' : ι → Set α := fun i => X i ∪ Y i
inhabited_h : Inhabited ι

⊢ 3 ≤ #(H default)
-- Mathlib/GroupTheory/CoprodI.lean
ι : Type u_1
M : ι → Type u_2
inst✝⁴ : (i : ι) → Monoid (M i)
N : Type u_3
inst✝³ : Monoid N
inst✝² : Nontrivial ι
G : Type u_1
inst✝¹ : Group G
a : ι → G
α : Type u_4
inst✝ : MulAction G α
X Y : ι → Set α
hXnonempty : ∀ (i : ι), (X i).Nonempty
hXdisj : Pairwise fun i j => Disjoint (X i) (X j)
hYdisj : Pairwise fun i j => Disjoint (Y i) (Y j)
hXYdisj : ∀ (i j : ι), Disjoint (X i) (Y j)
hX : ∀ (i : ι), a i • (Y i)ᶜ ⊆ X i
hY : ∀ (i : ι), a⁻¹ i • (X i)ᶜ ⊆ Y i
H : ι → Type := fun _i => FreeGroup Unit
f : (i : ι) → H i →* G := fun i => FreeGroup.lift fun x => a i
X' : ι → Set α := fun i => X i ∪ Y i
inhabited_h : Inhabited ι

⊢ 3 ≤ #(FreeGroup Unit)
-- Mathlib/GroupTheory/CoprodI.lean
ι : Type u_1
M : ι → Type u_2
inst✝⁴ : (i : ι) → Monoid (M i)
N : Type u_3
inst✝³ : Monoid N
inst✝² : Nontrivial ι
G : Type u_1
inst✝¹ : Group G
a : ι → G
α : Type u_4
inst✝ : MulAction G α
X Y : ι → Set α
hXnonempty : ∀ (i : ι), (X i).Nonempty
hXdisj : Pairwise fun i j => Disjoint (X i) (X j)
hYdisj : Pairwise fun i j => Disjoint (Y i) (Y j)
hXYdisj : ∀ (i j : ι), Disjoint (X i) (Y j)
hX : ∀ (i : ι), a i • (Y i)ᶜ ⊆ X i
hY : ∀ (i : ι), a⁻¹ i • (X i)ᶜ ⊆ Y i
H : ι → Type := fun _i => FreeGroup Unit
f : (i : ι) → H i →* G := fun i => FreeGroup.lift fun x => a i
X' : ι → Set α := fun i => X i ∪ Y i
inhabited_h : Inhabited ι

⊢ 3 ≤ ℵ₀
-- Mathlib/GroupTheory/CoprodI.lean
ι : Type u_1
M : ι → Type u_2
inst✝⁴ : (i : ι) → Monoid (M i)
N : Type u_3
inst✝³ : Monoid N
inst✝² : Nontrivial ι
G : Type u_1
inst✝¹ : Group G
a : ι → G
α : Type u_4
inst✝ : MulAction G α
X Y : ι → Set α
hXnonempty : ∀ (i : ι), (X i).Nonempty
hXdisj : Pairwise fun i j => Disjoint (X i) (X j)
hYdisj : Pairwise fun i j => Disjoint (Y i) (Y j)
hXYdisj : ∀ (i j : ι), Disjoint (X i) (Y j)
hX : ∀ (i : ι), a i • (Y i)ᶜ ⊆ X i
hY : ∀ (i : ι), a⁻¹ i • (X i)ᶜ ⊆ Y i
H : ι → Type := fun _i => FreeGroup Unit
f : (i : ι) → H i →* G := fun i => FreeGroup.lift fun x => a i
X' : ι → Set α := fun i => X i ∪ Y i
inhabited_h : Inhabited ι

⊢ 3 < ℵ₀
-- Mathlib/Analysis/InnerProductSpace/LaxMilgram.lean
V : Type u
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : CompleteSpace V
B : V →L[ℝ] V →L[ℝ] ℝ
coercive : IsCoercive B

