ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Analysis/NormedSpace/AddTorsor.lean
α : Type u_1
V : Type u_2
P : Type u_3
W : Type u_4
Q : Type u_5
inst✝⁸ : SeminormedAddCommGroup V
inst✝⁷ : PseudoMetricSpace P
inst✝⁶ : NormedAddTorsor V P
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : MetricSpace Q
inst✝³ : NormedAddTorsor W Q
𝕜 : Type u_6
inst✝² : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 V
inst✝ : NormedSpace 𝕜 W
p₁ p₂ : P
c : 𝕜
\u22a2 dist ((homothety p₁ c) p₂) p₁ = ‖c‖ * dist p₁ p₂
-- Mathlib/Control/Bitraversable/Lemmas.lean
t : Type u → Type u → Type u
inst✝⁵ : Bitraversable t
β✝ : Type u
F G : Type u → Type u
inst✝⁴ : Applicative F
inst✝³ : Applicative G
inst✝² : LawfulBitraversable t
inst✝¹ : LawfulApplicative F
inst✝ : LawfulApplicative G
α₀ α₁ α₂ β : Type u
f : α₀ → F α₁
f' : α₁ → G α₂
x : t α₀ β
\u22a2 Comp.mk (tfst f' <$> tfst f x) = tfst (Comp.mk ∘ map f' ∘ f) x
-- Mathlib/Control/Bitraversable/Lemmas.lean
t : Type u → Type u → Type u
inst✝⁵ : Bitraversable t
β✝ : Type u
F G : Type u → Type u
inst✝⁴ : Applicative F
inst✝³ : Applicative G
inst✝² : LawfulBitraversable t
inst✝¹ : LawfulApplicative F
inst✝ : LawfulApplicative G
α₀ α₁ α₂ β : Type u
f : α₀ → F α₁
f' : α₁ → G α₂
x : t α₀ β
\u22a2 bitraverse (Comp.mk ∘ map f' ∘ f) (Comp.mk ∘ map pure ∘ pure) x = tfst (Comp.mk ∘ map f' ∘ f) x
-- Mathlib/SetTheory/Cardinal/Basic.lean
α β : Type u
c : Cardinal.{u}
\u22a2 c + 1 ≤ succ c
-- Mathlib/SetTheory/Cardinal/Basic.lean
α β : Type u
c : Cardinal.{u}
this : {c' | c < c'}.Nonempty
\u22a2 c + 1 ≤ succ c
-- Mathlib/SetTheory/Cardinal/Basic.lean
α β : Type u
c : Cardinal.{u}
this : {c' | c < c'}.Nonempty
\u22a2 ∀ (b : Cardinal.{u}), c < b → c + 1 ≤ b
-- Mathlib/SetTheory/Cardinal/Basic.lean
α β✝ : Type u
c b : Cardinal.{u}
β γ : Type u
this : {c' | Quot.mk Setoid.r γ < c'}.Nonempty
hlt : Quot.mk Setoid.r γ < Quot.mk Setoid.r β
\u22a2 Quot.mk Setoid.r γ + 1 ≤ Quot.mk Setoid.r β
-- Mathlib/SetTheory/Cardinal/Basic.lean
α β✝ : Type u
c b : Cardinal.{u}
β γ : Type u
this : {c' | Quot.mk Setoid.r γ < c'}.Nonempty
hlt : Quot.mk Setoid.r γ < Quot.mk Setoid.r β
f : γ ↪ β
\u22a2 Quot.mk Setoid.r γ + 1 ≤ Quot.mk Setoid.r β
-- Mathlib/SetTheory/Cardinal/Basic.lean
α β✝ : Type u
c b : Cardinal.{u}
β γ : Type u
this✝ : {c' | Quot.mk Setoid.r γ < c'}.Nonempty
hlt : Quot.mk Setoid.r γ < Quot.mk Setoid.r β
f : γ ↪ β
this : ¬Surjective ⇑f
\u22a2 Quot.mk Setoid.r γ + 1 ≤ Quot.mk Setoid.r β
-- Mathlib/SetTheory/Cardinal/Basic.lean
α β✝ : Type u
c b : Cardinal.{u}
β γ : Type u
