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 |
|---|---|---|---|---|---|---|
HomogeneousLocalization.Away.eventually_smul_mem | Mathlib/RingTheory/GradedAlgebra/HomogeneousLocalization.lean | theorem Away.eventually_smul_mem {m} (hf : f ∈ 𝒜 m) (z : Away 𝒜 f) :
∀ᶠ n in Filter.atTop, f ^ n • z.val ∈ algebraMap _ _ '' (𝒜 (n • m) : Set A) | case neg
ι : Type u_1
R : Type u_2
A : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
𝒜 : ι → Submodule R A
f : A
inst✝² : AddCommMonoid ι
inst✝¹ : DecidableEq ι
inst✝ : GradedAlgebra 𝒜
m : ι
hf : f ∈ 𝒜 m
z : Away 𝒜 f
k : ℕ
hk : f ^ k = den z
k' : ℕ
hk' : k ≤ k'
hfk : ¬f ^ k = 0
⊢ ∃ x ∈ 𝒜 ((... | rw [← smul_eq_mul, add_smul,
DirectSum.degree_eq_of_mem_mem 𝒜 (SetLike.pow_mem_graded _ hf) (hk.symm ▸ z.den_mem_deg) hfk] | case neg
ι : Type u_1
R : Type u_2
A : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
𝒜 : ι → Submodule R A
f : A
inst✝² : AddCommMonoid ι
inst✝¹ : DecidableEq ι
inst✝ : GradedAlgebra 𝒜
m : ι
hf : f ∈ 𝒜 m
z : Away 𝒜 f
k : ℕ
hk : f ^ k = den z
k' : ℕ
hk' : k ≤ k'
hfk : ¬f ^ k = 0
⊢ ∃ x ∈ 𝒜 ((... | 856eec757b75e82a |
MeasureTheory.Filtration.filtrationOfSet_eq_natural | Mathlib/Probability/Process/Filtration.lean | theorem filtrationOfSet_eq_natural [MulZeroOneClass β] [Nontrivial β] {s : ι → Set Ω}
(hsm : ∀ i, MeasurableSet[m] (s i)) :
filtrationOfSet hsm = natural (fun i => (s i).indicator (fun _ => 1 : Ω → β)) fun i =>
stronglyMeasurable_one.indicator (hsm i) | Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁵ : TopologicalSpace β
inst✝⁴ : MetrizableSpace β
mβ : MeasurableSpace β
inst✝³ : BorelSpace β
inst✝² : Preorder ι
inst✝¹ : MulZeroOneClass β
inst✝ : Nontrivial β
s : ι → Set Ω
hsm : ∀ (i : ι), MeasurableSet (s i)
⊢ filtrationOfSet hsm = natural (fun i =... | simp only [filtrationOfSet, natural, measurableSpace_iSup_eq, exists_prop, mk.injEq] | Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁵ : TopologicalSpace β
inst✝⁴ : MetrizableSpace β
mβ : MeasurableSpace β
inst✝³ : BorelSpace β
inst✝² : Preorder ι
inst✝¹ : MulZeroOneClass β
inst✝ : Nontrivial β
s : ι → Set Ω
hsm : ∀ (i : ι), MeasurableSet (s i)
⊢ (fun i => MeasurableSpace.generateFrom... | 7ffbace60117795e |
MeasureTheory.hitting_mono | Mathlib/Probability/Process/HittingTime.lean | theorem hitting_mono {m₁ m₂ : ι} (hm : m₁ ≤ m₂) : hitting u s n m₁ ω ≤ hitting u s n m₂ ω | case pos.intro.intro.intro.intro
Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n : ι
ω : Ω
m₁ m₂ : ι
hm : m₁ ≤ m₂
h : ¬∃ j ∈ Set.Icc n m₁, u j ω ∈ s
j : ι
hj₁ : j ∈ Set.Icc n m₂
hj₂ : u j ω ∈ s
i : ι
hi₁ : i ∈ Set.Icc n m₂ ∩ {i | u i ω ∈ s}
hi₂ : i < m₁
⊢ Fals... | exact h ⟨i, ⟨hi₁.1.1, hi₂.le⟩, hi₁.2⟩ | no goals | 4352e36fcea59599 |
Int.le_of_add_le_add_left | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/Order.lean | theorem le_of_add_le_add_left {a b c : Int} (h : a + b ≤ a + c) : b ≤ c | a b c : Int
h : a + b ≤ a + c
⊢ b ≤ c | have : -a + (a + b) ≤ -a + (a + c) := Int.add_le_add_left h _ | a b c : Int
h : a + b ≤ a + c
this : -a + (a + b) ≤ -a + (a + c)
⊢ b ≤ c | b10c042dc3cb3efd |
Algebra.Transcendental.rank_eq_cardinalMk | Mathlib/RingTheory/AlgebraicIndependent/RankAndCardinality.lean | theorem Algebra.Transcendental.rank_eq_cardinalMk
(F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E] [Algebra.Transcendental F E] :
Module.rank F E = #E | case intro.intro
F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Algebra.Transcendental F E
ι : Type v
x : ι → E
hx : IsTranscendenceBasis F x
this : Nonempty ι
⊢ Module.rank F E = #E | simpa [← hx.lift_cardinalMk_eq_max_lift] using hx.lift_rank_eq_max_lift | no goals | 07f837883b0d32bf |
InnerProductSpaceable.real_prop | Mathlib/Analysis/InnerProductSpace/OfNorm.lean | theorem real_prop (r : ℝ) : innerProp' E (r : 𝕜) | 𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : InnerProductSpaceable E
r : ℝ
x y : E
⊢ inner_ 𝕜 (↑r • x) y = (starRingEnd 𝕜) ↑r * inner_ 𝕜 x y | revert r | 𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : InnerProductSpaceable E
x y : E
⊢ ∀ (r : ℝ), inner_ 𝕜 (↑r • x) y = (starRingEnd 𝕜) ↑r * inner_ 𝕜 x y | 24df77eefe369f5b |
Asymptotics.isBigO_iff_isBoundedUnder_le_div | Mathlib/Analysis/Asymptotics/Lemmas.lean | theorem isBigO_iff_isBoundedUnder_le_div (h : ∀ᶠ x in l, g'' x ≠ 0) :
f =O[l] g'' ↔ IsBoundedUnder (· ≤ ·) l fun x => ‖f x‖ / ‖g'' x‖ | α : Type u_1
E : Type u_3
F'' : Type u_10
inst✝¹ : Norm E
inst✝ : NormedAddCommGroup F''
f : α → E
g'' : α → F''
l : Filter α
h : ∀ᶠ (x : α) in l, g'' x ≠ 0
⊢ (∃ c, ∀ᶠ (x : α) in l, ‖f x‖ ≤ c * ‖g'' x‖) ↔ ∃ b, ∀ᶠ (a : α) in l, ‖f a‖ / ‖g'' a‖ ≤ b | exact
exists_congr fun c =>
eventually_congr <| h.mono fun x hx => (div_le_iff₀ <| norm_pos_iff.2 hx).symm | no goals | 81eaf763ade639e4 |
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_1
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 (Genera... | intro a p | case intro.intro.refine_1
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 (Genera... | 4742e81f46ca4ae4 |
Equiv.Perm.ofSubtype_swap_eq | Mathlib/GroupTheory/Perm/Support.lean | theorem ofSubtype_swap_eq {p : α → Prop} [DecidablePred p] (x y : Subtype p) :
ofSubtype (Equiv.swap x y) = Equiv.swap ↑x ↑y :=
Equiv.ext fun z => by
by_cases hz : p z
· rw [swap_apply_def, ofSubtype_apply_of_mem _ hz]
split_ifs with hzx hzy
· simp_rw [hzx, Subtype.coe_eta, swap_apply_left]
... | case pos
α : Type u_1
inst✝¹ : DecidableEq α
p : α → Prop
inst✝ : DecidablePred p
x y : Subtype p
z : α
hz : p z
hzx : ¬z = ↑x
hzy : z = ↑y
⊢ ↑((swap x y) ⟨z, hz⟩) = ↑x | simp_rw [hzy, Subtype.coe_eta, swap_apply_right] | no goals | 7c967a4cc58d1491 |
Vector.extract_size | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Lemmas.lean | theorem extract_size (as : Vector α n) : as.extract 0 n = as.cast (by simp) | case mk
α : Type u_1
as : Array α
⊢ { toArray := as, size_toArray := ⋯ }.extract = Vector.cast ⋯ { toArray := as, size_toArray := ⋯ } | simp | no goals | e2d5691232de6760 |
tendsto_integral_mulExpNegMulSq_comp | Mathlib/Analysis/SpecialFunctions/MulExpNegMulSqIntegral.lean | theorem tendsto_integral_mulExpNegMulSq_comp (g : E →ᵇ ℝ) :
Tendsto (fun ε => ∫ x, mulExpNegMulSq ε (g x) ∂P) (𝓝[>] 0) (𝓝 (∫ x, g x ∂P)) | case h
E : Type u_1
inst✝³ : TopologicalSpace E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
P : Measure E
inst✝ : IsFiniteMeasure P
g : E →ᵇ ℝ
u : ℕ → ℝ
hu : Tendsto u atTop (𝓝[>] 0)
N : ℕ
hupos : ∀ b ≥ N, u b ∈ Set.Ioi 0
⊢ ∃ a, ∀ b ≥ a, ∀ᵐ (ω : E) ∂P, ‖(u b).mulExpNegMulSq (g ω)‖ ≤ ‖g‖ | use N | case h
E : Type u_1
inst✝³ : TopologicalSpace E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
P : Measure E
inst✝ : IsFiniteMeasure P
g : E →ᵇ ℝ
u : ℕ → ℝ
hu : Tendsto u atTop (𝓝[>] 0)
N : ℕ
hupos : ∀ b ≥ N, u b ∈ Set.Ioi 0
⊢ ∀ b ≥ N, ∀ᵐ (ω : E) ∂P, ‖(u b).mulExpNegMulSq (g ω)‖ ≤ ‖g‖ | 9677e28f00b7be5e |
quotient_nhd_basis | Mathlib/Analysis/Normed/Group/Quotient.lean | theorem quotient_nhd_basis (S : AddSubgroup M) :
(𝓝 (0 : M ⧸ S)).HasBasis (fun ε ↦ 0 < ε) fun ε ↦ { x | ‖x‖ < ε } | M : Type u_1
inst✝ : SeminormedAddCommGroup M
S : AddSubgroup M
⊢ ∀ (ε : ℝ), mk '' ball 0 ε = {x | ‖x‖ < ε} | refine fun ε ↦ Set.ext <| forall_mk.2 fun x ↦ ?_ | M : Type u_1
inst✝ : SeminormedAddCommGroup M
S : AddSubgroup M
ε : ℝ
x : M
⊢ ↑x ∈ mk '' ball 0 ε ↔ ↑x ∈ {x | ‖x‖ < ε} | 9de1111f9e57bce3 |
Nat.not_exists_sq | Mathlib/Data/Nat/Sqrt.lean | /-- There are no perfect squares strictly between m² and (m+1)² -/
lemma not_exists_sq (hl : m * m < n) (hr : n < (m + 1) * (m + 1)) : ¬∃ t, t * t = n | m n : ℕ
hl : m * m < n
hr : n < (m + 1) * (m + 1)
⊢ ¬∃ t, t * t = n | rintro ⟨t, rfl⟩ | case intro
m t : ℕ
hl : m * m < t * t
hr : t * t < (m + 1) * (m + 1)