⊢ range (continuousLinearMapOfBilin B) = ⊤
-- Mathlib/Analysis/InnerProductSpace/LaxMilgram.lean
V : Type u
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : CompleteSpace V
B : V →L[ℝ] V →L[ℝ] ℝ
coercive : IsCoercive B
this : CompleteSpace ↑↑(range (continuousLinearMapOfBilin B))

⊢ range (continuousLinearMapOfBilin B) = ⊤
-- Mathlib/Analysis/InnerProductSpace/LaxMilgram.lean
V : Type u
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : CompleteSpace V
B : V →L[ℝ] V →L[ℝ] ℝ
coercive : IsCoercive B
this : CompleteSpace ↑↑(range (continuousLinearMapOfBilin B))

⊢ (range (continuousLinearMapOfBilin B))ᗮᗮ = ⊤
-- Mathlib/Analysis/InnerProductSpace/LaxMilgram.lean
V : Type u
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : CompleteSpace V
B : V →L[ℝ] V →L[ℝ] ℝ
coercive : IsCoercive B
this : CompleteSpace ↑↑(range (continuousLinearMapOfBilin B))

⊢ ∀ (x : V), x ∈ (range (continuousLinearMapOfBilin B))ᗮᗮ
-- Mathlib/Analysis/InnerProductSpace/LaxMilgram.lean
V : Type u
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : CompleteSpace V
B : V →L[ℝ] V →L[ℝ] ℝ
this : CompleteSpace ↑↑(range (continuousLinearMapOfBilin B))
v : V
C : ℝ
C_pos : 0 < C
coercivity : ∀ (u : V), C * ‖u‖ * ‖u‖ ≤ (B u) u
mem_w_orthogonal : 0 ∈ (range (continuousLinearMapOfBilin B))ᗮ

⊢ ⟪0, v⟫_ℝ = 0
-- Mathlib/Probability/Kernel/WithDensity.lean
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : Kernel α β
f : α → β → ℝ≥0∞
κ : Kernel α β
inst✝ : IsSFiniteKernel κ
hf_ne_top : ∀ (a : α) (b : β), f a b ≠ ⊤

⊢ IsSFiniteKernel (κ.withDensity f)
-- Mathlib/Probability/Kernel/WithDensity.lean
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : Kernel α β
f : α → β → ℝ≥0∞
κ : Kernel α β
inst✝ : IsSFiniteKernel κ
hf_ne_top : ∀ (a : α) (b : β), f a b ≠ ⊤
h_eq_sum : κ.withDensity f = Kernel.sum fun i => (κ.seq i).withDensity f

⊢ IsSFiniteKernel (κ.withDensity f)
-- Mathlib/Probability/Kernel/WithDensity.lean
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : Kernel α β
f : α → β → ℝ≥0∞
κ : Kernel α β
inst✝ : IsSFiniteKernel κ
hf_ne_top : ∀ (a : α) (b : β), f a b ≠ ⊤
h_eq_sum : κ.withDensity f = Kernel.sum fun i => (κ.seq i).withDensity f

⊢ IsSFiniteKernel (Kernel.sum fun i => (κ.seq i).withDensity f)
-- Mathlib/Analysis/Normed/Group/Basic.lean
𝓕 : Type u_1
𝕜 : Type u_2
α : Type u_3
ι : Type u_4
κ : Type u_5
E : Type u_6
F : Type u_7
G : Type u_8
inst✝² : SeminormedGroup E
inst✝¹ : SeminormedGroup F
inst✝ : SeminormedGroup G
s : Set E
a a₁ a₂ b b₁ b₂ : E
r r₁ r₂ : ℝ

