ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Topology/Algebra/Module/StrongTopology.lean
𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝⁹ : NormedField 𝕜₁
inst✝⁸ : NormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
E : Type u_3
F : Type u_4
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜₁ E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : AddCommGroup F
inst✝³ : Module 𝕜₂ F
inst✝² : TopologicalSpace F
inst✝¹ : TopologicalAddGroup F
inst✝ : T2Space F
𝔖 : Set (Set E)
h𝔖 : ⋃₀ 𝔖 = Set.univ
this✝ : UniformSpace F := TopologicalAddGroup.toUniformSpace F
this : UniformAddGroup F
\u22a2 T2Space (UniformConvergenceCLM σ F 𝔖)
-- Mathlib/Topology/Algebra/Module/StrongTopology.lean
𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝⁹ : NormedField 𝕜₁
inst✝⁸ : NormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
E : Type u_3
F : Type u_4
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜₁ E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : AddCommGroup F
inst✝³ : Module 𝕜₂ F
inst✝² : TopologicalSpace F
inst✝¹ : TopologicalAddGroup F
inst✝ : T2Space F
𝔖 : Set (Set E)
h𝔖 : ⋃₀ 𝔖 = Set.univ
this✝¹ : UniformSpace F := TopologicalAddGroup.toUniformSpace F
this✝ : UniformAddGroup F
this : T2Space (E →ᵤ[𝔖] F)
\u22a2 T2Space (UniformConvergenceCLM σ F 𝔖)
-- 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✝¹ : SeminormedCommGroup E
inst✝ : SeminormedCommGroup F
a a₁ a₂ b b₁ b₂ : E
r r₁ r₂ : ℝ
x✝ : E
\u22a2 x✝ ∈ a • ball b r ↔ x✝ ∈ ball (a • b) r
-- Mathlib/Topology/MetricSpace/Lipschitz.lean
α : Type u
β : Type v
γ : Type w
ι : Type x
inst✝² : PseudoMetricSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : PseudoMetricSpace γ
K : ℝ≥0
s : Set α
f : α → β
h : ∀ x ∈ s, ∀ y ∈ s, dist (f x) (f y) ≤ dist x y
\u22a2 ∀ x ∈ s, ∀ y ∈ s, dist (f x) (f y) ≤ ↑1 * dist x y
-- Mathlib/CategoryTheory/Shift/ShiftSequence.lean
C : Type u_1
A : Type u_2
inst✝⁵ : Category.{?u.225, u_1} C
inst✝⁴ : Category.{?u.229, u_2} A
F : C ⥤ A
M : Type u_3
inst✝³ : AddMonoid M
inst✝² : HasShift C M
G : Type u_4
inst✝¹ : AddGroup G
inst✝ : HasShift C G
sequence : M → C ⥤ A
isoZero : sequence 0 ≅ F
shiftIso : (n a a' : M) → n + a = a' → (shiftFunctor C n ⋙ sequence a ≅ sequence a')
shiftIso_zero : ∀ (a : M), shiftIso 0 a a ⋯ = isoWhiskerRight (shiftFunctorZero C M) (sequence a) ≪≫ (sequence a).leftUnitor
n m a a' a'' : M
ha' : n + a = a'
ha'' : m + a' = a''
\u22a2 m + n + a = a''
-- Mathlib/RingTheory/MvPowerSeries/Basic.lean
σ : Type u_1
R : Type u_2
inst✝¹ : Semiring R
m n : σ →₀ ℕ
φ ψ : MvPowerSeries σ R
inst✝ : DecidableEq σ
\u22a2 (coeff R n) (φ * ψ) = ∑ p ∈ antidiagonal n, (coeff R p.1) φ * (coeff R p.2) ψ
-- Mathlib/RingTheory/MvPowerSeries/Basic.lean
σ : Type u_1
R : Type u_2
inst✝¹ : Semiring R