⊢ False | c0988c9a124cf33a |
MeasureTheory.addHaar_image_le_lintegral_abs_det_fderiv_aux1 | Mathlib/MeasureTheory/Function/Jacobian.lean | theorem addHaar_image_le_lintegral_abs_det_fderiv_aux1 (hs : MeasurableSet s)
(hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) {ε : ℝ≥0} (εpos : 0 < ε) :
μ (f '' s) ≤ (∫⁻ x in s, ENNReal.ofReal |(f' x).det| ∂μ) + 2 * ε * μ s | case e_a
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
ε : ℝ≥0
εpos : 0... | abel | no goals | d569d7f4f24d982c |
Real.ediam_Icc | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | theorem ediam_Icc (a b : ℝ) : EMetric.diam (Icc a b) = ENNReal.ofReal (b - a) | case inr
a b : ℝ
h : b < a
⊢ EMetric.diam (Icc a b) = ENNReal.ofReal (b - a) | simp [h, h.le] | no goals | 79ca1c3df5e45d2d |
Bimod.AssociatorBimod.hom_inv_id | Mathlib/CategoryTheory/Monoidal/Bimod.lean | theorem hom_inv_id : hom P Q L ≫ inv P Q L = 𝟙 _ | case 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)
R S T U : Mon_ C
P : Bimod R S
Q : Bimod S T
L : Bimo... | slice_lhs 1 2 => rw [coequalizer.π_desc] | case 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)
R S T U : Mon_ C
P : Bimod R S
Q : Bimod S T
L : Bimo... | 4613b3e3da3fca9a |
CategoryTheory.PreGaloisCategory.comp_autMap_apply | Mathlib/CategoryTheory/Galois/GaloisObjects.lean | @[simp]
lemma comp_autMap_apply (F : C ⥤ FintypeCat.{w}) {A B : C} [IsConnected A] [IsGalois B]
(f : A ⟶ B) (σ : Aut A) (a : F.obj A) :
F.map (autMap f σ).hom (F.map f a) = F.map f (F.map σ.hom a) | C : Type u₁
inst✝³ : Category.{u₂, u₁} C
inst✝² : GaloisCategory C
F : C ⥤ FintypeCat
A B : C
inst✝¹ : IsConnected A
inst✝ : IsGalois B
f : A ⟶ B
σ : Aut A
a : (F.obj A).carrier
⊢ F.map (autMap f σ).hom (F.map f a) = F.map f (F.map σ.hom a) | simpa [-comp_autMap] using congrFun (F.congr_map (comp_autMap f σ)) a | no goals | c6b2ddd1dbe9e3c5 |
ProbabilityTheory.Kernel.iIndepFun_iff_measure_inter_preimage_eq_mul | Mathlib/Probability/Independence/Kernel.lean | theorem iIndepFun_iff_measure_inter_preimage_eq_mul {ι : Type*} {β : ι → Type*}
(m : ∀ x, MeasurableSpace (β x)) (f : ∀ i, Ω → β i) :
iIndepFun m f κ μ ↔
∀ (S : Finset ι) {sets : ∀ i : ι, Set (β i)} (_H : ∀ i, i ∈ S → MeasurableSet[m i] (sets i)),
∀ᵐ a ∂μ, κ a (⋂ i ∈ S, (f i) ⁻¹' (sets i)) = ∏ i ∈... | α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
ι : Type u_8
β : ι → Type u_9
m : (x : ι) → MeasurableSpace (β x)
f : (i : ι) → Ω → β i
h :
∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
(∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻... | exact (h_meas i hi_mem).choose_spec.1 | no goals | 7edd4dfb8fab9f13 |
MeasureTheory.ae_le_of_forall_setLIntegral_le_of_sigmaFinite₀ | Mathlib/MeasureTheory/Function/AEEqOfLIntegral.lean | theorem ae_le_of_forall_setLIntegral_le_of_sigmaFinite₀ [SigmaFinite μ]
{f g : α → ℝ≥0∞} (hf : AEMeasurable f μ)
(h : ∀ s, MeasurableSet s → μ s < ∞ → ∫⁻ x in s, f x ∂μ ≤ ∫⁻ x in s, g x ∂μ) :
f ≤ᵐ[μ] g | α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
h : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in s, g x ∂μ
A : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0
u : ℕ → ℝ≥0
left✝ : ... | simp only [μs, tsum_zero] | no goals | acef02babd9dc977 |
FirstOrder.Language.Formula.realize_iExs | Mathlib/ModelTheory/Semantics.lean | theorem _root_.FirstOrder.Language.Formula.realize_iExs
[Finite γ] {φ : L.Formula (α ⊕ γ)} {v : α → M} : (φ.iExs γ).Realize v ↔
∃ (i : γ → M), φ.Realize (Sum.elim v i) | L : Language
M : Type w
inst✝¹ : L.Structure M
α : Type u'
γ : Type u_3
inst✝ : Finite γ
φ : L.Formula (α ⊕ γ)
v : α → M
e : γ ≃ Fin (Classical.choose ⋯) := Classical.choice ⋯
x✝ : Fin (Classical.choose ⋯) → M
⊢ (fun v => v ∘ ⇑e.symm) ((fun v => v ∘ ⇑e) x✝) = x✝ | simp [Function.comp_def] | no goals | edaad082a0094423 |
integral_le_sum_mul_Ico_of_antitone_monotone | Mathlib/Analysis/SumIntegralComparisons.lean | lemma integral_le_sum_mul_Ico_of_antitone_monotone
(hab : a ≤ b) (hf : AntitoneOn f (Icc a b)) (hg : MonotoneOn g (Icc (a - 1) (b - 1)))
(fpos : 0 ≤ f b) (gpos : 0 ≤ g (a - 1)) :
∫ x in a..b, f x * g (x - 1) ≤ ∑ i ∈ Finset.Ico a b, f i * g i | case hg
a b : ℕ
f g : ℝ → ℝ
hab : a ≤ b
hf : AntitoneOn f (Icc ↑a ↑b)
hg : MonotoneOn g (Icc (↑a - 1) (↑b - 1))
fpos : 0 ≤ f ↑b
gpos : 0 ≤ g (↑a - 1)
⊢ IntegrableOn (fun x => f x * g (x - 1)) (Ico ↑a ↑b) volume | apply Integrable.mono_measure _ (Measure.restrict_mono_set _ Ico_subset_Icc_self) | a b : ℕ
f g : ℝ → ℝ
hab : a ≤ b
hf : AntitoneOn f (Icc ↑a ↑b)
hg : MonotoneOn g (Icc (↑a - 1) (↑b - 1))
fpos : 0 ≤ f ↑b
gpos : 0 ≤ g (↑a - 1)
⊢ Integrable (fun x => f x * g (x - 1)) (volume.restrict (Icc ↑a ↑b)) | 53aa89cbac366b1c |
ENNReal.pow_eq_top_iff | Mathlib/Data/ENNReal/Order.lean | theorem pow_eq_top_iff {n : ℕ} : a ^ n = ∞ ↔ a = ∞ ∧ n ≠ 0 | case inl
a : ℝ≥0∞
⊢ a ^ 0 = ⊤ ↔ a = ⊤ ∧ 0 ≠ 0 | simp | no goals | c18f975dfad5151a |
ProbabilityTheory.tsum_prob_mem_Ioi_lt_top | Mathlib/Probability/StrongLaw.lean | theorem tsum_prob_mem_Ioi_lt_top {X : Ω → ℝ} (hint : Integrable X) (hnonneg : 0 ≤ X) :
(∑' j : ℕ, ℙ {ω | X ω ∈ Set.Ioi (j : ℝ)}) < ∞ | case h
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : Ω → ℝ
hint : Integrable X ℙ
hnonneg : 0 ≤ X
K : ℕ
A : Tendsto (fun N => ∑ j ∈ range K, ℙ {ω | X ω ∈ Set.Ioc ↑j ↑N}) atTop (𝓝 (∑ j ∈ range K, ℙ {ω | X ω ∈ Set.Ioi ↑j}))
N : ℕ
hN : N ∈ Set.Ici K
⊢ ∑ j ∈ range K, ℙ {ω | X ω ∈ Set.Ioc ↑j ↑N} ≤ ... | exact sum_prob_mem_Ioc_le hint hnonneg hN | no goals | 104c70daf554c26d |
FreeGroup.reduce.min | Mathlib/GroupTheory/FreeGroup/Reduce.lean | theorem reduce.min (H : Red (reduce L₁) L₂) : reduce L₁ = L₂ | case tail
α : Type u_1
L₁ L₂ : List (α × Bool)
inst✝ : DecidableEq α
L1 L' : List (α × Bool)
L2 : Relation.ReflTransGen Red.Step (reduce L₁) L1
H1 : Red.Step L1 L'
H2 : reduce L₁ = L1
⊢ reduce L₁ = L' | obtain ⟨L4, L5, x, b⟩ := H1 | case tail.not
α : Type u_1
L₁ L₂ : List (α × Bool)
inst✝ : DecidableEq α
L₁✝ L₂✝ : List (α × Bool)
x✝ : α
b✝ : Bool
L2 : Relation.ReflTransGen Red.Step (reduce L₁) (L₁✝ ++ (x✝, b✝) :: (x✝, !b✝) :: L₂✝)
H2 : reduce L₁ = L₁✝ ++ (x✝, b✝) :: (x✝, !b✝) :: L₂✝
⊢ reduce L₁ = L₁✝ ++ L₂✝ | 2a1395f3d46bbfa5 |
PrimeSpectrum.basicOpen_eq_zeroLocus_of_isIdempotentElem | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | @[stacks 00EC]
lemma basicOpen_eq_zeroLocus_of_isIdempotentElem
(e : R) (he : IsIdempotentElem e) :
basicOpen e = zeroLocus {1 - e} | case h.mp
R : Type u
inst✝ : CommRing R
e : R
he : IsIdempotentElem e
p : PrimeSpectrum R
⊢ e ∉ p.asIdeal → 1 - e ∈ p.asIdeal | refine (p.2.mem_or_mem_of_mul_eq_zero ?_).resolve_left | case h.mp
R : Type u
inst✝ : CommRing R
e : R
he : IsIdempotentElem e
p : PrimeSpectrum R
⊢ e * (1 - e) = 0 | ce16bea54b1a9bbc |
Std.Sat.AIG.RefVec.map.go_decl_eq | Mathlib/.lake/packages/lean4/src/lean/Std/Sat/AIG/RefVecOperator/Map.lean | theorem map.go_decl_eq {aig : AIG α} (i) (hi)
(s : RefVec aig i) (input : RefVec aig len) (f : (aig : AIG α) → Ref aig → Entrypoint α)
[LawfulOperator α Ref f] [LawfulMapOperator α f] :
∀ (idx : Nat) (h1) (h2), (go aig i hi s input f).1.decls[idx]'h2 = aig.decls[idx]'h1 | α : Type
inst✝³ : Hashable α
inst✝² : DecidableEq α
len : Nat
aig : AIG α
i : Nat
hi : i ≤ len
s : aig.RefVec i
input : aig.RefVec len
f : (aig : AIG α) → aig.Ref → Entrypoint α
inst✝¹ : LawfulOperator α Ref f
inst✝ : LawfulMapOperator α f
res : RefVecEntry α len
hgo :
(if hidx : i < len then
let res := f aig (... | split at hgo | case isTrue
α : Type
inst✝³ : Hashable α
inst✝² : DecidableEq α
len : Nat
aig : AIG α
i : Nat
hi : i ≤ len
s : aig.RefVec i
input : aig.RefVec len
f : (aig : AIG α) → aig.Ref → Entrypoint α
inst✝¹ : LawfulOperator α Ref f
inst✝ : LawfulMapOperator α f