⊢ (𝓤 E).HasBasis (fun ε => 0 < ε) fun ε => {p | ‖p.1 / p.2‖ < ε}
-- Mathlib/Analysis/Normed/Group/Basic.lean
𝓕 : Type u_1
𝕜 : Type u_2
α : Type u_3
ι : Type u_4
κ : Type u_5
E : Type u_6
F : Type u_7
G : Type u_8
inst✝² : SeminormedGroup E
inst✝¹ : SeminormedGroup F
inst✝ : SeminormedGroup G
s : Set E
a a₁ a₂ b b₁ b₂ : E
r r₁ r₂ : ℝ

⊢ (fun ε => {p | ‖p.1 / p.2‖ < ε}) = fun ε => {p | dist p.1 p.2 < ε}
-- 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
s t : Set E

⊢ (convexHull R) (s - t) = (convexHull R) s - (convexHull R) t
-- Mathlib/AlgebraicGeometry/Morphisms/IsIso.lean

⊢ isomorphisms Scheme = (isomorphisms TopCat).inverseImage Scheme.forgetToTop ⊓ stalkwise fun {R S} [CommRing R] [CommRing S] f => Function.Bijective ⇑f
-- Mathlib/AlgebraicGeometry/Morphisms/IsIso.lean

⊢ (((topologically fun {α β} [TopologicalSpace α] [TopologicalSpace β] => OpenEmbedding) ⊓ topologically fun {α β} [TopologicalSpace α] [TopologicalSpace β] => Function.Surjective) ⊓ stalkwise fun {R S} [CommRing R] [CommRing S] f => Function.Bijective ⇑f) = (isomorphisms TopCat).inverseImage Scheme.forgetToTop ⊓ stalkwise fun {R S} [CommRing R] [CommRing S] f => Function.Bijective ⇑f
-- Mathlib/AlgebraicGeometry/Morphisms/IsIso.lean
X Y : Scheme
f : X ⟶ Y

⊢ ((topologically fun {α β} [TopologicalSpace α] [TopologicalSpace β] => OpenEmbedding) ⊓ topologically fun {α β} [TopologicalSpace α] [TopologicalSpace β] => Function.Surjective) f ↔ (isomorphisms TopCat).inverseImage Scheme.forgetToTop f
-- Mathlib/Analysis/Calculus/TangentCone.lean
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
x y : E
s t : Set E
a b : ℝ
hab : a < b

⊢ (interior (Ioc a b)).Nonempty
-- Mathlib/Analysis/Calculus/TangentCone.lean
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
x y : E
s t : Set E
a b : ℝ
hab : ¬a < b

⊢ UniqueDiffOn ℝ (Ioc a b)
-- .lake/packages/batteries/Batteries/Data/List/Lemmas.lean
α : Type u_1
a : α
n : Nat
l : List α
h : n < l.length

⊢ (l.set n a).get? n = some a
-- Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedAddCommGroup G
𝕜 : Type u_5
inst✝⁴ : NormedDivisionRing 𝕜
inst✝³ : MulActionWithZero 𝕜 E
inst✝² : Module 𝕜 F
inst✝¹ : BoundedSMul 𝕜 E
inst✝ : BoundedSMul 𝕜 F
c : 𝕜
f : α → F

⊢ eLpNorm (c • f) p μ = ↑‖c‖₊ * eLpNorm f p μ
-- Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedAddCommGroup G
𝕜 : Type u_5
inst✝⁴ : NormedDivisionRing 𝕜
inst✝³ : MulActionWithZero 𝕜 E
inst✝² : Module 𝕜 F
inst✝¹ : BoundedSMul 𝕜 E
inst✝ : BoundedSMul 𝕜 F
f : α → F

⊢ eLpNorm (0 • f) p μ = ↑‖0‖₊ * eLpNorm f p μ
-- Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedAddCommGroup G
𝕜 : Type u_5
inst✝⁴ : NormedDivisionRing 𝕜
inst✝³ : MulActionWithZero 𝕜 E
inst✝² : Module 𝕜 F
inst✝¹ : BoundedSMul 𝕜 E
inst✝ : BoundedSMul 𝕜 F
c : 𝕜
f : α → F
hc : c ≠ 0

