name stringlengths 3 112 | file stringlengths 21 116 | statement stringlengths 17 8.64k | state stringlengths 7 205k | tactic stringlengths 3 4.55k | result stringlengths 7 205k | id stringlengths 16 16 |
|---|---|---|---|---|---|---|
MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul_of_measurable | Mathlib/MeasureTheory/Integral/Layercake.lean | theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable (μ : Measure α)
(f_nn : 0 ≤ f) (f_mble : Measurable f)
(g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t) (g_mble : Measurable g)
(g_nn : ∀ t > 0, 0 ≤ g t) :
∫⁻ ω, ENNReal.ofReal (∫ t in (0)..f ω, g t) ∂μ =
∫⁻ t in Ioi 0, μ {a : α | ... | case intro.intro.intro
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᶠ[ae (volume.restrict (Ioi 0))] 0
H2 : ∀ s > 0, 0 < ... | exact ⟨⟨s⟩⟩ | no goals | 6098eac6b53bede3 |
ZMod.Ico_map_valMinAbs_natAbs_eq_Ico_map_id | Mathlib/NumberTheory/LegendreSymbol/GaussEisensteinLemmas.lean | theorem Ico_map_valMinAbs_natAbs_eq_Ico_map_id (p : ℕ) [hp : Fact p.Prime] (a : ZMod p)
(hap : a ≠ 0) : ((Ico 1 (p / 2).succ).1.map fun (x : ℕ) => (a * x).valMinAbs.natAbs) =
(Ico 1 (p / 2).succ).1.map fun a => a | p : ℕ
hp : Fact (Nat.Prime p)
a : ZMod p
hap : a ≠ 0
he : ∀ {x : ℕ}, x ∈ Ico 1 (p / 2).succ → x ≠ 0 ∧ x ≤ p / 2
hep : ∀ {x : ℕ}, x ∈ Ico 1 (p / 2).succ → x < p
hpe : ∀ {x : ℕ}, x ∈ Ico 1 (p / 2).succ → ¬p ∣ x
x : ℕ
hx : x ∈ Ico 1 (p / 2).succ
⊢ (a * ↑x).valMinAbs.natAbs ∈ Ico 1 (p / 2).succ | simp [hap, CharP.cast_eq_zero_iff (ZMod p) p, hpe hx, Nat.lt_succ_iff, succ_le_iff,
pos_iff_ne_zero, natAbs_valMinAbs_le _] | no goals | e252720ce4f19196 |
cfcₙ_neg | Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/NonUnital.lean | lemma cfcₙ_neg : cfcₙ (fun x ↦ - (f x)) a = - (cfcₙ f a) | case neg.inr.inr
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommRing R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalRing R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A... | rw [cfcₙ_apply_of_not_map_zero a h0, cfcₙ_apply_of_not_map_zero, neg_zero] | case neg.inr.inr.hf
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommRing R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalRing R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R ... | 67959a27fdf049de |
Function.Antiperiodic.const_inv_smul | Mathlib/Algebra/Ring/Periodic.lean | theorem Antiperiodic.const_inv_smul [AddMonoid α] [Neg β] [Group γ] [DistribMulAction γ α]
(h : Antiperiodic f c) (a : γ) : Antiperiodic (fun x => f (a⁻¹ • x)) (a • c) | α : Type u_1
β : Type u_2
γ : Type u_3
f : α → β
c : α
inst✝³ : AddMonoid α
inst✝² : Neg β
inst✝¹ : Group γ
inst✝ : DistribMulAction γ α
h : Antiperiodic f c
a : γ
⊢ Antiperiodic (fun x => f (a⁻¹ • x)) (a • c) | simpa only [inv_inv] using h.const_smul a⁻¹ | no goals | 8b641bea6e9a552c |
List.filterMap_eq_nil_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean | theorem filterMap_eq_nil_iff {l} : filterMap f l = [] ↔ ∀ a ∈ l, f a = none | case mp
α✝¹ : Type u_1
α✝ : Type u_2
f : α✝¹ → Option α✝
l : List α✝¹
⊢ filterMap f l = [] → ∀ (a : α✝¹), a ∈ l → f a = none | exact forall_none_of_filterMap_eq_nil | no goals | 076f9b7255d24be6 |
WeierstrassCurve.Jacobian.equiv_some_of_Z_ne_zero | Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean | lemma equiv_some_of_Z_ne_zero {P : Fin 3 → F} (hPz : P z ≠ 0) :
P ≈ ![P x / P z ^ 2, P y / P z ^ 3, 1] :=
equiv_of_X_eq_of_Y_eq hPz one_ne_zero
(by linear_combination (norm := (matrix_simp; ring1)) -P x * div_self (pow_ne_zero 2 hPz))
(by linear_combination (norm := (matrix_simp; ring1)) -P y * div_self (... | case a.a
F : Type u
inst✝ : Field F
P : Fin 3 → F
hPz : P z ≠ 0
⊢ P y * ![P x / P z ^ 2, P y / P z ^ 3, 1] z ^ 3 + -P y * 1 -
(![P x / P z ^ 2, P y / P z ^ 3, 1] y * P z ^ 3 + -P y * (P z ^ 3 / P z ^ 3)) =
0 | matrix_simp | case a.a
F : Type u
inst✝ : Field F
P : Fin 3 → F
hPz : P z ≠ 0
⊢ P y * 1 ^ 3 + -P y * 1 - (P y / P z ^ 3 * P z ^ 3 + -P y * (P z ^ 3 / P z ^ 3)) = 0 | 16c9fbaefaf47c75 |
CauSeq.abv_pos_of_not_limZero | Mathlib/Algebra/Order/CauSeq/Basic.lean | theorem abv_pos_of_not_limZero {f : CauSeq β abv} (hf : ¬LimZero f) :
∃ K > 0, ∃ i, ∀ j ≥ i, K ≤ abv (f j) | α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
hf : ¬f.LimZero
this : (a : Prop) → Decidable a
nk : ¬∃ K > 0, ∃ i, ∀ j ≥ i, K ≤ abv (↑f j)
⊢ False | refine hf fun ε ε0 => ?_ | α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : CauSeq β abv
hf : ¬f.LimZero
this : (a : Prop) → Decidable a
nk : ¬∃ K > 0, ∃ i, ∀ j ≥ i, K ≤ abv (↑f j)
ε : α
ε0 : ε > 0
⊢ ∃ i, ∀ j ≥ i, abv (↑f j) < ε | 425b1d6ff87ad48e |
List.map_pmap | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Attach.lean | theorem map_pmap {p : α → Prop} (g : β → γ) (f : ∀ a, p a → β) (l H) :
map g (pmap f l H) = pmap (fun a h => g (f a h)) l H | case nil
α : Type u_1
β : Type u_2
γ : Type u_3
p : α → Prop
g : β → γ
f : (a : α) → p a → β
H : ∀ (a : α), a ∈ [] → p a
⊢ map g (pmap f [] H) = pmap (fun a h => g (f a h)) [] H | rfl | no goals | 2c17ae1ed68d788e |
PadicSeq.norm_nonarchimedean | Mathlib/NumberTheory/Padics/PadicNumbers.lean | theorem norm_nonarchimedean (f g : PadicSeq p) : (f + g).norm ≤ max f.norm g.norm | p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hfg : ¬f + g ≈ 0
hf : f ≈ 0
hfg' : f + g ≈ g
hcfg : (f + g).norm = g.norm
⊢ (f + g).norm ≤ f.norm ⊔ g.norm | have hcl : f.norm = 0 := (norm_zero_iff f).2 hf | p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hfg : ¬f + g ≈ 0
hf : f ≈ 0
hfg' : f + g ≈ g
hcfg : (f + g).norm = g.norm
hcl : f.norm = 0
⊢ (f + g).norm ≤ f.norm ⊔ g.norm | 4a919ac1cede5d3f |
MeasureTheory.MemLp.mono_exponent_of_measure_support_ne_top | Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean | /-- If a function is supported on a finite-measure set and belongs to `ℒ^p`, then it belongs to
`ℒ^q` for any `q ≤ p`. -/
lemma MemLp.mono_exponent_of_measure_support_ne_top {p q : ℝ≥0∞} {f : α → E} (hfq : MemLp f q μ)
{s : Set α} (hf : ∀ x, x ∉ s → f x = 0) (hs : μ s ≠ ∞) (hpq : p ≤ q) : MemLp f p μ | α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
p q : ℝ≥0∞
f : α → E
hfq : MemLp f q μ
s : Set α
hf : ∀ x ∉ s, f x = 0
hs : μ s ≠ ⊤
hpq : p ≤ q
⊢ (toMeasurable μ s).indicator f = f | apply Set.indicator_eq_self.2 | α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
p q : ℝ≥0∞
f : α → E
hfq : MemLp f q μ
s : Set α
hf : ∀ x ∉ s, f x = 0
hs : μ s ≠ ⊤
hpq : p ≤ q
⊢ Function.support f ⊆ toMeasurable μ s | a58e9c9fe83a60b2 |
Polynomial.count_map_roots | Mathlib/Algebra/Polynomial/Roots.lean | theorem count_map_roots [IsDomain A] [DecidableEq B] {p : A[X]} {f : A →+* B} (hmap : map f p ≠ 0)
(b : B) :
(p.roots.map f).count b ≤ rootMultiplicity b (p.map f) | case h.e'_3
A : Type u_1
B : Type u_2
inst✝³ : CommRing A
inst✝² : CommRing B
inst✝¹ : IsDomain A
inst✝ : DecidableEq B
p : A[X]
f : A →+* B
hmap : map f p ≠ 0
b : B
⊢ (Multiset.map (fun a => X - C a) (Multiset.map (⇑f) p.roots)).prod =
map f (Multiset.map (fun a => X - C a) p.roots).prod | simp only [Polynomial.map_multiset_prod, Multiset.map_map] | case h.e'_3
A : Type u_1
B : Type u_2
inst✝³ : CommRing A
inst✝² : CommRing B
inst✝¹ : IsDomain A
inst✝ : DecidableEq B
p : A[X]
f : A →+* B
hmap : map f p ≠ 0
b : B
⊢ (Multiset.map ((fun a => X - C a) ∘ ⇑f) p.roots).prod = (Multiset.map (map f ∘ fun a => X - C a) p.roots).prod | 085f91995e22d071 |
Finsupp.linearIndependent_single | Mathlib/LinearAlgebra/Finsupp/VectorSpace.lean | theorem linearIndependent_single {φ : ι → Type*} {f : ∀ ι, φ ι → M}
(hf : ∀ i, LinearIndependent R (f i)) :
LinearIndependent R fun ix : Σi, φ i => single ix.1 (f ix.1 ix.2) | case hd.refine_2
R : Type u_1
M : Type u_2
ι : Type u_3
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
φ : ι → Type u_4
f : (ι : ι) → φ ι → M
hf : ∀ (i : ι), LinearIndependent R (f i)
i : ι
t : Set ι
a✝ : t.Finite
hit : i ∉ t
⊢ ⨆ i ∈ t, span R (Set.range fun x => single i (f i x)) ≤ ⨆ a ∈ t, LinearMap.range... | refine iSup₂_mono fun i hi => ?_ | case hd.refine_2
R : Type u_1
M : Type u_2
ι : Type u_3
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
φ : ι → Type u_4
f : (ι : ι) → φ ι → M
hf : ∀ (i : ι), LinearIndependent R (f i)
i✝ : ι
t : Set ι
a✝ : t.Finite
hit : i✝ ∉ t
i : ι
hi : i ∈ t
⊢ span R (Set.range fun x => single i (f i x)) ≤ LinearMap.rang... | 16f035d3b7a67f3d |
cfcₙ_integral | Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Integral.lean | /-- The non-unital continuous functional calculus commutes with integration. -/
lemma cfcₙ_integral [TopologicalSpace X] [OpensMeasurableSpace X] (f : X → 𝕜 → 𝕜)
(bound : X → ℝ) (a : A) [SecondCountableTopologyEither X C(quasispectrum 𝕜 a, 𝕜)₀]
(hf₁ : ∀ x, ContinuousOn (f x) (quasispectrum 𝕜 a))
(hf₂ :... | X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝¹² : RCLike 𝕜
inst✝¹¹ : MeasurableSpace X
μ : Measure X
inst✝¹⁰ : NonUnitalNormedRing A
inst✝⁹ : StarRing A
inst✝⁸ : CompleteSpace A
inst✝⁷ : NormedSpace 𝕜 A
inst✝⁶ : NormedSpace ℝ A
inst✝⁵ : IsScalarTower 𝕜 A A
inst✝⁴ : SMulCommClass 𝕜 A A
inst✝³ : NonUnita... | ext | case h
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝¹² : RCLike 𝕜
inst✝¹¹ : MeasurableSpace X