res : RefVecEntry α len
h✝ : i < len
hgo :
(let res := f aig (inpu... | 733cd3cf6f81bdb1 |
Algebra.adjoin_restrictScalars | Mathlib/RingTheory/Adjoin/Tower.lean | theorem adjoin_restrictScalars (C D E : Type*) [CommSemiring C] [CommSemiring D] [CommSemiring E]
[Algebra C D] [Algebra C E] [Algebra D E] [IsScalarTower C D E] (S : Set E) :
(Algebra.adjoin D S).restrictScalars C =
(Algebra.adjoin ((⊤ : Subalgebra C D).map (IsScalarTower.toAlgHom C D E)) S).restrictScal... | case h.mpr.intro.mk.intro.intro
C : Type u_1
D : Type u_2
E : Type u_3
inst✝⁶ : CommSemiring C
inst✝⁵ : CommSemiring D
inst✝⁴ : CommSemiring E
inst✝³ : Algebra C D
inst✝² : Algebra C E
inst✝¹ : Algebra D E
inst✝ : IsScalarTower C D E
S : Set E
x y : E
z : D
h0 : z ∈ ↑⊤.toSubsemiring
h1 : ↑(IsScalarTower.toAlgHom C D E)... | exact ⟨z, Eq.trans h1 h2⟩ | no goals | 15767984f0ae75a7 |
BaireMeasurableSet.residualEq_isOpen | Mathlib/Topology/Baire/BaireMeasurable.lean | theorem BaireMeasurableSet.residualEq_isOpen (h : BaireMeasurableSet s) :
∃ u : Set α, (IsOpen u) ∧ s =ᵇ u | case intro.intro
α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
this✝¹ : MeasurableSpace α := borel α
this✝ : BorelSpace α
t : Set α
ht : MeasurableSet t
hst : s =ᶠ[residual α] t
⊢ ∃ u, IsOpen u ∧ s =ᶠ[residual α] u | rcases ht.residualEq_isOpen with ⟨u, hu, htu⟩ | case intro.intro.intro.intro
α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
this✝¹ : MeasurableSpace α := borel α
this✝ : BorelSpace α
t : Set α
ht : MeasurableSet t
hst : s =ᶠ[residual α] t
u : Set α
hu : IsOpen u
htu : t =ᶠ[residual α] u
⊢ ∃ u, IsOpen u ∧ s =ᶠ[residual α] u | 236c01c0fa74852d |
PFunctor.M.ext_aux | Mathlib/Data/PFunctor/Univariate/M.lean | theorem ext_aux [Inhabited (M F)] [DecidableEq F.A] {n : ℕ} (x y z : M F) (hx : Agree' n z x)
(hy : Agree' n z y) (hrec : ∀ ps : Path F, n = ps.length → iselect ps x = iselect ps y) :
x.approx (n + 1) = y.approx (n + 1) | case succ.step.step.f.f.refl.refl.refl.refl.refl.e_a.h
F : PFunctor.{u}
inst✝ : Inhabited F.M
n_ih✝ : DecidableEq F.A
n : ℕ
n_ih :
∀ (x y z : F.M),
Agree' n z x →
Agree' n z y →
(∀ (ps : Path F), n = length ps → iselect ps x = iselect ps y) → x.approx (n + 1) = y.approx (n + 1)
a : F.A
f₃ f₂ : F.B a... | apply n_ih | case succ.step.step.f.f.refl.refl.refl.refl.refl.e_a.h.hx
F : PFunctor.{u}
inst✝ : Inhabited F.M
n_ih✝ : DecidableEq F.A
n : ℕ
n_ih :
∀ (x y z : F.M),
Agree' n z x →
Agree' n z y →
(∀ (ps : Path F), n = length ps → iselect ps x = iselect ps y) → x.approx (n + 1) = y.approx (n + 1)
a : F.A
f₃ f₂ : F.... | 4a16a3c615b0519f |
CategoryTheory.leftDistributor_ext_right | Mathlib/CategoryTheory/Monoidal/Preadditive.lean | theorem leftDistributor_ext_right {J : Type} [Fintype J] {X Y : C} {f : J → C} {g h : X ⟶ Y ⊗ ⨁ f}
(w : ∀ j, g ≫ (Y ◁ biproduct.π f j) = h ≫ (Y ◁ biproduct.π f j)) : g = h | case w
C : Type u_1
inst✝⁵ : Category.{u_2, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : MonoidalCategory C
inst✝² : MonoidalPreadditive C
inst✝¹ : HasFiniteBiproducts C
J : Type
inst✝ : Fintype J
X Y : C
f : J → C
g h : X ⟶ Y ⊗ ⨁ f
w : ∀ (j : J), g ≫ Y ◁ biproduct.π f j = h ≫ Y ◁ biproduct.π f j
j✝ : J
⊢ g ≫ Y ◁ biproduct.π... | apply w | no goals | 3631c312135153c0 |
isNilpotent_pi_of_bounded_class | Mathlib/GroupTheory/Nilpotent.lean | theorem isNilpotent_pi_of_bounded_class [∀ i, IsNilpotent (Gs i)] (n : ℕ)
(h : ∀ i, Group.nilpotencyClass (Gs i) ≤ n) : IsNilpotent (∀ i, Gs i) | η : Type u_2
Gs : η → Type u_3
inst✝¹ : (i : η) → Group (Gs i)
inst✝ : ∀ (i : η), IsNilpotent (Gs i)
n : ℕ
h : ∀ (i : η), nilpotencyClass (Gs i) ≤ n
⊢ (pi Set.univ fun i => lowerCentralSeries (Gs i) n) ≤ ⊥ | rw [← eq_bot_iff, pi_eq_bot_iff] | η : Type u_2
Gs : η → Type u_3
inst✝¹ : (i : η) → Group (Gs i)
inst✝ : ∀ (i : η), IsNilpotent (Gs i)
n : ℕ
h : ∀ (i : η), nilpotencyClass (Gs i) ≤ n
⊢ ∀ (i : η), lowerCentralSeries (Gs i) n = ⊥ | 8fc5a2d88b529828 |
SimpleGraph.IsHamiltonian.connected | Mathlib/Combinatorics/SimpleGraph/Hamiltonian.lean | lemma IsHamiltonian.connected (hG : G.IsHamiltonian) : G.Connected where
preconnected a b | α : Type u_1
inst✝¹ : DecidableEq α
G : SimpleGraph α
inst✝ : Fintype α
hG : G.IsHamiltonian
x✝ : IsEmpty α
⊢ False | simpa using hG <| by simp [@Fintype.card_eq_zero] | no goals | ea97fcbde159cdd1 |
CategoryTheory.exact_d_f | Mathlib/CategoryTheory/Abelian/Projective/Resolution.lean | theorem exact_d_f {X Y : C} (f : X ⟶ Y) :
(ShortComplex.mk (d f) f (by simp)).Exact | C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : EnoughProjectives C
X Y : C
f : X ⟶ Y
α : ShortComplex.mk (d f) f ⋯ ⟶ ShortComplex.mk (kernel.ι f) f ⋯ :=
{ τ₁ := π (kernel f), τ₂ := 𝟙 (ShortComplex.mk (d f) f ⋯).X₂, τ₃ := 𝟙 (ShortComplex.mk (d f) f ⋯).X₃, comm₁₂ := ⋯,
comm₂₃ := ⋯ }
this✝ : Epi ... | have : Mono α.τ₃ := by dsimp; infer_instance | C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : EnoughProjectives C
X Y : C
f : X ⟶ Y
α : ShortComplex.mk (d f) f ⋯ ⟶ ShortComplex.mk (kernel.ι f) f ⋯ :=
{ τ₁ := π (kernel f), τ₂ := 𝟙 (ShortComplex.mk (d f) f ⋯).X₂, τ₃ := 𝟙 (ShortComplex.mk (d f) f ⋯).X₃, comm₁₂ := ⋯,
comm₂₃ := ⋯ }
this✝¹ : Epi... | 37e4e95338054a45 |
Antitone.tendsto_setIntegral | Mathlib/MeasureTheory/Integral/DominatedConvergence.lean | theorem _root_.Antitone.tendsto_setIntegral (hsm : ∀ i, MeasurableSet (s i)) (h_anti : Antitone s)
(hfi : IntegrableOn f (s 0) μ) :
Tendsto (fun i => ∫ a in s i, f a ∂μ) atTop (𝓝 (∫ a in ⋂ n, s n, f a ∂μ)) | α : Type u_1
E : Type u_2
inst✝² : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s : ℕ → Set α
f : α → E
hsm : ∀ (i : ℕ), MeasurableSet (s i)
h_anti : Antitone s
hfi : IntegrableOn f (s 0) μ
⊢ Tendsto (fun i => ∫ (a : α) in s i, f a ∂μ) atTop (𝓝 (∫ (a : α) in ⋂ n, s n, f a ∂μ)) | let bound : α → ℝ := indicator (s 0) fun a => ‖f a‖ | α : Type u_1
E : Type u_2
inst✝² : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s : ℕ → Set α
f : α → E
hsm : ∀ (i : ℕ), MeasurableSet (s i)
h_anti : Antitone s
hfi : IntegrableOn f (s 0) μ
bound : α → ℝ := (s 0).indicator fun a => ‖f a‖
⊢ Tendsto (fun i => ∫ (a : α) in s i, f a... | 28378f6f1986a1f8 |
MeasureTheory.Lp.eLpNorm'_sum_norm_sub_le_tsum_of_cauchy_eLpNorm' | Mathlib/MeasureTheory/Function/LpSpace/Basic.lean | theorem eLpNorm'_sum_norm_sub_le_tsum_of_cauchy_eLpNorm' {f : ℕ → α → E}
(hf : ∀ n, AEStronglyMeasurable (f n) μ) {p : ℝ} (hp1 : 1 ≤ p) {B : ℕ → ℝ≥0∞}
(h_cau : ∀ N n m : ℕ, N ≤ n → N ≤ m → eLpNorm' (f n - f m) p μ < B N) (n : ℕ) :
eLpNorm' (fun x => ∑ i ∈ Finset.range (n + 1), ‖f (i + 1) x - f i x‖) p μ ≤ ∑... | α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → eLpNorm' (f n - f m) p μ < B N
n : ℕ
f_norm_diff : ℕ → α → ℝ := fun i x => ‖f (i + 1) x - f i x‖
hg... | refine (Finset.sum_le_sum ?_).trans (sum_le_tsum _ (fun m _ => zero_le _) ENNReal.summable) | α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → eLpNorm' (f n - f m) p μ < B N
n : ℕ
f_norm_diff : ℕ → α → ℝ := fun i x => ‖f (i + 1) x - f i x‖
hg... | 6fde102bdd3390d2 |
ENNReal.exists_inv_two_pow_lt | Mathlib/Data/ENNReal/Inv.lean | theorem exists_inv_two_pow_lt (ha : a ≠ 0) : ∃ n : ℕ, 2⁻¹ ^ n < a | case intro
a : ℝ≥0∞
ha : a ≠ 0
n : ℕ
hn : (↑n)⁻¹ < a
⊢ 2⁻¹ ^ n < (↑n)⁻¹ | rw [← ENNReal.inv_pow, ENNReal.inv_lt_inv] | case intro
a : ℝ≥0∞
ha : a ≠ 0
n : ℕ
hn : (↑n)⁻¹ < a
⊢ ↑n < 2 ^ n | 61ab4e8cda5ef6da |
Nat.dvd_iff_div_factorization_eq_tsub | Mathlib/Data/Nat/Factorization/Basic.lean | theorem dvd_iff_div_factorization_eq_tsub {d n : ℕ} (hd : d ≠ 0) (hdn : d ≤ n) :
d ∣ n ↔ (n / d).factorization = n.factorization - d.factorization | d n : ℕ
hd : d ≠ 0
hdn : d ≤ n
⊢ d ∣ n ↔ (n / d).factorization = n.factorization - d.factorization | refine ⟨factorization_div, ?_⟩ | d n : ℕ
