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 |
|---|---|---|---|---|---|---|
CategoryTheory.Localization.LeftBousfield.W_isoClosure | Mathlib/CategoryTheory/Localization/Bousfield.lean | lemma W_isoClosure : W P.isoClosure = W P | case h.mpr.intro.intro.intro
C : Type u_1
inst✝ : Category.{u_3, u_1} C
P : ObjectProperty C
X Y : C
f : X ⟶ Y
hf : W P f
Z Z' : C
hZ' : P Z'
e : Z ≅ Z'
⊢ Function.Bijective fun g => f ≫ g | constructor | case h.mpr.intro.intro.intro.left
C : Type u_1
inst✝ : Category.{u_3, u_1} C
P : ObjectProperty C
X Y : C
f : X ⟶ Y
hf : W P f
Z Z' : C
hZ' : P Z'
e : Z ≅ Z'
⊢ Function.Injective fun g => f ≫ g
case h.mpr.intro.intro.intro.right
C : Type u_1
inst✝ : Category.{u_3, u_1} C
P : ObjectProperty C
X Y : C
f : X ⟶ Y
hf : W P... | 905a2b1daf44ee1d |
CFC.exists_sqrt_of_isSelfAdjoint_of_quasispectrumRestricts | Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Instances.lean | lemma CFC.exists_sqrt_of_isSelfAdjoint_of_quasispectrumRestricts {A : Type*} [NonUnitalRing A]
[StarRing A] [TopologicalSpace A] [Module ℝ A] [IsScalarTower ℝ A A] [SMulCommClass ℝ A A ]
[NonUnitalContinuousFunctionalCalculus ℝ (IsSelfAdjoint : A → Prop)]
{a : A} (ha₁ : IsSelfAdjoint a) (ha₂ : Quasispectrum... | case right.right
A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module ℝ A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ IsSelfAdjoint
a : A
ha₁ : IsSelfAdjoint a
ha₂ : QuasispectrumRestricts a ⇑ContinuousMap.rea... | apply cfcₙ_congr fun x hx ↦ ?_ | A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module ℝ A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ IsSelfAdjoint
a : A
ha₁ : IsSelfAdjoint a
ha₂ : QuasispectrumRestricts a ⇑ContinuousMap.realToNNReal
x : ℝ
h... | 01ce0efe5b087e60 |
AdicCompletion.firstRow_exact | Mathlib/RingTheory/AdicCompletion/AsTensorProduct.lean | private lemma firstRow_exact : (firstRow I M f).Exact where
zero
| 0, _ => ModuleCat.hom_ext (tens_exact I M f hf).linearMap_comp_eq_zero
| 1, _ => ModuleCat.hom_ext (LinearMap.zero_comp _)
| 2, _ => ModuleCat.hom_ext (LinearMap.zero_comp 0)
exact k _ | R : Type u
inst✝² : CommRing R
I : Ideal R
M : Type u
inst✝¹ : AddCommGroup M
inst✝ : Module R M
ι : Type
f : (ι → R) →ₗ[R] M
hf : Function.Surjective ⇑f
k : ℕ
x✝¹ : autoParam (2 + 2 ≤ 4) _auto✝
x₂✝ : ↑((AdicCompletion.firstRow I M f).sc ⋯ 2 ⋯).X₂
x✝ : (ConcreteCategory.hom ((AdicCompletion.firstRow I M f).sc ⋯ 2 ⋯).g)... | exact ⟨0, rfl⟩ | no goals | 7e249886ede9ebfc |
List.infix_cons_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Sublist.lean | theorem infix_cons_iff : l₁ <:+: a :: l₂ ↔ l₁ <+: a :: l₂ ∨ l₁ <:+: l₂ | case mpr.inr
α✝ : Type u_1
l₁ : List α✝
a : α✝
l₂ : List α✝
hl₁ : l₁ <:+: l₂
⊢ l₁ <:+: a :: l₂ | exact infix_cons hl₁ | no goals | 098d2c28c44c7406 |
IsCyclic.normalizer_le_centralizer | Mathlib/GroupTheory/Transfer.lean | theorem normalizer_le_centralizer (hP : IsCyclic P) : P.normalizer ≤ centralizer (P : Set G) | case neg.intro.inr.refine_2
G : Type u_3
inst✝¹ : Group G
inst✝ : Finite G
P : Sylow (Nat.card G).minFac G
hP : IsCyclic ↥↑P
hn : ¬Nat.card G = 1
this : Fact (Nat.Prime (Nat.card G).minFac)
key : (centralizer ↑↑P).relindex (↑P).normalizer ∣ Nat.card (MulAut ↥↑P)
k : ℕ
hk : Nat.card ↥↑P = (Nat.card G).minFac ^ k
h0 : 0 ... | apply Nat.Coprime.coprime_dvd_left (card_subgroup_dvd_card P.normalizer) | case neg.intro.inr.refine_2
G : Type u_3
inst✝¹ : Group G
inst✝ : Finite G
P : Sylow (Nat.card G).minFac G
hP : IsCyclic ↥↑P
hn : ¬Nat.card G = 1
this : Fact (Nat.Prime (Nat.card G).minFac)
key : (centralizer ↑↑P).relindex (↑P).normalizer ∣ Nat.card (MulAut ↥↑P)
k : ℕ
hk : Nat.card ↥↑P = (Nat.card G).minFac ^ k
h0 : 0 ... | 319753e59f979dd9 |
AlgebraicGeometry.IsOpenImmersion.map_ΓIso_inv | Mathlib/AlgebraicGeometry/OpenImmersion.lean | @[reassoc, elementwise]
lemma map_ΓIso_inv {X Y : Scheme.{u}} (f : X ⟶ Y) [IsOpenImmersion f] (U : Y.Opens) :
Y.presheaf.map (homOfLE inf_le_right).op ≫ (ΓIso f U).inv = f.app U | X Y : Scheme
f : X ⟶ Y
inst✝ : IsOpenImmersion f
U : Y.Opens
⊢ Y.presheaf.map (homOfLE ⋯).op ≫ (ΓIso f U).inv = Scheme.Hom.app f U | simp [Scheme.Hom.appLE_eq_app] | no goals | 568000bd1dbe8b28 |
AlgebraicGeometry.Proj.awayMap_awayToSection | Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Basic.lean | @[reassoc]
lemma awayMap_awayToSection :
CommRingCat.ofHom (awayMap 𝒜 g_deg hx) ≫ awayToSection 𝒜 x =
awayToSection 𝒜 f ≫ (Proj 𝒜).presheaf.map (homOfLE (basicOpen_mono _ _ _ ⟨_, hx⟩)).op | case hf.a.a
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m' : ℕ
g : A
g_deg : g ∈ 𝒜 m'
x : A
hx : x = f * g
a : ↑(CommRingCat.of (Away 𝒜 f))
⊢ ↑((CommRingCat.Hom.hom (CommRingCat.ofHom (awayMap 𝒜 g_deg hx) ≫ awayToSection... | ext ⟨i, hi⟩ | case hf.a.a.h.mk.a
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m' : ℕ
g : A
g_deg : g ∈ 𝒜 m'
x : A
hx : x = f * g
a : ↑(CommRingCat.of (Away 𝒜 f))
i : ↑(ProjectiveSpectrum.top 𝒜)
hi : i ∈ Opposite.unop (Opposite.op (basi... | b5f9679f2160285c |
volume_regionBetween_eq_lintegral | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | theorem volume_regionBetween_eq_lintegral [SFinite μ] (hf : AEMeasurable f (μ.restrict s))
(hg : AEMeasurable g (μ.restrict s)) (hs : MeasurableSet s) :
μ.prod volume (regionBetween f g s) = ∫⁻ y in s, ENNReal.ofReal ((g - f) y) ∂μ | α : Type u_1
inst✝¹ : MeasurableSpace α
μ : Measure α
f g : α → ℝ
s : Set α
inst✝ : SFinite μ
hf : AEMeasurable f (μ.restrict s)
hg : AEMeasurable g (μ.restrict s)
hs : MeasurableSet s
h₁ :
(fun y => ofReal ((g - f) y)) =ᶠ[ae (μ.restrict s)] fun y => ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)
h₂ :
((μ... | convert h₂ using 1 | case h.e'_2
α : Type u_1
inst✝¹ : MeasurableSpace α
μ : Measure α
f g : α → ℝ
s : Set α
inst✝ : SFinite μ
hf : AEMeasurable f (μ.restrict s)
hg : AEMeasurable g (μ.restrict s)
hs : MeasurableSet s
h₁ :
(fun y => ofReal ((g - f) y)) =ᶠ[ae (μ.restrict s)] fun y => ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y... | 8d14176b7ee0e860 |
mellin_hasDerivAt_of_isBigO_rpow | Mathlib/Analysis/MellinTransform.lean | theorem mellin_hasDerivAt_of_isBigO_rpow [NormedSpace ℂ E] {a b : ℝ}
{f : ℝ → E} {s : ℂ} (hfc : LocallyIntegrableOn f (Ioi 0)) (hf_top : f =O[atTop] (· ^ (-a)))
(hs_top : s.re < a) (hf_bot : f =O[𝓝[>] 0] (· ^ (-b))) (hs_bot : b < s.re) :
MellinConvergent (fun t => log t • f t) s ∧
HasDerivAt (mellin ... | case intro.intro.intro.intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b : ℝ
f : ℝ → E
s : ℂ
hfc : LocallyIntegrableOn f (Ioi 0) volume
hf_top : f =O[atTop] fun x => x ^ (-a)
hs_top : s.re < a
hf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)
hs_bot : b < s.re
F : ℂ → ℝ → E := fun z t => ↑t ^ (z - 1) ... | exact
⟨min w w', lt_min hw1 hw1', (min_le_right _ _).trans_lt hw2', (min_le_left _ _).trans_lt hw2⟩ | no goals | 3ad5d9a87042b6f9 |
factorization_zero | Mathlib/RingTheory/UniqueFactorizationDomain/Finsupp.lean | theorem factorization_zero : factorization (0 : α) = 0 | α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : NormalizationMonoid α
inst✝ : DecidableEq α
⊢ factorization 0 = 0 | simp [factorization] | no goals | 480f5b693fc3b980 |
Prefunctor.pathStar_injective | Mathlib/Combinatorics/Quiver/Covering.lean | theorem Prefunctor.pathStar_injective (hφ : ∀ u, Injective (φ.star u)) (u : U) :
Injective (φ.pathStar u) | case mk.cons.mk.cons.intro
U : Type u_1
inst✝¹ : Quiver U
V : Type u_2
inst✝ : Quiver V
φ : U ⥤q V
hφ : ∀ (u : U), Injective (φ.star u)
u v₁ x₁ y₁ : U
p₁ : Path u x₁
e₁ : x₁ ⟶ y₁
ih :
∀ ⦃a₂ : PathStar u⦄,
(fun p => ⟨φ.obj p.fst, φ.mapPath p.snd⟩) ⟨x₁, p₁⟩ = (fun p => ⟨φ.obj p.fst, φ.mapPath p.snd⟩) a₂ → ⟨x₁, p₁⟩ ... | have hφx := Path.obj_eq_of_cons_eq_cons h' | case mk.cons.mk.cons.intro
U : Type u_1
inst✝¹ : Quiver U
V : Type u_2
inst✝ : Quiver V
φ : U ⥤q V
hφ : ∀ (u : U), Injective (φ.star u)
u v₁ x₁ y₁ : U
p₁ : Path u x₁
e₁ : x₁ ⟶ y₁
ih :
∀ ⦃a₂ : PathStar u⦄,
(fun p => ⟨φ.obj p.fst, φ.mapPath p.snd⟩) ⟨x₁, p₁⟩ = (fun p => ⟨φ.obj p.fst, φ.mapPath p.snd⟩) a₂ → ⟨x₁, p₁⟩ ... | 1252cc1c7c39ac57 |
MeromorphicOn.inv_iff | Mathlib/Analysis/Meromorphic/Basic.lean | @[simp] lemma inv_iff : MeromorphicOn s⁻¹ U ↔ MeromorphicOn s U :=
⟨fun h ↦ by simpa only [inv_inv] using h.inv, inv⟩
| 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
s : 𝕜 → 𝕜
U : Set 𝕜
h : MeromorphicOn s⁻¹ U
⊢ MeromorphicOn s U | simpa only [inv_inv] using h.inv | no goals | ff4e2a6c566ed665 |
integral_Ioi_cpow_of_lt | Mathlib/Analysis/SpecialFunctions/ImproperIntegrals.lean | theorem integral_Ioi_cpow_of_lt {a : ℂ} (ha : a.re < -1) {c : ℝ} (hc : 0 < c) :
(∫ t : ℝ in Ioi c, (t : ℂ) ^ a) = -(c : ℂ) ^ (a + 1) / (a + 1) | a : ℂ
ha : a.re < -1
c : ℝ
hc : 0 < c
this : Tendsto (fun x => (↑x ^ (a + 1) - ↑c ^ (a + 1)) / (a + 1)) atTop (𝓝 (-↑c ^ (a + 1) / (a + 1)))
