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 |
|---|---|---|---|---|---|---|
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.safe_insert_of_insertRup | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddSound.lean | theorem safe_insert_of_insertRup {n : Nat} (f : DefaultFormula n) (f_readyForRupAdd : ReadyForRupAdd f)
(c : DefaultClause n) :
(insertRupUnits f (negate c)).2 = true → Limplies (PosFin n) f (f.insert c) | n : Nat
f : DefaultFormula n
f_readyForRupAdd : f.ReadyForRupAdd
c : DefaultClause n
h : (f.insertRupUnits c.negate).snd = true
p : PosFin n → Bool
pf : p ⊨ f
⊢ p ⊨ f.insert c | simp only [formulaEntails_def, List.all_eq_true, decide_eq_true_eq] | n : Nat
f : DefaultFormula n
f_readyForRupAdd : f.ReadyForRupAdd
c : DefaultClause n
h : (f.insertRupUnits c.negate).snd = true
p : PosFin n → Bool
pf : p ⊨ f
⊢ ∀ (x : DefaultClause n), x ∈ (f.insert c).toList → p ⊨ x | 3eb652326c8fb9b8 |
MeasureTheory.condExp_zero | Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean | theorem condExp_zero : μ[(0 : α → E)|m] = 0 | case pos
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hm : m ≤ m₀
hμm : SigmaFinite (μ.trim hm)
⊢ μ[0|m] = 0 | exact condExp_of_stronglyMeasurable hm stronglyMeasurable_zero (integrable_zero _ _ _) | no goals | 9d142563cfa659e1 |
MeasureTheory.OuterMeasure.IsCaratheodory.biUnion_of_finite | Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean | lemma IsCaratheodory.biUnion_of_finite {ι : Type*} {s : ι → Set α} {t : Set ι} (ht : t.Finite)
(h : ∀ i ∈ t, m.IsCaratheodory (s i)) :
m.IsCaratheodory (⋃ i ∈ t, s i) | case intro.empty
α : Type u
m : OuterMeasure α
ι : Type u_1
s : ι → Set α
h : ∀ i ∈ ↑∅, m.IsCaratheodory (s i)
⊢ m.IsCaratheodory (⋃ i ∈ ↑∅, s i) | simp | no goals | cba06c9789e0e736 |
PrimeSpectrum.isIdempotentElemEquivClopens_symm_top | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | lemma isIdempotentElemEquivClopens_symm_top :
isIdempotentElemEquivClopens.symm ⊤ = ⟨(1 : R), .one⟩ :=
isIdempotentElemEquivClopens.symm_apply_eq.mpr <| Clopens.ext <| by
rw [coe_isIdempotentElemEquivClopens_apply, basicOpen_one]; rfl
| R : Type u
inst✝ : CommRing R
⊢ ↑⊤ = ↑⊤ | rfl | no goals | 86da9d8dbd28cea9 |
groupCohomology.resolution.forget₂ToModuleCatHomotopyEquiv_f_0_eq | Mathlib/RepresentationTheory/GroupCohomology/Resolution.lean | theorem forget₂ToModuleCatHomotopyEquiv_f_0_eq :
(forget₂ToModuleCatHomotopyEquiv k G).1.f 0 = (forget₂ (Rep k G) _).map (ε k G) | k G : Type u
inst✝¹ : CommRing k
inst✝ : Monoid G
⊢ ((HomotopyEquiv.ofIso (compForgetAugmentedIso k G).symm).hom ≫
(extraDegeneracyCompForgetAugmentedToModule k G).homotopyEquiv.hom ≫
(HomotopyEquiv.ofIso
((ChainComplex.single₀ (ModuleCat k)).mapIso
(LinearEquiv.f... | simp only [HomologicalComplex.comp_f] | k G : Type u
inst✝¹ : CommRing k
inst✝ : Monoid G
⊢ (HomotopyEquiv.ofIso (compForgetAugmentedIso k G).symm).hom.f 0 ≫
(extraDegeneracyCompForgetAugmentedToModule k G).homotopyEquiv.hom.f 0 ≫
(HomotopyEquiv.ofIso
((ChainComplex.single₀ (ModuleCat k)).mapIso
(LinearEquiv.fi... | ae31d1387602f432 |