m n : σ →₀ ℕ
φ ψ : MvPowerSeries σ R
inst✝ : DecidableEq σ
\u22a2 antidiagonal n = antidiagonal n
-- Mathlib/Data/Semiquot.lean
α : Type u_1
β : Type u_2
q : Semiquot α
f : α → Semiquot β
b : β
\u22a2 b ∈ q.bind f ↔ ∃ a ∈ q, b ∈ f a
-- Mathlib/Data/Semiquot.lean
α : Type u_1
β : Type u_2
q : Semiquot α
f : α → Semiquot β
b : β
\u22a2 b ∈ q.bind f ↔ ∃ a, ∃ (_ : a ∈ q), b ∈ f a
-- Mathlib/CategoryTheory/Monoidal/Subcategory.lean
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : MonoidalCategory C
P : C → Prop
inst✝² : MonoidalPredicate P
P' : C → Prop
inst✝¹ : MonoidalPredicate P'
inst✝ : BraidedCategory C
X Y : FullSubcategory P
\u22a2 (fullMonoidalSubcategoryInclusion P).map (β_ X Y).hom = inv ((fullMonoidalSubcategoryInclusion P).μ X Y) ≫ (β_ ((fullMonoidalSubcategoryInclusion P).obj X) ((fullMonoidalSubcategoryInclusion P).obj Y)).hom ≫ (fullMonoidalSubcategoryInclusion P).μ Y X
-- Mathlib/CategoryTheory/Monoidal/Subcategory.lean
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : MonoidalCategory C
P : C → Prop
inst✝² : MonoidalPredicate P
P' : C → Prop
inst✝¹ : MonoidalPredicate P'
inst✝ : BraidedCategory C
X Y : FullSubcategory P
\u22a2 (fullMonoidalSubcategoryInclusion P).μ X Y ≫ (fullMonoidalSubcategoryInclusion P).map (β_ X Y).hom = (β_ ((fullMonoidalSubcategoryInclusion P).obj X) ((fullMonoidalSubcategoryInclusion P).obj Y)).hom ≫ (fullMonoidalSubcategoryInclusion P).μ Y X
-- Mathlib/Data/Nat/Defs.lean
a b c d m n✝ k : ℕ
p q : ℕ → Prop
n : ℕ
hn : n ≠ 0
\u22a2 1 < 0 ^ n ↔ 1 < 0
-- Mathlib/Data/Nat/Defs.lean
a✝ b c d m n✝ k : ℕ
p q : ℕ → Prop
n : ℕ
hn : n ≠ 0
a : ℕ
\u22a2 1 < (a + 2) ^ n ↔ 1 < a + 2
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
x : ℝ
\u22a2 cos (π / 6) ^ 2 = 3 / 4
-- Mathlib/Computability/Primrec.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
σ : Type u_5
inst✝⁵ : Primcodable α
inst✝⁴ : Primcodable β
inst✝³ : Primcodable γ
inst✝² : Primcodable δ
inst✝¹ : Primcodable σ
c : α → Prop
inst✝ : DecidablePred c
f g : α → σ
hc : PrimrecPred c
hf : Primrec f
hg : Primrec g
\u22a2 Primrec fun a => if c a then f a else g a
-- Mathlib/Data/Ordmap/Ordset.lean
α : Type u_1
l : Ordnode α
x : α
hl : l.Sized
size✝ : ℕ
l✝ : Ordnode α
x✝ : α
r✝ : Ordnode α
hr : (node size✝ l✝ x✝ r✝).Sized
\u22a2 (l.rotateL x (node size✝ l✝ x✝ r✝)).Sized
-- Mathlib/Data/Ordmap/Ordset.lean
α : Type u_1
l : Ordnode α
x : α
hl : l.Sized
size✝ : ℕ
l✝ : Ordnode α
x✝ : α
r✝ : Ordnode α
hr : (node size✝ l✝ x✝ r✝).Sized
\u22a2 (if l✝.size < ratio * r✝.size then l.node3L x l✝ x✝ r✝ else l.node4L x l✝ x✝ r✝).Sized
-- Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : AffineSubspace k P
v : V
p : P
hp : p ∈ s
h : v +ᵥ p ∈ s
\u22a2 v ∈ s.direction
-- Mathlib/Topology/Basic.lean
X : Type u
Y : Type v