x : ℝ
hx : 0 < x
⊢ a.re ≠ (-1).re | rw [Complex.neg_re, Complex.one_re] | a : ℂ
ha : a.re < -1
c : ℝ
hc : 0 < c
this : Tendsto (fun x => (↑x ^ (a + 1) - ↑c ^ (a + 1)) / (a + 1)) atTop (𝓝 (-↑c ^ (a + 1) / (a + 1)))
x : ℝ
hx : 0 < x
⊢ a.re ≠ -1 | 548595cb49c80036 |
inner_gramSchmidtOrthonormalBasis_eq_zero | Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean | theorem inner_gramSchmidtOrthonormalBasis_eq_zero {f : ι → E} {i : ι}
(hi : gramSchmidtNormed 𝕜 f i = 0) (j : ι) : ⟪gramSchmidtOrthonormalBasis h f i, f j⟫ = 0 | 𝕜 : Type u_1
E : Type u_2
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝⁴ : LinearOrder ι
inst✝³ : LocallyFiniteOrderBot ι
inst✝² : WellFoundedLT ι
inst✝¹ : Fintype ι
inst✝ : FiniteDimensional 𝕜 E
h : finrank 𝕜 E = Fintype.card ι
f : ι → E
i : ι
hi : gramSchmidtNo... | rintro rfl | 𝕜 : Type u_1
E : Type u_2
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝⁴ : LinearOrder ι
inst✝³ : LocallyFiniteOrderBot ι
inst✝² : WellFoundedLT ι
inst✝¹ : Fintype ι
inst✝ : FiniteDimensional 𝕜 E
h : finrank 𝕜 E = Fintype.card ι
f : ι → E
j k : ι
left✝ : k ∈ Set.... | abe7d3d273f2cc63 |
MeasureTheory.L1.setToL1_smul_left' | Mathlib/MeasureTheory/Integral/SetToL1.lean | theorem setToL1_smul_left' (hT : DominatedFinMeasAdditive μ T C)
(hT' : DominatedFinMeasAdditive μ T' C') (c : ℝ)
(h_smul : ∀ s, MeasurableSet s → μ s < ∞ → T' s = c • T s) (f : α →₁[μ] E) :
setToL1 hT' f = c • setToL1 hT f | α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' : Set α → E →L[ℝ] F
C C' : ℝ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
c ... | refine ContinuousLinearMap.extend_unique (setToL1SCLM α E μ hT') _ _ _ _ ?_ | α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' : Set α → E →L[ℝ] F
C C' : ℝ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
c ... | 08b410fcebe15fcc |
Array.getElem_shrink_loop | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean | theorem getElem_shrink_loop (a : Array α) (n : Nat) (i : Nat) (h : i < (shrink.loop n a).size) :
(shrink.loop n a)[i] = a[i]'(by simp at h; omega) | case succ
α : Type u_1
n : Nat
ih : ∀ (a : Array α) (i : Nat) (h : i < (shrink.loop n a).size), (shrink.loop n a)[i] = a[i]
a : Array α
i : Nat
h : i < (shrink.loop (n + 1) a).size
⊢ (shrink.loop (n + 1) a)[i] = a[i] | simp [shrink.loop, ih] | no goals | e66f39a8885b2d19 |
MeasureTheory.Measure.InnerRegularWRT.weaklyRegular_of_finite | Mathlib/MeasureTheory/Measure/Regular.lean | theorem weaklyRegular_of_finite [BorelSpace α] (μ : Measure α) [IsFiniteMeasure μ]
(H : InnerRegularWRT μ IsClosed IsOpen) : WeaklyRegular μ | case bc.h
α : Type u_1
inst✝³ : MeasurableSpace α
inst✝² : TopologicalSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
H✝ : μ.InnerRegularWRT IsClosed IsOpen
hfin : ∀ {s : Set α}, μ s ≠ ⊤
s : ℕ → Set α
hsd : Pairwise (Function.onFun Disjoint s)
hsm : ∀ (i : ℕ), MeasurableSet (s i)
H : ∀ (i : ℕ) (ε ... | apply hF | no goals | 663400e1c8207734 |
ProperSpace.of_locallyCompactSpace | Mathlib/Analysis/Normed/Module/FiniteDimension.lean | /-- A locally compact normed vector space is proper. -/
lemma ProperSpace.of_locallyCompactSpace (𝕜 : Type*) [NontriviallyNormedField 𝕜]
{E : Type*} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] [LocallyCompactSpace E] :
ProperSpace E | case intro.intro
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : LocallyCompactSpace E
r : ℝ
rpos : 0 < r
hr : IsCompact (closedBall 0 r)
⊢ ProperSpace E | rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc⟩ | case intro.intro.intro
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : LocallyCompactSpace E
r : ℝ
rpos : 0 < r
hr : IsCompact (closedBall 0 r)
c : 𝕜
hc : 1 < ‖c‖
⊢ ProperSpace E | d9806832454f6786 |
AkraBazziRecurrence.GrowsPolynomially.eventually_atTop_nonneg_or_nonpos | Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean | lemma eventually_atTop_nonneg_or_nonpos (hf : GrowsPolynomially f) :
(∀ᶠ x in atTop, 0 ≤ f x) ∨ (∀ᶠ x in atTop, f x ≤ 0) | f : ℝ → ℝ
hf : GrowsPolynomially f
c₁ : ℝ
left✝¹ : c₁ > 0
c₂ : ℝ
left✝ : c₂ > 0
heq : c₁ = c₂
n₀ : ℝ
hn₀ : ∀ b ≥ n₀, ∀ u ∈ Set.Icc (1 / 2 * b) b, f u = c₂ * f b
n : ℕ
hn : n ≥ 1
hyp_ind : ∀ z ∈ Set.Ico (n₀ ⊔ 2) (2 ^ n * (n₀ ⊔ 2)), f z = f (n₀ ⊔ 2)
z : ℝ
hz : z ∈ Set.Ico (2 ^ n * (n₀ ⊔ 2)) (2 ^ (n + 1) * (n₀ ⊔ 2))
⊢ 0 ≤... | norm_num | no goals | 69674f7264744f71 |
MeasureTheory.upcrossingsBefore_eq_sum | Mathlib/Probability/Martingale/Upcrossing.lean | theorem upcrossingsBefore_eq_sum (hab : a < b) : upcrossingsBefore a b f N ω =
∑ i ∈ Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i | Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
N : ℕ
ω : Ω
hab : a < b
hN : ¬N = 0
⊢ ∀ k ∈ Finset.Ico 1 (upcrossingsBefore a b f N ω + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 k = 1 | rintro k hk | Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
N : ℕ
ω : Ω
hab : a < b
hN : ¬N = 0
k : ℕ
hk : k ∈ Finset.Ico 1 (upcrossingsBefore a b f N ω + 1)
⊢ {n | upperCrossingTime a b f N n ω < N}.indicator 1 k = 1 | 50c14e6e447fb83a |
Complex.Gamma_mul_Gamma_eq_betaIntegral | Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | theorem Gamma_mul_Gamma_eq_betaIntegral {s t : ℂ} (hs : 0 < re s) (ht : 0 < re t) :
Gamma s * Gamma t = Gamma (s + t) * betaIntegral s t | s t : ℂ
hs : 0 < s.re
ht : 0 < t.re
conv_int :
∫ (x : ℝ) in Ioi 0,
∫ (t_1 : ℝ) in 0 ..x, ↑(rexp (-t_1)) * ↑t_1 ^ (s - 1) * (↑(rexp (-(x - t_1))) * ↑(x - t_1) ^ (t - 1)) ∂volume =
(∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (s - 1) ∂volume) * ∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ (t - 1) ∂volume
hst : 0 < (s ... | ring | no goals | 18eb59fc7415b248 |
CategoryTheory.MorphismProperty.Over.pullbackCongr_hom_app_left_fst | Mathlib/CategoryTheory/MorphismProperty/OverAdjunction.lean | @[reassoc (attr := simp)]
lemma Over.pullbackCongr_hom_app_left_fst {X Y : T} {f g : X ⟶ Y} (h : f = g) (A : P.Over Q Y) :
((Over.pullbackCongr h).hom.app A).left ≫ pullback.fst A.hom g =
pullback.fst A.hom f | T : Type u_1
inst✝⁴ : Category.{u_2, u_1} T
P Q : MorphismProperty T
inst✝³ : Q.IsMultiplicative
inst✝² : HasPullbacks T
inst✝¹ : P.IsStableUnderBaseChange
inst✝ : Q.IsStableUnderBaseChange
X Y : T
f : X ⟶ Y
A : P.Over Q Y
⊢ ((pullbackCongr ⋯).hom.app A).left ≫ pullback.fst A.hom f = pullback.fst A.hom f | simp [pullbackCongr] | no goals | 537174f9a294a6b9 |
Std.DHashMap.Internal.List.perm_cons_getEntry | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/List/Associative.lean | theorem perm_cons_getEntry [BEq α] {l : List ((a : α) × β a)} {a : α} (h : containsKey a l) :
∃ l', Perm l (getEntry a l h :: l') | α : Type u
β : α → Type v
inst✝ : BEq α
a k' : α
v' : β k'
t : List ((a : α) × β a)
ih : ∀ (h : containsKey a t = true), ∃ l', t.Perm (getEntry a t h :: l')
h✝ : containsKey a (⟨k', v'⟩ :: t) = true
h : (k' == a) = true ∨ containsKey a t = true
hk : (k' == a) = true
⊢ (⟨k', v'⟩ :: t).Perm (getEntry a (⟨k', v'⟩ :: t) h✝... | rw [getEntry_cons_of_beq hk] | no goals | 806cedfca58fd363 |
CategoryTheory.Functor.uncurry_obj_curry_obj_flip_flip' | Mathlib/CategoryTheory/Functor/Currying.lean | lemma uncurry_obj_curry_obj_flip_flip' (F₁ : B ⥤ C) (F₂ : D ⥤ E) (G : C × E ⥤ H) :
uncurry.obj (F₁ ⋙ (F₂ ⋙ (curry.obj G).flip).flip) = (F₁.prod F₂) ⋙ G :=
Functor.ext (by simp) (fun ⟨x₁, x₂⟩ ⟨y₁, y₂⟩ ⟨f₁, f₂⟩ => by
dsimp
simp only [Category.id_comp, Category.comp_id, ← G.map_comp, prod_comp])
| B : Type u₁
inst✝⁴ : Category.{v₁, u₁} B
C : Type u₂
inst✝³ : Category.{v₂, u₂} C
D : Type u₃
inst✝² : Category.{v₃, u₃} D
E : Type u₄
inst✝¹ : Category.{v₄, u₄} E
H : Type u₅
inst✝ : Category.{v₅, u₅} H
F₁ : B ⥤ C
F₂ : D ⥤ E
G : C × E ⥤ H
x✝² x✝¹ : B × D
x₁ : B
x₂ : D
y₁ : B
y₂ : D
x✝ : (x₁, x₂) ⟶ (y₁, y₂)
f₁ : (x₁, x... | dsimp | B : Type u₁
inst✝⁴ : Category.{v₁, u₁} B
C : Type u₂
inst✝³ : Category.{v₂, u₂} C
D : Type u₃
inst✝² : Category.{v₃, u₃} D
E : Type u₄
inst✝¹ : Category.{v₄, u₄} E
H : Type u₅
inst✝ : Category.{v₅, u₅} H
F₁ : B ⥤ C
F₂ : D ⥤ E
G : C × E ⥤ H
x✝² x✝¹ : B × D
x₁ : B
x₂ : D
y₁ : B
y₂ : D
x✝ : (x₁, x₂) ⟶ (y₁, y₂)
f₁ : (x₁, x... | 24b3f8808a4e0b85 |
IsGalois.of_separable_splitting_field | Mathlib/FieldTheory/Galois/Basic.lean | theorem of_separable_splitting_field [sp : p.IsSplittingField F E] (hp : p.Separable) :
IsGalois F E | F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
p : F[X]
sp : Polynomial.IsSplittingField F E p
hp : p.Separable
hFE : FiniteDimensional F E
this : DecidableEq E := Classical.decEq E
s : Set E := p.rootSet E
adjoin_root : adjoin F s = ⊤
P : IntermediateField F E → Prop := fun K => Fintyp... | suffices P (IntermediateField.adjoin F s) by
rw [adjoin_root] at this
apply of_card_aut_eq_finrank
rw [← Eq.trans this (LinearEquiv.finrank_eq IntermediateField.topEquiv.toLinearEquiv)]