⊢ eLpNorm (c • f) p μ = ↑‖c‖₊ * eLpNorm f p μ
-- Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedAddCommGroup G
𝕜 : Type u_5
inst✝⁴ : NormedDivisionRing 𝕜
inst✝³ : MulActionWithZero 𝕜 E
inst✝² : Module 𝕜 F
inst✝¹ : BoundedSMul 𝕜 E
inst✝ : BoundedSMul 𝕜 F
c : 𝕜
f : α → F
hc : c ≠ 0

⊢ ↑‖c‖₊ * eLpNorm f p μ ≤ eLpNorm (c • f) p μ
-- Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedAddCommGroup G
𝕜 : Type u_5
inst✝⁴ : NormedDivisionRing 𝕜
inst✝³ : MulActionWithZero 𝕜 E
inst✝² : Module 𝕜 F
inst✝¹ : BoundedSMul 𝕜 E
inst✝ : BoundedSMul 𝕜 F
c : 𝕜
f : α → F
hc : c ≠ 0
this : eLpNorm (c⁻¹ • c • f) p μ ≤ ‖c⁻¹‖₊ • eLpNorm (c • f) p μ

⊢ ↑‖c‖₊ * eLpNorm f p μ ≤ eLpNorm (c • f) p μ
-- Mathlib/SetTheory/Surreal/Basic.lean
x : PGame
IHl : ∀ (i : x.LeftMoves) {y : PGame}, x.moveLeft i ≡r y → (x.moveLeft i).Numeric → y.Numeric
IHr : ∀ (j : x.RightMoves) {y : PGame}, x.moveRight j ≡r y → (x.moveRight j).Numeric → y.Numeric
y : PGame
r : x ≡r y
ox : x.Numeric

⊢ y.Numeric
-- Mathlib/SetTheory/Surreal/Basic.lean
x : PGame
IHl : ∀ (i : x.LeftMoves) {y : PGame}, x.moveLeft i ≡r y → (x.moveLeft i).Numeric → y.Numeric
IHr : ∀ (j : x.RightMoves) {y : PGame}, x.moveRight j ≡r y → (x.moveRight j).Numeric → y.Numeric
y : PGame
r : x ≡r y
ox : x.Numeric
i : y.LeftMoves
j : y.RightMoves

⊢ y.moveLeft i < y.moveRight j
-- Mathlib/SetTheory/Surreal/Basic.lean
x : PGame
IHl : ∀ (i : x.LeftMoves) {y : PGame}, x.moveLeft i ≡r y → (x.moveLeft i).Numeric → y.Numeric
IHr : ∀ (j : x.RightMoves) {y : PGame}, x.moveRight j ≡r y → (x.moveRight j).Numeric → y.Numeric
y : PGame
r : x ≡r y
ox : x.Numeric
i : y.LeftMoves

⊢ (y.moveLeft i).Numeric
-- Mathlib/SetTheory/Surreal/Basic.lean
x : PGame
IHl : ∀ (i : x.LeftMoves) {y : PGame}, x.moveLeft i ≡r y → (x.moveLeft i).Numeric → y.Numeric
IHr : ∀ (j : x.RightMoves) {y : PGame}, x.moveRight j ≡r y → (x.moveRight j).Numeric → y.Numeric
y : PGame
r : x ≡r y
ox : x.Numeric
j : y.RightMoves

⊢ (y.moveRight j).Numeric
-- Mathlib/Algebra/Divisibility/Prod.lean
ι : Type u_1
G₁ : Type u_2
G₂ : Type u_3
G : ι → Type u_4
inst✝² : Semigroup G₁
inst✝¹ : Semigroup G₂
inst✝ : (i : ι) → Semigroup (G i)
fst✝¹ : G₁
snd✝¹ : G₂
fst✝ : G₁
snd✝ : G₂