ι : Sort w
α : Type u_1
β : Type u_2
x✝¹ : X
s s₁ s₂ t : Set X
p p₁ p₂ : X → Prop
inst✝ : TopologicalSpace X
h₁ : IsClosed s
h₂ : interior t = ∅
x : X
x✝ : x ∈ interior (s ∪ t)
u : Set X
hu₁ : IsOpen u
hu₂ : u ⊆ s ∪ t
hx₁ : x ∈ u
hx₂ : x ∉ s
this : u \ s ⊆ t
\u22a2 u \ s ⊆ interior t
-- Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean
X : Type u_1
\u22a2 support 0 = ∅
-- Mathlib/CategoryTheory/Sites/Subsheaf.lean
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F F' F'' : Cᵒᵖ ⥤ Type w
G G' : Subpresheaf F
f : G.toPresheaf ⟶ F'
h : Presieve.IsSheaf J F'
U : Cᵒᵖ
s : G.toPresheaf.obj U
\u22a2 (homOfLe ⋯ ≫ G.sheafifyLift f h).app U s = f.app U s
-- Mathlib/CategoryTheory/Sites/Subsheaf.lean
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F F' F'' : Cᵒᵖ ⥤ Type w
G G' : Subpresheaf F
f : G.toPresheaf ⟶ F'
h : Presieve.IsSheaf J F'
U : Cᵒᵖ
s : G.toPresheaf.obj U
\u22a2 ∀ ⦃Y : C⦄ ⦃f_1 : Y ⟶ unop U⦄, (G.sieveOfSection ↑((homOfLe ⋯).app U s)).arrows f_1 → F'.map f_1.op ((homOfLe ⋯ ≫ G.sheafifyLift f h).app U s) = F'.map f_1.op (f.app U s)
-- Mathlib/CategoryTheory/Sites/Subsheaf.lean
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F F' F'' : Cᵒᵖ ⥤ Type w
G G' : Subpresheaf F
f : G.toPresheaf ⟶ F'
h : Presieve.IsSheaf J F'
U : Cᵒᵖ
s : G.toPresheaf.obj U
V : C
i : V ⟶ unop U
hi : (G.sieveOfSection ↑((homOfLe ⋯).app U s)).arrows i
\u22a2 F'.map i.op ((homOfLe ⋯ ≫ G.sheafifyLift f h).app U s) = F'.map i.op (f.app U s)
-- Mathlib/CategoryTheory/Sites/Subsheaf.lean
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
F F' F'' : Cᵒᵖ ⥤ Type w
G G' : Subpresheaf F
f : G.toPresheaf ⟶ F'
h : Presieve.IsSheaf J F'
U : Cᵒᵖ
s : G.toPresheaf.obj U
V : C
i : V ⟶ unop U
hi : (G.sieveOfSection ↑((homOfLe ⋯).app U s)).arrows i
this : ∀ ⦃X Y : Cᵒᵖ⦄ (f_1 : X ⟶ Y) (x : G.toPresheaf.obj X), f.app Y (G.toPresheaf.map f_1 x) = F'.map f_1 (f.app X x)
\u22a2 F'.map i.op ((homOfLe ⋯ ≫ G.sheafifyLift f h).app U s) = F'.map i.op (f.app U s)
-- Mathlib/LinearAlgebra/AffineSpace/Independent.lean
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
ι : Type u_4
s✝ : Finset ι
w✝ w₁✝ w₂✝ : ι → k
p✝ : ι → V
p : ι → P
h : AffineIndependent k p
w w₁ w₂ : ι → k
s : Finset ι
x✝ : ∑ i ∈ s, w i = 1
hw₁ : ∑ i ∈ s, w₁ i = 1
hw₂ : ∑ i ∈ s, w₂ i = 1
\u22a2 (affineCombination k s p) w ∈ affineSpan k {(affineCombination k s p) w₁, (affineCombination k s p) w₂} ↔ ∃ r, ∀ i ∈ s, w i = r * (w₂ i - w₁ i) + w₁ i
-- Mathlib/LinearAlgebra/AffineSpace/Independent.lean
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
ι : Type u_4
s✝ : Finset ι
w✝ w₁✝ w₂✝ : ι → k
p✝ : ι → V
p : ι → P
h : AffineIndependent k p
w w₁ w₂ : ι → k
s : Finset ι
x✝ : ∑ i ∈ s, w i = 1
hw₁ : ∑ i ∈ s, w₁ i = 1
hw₂ : ∑ i ∈ s, w₂ i = 1