exact Fintype.card_congr ((algEquivEquivAlgHom F E).toEquiv.trans
(IntermediateField.topEquiv.symm.arrowCongr AlgEquiv.ref... | F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
p : F[X]
sp : Polynomial.IsSplittingField F E p
hp : p.Separable
hFE : FiniteDimensional F E
this : DecidableEq E := Classical.decEq E
s : Set E := p.rootSet E
adjoin_root : adjoin F s = ⊤
P : IntermediateField F E → Prop := fun K => Fintyp... | d4b369ecd5955422 |
LSeries_tendsto_sub_mul_nhds_one_of_tendsto_sum_div_aux₁ | Mathlib/NumberTheory/LSeries/SumCoeff.lean | theorem LSeries_tendsto_sub_mul_nhds_one_of_tendsto_sum_div_aux₁
(hlim : Tendsto (fun n : ℕ ↦ (∑ k ∈ Icc 1 n, f k) / n) atTop (𝓝 l)) {ε : ℝ} (hε : 0 < ε) :
∀ᶠ t : ℝ in atTop, ‖(∑ k ∈ Icc 1 ⌊t⌋₊, f k) - l * t‖ < ε * t | f : ℕ → ℂ
l : ℂ
hlim : Tendsto (fun n => (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)
ε : ℝ
hε : 0 < ε
⊢ ∀ᶠ (t : ℝ) in atTop, ‖∑ k ∈ Icc 1 ⌊t⌋₊, f k - l * ↑t‖ < ε * t | have h_lim' : Tendsto (fun t : ℝ ↦ (∑ k ∈ Icc 1 ⌊t⌋₊, f k : ℂ) / t) atTop (𝓝 l) := by
refine (mul_one l ▸ ofReal_one ▸ ((hlim.comp tendsto_nat_floor_atTop).mul <|
tendsto_ofReal_iff.mpr <| tendsto_nat_floor_div_atTop)).congr' ?_
filter_upwards [eventually_ge_atTop 1] with t ht
simp [div_mul_div_cancel₀ (show... | f : ℕ → ℂ
l : ℂ
hlim : Tendsto (fun n => (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)
ε : ℝ
hε : 0 < ε
h_lim' : Tendsto (fun t => (∑ k ∈ Icc 1 ⌊t⌋₊, f k) / ↑t) atTop (𝓝 l)
⊢ ∀ᶠ (t : ℝ) in atTop, ‖∑ k ∈ Icc 1 ⌊t⌋₊, f k - l * ↑t‖ < ε * t | 231071717e1b68ef |
BoxIntegral.TaggedPrepartition.IsHenstock.card_filter_tag_eq_le | Mathlib/Analysis/BoxIntegral/Partition/Tagged.lean | theorem IsHenstock.card_filter_tag_eq_le [Fintype ι] (h : π.IsHenstock) (x : ι → ℝ) :
#{J ∈ π.boxes | π.tag J = x} ≤ 2 ^ Fintype.card ι | ι : Type u_1
I : Box ι
π : TaggedPrepartition I
inst✝ : Fintype ι
h : π.IsHenstock
x : ι → ℝ
J : Box ι
hJ : J ∈ π.boxes ∧ π.tag J = x
⊢ J ∈ π.boxes ∧ x ∈ Box.Icc J | rcases hJ with ⟨hJ, rfl⟩ | case intro
ι : Type u_1
I : Box ι
π : TaggedPrepartition I
inst✝ : Fintype ι
h : π.IsHenstock
J : Box ι
hJ : J ∈ π.boxes
⊢ J ∈ π.boxes ∧ π.tag J ∈ Box.Icc J | 80e0b7e46e65ef7c |
Matrix.coe_units_inv | Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | theorem coe_units_inv (A : (Matrix n n α)ˣ) : ↑A⁻¹ = (A⁻¹ : Matrix n n α) | n : Type u'
α : Type v
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A : (Matrix n n α)ˣ
⊢ ↑A⁻¹ = (↑A)⁻¹ | letI := A.invertible | n : Type u'
α : Type v
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A : (Matrix n n α)ˣ
this : Invertible ↑A := A.invertible
⊢ ↑A⁻¹ = (↑A)⁻¹ | ac999164f8d4f650 |
UniqueFactorizationMonoid.normalizedFactors_irreducible | Mathlib/RingTheory/UniqueFactorizationDomain/NormalizedFactors.lean | theorem normalizedFactors_irreducible {a : α} (ha : Irreducible a) :
normalizedFactors a = {normalize a} | case intro.intro
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
ha : Irreducible a
p : α
a_assoc : a ~ᵤ p
hp : normalizedFactors a = {p}
p_mem : p ∈ normalizedFactors a
⊢ normalizedFactors a = {normalize a} | convert hp | case h.e'_3.h.e'_4
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
ha : Irreducible a
p : α
a_assoc : a ~ᵤ p
hp : normalizedFactors a = {p}
p_mem : p ∈ normalizedFactors a
⊢ normalize a = p | 93db3636e3c8bc28 |
MeasureTheory.dirac_ne_dirac_iff_exists_measurableSet | Mathlib/MeasureTheory/Measure/Dirac.lean | /-- Dirac delta measures at two points are different if and only if there is a measurable set
containing one of the points but not the other. -/
lemma dirac_ne_dirac_iff_exists_measurableSet {x y : α} :
Measure.dirac x ≠ Measure.dirac y ↔ ∃ A, MeasurableSet A ∧ x ∈ A ∧ y ∉ A | α : Type u_1
inst✝ : MeasurableSpace α
x y : α
h : ∀ (A : Set α), MeasurableSet A → (x ∈ A ↔ y ∈ A)
A : Set α
A_mble : MeasurableSet A
⊢ x ∈ A → y ∈ A | simp only [h A A_mble, imp_self] | no goals | 621679eaf8c252cf |
CategoryTheory.Grothendieck.map_id_eq | Mathlib/CategoryTheory/Grothendieck.lean | theorem map_id_eq : map (𝟙 F) = 𝟙 (Cat.of <| Grothendieck <| F) | case h_obj
C : Type u
inst✝ : Category.{v, u} C
F : C ⥤ Cat
⊢ ∀ (X : Grothendieck F), (map (𝟙 F)).obj X = (𝟙 (Cat.of (Grothendieck F))).obj X | intro X | case h_obj
C : Type u
inst✝ : Category.{v, u} C
F : C ⥤ Cat
X : Grothendieck F
⊢ (map (𝟙 F)).obj X = (𝟙 (Cat.of (Grothendieck F))).obj X | 5867a0755a02dad4 |
Matroid.existsMaximalSubsetProperty_of_bdd | Mathlib/Data/Matroid/IndepAxioms.lean | theorem _root_.Matroid.existsMaximalSubsetProperty_of_bdd {P : Set α → Prop}
(hP : ∃ (n : ℕ), ∀ Y, P Y → Y.encard ≤ n) (X : Set α) : ExistsMaximalSubsetProperty P X | case intro
α : Type u_1
P : Set α → Prop
X : Set α
n : ℕ
hP : ∀ (Y : Set α), P Y → Y.encard ≤ ↑n
⊢ ExistsMaximalSubsetProperty P X | rintro I hI hIX | case intro
α : Type u_1
P : Set α → Prop
X : Set α
n : ℕ
hP : ∀ (Y : Set α), P Y → Y.encard ≤ ↑n
I : Set α
hI : P I
hIX : I ⊆ X
⊢ ∃ J, I ⊆ J ∧ Maximal (fun K => P K ∧ K ⊆ X) J | 25330014d05a8b6c |
ProbabilityTheory.deriv_cgf | Mathlib/Probability/Moments/MGFAnalytic.lean | lemma deriv_cgf (h : v ∈ interior (integrableExpSet X μ)) :
deriv (cgf X μ) v = μ[fun ω ↦ X ω * exp (v * X ω)] / mgf X μ v | case neg
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
v : ℝ
h : v ∈ interior (integrableExpSet X μ)
hμ : ¬μ = 0
⊢ deriv (cgf X μ) v = (∫ (x : Ω), (fun ω => X ω * rexp (v * X ω)) x ∂μ) / mgf X μ v | have hv : Integrable (fun ω ↦ exp (v * X ω)) μ := interior_subset (s := integrableExpSet X μ) h | case neg
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
v : ℝ
h : v ∈ interior (integrableExpSet X μ)
hμ : ¬μ = 0
hv : Integrable (fun ω => rexp (v * X ω)) μ
⊢ deriv (cgf X μ) v = (∫ (x : Ω), (fun ω => X ω * rexp (v * X ω)) x ∂μ) / mgf X μ v | 7a98681b0e228df0 |
MeasureTheory.IsFundamentalDomain.absolutelyContinuous_map | Mathlib/MeasureTheory/Measure/Haar/Quotient.lean | /-- Given a quotient space `G ⧸ Γ` where `Γ` is `Countable`, and the restriction,
`μ_𝓕`, of a right-invariant measure `μ` on `G` to a fundamental domain `𝓕`, a set
in the quotient which has `μ_𝓕`-measure zero, also has measure zero under the
folding of `μ` under the quotient. Note that, if `Γ` is infinite, the... | G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : MeasurableSpace G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : BorelSpace G
μ : Measure G
Γ : Subgroup G
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥Γ.op) 𝓕 μ
inst✝³ : Countable ↥Γ
inst✝² : MeasurableSpace (G ⧸ Γ)
inst✝¹ : BorelSpace (G ⧸ Γ)
inst✝ : μ.IsMulRightI... | set π : G → G ⧸ Γ := QuotientGroup.mk | G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : MeasurableSpace G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : BorelSpace G
μ : Measure G
Γ : Subgroup G
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥Γ.op) 𝓕 μ
inst✝³ : Countable ↥Γ
inst✝² : MeasurableSpace (G ⧸ Γ)
inst✝¹ : BorelSpace (G ⧸ Γ)
inst✝ : μ.IsMulRightI... | f3d9bf43ea92eda9 |
Equiv.Perm.cycle_is_cycleOf | Mathlib/GroupTheory/Perm/Cycle/Factors.lean | theorem cycle_is_cycleOf {f c : Equiv.Perm α} {a : α} (ha : a ∈ c.support)
(hc : c ∈ f.cycleFactorsFinset) : c = f.cycleOf a | α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f c : Perm α
a : α
ha : a ∈ c.support
hc : c ∈ f.cycleFactorsFinset
this : f.cycleOf a = c.cycleOf a
⊢ c = c.cycleOf a | apply symm | case a
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f c : Perm α
a : α
ha : a ∈ c.support
hc : c ∈ f.cycleFactorsFinset
this : f.cycleOf a = c.cycleOf a
⊢ c.cycleOf a = c | 51f66f0a14646a04 |
Std.Tactic.BVDecide.Reflect.BitVec.ult_congr | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Reflect.lean | theorem ult_congr (lhs rhs lhs' rhs' : BitVec w) (h1 : lhs' = lhs) (h2 : rhs' = rhs) :
(BitVec.ult lhs' rhs') = (BitVec.ult lhs rhs) | w : Nat
lhs rhs lhs' rhs' : BitVec w
h1 : lhs' = lhs
h2 : rhs' = rhs
⊢ lhs'.ult rhs' = lhs.ult rhs | simp[*] | no goals | b17ce039cdaaf6bc |
Ordnode.balance_eq_balance' | Mathlib/Data/Ordmap/Ordset.lean | theorem balance_eq_balance' {l x r} (hl : Balanced l) (hr : Balanced r) (sl : Sized l)
(sr : Sized r) : @balance α l x r = balance' l x r | α : Type u_1
l : Ordnode α
x : α
r : Ordnode α
hl : l.Balanced
hr : r.Balanced
sl : l.Sized
sr : r.Sized
⊢ l.balance x r = l.balance' x r | obtain - | ⟨ls, ll, lx, lr⟩ := l | case nil
α : Type u_1
x : α
r : Ordnode α
hr : r.Balanced
sr : r.Sized
hl : nil.Balanced
sl : nil.Sized
⊢ nil.balance x r = nil.balance' x r
case node
α : Type u_1
x : α
r : Ordnode α
hr : r.Balanced
sr : r.Sized
ls : ℕ
ll : Ordnode α
lx : α
lr : Ordnode α
hl : (node ls ll lx lr).Balanced
sl : (node ls ll lx lr).Sized... | 23a52200299a6569 |