μ : Measure X
inst✝¹⁰ : NonUnitalNormedRing A
inst✝⁹ : StarRing A
inst✝⁸ : CompleteSpace A
inst✝⁷ : NormedSpace 𝕜 A
inst✝⁶ : NormedSpace ℝ A
inst✝⁵ : IsScalarTower 𝕜 A A
inst✝⁴ : SMulCommClass 𝕜 A A
inst✝³ : N... | c8ceb0f73ee7838a |
Basis.det_smul_mk_coord_eq_det_update | Mathlib/LinearAlgebra/Determinant.lean | theorem Basis.det_smul_mk_coord_eq_det_update {v : ι → M} (hli : LinearIndependent R v)
(hsp : ⊤ ≤ span R (range v)) (i : ι) :
e.det v • (Basis.mk hli hsp).coord i = e.det.toMultilinearMap.toLinearMap v i | case inl
R : Type u_1
inst✝⁴ : CommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_4
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
e : Basis ι R M
v : ι → M
hli : LinearIndependent R v
hsp : ⊤ ≤ span R (Set.range v)
k : ι
⊢ e.det v * ((Basis.mk hli hsp).coord k) (v k) = ↑e.det (update v k (v k)) | rw [Basis.mk_coord_apply_eq, mul_one, update_eq_self] | case inl
R : Type u_1
inst✝⁴ : CommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_4
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
e : Basis ι R M
v : ι → M
hli : LinearIndependent R v
hsp : ⊤ ≤ span R (Set.range v)
k : ι
⊢ e.det v = ↑e.det v | 75d3b80f5c626c98 |
Complex.verticalSegment_eq | Mathlib/Data/Complex/Basic.lean | /-- A vertical segment `[b₁, b₂]` translated by `a` is the complex line segment. -/
lemma verticalSegment_eq (a b₁ b₂ : ℝ) :
(fun (y : ℝ) ↦ a + y * I) '' [[b₁, b₂]] = {a} ×ℂ [[b₁, b₂]] | case h.mpr
a b₁ b₂ : ℝ
x : ℂ
⊢ x ∈ ⇑equivRealProd ⁻¹' {a} ×ˢ [[b₁, b₂]] → x ∈ (fun y => ↑a + ↑y * I) '' [[b₁, b₂]] | intro hx | case h.mpr
a b₁ b₂ : ℝ
x : ℂ
hx : x ∈ ⇑equivRealProd ⁻¹' {a} ×ˢ [[b₁, b₂]]
⊢ x ∈ (fun y => ↑a + ↑y * I) '' [[b₁, b₂]] | f9022af6a3d4704d |
ConvexOn.continuousOn_tfae | Mathlib/Analysis/Convex/Continuous.lean | lemma ConvexOn.continuousOn_tfae (hC : IsOpen C) (hC' : C.Nonempty) (hf : ConvexOn ℝ C f) : TFAE [
LocallyLipschitzOn C f,
ContinuousOn f C,
∃ x₀ ∈ C, ContinuousAt f x₀,
∃ x₀ ∈ C, (𝓝 x₀).IsBoundedUnder (· ≤ ·) f,
∀ ⦃x₀⦄, x₀ ∈ C → (𝓝 x₀).IsBoundedUnder (· ≤ ·) f,
∀ ⦃x₀⦄, x₀ ∈ C → (𝓝 x₀).IsBoun... | E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
C : Set E
f : E → ℝ
hC : IsOpen C
hC' : C.Nonempty
hf : ConvexOn ℝ C f
tfae_1_to_2 : LocallyLipschitzOn C f → ContinuousOn f C
tfae_2_to_3 : ContinuousOn f C → ∃ x₀ ∈ C, ContinuousAt f x₀
tfae_3_to_4 : (∃ x₀ ∈ C, ContinuousAt f x₀) → ∃ x₀ ∈ C, Filter.Is... | simpa | no goals | d7a82928a7cc2bf5 |
IsFreeGroupoid.SpanningTree.endIsFree | Mathlib/GroupTheory/FreeGroup/NielsenSchreier.lean | /-- Given a free groupoid and an arborescence of its generating quiver, the vertex
group at the root is freely generated by loops coming from generating arrows
in the complement of the tree. -/
lemma endIsFree : IsFreeGroup (End (root' T)) :=
IsFreeGroup.ofUniqueLift ((wideSubquiverEquivSetTotal <| wideSubqui... | case intro.intro.refine_2.h
G : Type u
inst✝³ : Groupoid G
inst✝² : IsFreeGroupoid G
T : WideSubquiver (Symmetrify (Generators G))
inst✝¹ : Arborescence (WideSubquiver.toType (Symmetrify (Generators G)) T)
X : Type u
inst✝ : Group X
f : ↑(wideSubquiverEquivSetTotal (wideSubquiverSymmetrify T))ᶜ → X
f' : Labelling (Gene... | suffices (functorOfMonoidHom T E).map x = F'.map x by
simpa only [loopOfHom, functorOfMonoidHom, IsIso.inv_id, treeHom_root,
Category.id_comp, Category.comp_id] using this | case intro.intro.refine_2.h
G : Type u
inst✝³ : Groupoid G
inst✝² : IsFreeGroupoid G
T : WideSubquiver (Symmetrify (Generators G))
inst✝¹ : Arborescence (WideSubquiver.toType (Symmetrify (Generators G)) T)
X : Type u
inst✝ : Group X
f : ↑(wideSubquiverEquivSetTotal (wideSubquiverSymmetrify T))ᶜ → X
f' : Labelling (Gene... | 4742e81f46ca4ae4 |
Nat.Linear.Poly.denote_eq_cancelAux | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Linear.lean | theorem Poly.denote_eq_cancelAux (ctx : Context) (fuel : Nat) (m₁ m₂ r₁ r₂ : Poly)
(h : denote_eq ctx (r₁.reverse ++ m₁, r₂.reverse ++ m₂)) : denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) | case succ
ctx : Context
fuel : Nat
ih :
∀ (m₁ m₂ r₁ r₂ : Poly),
denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂) → denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂)
m₁ m₂ r₁ r₂ : Poly
h : denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)
⊢ denote_eq ctx (cancelAux (fuel + 1) m₁ m₂ r₁ r₂) | simp | case succ
ctx : Context
fuel : Nat
ih :
∀ (m₁ m₂ r₁ r₂ : Poly),
denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂) → denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂)
m₁ m₂ r₁ r₂ : Poly
h : denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)
⊢ denote_eq ctx
(match m₁, m₂ with
| m₁, [] => (List... | 9bfcd466b3bdd638 |
PseudoMetricSpace.le_two_mul_dist_ofPreNNDist | Mathlib/Topology/Metrizable/Uniformity.lean | theorem le_two_mul_dist_ofPreNNDist (d : X → X → ℝ≥0) (dist_self : ∀ x, d x x = 0)
(dist_comm : ∀ x y, d x y = d y x)
(hd : ∀ x₁ x₂ x₃ x₄, d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))) (x y : X) :
↑(d x y) ≤ 2 * @dist X
(@PseudoMetricSpace.toDist X (PseudoMetricSpace.ofPreNNDist d dist_self d... | X : Type u_1
d : X → X → ℝ≥0
dist_self : ∀ (x : X), d x x = 0
dist_comm : ∀ (x y : X), d x y = d y x
hd : ∀ (x₁ x₂ x₃ x₄ : X), d x₁ x₄ ≤ 2 * (d x₁ x₂ ⊔ (d x₂ x₃ ⊔ d x₃ x₄))
hd₀_trans : Transitive fun x y => d x y = 0
this : IsTrans X fun x y => d x y = 0
x y : X
l : List X
ihn : ∀ m < l.length, ∀ (x y : X) (l : List X)... | rw [← not_lt, Nat.lt_iff_add_one_le, ← hL_len] | X : Type u_1
d : X → X → ℝ≥0
dist_self : ∀ (x : X), d x x = 0
dist_comm : ∀ (x y : X), d x y = d y x
hd : ∀ (x₁ x₂ x₃ x₄ : X), d x₁ x₄ ≤ 2 * (d x₁ x₂ ⊔ (d x₂ x₃ ⊔ d x₃ x₄))
hd₀_trans : Transitive fun x y => d x y = 0
this : IsTrans X fun x y => d x y = 0
x y : X
l : List X
ihn : ∀ m < l.length, ∀ (x y : X) (l : List X)... | a217eb6776c388b0 |
MonomialOrder.degLex_single_lt_iff | Mathlib/Data/Finsupp/MonomialOrder/DegLex.lean | theorem degLex_single_lt_iff {a b : σ} :
single a 1 ≺[degLex] single b 1 ↔ b < a | σ : Type u_2
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
a b : σ
⊢ degLex.toSyn (single a 1) < degLex.toSyn (single b 1) ↔ b < a | rw [MonomialOrder.degLex_lt_iff, DegLex.single_lt_iff] | no goals | 58b9c1384731043a |
Cardinal.mul_eq_left_iff | Mathlib/SetTheory/Cardinal/Arithmetic.lean | theorem mul_eq_left_iff {a b : Cardinal} : a * b = a ↔ max ℵ₀ b ≤ a ∧ b ≠ 0 ∨ b = 1 ∨ a = 0 | n m : ℕ
h2a : 1 ≤ n
hb : 1 ≤ m
h : n * m = n
⊢ ¬1 < m | apply fun h2b => ne_of_gt _ h | n m : ℕ
h2a : 1 ≤ n
hb : 1 ≤ m
h : n * m = n
⊢ 1 < m → n < n * m | ce26cd2fc398ff3e |
MeasureTheory.setIntegral_tilted | Mathlib/MeasureTheory/Measure/Tilted.lean | lemma setIntegral_tilted [SFinite μ] (f : α → ℝ) (g : α → E) (s : Set α) :
∫ x in s, g x ∂(μ.tilted f) = ∫ x in s, (exp (f x) / ∫ x, exp (f x) ∂μ) • (g x) ∂μ | α : Type u_1
mα : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : SFinite μ
f : α → ℝ
g : α → E
s : Set α
hf : AEMeasurable f μ
this : AEMeasurable (fun x => rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) μ
⊢ AEMeasurable (fun x => ⟨rexp (f x) / ∫ (x : α), rexp (f x) ... | rw [← aemeasurable_coe_nnreal_real_iff] | α : Type u_1
mα : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : SFinite μ
f : α → ℝ
g : α → E
s : Set α
hf : AEMeasurable f μ
this : AEMeasurable (fun x => rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) μ
⊢ AEMeasurable (fun x => ↑⟨rexp (f x) / ∫ (x : α), rexp (f x)... | 9479368488db4a2b |
ApproximatesLinearOn.surjOn_closedBall_of_nonlinearRightInverse | Mathlib/Analysis/Calculus/InverseFunctionTheorem/ApproximatesLinearOn.lean | theorem surjOn_closedBall_of_nonlinearRightInverse
(hf : ApproximatesLinearOn f f' s c)
(f'symm : f'.NonlinearRightInverse) {ε : ℝ} {b : E} (ε0 : 0 ≤ ε) (hε : closedBall b ε ⊆ s) :
SurjOn f (closedBall b ε) (closedBall (f b) (((f'symm.nnnorm : ℝ)⁻¹ - c) * ε)) | 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E → F
inst✝ : CompleteSpace E
s : Set E
c : ℝ≥0
f' : E →L[𝕜] F
hf : ApproximatesLinearOn f f' s c
f'symm : f'.NonlinearRightInv... | linarith | no goals | dd30cb34cb471f42 |
EuclideanGeometry.OrthocentricSystem.eq_insert_orthocenter | Mathlib/Geometry/Euclidean/MongePoint.lean | theorem OrthocentricSystem.eq_insert_orthocenter {s : Set P} (ho : OrthocentricSystem s)
{t : Triangle ℝ P} (ht : Set.range t.points ⊆ s) :
s = insert t.orthocenter (Set.range t.points) | case intro.intro.inl.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
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
s : Set P
t t₀ : Triangle ℝ P
ht : Set.range t.points ⊆ insert t₀.orthocenter (Set.range ... | obtain ⟨j₁, hj₁₂, hj₁₃, hj₁₂₃⟩ :
∃ j₁ : Fin 3, j₁ ≠ j₂ ∧ j₁ ≠ j₃ ∧ ∀ j : Fin 3, j = j₁ ∨ j = j₂ ∨ j = j₃ := by
clear h₂ h₃
decide +revert | case intro.intro.inl.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
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
s : Set P
t t₀ : Triangle ℝ P
ht : Set.range t.points ⊆ insert t₀.ortho... | 727f37cad368951d |
Module.End.disjoint_genEigenspace | Mathlib/LinearAlgebra/Eigenspace/Basic.lean | lemma disjoint_genEigenspace [NoZeroSMulDivisors R M]
(f : End R M) {μ₁ μ₂ : R} (hμ : μ₁ ≠ μ₂) (k l : ℕ∞) :