\u22a2 (∃ r, ∀ i ∈ s, (w - w₁) i = r * (w₂ i - w₁ i)) ↔ ∃ r, ∀ i ∈ s, w i = r * (w₂ i - w₁ i) + w₁ i
-- Mathlib/LinearAlgebra/AffineSpace/Independent.lean
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
ι : Type u_4
s✝ : Finset ι
w✝ w₁✝ w₂✝ : ι → k
p✝ : ι → V
p : ι → P
h : AffineIndependent k p
w w₁ w₂ : ι → k
s : Finset ι
x✝ : ∑ i ∈ s, w i = 1
hw₁ : ∑ i ∈ s, w₁ i = 1
hw₂ : ∑ i ∈ s, w₂ i = 1
\u22a2 ∑ i ∈ s, (w - w₁) i = 0
-- Mathlib/Topology/Sets/Closeds.lean
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
s : Closeds α
\u22a2 (Opens.compl ∘ ⇑ofDual) ((⇑toDual ∘ compl) s) = s
-- Mathlib/Topology/Sets/Closeds.lean
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
s : (Opens α)ᵒᵈ
\u22a2 (⇑toDual ∘ compl) ((Opens.compl ∘ ⇑ofDual) s) = s
-- Mathlib/Data/Finsupp/Antidiagonal.lean
α : Type u
inst✝ : DecidableEq α
a : α
n : ℕ
x y : α →₀ ℕ
\u22a2 (x, y) ∈ antidiagonal (single a n) ↔ (x, y) ∈ map ({ toFun := single a, inj' := ⋯ }.prodMap { toFun := single a, inj' := ⋯ }) (antidiagonal n)
-- Mathlib/Data/Finsupp/Antidiagonal.lean
α : Type u
inst✝ : DecidableEq α
a : α
n : ℕ
x y : α →₀ ℕ
\u22a2 x + y = single a n ↔ ∃ a_1 b, a_1 + b = n ∧ single a a_1 = x ∧ single a b = y
-- Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ
f : InfinitePlace K → ℝ≥0
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
B : ℝ
h : minkowskiBound K I ≤ volume (convexBodySum K B)
\u22a2 ∃ a ∈ ↑I, a ≠ 0 ∧ ↑|(Algebra.norm ℚ) a| ≤ (B / ↑(finrank ℚ K)) ^ finrank ℚ K
-- Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ
f : InfinitePlace K → ℝ≥0
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
B : ℝ
h : minkowskiBound K I ≤ volume (convexBodySum K B)
hB : 0 ≤ B
\u22a2 ∃ a ∈ ↑I, a ≠ 0 ∧ ↑|(Algebra.norm ℚ) a| ≤ (B / ↑(finrank ℚ K)) ^ finrank ℚ K
-- Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ
f : InfinitePlace K → ℝ≥0
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
B : ℝ
h : minkowskiBound K I ≤ volume (convexBodySum K B)
hB : 0 ≤ B
h1 : 0 < (↑(finrank ℚ K))⁻¹
\u22a2 ∃ a ∈ ↑I, a ≠ 0 ∧ ↑|(Algebra.norm ℚ) a| ≤ (B / ↑(finrank ℚ K)) ^ finrank ℚ K
-- Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ
f : InfinitePlace K → ℝ≥0
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
B : ℝ
h : minkowskiBound K I ≤ volume (convexBodySum K B)
hB : 0 ≤ B
h1 : 0 < (↑(finrank ℚ K))⁻¹
h2 : 0 ≤ B / ↑(finrank ℚ K)
\u22a2 ∃ a ∈ ↑I, a ≠ 0 ∧ ↑|(Algebra.norm ℚ) a| ≤ (B / ↑(finrank ℚ K)) ^ finrank ℚ K
-- Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ
f : InfinitePlace K → ℝ≥0
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
B : ℝ
h : minkowskiBound K I ≤ volume (convexBodySum K B)
hB : 0 ≤ B
h1 : 0 < (↑(finrank ℚ K))⁻¹
h2 : 0 ≤ B / ↑(finrank ℚ K)
h_fund : IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup) (fundamentalDomain (fractionalIdealLatticeBasis K I)) volume