Finset.lcm_image | Mathlib/Algebra/GCDMonoid/Finset.lean | theorem lcm_image [DecidableEq β] {g : γ → β} (s : Finset γ) :
(s.image g).lcm f = s.lcm (f ∘ g) | α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : NormalizedGCDMonoid α
f : β → α
inst✝ : DecidableEq β
g : γ → β
s : Finset γ
⊢ (image g s).lcm f = s.lcm (f ∘ g) | induction s using Finset.induction <;> simp [*] | no goals | e8222f0889deb6ae |
StrictMono.div_const | Mathlib/Algebra/Order/Field/Basic.lean | theorem StrictMono.div_const {β : Type*} [Preorder β] {f : β → α} (hf : StrictMono f) {c : α}
(hc : 0 < c) : StrictMono fun x => f x / c | α : Type u_2
inst✝¹ : LinearOrderedSemifield α
β : Type u_4
inst✝ : Preorder β
f : β → α
hf : StrictMono f
c : α
hc : 0 < c
⊢ StrictMono fun x => f x / c | simpa only [div_eq_mul_inv] using hf.mul_const (inv_pos.2 hc) | no goals | cd6dde90db842f14 |
IsSuccArchimedean.of_orderIso | Mathlib/Order/SuccPred/Archimedean.lean | /-- `IsSuccArchimedean` transfers across equivalences between `SuccOrder`s. -/
protected lemma IsSuccArchimedean.of_orderIso [SuccOrder X] [IsSuccArchimedean X] [SuccOrder Y]
(f : X ≃o Y) : IsSuccArchimedean Y where
exists_succ_iterate_of_le {a b} h | X : Type u_3
Y : Type u_4
inst✝⁴ : PartialOrder X
inst✝³ : PartialOrder Y
inst✝² : SuccOrder X
inst✝¹ : IsSuccArchimedean X
inst✝ : SuccOrder Y
f : X ≃o Y
a b : Y
h : a ≤ b
⊢ ∀ (a_1 : ℕ), succ^[a_1] (EquivLike.inv f a) = EquivLike.inv f b → succ^[a_1] a = b | intro n | X : Type u_3
Y : Type u_4
inst✝⁴ : PartialOrder X
inst✝³ : PartialOrder Y
inst✝² : SuccOrder X
inst✝¹ : IsSuccArchimedean X
inst✝ : SuccOrder Y
f : X ≃o Y
a b : Y
h : a ≤ b
n : ℕ
⊢ succ^[n] (EquivLike.inv f a) = EquivLike.inv f b → succ^[n] a = b | 2c19e01a1dfddaf9 |
Array.filterMap_subtype | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Attach.lean | theorem filterMap_subtype {p : α → Prop} {l : Array { x // p x }}
{f : { x // p x } → Option β} {g : α → Option β} (hf : ∀ x h, f ⟨x, h⟩ = g x) :
l.filterMap f = l.unattach.filterMap g | case mk
α : Type u_1
β : Type u_2
p : α → Prop
f : { x // p x } → Option β
g : α → Option β
hf : ∀ (x : α) (h : p x), f ⟨x, h⟩ = g x
toList✝ : List { x // p x }
⊢ filterMap f { toList := toList✝ } = filterMap g { toList := toList✝ }.unattach | simp only [size_toArray, List.filterMap_toArray', List.unattach_toArray, List.length_unattach,
mk.injEq] | case mk
α : Type u_1
β : Type u_2
p : α → Prop
f : { x // p x } → Option β
g : α → Option β
hf : ∀ (x : α) (h : p x), f ⟨x, h⟩ = g x
toList✝ : List { x // p x }
⊢ List.filterMap f toList✝ = List.filterMap g toList✝.unattach | dce3ec18d0437417 |
MvPolynomial.prod_C_add_X_eq_sum_esymm | Mathlib/RingTheory/Polynomial/Vieta.lean | theorem MvPolynomial.prod_C_add_X_eq_sum_esymm :
(∏ i : σ, (Polynomial.X + Polynomial.C (MvPolynomial.X i))) =
∑ j ∈ range (card σ + 1), Polynomial.C
(MvPolynomial.esymm σ R j) * Polynomial.X ^ (card σ - j) | R : Type u_1
σ : Type u_2
inst✝¹ : CommSemiring R
inst✝ : Fintype σ
s : Multiset (MvPolynomial σ R) := Multiset.map (fun i => X i) univ.val
⊢ Fintype.card σ = s.card | rw [Multiset.card_map, ← Finset.card_univ, Finset.card_def] | no goals | 4b1054fbcc0b7548 |