hd : d ≠ 0
hdn : d ≤ n
⊢ (n / d).factorization = n.factorization - d.factorization → d ∣ n | eca1d313a671b53a |
EisensteinSeries.auxbound2 | Mathlib/NumberTheory/ModularForms/EisensteinSeries/UniformConvergence.lean | lemma auxbound2 (c : ℝ) {d : ℝ} (hd : 1 ≤ d ^ 2) : r z ≤ ‖c * (z : ℂ) + d‖ | z : ℍ
c d : ℝ
hd : 1 ≤ d ^ 2
H1 : √(r1 z) ≤ √((c * z.re + d) ^ 2 + (c * z.im) ^ 2)
⊢ r z ≤ ‖↑c * ↑z + ↑d‖ | simpa only [r, norm_def, normSq_apply, add_re, re_ofReal_mul, coe_re, ofReal_re, ← pow_two,
add_im, im_ofReal_mul, coe_im, ofReal_im, add_zero, min_le_iff] using Or.inr H1 | no goals | d211823d5b2608a3 |
AlgebraicGeometry.Scheme.IdealSheafData.ideal_inf | Mathlib/AlgebraicGeometry/IdealSheaf.lean | @[simp]
lemma ideal_inf {I J : IdealSheafData X} : (I ⊓ J).ideal = I.ideal ⊓ J.ideal | X : Scheme
I J : X.IdealSheafData
⊢ ∀ (U : ↑X.affineOpens) (f : ↑Γ(X, ↑U)),
Ideal.map (CommRingCat.Hom.hom (X.presheaf.map (homOfLE ⋯).op)) ((I.ideal ⊓ J.ideal) U) =
(I.ideal ⊓ J.ideal) (X.affineBasicOpen f) | intro U f | X : Scheme
I J : X.IdealSheafData
U : ↑X.affineOpens
f : ↑Γ(X, ↑U)
⊢ Ideal.map (CommRingCat.Hom.hom (X.presheaf.map (homOfLE ⋯).op)) ((I.ideal ⊓ J.ideal) U) =
(I.ideal ⊓ J.ideal) (X.affineBasicOpen f) | 5b4190dee859ad66 |
hasFTaylorSeriesUpToOn_succ_nat_iff_right | Mathlib/Analysis/Calculus/ContDiff/FTaylorSeries.lean | theorem hasFTaylorSeriesUpToOn_succ_nat_iff_right {n : ℕ} :
HasFTaylorSeriesUpToOn (n + 1 : ℕ) f p s ↔
(∀ x ∈ s, (p x 0).curry0 = f x) ∧
(∀ x ∈ s, HasFDerivWithinAt (fun y => p y 0) (p x 1).curryLeft s x) ∧
HasFTaylorSeriesUpToOn n (fun x => continuousMultilinearCurryFin1 𝕜 E F (p x 1))
... | case mp
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
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
H : HasFTaylorSeriesUpToOn (↑(n + 1)) f p s
⊢ (∀ x ∈ s, (p x 0).... | refine ⟨H.zero_eq, H.fderivWithin 0 (Nat.cast_lt.2 (Nat.succ_pos n)), ?_⟩ | case mp
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
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
H : HasFTaylorSeriesUpToOn (↑(n + 1)) f p s
⊢ HasFTaylorSeriesUp... | d9b18c3ba0915ffe |
AkraBazziRecurrence.isTheta_smoothingFn_sub_self | Mathlib/Computability/AkraBazzi/AkraBazzi.lean | lemma isTheta_smoothingFn_sub_self (i : α) :
(fun (n : ℕ) => ε (b i * n) - ε n) =Θ[atTop] fun n => 1 / (log n)^2 | α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
i : α
⊢ (fun n => ε (b i * ↑n) - ε ↑n) =Θ[atTop] fun n => -log (b i) / log ↑n ^ 2 | exact (R.isEquivalent_smoothingFn_sub_self i).isTheta | no goals | 12a47199e3345c4b |
BumpCovering.exists_isSubordinate_of_locallyFinite_of_prop | Mathlib/Topology/PartitionOfUnity.lean | theorem exists_isSubordinate_of_locallyFinite_of_prop [NormalSpace X] (p : (X → ℝ) → Prop)
(h01 : ∀ s t, IsClosed s → IsClosed t → Disjoint s t →
∃ f : C(X, ℝ), p f ∧ EqOn f 0 s ∧ EqOn f 1 t ∧ ∀ x, f x ∈ Icc (0 : ℝ) 1)
(hs : IsClosed s) (U : ι → Set X) (ho : ∀ i, IsOpen (U i)) (hf : LocallyFinite U)
(... | case intro.intro.intro
ι : Type u
X : Type v
inst✝¹ : TopologicalSpace X
s : Set X
inst✝ : NormalSpace X
p : (X → ℝ) → Prop
h01 :
∀ (s t : Set X),
IsClosed s → IsClosed t → Disjoint s t → ∃ f, p ⇑f ∧ EqOn (⇑f) 0 s ∧ EqOn (⇑f) 1 t ∧ ∀ (x : X), f x ∈ Icc 0 1
hs : IsClosed s
U : ι → Set X
ho : ∀ (i : ι), IsOpen (U i... | rcases exists_subset_iUnion_closure_subset hs hVo (fun x _ => (hf.subset hVU').point_finite x)
hsV with
⟨W, hsW, hWo, hWV⟩ | case intro.intro.intro.intro.intro.intro
ι : Type u
X : Type v
inst✝¹ : TopologicalSpace X
s : Set X
inst✝ : NormalSpace X
p : (X → ℝ) → Prop
h01 :
∀ (s t : Set X),
IsClosed s → IsClosed t → Disjoint s t → ∃ f, p ⇑f ∧ EqOn (⇑f) 0 s ∧ EqOn (⇑f) 1 t ∧ ∀ (x : X), f x ∈ Icc 0 1
hs : IsClosed s
U : ι → Set X
ho : ∀ (i... | 2d21855e51c18b8e |
eVariationOn.add_le_union | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | theorem add_le_union (f : α → E) {s t : Set α} (h : ∀ x ∈ s, ∀ y ∈ t, x ≤ y) :
eVariationOn f s + eVariationOn f t ≤ eVariationOn f (s ∪ t) | case neg
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s t : Set α
h : ∀ x ∈ s, ∀ y ∈ t, x ≤ y
hs : ¬s = ∅
this : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ s }
⊢ eVariationOn f s + eVariationOn f t ≤ eVariationOn f (s ∪ t) | by_cases ht : t = ∅ | case pos
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s t : Set α
h : ∀ x ∈ s, ∀ y ∈ t, x ≤ y
hs : ¬s = ∅
this : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ s }
ht : t = ∅
⊢ eVariationOn f s + eVariationOn f t ≤ eVariationOn f (s ∪ t)
case neg
α : Type u_1
inst✝¹ : LinearOr... | 4878120f0018b659 |
ContinuousLinearMap.norm_iteratedFDerivWithin_le_of_bilinear_aux | Mathlib/Analysis/Calculus/ContDiff/Bounds.lean | theorem ContinuousLinearMap.norm_iteratedFDerivWithin_le_of_bilinear_aux {Du Eu Fu Gu : Type u}
[NormedAddCommGroup Du] [NormedSpace 𝕜 Du] [NormedAddCommGroup Eu] [NormedSpace 𝕜 Eu]
[NormedAddCommGroup Fu] [NormedSpace 𝕜 Fu] [NormedAddCommGroup Gu] [NormedSpace 𝕜 Gu]
(B : Eu →L[𝕜] Fu →L[𝕜] Gu) {f : Du... | case succ
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
Du : Type u
inst✝⁷ : NormedAddCommGroup Du
inst✝⁶ : NormedSpace 𝕜 Du
s : Set Du
x : Du
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
IH :
∀ {Eu Fu Gu : Type u} [inst : NormedAddCommGroup Eu] [inst_1 : NormedSpace 𝕜 Eu] [inst_2 : NormedAddCommGroup Fu]
[inst_... | have A : ContDiffOn 𝕜 n (fun y => B.precompR Du (f y) (fderivWithin 𝕜 g s y)) s :=
(B.precompR Du).isBoundedBilinearMap.contDiff.comp₂_contDiffOn
(hf.of_le (Nat.cast_le.2 (Nat.le_succ n))) (hg.fderivWithin hs In) | case succ
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
Du : Type u
inst✝⁷ : NormedAddCommGroup Du
inst✝⁶ : NormedSpace 𝕜 Du
s : Set Du
x : Du
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
IH :
∀ {Eu Fu Gu : Type u} [inst : NormedAddCommGroup Eu] [inst_1 : NormedSpace 𝕜 Eu] [inst_2 : NormedAddCommGroup Fu]
[inst_... | 3f7cf0e0e429d040 |
List.enumFrom_eq_cons_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/Range.lean | theorem enumFrom_eq_cons_iff {l : List α} {n : Nat} :
l.enumFrom n = x :: l' ↔ ∃ a as, l = a :: as ∧ x = (n, a) ∧ l' = enumFrom (n + 1) as | case mp.intro.intro.intro.intro
α : Type u_1
x : Nat × α
n : Nat
l₁ : List Nat
l₂ : List α
h : n = x.fst ∧ 0 < (x.snd :: l₂).length ∧ l₁ = range' (x.fst + 1) ((x.snd :: l₂).length - 1)
⊢ ∃ a as, x.snd :: l₂ = a :: as ∧ x = (n, a) ∧ l₁.zip l₂ = enumFrom (n + 1) as | obtain ⟨rfl, -, rfl⟩ := h | case mp.intro.intro.intro.intro.intro.intro
α : Type u_1
x : Nat × α
l₂ : List α
⊢ ∃ a as,
x.snd :: l₂ = a :: as ∧
x = (x.fst, a) ∧ (range' (x.fst + 1) ((x.snd :: l₂).length - 1)).zip l₂ = enumFrom (x.fst + 1) as | 20266ce10a4cfd06 |
MeasureTheory.hahn_decomposition | Mathlib/MeasureTheory/Decomposition/UnsignedHahn.lean | theorem hahn_decomposition (μ ν : Measure α) [IsFiniteMeasure μ] [IsFiniteMeasure ν] :
∃ s, MeasurableSet s ∧ (∀ t, MeasurableSet t → t ⊆ s → ν t ≤ μ t) ∧
∀ t, MeasurableSet t → t ⊆ sᶜ → μ t ≤ ν t | α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(μ s).toNNReal - ↑(ν s).toNNReal
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), μ s ≠ ⊤
hν : ∀ (s : Set α), ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ↑(μ s).toNNRea... | refine Nat.le_induction ?_ ?_ n h | case refine_1
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s => ↑(μ s).toNNReal - ↑(ν s).toNNReal
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), μ s ≠ ⊤
hν : ∀ (s : Set α), ν s ≠ ⊤
to_nnreal_μ : ∀ (s : Set α), ... | b74e26e9476d0e19 |
AlgebraicGeometry.HasRingHomProperty.stalkMap | Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean | /-- Let `Q` be a property of ring maps that is stable under localization.