\u22a2 ∃ a ∈ ↑I, a ≠ 0 ∧ ↑|(Algebra.norm ℚ) a| ≤ (B / ↑(finrank ℚ K)) ^ finrank ℚ K
-- Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ
f : InfinitePlace K → ℝ≥0
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
B : ℝ
h : minkowskiBound K I ≤ volume (convexBodySum K B)
hB : 0 ≤ B
h1 : 0 < (↑(finrank ℚ K))⁻¹
h2 : 0 ≤ B / ↑(finrank ℚ K)
h_fund : IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup) (fundamentalDomain (fractionalIdealLatticeBasis K I)) volume
this : Countable ↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup
\u22a2 ∃ a ∈ ↑I, a ≠ 0 ∧ ↑|(Algebra.norm ℚ) a| ≤ (B / ↑(finrank ℚ K)) ^ finrank ℚ K
-- Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ
f : InfinitePlace K → ℝ≥0
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
B : ℝ
h : minkowskiBound K I ≤ volume (convexBodySum K B)
hB : 0 ≤ B
h1 : 0 < (↑(finrank ℚ K))⁻¹
h2 : 0 ≤ B / ↑(finrank ℚ K)
h_fund : IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup) (fundamentalDomain (fractionalIdealLatticeBasis K I)) volume
this : Countable ↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup
x : ({ w // w.IsReal } → ℝ) × ({ w // w.IsComplex } → ℂ)
hx : x ∈ (span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup
h_nz : ⟨x, hx⟩ ≠ 0
h_mem : ↑⟨x, hx⟩ ∈ convexBodySum K B
\u22a2 ∃ a ∈ ↑I, a ≠ 0 ∧ ↑|(Algebra.norm ℚ) a| ≤ (B / ↑(finrank ℚ K)) ^ finrank ℚ K
-- Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ
f : InfinitePlace K → ℝ≥0
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
B : ℝ
h : minkowskiBound K I ≤ volume (convexBodySum K B)
hB : 0 ≤ B
h1 : 0 < (↑(finrank ℚ K))⁻¹
h2 : 0 ≤ B / ↑(finrank ℚ K)
h_fund : IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup) (fundamentalDomain (fractionalIdealLatticeBasis K I)) volume
this : Countable ↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup
x : ({ w // w.IsReal } → ℝ) × ({ w // w.IsComplex } → ℂ)
hx✝ : x ∈ (span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup
hx : x ∈ ⇑(mixedEmbedding K) '' ↑↑I
h_nz : ⟨x, hx✝⟩ ≠ 0
h_mem : ↑⟨x, hx✝⟩ ∈ convexBodySum K B
\u22a2 ∃ a ∈ ↑I, a ≠ 0 ∧ ↑|(Algebra.norm ℚ) a| ≤ (B / ↑(finrank ℚ K)) ^ finrank ℚ K
-- Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ
f : InfinitePlace K → ℝ≥0
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
B : ℝ
h : minkowskiBound K I ≤ volume (convexBodySum K B)
hB : 0 ≤ B
h1 : 0 < (↑(finrank ℚ K))⁻¹
h2 : 0 ≤ B / ↑(finrank ℚ K)
h_fund : IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup) (fundamentalDomain (fractionalIdealLatticeBasis K I)) volume
this : Countable ↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup
a : K
ha : a ∈ ↑↑I
hx : (mixedEmbedding K) a ∈ (span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup
h_nz : ⟨(mixedEmbedding K) a, hx⟩ ≠ 0