Disjoint (f.genEigenspace μ₁ k) (f.genEigenspace μ₂ l) | R : Type v
M : Type w
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : NoZeroSMulDivisors R M
f : End R M
μ₁ μ₂ : R
hμ : μ₁ ≠ μ₂
k✝ l✝ : ℕ∞
k : ℕ
property✝¹ : ↑k ≤ k✝
l : ℕ
property✝ : ↑l ≤ l✝
a✝ : Nontrivial M
this : IsReduced R
p : Submodule R M := (f.genEigenspace μ₁) ↑k ⊓ (f.genEigenspace μ₂)... | apply mapsTo_genEigenspace_of_comm (Algebra.mul_sub_algebraMap_commutes f _) | R : Type v
M : Type w
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : NoZeroSMulDivisors R M
f : End R M
μ₁ μ₂ : R
hμ : μ₁ ≠ μ₂
k✝ l✝ : ℕ∞
k : ℕ
property✝¹ : ↑k ≤ k✝
l : ℕ
property✝ : ↑l ≤ l✝
a✝ : Nontrivial M
this : IsReduced R
p : Submodule R M := (f.genEigenspace μ₁) ↑k ⊓ (f.genEigenspace μ₂)... | 9dce4b617b24cb5b |
MeasureTheory.Measure.MeasureDense.of_generateFrom_isSetAlgebra_sigmaFinite | Mathlib/MeasureTheory/Measure/SeparableMeasure.lean | theorem Measure.MeasureDense.of_generateFrom_isSetAlgebra_sigmaFinite (h𝒜 : IsSetAlgebra 𝒜)
(S : μ.FiniteSpanningSetsIn 𝒜) (hgen : m = MeasurableSpace.generateFrom 𝒜) :
μ.MeasureDense 𝒜 where
measurable s hs := hgen ▸ measurableSet_generateFrom hs
approx s ms hμs ε ε_pos | X : Type u_1
m : MeasurableSpace X
μ : Measure X
𝒜 : Set (Set X)
h𝒜 : IsSetAlgebra 𝒜
S : μ.FiniteSpanningSetsIn 𝒜
hgen : m = MeasurableSpace.generateFrom 𝒜
s : Set X
ms : MeasurableSet s
hμs : μ s ≠ ⊤
ε : ℝ
ε_pos : 0 < ε
T : ℕ → Set X := Accumulate S.set
T_mem : ∀ (n : ℕ), T n ∈ 𝒜
T_finite : ∀ (n : ℕ), μ (T n) < ... | linarith [ε_pos] | no goals | 9296ab2335ceb925 |
biSup_prod | Mathlib/Order/CompleteLattice.lean | theorem biSup_prod {f : β × γ → α} {s : Set β} {t : Set γ} :
⨆ x ∈ s ×ˢ t, f x = ⨆ (a ∈ s) (b ∈ t), f (a, b) | α : Type u_1
β : Type u_2
γ : Type u_3
inst✝ : CompleteLattice α
f : β × γ → α
s : Set β
t : Set γ
⊢ ⨆ i, ⨆ j, ⨆ (_ : i ∈ s), ⨆ (_ : j ∈ t), f (i, j) = ⨆ a ∈ s, ⨆ b ∈ t, f (a, b) | exact iSup_congr fun _ => iSup_comm | no goals | f13b21bdea2248ff |
ProbabilityTheory.Kernel.singularPart_eq_zero_iff_measure_eq_zero | Mathlib/Probability/Kernel/RadonNikodym.lean | lemma singularPart_eq_zero_iff_measure_eq_zero (κ η : Kernel α γ)
[IsFiniteKernel κ] [IsFiniteKernel η] (a : α) :
singularPart κ η a = 0 ↔ κ a (mutuallySingularSetSlice κ η a) = 0 | α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : Kernel α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
h_eq_add : η.withDensity (κ.rnDeriv η) + κ.singularPart η = κ
⊢ (κ.singularPart η) a = 0 ↔ (κ a) (κ.mutuallySingularSe... | simp_rw [Kernel.ext_iff, Measure.ext_iff] at h_eq_add | α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : Kernel α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
h_eq_add : ∀ (a : α) (s : Set γ), MeasurableSet s → ((η.withDensity (κ.rnDeriv η) + κ.singularPart η) a) s = (κ a) s
⊢... | 7fe379540d5278c6 |
Besicovitch.TauPackage.color_lt | Mathlib/MeasureTheory/Covering/Besicovitch.lean | theorem color_lt {i : Ordinal.{u}} (hi : i < p.lastStep) {N : ℕ}
(hN : IsEmpty (SatelliteConfig α N p.τ)) : p.color i < N | α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
⋃ j,
⋃ (_ :
(closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.i... | refine ⟨p.r_bound, fun t ht => ?_⟩ | α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
⋃ j,
⋃ (_ :
(closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.i... | 70d4d5d8aa8a7de2 |
Matrix.det_updateCol_sum | Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean | theorem det_updateCol_sum (A : Matrix n n R) (j : n) (c : n → R) :
(A.updateCol j (fun k ↦ ∑ i, (c i) • A k i)).det = (c j) • A.det | n : Type u_2
inst✝² : DecidableEq n
inst✝¹ : Fintype n
R : Type v
inst✝ : CommRing R
A : Matrix n n R
j : n
c : n → R
⊢ (A.updateCol j fun k => ∑ i : n, c i • A k i).det = c j • A.det | rw [← det_transpose, ← updateRow_transpose, ← det_transpose A] | n : Type u_2
inst✝² : DecidableEq n
inst✝¹ : Fintype n
R : Type v
inst✝ : CommRing R
A : Matrix n n R
j : n
c : n → R
⊢ (Aᵀ.updateRow j fun k => ∑ i : n, c i • A k i).det = c j • Aᵀ.det | ca15b90f65b684f6 |
SimpleGraph.colorable_iff_exists_bdd_nat_coloring | Mathlib/Combinatorics/SimpleGraph/Coloring.lean | theorem colorable_iff_exists_bdd_nat_coloring (n : ℕ) :
G.Colorable n ↔ ∃ C : G.Coloring ℕ, ∀ v, C v < n | case mpr
V : Type u
G : SimpleGraph V
n : ℕ
⊢ (∃ C, ∀ (v : V), C v < n) → G.Colorable n | rintro ⟨C, Cf⟩ | case mpr.intro
V : Type u
G : SimpleGraph V
n : ℕ
C : G.Coloring ℕ
Cf : ∀ (v : V), C v < n
⊢ G.Colorable n | d1efc174a6faab8a |
iteratedFDerivWithin_succ_const | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | theorem iteratedFDerivWithin_succ_const (n : ℕ) (c : F) :
iteratedFDerivWithin 𝕜 (n + 1) (fun _ : E ↦ c) s = 0 | case succ
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
c : F
n : ℕ
IH : iteratedFDerivWithin 𝕜 (n + 1) (fun x => c) s = 0
⊢ ⇑(continuousMultilinearCurryLeftEquiv 𝕜 (fun... | simp only [Pi.zero_def, comp_def, fderivWithin_const, map_zero] | no goals | 4467b7a722aa0511 |
CategoryTheory.HasLiftingProperty.unop | Mathlib/CategoryTheory/LiftingProperties/Basic.lean | theorem unop {A B X Y : Cᵒᵖ} {i : A ⟶ B} {p : X ⟶ Y} (h : HasLiftingProperty i p) :
HasLiftingProperty p.unop i.unop :=
⟨fun {f} {g} sq => by
rw [CommSq.HasLift.iff_op]
simp only [Quiver.Hom.op_unop]
infer_instance⟩
| C : Type u_1
inst✝ : Category.{u_2, u_1} C
A B X Y : Cᵒᵖ
i : A ⟶ B
p : X ⟶ Y
h : HasLiftingProperty i p
f : Opposite.unop Y ⟶ Opposite.unop B
g : Opposite.unop X ⟶ Opposite.unop A
sq : CommSq f p.unop i.unop g
⊢ ⋯.HasLift | infer_instance | no goals | 2aecf067f83edbe4 |
FirstOrder.Language.Theory.imp_top | Mathlib/ModelTheory/Equivalence.lean | lemma imp_top (φ : L.BoundedFormula α n) : φ ⟹[T] ⊤ := fun M v xs => by
simp only [BoundedFormula.realize_imp, BoundedFormula.realize_top, implies_true]
| L : Language
T : L.Theory
α : Type w
n : ℕ
φ : L.BoundedFormula α n
M : T.ModelType
v : α → ↑M
xs : Fin n → ↑M
⊢ (φ ⟹ ⊤).Realize v xs | simp only [BoundedFormula.realize_imp, BoundedFormula.realize_top, implies_true] | no goals | 3c45870ac0b32034 |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.confirmRupHint_preserves_invariant_helper | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddResult.lean | theorem confirmRupHint_preserves_invariant_helper {n : Nat} (f : DefaultFormula n)
(f_assignments_size : f.assignments.size = n)
(acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool) (hsize : acc.1.size = n)
(l : Literal (PosFin n)) (ih : DerivedLitsInvariant f f_assignments_size acc.1 hsize acc.2.... | n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool
hsize : acc.fst.size = n
l : Literal (PosFin n)
ih : f.DerivedLitsInvariant f_assignments_size acc.fst hsize acc.snd.fst
h : ¬hasAssignment l.snd acc.fst[l.fst.val]! = true
hsize'✝ : (ac... | apply Or.inr ∘ Or.inl ∘ Exists.intro ⟨0, zero_lt_length_list⟩ | n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool
hsize : acc.fst.size = n
l : Literal (PosFin n)
ih : f.DerivedLitsInvariant f_assignments_size acc.fst hsize acc.snd.fst
h : ¬hasAssignment l.snd acc.fst[l.fst.val]! = true
hsize'✝ : (ac... | 02ca1706b397b254 |
IsFractionRing.mk'_eq_one_iff_eq | Mathlib/RingTheory/Localization/FractionRing.lean | theorem mk'_eq_one_iff_eq {x : A} {y : nonZeroDivisors A} : mk' K x y = 1 ↔ x = y | A : Type u_4
inst✝³ : CommRing A
K : Type u_5
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
x : A
y : ↥(nonZeroDivisors A)
this : Nontrivial A
⊢ mk' K x y = 1 ↔ x = ↑y | refine ⟨?_, fun hxy => by rw [hxy, mk'_self']⟩ | A : Type u_4
inst✝³ : CommRing A
K : Type u_5
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
x : A
y : ↥(nonZeroDivisors A)
this : Nontrivial A
⊢ mk' K x y = 1 → x = ↑y | 1acd3476fc13544b |
Ordinal.opow_le_opow_left | Mathlib/SetTheory/Ordinal/Exponential.lean | theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a ≤ b) : a ^ c ≤ b ^ c | case pos
a b c : Ordinal.{u_1}
ab : a ≤ b
a0 : a = 0
⊢ a ^ c ≤ b ^ c | subst a | case pos
b c : Ordinal.{u_1}
ab : 0 ≤ b
⊢ 0 ^ c ≤ b ^ c | acbd3ac5d04a4f70 |
CategoryTheory.Triangulated.TStructure.exists_triangle | Mathlib/CategoryTheory/Triangulated/TStructure/Basic.lean | lemma exists_triangle (A : C) (n₀ n₁ : ℤ) (h : n₀ + 1 = n₁) :
∃ (X Y : C) (_ : t.le n₀ X) (_ : t.ge n₁ Y) (f : X ⟶ A) (g : A ⟶ Y)
(h : Y ⟶ X⟦(1 : ℤ)⟧), Triangle.mk f g h ∈ distTriang C | case intro.intro.intro.intro.intro.intro.intro
C : Type u_1
inst✝⁵ : Category.{u_2, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
A : C
n₀ n₁ : ℤ
h✝ : n₀ + 1 = n₁
X Y : C
hX : t.le 0 X
hY : t.ge 1 Y... | exact ⟨_, _, t.le_shift _ _ _ (neg_add_cancel n₀) _ hX,
t.ge_shift _ _ _ (by omega) _ hY, _, _, _, hT'⟩ | no goals | 1024dfa167963c4d |
IsClosed.exists_minimal_nonempty_closed_subset | Mathlib/Topology/Compactness/Compact.lean | theorem IsClosed.exists_minimal_nonempty_closed_subset [CompactSpace X] {S : Set X}
(hS : IsClosed S) (hne : S.Nonempty) :
∃ V : Set X, V ⊆ S ∧ V.Nonempty ∧ IsClosed V ∧
∀ V' : Set X, V' ⊆ V → V'.Nonempty → IsClosed V' → V' = V | case h.refine_3.htd.mk.mk
X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
S : Set X
hS : IsClosed S
hne : S.Nonempty