MeasureTheory.OuterMeasure.iUnion_nat_of_monotone_of_tsum_ne_top | Mathlib/MeasureTheory/OuterMeasure/Basic.lean | theorem iUnion_nat_of_monotone_of_tsum_ne_top (m : OuterMeasure α) {s : ℕ → Set α}
(h_mono : ∀ n, s n ⊆ s (n + 1)) (h0 : (∑' k, m (s (k + 1) \ s k)) ≠ ∞) :
m (⋃ n, s n) = ⨆ n, m (s n) | α : Type u_1
m : OuterMeasure α
s : ℕ → Set α
h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)
h0 : ∑' (k : ℕ), m (s (k + 1) \ s k) ≠ ⊤
n : ℕ
h' : Monotone s
⊢ ∀ (i : ℕ), s i ⊆ s n ∪ ⋃ i, s (i + n + 1) \ s (i + n) | intro i x hx | α : Type u_1
m : OuterMeasure α
s : ℕ → Set α
h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)
h0 : ∑' (k : ℕ), m (s (k + 1) \ s k) ≠ ⊤
n : ℕ
h' : Monotone s
i : ℕ
x : α
hx : x ∈ s i
⊢ x ∈ s n ∪ ⋃ i, s (i + n + 1) \ s (i + n) | a03a31d80368140b |
CliffordAlgebra.evenOdd_induction | Mathlib/LinearAlgebra/CliffordAlgebra/Grading.lean | theorem evenOdd_induction (n : ZMod 2) {motive : ∀ x, x ∈ evenOdd Q n → Prop}
(range_ι_pow : ∀ (v) (h : v ∈ LinearMap.range (ι Q) ^ n.val),
motive v (Submodule.mem_iSup_of_mem ⟨n.val, n.natCast_zmod_val⟩ h))
(add : ∀ x y hx hy, motive x hx → motive y hy → motive (x + y) (Submodule.add_mem _ hx hy))
... | case intro.mem_mul_mem.add
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
n✝ : ZMod 2
motive : (x : CliffordAlgebra Q) → x ∈ evenOdd Q n✝ → Prop
range_ι_pow : ∀ (v : CliffordAlgebra Q) (h : v ∈ LinearMap.range (ι Q) ^ n✝.val), motive v ⋯
add :
∀ (x y : C... | simp_rw [add_mul] | case intro.mem_mul_mem.add
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
n✝ : ZMod 2
motive : (x : CliffordAlgebra Q) → x ∈ evenOdd Q n✝ → Prop
range_ι_pow : ∀ (v : CliffordAlgebra Q) (h : v ∈ LinearMap.range (ι Q) ^ n✝.val), motive v ⋯
add :
∀ (x y : C... | 3b3d6ba6e257fdcc |
Module.FinitePresentation.fg_ker | Mathlib/Algebra/Module/FinitePresentation.lean | lemma Module.FinitePresentation.fg_ker [Module.Finite R M]
[h : Module.FinitePresentation R N] (l : M →ₗ[R] N) (hl : Function.Surjective l) :
(LinearMap.ker l).FG | case mk.intro.intro.intro.hs2
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁵ : Ring R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : Module.Finite R M
l : M →ₗ[R] N
hl : Function.Surjective ⇑l
s : Finset N
hs : Submodule.span R ↑s = ⊤
hs' : (LinearMap.ker (linearCombinati... | exact hs'.map f | no goals | 1eeff1d09d79515c |
Ordinal.eq_zero_or_opow_omega0_le_of_mul_eq_right | Mathlib/SetTheory/Ordinal/FixedPoint.lean | theorem eq_zero_or_opow_omega0_le_of_mul_eq_right {a b : Ordinal} (hab : a * b = b) :
b = 0 ∨ a ^ ω ≤ b | case inl
a b : Ordinal.{u_1}
hab : a * b = b
ha : a = 0
⊢ b = 0 ∨ 0 ≤ b | exact Or.inr (Ordinal.zero_le b) | no goals | 05688dbb400f4575 |
Int.Matrix.card_S_eq | Mathlib/NumberTheory/SiegelsLemma.lean | private lemma card_S_eq [DecidableEq α] : #(Finset.Icc N P) = ∏ i : α, (P i - N i + 1) | case e_f
α : Type u_1
β : Type u_2
inst✝² : Fintype α
inst✝¹ : Fintype β
A : Matrix α β ℤ
inst✝ : DecidableEq α
⊢ (fun i => ↑(#(Icc (∑ j : β, ↑B * -(A i j)⁻) (∑ j : β, ↑B * (A i j)⁺)))) = fun i => P i - N i + 1 | ext i | case e_f.h
α : Type u_1
β : Type u_2
inst✝² : Fintype α
inst✝¹ : Fintype β
A : Matrix α β ℤ
inst✝ : DecidableEq α
i : α
⊢ ↑(#(Icc (∑ j : β, ↑B * -(A i j)⁻) (∑ j : β, ↑B * (A i j)⁺))) = P i - N i + 1 | fdc73b569700261b |
tendsto_measure_Icc_nhdsWithin_right' | Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean | lemma tendsto_measure_Icc_nhdsWithin_right' (b : ℝ) :
Tendsto (fun δ ↦ μ (Icc (b - δ) (b + δ))) (𝓝[>] (0 : ℝ)) (𝓝 (μ {b})) | μ : Measure ℝ
inst✝ : IsFiniteMeasureOnCompacts μ
b r s : ℝ
_rpos : 0 < r
hrs : r ≤ s
⊢ b + r ≤ b + s | linarith | no goals | 234976a60451ee77 |
le_iff_oneLePart_leOnePart | Mathlib/Algebra/Order/Group/PosPart.lean | @[to_additive] lemma le_iff_oneLePart_leOnePart (a b : α) : a ≤ b ↔ a⁺ᵐ ≤ b⁺ᵐ ∧ b⁻ᵐ ≤ a⁻ᵐ | α : Type u_1
inst✝² : Lattice α
inst✝¹ : CommGroup α
inst✝ : MulLeftMono α
a b : α
⊢ a ≤ b ↔ a⁺ᵐ ≤ b⁺ᵐ ∧ b⁻ᵐ ≤ a⁻ᵐ | refine ⟨fun h ↦ ⟨oneLePart_mono h, leOnePart_anti h⟩, fun h ↦ ?_⟩ | α : Type u_1
inst✝² : Lattice α
inst✝¹ : CommGroup α
inst✝ : MulLeftMono α
a b : α
h : a⁺ᵐ ≤ b⁺ᵐ ∧ b⁻ᵐ ≤ a⁻ᵐ
⊢ a ≤ b | cafd06bafc8fb100 |
Polynomial.cyclotomic_eval_lt_add_one_pow_totient | Mathlib/RingTheory/Polynomial/Cyclotomic/Eval.lean | theorem cyclotomic_eval_lt_add_one_pow_totient {n : ℕ} {q : ℝ} (hn' : 3 ≤ n) (hq' : 1 < q) :
(cyclotomic n ℝ).eval q < (q + 1) ^ totient n | case h.e'_4
n : ℕ
q : ℝ
hn' : 3 ≤ n
hq' : 1 < q
hn : 0 < n
hq : 0 < q
hfor : ∀ ζ' ∈ primitiveRoots n ℂ, ‖↑q - ζ'‖ ≤ q + 1
ζ : ℂ := Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)
hζ : IsPrimitiveRoot ζ n
hex : ∃ ζ' ∈ primitiveRoots n ℂ, ‖↑q - ζ'‖ < q + 1
this : ¬eval (↑q) (cyclotomic n ℂ) = 0
⊢ Units.mk0 (q + 1).toNNReal ⋯... | simp [Complex.card_primitiveRoots] | no goals | 6a3a0563e7ff7373 |
Submonoid.prod_le_iff | Mathlib/Algebra/Group/Submonoid/Operations.lean | theorem prod_le_iff {s : Submonoid M} {t : Submonoid N} {u : Submonoid (M × N)} :
s.prod t ≤ u ↔ s.map (inl M N) ≤ u ∧ t.map (inr M N) ≤ u | N : Type u_2
inst✝¹ : MulOneClass N
M : Type u_5
inst✝ : MulOneClass M
s : Submonoid M
t : Submonoid N
u : Submonoid (M × N)
⊢ s.prod t ≤ u ↔ map (inl M N) s ≤ u ∧ map (inr M N) t ≤ u | constructor | case mp
N : Type u_2
inst✝¹ : MulOneClass N
M : Type u_5
inst✝ : MulOneClass M
s : Submonoid M
t : Submonoid N
u : Submonoid (M × N)
⊢ s.prod t ≤ u → map (inl M N) s ≤ u ∧ map (inr M N) t ≤ u
case mpr
N : Type u_2
inst✝¹ : MulOneClass N
M : Type u_5
inst✝ : MulOneClass M
s : Submonoid M
t : Submonoid N
u : Submonoid (... | 1fa538bbe53be702 |
exists_seq_forall_proj_of_forall_finite | Mathlib/Order/KonigLemma.lean | theorem exists_seq_forall_proj_of_forall_finite {α : ℕ → Type*} [Finite (α 0)] [∀ i, Nonempty (α i)]
(π : {i j : ℕ} → (hij : i ≤ j) → α j → α i)
(π_refl : ∀ ⦃i⦄ (a : α i), π rfl.le a = a)
(π_trans : ∀ ⦃i j k⦄ (hij : i ≤ j) (hjk : j ≤ k) a, π hij (π hjk a) = π (hij.trans hjk) a)
(hfin : ∀ i a, {b : α (i+... | case intro.refine_2.intro.intro
α : ℕ → Type u_1
inst✝¹ : Finite (α 0)
inst✝ : ∀ (i : ℕ), Nonempty (α i)
π : {i j : ℕ} → i ≤ j → α j → α i
π_refl : ∀ ⦃i : ℕ⦄ (a : α i), π ⋯ a = a
π_trans : ∀ ⦃i j k : ℕ⦄ (hij : i ≤ j) (hjk : j ≤ k) (a : α k), π hij (π hjk a) = π ⋯ a
hfin : ∀ (i : ℕ) (a : α i), {b | π ⋯ b = a}.Finite
αs ... | rw [π_refl] | no goals | 271fcb2e96bcf388 |
BitVec.umod_eq_of_mul_add_toNat | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Bitblast.lean | theorem umod_eq_of_mul_add_toNat {d n q r : BitVec w} (hrd : r < d)
(hdqnr : d.toNat * q.toNat + r.toNat = n.toNat) :
n % d = r | case a
w : Nat
d n q r : BitVec w
hdqnr : r.toNat % d.toNat = n.toNat % d.toNat
hrd : r.toNat < d.toNat
⊢ n.toNat % d.toNat = r.toNat | rw [Nat.mod_eq_of_lt hrd] at hdqnr | case a
w : Nat
d n q r : BitVec w
hdqnr : r.toNat = n.toNat % d.toNat
hrd : r.toNat < d.toNat
⊢ n.toNat % d.toNat = r.toNat | 599854794cefda29 |
swap_mul_swap_mul_swap | Mathlib/Algebra/Group/End.lean | theorem swap_mul_swap_mul_swap {x y z : α} (hxy : x ≠ y) (hxz : x ≠ z) :
swap y z * swap x y * swap y z = swap z x | α : Type u_4
inst✝ : DecidableEq α
x y z : α
hxy : x ≠ y
hxz : x ≠ z
⊢ swap y z * swap x y * (swap y z)⁻¹ = swap z x | rw [← swap_apply_apply, swap_apply_left, swap_apply_of_ne_of_ne hxy hxz, swap_comm] | no goals | a6fbb5a9b4adfbc4 |
MeasureTheory.exists_upperSemicontinuous_le_lintegral_le | Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean | theorem exists_upperSemicontinuous_le_lintegral_le (f : α → ℝ≥0) (int_f : (∫⁻ x, f x ∂μ) ≠ ∞)
{ε : ℝ≥0∞} (ε0 : ε ≠ 0) :
∃ g : α → ℝ≥0, (∀ x, g x ≤ f x) ∧ UpperSemicontinuous g ∧
(∫⁻ x, f x ∂μ) ≤ (∫⁻ x, g x ∂μ) + ε | case intro.intro.intro.intro.intro
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
f : α → ℝ≥0
int_f : ∫⁻ (x : α), ↑(f x) ∂μ ≠ ⊤
ε : ℝ≥0∞
ε0 : ε ≠ 0
fs : α →ₛ ℝ≥0
fs_le_f : ∀ (x : α), fs x ≤ f x
int_fs : ∫⁻ (x : α), ↑(f x) ∂μ ≤ ∫⁻ (x : α), ... | calc
(∫⁻ x, f x ∂μ) ≤ (∫⁻ x, fs x ∂μ) + ε / 2 := int_fs
_ ≤ (∫⁻ x, g x ∂μ) + ε / 2 + ε / 2 := add_le_add gint le_rfl
_ = (∫⁻ x, g x ∂μ) + ε := by rw [add_assoc, ENNReal.add_halves] | no goals | 41492b8d4d821475 |
Finset.Shatters.of_compression | Mathlib/Combinatorics/SetFamily/Shatter.lean | lemma Shatters.of_compression (hs : (𝓓 a 𝒜).Shatters s) : 𝒜.Shatters s | α : Type u_1
inst✝ : DecidableEq α
𝒜 : Finset (Finset α)
s : Finset α
a : α
hs : (𝓓 a 𝒜).Shatters s
u : Finset α
ht : s ∩ u ⊆ s
hu : u ∉ 𝒜 ∧ insert a u ∈ 𝒜
ha : a ∈ s
v : Finset α
hsv : s ∩ v = insert a (s ∩ u)
hv : v ∉ 𝒜 ∧ insert a v ∈ 𝒜
⊢ a ∈ insert a (s ∩ u) | exact mem_insert_self _ _ | no goals | 49693a05561fee5b |
FirstOrder.Language.DirectLimit.unify_sigma_mk_self | Mathlib/ModelTheory/DirectLimit.lean | theorem unify_sigma_mk_self {α : Type*} {i : ι} {x : α → G i} :
(unify f (fun a => .mk f i (x a)) i fun _ ⟨_, hj⟩ =>