h_mem : ↑⟨(mixedEmbedding K) a, hx⟩ ∈ convexBodySum K B
\u22a2 ↑|(Algebra.norm ℚ) a| ≤ (B / ↑(finrank ℚ K)) ^ finrank ℚ K
-- Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ
f : InfinitePlace K → ℝ≥0
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
B : ℝ
h : minkowskiBound K I ≤ volume (convexBodySum K B)
hB : 0 ≤ B
h1 : 0 < (↑(finrank ℚ K))⁻¹
h2 : 0 ≤ B / ↑(finrank ℚ K)
h_fund : IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup) (fundamentalDomain (fractionalIdealLatticeBasis K I)) volume
this : Countable ↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup
a : K
ha : a ∈ ↑↑I
hx : (mixedEmbedding K) a ∈ (span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup
h_nz : ⟨(mixedEmbedding K) a, hx⟩ ≠ 0
h_mem : ↑⟨(mixedEmbedding K) a, hx⟩ ∈ convexBodySum K B
\u22a2 ↑(finrank ℚ K) * ↑|(Algebra.norm ℚ) a| ^ (↑(finrank ℚ K))⁻¹ ≤ B
-- Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ
f : InfinitePlace K → ℝ≥0
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
B : ℝ
h : minkowskiBound K I ≤ volume (convexBodySum K B)
hB : 0 ≤ B
h1 : 0 < (↑(finrank ℚ K))⁻¹
h2 : 0 ≤ B / ↑(finrank ℚ K)
h_fund : IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup) (fundamentalDomain (fractionalIdealLatticeBasis K I)) volume
this : Countable ↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup
a : K
ha : a ∈ ↑↑I
hx : (mixedEmbedding K) a ∈ (span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup
h_nz : ⟨(mixedEmbedding K) a, hx⟩ ≠ 0
h_mem : ↑⟨(mixedEmbedding K) a, hx⟩ ∈ convexBodySum K B
\u22a2 ↑(finrank ℚ K) * ↑|(Algebra.norm ℚ) a| ^ (↑(finrank ℚ K))⁻¹ ≤ ∑ w : InfinitePlace K, ↑w.mult * ↑w a
-- Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ
f : InfinitePlace K → ℝ≥0
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
B : ℝ
h : minkowskiBound K I ≤ volume (convexBodySum K B)
hB : 0 ≤ B
h1 : 0 < (↑(finrank ℚ K))⁻¹
h2 : 0 ≤ B / ↑(finrank ℚ K)
h_fund : IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup) (fundamentalDomain (fractionalIdealLatticeBasis K I)) volume
this : Countable ↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup
a : K
ha : a ∈ ↑↑I
hx : (mixedEmbedding K) a ∈ (span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup
h_nz : ⟨(mixedEmbedding K) a, hx⟩ ≠ 0
h_mem : ↑⟨(mixedEmbedding K) a, hx⟩ ∈ convexBodySum K B
\u22a2 ↑|(Algebra.norm ℚ) a| ^ (∑ x : InfinitePlace K, ↑x.mult)⁻¹ ≤ (∑ w : InfinitePlace K, ↑w.mult * ↑w a) / ∑ x : InfinitePlace K, ↑x.mult
-- Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ
f : InfinitePlace K → ℝ≥0
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
B : ℝ
h : minkowskiBound K I ≤ volume (convexBodySum K B)
hB : 0 ≤ B
h1 : 0 < (↑(finrank ℚ K))⁻¹
h2 : 0 ≤ B / ↑(finrank ℚ K)