opens : Set (Set X) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Uᶜ.Nonempty}
c : Set (Set X)
hc : c ⊆ opens
hz : IsChain (fun x1 x2 => x1 ⊆ x2) c
U₀ : Set X
hU₀ : U₀ ∈ c
this : Nonempty { U // U ∈ c }
U₀co... | obtain ⟨V, hVc, hVU, hVU'⟩ := hz.directedOn U hU U' hU' | case h.refine_3.htd.mk.mk.intro.intro.intro
X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
S : Set X
hS : IsClosed S
hne : S.Nonempty
opens : Set (Set X) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Uᶜ.Nonempty}
c : Set (Set X)
hc : c ⊆ opens
hz : IsChain (fun x1 x2 => x1 ⊆ x2) c
U₀ : Set X
hU₀ : U₀ ∈ c
this : Nonempty {... | b6749466be1a99f8 |
tendsto_setIntegral_peak_smul_of_integrableOn_of_tendsto | Mathlib/MeasureTheory/Integral/PeakFunction.lean | theorem tendsto_setIntegral_peak_smul_of_integrableOn_of_tendsto
(hs : MeasurableSet s) {t : Set α} (ht : MeasurableSet t) (hts : t ⊆ s) (h'ts : t ∈ 𝓝[s] x₀)
(h't : μ t ≠ ∞) (hnφ : ∀ᶠ i in l, ∀ x ∈ s, 0 ≤ φ i x)
(hlφ : ∀ u : Set α, IsOpen u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u))
(hiφ : Tendsto (f... | α : Type u_1
E : Type u_2
ι : Type u_3
hm : MeasurableSpace α
μ : Measure α
inst✝⁴ : TopologicalSpace α
inst✝³ : BorelSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
g : α → E
l : Filter ι
x₀ : α
s : Set α
φ : ι → α → ℝ
a : E
inst✝ : CompleteSpace E
hs : MeasurableSet s
t : Set α
ht : MeasurableSet t
hts... | refine Tendsto.add ?_ (Tendsto.smul hiφ tendsto_const_nhds) | α : Type u_1
E : Type u_2
ι : Type u_3
hm : MeasurableSpace α
μ : Measure α
inst✝⁴ : TopologicalSpace α
inst✝³ : BorelSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
g : α → E
l : Filter ι
x₀ : α
s : Set α
φ : ι → α → ℝ
a : E
inst✝ : CompleteSpace E
hs : MeasurableSet s
t : Set α
ht : MeasurableSet t
hts... | fcf9a03d4108054c |
HurwitzZeta.continuousOn_cosKernel | Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean | lemma continuousOn_cosKernel (a : UnitAddCircle) : ContinuousOn (cosKernel a) (Ioi 0) | a' x : ℝ
hx : x ∈ Ioi 0
⊢ ContinuousAt (fun u => (↑a', I * ↑u)) x | fun_prop | no goals | ad5ecbe9d916a8ac |
Computation.destruct_eq_think | Mathlib/Data/Seq/Computation.lean | theorem destruct_eq_think {s : Computation α} {s'} : destruct s = Sum.inr s' → s = think s' | α : Type u
s s' : Computation α
⊢ (match ↑s 0 with
| none => Sum.inr s.tail
| some a => Sum.inl a) =
Sum.inr s' →
s = s'.think | induction' f0 : s.1 0 with a' <;> intro h | case none
α : Type u
s s' : Computation α
f0 : ↑s 0 = none
h :
(match none with
| none => Sum.inr s.tail
| some a => Sum.inl a) =
Sum.inr s'
⊢ s = s'.think
case some
α : Type u
s s' : Computation α
a' : α
f0 : ↑s 0 = some a'
h :
(match some a' with
| none => Sum.inr s.tail
| some a => Sum.inl a... | 06ee7295d82d06ea |
Bimod.whisker_assoc_bimod | Mathlib/CategoryTheory/Monoidal/Bimod.lean | theorem whisker_assoc_bimod {W X Y Z : Mon_ C} (M : Bimod W X) {N N' : Bimod X Y} (f : N ⟶ N')
(P : Bimod Y Z) :
whiskerRight (whiskerLeft M f) P =
(associatorBimod M N P).hom ≫
whiskerLeft M (whiskerRight f P) ≫ (associatorBimod M N' P).inv | case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
W X Y Z : Mon_ C
M : Bimod W X
N N' : Bimod X Y
f :... | slice_rhs 3 5 => rw [id_tensor_π_preserves_coequalizer_inv_desc] | case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
W X Y Z : Mon_ C
M : Bimod W X
N N' : Bimod X Y
f :... | 64d3d49872bb8a8d |
MonoidHom.map_cyclic | Mathlib/GroupTheory/SpecificGroups/Cyclic.lean | theorem MonoidHom.map_cyclic [h : IsCyclic G] (σ : G →* G) :
∃ m : ℤ, ∀ g : G, σ g = g ^ m | case intro.intro.intro
G : Type u_2
inst✝ : Group G
h✝ : IsCyclic G
σ : G →* G
h : G
hG : ∀ (x : G), x ∈ zpowers h
m : ℤ
hm : (fun x => h ^ x) m = σ h
n : ℤ
⊢ σ ((fun x => h ^ x) n) = (fun x => h ^ x) n ^ m | rw [MonoidHom.map_zpow, ← hm, ← zpow_mul, ← zpow_mul'] | no goals | f2409e53177b2ec7 |
Vector.forIn'_map | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Monadic.lean | theorem forIn'_map [Monad m] [LawfulMonad m]
(l : Vector α n) (g : α → β) (f : (b : β) → b ∈ l.map g → γ → m (ForInStep γ)) :
forIn' (l.map g) init f = forIn' l init fun a h y => f (g a) (mem_map_of_mem g h) y | case mk
m : Type u_1 → Type u_2
α : Type u_3
n : Nat
β : Type u_4
γ : Type u_1
init : γ
inst✝¹ : Monad m
inst✝ : LawfulMonad m
g : α → β
toArray✝ : Array α
size_toArray✝ : toArray✝.size = n
f : (b : β) → b ∈ map g { toArray := toArray✝, size_toArray := size_toArray✝ } → γ → m (ForInStep γ)
⊢ forIn' (map g { toArray := ... | simp | no goals | bd5311b58c43ed9a |
WeierstrassCurve.Affine.CoordinateRing.XYIdeal_mul_XYIdeal | Mathlib/AlgebraicGeometry/EllipticCurve/Group.lean | lemma XYIdeal_mul_XYIdeal {x₁ x₂ y₁ y₂ : F} (h₁ : W.Equation x₁ y₁) (h₂ : W.Equation x₂ y₂)
(hxy : ¬(x₁ = x₂ ∧ y₁ = W.negY x₂ y₂)) :
XIdeal W (W.addX x₁ x₂ <| W.slope x₁ x₂ y₁ y₂) * (XYIdeal W x₁ (C y₁) * XYIdeal W x₂ (C y₂)) =
YIdeal W (linePolynomial x₁ y₁ <| W.slope x₁ x₂ y₁ y₂) *
XYIdeal W (W.... | F : Type u
inst✝ : Field F
W : Affine F
x₁ x₂ y₁ y₂ : F
h₁ : W.Equation x₁ y₁
h₂ : W.Equation x₂ y₂
hxy : ¬(x₁ = x₂ ∧ y₁ = W.negY x₂ y₂)
x✝² x✝¹ c x✝ : Ideal W.CoordinateRing
⊢ x✝² ⊔ (x✝¹ ⊔ (c ⊔ x✝)) = x✝² ⊔ x✝ ⊔ x✝¹ ⊔ c | rw [← sup_assoc, sup_comm c, sup_sup_sup_comm, ← sup_assoc] | no goals | cdd19a65764964f5 |
tsum_of_norm_bounded | Mathlib/Analysis/Normed/Group/InfiniteSum.lean | theorem tsum_of_norm_bounded {f : ι → E} {g : ι → ℝ} {a : ℝ} (hg : HasSum g a)
(h : ∀ i, ‖f i‖ ≤ g i) : ‖∑' i : ι, f i‖ ≤ a | case neg
ι : Type u_1
E : Type u_3
inst✝ : SeminormedAddCommGroup E
f : ι → E
g : ι → ℝ
a : ℝ
hg : HasSum g a
h : ∀ (i : ι), ‖f i‖ ≤ g i
hf : ¬Summable f
⊢ ‖∑' (i : ι), f i‖ ≤ a | rw [tsum_eq_zero_of_not_summable hf, norm_zero] | case neg
ι : Type u_1
E : Type u_3
inst✝ : SeminormedAddCommGroup E
f : ι → E
g : ι → ℝ
a : ℝ
hg : HasSum g a
h : ∀ (i : ι), ‖f i‖ ≤ g i
hf : ¬Summable f
⊢ 0 ≤ a | 933cc7d466c00692 |
Std.Tactic.BVDecide.BVExpr.bitblast.blastShiftRight.twoPowShift_eq | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/ShiftRight.lean | theorem twoPowShift_eq (aig : AIG α) (target : TwoPowShiftTarget aig w) (lhs : BitVec w)
(rhs : BitVec target.n) (assign : α → Bool)
(hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, target.lhs.get idx hidx, assign⟧ = lhs.getLsbD idx)
(hright : ∀ (idx : Nat) (hidx : idx < target.n), ⟦aig, target.rhs.get idx hi... | α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
lhs : BitVec w
assign : α → Bool
idx : Nat
hidx : idx < w
res : RefVecEntry α w
n : Nat
lvec : aig.RefVec w
rvec : aig.RefVec n
pow : Nat
rhs : BitVec { n := n, lhs := lvec, rhs := rvec, pow := pow }.n
hleft :
∀ (idx : Nat) (hidx : idx < w),
⟦... | simp only [RefVec.denote_ite, RefVec.get_cast, Ref.cast_eq,
denote_blastShiftRightConst] | α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
lhs : BitVec w
assign : α → Bool
idx : Nat
hidx : idx < w
res : RefVecEntry α w
n : Nat
lvec : aig.RefVec w
rvec : aig.RefVec n
pow : Nat
rhs : BitVec { n := n, lhs := lvec, rhs := rvec, pow := pow }.n
hleft :
∀ (idx : Nat) (hidx : idx < w),
⟦... | a2117c7fa85ff087 |
continuousOn_Ico_extendFrom_Ioo | Mathlib/Topology/Order/ExtendFrom.lean | theorem continuousOn_Ico_extendFrom_Ioo [TopologicalSpace α] [LinearOrder α] [DenselyOrdered α]
[OrderTopology α] [TopologicalSpace β] [RegularSpace β] {f : α → β} {a b : α} {la : β}
(hab : a < b) (hf : ContinuousOn f (Ioo a b)) (ha : Tendsto f (𝓝[>] a) (𝓝 la)) :
ContinuousOn (extendFrom (Ioo a b) f) (Ico... | case hf
α : Type u_1
β : Type u_2
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : DenselyOrdered α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
inst✝ : RegularSpace β
f : α → β
a b : α
la : β
hab : a < b
hf : ContinuousOn f (Ioo a b)
ha : Tendsto f (𝓝[>] a) (𝓝 la)
x : α
x_in : x ∈ Ico a b
⊢ ∃ y, T... | rcases eq_left_or_mem_Ioo_of_mem_Ico x_in with (rfl | h) | case hf.inl
α : Type u_1
β : Type u_2
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : DenselyOrdered α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
inst✝ : RegularSpace β
f : α → β
b : α
la : β
x : α
hab : x < b
hf : ContinuousOn f (Ioo x b)
ha : Tendsto f (𝓝[>] x) (𝓝 la)
x_in : x ∈ Ico x b
⊢ ∃ y,... | 3347b2c7738dfd3f |
HomologicalComplex.homotopyCofiber.ext_from_X | Mathlib/Algebra/Homology/HomotopyCofiber.lean | lemma ext_from_X (i j : ι) (hij : c.Rel j i) {A : C} {f g : X φ j ⟶ A}
(h₁ : inlX φ i j hij ≫ f = inlX φ i j hij ≫ g) (h₂ : inrX φ j ≫ f = inrX φ j ≫ g) :
f = g | case h₀
C : Type u_1
inst✝³ : Category.{u_3, u_1} C
inst✝² : Preadditive C
ι : Type u_2
c : ComplexShape ι
F G : HomologicalComplex C c
φ : F ⟶ G
inst✝¹ : HasHomotopyCofiber φ
inst✝ : DecidableRel c.Rel
i j : ι
hij : c.Rel j i
A : C
f g : X φ j ⟶ A
h₁ : inlX φ i j hij ≫ f = inlX φ i j hij ≫ g
h₂ : inrX φ j ≫ f = inrX φ... | simpa [inlX] using h₁ | no goals | 9e067f48861194cf |
NNReal.tsum_eq_toNNReal_tsum | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | theorem tsum_eq_toNNReal_tsum {f : β → ℝ≥0} : ∑' b, f b = (∑' b, (f b : ℝ≥0∞)).toNNReal | case pos
β : Type u_2
f : β → ℝ≥0
h : Summable f
⊢ ∑' (b : β), f b = (∑' (b : β), ↑(f b)).toNNReal | rw [← ENNReal.coe_tsum h, ENNReal.toNNReal_coe] | no goals | 833b04708db8631a |