Then if the associated property of scheme morphisms holds for `f`, `Q` holds on all stalks. -/
lemma stalkMap
(hQ : ∀ {R S : Type u} [CommRing R] [CommRing S] (f : R →+* S) (_ : Q f)
(J : Ideal S) (_ : J.IsPrime), Q (Localization.localR... | case inr.intro.intro.intro.intro.intro.intro
P : MorphismProperty Scheme
Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
inst✝ : HasRingHomProperty P Q
X Y : Scheme
f : X ⟶ Y
hQ :
∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] (f : R →+* S),
Q f → ∀ (J : Ideal S) (x... | exact this (IsLocalAtSource.resLE _ hf) _ ⟨hV, hU⟩ | no goals | 0ddf35d0417ee27e |
MvPolynomial.eval_indicator_apply_eq_zero | Mathlib/FieldTheory/Finite/Polynomial.lean | theorem eval_indicator_apply_eq_zero (a b : σ → K) (h : a ≠ b) : eval a (indicator b) = 0 | K : Type u_1
σ : Type u_2
inst✝² : Fintype K
inst✝¹ : Fintype σ
inst✝ : Field K
a b : σ → K
h : a ≠ b
⊢ ∃ i, a i ≠ b i | rwa [Ne, funext_iff, not_forall] at h | no goals | 7e49c2f905c6cd32 |
ExistsContDiffBumpBase.y_pos_of_mem_ball | Mathlib/Analysis/Calculus/BumpFunction/FiniteDimension.lean | theorem y_pos_of_mem_ball {D : ℝ} {x : E} (Dpos : 0 < D) (D_lt_one : D < 1)
(hx : x ∈ ball (0 : E) (1 + D)) : 0 < y D x | E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
Dpos : 0 < D
D_lt_one : D < 1
hx : ‖x‖ < 1 + D
z : E := (D / (1 + D)) • x
hz : z = (D / (1 + D)) • x
B : 0 < 1 + D
y : E
hy : y ∈ ball z (D * (1 + D - ‖x‖) / (1 ... | nlinarith only [hx, D_lt_one] | no goals | 646f97b7cc0fbf51 |
Topology.IsOpenMap.preimage_closure_subset_closure_preimage | Mathlib/Topology/Maps/Basic.lean | theorem preimage_closure_subset_closure_preimage (hf : IsOpenMap f) {s : Set Y} :
f ⁻¹' closure s ⊆ closure (f ⁻¹' s) | X : Type u_1
Y : Type u_2
f : X → Y
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
hf : IsOpenMap f
s : Set Y
⊢ (closure (f ⁻¹' s))ᶜ ⊆ (f ⁻¹' closure s)ᶜ | simp only [← interior_compl, ← preimage_compl, hf.interior_preimage_subset_preimage_interior] | no goals | da6fd42aa0cbb097 |
Complex.tendsto_tsum_powerSeries_nhdsWithin_stolzSet | Mathlib/Analysis/Complex/AbelLimit.lean | theorem tendsto_tsum_powerSeries_nhdsWithin_stolzSet
(h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)) {M : ℝ} :
Tendsto (fun z ↦ ∑' n, f n * z ^ n) (𝓝[stolzSet M] 1) (𝓝 l) | case inr
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
hm : ∀ ε > 0, ∃ N, ∀ n ≥ N, ‖∑ i ∈ range n, f i - l‖ < ε
⊢ ∀ ε > 0, ∃ δ > 0, ∀ ⦃x : ℂ⦄, x ∈ stolzSet M → ‖x - 1‖ < δ → ‖∑' (n : ℕ), f n ... | intro ε εpos | case inr
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
hm : ∀ ε > 0, ∃ N, ∀ n ≥ N, ‖∑ i ∈ range n, f i - l‖ < ε
ε : ℝ
εpos : ε > 0
⊢ ∃ δ > 0, ∀ ⦃x : ℂ⦄, x ∈ stolzSet M → ‖x - 1‖ < δ → ‖∑' (n ... | e1f8d437480849f9 |
MeasureTheory.tendsto_of_lintegral_tendsto_of_monotone_aux | 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. Auxiliary version assuming moreover that the
functions in the sequence are ae measurable. -... | case h
α : Type u_5
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hf_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
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 ... | have h_tendsto : Tendsto (fun i ↦ f i a) atTop atTop ∨
∃ l, Tendsto (fun i ↦ f i a) atTop (𝓝 l) := tendsto_of_monotone h_mono | case h
α : Type u_5
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hf_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
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 ... | 99ba951a0bb94bbf |
MeromorphicAt.order_eq_int_iff | Mathlib/Analysis/Meromorphic/Order.lean | /-- The order of a meromorphic function `f` at `z₀` equals an integer `n` iff `f` can locally be
written as `f z = (z - z₀) ^ n • g z`, where `g` is analytic and does not vanish at `z₀`. -/
lemma order_eq_int_iff {f : 𝕜 → E} {x : 𝕜} (hf : MeromorphicAt f x) (n : ℤ) : hf.order = n ↔
∃ g : 𝕜 → E, AnalyticAt 𝕜 g x... | case pos
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
hf : MeromorphicAt f x
n : ℤ
h : ∀ᶠ (z : 𝕜) in 𝓝 x, (z - x) ^ Exists.choose hf • f z = 0
x✝ : ∃ g, AnalyticAt 𝕜 g x ∧ g x ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = (z - x) ^ n • ... | apply EventuallyEq.eq_of_nhds | case pos.h
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
hf : MeromorphicAt f x
n : ℤ
h : ∀ᶠ (z : 𝕜) in 𝓝 x, (z - x) ^ Exists.choose hf • f z = 0
x✝ : ∃ g, AnalyticAt 𝕜 g x ∧ g x ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = (z - x) ^ n ... | 3ef297e0d0c9643e |
Path.Homotopic.hpath_hext | Mathlib/Topology/Homotopy/Path.lean | theorem hpath_hext {p₁ : Path x₀ x₁} {p₂ : Path x₂ x₃} (hp : ∀ t, p₁ t = p₂ t) :
@HEq (Path.Homotopic.Quotient _ _) ⟦p₁⟧ (Path.Homotopic.Quotient _ _) ⟦p₂⟧ | X : Type u
inst✝ : TopologicalSpace X
x₀ x₁ x₃ : X
p₁ : Path x₀ x₁
p₂ : Path x₀ x₃
hp : ∀ (t : ↑I), p₁ t = p₂ t
⊢ x₁ = x₃ | convert hp 1 <;> simp | no goals | 61c4e4eef9cc37e4 |
SimpleGraph.Walk.IsPath.of_cons | Mathlib/Combinatorics/SimpleGraph/Path.lean | theorem IsPath.of_cons {u v w : V} {h : G.Adj u v} {p : G.Walk v w} :
(cons h p).IsPath → p.IsPath | V : Type u
G : SimpleGraph V
u v w : V
h : G.Adj u v
p : G.Walk v w
⊢ (cons h p).IsPath → p.IsPath | simp [isPath_def] | no goals | 1e56d6acab169257 |
EReal.continuousAt_mul_top_top | Mathlib/Topology/Instances/EReal/Lemmas.lean | private lemma continuousAt_mul_top_top :
ContinuousAt (fun p : EReal × EReal ↦ p.1 * p.2) (⊤, ⊤) | case h.refine_1
x : ℝ
p : EReal × EReal
p_in_prod : (↑x < p.1 ∧ 0 < p.1) ∧ 1 < p.2
⊢ ↑x < p.1 * p.2 | rcases p_in_prod with ⟨⟨p1_gt_x, p1_pos⟩, p2_gt_1⟩ | case h.refine_1.intro.intro
x : ℝ
p : EReal × EReal
p2_gt_1 : 1 < p.2
p1_gt_x : ↑x < p.1
p1_pos : 0 < p.1
⊢ ↑x < p.1 * p.2 | 14ad4956648676bd |
SSet.StrictSegal.isPointwiseRightKanExtensionAt.fac_aux₂ | Mathlib/AlgebraicTopology/SimplicialSet/Coskeletal.lean | lemma fac_aux₂ {n : ℕ}
(s : Cone (proj (op ⦋n⦌) (Truncated.inclusion 2).op ⋙ (Truncated.inclusion 2).op ⋙ X))
(x : s.pt) (i j : ℕ) (hij : i ≤ j) (hj : j ≤ n) :
X.map (mkOfLe ⟨i, by omega⟩ ⟨j, by omega⟩ hij).op (lift s x) =
s.π.app (strArrowMk₂ (mkOfLe ⟨i, by omega⟩ ⟨j, by omega⟩ hij)) x | X : SSet
inst✝ : X.StrictSegal
n : ℕ
s : Cone (proj (op ⦋n⦌) (Truncated.inclusion 2).op ⋙ (Truncated.inclusion 2).op ⋙ X)
x : s.pt
k : ℕ
hk :
∀ (i j : ℕ) (hij : i ≤ j) (hj : j ≤ n),
i + k = j → X.map (mkOfLe ⟨i, ⋯⟩ ⟨j, ⋯⟩ hij).op (lift s x) = s.π.app (strArrowMk₂ (mkOfLe ⟨i, ⋯⟩ ⟨j, ⋯⟩ hij) ⋯) x
i j : ℕ
hij : i ≤ ... | ext x | case a.h.h.h
X : SSet
inst✝ : X.StrictSegal
n : ℕ
s : Cone (proj (op ⦋n⦌) (Truncated.inclusion 2).op ⋙ (Truncated.inclusion 2).op ⋙ X)
x✝ : s.pt
k : ℕ
hk :
∀ (i j : ℕ) (hij : i ≤ j) (hj : j ≤ n),
i + k = j → X.map (mkOfLe ⟨i, ⋯⟩ ⟨j, ⋯⟩ hij).op (lift s x✝) = s.π.app (strArrowMk₂ (mkOfLe ⟨i, ⋯⟩ ⟨j, ⋯⟩ hij) ⋯) x✝
i ... | 764b0c7acb432811 |
mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt | Mathlib/RingTheory/Polynomial/Eisenstein/IsIntegral.lean | theorem mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt {B : PowerBasis K L}
(hp : Prime p) (hBint : IsIntegral R B.gen) {z : L} (hzint : IsIntegral R z)
(hz : p • z ∈ adjoin R ({B.gen} : Set L)) (hei : (minpoly R B.gen).IsEisensteinAt 𝓟) :
z ∈ adjoin R ({B.gen} : Set L) | case neg
R : Type u
K : Type v
L : Type z
p : R
inst✝¹⁰ : CommRing R
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra R L
inst✝⁵ : Algebra R K
inst✝⁴ : IsScalarTower R K L
inst✝³ : Algebra.IsSeparable K L
inst✝² : IsDomain R
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
B : PowerBasis ... | induction' i using Nat.case_strong_induction_on with j hind | case neg.hz
R : Type u
K : Type v
L : Type z
p : R
inst✝¹⁰ : CommRing R
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra R L
inst✝⁵ : Algebra R K
inst✝⁴ : IsScalarTower R K L
inst✝³ : Algebra.IsSeparable K L
inst✝² : IsDomain R
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
B : PowerBas... | fa4f10e8cbdbb4c7 |
IsLinearTopology.hasBasis_subbimodule | Mathlib/Topology/Algebra/LinearTopology.lean | /-- Assume that `M` is a module over two rings `R` and `R'`, and that its topology
is linear with respect to each of these rings. Then, it has a basis of neighborhoods of zero
made of sub-`(R, R')`-bimodules.
The proof is inspired by lemma 9 in [I. Kaplansky, *Topological Rings*](kaplansky_topological_1947).