_root_.trans (le_of_eq hj.symm) (refl _)) = x | case h
L : Language
ι : Type v
inst✝² : Preorder ι
G : ι → Type w
inst✝¹ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
inst✝ : DirectedSystem G fun i j h => ⇑(f i j h)
α : Type u_1
i : ι
x : α → G i
a : α
⊢ unify f (fun a => Structure.Sigma.mk f i (x a)) i ⋯ a = x a | rw [unify] | case h
L : Language
ι : Type v
inst✝² : Preorder ι
G : ι → Type w
inst✝¹ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
inst✝ : DirectedSystem G fun i j h => ⇑(f i j h)
α : Type u_1
i : ι
x : α → G i
a : α
⊢ (f (Structure.Sigma.mk f i (x a)).fst i ⋯) (Structure.Sigma.mk f i (x a)).snd = x a | cdfcdd3e919999ce |
Int.bodd_add_div2 | Mathlib/Data/Int/Bitwise.lean | theorem bodd_add_div2 : ∀ n, cond (bodd n) 1 0 + 2 * div2 n = n
| (n : ℕ) => by
rw [show (cond (bodd n) 1 0 : ℤ) = (cond (bodd n) 1 0 : ℕ) by cases bodd n <;> rfl]
exact congr_arg ofNat n.bodd_add_div2
| -[n+1] => by
refine Eq.trans ?_ (congr_arg negSucc n.bodd_add_div2)
dsimp [bodd]; cases Nat.bodd... | case true
n : ℕ
⊢ 0 + 2 * -[n.div2+1] = -[1 + 2 * n.div2+1] | rw [zero_add, add_comm] | case true
n : ℕ
⊢ 2 * -[n.div2+1] = -[2 * n.div2 + 1+1] | b52aac4a92c4fde8 |
LieAlgebra.IsKilling.eq_neg_one_or_eq_zero_or_eq_one_of_eq_smul | Mathlib/Algebra/Lie/Weights/RootSystem.lean | lemma eq_neg_one_or_eq_zero_or_eq_one_of_eq_smul
(hα : α.IsNonZero) (k : K) (h : (β : H → K) = k • α) :
k = -1 ∨ k = 0 ∨ k = 1 | case inr.inl.intro
K : Type u_1
L : Type u_2
inst✝⁷ : Field K
inst✝⁶ : CharZero K
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra K L
inst✝³ : IsKilling K L
inst✝² : FiniteDimensional K L
H✝ : LieSubalgebra K L
inst✝¹ : H✝.IsCartanSubalgebra
inst✝ : IsTriangularizable K (↥H✝) L
α β : Weight K (↥H✝) L
hα : α.IsNonZero
k : K
h : ... | have := (rootSpace_zsmul_add_ne_bot_iff_mem α 0 hα (n - chainTopCoeff α β)).mp
(by rw [← Int.cast_smul_eq_zsmul K, ← H, ← h, Weight.coe_zero, add_zero]; exact β.2) | case inr.inl.intro
K : Type u_1
L : Type u_2
inst✝⁷ : Field K
inst✝⁶ : CharZero K
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra K L
inst✝³ : IsKilling K L
inst✝² : FiniteDimensional K L
H✝ : LieSubalgebra K L
inst✝¹ : H✝.IsCartanSubalgebra
inst✝ : IsTriangularizable K (↥H✝) L
α β : Weight K (↥H✝) L
hα : α.IsNonZero
k : K
h : ... | 1279a8c2c7f69b11 |
Polynomial.splits_prod_iff | Mathlib/Algebra/Polynomial/Splits.lean | theorem splits_prod_iff {ι : Type u} {s : ι → K[X]} {t : Finset ι} :
(∀ j ∈ t, s j ≠ 0) → ((∏ x ∈ t, s x).Splits i ↔ ∀ j ∈ t, (s j).Splits i) | K : Type v
L : Type w
inst✝¹ : Field K
inst✝ : Field L
i : K →+* L
ι : Type u
s : ι → K[X]
t✝ : Finset ι
a : ι
t : Finset ι
hat : a ∉ t
ih : (∀ j ∈ t, s j ≠ 0) → (Splits i (∏ x ∈ t, s x) ↔ ∀ j ∈ t, Splits i (s j))
ht : ∀ j ∈ insert a t, s j ≠ 0
⊢ Splits i (∏ x ∈ insert a t, s x) ↔ ∀ j ∈ insert a t, Splits i (s j) | rw [Finset.forall_mem_insert] at ht ⊢ | K : Type v
L : Type w
inst✝¹ : Field K
inst✝ : Field L
i : K →+* L
ι : Type u
s : ι → K[X]
t✝ : Finset ι
a : ι
t : Finset ι
hat : a ∉ t
ih : (∀ j ∈ t, s j ≠ 0) → (Splits i (∏ x ∈ t, s x) ↔ ∀ j ∈ t, Splits i (s j))
ht : s a ≠ 0 ∧ ∀ x ∈ t, s x ≠ 0
⊢ Splits i (∏ x ∈ insert a t, s x) ↔ Splits i (s a) ∧ ∀ x ∈ t, Splits i (s... | 72d77f00dc774bee |
BitVec.getLsbD_last | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem getLsbD_last (x : BitVec w) :
x.getLsbD (w-1) = decide (2 ^ (w-1) ≤ x.toNat) | case zero
x : BitVec 0
⊢ x.getLsbD (0 - 1) = decide (2 ^ (0 - 1) ≤ x.toNat) | simp [toNat_of_zero_length] | no goals | 61f42aafb499ffda |
AlgebraicGeometry.sigmaMk_mk | Mathlib/AlgebraicGeometry/Limits.lean | @[simp]
lemma sigmaMk_mk (i) (x : f i) :
sigmaMk f (.mk i x) = (Sigma.ι f i).base x | ι : Type u
f : ι → Scheme
i : ι
x : ↑↑(f i).toPresheafedSpace
⊢ (ConcreteCategory.hom
((TopCat.sigmaCofan fun x => (f x).toTopCat).inj i ≫
(colimit.isoColimitCocone
{ cocone := TopCat.sigmaCofan fun x => (f x).toTopCat,
isColimit := TopCat.sigmaCofanIsColimit fun x =>... | congr 2 | case e_a.e_a
ι : Type u
f : ι → Scheme
i : ι
x : ↑↑(f i).toPresheafedSpace
⊢ (TopCat.sigmaCofan fun x => (f x).toTopCat).inj i ≫
(colimit.isoColimitCocone
{ cocone := TopCat.sigmaCofan fun x => (f x).toTopCat,
isColimit := TopCat.sigmaCofanIsColimit fun x => (f x).toTopCat }).inv ≫
... | 40c892222485d21f |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.reduce_postcondition | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddSound.lean | theorem reduce_postcondition {n : Nat} (c : DefaultClause n) (assignment : Array Assignment) :
(reduce c assignment = reducedToEmpty → Incompatible (PosFin n) c assignment) ∧
(∀ l : Literal (PosFin n), reduce c assignment = reducedToUnit l → ∀ (p : (PosFin n) → Bool), p ⊨ assignment → p ⊨ c → p ⊨ l) | case intro.left
n : Nat
c : DefaultClause n
assignment : Array Assignment
c_arr : Array (Literal (PosFin n)) := List.toArray c.clause
c_clause_rw : c.clause = c_arr.toList
motive : Nat → ReduceResult (PosFin n) → Prop := ReducePostconditionInductionMotive c_arr assignment
h_base : motive 0 reducedToEmpty
h_inductive :
... | rcases h1 with h1 | h1 | case intro.left.inl
n : Nat
c : DefaultClause n
assignment : Array Assignment
c_arr : Array (Literal (PosFin n)) := List.toArray c.clause
c_clause_rw : c.clause = c_arr.toList
motive : Nat → ReduceResult (PosFin n) → Prop := ReducePostconditionInductionMotive c_arr assignment
h_base : motive 0 reducedToEmpty
h_inductiv... | bc01533c0ab5cdb8 |
toIcoDiv_neg' | Mathlib/Algebra/Order/ToIntervalMod.lean | theorem toIcoDiv_neg' (a b : α) : toIcoDiv hp (-a) b = -(toIocDiv hp a (-b) + 1) | α : Type u_1
inst✝ : LinearOrderedAddCommGroup α
hα : Archimedean α
p : α
hp : 0 < p
a b : α
⊢ toIcoDiv hp (-a) b = -(toIocDiv hp a (-b) + 1) | simpa only [neg_neg] using toIcoDiv_neg hp (-a) (-b) | no goals | e0280e37c60a849a |
MeasureTheory.Measure.restrict_sub_eq_restrict_sub_restrict | Mathlib/MeasureTheory/Measure/Sub.lean | theorem restrict_sub_eq_restrict_sub_restrict (h_meas_s : MeasurableSet s) :
(μ - ν).restrict s = μ.restrict s - ν.restrict s | case a
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
h_meas_s : MeasurableSet s
h_nonempty : {d | μ ≤ d + ν}.Nonempty
ν' : Measure α
h_ν'_in : μ.restrict s ≤ ν' + ν.restrict s
⊢ ν'.restrict s ∈ (fun μ => μ.restrict s) '' {d | μ ≤ d + ν} | refine ⟨ν' + (⊤ : Measure α).restrict sᶜ, ?_, ?_⟩ | case a.refine_1
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
h_meas_s : MeasurableSet s
h_nonempty : {d | μ ≤ d + ν}.Nonempty
ν' : Measure α
h_ν'_in : μ.restrict s ≤ ν' + ν.restrict s
⊢ ν' + ⊤.restrict sᶜ ∈ {d | μ ≤ d + ν}
case a.refine_2
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
h_m... | 2b225f2b24353253 |
ZMod.erdos_ginzburg_ziv_prime | Mathlib/Combinatorics/Additive/ErdosGinzburgZiv.lean | theorem ZMod.erdos_ginzburg_ziv_prime (a : ι → ZMod p) (hs : #s = 2 * p - 1) :
∃ t ⊆ s, #t = p ∧ ∑ i ∈ t, a i = 0 | case intro.refine_3
ι : Type u_1
p : ℕ
inst✝ : Fact (Nat.Prime p)
s : Finset ι
a : ι → ZMod p
hs : #s = 2 * p - 1
this : NeZero p
N : ℕ := Fintype.card { x // (eval x) (f₁ s a) = 0 ∧ (eval x) (f₂ s a) = 0 }
zero_sol : { x // (eval x) (f₁ s a) = 0 ∧ (eval x) (f₂ s a) = 0 } := ⟨0, ⋯⟩
hN₀ : 0 < N
hs' : 2 * p - 1 = Fintype... | simpa [f₂, ZMod.pow_card_sub_one, Finset.sum_filter] using x.2.2 | no goals | 3ec57b8d04698567 |
VitaliFamily.ae_tendsto_lintegral_enorm_sub_div'_of_integrable | Mathlib/MeasureTheory/Covering/Differentiation.lean | theorem ae_tendsto_lintegral_enorm_sub_div'_of_integrable {f : α → E} (hf : Integrable f μ)
(h'f : StronglyMeasurable f) :
∀ᵐ x ∂μ, Tendsto (fun a => (∫⁻ y in a, ‖f y - f x‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 0) | α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
f : α → E
hf : Integrable f μ
h'f : StronglyMeasurable f
A : μ.FiniteSpanningSetsIn {K | ... | have : f x ∈ closure t := ht (mem_range_self _) | α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
f : α → E
hf : Integrable f μ
h'f : StronglyMeasurable f
A : μ.FiniteSpanningSetsIn {K | ... | 38b9ca590c2fbd74 |
Order.length_le_height | Mathlib/Order/KrullDimension.lean | lemma length_le_height {p : LTSeries α} {x : α} (hlast : p.last ≤ x) :
p.length ≤ height x | case pos
α : Type u_1
inst✝ : Preorder α
p : LTSeries α
x : α
hlast : RelSeries.last p ≤ x
hlen0 : p.length ≠ 0
p' : RelSeries fun x1 x2 => x1 < x2 := (RelSeries.eraseLast p).snoc x ⋯
⊢ ↑p'.length ≤ height x | refine le_iSup₂_of_le p' ?_ le_rfl | case pos
α : Type u_1
inst✝ : Preorder α
p : LTSeries α
x : α
hlast : RelSeries.last p ≤ x
hlen0 : p.length ≠ 0
p' : RelSeries fun x1 x2 => x1 < x2 := (RelSeries.eraseLast p).snoc x ⋯
⊢ p'.last ≤ x | de1d1b5b1d97e68b |
Equiv.Perm.IsCycle.commute_iff | Mathlib/GroupTheory/Perm/Cycle/Basic.lean | theorem IsCycle.commute_iff {g c : Perm α} (hc : c.IsCycle) :
Commute g c ↔