RingHom.RespectsIso.cancel_right_isIso | Mathlib/RingTheory/RingHomProperties.lean | theorem RespectsIso.cancel_right_isIso (hP : RespectsIso @P) {R S T : CommRingCat} (f : R ⟶ S)
(g : S ⟶ T) [IsIso g] : P (g.hom.comp f.hom) ↔ P f.hom :=
⟨fun H => by
convert hP.1 (f ≫ g).hom (asIso g).symm.commRingCatIsoToRingEquiv H
simp [← CommRingCat.hom_comp],
hP.1 f.hom (asIso g).commRingCatIsoToR... | case h.e'_5
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : RespectsIso P
R S T : CommRingCat
f : R ⟶ S
g : S ⟶ T
inst✝ : IsIso g
H : P ((CommRingCat.Hom.hom g).comp (CommRingCat.Hom.hom f))
⊢ CommRingCat.Hom.hom f = (asIso g).symm.commRingCatIsoToRingEquiv.toRingHom.comp (CommR... | simp [← CommRingCat.hom_comp] | no goals | 95c7f3857952c279 |
maximal_orthonormal_iff_orthogonalComplement_eq_bot | Mathlib/Analysis/InnerProductSpace/Projection.lean | theorem maximal_orthonormal_iff_orthogonalComplement_eq_bot (hv : Orthonormal 𝕜 ((↑) : v → E)) :
(∀ u ⊇ v, Orthonormal 𝕜 ((↑) : u → E) → u = v) ↔ (span 𝕜 v)ᗮ = ⊥ | case mp.intro.intro.refine_2.mk.inl.mk
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
v : Set E
hv : Orthonormal 𝕜 Subtype.val
x : E
hx' : x ∈ (span 𝕜 v)ᗮ
hx : x ≠ 0
e : E := (↑‖x‖)⁻¹ • x
he : ‖e‖ = 1
he' : e ∈ (span 𝕜 v)ᗮ
he'' : e ∉ v
h_end : ∀ a ∈ v, ⟪a, ... | rw [inner_eq_zero_symm] | case mp.intro.intro.refine_2.mk.inl.mk
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
v : Set E
hv : Orthonormal 𝕜 Subtype.val
x : E
hx' : x ∈ (span 𝕜 v)ᗮ
hx : x ≠ 0
e : E := (↑‖x‖)⁻¹ • x
he : ‖e‖ = 1
he' : e ∈ (span 𝕜 v)ᗮ
he'' : e ∉ v
h_end : ∀ a ∈ v, ⟪a, ... | ce0a185108c33220 |
Real.expNear_succ | Mathlib/Data/Complex/Exponential.lean | theorem expNear_succ (n x r) : expNear (n + 1) x r = expNear n x (1 + x / (n + 1) * r) | n : ℕ
x r : ℝ
⊢ expNear (n + 1) x r = expNear n x (1 + x / (↑n + 1) * r) | simp [expNear, range_succ, mul_add, add_left_comm, add_assoc, pow_succ, div_eq_mul_inv,
mul_inv, Nat.factorial] | n : ℕ
x r : ℝ
⊢ x ^ n * x * ((↑n.factorial)⁻¹ * (↑n + 1)⁻¹) * r = x ^ n * (↑n.factorial)⁻¹ * (x * (↑n + 1)⁻¹ * r) | a822a9ca295a465e |
LittleWedderburn.InductionHyp.center_eq_top | Mathlib/RingTheory/LittleWedderburn.lean | theorem center_eq_top [Finite D] (hD : InductionHyp D) : Subring.center D = ⊤ | case intro
D : Type u_1
inst✝¹ : DivisionRing D
inst✝ : Finite D
hD : LittleWedderburn.InductionHyp D
val✝ : Fintype D
Z : Subring D := Subring.center D
hZ : Z ≠ ⊤
this : Field ↥Z := hD.field ⋯
q : ℕ := card ↥Z
card_Z : q = card ↥Z
hq : 1 < q
n : ℕ := finrank (↥Z) D
card_D : card D = q ^ n
h1qn : 1 ≤ q ^ n
Φₙ : ℤ[X] :=... | rw [Nat.cast_add, Nat.cast_sub h1qn, Nat.cast_sub hq.le, Nat.cast_one, Nat.cast_pow] at key | case intro
D : Type u_1
inst✝¹ : DivisionRing D
inst✝ : Finite D
hD : LittleWedderburn.InductionHyp D
val✝ : Fintype D
Z : Subring D := Subring.center D
hZ : Z ≠ ⊤
this : Field ↥Z := hD.field ⋯
q : ℕ := card ↥Z
card_Z : q = card ↥Z
hq : 1 < q
n : ℕ := finrank (↥Z) D
card_D : card D = q ^ n
h1qn : 1 ≤ q ^ n
Φₙ : ℤ[X] :=... | abd28bec54f62bc4 |
Submodule.map_le_smul_top | Mathlib/RingTheory/Ideal/Operations.lean | theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) :
Submodule.map f I ≤ I • (⊤ : Submodule R M) | R : Type u
M : Type v
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I : Ideal R
f : R →ₗ[R] M
⊢ map f I ≤ I • ⊤ | rintro _ ⟨y, hy, rfl⟩ | case intro.intro
R : Type u
M : Type v
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I : Ideal R
f : R →ₗ[R] M
y : R
hy : y ∈ ↑I
⊢ f y ∈ I • ⊤ | 7626369fba0eb45f |
CategoryTheory.constantSheafAdj_counit_w | Mathlib/CategoryTheory/Sites/ConstantSheaf.lean | /-- The counit of `constantSheafAdj` factors through the isomorphism `constantCommuteCompose`. -/
lemma constantSheafAdj_counit_w {T : C} (hT : IsTerminal T) :
((constantCommuteCompose J U).hom.app (F.val.obj ⟨T⟩)) ≫
((constantSheafAdj J B hT).counit.app ((sheafCompose J U).obj F)) =
((sheafCompose J ... | case h.w.h
C : Type u_1
inst✝⁶ : Category.{u_4, u_1} C
J : GrothendieckTopology C
D : Type u_2
inst✝⁵ : Category.{u_6, u_2} D
inst✝⁴ : HasWeakSheafify J D
B : Type u_3
inst✝³ : Category.{u_5, u_3} B
U : D ⥤ B
inst✝² : HasWeakSheafify J B
inst✝¹ : J.PreservesSheafification U
inst✝ : J.HasSheafCompose U
F : Sheaf J D
T :... | simp [← map_comp, ← NatTrans.comp_app] | no goals | a5cd496bde667b30 |
ArithmeticFunction.vonMangoldt.continuousOn_LFunctionResidueClassAux' | Mathlib/NumberTheory/LSeries/PrimesInAP.lean | /-- The auxiliary function is continuous away from the zeros of the L-functions of the Dirichlet
characters mod `q` (including at `s = 1`). -/
lemma continuousOn_LFunctionResidueClassAux' :
ContinuousOn (LFunctionResidueClassAux a)
{s | s = 1 ∨ ∀ χ : DirichletCharacter ℂ q, LFunction χ s ≠ 0} | case refine_2
q : ℕ
a : ZMod q
inst✝ : NeZero q
⊢ ContinuousOn (fun s => ∑ x ∈ {1}ᶜ, x a⁻¹ * (-deriv (LFunction x) s / LFunction x s))
{s | s = 1 ∨ ∀ (χ : DirichletCharacter ℂ q), LFunction χ s ≠ 0} | refine continuousOn_finset_sum _ fun χ hχ ↦ continuousOn_const.mul ?_ | case refine_2
q : ℕ
a : ZMod q
inst✝ : NeZero q
χ : DirichletCharacter ℂ q
hχ : χ ∈ {1}ᶜ
⊢ ContinuousOn (fun s => -deriv (LFunction χ) s / LFunction χ s)
{s | s = 1 ∨ ∀ (χ : DirichletCharacter ℂ q), LFunction χ s ≠ 0} | 12ce344008cdf864 |
HurwitzKernelBounds.isBigO_atTop_F_int_zero_sub | Mathlib/NumberTheory/ModularForms/JacobiTheta/Bounds.lean | lemma isBigO_atTop_F_int_zero_sub (a : UnitAddCircle) : ∃ p, 0 < p ∧
(fun t ↦ F_int 0 a t - (if a = 0 then 1 else 0)) =O[atTop] fun t ↦ exp (-p * t) | case intro.intro
a : ℝ
ha : a ∈ Ico 0 1
⊢ ∃ p, 0 < p ∧ (fun t => F_int 0 (↑a) t - if ↑a = 0 then 1 else 0) =O[atTop] fun t => rexp (-p * t) | obtain ⟨p, hp, hp'⟩ := isBigO_atTop_F_nat_zero_sub ha.1 | case intro.intro.intro.intro
a : ℝ
ha : a ∈ Ico 0 1
p : ℝ
hp : 0 < p
hp' : (fun t => F_nat 0 a t - if a = 0 then 1 else 0) =O[atTop] fun t => rexp (-p * t)
⊢ ∃ p, 0 < p ∧ (fun t => F_int 0 (↑a) t - if ↑a = 0 then 1 else 0) =O[atTop] fun t => rexp (-p * t) | 216d869dd795975f |
Ordinal.cof_bsup_le | Mathlib/SetTheory/Cardinal/Cofinality.lean | theorem cof_bsup_le {o : Ordinal} {f : ∀ a < o, Ordinal} :
(∀ i h, f i h < bsup.{u, u} o f) → cof (bsup.{u, u} o f) ≤ o.card | o : Ordinal.{u}
f : (a : Ordinal.{u}) → a < o → Ordinal.{u}
⊢ (∀ (i : Ordinal.{u}) (h : i < o), f i h < o.bsup f) → (o.bsup f).cof ≤ o.card | rw [← o.card.lift_id] | o : Ordinal.{u}
f : (a : Ordinal.{u}) → a < o → Ordinal.{u}
⊢ (∀ (i : Ordinal.{u}) (h : i < o), f i h < o.bsup f) → (o.bsup f).cof ≤ Cardinal.lift.{u, u} o.card | 02da607a4de133fe |
MeasureTheory.lintegral_iSup | Mathlib/MeasureTheory/Integral/Lebesgue.lean | theorem lintegral_iSup {f : ℕ → α → ℝ≥0∞} (hf : ∀ n, Measurable (f n)) (h_mono : Monotone f) :
∫⁻ a, ⨆ n, f n a ∂μ = ⨆ n, ∫⁻ a, f n a ∂μ | case h
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
hf : ∀ (n : ℕ), Measurable (f n)
h_mono : Monotone f
c : ℝ≥0 → ℝ≥0∞ := ofNNReal
F : α → ℝ≥0∞ := fun a => ⨆ n, f n a
s : α →ₛ ℝ≥0
hsf : ∀ (x : α), ↑(s x) ≤ ⨆ n, f n x
r✝ : ℝ≥0
right✝ : ↑r✝ < 1
ha✝ : ↑r✝ < 1
ha : r✝ < 1
rs : α →ₛ ℝ≥0 := SimpleFunc.m... | congr 2 with a | case h.e_a.h.e_6.h.h
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
hf : ∀ (n : ℕ), Measurable (f n)
h_mono : Monotone f
c : ℝ≥0 → ℝ≥0∞ := ofNNReal
F : α → ℝ≥0∞ := fun a => ⨆ n, f n a
s : α →ₛ ℝ≥0
hsf : ∀ (x : α), ↑(s x) ≤ ⨆ n, f n x
r✝ : ℝ≥0
right✝ : ↑r✝ < 1
ha✝ : ↑r✝ < 1
ha : r✝ < 1
rs : α →ₛ ℝ≥0 :... | e37d047cd3bb7874 |
hasDerivAt_of_tendstoUniformlyOnFilter | Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean | theorem hasDerivAt_of_tendstoUniformlyOnFilter [NeBot l]
(hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x))
(hf : ∀ᶠ n : ι × 𝕜 in l ×ˢ 𝓝 x, HasDerivAt (f n.1) (f' n.1 n.2) n.2)
(hfg : ∀ᶠ y in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))) : HasDerivAt g (g' x) x | case intro.intro
ι : Type u_1
l : Filter ι
𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
G : Type u_3
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
f : ι → 𝕜 → G
g : 𝕜 → G
f' : ι → 𝕜 → G
g' : 𝕜 → G
x : 𝕜
inst✝¹ : IsRCLikeNormedField 𝕜
inst✝ : l.NeBot
hf' : ∀ ε > 0, ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, dist... | apply (hf' q hq).mono | case intro.intro
ι : Type u_1
l : Filter ι
𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
G : Type u_3
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
f : ι → 𝕜 → G
g : 𝕜 → G
f' : ι → 𝕜 → G
g' : 𝕜 → G
x : 𝕜
inst✝¹ : IsRCLikeNormedField 𝕜
inst✝ : l.NeBot
hf' : ∀ ε > 0, ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, dist... | 3364bb7f5ca1a8f4 |
bernoulli'_odd_eq_zero | Mathlib/NumberTheory/Bernoulli.lean | theorem bernoulli'_odd_eq_zero {n : ℕ} (h_odd : Odd n) (hlt : 1 < n) : bernoulli' n = 0 | n : ℕ
h_odd : Odd n
hlt : 1 < n
B : ℚ⟦X⟧ := PowerSeries.mk fun n => bernoulli' n / ↑n !
this : (B - evalNegHom B) * (exp ℚ - 1) = X * (exp ℚ - 1)
⊢ bernoulli' n = 0 | rcases mul_eq_mul_right_iff.mp this with h | h <;>
simp only [PowerSeries.ext_iff, evalNegHom, coeff_X] at h | case inl
n : ℕ
h_odd : Odd n
hlt : 1 < n
B : ℚ⟦X⟧ := PowerSeries.mk fun n => bernoulli' n / ↑n !