h_fund : IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup) (fundamentalDomain (fractionalIdealLatticeBasis K I)) volume
this : Countable ↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup
a : K
ha : a ∈ ↑↑I
hx : (mixedEmbedding K) a ∈ (span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup
h_nz : ⟨(mixedEmbedding K) a, hx⟩ ≠ 0
h_mem : ↑⟨(mixedEmbedding K) a, hx⟩ ∈ convexBodySum K B
\u22a2 ↑|(Algebra.norm ℚ) a| ^ (∑ x : InfinitePlace K, ↑x.mult)⁻¹ ≤ (∏ i : InfinitePlace K, ↑i a ^ ↑i.mult) ^ (∑ i : InfinitePlace K, ↑i.mult)⁻¹
-- Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ
f : InfinitePlace K → ℝ≥0
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
B : ℝ
h : minkowskiBound K I ≤ volume (convexBodySum K B)
hB : 0 ≤ B
h1 : 0 < (↑(finrank ℚ K))⁻¹
h2 : 0 ≤ B / ↑(finrank ℚ K)
h_fund : IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup) (fundamentalDomain (fractionalIdealLatticeBasis K I)) volume
this : Countable ↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup
a : K
ha : a ∈ ↑↑I
hx : (mixedEmbedding K) a ∈ (span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup
h_nz : ⟨(mixedEmbedding K) a, hx⟩ ≠ 0
h_mem : ↑⟨(mixedEmbedding K) a, hx⟩ ∈ convexBodySum K B
\u22a2 0 < ∑ i : InfinitePlace K, ↑i.mult
-- Mathlib/MeasureTheory/Measure/MeasureSpace.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
c : ℝ≥0∞
hc : c ≠ 0
s✝ : Set α
\u22a2 s✝ ∈ ae (c • μ) ↔ s✝ ∈ ae μ
-- Mathlib/Control/Applicative.lean
F✝ : Type u → Type w
G : Type v → Type u
inst✝⁴ : Applicative F✝
inst✝³ : Applicative G
inst✝² : LawfulApplicative F✝
inst✝¹ : LawfulApplicative G
α β γ : Type v
F : Type u_1 → Type u_2
AF : Applicative F
inst✝ : LawfulApplicative F
α✝ β✝ : Type u_1
f : F (α✝ → β✝)
x : F α✝
\u22a2 (Seq.seq (id <$> f) fun x_1 => x) = Seq.seq f fun x_1 => x
-- Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
k : Type u_1
V : Type u_2
P : Type u_3
inst✝² : Ring k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
S : AffineSpace V P
p₁ p₂ : P
s1 s2 : AffineSubspace k P
he : ↑s1 ∩ ↑s2 = ∅
p1 : P
hp1 : p1 ∈ ↑s1
p2 : P
hp2 : p2 ∈ ↑s2
\u22a2 s1.direction ⊔ s2.direction < (s1 ⊔ s2).direction
-- Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
k : Type u_1
V : Type u_2
P : Type u_3
inst✝² : Ring k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
S : AffineSpace V P
p₁ p₂ : P
s1 s2 : AffineSubspace k P
he : ↑s1 ∩ ↑s2 = ∅
p1 : P
hp1 : p1 ∈ ↑s1
p2 : P
hp2 : p2 ∈ ↑s2
\u22a2 s1.direction ⊔ s2.direction ≤ (s1 ⊔ s2).direction ∧ ∃ x ∈ (s1 ⊔ s2).direction, x ∉ s1.direction ⊔ s2.direction
-- Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
k : Type u_1
V : Type u_2
P : Type u_3
inst✝² : Ring k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
S : AffineSpace V P
p₁ p₂ : P
s1 s2 : AffineSubspace k P
he : ↑s1 ∩ ↑s2 = ∅
p1 : P