TODO: For... | case intro.intro
R : Type u_1
R' : Type u_2
M : Type u_3
inst✝⁸ : Ring R
inst✝⁷ : Ring R'
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : Module R' M
inst✝³ : SMulCommClass R R' M
inst✝² : TopologicalSpace M
inst✝¹ : IsLinearTopology R M
inst✝ : IsLinearTopology R' M
I : Submodule R M
hI : ↑I ∈ 𝓝 0
J : Submodule ... | set uR : Set R := univ | case intro.intro
R : Type u_1
R' : Type u_2
M : Type u_3
inst✝⁸ : Ring R
inst✝⁷ : Ring R'
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : Module R' M
inst✝³ : SMulCommClass R R' M
inst✝² : TopologicalSpace M
inst✝¹ : IsLinearTopology R M
inst✝ : IsLinearTopology R' M
I : Submodule R M
hI : ↑I ∈ 𝓝 0
J : Submodule ... | 61e860eef2501f92 |
Std.Sat.AIG.RelabelNat.State.Inv1.property | Mathlib/.lake/packages/lean4/src/lean/Std/Sat/AIG/RelabelNat.lean | theorem Inv1.property [EquivBEq α] {n m : Nat} (x y : α) (map : HashMap α Nat) (hinv : Inv1 n map)
(hfound1 : map[x]? = some m) (hfound2 : map[y]? = some m) : x = y | α : Type
inst✝² : DecidableEq α
inst✝¹ : Hashable α
inst✝ : EquivBEq α
n m : Nat
x y : α
map : HashMap α Nat
n✝ : Nat
map✝ : HashMap α Nat
z : α
ih1 : Inv1 n✝ map✝
_ih2 : map✝[z]? = none
ih3 : map✝[x]? = some m → map✝[y]? = some m → x = y
hfound1 : (if (z == x) = true then some n✝ else map✝[x]?) = some m
hfound2 : (if ... | simp only [beq_iff_eq, hx, ↓reduceIte] at hfound1 | α : Type
inst✝² : DecidableEq α
inst✝¹ : Hashable α
inst✝ : EquivBEq α
n m : Nat
x y : α
map : HashMap α Nat
n✝ : Nat
map✝ : HashMap α Nat
z : α
ih1 : Inv1 n✝ map✝
_ih2 : map✝[z]? = none
ih3 : map✝[x]? = some m → map✝[y]? = some m → x = y
hfound2 : (if (z == y) = true then some n✝ else map✝[y]?) = some m
hx : ¬z = x
hf... | 9c2d63c0f1474e48 |
BitVec.getLsbD_eq_getMsbD | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem getLsbD_eq_getMsbD (x : BitVec w) (i : Nat) : x.getLsbD i = (decide (i < w) && x.getMsbD (w - 1 - i)) | case pos
w : Nat
x : BitVec w
i : Nat
h₁ : i < w
h₂ : w - 1 - i < w
⊢ x.getLsbD i = x.getLsbD (w - 1 - (w - 1 - i)) | congr | case pos.e_i
w : Nat
x : BitVec w
i : Nat
h₁ : i < w
h₂ : w - 1 - i < w
⊢ i = w - 1 - (w - 1 - i) | f9a0bc79b2f20e50 |
MeasureTheory.lintegral_iSup' | Mathlib/MeasureTheory/Integral/Lebesgue.lean | theorem lintegral_iSup' {f : ℕ → α → ℝ≥0∞} (hf : ∀ n, AEMeasurable (f n) μ)
(h_mono : ∀ᵐ x ∂μ, Monotone fun n => f n x) : ∫⁻ a, ⨆ n, f n a ∂μ = ⨆ n, ∫⁻ a, f n a ∂μ | α : Type u_1
m✝ : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
hf : ∀ (n : ℕ), AEMeasurable (f n) μ
h_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n => f n x
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' => Monotone f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
n m : ℕ
hnm : n ≤ m
x : α
⊢ aeSeq hf p n x ≤ aeSeq hf p m x | by_cases hx : x ∈ aeSeqSet hf p | case pos
α : Type u_1
m✝ : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
hf : ∀ (n : ℕ), AEMeasurable (f n) μ
h_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n => f n x
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' => Monotone f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
n m : ℕ
hnm : n ≤ m
x : α
hx : x ∈ aeSeqSet hf p
⊢ aeSeq hf p n x ≤... | 4311032ba3184512 |
CompleteLattice.isCompactElement_iff_le_of_directed_sSup_le | Mathlib/Order/CompactlyGenerated/Basic.lean | theorem isCompactElement_iff_le_of_directed_sSup_le (k : α) :
IsCompactElement k ↔
∀ s : Set α, s.Nonempty → DirectedOn (· ≤ ·) s → k ≤ sSup s → ∃ x : α, x ∈ s ∧ k ≤ x | case mp
α : Type u_2
inst✝ : CompleteLattice α
k : α
⊢ IsCompactElement k → ∀ (s : Set α), s.Nonempty → DirectedOn (fun x1 x2 => x1 ≤ x2) s → k ≤ sSup s → ∃ x ∈ s, k ≤ x | intro hk s hne hdir hsup | case mp
α : Type u_2
inst✝ : CompleteLattice α
k : α
hk : IsCompactElement k
s : Set α
hne : s.Nonempty
hdir : DirectedOn (fun x1 x2 => x1 ≤ x2) s
hsup : k ≤ sSup s
⊢ ∃ x ∈ s, k ≤ x | 445bf3d7fb93c3e6 |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.assignmentsInvariant_insertRupUnits_of_assignmentsInvariant | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddSound.lean | theorem assignmentsInvariant_insertRupUnits_of_assignmentsInvariant {n : Nat} (f : DefaultFormula n) (f_readyForRupAdd : ReadyForRupAdd f)
(units : CNF.Clause (PosFin n)) :
AssignmentsInvariant (insertRupUnits f units).1 | n : Nat
f : DefaultFormula n
f_readyForRupAdd : f.ReadyForRupAdd
units : CNF.Clause (PosFin n)
h :
let assignments := (f.insertRupUnits units).fst.assignments;
let_fun hsize := ⋯;
let rupUnits := (f.insertRupUnits units).fst.rupUnits;
InsertUnitInvariant f.assignments ⋯ rupUnits assignments hsize
hsize : (f.ins... | simp only [Prod.mk.injEq, and_true] | n : Nat
f : DefaultFormula n
f_readyForRupAdd : f.ReadyForRupAdd
units : CNF.Clause (PosFin n)
h :
let assignments := (f.insertRupUnits units).fst.assignments;
let_fun hsize := ⋯;
let rupUnits := (f.insertRupUnits units).fst.rupUnits;
InsertUnitInvariant f.assignments ⋯ rupUnits assignments hsize
hsize : (f.ins... | 7537b9cac0fe9952 |
algebraicIndependent_empty_iff | Mathlib/RingTheory/AlgebraicIndependent/Basic.lean | theorem algebraicIndependent_empty_iff :
AlgebraicIndependent R ((↑) : (∅ : Set A) → A) ↔ Injective (algebraMap R A) | R : Type u_3
A : Type u_5
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
⊢ AlgebraicIndependent R Subtype.val ↔ Injective ⇑(algebraMap R A) | simp | no goals | 70e4e9c214d99940 |
ContinuousLinearMap.coe_toContinuousLinearEquivOfDetNeZero | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | theorem coe_toContinuousLinearEquivOfDetNeZero (f : E →L[𝕜] E) (hf : f.det ≠ 0) :
(f.toContinuousLinearEquivOfDetNeZero hf : E →L[𝕜] E) = f | case h
𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
E : Type v
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : ContinuousSMul 𝕜 E
inst✝² : CompleteSpace 𝕜
inst✝¹ : T2Space E
inst✝ : FiniteDimensional 𝕜 E
f : E →L[𝕜] E
hf : f.det ≠ 0
x : E
⊢ ↑(f.to... | rfl | no goals | 8ec54aca9c1503af |
Matroid.map_dual | Mathlib/Data/Matroid/Map.lean | @[simp] lemma map_dual {hf} : (M.map f hf)✶ = M✶.map f hf | α : Type u_1
β : Type u_2
f : α → β
M : Matroid α
hf : InjOn f M.E
B : Set α
hB : B ⊆ M.E
⊢ M.IsBase (M.E \ B) → ∃ u ⊆ M.E, f '' u = f '' B | exact fun _ ↦ ⟨B, hB, rfl⟩ | no goals | 29df05518640e03f |
Wbtw.swap_right_iff | Mathlib/Analysis/Convex/Between.lean | theorem Wbtw.swap_right_iff [NoZeroSMulDivisors R V] {x y z : P} (h : Wbtw R x y z) :
Wbtw R x z y ↔ y = z | R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁴ : OrderedRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AddTorsor V P
inst✝ : NoZeroSMulDivisors R V
x y z : P
h : Wbtw R x y z
⊢ Wbtw R x z y ↔ y = z | rw [← wbtw_swap_right_iff R x, and_iff_right h] | no goals | caedc9860decd5de |
WithTop.forall_gt_iff_eq_top | Mathlib/Order/WithBot.lean | lemma forall_gt_iff_eq_top : (∀ a : α, a < y) ↔ y = ⊤ | case coe
α : Type u_1
inst✝ : Preorder α
a✝ : α
⊢ ∃ x, ¬x < a✝ | simpa using ⟨_, lt_irrefl _⟩ | no goals | e378fb13fa66897a |
List.rtakeWhile_eq_nil_iff | Mathlib/Data/List/DropRight.lean | theorem rtakeWhile_eq_nil_iff : rtakeWhile p l = [] ↔ ∀ hl : l ≠ [], ¬p (l.getLast hl) | case nil
α : Type u_1
p : α → Bool
l : List α
f : [] ≠ []
⊢ ¬p ([].getLast f) = true | contradiction | no goals | 76d022d188e44678 |
Set.inv_range | Mathlib/Algebra/Group/Pointwise/Set/Basic.lean | theorem inv_range {ι : Sort*} {f : ι → α} : (range f)⁻¹ = range fun i => (f i)⁻¹ | α : Type u_2
inst✝ : InvolutiveInv α
ι : Sort u_5
f : ι → α
⊢ (range f)⁻¹ = range fun i => (f i)⁻¹ | rw [← image_inv_eq_inv] | α : Type u_2
inst✝ : InvolutiveInv α
ι : Sort u_5
f : ι → α
⊢ (fun x => x⁻¹) '' range f = range fun i => (f i)⁻¹ | ee931b95f8849272 |
iteratedFDerivWithin_comp_add_left' | Mathlib/Analysis/Calculus/ContDiff/FTaylorSeries.lean | /-- The iterated derivative commutes with shifting the function by a constant on the left. -/
lemma iteratedFDerivWithin_comp_add_left' (n : ℕ) (a : E) :
iteratedFDerivWithin 𝕜 n (fun z ↦ f (a + z)) s =
fun x ↦ iteratedFDerivWithin 𝕜 n f (a +ᵥ s) (a + x) | 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 : ℕ
a : E
⊢ iteratedFDerivWithin 𝕜 n (fun z => f (a + z)) s = fun x => iteratedFDerivWithin 𝕜 n f (a +ᵥ s) (a + x... | induction n with
| zero => simp [iteratedFDerivWithin]
| succ n IH =>
ext v
rw [iteratedFDerivWithin_succ_eq_comp_left, iteratedFDerivWithin_succ_eq_comp_left]
simp only [Nat.succ_eq_add_one, IH, comp_apply, continuousMultilinearCurryLeftEquiv_symm_apply]
congr 2
rw [fderivWithin_comp_add_left] | no goals | c164319e4fbfc303 |
MeasureTheory.hitting_le | Mathlib/Probability/Process/HittingTime.lean | theorem hitting_le {m : ι} (ω : Ω) : hitting u s n m ω ≤ m | Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n m : ι
ω : Ω
⊢ hitting u s n m ω ≤ m | simp only [hitting] | Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : ConditionallyCompleteLinearOrder ι
u : ι → Ω → β
s : Set β
n m : ι
ω : Ω
⊢ (if ∃ j ∈ Set.Icc n m, u j ω ∈ s then sInf (Set.Icc n m ∩ {i | u i ω ∈ s}) else m) ≤ m | 98e6c15cf2e650e0 |
ProbabilityTheory.IndepFun.cgf_add | Mathlib/Probability/Moments/Basic.lean | theorem IndepFun.cgf_add {X Y : Ω → ℝ} (h_indep : IndepFun X Y μ)
(h_int_X : Integrable (fun ω => exp (t * X ω)) μ)
(h_int_Y : Integrable (fun ω => exp (t * Y ω)) μ) :