this : (B - evalNegHom B) * (exp ℚ - 1) = X * (exp ℚ - 1)
h : ∀ (n : ℕ), (coeff ℚ n) (B - (rescale (-1)) B) = if n = 1 then 1 else 0
⊢ bernoulli' n = 0
case inr
n : ℕ
h_odd : Odd n
hlt : 1 < n
B : ℚ⟦X⟧ := PowerSeries.mk fu... | e42772726b5421ab |
HomologicalComplex.Hom.isoOfComponents_app | Mathlib/Algebra/Homology/HomologicalComplex.lean | theorem isoOfComponents_app (f : ∀ i, C₁.X i ≅ C₂.X i)
(hf : ∀ i j, c.Rel i j → (f i).hom ≫ C₂.d i j = C₁.d i j ≫ (f j).hom) (i : ι) :
isoApp (isoOfComponents f hf) i = f i | case w
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
c : ComplexShape ι
C₁ C₂ : HomologicalComplex V c
f : (i : ι) → C₁.X i ≅ C₂.X i
hf : ∀ (i j : ι), c.Rel i j → (f i).hom ≫ C₂.d i j = C₁.d i j ≫ (f j).hom
i : ι
⊢ (isoApp (isoOfComponents f hf) i).hom = (f i).hom | simp | no goals | 599aadc8e09699fd |
MeasureTheory.integral_union_eq_left_of_ae_aux | Mathlib/MeasureTheory/Integral/SetIntegral.lean | theorem integral_union_eq_left_of_ae_aux (ht_eq : ∀ᵐ x ∂μ.restrict t, f x = 0)
(haux : StronglyMeasurable f) (H : IntegrableOn f (s ∪ t) μ) :
∫ x in s ∪ t, f x ∂μ = ∫ x in s, f x ∂μ | X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
s t : Set X
μ : Measure X
ht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0
haux : StronglyMeasurable f
H : IntegrableOn f (s ∪ t) μ
k : Set X := f ⁻¹' {0}
⊢ ∫ (x : X) in s ∪ t, f x ∂μ = ∫ (x : X) in s, f x ∂μ | have hk : MeasurableSet k := by borelize E; exact haux.measurable (measurableSet_singleton _) | X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
s t : Set X
μ : Measure X
ht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0
haux : StronglyMeasurable f
H : IntegrableOn f (s ∪ t) μ
k : Set X := f ⁻¹' {0}
hk : MeasurableSet k
⊢ ∫ (x : X) in s ∪ t, f x ∂μ = ∫ (x : ... | 6a4ba88f5f0d64e7 |
CategoryTheory.Presieve.FamilyOfElements.isCompatible_map_smul | Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean | lemma isCompatible_map_smul : ((r₀.smul m₀).map (whiskerRight φ (forget _))).Compatible | C : Type u₁
inst✝² : Category.{v₁, u₁} C
J : GrothendieckTopology C
R₀ R : Cᵒᵖ ⥤ RingCat
α : R₀ ⟶ R
inst✝¹ : Presheaf.IsLocallyInjective J α
M₀ : PresheafOfModules R₀
A : Cᵒᵖ ⥤ AddCommGrp
φ : M₀.presheaf ⟶ A
inst✝ : Presheaf.IsLocallyInjective J φ
hA : Presheaf.IsSeparated J A
X : C
r : ↑(R.obj (Opposite.op X))
m : ↑(A... | rw [← RingCat.comp_apply, NatTrans.naturality, RingCat.comp_apply, ha₁, Functor.map_comp,
RingCat.comp_apply] | no goals | c31ed53355a81817 |
Finset.Colex.shadow_initSeg | Mathlib/Combinatorics/SetFamily/KruskalKatona.lean | /-- This is important for iterating Kruskal-Katona: the shadow of an initial segment is also an
initial segment. -/
lemma shadow_initSeg [Fintype α] (hs : s.Nonempty) :
∂ (initSeg s) = initSeg (erase s <| min' s hs) | case h.mpr.inr.intro.intro.intro.inr.inl.h
α : Type u_1
inst✝¹ : LinearOrder α
s : Finset α
inst✝ : Fintype α
hs : s.Nonempty
t : Finset α
cards' : #(s.erase (s.min' hs)) = #t
k : α
hks : k ∈ s.erase (s.min' hs)
hkt : k ∉ t
z : ∀ ⦃a : α⦄, k < a → (a ∈ t ↔ a ∈ s.erase (s.min' hs))
j : α := tᶜ.min' ⋯
hjk : j ≤ k
this : j... | apply min'_le _ _ (mem_of_mem_erase ‹_›) | no goals | a9c55e64e04a6093 |
Array.all_iff_forall | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean | theorem all_iff_forall {p : α → Bool} {as : Array α} {start stop} :
as.all p start stop ↔ ∀ (i : Nat) (_ : i < as.size), start ≤ i ∧ i < stop → p as[i] | α : Type u_1
p : α → Bool
as : Array α
start stop : Nat
this : ¬as.any (fun x => !p x) start stop = true ↔ ∀ (i : Nat) (x : i < as.size), start ≤ i ∧ i < stop → p as[i] = true
⊢ (!as.any (fun x => !p x) start stop) = true ↔ ∀ (i : Nat) (x : i < as.size), start ≤ i ∧ i < stop → p as[i] = true | simp_all | no goals | 421598491cc71d66 |
exists_forall_closed_ball_dist_add_le_two_mul_sub | Mathlib/Analysis/Convex/Uniform.lean | theorem exists_forall_closed_ball_dist_add_le_two_mul_sub (hε : 0 < ε) (r : ℝ) :
∃ δ, 0 < δ ∧ ∀ ⦃x : E⦄, ‖x‖ ≤ r → ∀ ⦃y⦄, ‖y‖ ≤ r → ε ≤ ‖x - y‖ → ‖x + y‖ ≤ 2 * r - δ | case inr.intro.intro
E : Type u_1
inst✝² : SeminormedAddCommGroup E
inst✝¹ : UniformConvexSpace E
ε : ℝ
inst✝ : NormedSpace ℝ E
hε : 0 < ε
r : ℝ
hr : 0 < r
δ : ℝ
hδ : 0 < δ
h : ∀ ⦃x : E⦄, ‖x‖ ≤ 1 → ∀ ⦃y : E⦄, ‖y‖ ≤ 1 → ε / r ≤ ‖x - y‖ → ‖x + y‖ ≤ 2 - δ
x : E
hx : ‖r⁻¹ • x‖ ≤ 1
y : E
hy : ‖r⁻¹ • y‖ ≤ 1
hxy : ε ≤ ‖x - y‖... | simp_rw [← smul_add, ← smul_sub, norm_smul_of_nonneg (inv_nonneg.2 hr.le), ← div_eq_inv_mul,
div_le_div_iff_of_pos_right hr, div_le_iff₀ hr, sub_mul] at this | case inr.intro.intro
E : Type u_1
inst✝² : SeminormedAddCommGroup E
inst✝¹ : UniformConvexSpace E
ε : ℝ
inst✝ : NormedSpace ℝ E
hε : 0 < ε
r : ℝ
hr : 0 < r
δ : ℝ
hδ : 0 < δ
h : ∀ ⦃x : E⦄, ‖x‖ ≤ 1 → ∀ ⦃y : E⦄, ‖y‖ ≤ 1 → ε / r ≤ ‖x - y‖ → ‖x + y‖ ≤ 2 - δ
x : E
hx : ‖r⁻¹ • x‖ ≤ 1
y : E
hy : ‖r⁻¹ • y‖ ≤ 1
hxy : ε ≤ ‖x - y‖... | f878a0f9eb4a1c37 |
ProbabilityTheory.IndepFun.integral_mul | Mathlib/Probability/Integration.lean | theorem IndepFun.integral_mul (hXY : IndepFun X Y μ) (hX : AEStronglyMeasurable X μ)
(hY : AEStronglyMeasurable Y μ) : integral μ (X * Y) = integral μ X * integral μ Y | Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X Y : Ω → ℝ
hXY : IndepFun X Y μ
hX : AEStronglyMeasurable X μ
hY : AEStronglyMeasurable Y μ
h'X : X =ᶠ[ae μ] 0
⊢ X * Y =ᶠ[ae μ] 0 | filter_upwards [h'X] with ω hω | case h
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X Y : Ω → ℝ
hXY : IndepFun X Y μ
hX : AEStronglyMeasurable X μ
hY : AEStronglyMeasurable Y μ
h'X : X =ᶠ[ae μ] 0
ω : Ω
hω : X ω = 0 ω
⊢ (X * Y) ω = 0 ω | d5b07299b613cdd4 |
Submodule.quotientPi_aux.left_inv | Mathlib/LinearAlgebra/Quotient/Pi.lean | theorem left_inv : Function.LeftInverse (invFun p) (toFun p) := fun x =>
Submodule.Quotient.induction_on _ x fun x' => by
dsimp only [toFun, invFun]
rw [quotientPiLift_mk p, funext fun i => (mkQ_apply (p i) (x' i)), piQuotientLift_mk p,
lsum_single, id_apply]
| ι : Type u_1
R : Type u_2
inst✝⁴ : CommRing R
Ms : ι → Type u_3
inst✝³ : (i : ι) → AddCommGroup (Ms i)
inst✝² : (i : ι) → Module R (Ms i)
p : (i : ι) → Submodule R (Ms i)
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
x : ((i : ι) → Ms i) ⧸ pi Set.univ p
x' : (i : ι) → Ms i
⊢ invFun p (toFun p (Quotient.mk x')) = Quotient.mk... | dsimp only [toFun, invFun] | ι : Type u_1
R : Type u_2
inst✝⁴ : CommRing R
Ms : ι → Type u_3
inst✝³ : (i : ι) → AddCommGroup (Ms i)
inst✝² : (i : ι) → Module R (Ms i)
p : (i : ι) → Submodule R (Ms i)
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
x : ((i : ι) → Ms i) ⧸ pi Set.univ p
x' : (i : ι) → Ms i
⊢ (piQuotientLift p (pi Set.univ p) (single R Ms) ⋯... | a186def17e9e3acf |
Bimod.pentagon_bimod | Mathlib/CategoryTheory/Monoidal/Bimod.lean | theorem pentagon_bimod {V W X Y Z : Mon_ C} (M : Bimod V W) (N : Bimod W X) (P : Bimod X Y)
(Q : Bimod Y Z) :
whiskerRight (associatorBimod M N P).hom Q ≫
(associatorBimod M (N.tensorBimod P) Q).hom ≫
whiskerLeft M (associatorBimod N P Q).hom =
(associatorBimod (M.tensorBimod N) P Q).hom ≫
... | case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : ... | slice_lhs 3 4 =>
rw [← MonoidalCategory.whiskerLeft_comp, π_tensor_id_preserves_coequalizer_inv_desc,
MonoidalCategory.whiskerLeft_comp, MonoidalCategory.whiskerLeft_comp] | case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : ... | dad18f8e30f0e6d9 |
geom_sum_inv | Mathlib/Algebra/GeomSum.lean | theorem geom_sum_inv [DivisionRing α] {x : α} (hx1 : x ≠ 1) (hx0 : x ≠ 0) (n : ℕ) :
∑ i ∈ range n, x⁻¹ ^ i = (x - 1)⁻¹ * (x - x⁻¹ ^ n * x) | α : Type u
inst✝ : DivisionRing α
x : α
hx1 : x ≠ 1
hx0 : x ≠ 0
n : ℕ
⊢ x⁻¹ ≠ 1 | rwa [inv_eq_one_div, Ne, div_eq_iff_mul_eq hx0, one_mul] | no goals | 90f187432f831566 |
LSeries_tendsto_sub_mul_nhds_one_of_tendsto_sum_div_aux₂ | Mathlib/NumberTheory/LSeries/SumCoeff.lean | theorem LSeries_tendsto_sub_mul_nhds_one_of_tendsto_sum_div_aux₂ {s T ε : ℝ} {S : ℝ → ℂ}
(hS : LocallyIntegrableOn (fun t ↦ S t - l * t) (Set.Ici 1)) (hε : 0 < ε)
(hs : 1 < s) (hT₁ : 1 ≤ T) (hT : ∀ t ≥ T, ‖S t - l * t‖ ≤ ε * t) :
(s - 1) * ∫ (t : ℝ) in Set.Ioi T, ‖S t - l * t‖ * t ^ (-s - 1) ≤ ε | l : ℂ
s T ε : ℝ
S : ℝ → ℂ
hS : LocallyIntegrableOn (fun t => S t - l * ↑t) (Set.Ici 1) volume
hε : 0 < ε
hs : 1 < s
hT₁ : 1 ≤ T
hT : ∀ t ≥ T, ‖S t - l * ↑t‖ ≤ ε * t
hT₀ : 0 < T
h : ∀ {t : ℝ}, 0 < t → t ^ (-s) = t * t ^ (-s - 1)
⊢ ε * ((s - 1) * (-1 / (-s + 1))) = ε | field_simp [show -s + 1 ≠ 0 by linarith, hε.ne'] | l : ℂ
s T ε : ℝ
S : ℝ → ℂ
hS : LocallyIntegrableOn (fun t => S t - l * ↑t) (Set.Ici 1) volume
hε : 0 < ε
hs : 1 < s
hT₁ : 1 ≤ T
hT : ∀ t ≥ T, ‖S t - l * ↑t‖ ≤ ε * t
hT₀ : 0 < T
h : ∀ {t : ℝ}, 0 < t → t ^ (-s) = t * t ^ (-s - 1)
⊢ 1 - s = -s + 1 | f4c86713eaa4d22c |
HahnSeries.SummableFamily.hsum_ofFinsupp | Mathlib/RingTheory/HahnSeries/Summable.lean | theorem hsum_ofFinsupp {f : α →₀ HahnSeries Γ R} : (ofFinsupp f).hsum = f.sum fun _ => id | case coeff.h
Γ : Type u_1
R : Type u_3
inst✝¹ : PartialOrder Γ
inst✝ : AddCommMonoid R
α : Type u_7
f : α →₀ HahnSeries Γ R
g : Γ
⊢ ∑ᶠ (i : α), (f i).coeff g = (∑ x ∈ f.support, id (f x)).coeff g | simp_rw [← coeff.addMonoidHom_apply, id] | case coeff.h
Γ : Type u_1
R : Type u_3
inst✝¹ : PartialOrder Γ
inst✝ : AddCommMonoid R
α : Type u_7
f : α →₀ HahnSeries Γ R
g : Γ