this✝ : {c' | Quot.mk Setoid.r γ < c'}.Nonempty
hlt : Quot.mk Setoid.r γ < Quot.mk Setoid.r β
f : γ ↪ β
this : ∃ x, ¬∃ a, f a = x
\u22a2 Quot.mk Setoid.r γ + 1 ≤ Quot.mk Setoid.r β
-- Mathlib/SetTheory/Cardinal/Basic.lean
α β✝ : Type u
c b✝ : Cardinal.{u}
β γ : Type u
this : {c' | Quot.mk Setoid.r γ < c'}.Nonempty
hlt : Quot.mk Setoid.r γ < Quot.mk Setoid.r β
f : γ ↪ β
b : β
hb : ¬∃ a, f a = b
\u22a2 Quot.mk Setoid.r γ + 1 ≤ Quot.mk Setoid.r β
-- Mathlib/Data/Nat/Cast/Defs.lean
R : Type u_1
inst✝ : AddMonoidWithOne R
\u22a2 ↑1 = 1
-- Mathlib/RingTheory/Polynomial/Eisenstein/Basic.lean
R : Type u
inst✝¹ : CommSemiring R
𝓟 : Ideal R
f : R[X]
hf : f.IsWeaklyEisensteinAt 𝓟
A : Type v
inst✝ : CommRing A
φ : R →+* A
\u22a2 (Polynomial.map φ f).IsWeaklyEisensteinAt (Ideal.map φ 𝓟)
-- Mathlib/RingTheory/Polynomial/Eisenstein/Basic.lean
R : Type u
inst✝¹ : CommSemiring R
𝓟 : Ideal R
f : R[X]
hf : f.IsWeaklyEisensteinAt 𝓟
A : Type v
inst✝ : CommRing A
φ : R →+* A
n✝ : ℕ
hn : n✝ < (Polynomial.map φ f).natDegree
\u22a2 (Polynomial.map φ f).coeff n✝ ∈ Ideal.map φ 𝓟
-- Mathlib/RingTheory/Polynomial/Eisenstein/Basic.lean
R : Type u
inst✝¹ : CommSemiring R
𝓟 : Ideal R
f : R[X]
hf : f.IsWeaklyEisensteinAt 𝓟
A : Type v
inst✝ : CommRing A
φ : R →+* A
n✝ : ℕ
hn : n✝ < (Polynomial.map φ f).natDegree
\u22a2 φ (f.coeff n✝) ∈ Ideal.map φ 𝓟
-- Mathlib/Data/Set/Image.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s✝ t : Set α
f : α → β
s : Set β
hs : s ⊆ range f
\u22a2 f '' (f ⁻¹' s) = s
-- Mathlib/Topology/MetricSpace/HausdorffDistance.lean
ι : Sort u_1
α : Type u
β : Type v
inst✝² : PseudoMetricSpace α
inst✝¹ : PseudoMetricSpace β
s t u : Set α
x✝ y : α
Φ : α → β
inst✝ : ProperSpace α
hne : s.Nonempty
x : α
\u22a2 ∃ y ∈ closure s, infDist x s = dist x y
-- Mathlib/Analysis/SpecificLimits/Basic.lean
α : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : PseudoEMetricSpace α
C : ℝ≥0∞
hC : C ≠ ⊤
f : ℕ → α
hu : ∀ (n : ℕ), edist (f n) (f (n + 1)) ≤ C / 2 ^ n
a : α
ha : Tendsto f atTop (𝓝 a)
n : ℕ
\u22a2 edist (f n) a ≤ 2 * C / 2 ^ n
-- Mathlib/Analysis/SpecificLimits/Basic.lean
α : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : PseudoEMetricSpace α
C : ℝ≥0∞
hC : C ≠ ⊤
f : ℕ → α
a : α
ha : Tendsto f atTop (𝓝 a)
n : ℕ
hu : ∀ (n : ℕ), edist (f n) (f (n + 1)) ≤ C * 2⁻¹ ^ n
\u22a2 edist (f n) a ≤ 2 * C * 2⁻¹ ^ n
-- Mathlib/Analysis/SpecificLimits/Basic.lean
α : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : PseudoEMetricSpace α
C : ℝ≥0∞
hC : C ≠ ⊤
f : ℕ → α
a : α
ha : Tendsto f atTop (𝓝 a)
n : ℕ