cgf (X + Y) μ t = cgf X μ t + cgf Y μ t | case pos
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
t : ℝ
X Y : Ω → ℝ
h_indep : IndepFun X Y μ
h_int_X : Integrable (fun ω => rexp (t * X ω)) μ
h_int_Y : Integrable (fun ω => rexp (t * Y ω)) μ
hμ : μ = 0
⊢ cgf (X + Y) μ t = cgf X μ t + cgf Y μ t | simp [hμ] | no goals | 692da647c7c67695 |
WittVector.map_frobeniusPoly | Mathlib/RingTheory/WittVector/Frobenius.lean | theorem map_frobeniusPoly (n : ℕ) :
MvPolynomial.map (Int.castRingHom ℚ) (frobeniusPoly p n) = frobeniusPolyRat p n | p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
h1 : ↑p ^ n * ⅟↑p ^ n = 1
i : ℕ
hi : i < n
j : ℕ
hj : j < p ^ (n - i)
this :
↑((p ^ (n - i)).choose (j + 1)) * ↑p ^ (j - v p (j + 1)) * ↑p * ↑p ^ n =
↑p ^ j * ↑p * ↑((p ^ (n - i)).choose (j + 1) * p ^ i) * ↑p ^ (n - i - v p (j + 1))
k✝ : ℕ
⊢ ↑p ^ k✝ ≠ 0 | apply pow_ne_zero | case h
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
h1 : ↑p ^ n * ⅟↑p ^ n = 1
i : ℕ
hi : i < n
j : ℕ
hj : j < p ^ (n - i)
this :
↑((p ^ (n - i)).choose (j + 1)) * ↑p ^ (j - v p (j + 1)) * ↑p * ↑p ^ n =
↑p ^ j * ↑p * ↑((p ^ (n - i)).choose (j + 1) * p ^ i) * ↑p ^ (n - i - v p (j + 1))
k✝ : ℕ
⊢ ↑p ≠ 0 | 1eb072545457e9c3 |
PowerSeries.order_monomial | Mathlib/RingTheory/PowerSeries/Order.lean | theorem order_monomial (n : ℕ) (a : R) [Decidable (a = 0)] :
order (monomial R n a) = if a = 0 then (⊤ : ℕ∞) else n | case neg.right
R : Type u_1
inst✝¹ : Semiring R
n : ℕ
a : R
inst✝ : Decidable (a = 0)
h : ¬a = 0
i : ℕ
hi : i < n
⊢ (coeff R i) ((monomial R n) a) = 0 | rw [coeff_monomial, if_neg] | case neg.right.hnc
R : Type u_1
inst✝¹ : Semiring R
n : ℕ
a : R
inst✝ : Decidable (a = 0)
h : ¬a = 0
i : ℕ
hi : i < n
⊢ ¬i = n | a282ebf719a1b402 |
MeasureTheory.tendsto_Lp_of_tendsto_ae_of_meas | Mathlib/MeasureTheory/Function/UnifTight.lean | theorem tendsto_Lp_of_tendsto_ae_of_meas (hp : 1 ≤ p) (hp' : p ≠ ∞)
{f : ℕ → α → β} {g : α → β} (hf : ∀ n, StronglyMeasurable (f n)) (hg : StronglyMeasurable g)
(hg' : MemLp g p μ) (hui : UnifIntegrable f p μ) (hut : UnifTight f p μ)
(hfg : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))) :
Tendsto (... | α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : MemLp g p μ
hui : UnifIntegrable f p μ
hut : UnifTight f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => ... | apply eLpNorm_sub_le (by assumption) (by assumption) hp | no goals | 77334843f098ce11 |
Array.filterMap_push_none | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean | theorem filterMap_push_none {f : α → Option β} {a : α} {l : Array α}
(h : f a = none) (w : stop = l.size + 1) :
filterMap f (l.push a) 0 stop = filterMap f l | α : Type u_1
β : Type u_2
stop : Nat
f : α → Option β
a : α
l : Array α
h : f a = none
w : stop = l.size + 1
⊢ filterMap f (l.push a) 0 stop = filterMap f l | subst w | α : Type u_1
β : Type u_2
f : α → Option β
a : α
l : Array α
h : f a = none
⊢ filterMap f (l.push a) 0 (l.size + 1) = filterMap f l | 21c9bb055d8d21dd |
SetTheory.PGame.birthday_moveLeft_lt | Mathlib/SetTheory/Game/Birthday.lean | theorem birthday_moveLeft_lt {x : PGame} (i : x.LeftMoves) :
(x.moveLeft i).birthday < x.birthday | case mk
α✝ β✝ : Type u_1
a✝¹ : α✝ → PGame
a✝ : β✝ → PGame
i : (mk α✝ β✝ a✝¹ a✝).LeftMoves
⊢ ((mk α✝ β✝ a✝¹ a✝).moveLeft i).birthday < (lsub fun i => (a✝¹ i).birthday) ⊔ lsub fun i => (a✝ i).birthday | exact lt_max_of_lt_left (lt_lsub _ i) | no goals | 0d9a8a60714b8041 |
CategoryTheory.isFiltered_of_isFiltered_costructuredArrow_small | Mathlib/CategoryTheory/Filtered/CostructuredArrow.lean | private lemma isFiltered_of_isFiltered_costructuredArrow_small (L : A ⥤ T) (R : B ⥤ T)
[IsFiltered B] [Final R] [∀ b, IsFiltered (CostructuredArrow L (R.obj b))] : IsFiltered A | A : Type u₁
inst✝⁵ : SmallCategory A
B : Type u₁
inst✝⁴ : SmallCategory B
T : Type u₁
inst✝³ : SmallCategory T
L : A ⥤ T
R : B ⥤ T
inst✝² : IsFiltered B
inst✝¹ : R.Final
inst✝ : ∀ (b : B), IsFiltered (CostructuredArrow L (R.obj b))
J : Type u₁
x✝¹ : SmallCategory J
x✝ : FinCategory J
F : J ⥤ A ⥤ Type u₁
R' : Grothendie... | exact filtered_colim_preservesFiniteLimits | no goals | 40b92f5a714e8e8d |
BoxIntegral.unitPartition.tendsto_card_div_pow₂ | Mathlib/Analysis/BoxIntegral/UnitPartition.lean | theorem tendsto_card_div_pow₂ (hs₁ : IsBounded s)
(hs₄ : ∀ ⦃x y : ℝ⦄, 0 < x → x ≤ y → x • s ⊆ y • s) {x y : ℝ} (hx : 0 < x) (hy : x ≤ y) :
Nat.card ↑(s ∩ x⁻¹ • L) ≤ Nat.card ↑(s ∩ y⁻¹ • L) | ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
hs₁ : Bornology.IsBounded s
hs₄ : ∀ ⦃x y : ℝ⦄, 0 < x → x ≤ y → x • s ⊆ y • s
x y : ℝ
hx : 0 < x
hy : x ≤ y
⊢ Nat.card ↑(x • s ∩ ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι)))) ≤
Nat.card ↑(y • s ∩ ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι)))) | refine Nat.card_mono ?_ ?_ | case refine_1
ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
hs₁ : Bornology.IsBounded s
hs₄ : ∀ ⦃x y : ℝ⦄, 0 < x → x ≤ y → x • s ⊆ y • s
x y : ℝ
hx : 0 < x
hy : x ≤ y
⊢ (y • s ∩ ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι)))).Finite
case refine_2
ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
hs₁ : Bornology.IsBounded s
hs... | 5e10a1a1aca1cc38 |
CategoryTheory.Subobject.inf_map | Mathlib/CategoryTheory/Subobject/Lattice.lean | theorem inf_map {X Y : C} (g : Y ⟶ X) [Mono g] (f₁ f₂) :
(map g).obj (f₁ ⊓ f₂) = (map g).obj f₁ ⊓ (map g).obj f₂ | case h
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : HasPullbacks C
X Y : C
g : Y ⟶ X
inst✝ : Mono g
f₂ : Subobject Y
f₁ : MonoOver Y
⊢ (map (f₁.arrow ≫ g)).obj ((pullback f₁.arrow).obj f₂) =
(map (f₁.arrow ≫ g)).obj ((pullback (f₁.arrow ≫ g)).obj ((map g).obj f₂)) | rw [pullback_comp, pullback_map_self] | no goals | bfa22f20b0f93c13 |
CategoryTheory.Localization.LeftBousfield.W_iff_isIso_map | Mathlib/CategoryTheory/Localization/Bousfield.lean | lemma W_iff_isIso_map {X Y : D} (f : X ⟶ Y) :
W (· ∈ Set.range F.obj) f ↔ IsIso (G.map f) | case mp
C : Type u_1
D : Type u_2
inst✝³ : Category.{u_4, u_1} C
inst✝² : Category.{u_3, u_2} D
F : C ⥤ D
G : D ⥤ C
adj : G ⊣ F
inst✝¹ : F.Full
inst✝ : F.Faithful
X Y : D
f : X ⟶ Y
h : IsIso ((G ⋙ F).map f)
⊢ IsIso (G.map f) | dsimp at h | case mp
C : Type u_1
D : Type u_2
inst✝³ : Category.{u_4, u_1} C
inst✝² : Category.{u_3, u_2} D
F : C ⥤ D
G : D ⥤ C
adj : G ⊣ F
inst✝¹ : F.Full
inst✝ : F.Faithful
X Y : D
f : X ⟶ Y
h : IsIso (F.map (G.map f))
⊢ IsIso (G.map f) | f5874bf39d1e1a07 |
EqualCharZero.of_algebraRat | Mathlib/Algebra/CharP/MixedCharZero.lean | theorem of_algebraRat [Algebra ℚ R] : ∀ I : Ideal R, I ≠ ⊤ → CharZero (R ⧸ I) | case cast_injective.refine_2
R : Type u_1
inst✝¹ : CommRing R
inst✝ : Algebra ℚ R
I : Ideal R
a b : ℕ
h_ab : ↑a = ↑b
hI : a ≠ b
⊢ ↑a - ↑b ≠ 0 | simpa only [Ne, sub_eq_zero] using (@Nat.cast_injective ℚ _ _).ne hI | no goals | a2aa33cbf4959196 |
MeasureTheory.Measure.haveLebesgueDecomposition_of_finiteMeasure | Mathlib/MeasureTheory/Decomposition/Lebesgue.lean | theorem haveLebesgueDecomposition_of_finiteMeasure [IsFiniteMeasure μ] [IsFiniteMeasure ν] :
HaveLebesgueDecomposition μ ν where
lebesgue_decomposition | case refine_1.intro.intro.intro.intro.intro
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
g : ℕ → ℝ≥0∞
h✝ : Monotone g
hg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))
f : ℕ → α → ℝ≥0∞
hf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ
hf₂ : ∀ (n : ... | have : (∫⁻ a, ξ a + E.indicator (fun _ ↦ (ε : ℝ≥0∞)) a ∂ν) ≤ sSup (measurableLEEval ν μ) :=
le_sSup ⟨ξ + E.indicator fun _ ↦ (ε : ℝ≥0∞), hξε, rfl⟩ | case refine_1.intro.intro.intro.intro.intro
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
g : ℕ → ℝ≥0∞
h✝ : Monotone g
hg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))
f : ℕ → α → ℝ≥0∞
hf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ
hf₂ : ∀ (n : ... | 0e340e69a0abebdf |
CategoryTheory.lift_comp_preservesLimitIso_hom | Mathlib/CategoryTheory/Limits/Preserves/Limits.lean | theorem lift_comp_preservesLimitIso_hom (t : Cone F) :
G.map (limit.lift _ t) ≫ (preservesLimitIso G F).hom =
limit.lift (F ⋙ G) (G.mapCone _) | C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
G : C ⥤ D
J : Type w
inst✝² : Category.{w', w} J
F : J ⥤ C
inst✝¹ : PreservesLimit F G
inst✝ : HasLimit F
t : Cone F
⊢ G.map (limit.lift F t) ≫ (preservesLimitIso G F).hom = limit.lift (F ⋙ G) (G.mapCone t) | ext | case w
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
G : C ⥤ D
J : Type w
inst✝² : Category.{w', w} J
F : J ⥤ C
inst✝¹ : PreservesLimit F G
inst✝ : HasLimit F
t : Cone F
j✝ : J
⊢ (G.map (limit.lift F t) ≫ (preservesLimitIso G F).hom) ≫ limit.π (F ⋙ G) j✝ =
limit.lift (F ⋙ G) (G.m... | 31314c3b9d2b26b4 |
convexHull_range_eq_exists_affineCombination | Mathlib/Analysis/Convex/Combination.lean | theorem convexHull_range_eq_exists_affineCombination (v : ι → E) : convexHull R (range v) =
{ x | ∃ (s : Finset ι) (w : ι → R), (∀ i ∈ s, 0 ≤ w i) ∧ s.sum w = 1 ∧
s.affineCombination R v w = x } | case refine_2.intro.intro.intro.intro.intro.intro.intro.intro.refine_2
R : Type u_1
E : Type u_3
ι : Type u_5
inst✝² : LinearOrderedField R
inst✝¹ : AddCommGroup E
inst✝ : Module R E
v : ι → E
s : Finset ι
w : ι → R
hw₀ : ∀ i ∈ s, 0 ≤ w i
hw₁ : s.sum w = 1
s' : Finset ι
w' : ι → R
hw₀' : ∀ i ∈ s', 0 ≤ w' i