hp1 : p1 ∈ ↑s1
p2 : P
hp2 : p2 ∈ ↑s2
\u22a2 p2 -ᵥ p1 ∉ s1.direction ⊔ s2.direction
-- Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
k : Type u_1
V : Type u_2
P : Type u_3
inst✝² : Ring k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
S : AffineSpace V P
p₁ p₂ : P
s1 s2 : AffineSubspace k P
he : ↑s1 ∩ ↑s2 = ∅
p1 : P
hp1 : p1 ∈ ↑s1
p2 : P
hp2 : p2 ∈ ↑s2
h : p2 -ᵥ p1 ∈ s1.direction ⊔ s2.direction
\u22a2 False
-- Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
k : Type u_1
V : Type u_2
P : Type u_3
inst✝² : Ring k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
S : AffineSpace V P
p₁ p₂ : P
s1 s2 : AffineSubspace k P
he : ↑s1 ∩ ↑s2 = ∅
p1 : P
hp1 : p1 ∈ ↑s1
p2 : P
hp2 : p2 ∈ ↑s2
h : ∃ y ∈ s1.direction, ∃ z ∈ s2.direction, y + z = p2 -ᵥ p1
\u22a2 False
-- Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
k : Type u_1
V : Type u_2
P : Type u_3
inst✝² : Ring k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
S : AffineSpace V P
p₁ p₂ : P
s1 s2 : AffineSubspace k P
he : ↑s1 ∩ ↑s2 = ∅
p1 : P
hp1 : p1 ∈ ↑s1
p2 : P
hp2 : p2 ∈ ↑s2
v1 : V
hv1 : v1 ∈ s1.direction
v2 : V
hv2 : v2 ∈ s2.direction
hv1v2 : v1 + v2 = p2 -ᵥ p1
\u22a2 False
-- Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
k : Type u_1
V : Type u_2
P : Type u_3
inst✝² : Ring k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
S : AffineSpace V P
p₁ p₂ : P
s1 s2 : AffineSubspace k P
he : ↑s1 ∩ ↑s2 = ∅
p1 : P
hp1 : p1 ∈ ↑s1
p2 : P
hp2 : p2 ∈ ↑s2
v1 : V
hv1 : v1 ∈ s1.direction
v2 : V
hv2 : v2 ∈ s2.direction
hv1v2 : v1 +ᵥ p1 = -v2 +ᵥ p2
\u22a2 False
-- Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
k : Type u_1
V : Type u_2
P : Type u_3
inst✝² : Ring k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
S : AffineSpace V P
p₁ p₂ : P
s1 s2 : AffineSubspace k P
he : ↑s1 ∩ ↑s2 = ∅
p1 : P
hp1 : p1 ∈ ↑s1
p2 : P
hp2 : p2 ∈ ↑s2
v1 : V
hv1 : v1 ∈ s1.direction
v2 : V
hv2 : v2 ∈ s2.direction
hv1v2 : v1 +ᵥ p1 = -v2 +ᵥ p2
\u22a2 (↑s1 ∩ ↑s2).Nonempty
-- Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
k : Type u_1
V : Type u_2
P : Type u_3
inst✝² : Ring k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
S : AffineSpace V P
p₁ p₂ : P
s1 s2 : AffineSubspace k P
he : ↑s1 ∩ ↑s2 = ∅
p1 : P
hp1 : p1 ∈ ↑s1
p2 : P
hp2 : p2 ∈ ↑s2
v1 : V
hv1 : v1 ∈ s1.direction
v2 : V
hv2 : v2 ∈ s2.direction
hv1v2 : v1 +ᵥ p1 = -v2 +ᵥ p2
\u22a2 v1 +ᵥ p1 ∈ ↑s2
-- Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
k : Type u_1
V : Type u_2
P : Type u_3
inst✝² : Ring k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
S : AffineSpace V P
p₁ p₂ : P
s1 s2 : AffineSubspace k P
he : ↑s1 ∩ ↑s2 = ∅
p1 : P
hp1 : p1 ∈ ↑s1
p2 : P
hp2 : p2 ∈ ↑s2
v1 : V
hv1 : v1 ∈ s1.direction
v2 : V
hv2 : v2 ∈ s2.direction
hv1v2 : v1 +ᵥ p1 = -v2 +ᵥ p2
\u22a2 -v2 +ᵥ p2 ∈ ↑s2