hu : ∀ (n : ℕ), edist (f n) (f (n + 1)) ≤ C * 2⁻¹ ^ n
\u22a2 edist (f n) a ≤ C * 2⁻¹ ^ n * 2
-- Mathlib/Analysis/SpecificLimits/Basic.lean
α : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : PseudoEMetricSpace α
C : ℝ≥0∞
hC : C ≠ ⊤
f : ℕ → α
a : α
ha : Tendsto f atTop (𝓝 a)
n : ℕ
hu : ∀ (n : ℕ), edist (f n) (f (n + 1)) ≤ C * 2⁻¹ ^ n
\u22a2 C * 2⁻¹ ^ n * 2 = C * 2⁻¹ ^ n / (1 - 2⁻¹)
-- Mathlib/Topology/MetricSpace/Bounded.lean
α : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst✝¹ : PseudoMetricSpace α
x : α
s t : Set α
r : ℝ
inst✝ : ProperSpace α
hc : IsClosed s
hb : IsBounded s
\u22a2 IsCompact s
-- Mathlib/Topology/MetricSpace/Bounded.lean
α : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst✝¹ : PseudoMetricSpace α
x : α
t : Set α
r : ℝ
inst✝ : ProperSpace α
hc : IsClosed ∅
hb : IsBounded ∅
\u22a2 IsCompact ∅
-- Mathlib/Topology/MetricSpace/Bounded.lean
α : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst✝¹ : PseudoMetricSpace α
x✝ : α
s t : Set α
r : ℝ
inst✝ : ProperSpace α
hc : IsClosed s
hb : IsBounded s
x : α
\u22a2 IsCompact s
-- Mathlib/Data/DFinsupp/Basic.lean
ι : Type u
γ : Type w
β : ι → Type v
β₁ : ι → Type v₁
β₂ : ι → Type v₂
inst✝² : DecidableEq ι
inst✝¹ : (i : ι) → Zero (β i)
inst✝ : (i : ι) → (x : β i) → Decidable (x ≠ 0)
s : Finset ι
f : (i : { x // x ∈ s }) → { x // x ≠ 0 }
i : ι
\u22a2 i ∈ (mk s fun i => ↑(f i)).support ↔ i ∈ s
-- Mathlib/Data/DFinsupp/Basic.lean
ι : Type u
γ : Type w
β : ι → Type v
β₁ : ι → Type v₁
β₂ : ι → Type v₂
inst✝² : DecidableEq ι
inst✝¹ : (i : ι) → Zero (β i)
inst✝ : (i : ι) → (x : β i) → Decidable (x ≠ 0)
f : Π₀ (i : ι), β i
i : ι
\u22a2 ↑((fun f_1 => ⟨mk f.support fun i => ↑(f_1 i), ⋯⟩) ((fun x => match (motive := { f_1 // f_1.support = f.support } → (i : { x // x ∈ f.support }) → { x // x ≠ 0 }) x with | ⟨f_1, hf⟩ => fun x => match x with | ⟨i, hi⟩ => ⟨f_1 i, ⋯⟩) ⟨f, ⋯⟩)) i = ↑⟨f, ⋯⟩ i
-- Mathlib/Data/DFinsupp/Basic.lean
ι : Type u
γ : Type w
β : ι → Type v
β₁ : ι → Type v₁
β₂ : ι → Type v₂
inst✝² : DecidableEq ι
inst✝¹ : (i : ι) → Zero (β i)
inst✝ : (i : ι) → (x : β i) → Decidable (x ≠ 0)
s : Finset ι
f : (i : { x // x ∈ s }) → { x // x ≠ 0 }
x✝ : { x // x ∈ s }
\u22a2 (fun x => match (motive := { f // f.support = s } → (i : { x // x ∈ s }) → { x // x ≠ 0 }) x with | ⟨f, hf⟩ => fun x => match x with | ⟨i, hi⟩ => ⟨f i, ⋯⟩) ((fun f => ⟨mk s fun i => ↑(f i), ⋯⟩) f) x✝ = f x✝
-- Mathlib/Data/DFinsupp/Basic.lean
ι : Type u
γ : Type w
β : ι → Type v
β₁ : ι → Type v₁
β₂ : ι → Type v₂
inst✝² : DecidableEq ι
inst✝¹ : (i : ι) → Zero (β i)