⊢ ∑ᶠ (i : α), (coeff.addMonoidHom g) (f i) = (coeff.addMonoidHom g) (∑ x ∈ f.support, f x) | a44e6b7c05b9c84b |
eVariationOn.add_point | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | theorem add_point (f : α → E) {s : Set α} {x : α} (hx : x ∈ s) (u : ℕ → α) (hu : Monotone u)
(us : ∀ i, u i ∈ s) (n : ℕ) :
∃ (v : ℕ → α) (m : ℕ), Monotone v ∧ (∀ i, v i ∈ s) ∧ x ∈ v '' Iio m ∧
(∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i))) ≤
∑ j ∈ Finset.range m, edist (f (v (j + 1))) (f ... | α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
x : α
hx : x ∈ s
u : ℕ → α
hu : Monotone u
us : ∀ (i : ℕ), u i ∈ s
n : ℕ
h : u n ≤ x
v : ℕ → α := fun i => if i ≤ n then u i else x
i : ℕ
⊢ v i ∈ s | simp only [v] | α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
x : α
hx : x ∈ s
u : ℕ → α
hu : Monotone u
us : ∀ (i : ℕ), u i ∈ s
n : ℕ
h : u n ≤ x
v : ℕ → α := fun i => if i ≤ n then u i else x
i : ℕ
⊢ (if i ≤ n then u i else x) ∈ s | 760e94573f7a043d |
continuous_parametric_integral_of_continuous | Mathlib/MeasureTheory/Integral/SetIntegral.lean | theorem continuous_parametric_integral_of_continuous
[FirstCountableTopology X] [LocallyCompactSpace X]
[SecondCountableTopologyEither Y E] [IsLocallyFiniteMeasure μ]
{f : X → Y → E} (hf : Continuous f.uncurry) {s : Set Y} (hs : IsCompact s) :
Continuous (∫ y in s, f · y ∂μ) | Y : Type u_2
E : Type u_3
X : Type u_5
inst✝⁹ : TopologicalSpace X
inst✝⁸ : TopologicalSpace Y
inst✝⁷ : MeasurableSpace Y
inst✝⁶ : OpensMeasurableSpace Y
μ : Measure Y
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FirstCountableTopology X
inst✝² : LocallyCompactSpace X
inst✝¹ : SecondCountableTopology... | fun_prop | no goals | 77c93b2e7b9a838f |
not_IntegrableOn_Ici_inv | Mathlib/Analysis/SpecialFunctions/NonIntegrable.lean | theorem not_IntegrableOn_Ici_inv {a : ℝ} :
¬ IntegrableOn (fun x => x⁻¹) (Ici a) | a : ℝ
A : ∀ᶠ (x : ℝ) in atTop, HasDerivAt (fun x => Real.log x) x⁻¹ x
B : Tendsto (fun x => ‖Real.log x‖) atTop atTop
⊢ ¬IntegrableOn (fun x => x⁻¹) (Ici a) volume | exact not_integrableOn_of_tendsto_norm_atTop_of_deriv_isBigO_filter atTop (Ici_mem_atTop a)
(A.mono (fun x hx ↦ hx.differentiableAt)) B
(Filter.EventuallyEq.isBigO (A.mono (fun x hx ↦ hx.deriv))) | no goals | 235cf6d4794b742f |
NumberField.InfinitePlace.card_isUnramified_compl | Mathlib/NumberTheory/NumberField/Embeddings.lean | lemma card_isUnramified_compl [NumberField k] [IsGalois k K] :
#({w : InfinitePlace K | w.IsUnramified k} : Finset _)ᶜ =
#({w : InfinitePlace k | w.IsUnramifiedIn K} : Finset _)ᶜ * (finrank k K / 2) | k : Type u_1
inst✝⁵ : Field k
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra k K
inst✝² : NumberField K
inst✝¹ : NumberField k
inst✝ : IsGalois k K
this : Module.Finite k K := Finite.of_restrictScalars_finite ℚ k K
w : InfinitePlace K
hw : ¬IsUnramifiedIn K (w.comap (algebraMap k K))
⊢ #(MulAction.orbit (K ≃ₐ[k] K) w).... | rw [← MulAction.card_orbit_mul_card_stabilizer_eq_card_group _ w,
← Nat.card_eq_fintype_card (α := Stab w), InfinitePlace.card_stabilizer, if_neg,
Nat.mul_div_cancel _ zero_lt_two, Set.toFinset_card] | case hnc
k : Type u_1
inst✝⁵ : Field k
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra k K
inst✝² : NumberField K
inst✝¹ : NumberField k
inst✝ : IsGalois k K
this : Module.Finite k K := Finite.of_restrictScalars_finite ℚ k K
w : InfinitePlace K
hw : ¬IsUnramifiedIn K (w.comap (algebraMap k K))
⊢ ¬IsUnramified k w | d92790b65480f1b5 |
Fin.addRothNumber_le_rothNumberNat | Mathlib/Combinatorics/Additive/AP/Three/Defs.lean | lemma Fin.addRothNumber_le_rothNumberNat (k n : ℕ) (hkn : k ≤ n) :
addRothNumber (Iio k : Finset (Fin n.succ)) ≤ rothNumberNat k | case refine_1
k n : ℕ
hkn : k ≤ n
⊢ Set.MapsTo Nat.cast ↑(range k) ↑(Iio ↑k) | simpa using fun x ↦ natCast_strictMono hkn | no goals | 7e145f3c7da155cb |
Scott.isOpen_sUnion | Mathlib/Topology/OmegaCompletePartialOrder.lean | theorem isOpen_sUnion (s : Set (Set α)) (hs : ∀ t ∈ s, IsOpen α t) : IsOpen α (⋃₀ s) | α : Type u
inst✝ : OmegaCompletePartialOrder α
s : Set (Set α)
hs : ∀ t ∈ s, IsOpen α t
⊢ IsOpen α (⋃₀ s) | simp only [IsOpen] at hs ⊢ | α : Type u
inst✝ : OmegaCompletePartialOrder α
s : Set (Set α)
hs : ∀ t ∈ s, ωScottContinuous fun x => x ∈ t
⊢ ωScottContinuous fun x => x ∈ ⋃₀ s | 770b4bce19d0f20c |
IsLocalizedModule.exist_integer_multiples | Mathlib/Algebra/Module/LocalizedModule/Int.lean | theorem exist_integer_multiples {ι : Type*} (s : Finset ι) (g : ι → M') :
∃ b : S, ∀ i ∈ s, IsInteger f (b.val • g i) | case refine_2.e_a
R : Type u_1
inst✝⁵ : CommSemiring R
S : Submonoid R
M : Type u_2
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
M' : Type u_3
inst✝² : AddCommMonoid M'
inst✝¹ : Module R M'
f : M →ₗ[R] M'
inst✝ : IsLocalizedModule S f
ι : Type u_4
s : Finset ι
g : ι → M'
sec : ι → M × ↥S
hsec : ∀ (i : ι), (sec i).2 • g... | simp only [Submonoid.coe_mul, Submonoid.coe_finset_prod, mul_comm] | case refine_2.e_a
R : Type u_1
inst✝⁵ : CommSemiring R
S : Submonoid R
M : Type u_2
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
M' : Type u_3
inst✝² : AddCommMonoid M'
inst✝¹ : Module R M'
f : M →ₗ[R] M'
inst✝ : IsLocalizedModule S f
ι : Type u_4
s : Finset ι
g : ι → M'
sec : ι → M × ↥S
hsec : ∀ (i : ι), (sec i).2 • g... | 0d28f5679ff37f99 |
AkraBazziRecurrence.growsPolynomially_log | Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean | lemma growsPolynomially_log : GrowsPolynomially Real.log | ⊢ GrowsPolynomially log | intro b hb | b : ℝ
hb : b ∈ Set.Ioo 0 1
⊢ ∃ c₁ > 0, ∃ c₂ > 0, ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, log u ∈ Set.Icc (c₁ * log x) (c₂ * log x) | 441cf9a25600ec9c |
QuasispectrumRestricts.isClosedEmbedding_nonUnitalStarAlgHom | Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Restrict.lean | lemma isClosedEmbedding_nonUnitalStarAlgHom {a : A} {φ : C(σₙ S a, S)₀ →⋆ₙₐ[S] A}
(hφ : IsClosedEmbedding φ) {f : C(S, R)} (h : QuasispectrumRestricts a f)
(halg : IsUniformEmbedding (algebraMap R S)) :
IsClosedEmbedding (h.nonUnitalStarAlgHom φ) | R : Type u_1
S : Type u_2
A : Type u_3
inst✝²¹ : Semifield R
inst✝²⁰ : StarRing R
inst✝¹⁹ : MetricSpace R
inst✝¹⁸ : IsTopologicalSemiring R
inst✝¹⁷ : ContinuousStar R
inst✝¹⁶ : Field S
inst✝¹⁵ : StarRing S
inst✝¹⁴ : MetricSpace S
inst✝¹³ : IsTopologicalRing S
inst✝¹² : ContinuousStar S
inst✝¹¹ : NonUnitalRing A
inst✝¹⁰... | refine hφ.comp <| IsUniformEmbedding.isClosedEmbedding <| .comp
(ContinuousMapZero.isUniformEmbedding_comp _ halg)
(UniformEquiv.arrowCongrLeft₀ h.homeomorph.symm this |>.isUniformEmbedding) | no goals | 67289aebe6f41414 |
ContDiffOn.ftaylorSeriesWithin | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | theorem ContDiffOn.ftaylorSeriesWithin
(h : ContDiffOn 𝕜 n f s) (hs : UniqueDiffOn 𝕜 s) :
HasFTaylorSeriesUpToOn n f (ftaylorSeriesWithin 𝕜 f s) s | 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
n : WithTop ℕ∞
h : ContDiffOn 𝕜 n f s
hs : UniqueDiffOn 𝕜 s
m : ℕ
hm : ↑m < n
x : E
hx : x ∈ s
this : ↑(m + 1) ≤ n
... | exact (Hp.mono ho).eq_iteratedFDerivWithin_of_uniqueDiffOn le_rfl (hs.inter o_open) ⟨hx, xo⟩ | no goals | b494b7e6e18309e9 |
AffineSubspace.wOppSide_iff_exists_wbtw | Mathlib/Analysis/Convex/Side.lean | theorem wOppSide_iff_exists_wbtw {s : AffineSubspace R P} {x y : P} :
s.WOppSide x y ↔ ∃ p ∈ s, Wbtw R x p y | case intro.intro.intro.intro.inr.inl
R : Type u_1
V : Type u_2
P : Type u_4
inst✝³ : LinearOrderedField R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x y p₁ : P
hp₁ : p₁ ∈ s
p₂ : P
hp₂ : p₂ ∈ s
h : p₂ -ᵥ y = 0
⊢ ∃ p ∈ s, Wbtw R x p y | rw [vsub_eq_zero_iff_eq] at h | case intro.intro.intro.intro.inr.inl
R : Type u_1
V : Type u_2
P : Type u_4
inst✝³ : LinearOrderedField R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x y p₁ : P
hp₁ : p₁ ∈ s
p₂ : P
hp₂ : p₂ ∈ s
h : p₂ = y
⊢ ∃ p ∈ s, Wbtw R x p y | e5c82043c73c8683 |
MeasureTheory.upcrossingsBefore_mono | Mathlib/Probability/Martingale/Upcrossing.lean | theorem upcrossingsBefore_mono (hab : a < b) : Monotone fun N ω => upcrossingsBefore a b f N ω | case pos
Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
hab : a < b
N M : ℕ
hNM : N ≤ M
ω : Ω
hemp : {n | upperCrossingTime a b f N n ω < N}.Nonempty
n : ℕ
hn : n ∈ {n | upperCrossingTime a b f N n ω < N}
⊢ n ∈ {n | upperCrossingTime a b f M n ω < M} | rw [Set.mem_setOf_eq, upperCrossingTime_eq_upperCrossingTime_of_lt hNM hn] | case pos
Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
hab : a < b
N M : ℕ
hNM : N ≤ M
ω : Ω
hemp : {n | upperCrossingTime a b f N n ω < N}.Nonempty
n : ℕ
hn : n ∈ {n | upperCrossingTime a b f N n ω < N}
⊢ upperCrossingTime a b f N n ω < M | 773bf71e1e9928ea |
MeasureTheory.IsStoppingTime.measurableSet_le_stopping_time | Mathlib/Probability/Process/Stopping.lean | theorem measurableSet_le_stopping_time [TopologicalSpace ι] [SecondCountableTopology ι]
[OrderTopology ι] [MeasurableSpace ι] [BorelSpace ι] (hτ : IsStoppingTime f τ)
(hπ : IsStoppingTime f π) : MeasurableSet[hτ.measurableSpace] {ω | τ ω ≤ π ω} | Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁵ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : SecondCountableTopology ι
inst✝² : OrderTopology ι
inst✝¹ : MeasurableSpace ι
inst✝ : BorelSpace ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