⊢ (fst✝¹, snd✝¹) ∣ (fst✝, snd✝) ↔ (fst✝¹, snd✝¹).1 ∣ (fst✝, snd✝).1 ∧ (fst✝¹, snd✝¹).2 ∣ (fst✝, snd✝).2
-- Mathlib/RingTheory/LaurentSeries.lean
R : Type u_1
inst✝ : Semiring R
x : LaurentSeries R

⊢ (ofPowerSeries ℤ R) x.powerSeriesPart = (single (-order x)) 1 * x
-- Mathlib/RingTheory/LaurentSeries.lean
R : Type u_1
inst✝ : Semiring R
x : LaurentSeries R

⊢ (ofPowerSeries ℤ R) x.powerSeriesPart = (single (-order x)) 1 * ((single (order x)) 1 * (ofPowerSeries ℤ R) x.powerSeriesPart)
-- Mathlib/Data/Fin/Basic.lean
n m : ℕ
a : Fin n
b : Fin (n + 1)

⊢ a.succ.predAbove b.succ = (a.predAbove b).succ
-- Mathlib/Data/Fin/Basic.lean
n m : ℕ
a : Fin n
b : Fin (n + 1)
h : a.succ ≤ b

⊢ a.succ.predAbove b.succ = (a.predAbove b).succ
-- Mathlib/Data/Fin/Basic.lean
n m : ℕ
a : Fin n
b : Fin (n + 1)
h : b < a.succ

⊢ a.succ.predAbove b.succ = (a.predAbove b).succ
-- Mathlib/NumberTheory/SmoothNumbers.lean

⊢ smoothNumbers 1 = {1}
-- Mathlib/AlgebraicTopology/SimplicialObject.lean
C : Type u
inst✝ : Category.{v, u} C
X✝ Y✝ : (SimplicialObject.Augmented C)ᵒᵖ
f : X✝ ⟶ Y✝
x : SimplexCategory

⊢ f.unop.right.op ≫ ((unop Y✝).hom.app (op x)).op = ((unop X✝).hom.app (op x)).op ≫ (f.unop.left.app (op x)).op
-- Mathlib/AlgebraicTopology/SimplicialObject.lean
C : Type u
inst✝ : Category.{v, u} C
X✝ Y✝ : (SimplicialObject.Augmented C)ᵒᵖ
f : X✝ ⟶ Y✝
x : SimplexCategory

⊢ ((unop Y✝).hom.app (op x) ≫ f.unop.right).op = (f.unop.left.app (op x) ≫ (unop X✝).hom.app (op x)).op
-- Mathlib/AlgebraicTopology/SimplicialObject.lean
C : Type u
inst✝ : Category.{v, u} C
X✝ Y✝ : (SimplicialObject.Augmented C)ᵒᵖ
f : X✝ ⟶ Y✝
x : SimplexCategory

⊢ (unop Y✝).hom.app (op x) ≫ f.unop.right = f.unop.left.app (op x) ≫ (unop X✝).hom.app (op x)
-- Mathlib/LinearAlgebra/Orientation.lean
R : Type u_1
inst✝⁵ : LinearOrderedCommRing R
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
ι : Type u_3
inst✝² : Fintype ι
inst✝¹ : DecidableEq ι
inst✝ : Nonempty ι
e : Basis ι R M
x : Orientation R M ι
v : ι → M

⊢ |(e.adjustToOrientation x).det v| = |e.det v|
-- Mathlib/CategoryTheory/Closed/Functor.lean
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u'
inst✝⁵ : Category.{v, u'} D
inst✝⁴ : HasFiniteProducts C
inst✝³ : HasFiniteProducts D
F : C ⥤ D
L : D ⥤ C
inst✝² : CartesianClosed C
inst✝¹ : CartesianClosed D
inst✝ : PreservesLimitsOfShape (Discrete WalkingPair) F
A A' : C
f : A' ⟶ A