inst✝ : (i : ι) → (x : β i) → Decidable (x ≠ 0)
s : Finset ι
f : (i : { x // x ∈ s }) → { x // x ≠ 0 }
x✝ : { x // x ∈ s }
\u22a2 f ⟨↑x✝, ⋯⟩ = f x✝
-- Mathlib/Data/QPF/Univariate/Basic.lean
F : Type u → Type u
q : QPF F
h : IsUniform
α β : Type u
g : α → β
x : F α
\u22a2 supp (g <$> x) = g '' supp x
-- Mathlib/Data/QPF/Univariate/Basic.lean
F : Type u → Type u
q : QPF F
h : IsUniform
α β : Type u
g : α → β
x : F α
\u22a2 supp (g <$> abs (repr x)) = g '' supp (abs (repr x))
-- Mathlib/Data/QPF/Univariate/Basic.lean
F : Type u → Type u
q : QPF F
h : IsUniform
α β : Type u
g : α → β
x : F α
a : (P F).A
f : (P F).B a → α
\u22a2 supp (g <$> abs ⟨a, f⟩) = g '' supp (abs ⟨a, f⟩)
-- Mathlib/Data/QPF/Univariate/Basic.lean
F : Type u → Type u
q : QPF F
h : IsUniform
α β : Type u
g : α → β
x : F α
a : (P F).A
f : (P F).B a → α
\u22a2 supp (abs ⟨a, g ∘ f⟩) = g '' supp (abs ⟨a, f⟩)
-- Mathlib/Data/Set/Basic.lean
α : Type u
β : Type v
γ : Type w
ι : Sort x
a b : α
s✝ s₁ s₂ t✝ t₁ t₂ u s t : Set α
\u22a2 s \ t = tᶜ ∩ s
-- Mathlib/Logic/Denumerable.lean
α : Type u_1
β : Type u_2
s : Set ℕ
inst✝¹ : Infinite ↑s
inst✝ : DecidablePred fun x => x ∈ s
x✝ : ℕ
x : ℕ := x✝
hx : x✝ ∈ s
\u22a2 ∃ n, ofNat s n = ⟨x✝, hx⟩
-- Mathlib/Logic/Denumerable.lean
α : Type u_1
β : Type u_2
s : Set ℕ
inst✝¹ : Infinite ↑s
inst✝ : DecidablePred fun x => x ∈ s
x✝ : ℕ
x : ℕ := x✝
hx : x✝ ∈ s
t : List ↑s := List.pmap (fun y hy => ⟨y, hy⟩) (List.filter (fun y => decide (y ∈ s)) (List.range x)) ⋯
ht : t = List.pmap (fun y hy => ⟨y, hy⟩) (List.filter (fun y => decide (y ∈ s)) (List.range x)) ⋯
\u22a2 ∃ n, ofNat s n = ⟨x✝, hx⟩
-- Mathlib/Logic/Denumerable.lean
α : Type u_1
β : Type u_2
s : Set ℕ
inst✝¹ : Infinite ↑s
inst✝ : DecidablePred fun x => x ∈ s
x✝ : ℕ
x : ℕ := x✝
hx : x✝ ∈ s
t : List ↑s := List.pmap (fun y hy => ⟨y, hy⟩) (List.filter (fun y => decide (y ∈ s)) (List.range x)) ⋯
ht : t = List.pmap (fun y hy => ⟨y, hy⟩) (List.filter (fun y => decide (y ∈ s)) (List.range x)) ⋯
hmt : ∀ {y : ↑s}, y ∈ t ↔ y < ⟨x, hx⟩
\u22a2 ∃ n, ofNat s n = ⟨x✝, hx⟩
-- Mathlib/Logic/Denumerable.lean
α : Type u_1
β : Type u_2
s : Set ℕ
inst✝¹ : Infinite ↑s
inst✝ : DecidablePred fun x => x ∈ s
x✝ : ℕ
x : ℕ := x✝
hx : x✝ ∈ s
t : List ↑s := List.pmap (fun y hy => ⟨y, hy⟩) (List.filter (fun y => decide (y ∈ s)) (List.range x)) ⋯
ht : t = List.pmap (fun y hy => ⟨y, hy⟩) (List.filter (fun y => decide (y ∈ s)) (List.range x)) ⋯
hmt : ∀ {y : ↑s}, y ∈ t ↔ y < ⟨x, hx⟩
wf : ∀ (m : ↑s), t.maximum = ↑m → ↑m < x
\u22a2 ∃ n, ofNat s n = ⟨x✝, hx⟩
-- Mathlib/Logic/Denumerable.lean
α : Type u_1
β : Type u_2