∃ hc' : ∀ x : α, x ∈ c.support ↔ g x ∈ c.support,
ofSubtype (subtypePerm g hc') ∈ Subgroup.zpowers c | α : Type u_2
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g c : Perm α
hc : c.IsCycle
hc' : ∀ (x : α), x ∈ c.support ↔ g x ∈ c.support
k : ℤ
⊢ (c ^ k).subtypePerm ⋯ = g.subtypePerm ⋯ ↔ c ^ k = ofSubtype (g.subtypePerm ⋯) | simp only [Perm.ext_iff, subtypePerm_apply, Subtype.mk.injEq, Subtype.forall] | α : Type u_2
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g c : Perm α
hc : c.IsCycle
hc' : ∀ (x : α), x ∈ c.support ↔ g x ∈ c.support
k : ℤ
⊢ (∀ a ∈ c.support, (c ^ k) a = g a) ↔ ∀ (x : α), (c ^ k) x = (ofSubtype (g.subtypePerm ⋯)) x | 6e46ea6e7f64d962 |
aemeasurable_Ioi_of_forall_Ioc | Mathlib/MeasureTheory/Measure/AEMeasurable.lean | theorem aemeasurable_Ioi_of_forall_Ioc {β} {mβ : MeasurableSpace β} [LinearOrder α]
[(atTop : Filter α).IsCountablyGenerated] {x : α} {g : α → β}
(g_meas : ∀ t > x, AEMeasurable g (μ.restrict (Ioc x t))) :
AEMeasurable g (μ.restrict (Ioi x)) | case intro
α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
β : Type u_7
mβ : MeasurableSpace β
inst✝¹ : LinearOrder α
inst✝ : atTop.IsCountablyGenerated
x : α
g : α → β
g_meas : ∀ t > x, AEMeasurable g (μ.restrict (Ioc x t))
this : Nonempty α
u : ℕ → α
hu_tendsto : Tendsto u atTop atTop
Ioi_eq_iUnion : Ioi x = ⋃ n, I... | intro n | case intro
α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
β : Type u_7
mβ : MeasurableSpace β
inst✝¹ : LinearOrder α
inst✝ : atTop.IsCountablyGenerated
x : α
g : α → β
g_meas : ∀ t > x, AEMeasurable g (μ.restrict (Ioc x t))
this : Nonempty α
u : ℕ → α
hu_tendsto : Tendsto u atTop atTop
Ioi_eq_iUnion : Ioi x = ⋃ n, I... | 51ec0b905d628fc2 |
List.forIn'_loop_toArray | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/ToArray.lean | theorem forIn'_loop_toArray [Monad m] (l : List α) (f : (a : α) → a ∈ l.toArray → β → m (ForInStep β)) (i : Nat)
(h : i ≤ l.length) (b : β) :
Array.forIn'.loop l.toArray f i h b =
forIn' (l.drop (l.length - i)) b (fun a m b => f a (by simpa using mem_of_mem_drop m) b) | case succ
m : Type u_1 → Type u_2
α : Type u_3
β : Type u_1
inst✝ : Monad m
i : Nat
ih :
∀ (l : List α) (f : (a : α) → a ∈ l.toArray → β → m (ForInStep β)) (h : i ≤ l.length) (b : β),
Array.forIn'.loop l.toArray f i h b = forIn' (drop (l.length - i) l) b fun a m b => f a ⋯ b
l : List α
f : (a : α) → a ∈ l.toArray... | simp only [Array.forIn'.loop, size_toArray, getElem_toArray, ih] | case succ
m : Type u_1 → Type u_2
α : Type u_3
β : Type u_1
inst✝ : Monad m
i : Nat
ih :
∀ (l : List α) (f : (a : α) → a ∈ l.toArray → β → m (ForInStep β)) (h : i ≤ l.length) (b : β),
Array.forIn'.loop l.toArray f i h b = forIn' (drop (l.length - i) l) b fun a m b => f a ⋯ b
l : List α
f : (a : α) → a ∈ l.toArray... | d007f422d6d3b9ca |
CategoryTheory.FreeMonoidalCategory.normalizeIsoApp_eq | Mathlib/CategoryTheory/Monoidal/Free/Coherence.lean | theorem normalizeIsoApp_eq :
∀ (X : F C) (n : N C), normalizeIsoApp C X n = normalizeIsoApp' C X n.as
| of _, _ => rfl
| unit, _ => rfl
| tensor X Y, n => by
rw [normalizeIsoApp, normalizeIsoApp']
rw [normalizeIsoApp_eq X n]
rw [normalizeIsoApp_eq Y ⟨normalizeObj X n.as⟩]
rfl
| C : Type u
X Y : F C
n : (Discrete ∘ NormalMonoidalObject) C
⊢ normalizeIsoApp C (X.tensor Y) n = normalizeIsoApp' C (X.tensor Y) n.as | rw [normalizeIsoApp, normalizeIsoApp'] | C : Type u
X Y : F C
n : (Discrete ∘ NormalMonoidalObject) C
⊢ (α_ (inclusion.obj { as := n.as }) X Y).symm ≪≫
whiskerRightIso (normalizeIsoApp C X n) Y ≪≫ normalizeIsoApp C Y { as := X.normalizeObj n.as } =
(α_ (inclusionObj n.as) X Y).symm ≪≫
whiskerRightIso (normalizeIsoApp' C X n.as) Y ≪≫ normalizeI... | a4f3a9004ff4ba6a |
WeierstrassCurve.Affine.Point.toClass_eq_zero | Mathlib/AlgebraicGeometry/EllipticCurve/Group.lean | lemma toClass_eq_zero (P : W.Point) : toClass P = 0 ↔ P = 0 | case mpr
F : Type u
inst✝ : Field F
W : Affine F
P : W.Point
⊢ P = 0 → toClass P = 0 | exact congr_arg toClass | no goals | 110279bc1e8b3f89 |
LinearMap.BilinForm.dualSubmodule_span_of_basis | Mathlib/LinearAlgebra/BilinearForm/DualLattice.lean | lemma dualSubmodule_span_of_basis {ι} [Finite ι] [DecidableEq ι]
(hB : B.Nondegenerate) (b : Basis ι S M) :
B.dualSubmodule (Submodule.span R (Set.range b)) =
Submodule.span R (Set.range <| B.dualBasis hB b) | case intro.a.intro.intro
R : Type u_4
S : Type u_2
M : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : Field S
inst✝⁶ : AddCommGroup M
inst✝⁵ : Algebra R S
inst✝⁴ : Module R M
inst✝³ : Module S M
inst✝² : IsScalarTower R S M
B : BilinForm S M
ι : Type u_1
inst✝¹ : Finite ι
inst✝ : DecidableEq ι
hB : B.Nondegenerate
b : Basis ι S... | simp only [map_sum, map_smul] | case intro.a.intro.intro
R : Type u_4
S : Type u_2
M : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : Field S
inst✝⁶ : AddCommGroup M
inst✝⁵ : Algebra R S
inst✝⁴ : Module R M
inst✝³ : Module S M
inst✝² : IsScalarTower R S M
B : BilinForm S M
ι : Type u_1
inst✝¹ : Finite ι
inst✝ : DecidableEq ι
hB : B.Nondegenerate
b : Basis ι S... | 7056a00b0b948261 |
IsCompact.finite_of_discrete | Mathlib/Topology/Compactness/Compact.lean | theorem IsCompact.finite_of_discrete [DiscreteTopology X] (hs : IsCompact s) : s.Finite | X : Type u
inst✝¹ : TopologicalSpace X
s : Set X
inst✝ : DiscreteTopology X
hs : IsCompact s
this : ∀ (x : X), {x} ∈ 𝓝 x
⊢ s.Finite | rcases hs.elim_nhds_subcover (fun x => {x}) fun x _ => this x with ⟨t, _, hst⟩ | case intro.intro
X : Type u
inst✝¹ : TopologicalSpace X
s : Set X
inst✝ : DiscreteTopology X
hs : IsCompact s
this : ∀ (x : X), {x} ∈ 𝓝 x
t : Finset X
left✝ : ∀ x ∈ t, x ∈ s
hst : s ⊆ ⋃ x ∈ t, {x}
⊢ s.Finite | 9c90eb234e0887ea |
RCLike.nonUnitalContinuousFunctionalCalculus | Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Instances.lean | theorem RCLike.nonUnitalContinuousFunctionalCalculus :
NonUnitalContinuousFunctionalCalculus 𝕜 (p : A → Prop) where
predicate_zero | case map_spec
𝕜 : Type u_1
A : Type u_2
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace 𝕜 A
inst✝⁵ : IsScalarTower 𝕜 A A
inst✝⁴ : SMulCommClass 𝕜 A A
inst✝³ : StarModule 𝕜 A
p : A → Prop
p₁ : Unitization 𝕜 A → Prop
hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x
inst✝² : ContinuousFunction... | case map_spec =>
exact quasispectrum_eq_spectrum_inr' 𝕜 𝕜 (ψ f) ▸ coe_ψ _ ▸ spec_cfcₙAux hp₁ a ha f | case isStarNormal
𝕜 : Type u_1
A : Type u_2
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace 𝕜 A
inst✝⁵ : IsScalarTower 𝕜 A A
inst✝⁴ : SMulCommClass 𝕜 A A
inst✝³ : StarModule 𝕜 A
p : A → Prop
p₁ : Unitization 𝕜 A → Prop
hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x
inst✝² : ContinuousFunc... | 95b319408dc7d905 |
List.lex_eq_true_iff_exists | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lex.lean | theorem lex_eq_true_iff_exists [BEq α] (lt : α → α → Bool) :
lex l₁ l₂ lt = true ↔
(l₁.isEqv (l₂.take l₁.length) (· == ·) ∧ l₁.length < l₂.length) ∨
(∃ (i : Nat) (h₁ : i < l₁.length) (h₂ : i < l₂.length),
(∀ j, (hj : j < i) →
l₁[j]'(Nat.lt_trans hj h₁) == l₂[j]'(Nat.lt_trans hj h... | α : Type u_1
inst✝ : BEq α
lt : α → α → Bool
a : α
l₁ : List α
ih :
∀ {l₂ : List α},
l₁.lex l₂ lt = true ↔
(l₁.isEqv (take l₁.length l₂) fun x1 x2 => x1 == x2) = true ∧ l₁.length < l₂.length ∨
∃ i h₁ h₂, (∀ (j : Nat) (hj : j < i), (l₁[j] == l₂[j]) = true) ∧ lt l₁[i] l₂[i] = true
b : α
l₂ : List α
i ... | simpa using h₁ | no goals | 98cd2b86355fa73f |
OrderedFinpartition.one_lt_partSize_index_zero | Mathlib/Analysis/Calculus/ContDiff/FaaDiBruno.lean | /-- If the left-most part is not `{0}`, then the part containing `0` has at least two elements:
either because it's the left-most part, and then it's not just `0` by assumption, or because it's
not the left-most part and then, by increasingness of maximal elements in parts, it contains
a positive element. -/
lemma one_... | case h.e'_3
n : ℕ
c : OrderedFinpartition (n + 1)
hc : range (c.emb 0) ≠ {0}
this✝ : c.partSize (c.index 0) = Nat.card ↑(range (c.emb (c.index 0)))
h : c.index 0 ≠ 0
this : {0, c.emb (c.index 0) ⟨c.partSize (c.index 0) - 1, ⋯⟩} ⊆ range (c.emb (c.index 0))
⊢ 2 = {0, c.emb (c.index 0) ⟨c.partSize (c.index 0) - 1, ⋯⟩}.car... | apply (Finset.card_pair ?_).symm | n : ℕ
c : OrderedFinpartition (n + 1)
hc : range (c.emb 0) ≠ {0}
this✝ : c.partSize (c.index 0) = Nat.card ↑(range (c.emb (c.index 0)))
h : c.index 0 ≠ 0
this : {0, c.emb (c.index 0) ⟨c.partSize (c.index 0) - 1, ⋯⟩} ⊆ range (c.emb (c.index 0))
⊢ 0 ≠ c.emb (c.index 0) ⟨c.partSize (c.index 0) - 1, ⋯⟩ | 91216df49668e0d3 |
Fin.prod_univ_eq_prod_range | Mathlib/Data/Fintype/BigOperators.lean | theorem Fin.prod_univ_eq_prod_range [CommMonoid α] (f : ℕ → α) (n : ℕ) :
∏ i : Fin n, f i = ∏ i ∈ range n, f i :=
calc
∏ i : Fin n, f i = ∏ i : { x // x ∈ range n }, f i :=