⊢ (mateEquiv (exp.adjunction A) (exp.adjunction (F.obj A))) (prodComparisonNatIso F A).inv ≫ whiskerLeft F ((conjugateEquiv (exp.adjunction (F.obj A)) (exp.adjunction (F.obj A'))) (prod.functor.map (F.map f))) = whiskerRight ((conjugateEquiv (exp.adjunction A) (exp.adjunction A')) (prod.functor.map f)) F ≫ (mateEquiv (exp.adjunction A') (exp.adjunction (F.obj A'))) (prodComparisonNatIso F A').inv
-- Mathlib/CategoryTheory/Closed/Functor.lean
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u'
inst✝⁵ : Category.{v, u'} D
inst✝⁴ : HasFiniteProducts C
inst✝³ : HasFiniteProducts D
F : C ⥤ D
L : D ⥤ C
inst✝² : CartesianClosed C
inst✝¹ : CartesianClosed D
inst✝ : PreservesLimitsOfShape (Discrete WalkingPair) F
A A' : C
f : A' ⟶ A
vcomp1 : (mateEquiv (exp.adjunction A) (exp.adjunction (F.obj A'))) (leftAdjointSquareConjugate.vcomp (prodComparisonNatIso F A).inv (prod.functor.map (F.map f))) = rightAdjointSquareConjugate.vcomp ((mateEquiv (exp.adjunction A) (exp.adjunction (F.obj A))) (prodComparisonNatIso F A).inv) ((conjugateEquiv (exp.adjunction (F.obj A)) (exp.adjunction (F.obj A'))) (prod.functor.map (F.map f)))

⊢ (mateEquiv (exp.adjunction A) (exp.adjunction (F.obj A))) (prodComparisonNatIso F A).inv ≫ whiskerLeft F ((conjugateEquiv (exp.adjunction (F.obj A)) (exp.adjunction (F.obj A'))) (prod.functor.map (F.map f))) = whiskerRight ((conjugateEquiv (exp.adjunction A) (exp.adjunction A')) (prod.functor.map f)) F ≫ (mateEquiv (exp.adjunction A') (exp.adjunction (F.obj A'))) (prodComparisonNatIso F A').inv
-- Mathlib/CategoryTheory/Closed/Functor.lean
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u'
inst✝⁵ : Category.{v, u'} D
inst✝⁴ : HasFiniteProducts C
inst✝³ : HasFiniteProducts D
F : C ⥤ D
L : D ⥤ C
inst✝² : CartesianClosed C
inst✝¹ : CartesianClosed D
inst✝ : PreservesLimitsOfShape (Discrete WalkingPair) F
A A' : C
f : A' ⟶ A
vcomp1 : (mateEquiv (exp.adjunction A) (exp.adjunction (F.obj A'))) (leftAdjointSquareConjugate.vcomp (prodComparisonNatIso F A).inv (prod.functor.map (F.map f))) = rightAdjointSquareConjugate.vcomp ((mateEquiv (exp.adjunction A) (exp.adjunction (F.obj A))) (prodComparisonNatIso F A).inv) ((conjugateEquiv (exp.adjunction (F.obj A)) (exp.adjunction (F.obj A'))) (prod.functor.map (F.map f)))
vcomp2 : (mateEquiv (exp.adjunction A) (exp.adjunction (F.obj A'))) (leftAdjointConjugateSquare.vcomp (prod.functor.map f) (prodComparisonNatIso F A').inv) = rightAdjointConjugateSquare.vcomp ((conjugateEquiv (exp.adjunction A) (exp.adjunction A')) (prod.functor.map f)) ((mateEquiv (exp.adjunction A') (exp.adjunction (F.obj A'))) (prodComparisonNatIso F A').inv)