s : Set ℕ
inst✝¹ : Infinite ↑s
inst✝ : DecidablePred fun x => x ∈ s
x✝ : ℕ
x : ℕ := x✝
hx : x✝ ∈ s
t : List ↑s := List.pmap (fun y hy => ⟨y, hy⟩) (List.filter (fun y => decide (y ∈ s)) (List.range x)) ⋯
ht : t = List.pmap (fun y hy => ⟨y, hy⟩) (List.filter (fun y => decide (y ∈ s)) (List.range x)) ⋯
hmt : ∀ {y : ↑s}, y ∈ t ↔ y < ⟨x, hx⟩
wf : ∀ (m : ↑s), t.maximum = ↑m → ↑m < x
m : ↑s
hmax : t.maximum = ↑m
\u22a2 ∃ n, ofNat s n = ⟨x✝, hx⟩
-- Mathlib/Logic/Denumerable.lean
α : Type u_1
β : Type u_2
s : Set ℕ
inst✝¹ : Infinite ↑s
inst✝ : DecidablePred fun x => x ∈ s
x✝ : ℕ
x : ℕ := x✝
hx : x✝ ∈ s
t : List ↑s := List.pmap (fun y hy => ⟨y, hy⟩) (List.filter (fun y => decide (y ∈ s)) (List.range x)) ⋯
ht : t = List.pmap (fun y hy => ⟨y, hy⟩) (List.filter (fun y => decide (y ∈ s)) (List.range x)) ⋯
hmt : ∀ {y : ↑s}, y ∈ t ↔ y < ⟨x, hx⟩
wf : ∀ (m : ↑s), t.maximum = ↑m → ↑m < x
m : ↑s
hmax : t.maximum = ↑m
a : ℕ
ha : ofNat s a = ⟨↑m, ⋯⟩
\u22a2 ∃ n, ofNat s n = ⟨x✝, hx⟩
-- Mathlib/Logic/Denumerable.lean
α : Type u_1
β : Type u_2
s : Set ℕ
inst✝¹ : Infinite ↑s
inst✝ : DecidablePred fun x => x ∈ s
x✝ : ℕ
x : ℕ := x✝
hx : x✝ ∈ s
t : List ↑s := List.pmap (fun y hy => ⟨y, hy⟩) (List.filter (fun y => decide (y ∈ s)) (List.range x)) ⋯
ht : t = List.pmap (fun y hy => ⟨y, hy⟩) (List.filter (fun y => decide (y ∈ s)) (List.range x)) ⋯
hmt : ∀ {y : ↑s}, y ∈ t ↔ y < ⟨x, hx⟩
wf : ∀ (m : ↑s), t.maximum = ↑m → ↑m < x
m : ↑s
hmax : t.maximum = ↑m
a : ℕ
ha : ofNat s a = ⟨↑m, ⋯⟩
\u22a2 succ (ofNat s a) ≤ ⟨x✝, hx⟩
-- Mathlib/Logic/Denumerable.lean
α : Type u_1
β : Type u_2
s : Set ℕ
inst✝¹ : Infinite ↑s
inst✝ : DecidablePred fun x => x ∈ s
x✝ : ℕ
x : ℕ := x✝
hx : x✝ ∈ s
t : List ↑s := List.pmap (fun y hy => ⟨y, hy⟩) (List.filter (fun y => decide (y ∈ s)) (List.range x)) ⋯
ht : t = List.pmap (fun y hy => ⟨y, hy⟩) (List.filter (fun y => decide (y ∈ s)) (List.range x)) ⋯
hmt : ∀ {y : ↑s}, y ∈ t ↔ y < ⟨x, hx⟩
wf : ∀ (m : ↑s), t.maximum = ↑m → ↑m < x
m : ↑s
hmax : t.maximum = ↑m
a : ℕ
ha : ofNat s a = ⟨↑m, ⋯⟩
\u22a2 ⟨x✝, hx⟩ ≤ succ (ofNat s a)
-- Mathlib/Logic/Denumerable.lean
s : Set ℕ
inst✝¹ : Infinite ↑s
inst✝ : DecidablePred fun x => x ∈ s
x✝ : ℕ
a✝ : ∀ (y : ℕ), (invImage (fun x => x) instWellFoundedRelationOfSizeOf).1 y x✝ → ∀ (hx : y ∈ s), ∃ n, ofNat s n = ⟨y, hx⟩
x : ℕ := x✝
hx : x✝ ∈ s
t : List ↑s := List.pmap (fun y hy => ⟨y, hy⟩) (List.filter (fun y => decide (y ∈ s)) (List.range x)) ⋯
ht : t = List.pmap (fun y hy => ⟨y, hy⟩) (List.filter (fun y => decide (y ∈ s)) (List.range x)) ⋯
hmt : ∀ {y : ↑s}, y ∈ t ↔ y < ⟨x, hx⟩