j : ι
this : {ω | τ ω ≤ π ω} ∩ {ω ... | refine MeasurableSet.inter ?_ (hτ.measurableSet_le j) | Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁵ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : SecondCountableTopology ι
inst✝² : OrderTopology ι
inst✝¹ : MeasurableSpace ι
inst✝ : BorelSpace ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
j : ι
this : {ω | τ ω ≤ π ω} ∩ {ω ... | d8297f544f076b41 |
ZNum.ofInt'_eq | Mathlib/Data/Num/Lemmas.lean | theorem ofInt'_eq : ∀ n : ℤ, ZNum.ofInt' n = n
| (n : ℕ) => rfl
| -[n+1] => by
show Num.toZNumNeg (n + 1 : ℕ) = -(n + 1 : ℕ)
rw [← neg_inj, neg_neg, Nat.cast_succ, Num.add_one, Num.zneg_toZNumNeg, Num.toZNum_succ,
Nat.cast_succ, ZNum.add_one]
rfl
| n : ℕ
⊢ ofInt' -[n+1] = ↑-[n+1] | show Num.toZNumNeg (n + 1 : ℕ) = -(n + 1 : ℕ) | n : ℕ
⊢ (↑(n + 1)).toZNumNeg = -↑(n + 1) | 463dd2f005417e04 |
LinearMap.charpoly_baseChange | Mathlib/LinearAlgebra/Charpoly/BaseChange.lean | @[simp]
lemma LinearMap.charpoly_baseChange {R M} [CommRing R] [AddCommGroup M] [Module R M]
[Module.Free R M] [Module.Finite R M] (f : M →ₗ[R] M)
(A) [CommRing A] [Algebra R A] :
(f.baseChange A).charpoly = f.charpoly.map (algebraMap R A) | R : Type u_1
M : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : Module.Free R M
inst✝² : Module.Finite R M
f : M →ₗ[R] M
A : Type u_3
inst✝¹ : CommRing A
inst✝ : Algebra R A
a✝ : Nontrivial A
⊢ (baseChange A f).charpoly = Polynomial.map (algebraMap R A) f.charpoly | have := (algebraMap R A).domain_nontrivial | R : Type u_1
M : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : Module.Free R M
inst✝² : Module.Finite R M
f : M →ₗ[R] M
A : Type u_3
inst✝¹ : CommRing A
inst✝ : Algebra R A
a✝ : Nontrivial A
this : Nontrivial R
⊢ (baseChange A f).charpoly = Polynomial.map (algebraMap R A) f.charpoly | c97f3623cdea36a4 |
LieAlgebra.hasTrivialRadical_and_of_isIrreducible_of_isFaithful | Mathlib/Algebra/Lie/Semisimple/Lemmas.lean | theorem hasTrivialRadical_and_of_isIrreducible_of_isFaithful
(h : ∀ x, LinearMap.trace k _ (toEnd k L M x) = 0) : HasTrivialRadical k L | case intro
k : Type u_1
L : Type u_2
M : Type u_3
inst✝¹² : Field k
inst✝¹¹ : CharZero k
inst✝¹⁰ : LieRing L
inst✝⁹ : LieAlgebra k L
inst✝⁸ : Module.Finite k L
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module k M
inst✝⁵ : LieRingModule L M
inst✝⁴ : LieModule k L M
inst✝³ : Module.Finite k M
inst✝² : LieModule.IsIrreducible k L ... | intro x hx | case intro
k : Type u_1
L : Type u_2
M : Type u_3
inst✝¹² : Field k
inst✝¹¹ : CharZero k
inst✝¹⁰ : LieRing L
inst✝⁹ : LieAlgebra k L
inst✝⁸ : Module.Finite k L
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module k M
inst✝⁵ : LieRingModule L M
inst✝⁴ : LieModule k L M
inst✝³ : Module.Finite k M
inst✝² : LieModule.IsIrreducible k L ... | a1ed9c83290ed046 |
LSeries_eventually_eq_zero_iff' | Mathlib/NumberTheory/LSeries/Injectivity.lean | /-- The `LSeries` of `f` is zero for large real arguments if and only if either `f n = 0`
for all `n ≠ 0` or the L-series converges nowhere. -/
lemma LSeries_eventually_eq_zero_iff' {f : ℕ → ℂ} :
(fun x : ℝ ↦ LSeries f x) =ᶠ[atTop] 0 ↔ (∀ n ≠ 0, f n = 0) ∨ abscissaOfAbsConv f = ⊤ | f : ℕ → ℂ
h : ¬abscissaOfAbsConv f = ⊤
H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0
F : ℕ → ℂ := fun n => if n = 0 then 0 else f n
hF₀ : F 0 = 0
hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n
ha : ¬abscissaOfAbsConv F = ⊤
h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x
H' : ∀ (n : ℕ), (fun x => ↑n ^ ↑x * LSeries F ↑x) =ᶠ[atTop] fun x => 0
n... | replace this := this.congr' <| H' n | f : ℕ → ℂ
h : ¬abscissaOfAbsConv f = ⊤
H : (fun x => LSeries f ↑x) =ᶠ[atTop] 0
F : ℕ → ℂ := fun n => if n = 0 then 0 else f n
hF₀ : F 0 = 0
hF : ∀ {n : ℕ}, n ≠ 0 → F n = f n
ha : ¬abscissaOfAbsConv F = ⊤
h' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x
H' : ∀ (n : ℕ), (fun x => ↑n ^ ↑x * LSeries F ↑x) =ᶠ[atTop] fun x => 0
n... | 7103bb783fd652c6 |
extChartAt_inl_apply | Mathlib/Geometry/Manifold/ContMDiff/Constructions.lean | lemma extChartAt_inl_apply {x y : M} :
(extChartAt I (.inl x : M ⊕ M')) (Sum.inl y) = (extChartAt I x) y | 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
M' : Type u_16
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace ... | simp | no goals | 2dbef5c77e702595 |
MvPowerSeries.WithPiTopology.tendsto_pow_of_constantCoeff_nilpotent_iff | Mathlib/RingTheory/MvPowerSeries/PiTopology.lean | theorem tendsto_pow_of_constantCoeff_nilpotent_iff [CommRing R] [DiscreteTopology R] (f) :
Tendsto (fun n : ℕ => f ^ n) atTop (nhds 0) ↔
IsNilpotent (constantCoeff σ R f) | σ : Type u_1
R : Type u_2
inst✝² : TopologicalSpace R
inst✝¹ : CommRing R
inst✝ : DiscreteTopology R
f : MvPowerSeries σ R
h : Tendsto (fun n => f ^ n) atTop (nhds 0)
this : ∃ a, ∀ (b : ℕ), a ≤ b → (constantCoeff σ R) f ^ b = 0
⊢ IsNilpotent ((constantCoeff σ R) f) | obtain ⟨m, hm⟩ := this | case intro
σ : Type u_1
R : Type u_2
inst✝² : TopologicalSpace R
inst✝¹ : CommRing R
inst✝ : DiscreteTopology R
f : MvPowerSeries σ R
h : Tendsto (fun n => f ^ n) atTop (nhds 0)
m : ℕ
hm : ∀ (b : ℕ), m ≤ b → (constantCoeff σ R) f ^ b = 0
⊢ IsNilpotent ((constantCoeff σ R) f) | 75b216f0c0004b82 |
CategoryTheory.Arrow.mk_injective | Mathlib/CategoryTheory/Comma/Arrow.lean | theorem mk_injective (A B : T) :
Function.Injective (Arrow.mk : (A ⟶ B) → Arrow T) := fun f g h => by
cases h
rfl
| T : Type u
inst✝ : Category.{v, u} T
A B : T
f g : A ⟶ B
h : mk f = mk g
⊢ f = g | cases h | case refl
T : Type u
inst✝ : Category.{v, u} T
A B : T
f : A ⟶ B
⊢ f = f | 7a8aede1e26406a1 |
levenshtein_nil_nil | Mathlib/Data/List/EditDistance/Defs.lean | theorem levenshtein_nil_nil : levenshtein C [] [] = 0 | α : Type u_1
β : Type u_2
δ : Type u_3
inst✝¹ : AddZeroClass δ
inst✝ : Min δ
C : Cost α β δ
⊢ levenshtein C [] [] = 0 | simp [levenshtein, suffixLevenshtein] | no goals | 40864859ce7dd54c |
Bornology.IsVonNBounded.add | Mathlib/Analysis/LocallyConvex/Bounded.lean | theorem IsVonNBounded.add (hs : IsVonNBounded 𝕜 s) (ht : IsVonNBounded 𝕜 t) :
IsVonNBounded 𝕜 (s + t) := fun U hU ↦ by
rcases exists_open_nhds_zero_add_subset hU with ⟨V, hVo, hV, hVU⟩
exact ((hs <| hVo.mem_nhds hV).add (ht <| hVo.mem_nhds hV)).mono_left hVU
| case intro.intro.intro
𝕜 : Type u_1
E : Type u_3
inst✝⁴ : SeminormedRing 𝕜
inst✝³ : AddZeroClass E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
inst✝ : DistribSMul 𝕜 E
s t : Set E
hs : IsVonNBounded 𝕜 s
ht : IsVonNBounded 𝕜 t
U : Set E
hU : U ∈ 𝓝 0
V : Set E
hVo : IsOpen V
hV : 0 ∈ V
hVU : V + V ⊆ U
⊢ Abs... | exact ((hs <| hVo.mem_nhds hV).add (ht <| hVo.mem_nhds hV)).mono_left hVU | no goals | 888e85e128c78ea7 |
zorn_le₀ | Mathlib/Order/Zorn.lean | theorem zorn_le₀ (s : Set α) (ih : ∀ c ⊆ s, IsChain (· ≤ ·) c → ∃ ub ∈ s, ∀ z ∈ c, z ≤ ub) :
∃ m, Maximal (· ∈ s) m :=
let ⟨⟨m, hms⟩, h⟩ :=
@zorn_le s _ fun c hc =>
let ⟨ub, hubs, hub⟩ :=
ih (Subtype.val '' c) (fun _ ⟨⟨_, hx⟩, _, h⟩ => h ▸ hx)
(by
rintro _ ⟨p, hpc, rfl⟩ _ ⟨... | case intro.intro.intro.intro
α : Type u_1
inst✝ : Preorder α
s : Set α
ih : ∀ c ⊆ s, IsChain (fun x1 x2 => x1 ≤ x2) c → ∃ ub ∈ s, ∀ z ∈ c, z ≤ ub
c : Set ↑s
hc : IsChain (fun x1 x2 => x1 ≤ x2) c
p : { x // x ∈ s }
hpc : p ∈ c
q : { x // x ∈ s }
hqc : q ∈ c
hpq : ↑p ≠ ↑q
⊢ (fun x1 x2 => x1 ≤ x2) ↑p ↑q ∨ (fun x1 x2 => x1... | exact hc hpc hqc fun t => hpq (Subtype.ext_iff.1 t) | no goals | fc5ad337988981ad |
IsPrimitiveRoot.norm_toInteger_pow_sub_one_of_prime_pow_ne_two | Mathlib/NumberTheory/Cyclotomic/Rat.lean | /-- The norm, relative to `ℤ`, of `ζ ^ p ^ s - 1` in a `p ^ (k + 1)`-th cyclotomic extension of `ℚ`
is p ^ p ^ s` if `s ≤ k` and `p ^ (k - s + 1) ≠ 2`. -/
lemma norm_toInteger_pow_sub_one_of_prime_pow_ne_two [IsCyclotomicExtension {p ^ (k + 1)} ℚ K]
(hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))) {s : ℕ} (hs : s ≤ k) (htwo... | p : ℕ+
k : ℕ
K : Type u
inst✝² : Field K
ζ : K
hp : Fact (Nat.Prime ↑p)
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))
s : ℕ
hs : s ≤ k
htwo : p ^ (k - s + 1) ≠ 2
⊢ (Algebra.norm ℤ) (hζ.toInteger ^ ↑p ^ s - 1) = ↑↑p ^ ↑p ^ s | have : NumberField K := IsCyclotomicExtension.numberField {p ^ (k + 1)} ℚ K | p : ℕ+
k : ℕ
K : Type u
inst✝² : Field K
ζ : K
hp : Fact (Nat.Prime ↑p)
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))
s : ℕ
hs : s ≤ k
htwo : p ^ (k - s + 1) ≠ 2
this : NumberField K
⊢ (Algebra.norm ℤ) (hζ.toInteger ^ ↑p ^ s - 1) = ↑↑p ^ ↑p ^ s | 6a689a8ec305f175 |
AEMeasurable.mul_iff_right | Mathlib/MeasureTheory/Group/Arithmetic.lean | theorem AEMeasurable.mul_iff_right {G : Type*} [MeasurableSpace G] [MeasurableSpace α] [CommGroup G]
[MeasurableMul₂ G] [MeasurableInv G] {μ : Measure α} {f g : α → G} (hf : AEMeasurable f μ) :
AEMeasurable (f * g) μ ↔ AEMeasurable g μ :=
⟨fun h ↦ show g = f * g * f⁻¹ by simp only [mul_inv_cancel_comm] ▸ h.mu... | α : Type u_1
G : Type u_2
inst✝⁴ : MeasurableSpace G
inst✝³ : MeasurableSpace α
inst✝² : CommGroup G
inst✝¹ : MeasurableMul₂ G
inst✝ : MeasurableInv G
μ : Measure α
f g : α → G
hf : AEMeasurable f μ
h : AEMeasurable (f * g) μ
⊢ g = f * g * f⁻¹ | simp only [mul_inv_cancel_comm] | no goals | 59d0d98dbc9565df |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.