⊢ (mateEquiv (exp.adjunction A) (exp.adjunction (F.obj A))) (prodComparisonNatIso F A).inv ≫ whiskerLeft F ((conjugateEquiv (exp.adjunction (F.obj A)) (exp.adjunction (F.obj A'))) (prod.functor.map (F.map f))) = whiskerRight ((conjugateEquiv (exp.adjunction A) (exp.adjunction A')) (prod.functor.map f)) F ≫ (mateEquiv (exp.adjunction A') (exp.adjunction (F.obj A'))) (prodComparisonNatIso F A').inv
-- Mathlib/CategoryTheory/Closed/Functor.lean
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u'
inst✝⁵ : Category.{v, u'} D
inst✝⁴ : HasFiniteProducts C
inst✝³ : HasFiniteProducts D
F : C ⥤ D
L : D ⥤ C
inst✝² : CartesianClosed C
inst✝¹ : CartesianClosed D
inst✝ : PreservesLimitsOfShape (Discrete WalkingPair) F
A A' : C
f : A' ⟶ A
vcomp1 : (mateEquiv (exp.adjunction A) (exp.adjunction (F.obj A'))) (whiskerLeft F (prod.functor.map (F.map f)) ≫ (prodComparisonNatIso F A).inv) = (mateEquiv (exp.adjunction A) (exp.adjunction (F.obj A))) (prodComparisonNatIso F A).inv ≫ whiskerLeft F ((conjugateEquiv (exp.adjunction (F.obj A)) (exp.adjunction (F.obj A'))) (prod.functor.map (F.map f)))
vcomp2 : (mateEquiv (exp.adjunction A) (exp.adjunction (F.obj A'))) ((prodComparisonNatIso F A').inv ≫ whiskerRight (prod.functor.map f) F) = whiskerRight ((conjugateEquiv (exp.adjunction A) (exp.adjunction A')) (prod.functor.map f)) F ≫ (mateEquiv (exp.adjunction A') (exp.adjunction (F.obj A'))) (prodComparisonNatIso F A').inv
B : C

⊢ (whiskerLeft F (prod.functor.map (F.map f)) ≫ (prodComparisonNatIso F A).inv).app B = ((prodComparisonNatIso F A').inv ≫ whiskerRight (prod.functor.map f) F).app B
-- Mathlib/CategoryTheory/Closed/Functor.lean
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u'
inst✝⁵ : Category.{v, u'} D
inst✝⁴ : HasFiniteProducts C
inst✝³ : HasFiniteProducts D
F : C ⥤ D
L : D ⥤ C
inst✝² : CartesianClosed C
inst✝¹ : CartesianClosed D
inst✝ : PreservesLimitsOfShape (Discrete WalkingPair) F
A A' : C
f : A' ⟶ A
vcomp1 : (mateEquiv (exp.adjunction A) (exp.adjunction (F.obj A'))) (whiskerLeft F (prod.functor.map (F.map f)) ≫ (prodComparisonNatIso F A).inv) = (mateEquiv (exp.adjunction A) (exp.adjunction (F.obj A))) (prodComparisonNatIso F A).inv ≫ whiskerLeft F ((conjugateEquiv (exp.adjunction (F.obj A)) (exp.adjunction (F.obj A'))) (prod.functor.map (F.map f)))
vcomp2 : (mateEquiv (exp.adjunction A) (exp.adjunction (F.obj A'))) ((prodComparisonNatIso F A').inv ≫ whiskerRight (prod.functor.map f) F) = whiskerRight ((conjugateEquiv (exp.adjunction A) (exp.adjunction A')) (prod.functor.map f)) F ≫ (mateEquiv (exp.adjunction A') (exp.adjunction (F.obj A'))) (prodComparisonNatIso F A').inv
B : C