wf : ∀ (m : ↑s), t.maximum = ↑m → ↑m < x
m : ↑s
h✝ hmax : t.maximum = ↑m
\u22a2 (invImage (fun x => x) instWellFoundedRelationOfSizeOf).1 (↑m) x✝
-- Mathlib/Data/Real/Archimedean.lean
s : Set ℝ
h : BddBelow s
h' : s.Nonempty
a : ℝ
\u22a2 sInf s ≤ a ↔ ∀ (ε : ℝ), 0 < ε → ∃ x ∈ s, x < a + ε
-- Mathlib/Data/Real/Archimedean.lean
s : Set ℝ
h : BddBelow s
h' : s.Nonempty
a : ℝ
\u22a2 (∀ (ε : ℝ), 0 < ε → sInf s < a + ε) ↔ ∀ (ε : ℝ), 0 < ε → ∃ x ∈ s, x < a + ε
-- Mathlib/Geometry/Euclidean/MongePoint.lean
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
i₁ i₂ : Fin (n + 3)
h : i₁ ≠ i₂
\u22a2 ∑ i : PointsWithCircumcenterIndex (n + 2), mongePointVSubFaceCentroidWeightsWithCircumcenter i₁ i₂ i = 0
-- Mathlib/Geometry/Euclidean/MongePoint.lean
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
i₁ i₂ : Fin (n + 3)
h : i₁ ≠ i₂
\u22a2 ∑ i : PointsWithCircumcenterIndex (n + 2), (mongePointWeightsWithCircumcenter n - centroidWeightsWithCircumcenter {i₁, i₂}ᶜ) i = 0
-- Mathlib/Geometry/Euclidean/MongePoint.lean
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
i₁ i₂ : Fin (n + 3)
h : i₁ ≠ i₂
\u22a2 1 - ∑ x : PointsWithCircumcenterIndex (n + 2), centroidWeightsWithCircumcenter {i₁, i₂}ᶜ x = 0
-- Mathlib/Geometry/Euclidean/MongePoint.lean
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
i₁ i₂ : Fin (n + 3)
h : i₁ ≠ i₂
\u22a2 {i₁, i₂}ᶜ.Nonempty
-- 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
\u22a2 (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
\u22a2 ((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
\u22a2 ((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
\u22a2 (((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
\u22a2 (((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/GroupTheory/GroupAction/SubMulAction.lean
S : Type u'
T : Type u''
R : Type u
M : Type v
inst✝¹ : Monoid R
inst✝ : MulAction R M
p : SubMulAction R M
m : ↥p
\u22a2 Subtype.val ⁻¹' MulAction.orbit R ↑m = MulAction.orbit R m
-- Mathlib/Data/Matrix/Kronecker.lean
R : Type u_1
α : Type u_2
α' : Type u_3
β : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
l : Type u_8
m : Type u_9
n : Type u_10
p : Type u_11
q : Type u_12
r : Type u_13
l' : Type u_14
m' : Type u_15
n' : Type u_16
p' : Type u_17
inst✝² : NonAssocSemiring α
inst✝¹ : DecidableEq m
inst✝ : DecidableEq n
a b : ℕ
\u22a2 (diagonal fun mn => ↑a * ↑b) = ↑(a * b)
-- Mathlib/Data/Matrix/Kronecker.lean
R : Type u_1
α : Type u_2
α' : Type u_3
β : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
l : Type u_8
m : Type u_9
n : Type u_10