hw₁' : s'.su... | rw [← sum_subset subset_union_left, ← sum_subset subset_union_right] | case refine_2.intro.intro.intro.intro.intro.intro.intro.intro.refine_2
R : Type u_1
E : Type u_3
ι : Type u_5
inst✝² : LinearOrderedField R
inst✝¹ : AddCommGroup E
inst✝ : Module R E
v : ι → E
s : Finset ι
w : ι → R
hw₀ : ∀ i ∈ s, 0 ≤ w i
hw₁ : s.sum w = 1
s' : Finset ι
w' : ι → R
hw₀' : ∀ i ∈ s', 0 ≤ w' i
hw₁' : s'.su... | c246f547060b42da |
Finset.not_mem_of_mem_powerset_of_not_mem | Mathlib/Data/Finset/Powerset.lean | theorem not_mem_of_mem_powerset_of_not_mem {s t : Finset α} {a : α} (ht : t ∈ s.powerset)
(h : a ∉ s) : a ∉ t | α : Type u_1
s t : Finset α
a : α
ht : t ∈ s.powerset
h : a ∉ s
⊢ a ∉ t | apply mt _ h | α : Type u_1
s t : Finset α
a : α
ht : t ∈ s.powerset
h : a ∉ s
⊢ a ∈ t → a ∈ s | 3a56f3984a651c3c |
CategoryTheory.TwoSquare.GuitartExact.whiskerVertical | Mathlib/CategoryTheory/GuitartExact/VerticalComposition.lean | /-- A 2-square stays Guitart exact if we replace the left and right functors
by isomorphic functors. See also `whiskerVertical_iff`. -/
lemma whiskerVertical [w.GuitartExact] (α : L ≅ L') (β : R ≅ R') :
(w.whiskerVertical α.hom β.inv).GuitartExact | C₁ : Type u_1
C₂ : Type u_2
D₁ : Type u_4
D₂ : Type u_5
inst✝⁴ : Category.{u_7, u_1} C₁
inst✝³ : Category.{u_9, u_2} C₂
inst✝² : Category.{u_8, u_4} D₁
inst✝¹ : Category.{u_10, u_5} D₂
T : C₁ ⥤ D₁
L : C₁ ⥤ C₂
R : D₁ ⥤ D₂
B : C₂ ⥤ D₂
w : TwoSquare T L R B
L' : C₁ ⥤ C₂
R' : D₁ ⥤ D₂
inst✝ : w.GuitartExact
α : L ≅ L'
β : R... | simp only [NatTrans.naturality_assoc, assoc, NatIso.cancel_natIso_inv_left, ← B.map_comp,
Iso.hom_inv_id_app, B.map_id, comp_id] | no goals | f2a5c6dddf567542 |
MeasureTheory.convolution_assoc | Mathlib/Analysis/Convolution.lean | theorem convolution_assoc (hL : ∀ (x : E) (y : E') (z : E''), L₂ (L x y) z = L₃ x (L₄ y z)) {x₀ : G}
(hf : AEStronglyMeasurable f ν) (hg : AEStronglyMeasurable g μ) (hk : AEStronglyMeasurable k μ)
(hfg : ∀ᵐ y ∂μ, ConvolutionExistsAt f g y L ν)
(hgk : ∀ᵐ x ∂ν, ConvolutionExistsAt (fun x => ‖g x‖) (fun x => ‖... | 𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
E'' : Type uE''
F : Type uF
F' : Type uF'
F'' : Type uF''
inst✝²⁶ : NormedAddCommGroup E
inst✝²⁵ : NormedAddCommGroup E'
inst✝²⁴ : NormedAddCommGroup E''
inst✝²³ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝²² : RCLike 𝕜
inst✝²¹ : NormedSpace 𝕜 E
inst✝²⁰ : Norme... | refine L₄.aestronglyMeasurable_comp₂ hg.fst ?_ | 𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
E'' : Type uE''
F : Type uF
F' : Type uF'
F'' : Type uF''
inst✝²⁶ : NormedAddCommGroup E
inst✝²⁵ : NormedAddCommGroup E'
inst✝²⁴ : NormedAddCommGroup E''
inst✝²³ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝²² : RCLike 𝕜
inst✝²¹ : NormedSpace 𝕜 E
inst✝²⁰ : Norme... | f21dcf6a44458531 |
CategoryTheory.IsGrothendieckAbelian.generatingMonomorphisms.exists_transfiniteCompositionOfShape | Mathlib/CategoryTheory/Abelian/GrothendieckCategory/EnoughInjectives.lean | /-- Let `C` be a Grothendieck abelian category. Assume that `G : C` is a generator
of `C`. Then, any morphism in `C` is a transfinite composition of pushouts
of monomorphisms in the family `generatingMonomorphisms G` which consists
of the inclusions of the subobjects of `G`. -/
lemma exists_transfiniteCompositionOfShap... | case intro.intro
C : Type u
inst✝³ : Category.{v, u} C
G : C
inst✝² : Abelian C
hG : IsSeparator G
X : C
inst✝¹ : IsGrothendieckAbelian.{w, v, u} C
A : C
f : A ⟶ X
inst✝ : Mono f
o : Ordinal.{w}
j : o.toType
hj : transfiniteIterate (largerSubobject hG) j (Subobject.mk f) = ⊤
this : OrderBot o.toType := Ordinal.toTypeOr... | exact ⟨_, _, _, _, _, ⟨transfiniteCompositionOfShapeOfEqTop hG hj⟩⟩ | no goals | 628936178606867e |
ascPochhammer_eval_neg_eq_descPochhammer | Mathlib/RingTheory/Polynomial/Pochhammer.lean | theorem ascPochhammer_eval_neg_eq_descPochhammer (r : R) : ∀ (k : ℕ),
(ascPochhammer R k).eval (-r) = (-1)^k * (descPochhammer R k).eval r
| 0 => by
rw [ascPochhammer_zero, descPochhammer_zero]
simp only [eval_one, pow_zero, mul_one]
| (k+1) => by
rw [ascPochhammer_succ_right, mul_add, eval_add, eva... | R : Type u
inst✝ : Ring R
r : R
⊢ eval (-r) 1 = (-1) ^ 0 * eval r 1 | simp only [eval_one, pow_zero, mul_one] | no goals | 91e878a9eb30688b |
Module.Baer.ExtensionOfMaxAdjoin.extensionToFun_wd | Mathlib/Algebra/Module/Injective.lean | theorem ExtensionOfMaxAdjoin.extensionToFun_wd (h : Module.Baer R Q) {y : N}
(x : supExtensionOfMaxSingleton i f y) (a : (extensionOfMax i f).domain)
(r : R) (eq1 : ↑x = ↑a + r • y) :
ExtensionOfMaxAdjoin.extensionToFun i f h x =
(extensionOfMax i f).toLinearPMap a + ExtensionOfMaxAdjoin.extendIdealTo... | case mk
R : Type u
inst✝⁷ : Ring R
Q : Type v
inst✝⁶ : AddCommGroup Q
inst✝⁵ : Module R Q
M : Type u_1
N : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : AddCommGroup N
inst✝² : Module R M
inst✝¹ : Module R N
i : M →ₗ[R] N
f : M →ₗ[R] Q
inst✝ : Fact (Function.Injective ⇑i)
h : Baer R Q
y : N
x : ↥(supExtensionOfMaxSingleton... | simp only [map_sub, sub_smul, sub_eq_iff_eq_add] at eq3 | case mk
R : Type u
inst✝⁷ : Ring R
Q : Type v
inst✝⁶ : AddCommGroup Q
inst✝⁵ : Module R Q
M : Type u_1
N : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : AddCommGroup N
inst✝² : Module R M
inst✝¹ : Module R N
i : M →ₗ[R] N
f : M →ₗ[R] Q
inst✝ : Fact (Function.Injective ⇑i)
h : Baer R Q
y : N
x : ↥(supExtensionOfMaxSingleton... | be5bc40bd455ceb2 |
CategoryTheory.CosimplicialObject.hom_ext | Mathlib/AlgebraicTopology/SimplicialObject/Basic.lean | @[ext]
lemma hom_ext {X Y : CosimplicialObject C} (f g : X ⟶ Y)
(h : ∀ (n : SimplexCategory), f.app n = g.app n) : f = g :=
NatTrans.ext (by ext; apply h)
| case h
C : Type u
inst✝ : Category.{v, u} C
X Y : CosimplicialObject C
f g : X ⟶ Y
h : ∀ (n : SimplexCategory), f.app n = g.app n
x✝ : SimplexCategory
⊢ f.app x✝ = g.app x✝ | apply h | no goals | a0a891c1d68e2274 |
conjneg_sub | Mathlib/Algebra/Star/Conjneg.lean | @[simp] lemma conjneg_sub (f g : G → R) : conjneg (f - g) = conjneg f - conjneg g | G : Type u_2
R : Type u_3
inst✝² : AddGroup G
inst✝¹ : CommRing R
inst✝ : StarRing R
f g : G → R
⊢ conjneg (f - g) = conjneg f - conjneg g | ext | case h
G : Type u_2
R : Type u_3
inst✝² : AddGroup G
inst✝¹ : CommRing R
inst✝ : StarRing R
f g : G → R
x✝ : G
⊢ conjneg (f - g) x✝ = (conjneg f - conjneg g) x✝ | fdd9eb9a553c3338 |
List.map_set | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean | theorem map_set {f : α → β} {l : List α} {i : Nat} {a : α} :
(l.set i a).map f = (l.map f).set i (f a) | α : Type u_1
β : Type u_2
f : α → β
l : List α
i : Nat
a : α
⊢ map f (l.set i a) = (map f l).set i (f a) | induction l generalizing i with
| nil => simp
| cons b l ih => cases i <;> simp_all | no goals | f232fdd07fb65da8 |
MeasureTheory.ae_bdd_condExp_of_ae_bdd | Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean | theorem ae_bdd_condExp_of_ae_bdd {R : ℝ≥0} {f : α → ℝ} (hbdd : ∀ᵐ x ∂μ, |f x| ≤ R) :
∀ᵐ x ∂μ, |(μ[f|m]) x| ≤ R | case pos.refine_2.refine_2
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
h : 0 < μ {x | ↑R < |(μ[f|m]) x|}
x : α
hx : x ∈ {x | ↑R < |(μ[f|m]) x|}
⊢ ⟨↑R, ⋯⟩ ≤ ‖(μ[f|m]) x‖₊ | exact Subtype.mk_le_mk.2 (le_of_lt hx) | no goals | 8f98e6efac06e3d2 |
isSimpleOrder_iff_isSimpleOrder_orderDual | Mathlib/Order/Atoms.lean | theorem isSimpleOrder_iff_isSimpleOrder_orderDual [LE α] [BoundedOrder α] :
IsSimpleOrder α ↔ IsSimpleOrder αᵒᵈ | α : Type u_2
inst✝¹ : LE α
inst✝ : BoundedOrder α
⊢ IsSimpleOrder α ↔ IsSimpleOrder αᵒᵈ | constructor <;> intro i <;> haveI := i | case mp
α : Type u_2
inst✝¹ : LE α
inst✝ : BoundedOrder α
i this : IsSimpleOrder α
⊢ IsSimpleOrder αᵒᵈ
case mpr
α : Type u_2
inst✝¹ : LE α
inst✝ : BoundedOrder α
i this : IsSimpleOrder αᵒᵈ
⊢ IsSimpleOrder α | a78ad2526e8ee886 |
Grp.SurjectiveOfEpiAuxs.g_ne_h | Mathlib/Algebra/Category/Grp/EpiMono.lean | theorem g_ne_h (x : B) (hx : x ∉ f.hom.range) : g ≠ h | A B : Grp
f : A ⟶ B
x : ↑B
hx : x ∉ (Hom.hom f).range
r :
fromCoset ⟨x • ↑⟨↑(Hom.hom f).range, ⋯⟩, ⋯⟩ =
(((Equiv.symm τ).trans (g x)).trans τ) (fromCoset ⟨↑(Hom.hom f).range, ⋯⟩)
⊢ False | simp only [MonoidHom.coe_range, Subtype.coe_mk, Equiv.symm_swap, Equiv.toFun_as_coe,
Equiv.coe_trans, Function.comp_apply] at r | A B : Grp
f : A ⟶ B
x : ↑B
hx : x ∉ (Hom.hom f).range
r : fromCoset ⟨x • Set.range ⇑(Hom.hom f), ⋯⟩ = τ ((g x) ((Equiv.symm τ) (fromCoset ⟨Set.range ⇑(Hom.hom f), ⋯⟩)))
⊢ False | 93413079f037de8f |
Std.Tactic.BVDecide.BVExpr.bitblast.blastShiftLeftConst.go_denote_eq | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/ShiftLeft.lean | theorem go_denote_eq (aig : AIG α) (distance : Nat) (input : AIG.RefVec aig w)
(assign : α → Bool) (curr : Nat) (hcurr : curr ≤ w) (s : AIG.RefVec aig curr) :
∀ (idx : Nat) (hidx1 : idx < w),
curr ≤ idx
→
⟦
(go aig input distance curr hcurr s).aig,
(go aig input dis... | case a
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
distance : Nat
input : aig.RefVec w
assign : α → Bool
curr : Nat
hcurr : curr ≤ w
s : aig.RefVec curr
idx : Nat
hidx1 : idx < w
hidx2 : curr ≤ idx
res : RefVecEntry α w
h✝² : curr < w
h✝¹ : curr < idx
h✝ : curr < distance
hgo :
go (aig.mkCo... | omega | no goals | 625f9e09be737d85 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.