⊢ prod.map (F.map f) (𝟙 (F.obj B)) ≫ inv (prodComparison F A B) = inv (prodComparison F A' B) ≫ F.map (prod.map f (𝟙 B))
-- Mathlib/CategoryTheory/Closed/Functor.lean
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u'
inst✝⁵ : Category.{v, u'} D
inst✝⁴ : HasFiniteProducts C
inst✝³ : HasFiniteProducts D
F : C ⥤ D
L : D ⥤ C
inst✝² : CartesianClosed C
inst✝¹ : CartesianClosed D
inst✝ : PreservesLimitsOfShape (Discrete WalkingPair) F
A A' : C
f : A' ⟶ A
vcomp1 : (mateEquiv (exp.adjunction A) (exp.adjunction (F.obj A'))) (whiskerLeft F (prod.functor.map (F.map f)) ≫ (prodComparisonNatIso F A).inv) = (mateEquiv (exp.adjunction A) (exp.adjunction (F.obj A))) (prodComparisonNatIso F A).inv ≫ whiskerLeft F ((conjugateEquiv (exp.adjunction (F.obj A)) (exp.adjunction (F.obj A'))) (prod.functor.map (F.map f)))
vcomp2 : (mateEquiv (exp.adjunction A) (exp.adjunction (F.obj A'))) ((prodComparisonNatIso F A').inv ≫ whiskerRight (prod.functor.map f) F) = whiskerRight ((conjugateEquiv (exp.adjunction A) (exp.adjunction A')) (prod.functor.map f)) F ≫ (mateEquiv (exp.adjunction A') (exp.adjunction (F.obj A'))) (prodComparisonNatIso F A').inv
B : C

⊢ prod.map (F.map f) (F.map (𝟙 B)) ≫ inv (prodComparison F A B) = inv (prodComparison F A' B) ≫ F.map (prod.map f (𝟙 B))
-- Mathlib/AlgebraicGeometry/AffineScheme.lean
X Y : Scheme
U : X.Opens
hU : IsAffineOpen U
f : ↑Γ(X, U)

⊢ Set.range ((⇑U.ι.val.base ∘ ⇑(↑U).isoSpec.inv.val.base) ∘ ⇑(Spec.map (X.presheaf.map (eqToHom ⋯).op)).val.base) = ↑U
-- Mathlib/AlgebraicGeometry/AffineScheme.lean
X Y : Scheme
U : X.Opens
hU : IsAffineOpen U
f : ↑Γ(X, U)

⊢ Set.range ⇑U.ι.val.base = ↑U
-- Mathlib/AlgebraicGeometry/AffineScheme.lean
X Y : Scheme
U : X.Opens
hU : IsAffineOpen U
f : ↑Γ(X, U)

⊢ Function.Surjective (⇑(↑U).isoSpec.inv.val.base ∘ ⇑(Spec.map (X.presheaf.map (eqToHom ⋯).op)).val.base)
-- Mathlib/AlgebraicGeometry/AffineScheme.lean
X Y : Scheme
U : X.Opens
hU : IsAffineOpen U
f : ↑Γ(X, U)

⊢ Epi ((Spec.map (X.presheaf.map (eqToHom ⋯).op)).val.base ≫ (↑U).isoSpec.inv.val.base)
-- Mathlib/Algebra/Polynomial/Expand.lean
R : Type u
inst✝¹ : CommSemiring R
S : Type v
inst✝ : CommSemiring S
p q : ℕ
f : R[X]

⊢ (expand R (p ^ Nat.zero)) f = (⇑(expand R p))^[Nat.zero] f
-- Mathlib/Algebra/Polynomial/Expand.lean
R : Type u
inst✝¹ : CommSemiring R
S : Type v
inst✝ : CommSemiring S
p q : ℕ
f : R[X]
n : ℕ
ih : (expand R (p ^ n)) f = (⇑(expand R p))^[n] f

⊢ (expand R (p ^ n.succ)) f = (⇑(expand R p))^[n.succ] f
-- Mathlib/MeasureTheory/Measure/Haar/Unique.lean
G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : Group G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : LocallyCompactSpace G
μ' μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : IsFiniteMeasureOnCompacts μ'
inst✝ : μ'.IsMulLeftInvariant
s : Set G
h's : IsCompact (closure s)
ν : Measure G := μ'.haarScalarFactor μ • μ

⊢ μ' s = μ'.haarScalarFactor μ • μ s