Fintype.prod_equiv (Fin.equivSubtype.trans (Equiv.subtypeEquivRight (by simp))) _ _ (by simp)
_ = ∏ i ∈ range n, f i | α : Type u_1
inst✝ : CommMonoid α
f : ℕ → α
n : ℕ
⊢ ∀ (x : ℕ), x < n ↔ x ∈ range n | simp | no goals | c5e43c757054c2b9 |
Std.DHashMap.Internal.AssocList.toList_map | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/AssocList/Lemmas.lean | theorem toList_map {f : (a : α) → β a → γ a} {l : AssocList α β} :
Perm (l.map f).toList (l.toList.map fun p => ⟨p.1, f p.1 p.2⟩) | α : Type u
β : α → Type v
γ : α → Type w
f : (a : α) → β a → γ a
l : AssocList α β
this :
∀ (l : AssocList α γ) (l' : AssocList α β),
(map.go f l l').toList.Perm (l.toList ++ List.map (fun p => ⟨p.fst, f p.fst p.snd⟩) l'.toList)
⊢ (map.go f nil l).toList.Perm (List.map (fun p => ⟨p.fst, f p.fst p.snd⟩) l.toList) | simpa using this .nil l | no goals | e5fcdeb8a937c7c6 |
exists_le_isAssociatedPrime_of_isNoetherianRing | Mathlib/RingTheory/Ideal/AssociatedPrime.lean | theorem exists_le_isAssociatedPrime_of_isNoetherianRing [H : IsNoetherianRing R] (x : M)
(hx : x ≠ 0) : ∃ P : Ideal R, IsAssociatedPrime P M ∧ ker (toSpanSingleton R M x) ≤ P | case intro.intro.intro.intro.intro
R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
H : IsNoetherianRing R
x : M
hx : x ≠ 0
this : ker (toSpanSingleton R M x) ≠ ⊤
y : M
h₃ :
∀ I ∈ {P | ker (toSpanSingleton R M x) ≤ P ∧ P ≠ ⊤ ∧ ∃ y, P = ker (toSpanSingleton R M y)},
¬ker (to... | rwa [H₁.eq_of_not_lt (h₃ _ ⟨l.trans H₁, H₂, _, rfl⟩),
mem_ker, toSpanSingleton_apply, smul_comm, smul_smul] | no goals | 8ef56feb72080758 |
MeasureTheory.FiniteMeasure.map_snd_prod | Mathlib/MeasureTheory/Measure/FiniteMeasureProd.lean | @[simp] lemma map_snd_prod : (μ.prod ν).map Prod.snd = μ univ • ν | α : Type u_1
inst✝¹ : MeasurableSpace α
β : Type u_2
inst✝ : MeasurableSpace β
μ : FiniteMeasure α
ν : FiniteMeasure β
⊢ (μ.prod ν).map Prod.snd = μ univ • ν | ext | case h
α : Type u_1
inst✝¹ : MeasurableSpace α
β : Type u_2
inst✝ : MeasurableSpace β
μ : FiniteMeasure α
ν : FiniteMeasure β
s✝ : Set β
a✝ : MeasurableSet s✝
⊢ ↑((μ.prod ν).map Prod.snd) s✝ = ↑(μ univ • ν) s✝ | 3bbe173be6f43f12 |
MeasureTheory.tendsto_of_lintegral_tendsto_of_monotone | Mathlib/MeasureTheory/Integral/Lebesgue.lean | /-- If a monotone sequence of functions has an upper bound and the sequence of integrals of these
functions tends to the integral of the upper bound, then the sequence of functions converges
almost everywhere to the upper bound. -/
lemma tendsto_of_lintegral_tendsto_of_monotone {α : Type*} {mα : MeasurableSpace α}
... | case a
α : Type u_5
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hF_meas : AEMeasurable F μ
hf_tendsto : Tendsto (fun i => ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i => f i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
h_int_finite : ∫⁻ (a : α... | apply lintegral_mono_ae | case a.h
α : Type u_5
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hF_meas : AEMeasurable F μ
hf_tendsto : Tendsto (fun i => ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i => f i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
h_int_finite : ∫⁻ (a :... | 1cdd224ac57038c3 |
Real.Angle.two_zsmul_eq_iff | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | theorem two_zsmul_eq_iff {ψ θ : Angle} : (2 : ℤ) • ψ = (2 : ℤ) • θ ↔ ψ = θ ∨ ψ = θ + ↑π | ψ θ : Angle
this : Int.natAbs 2 = 2
⊢ 2 • ψ = 2 • θ ↔ ψ = θ ∨ ψ = θ + ↑π | rw [zsmul_eq_iff two_ne_zero, this, Fin.exists_fin_two, Fin.val_zero,
Fin.val_one, zero_smul, add_zero, one_smul, Int.cast_two,
mul_div_cancel_left₀ (_ : ℝ) two_ne_zero] | no goals | db0c51dccf7bc329 |
Option.casesOn'_eq_elim | Mathlib/Data/Option/Basic.lean | lemma casesOn'_eq_elim (b : β) (f : α → β) (a : Option α) :
Option.casesOn' a b f = Option.elim a b f | α : Type u_1
β : Type u_2
b : β
f : α → β
a : Option α
⊢ a.casesOn' b f = a.elim b f | cases a <;> rfl | no goals | 95b9b288786797d1 |
Alexandrov.isSheaf_principalsKanExtension | Mathlib/Topology/Sheaves/Alexandrov.lean | theorem isSheaf_principalsKanExtension
{X : TopCat.{v}} [Preorder X] [Topology.IsUpperSet X] (F : X ⥤ C) :
Presheaf.IsSheaf (principalsKanExtension F) | case val
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasLimits C
X : TopCat
inst✝¹ : Preorder ↑X
inst✝ : Topology.IsUpperSet ↑X
F : ↑X ⥤ C
ι : Type v
Us : ι → Opens ↑X
⊢ IsLimit ((principalsKanExtension F).mapCone (opensLeCoverCocone Us).op) | apply isLimit | no goals | a2ff4b2978172368 |
Complex.norm_mul_exp_arg_mul_I | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | theorem norm_mul_exp_arg_mul_I (x : ℂ) : ‖x‖ * exp (arg x * I) = x | x : ℂ
⊢ ↑‖x‖ * cexp (↑x.arg * I) = x | rcases eq_or_ne x 0 with (rfl | hx) | case inl
⊢ ↑‖0‖ * cexp (↑(arg 0) * I) = 0
case inr
x : ℂ
hx : x ≠ 0
⊢ ↑‖x‖ * cexp (↑x.arg * I) = x | 0fca563789ed4990 |
Nat.gcd_mul_right | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean | theorem gcd_mul_right (m n k : Nat) : gcd (m * n) (k * n) = gcd m k * n | m n k : Nat
⊢ (m * n).gcd (k * n) = m.gcd k * n | rw [Nat.mul_comm m n, Nat.mul_comm k n, Nat.mul_comm (gcd m k) n, gcd_mul_left] | no goals | e836cd9703c694da |
Polynomial.dickson_one_one_zmod_p | Mathlib/RingTheory/Polynomial/Dickson.lean | theorem dickson_one_one_zmod_p (p : ℕ) [Fact p.Prime] : dickson 1 (1 : ZMod p) p = X ^ p | p : ℕ
inst✝ : Fact (Nat.Prime p)
K : Type
w✝¹ : Field K
w✝ : CharP K p
H : Set.univ.Infinite
h : {x | ∃ y, x = y + y⁻¹ ∧ y ≠ 0}.Finite
this : Set.univ = ⋃ x ∈ {x | ∃ y, x = y + y⁻¹ ∧ y ≠ 0}, {y | x = y + y⁻¹ ∨ y = 0}
⊢ (⋃ x ∈ {x | ∃ y, x = y + y⁻¹ ∧ y ≠ 0}, {y | x = y + y⁻¹ ∨ y = 0}).Finite | clear this | p : ℕ
inst✝ : Fact (Nat.Prime p)
K : Type
w✝¹ : Field K
w✝ : CharP K p
H : Set.univ.Infinite
h : {x | ∃ y, x = y + y⁻¹ ∧ y ≠ 0}.Finite
⊢ (⋃ x ∈ {x | ∃ y, x = y + y⁻¹ ∧ y ≠ 0}, {y | x = y + y⁻¹ ∨ y = 0}).Finite | 84ee1673e653698a |
LowerSemicontinuousWithinAt.add' | Mathlib/Topology/Semicontinuous.lean | theorem LowerSemicontinuousWithinAt.add' {f g : α → γ} (hf : LowerSemicontinuousWithinAt f s x)
(hg : LowerSemicontinuousWithinAt g s x)
(hcont : ContinuousAt (fun p : γ × γ => p.1 + p.2) (f x, g x)) :
LowerSemicontinuousWithinAt (fun z => f z + g z) s x | case pos
α : Type u_1
inst✝³ : TopologicalSpace α
x : α
s : Set α
γ : Type u_4
inst✝² : LinearOrderedAddCommMonoid γ
inst✝¹ : TopologicalSpace γ
inst✝ : OrderTopology γ
f g : α → γ
hf : LowerSemicontinuousWithinAt f s x
hg : LowerSemicontinuousWithinAt g s x
hcont : ContinuousAt (fun p => p.1 + p.2) (f x, g x)
y : γ
hy... | obtain ⟨z₂, z₂lt, h₂⟩ : ∃ z₂ < g x, Ioc z₂ (g x) ⊆ v :=
exists_Ioc_subset_of_mem_nhds (v_open.mem_nhds xv) hx₂ | case pos.intro.intro
α : Type u_1
inst✝³ : TopologicalSpace α
x : α
s : Set α
γ : Type u_4
inst✝² : LinearOrderedAddCommMonoid γ
inst✝¹ : TopologicalSpace γ
inst✝ : OrderTopology γ
f g : α → γ
hf : LowerSemicontinuousWithinAt f s x
hg : LowerSemicontinuousWithinAt g s x
hcont : ContinuousAt (fun p => p.1 + p.2) (f x, g... | 4df49a8a802d230c |
ContinuousMap.induction_on | Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean | theorem ContinuousMap.induction_on {𝕜 : Type*} [RCLike 𝕜] {s : Set 𝕜}
{p : C(s, 𝕜) → Prop} (const : ∀ r, p (.const s r)) (id : p (.restrict s <| .id 𝕜))
(star_id : p (star (.restrict s <| .id 𝕜)))
(add : ∀ f g, p f → p g → p (f + g)) (mul : ∀ f g, p f → p g → p (f * g))
(closure : (∀ f ∈ (polynomi... | 𝕜 : Type u_1
inst✝ : RCLike 𝕜
s : Set 𝕜
p : C(↑s, 𝕜) → Prop
const : ∀ (r : 𝕜), p (ContinuousMap.const (↑s) r)
id : p (restrict s (ContinuousMap.id 𝕜))
star_id : p (star (restrict s (ContinuousMap.id 𝕜)))
add : ∀ (f g : C(↑s, 𝕜)), p f → p g → p (f + g)
mul : ∀ (f g : C(↑s, 𝕜)), p f → p g → p (f * g)
closure : (... | induction hf using Algebra.adjoin_induction with
| mem f hf =>
simp only [Set.mem_union, Set.mem_singleton_iff, Set.mem_star] at hf
rw [star_eq_iff_star_eq, eq_comm (b := f)] at hf
obtain (rfl | rfl) := hf
all_goals simpa only [toContinuousMapOnAlgHom_apply, toContinuousMapOn_X_eq_restrict_id]
| algebraMap r =>... | no goals | 01b202580110cbda |
IsClosed.isClopenable | Mathlib/Topology/MetricSpace/Polish.lean | theorem _root_.IsClosed.isClopenable [TopologicalSpace α] [PolishSpace α] {s : Set α}
(hs : IsClosed s) : IsClopenable s | case refine_2
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : PolishSpace α
s : Set α
hs : IsClosed s
this✝ : PolishSpace ↑s
t : Set α := sᶜ
this : PolishSpace ↑t
f : ↑s ⊕ ↑t ≃ α := Equiv.Set.sumCompl s
hle : coinduced (⇑f) instTopologicalSpaceSum ≤ inst✝¹
inl : ∀ (x : ↑s), (Equiv.Set.sumCompl s) (Sum.inl x) = ↑x
inr ... | simp only [isOpen_univ, true_and] | case refine_2
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : PolishSpace α
s : Set α
hs : IsClosed s
this✝ : PolishSpace ↑s