p : Type u_11
q : Type u_12
r : Type u_13
l' : Type u_14
m' : Type u_15
n' : Type u_16
p' : Type u_17
inst✝² : NonAssocSemiring α
inst✝¹ : DecidableEq m
inst✝ : DecidableEq n
a b : ℕ
\u22a2 (diagonal fun mn => ↑(a * b)) = ↑(a * b)
-- Mathlib/AlgebraicGeometry/Morphisms/Constructors.lean
P : {α β : Type u} → [inst : TopologicalSpace α] → [inst : TopologicalSpace β] → (α → β) → Prop
inst✝ : (topologically fun {α β} [TopologicalSpace α] [TopologicalSpace β] => P).RespectsIso
hP₂ : ∀ {α β : Type u} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] (f : α → β) (s : Set β), P f → P (s.restrictPreimage f)
hP₃ : ∀ {α β : Type u} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] (f : α → β) {ι : Type u} (U : ι → Opens β), iSup U = ⊤ → Continuous f → (∀ (i : ι), P ((U i).carrier.restrictPreimage f)) → P f
\u22a2 IsLocalAtTarget (topologically fun {α β} [TopologicalSpace α] [TopologicalSpace β] => P)
-- Mathlib/AlgebraicGeometry/Morphisms/Constructors.lean
P : {α β : Type u} → [inst : TopologicalSpace α] → [inst : TopologicalSpace β] → (α → β) → Prop
inst✝ : (topologically fun {α β} [TopologicalSpace α] [TopologicalSpace β] => P).RespectsIso
hP₂ : ∀ {α β : Type u} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] (f : α → β) (s : Set β), P f → P (s.restrictPreimage f)
hP₃ : ∀ {α β : Type u} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] (f : α → β) {ι : Type u} (U : ι → Opens β), iSup U = ⊤ → Continuous f → (∀ (i : ι), P ((U i).carrier.restrictPreimage f)) → P f
\u22a2 ∀ {X Y : Scheme} (f : X ⟶ Y) (U : Y.Opens), topologically (fun {α β} [TopologicalSpace α] [TopologicalSpace β] => P) f → topologically (fun {α β} [TopologicalSpace α] [TopologicalSpace β] => P) (f ∣_ U)
-- Mathlib/AlgebraicGeometry/Morphisms/Constructors.lean
P : {α β : Type u} → [inst : TopologicalSpace α] → [inst : TopologicalSpace β] → (α → β) → Prop
inst✝ : (topologically fun {α β} [TopologicalSpace α] [TopologicalSpace β] => P).RespectsIso
hP₂ : ∀ {α β : Type u} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] (f : α → β) (s : Set β), P f → P (s.restrictPreimage f)
hP₃ : ∀ {α β : Type u} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] (f : α → β) {ι : Type u} (U : ι → Opens β), iSup U = ⊤ → Continuous f → (∀ (i : ι), P ((U i).carrier.restrictPreimage f)) → P f
\u22a2 ∀ {X Y : Scheme} (f : X ⟶ Y) {ι : Type u} (U : ι → Y.Opens), iSup U = ⊤ → (∀ (i : ι), topologically (fun {α β} [TopologicalSpace α] [TopologicalSpace β] => P) (f ∣_ U i)) → topologically (fun {α β} [TopologicalSpace α] [TopologicalSpace β] => P) f
-- Mathlib/Algebra/Algebra/Bilinear.lean
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
a : A
n : ℕ
\u22a2 mulLeft R a ^ n = mulLeft R (a ^ n)
-- Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀ x₁ : X
p : Path x₀ x₁
x✝ : ↑I
\u22a2 (p.trans (Path.refl x₁)) x✝ = (p.reparam (fun t => ⟨if ↑t ≤ 1 / 2 then 2 * ↑t else 1, ⋯⟩) ⋯ ⋯ ⋯) x✝
-- Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀ x₁ : X
p : Path x₀ x₁
x✝ : ↑I
\u22a2 (if h : ↑x✝ ≤ 2⁻¹ then p ⟨2 * ↑x✝, ⋯⟩ else x₁) = p ⟨if ↑x✝ ≤ 2⁻¹ then 2 * ↑x✝ else 1, ⋯⟩
-- Mathlib/RingTheory/Generators.lean
R : Type u
S : Type v
inst✝²⁵ : CommRing R
inst✝²⁴ : CommRing S
inst✝²³ : Algebra R S
P : Generators R S
R' : Type ?u.323650
S' : Type ?u.323653
inst✝²² : CommRing R'
inst✝²¹ : CommRing S'
inst✝²⁰ : Algebra R' S'
P' : Generators R' S'
inst✝¹⁹ : Algebra R R'
inst✝¹⁸ : Algebra S S'
inst✝¹⁷ : Algebra R S'
inst✝¹⁶ : IsScalarTower R R' S'
inst✝¹⁵ : IsScalarTower R S S'
R'' : Type ?u.326084
S'' : Type ?u.326087
inst✝¹⁴ : CommRing R''
inst✝¹³ : CommRing S''
inst✝¹² : Algebra R'' S''
P'' : Generators R'' S''
inst✝¹¹ : Algebra R R''
inst✝¹⁰ : Algebra S S''
inst✝⁹ : Algebra R S''
inst✝⁸ : IsScalarTower R R'' S''
inst✝⁷ : IsScalarTower R S S''
inst✝⁶ : Algebra R' R''
inst✝⁵ : Algebra S' S''
inst✝⁴ : Algebra R' S''
inst✝³ : IsScalarTower R' R'' S''
inst✝² : IsScalarTower R' S' S''
inst✝¹ : IsScalarTower R R' R''
inst✝ : IsScalarTower S S' S''
f : P'.Hom P''
g : P.Hom P'
x : P.vars
\u22a2 (aeval P''.val) ((aeval f.val) (g.val x)) = (algebraMap S S'') (P.val x)
-- Mathlib/RingTheory/Generators.lean
R : Type u
S : Type v
inst✝²⁵ : CommRing R
inst✝²⁴ : CommRing S
inst✝²³ : Algebra R S
P : Generators R S
R' : Type ?u.323650
S' : Type ?u.323653
inst✝²² : CommRing R'
inst✝²¹ : CommRing S'
inst✝²⁰ : Algebra R' S'
P' : Generators R' S'
inst✝¹⁹ : Algebra R R'
inst✝¹⁸ : Algebra S S'
inst✝¹⁷ : Algebra R S'
inst✝¹⁶ : IsScalarTower R R' S'
inst✝¹⁵ : IsScalarTower R S S'
R'' : Type ?u.326084
S'' : Type ?u.326087
inst✝¹⁴ : CommRing R''
inst✝¹³ : CommRing S''
inst✝¹² : Algebra R'' S''
P'' : Generators R'' S''
inst✝¹¹ : Algebra R R''
inst✝¹⁰ : Algebra S S''
inst✝⁹ : Algebra R S''
inst✝⁸ : IsScalarTower R R'' S''
inst✝⁷ : IsScalarTower R S S''
inst✝⁶ : Algebra R' R''
inst✝⁵ : Algebra S' S''
inst✝⁴ : Algebra R' S''
inst✝³ : IsScalarTower R' R'' S''
inst✝² : IsScalarTower R' S' S''
inst✝¹ : IsScalarTower R R' R''
inst✝ : IsScalarTower S S' S''
f : P'.Hom P''
g : P.Hom P'
x : P.vars
\u22a2 (aeval P''.val) ((aeval f.val) (g.val x)) = (algebraMap S' S'') ((aeval P'.val) (g.val x))