t : Set α := sᶜ
this : PolishSpace ↑t
f : ↑s ⊕ ↑t ≃ α := Equiv.Set.sumCompl s
hle : coinduced (⇑f) instTopologicalSpaceSum ≤ inst✝¹
inl : ∀ (x : ↑s), (Equiv.Set.sumCompl s) (Sum.inl x) = ↑x
inr ... | 86268e182c814247 |
exists_continuous_one_zero_of_isCompact | Mathlib/Topology/UrysohnsLemma.lean | theorem exists_continuous_one_zero_of_isCompact [RegularSpace X] [LocallyCompactSpace X]
{s t : Set X} (hs : IsCompact s) (ht : IsClosed t) (hd : Disjoint s t) :
∃ f : C(X, ℝ), EqOn f 1 s ∧ EqOn f 0 t ∧ HasCompactSupport f ∧ ∀ x, f x ∈ Icc (0 : ℝ) 1 | X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : LocallyCompactSpace X
s t : Set X
hs : IsCompact s
ht : IsClosed t
hd : Disjoint s t
k : Set X
k_comp : IsCompact k
k_closed : IsClosed k
sk : s ⊆ interior k
kt : k ⊆ tᶜ
f : X → ℝ
hf : Continuous f
hfs : EqOn (⇑{ toFun := f, continuous_toFun := hf... | simpa using hfs hx | no goals | d39f921b2dca6dfe |
MeasureTheory.LevyProkhorov.continuous_equiv_symm_probabilityMeasure | Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean | lemma LevyProkhorov.continuous_equiv_symm_probabilityMeasure :
Continuous (LevyProkhorov.equiv (α := ProbabilityMeasure Ω)).symm | Ω : Type u_1
inst✝³ : PseudoMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
inst✝ : SeparableSpace Ω
P : ProbabilityMeasure Ω
ε : ℝ
ε_pos : ε > 0
third_ε_pos : 0 < ε / 3
third_ε_pos' : 0 < ENNReal.ofReal (ε / 3)
Es : ℕ → Set Ω
Es_mble : ∀ (n : ℕ), MeasurableSet (Es n)
Es_bdd : ∀ (n : ℕ), Bornol... | obtain ⟨i, i_large, ω_in_Esi⟩ := hω | case intro.intro
Ω : Type u_1
inst✝³ : PseudoMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
inst✝ : SeparableSpace Ω
P : ProbabilityMeasure Ω
ε : ℝ
ε_pos : ε > 0
third_ε_pos : 0 < ε / 3
third_ε_pos' : 0 < ENNReal.ofReal (ε / 3)
Es : ℕ → Set Ω
Es_mble : ∀ (n : ℕ), MeasurableSet (Es n)
Es_bdd : ... | 1196bd5a9632bf03 |
LucasLehmer.norm_num_ext.sModNatTR_eq_sModNat | Mathlib/NumberTheory/LucasLehmer.lean | theorem sModNatTR_eq_sModNat (q : ℕ) (i : ℕ) : sModNatTR q i = sModNat q i | q i : ℕ
⊢ sModNatTR q i = sModNat q i | rw [sModNatTR, helper, sModNat_aux_eq] | no goals | a039f2f8f7c70cf9 |
Equiv.Perm.support_swap_mul_eq | Mathlib/GroupTheory/Perm/Support.lean | theorem support_swap_mul_eq (f : Perm α) (x : α) (h : f (f x) ≠ x) :
(swap x (f x) * f).support = f.support \ {x} | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f : Perm α
x : α
h : f (f x) ≠ x
hx : ¬f x = x
z : α
hzx : ¬z = x
hzf : z = f x
⊢ z ∈ (swap x (f x) * f).support ↔ z ∈ f.support \ {x} | simp [hzf, hx, h, swap_apply_of_ne_of_ne] | no goals | b3af2efcbf0301e0 |
Finset.update_piecewise_of_not_mem | Mathlib/Data/Finset/Piecewise.lean | lemma update_piecewise_of_not_mem [DecidableEq ι] {i : ι} (hi : i ∉ s) (v : π i) :
update (s.piecewise f g) i v = s.piecewise f (update g i v) | ι : Type u_1
π : ι → Sort u_2
s : Finset ι
f g : (i : ι) → π i
inst✝¹ : (j : ι) → Decidable (j ∈ s)
inst✝ : DecidableEq ι
i : ι
hi : i ∉ s
v : π i
j : ι
hj : j ∈ s
⊢ j ≠ i | exact fun h => hi (h ▸ hj) | no goals | 21cc099f8c434712 |
Array.append_eq_empty_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean | theorem append_eq_empty_iff : p ++ q = #[] ↔ p = #[] ∧ q = #[] | α✝ : Type u_1
p q : Array α✝
⊢ p ++ q = #[] ↔ p = #[] ∧ q = #[] | cases p <;> simp | no goals | 8a0f10d280c8f287 |
LieAlgebra.engel_isBot_of_isMin.lieCharpoly_map_eval | Mathlib/Algebra/Lie/CartanExists.lean | lemma lieCharpoly_map_eval (r : R) :
(lieCharpoly R M x y).map (evalRingHom r) = (φ (r • y + x)).charpoly | R : Type u_2
L : Type u_3
M : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : LieRing L
inst✝⁸ : LieAlgebra R L
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : LieRingModule L M
inst✝⁴ : LieModule R L M
inst✝³ : Module.Finite R L
inst✝² : Free R L
inst✝¹ : Module.Finite R M
inst✝ : Free R M
x y : L
r : R
⊢ Polynomial.map (... | rw [lieCharpoly, map_map] | R : Type u_2
L : Type u_3
M : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : LieRing L
inst✝⁸ : LieAlgebra R L
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : LieRingModule L M
inst✝⁴ : LieModule R L M
inst✝³ : Module.Finite R L
inst✝² : Free R L
inst✝¹ : Module.Finite R M
inst✝ : Free R M
x y : L
r : R
⊢ Polynomial.map
... | 26152cf20c663719 |
BitVec.sshiftRightRec_eq | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Bitblast.lean | theorem sshiftRightRec_eq (x : BitVec w₁) (y : BitVec w₂) (n : Nat) :
sshiftRightRec x y n = x.sshiftRight' ((y.setWidth (n + 1)).setWidth w₂) | w₁ w₂ n : Nat
ih : ∀ (x : BitVec w₁) (y : BitVec w₂), x.sshiftRightRec y n = x.sshiftRight' (setWidth w₂ (setWidth (n + 1) y))
x : BitVec w₁
y : BitVec w₂
h : ¬y.getLsbD (n + 1) = true
⊢ y.getLsbD (n + 1) = false | simp [h] | no goals | b0ce95334bc54cbb |
Submodule.mem_sSup_of_mem | Mathlib/Algebra/Module/Submodule/Lattice.lean | theorem mem_sSup_of_mem {S : Set (Submodule R M)} {s : Submodule R M} (hs : s ∈ S) :
∀ {x : M}, x ∈ s → x ∈ sSup S | R : Type u_1
M : Type u_3
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
S : Set (Submodule R M)
s : Submodule R M
hs : s ∈ S
⊢ ∀ {x : M}, x ∈ s → x ∈ sSup S | have := le_sSup hs | R : Type u_1
M : Type u_3
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
S : Set (Submodule R M)
s : Submodule R M
hs : s ∈ S
this : s ≤ sSup S
⊢ ∀ {x : M}, x ∈ s → x ∈ sSup S | 97d38de760144cae |
Asymptotics.isBigOWith_const_const | Mathlib/Analysis/Asymptotics/Defs.lean | theorem isBigOWith_const_const (c : E) {c' : F''} (hc' : c' ≠ 0) (l : Filter α) :
IsBigOWith (‖c‖ / ‖c'‖) l (fun _x : α => c) fun _x => c' | case h
α : Type u_1
E : Type u_3
F'' : Type u_10
inst✝¹ : Norm E
inst✝ : NormedAddCommGroup F''
c : E
c' : F''
hc' : c' ≠ 0
l : Filter α
⊢ ∀ (a : α), a ∈ {x | (fun x => ‖c‖ ≤ ‖c‖ / ‖c'‖ * ‖c'‖) x} | intro x | case h
α : Type u_1
E : Type u_3
F'' : Type u_10
inst✝¹ : Norm E
inst✝ : NormedAddCommGroup F''
c : E
c' : F''
hc' : c' ≠ 0
l : Filter α
x : α
⊢ x ∈ {x | (fun x => ‖c‖ ≤ ‖c‖ / ‖c'‖ * ‖c'‖) x} | 3e498fa7ccd81cc8 |
Finset.upShadow_compls | Mathlib/Combinatorics/SetFamily/Shadow.lean | @[simp] lemma upShadow_compls : ∂⁺ 𝒜ᶜˢ = (∂ 𝒜)ᶜˢ | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
s : Finset α
⊢ (∃ s_1, s_1ᶜ ∈ 𝒜 ∧ ∃ a ∉ s_1, insert a s_1 = s) ↔ ∃ s_1 ∈ 𝒜, ∃ a ∈ s_1, s_1.erase a = sᶜ | refine (compl_involutive.toPerm _).exists_congr_left.trans ?_ | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
𝒜 : Finset (Finset α)
s : Finset α
⊢ (∃ b,
((Equiv.symm (Function.Involutive.toPerm compl ⋯)) b)ᶜ ∈ 𝒜 ∧
∃ a ∉ (Equiv.symm (Function.Involutive.toPerm compl ⋯)) b,
insert a ((Equiv.symm (Function.Involutive.toPerm compl ⋯)) b) = s) ↔
... | 11f651cd81197b17 |
CategoryTheory.regularTopology.parallelPair_pullback_initial | Mathlib/CategoryTheory/Sites/Coherent/RegularSheaves.lean | theorem parallelPair_pullback_initial {X B : C} (π : X ⟶ B)
(c : PullbackCone π π) (hc : IsLimit c) :
(parallelPair (C := (Sieve.ofArrows (fun (_ : Unit) => X) (fun _ => π)).arrows.categoryᵒᵖ)
(Y := op ((Presieve.categoryMk _ (c.fst ≫ π) ⟨_, c.fst, π, ofArrows.mk (), rfl⟩)))
(X := op ((Presieve.category... | case h₂
C : Type u_1
inst✝ : Category.{u_4, u_1} C
X B : C
π : X ⟶ B
c : PullbackCone π π
hc : IsLimit c
Z : (Sieve.ofArrows (fun x => X) fun x => π).arrows.category
i j : unop (op Z) ⟶ unop (op ((Sieve.ofArrows (fun x => X) fun x => π).arrows.categoryMk π ⋯))
ij : (unop (op Z)).obj.left ⟶ c.pt := PullbackCone.IsLimit.... | refine ⟨Quiver.Hom.op (Over.homMk ij (by simpa [ij] using i.w)), ?_, ?_⟩ | case h₂.refine_1
C : Type u_1
inst✝ : Category.{u_4, u_1} C
X B : C
π : X ⟶ B
c : PullbackCone π π
hc : IsLimit c
Z : (Sieve.ofArrows (fun x => X) fun x => π).arrows.category
i j : unop (op Z) ⟶ unop (op ((Sieve.ofArrows (fun x => X) fun x => π).arrows.categoryMk π ⋯))
ij : (unop (op Z)).obj.left ⟶ c.pt := PullbackCone... | d548d8c18bc207b1 |
ProbabilityTheory.Kernel.measurable_kernel_prod_mk_left_of_finite | Mathlib/Probability/Kernel/MeasurableLIntegral.lean | theorem measurable_kernel_prod_mk_left_of_finite {t : Set (α × β)} (ht : MeasurableSet t)
(hκs : ∀ a, IsFiniteMeasure (κ a)) : Measurable fun a => κ a (Prod.mk a ⁻¹' t) | case basic.intro.intro.intro.intro
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
t : Set (α × β)
hκs : ∀ (a : α), IsFiniteMeasure (κ a)
t₁ : Set α
ht₁ : MeasurableSet t₁
t₂ : Set β
ht₂ : MeasurableSet t₂
⊢ Measurable fun a => (κ a) (Prod.mk a ⁻¹' t₁ ×ˢ t₂) | simp_rw [mk_preimage_prod_right_eq_if] | case basic.intro.intro.intro.intro
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
t : Set (α × β)
hκs : ∀ (a : α), IsFiniteMeasure (κ a)
t₁ : Set α
ht₁ : MeasurableSet t₁
t₂ : Set β
ht₂ : MeasurableSet t₂
⊢ Measurable fun a => (κ a) (if a ∈ t₁ then t₂ else ∅) | 4c14e8ca6e5d1721 |
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