exists_msmooth_support_eq_eq_one_iff | Mathlib/Geometry/Manifold/PartitionOfUnity.lean | theorem exists_msmooth_support_eq_eq_one_iff
{s t : Set M} (hs : IsOpen s) (ht : IsClosed t) (h : t ⊆ s) :
∃ f : M → ℝ, ContMDiff I 𝓘(ℝ) ∞ f ∧ range f ⊆ Icc 0 1 ∧ support f = s
∧ (∀ x, x ∈ t ↔ f x = 1) | E : Type uE
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type uH
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : FiniteDimensional ℝ E
inst✝² : IsManifold I ∞ M
inst✝¹ : SigmaCompactSpace M
inst✝ : T2Space M
s t : Set M
hs :... | simp only [mem_compl_iff, Decidable.not_not] at xs | E : Type uE
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type uH
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : FiniteDimensional ℝ E
inst✝² : IsManifold I ∞ M
inst✝¹ : SigmaCompactSpace M
inst✝ : T2Space M
s t : Set M
hs :... | 2508fb22dbe6e145 |
HomogeneousLocalization.val_awayMap_eq_aux | Mathlib/RingTheory/GradedAlgebra/HomogeneousLocalization.lean | lemma val_awayMap_eq_aux (a) : (awayMap 𝒜 hg hx a).val = awayMapAux 𝒜 ⟨_, hx⟩ a | case convert_2
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
inst✝² : AddCommMonoid ι
inst✝¹ : DecidableEq ι
inst✝ : GradedAlgebra 𝒜
e✝ : ι
f g : A
hg : g ∈ 𝒜 e✝
x : A
hx : x = f * g
a : Away 𝒜 f
e : Away 𝒜 x ≃+* ↥(algebraMap (Away 𝒜 x) (... | rw [e.apply_symm_apply] | no goals | f15c4398efada206 |
ContinuousLinearMap.isInvertible_zero_iff | Mathlib/Topology/Algebra/Module/Equiv.lean | theorem isInvertible_zero_iff :
IsInvertible (0 : M →L[R] M₂) ↔ Subsingleton M ∧ Subsingleton M₂ | case refine_2.intro
R : Type u_3
M : Type u_4
M₂ : Type u_5
inst✝⁶ : TopologicalSpace M
inst✝⁵ : TopologicalSpace M₂
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : AddCommMonoid M₂
inst✝ : Module R M₂
hM : Subsingleton M
hM₂ : Subsingleton M₂
e : M ≃L[R] M₂ :=
{ toFun := 0, map_add' := ⋯, m... | ext x | case refine_2.intro.h
R : Type u_3
M : Type u_4
M₂ : Type u_5
inst✝⁶ : TopologicalSpace M
inst✝⁵ : TopologicalSpace M₂
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : AddCommMonoid M₂
inst✝ : Module R M₂
hM : Subsingleton M
hM₂ : Subsingleton M₂
e : M ≃L[R] M₂ :=
{ toFun := 0, map_add' := ⋯,... | c1ca51f554bb9258 |
neg_dotProduct | Mathlib/Data/Matrix/Mul.lean | theorem neg_dotProduct : -v ⬝ᵥ w = -(v ⬝ᵥ w) | m : Type u_2
α : Type v
inst✝¹ : Fintype m
inst✝ : NonUnitalNonAssocRing α
v w : m → α
⊢ -v ⬝ᵥ w = -(v ⬝ᵥ w) | simp [dotProduct] | no goals | da2fc11f780bafdc |
PythagoreanTriple.classification | Mathlib/NumberTheory/PythagoreanTriples.lean | theorem classification :
PythagoreanTriple x y z ↔
∃ k m n,
(x = k * (m ^ 2 - n ^ 2) ∧ y = k * (2 * m * n) ∨
x = k * (2 * m * n) ∧ y = k * (m ^ 2 - n ^ 2)) ∧
(z = k * (m ^ 2 + n ^ 2) ∨ z = -k * (m ^ 2 + n ^ 2)) | case h.intro.inr.intro
z k m n : ℤ
right✝ : m.gcd n = 1
h : PythagoreanTriple (k * (2 * m * n)) (k * (m ^ 2 - n ^ 2)) z
this : z ^ 2 = (k * (m ^ 2 + n ^ 2)) ^ 2
⊢ z = k * (m ^ 2 + n ^ 2) ∨ z = -k * (m ^ 2 + n ^ 2) | simpa using eq_or_eq_neg_of_sq_eq_sq _ _ this | no goals | 09f1815ea86a75e0 |
CompactExhaustion.find_shiftr | Mathlib/Topology/Compactness/SigmaCompact.lean | theorem find_shiftr (x : X) : K.shiftr.find x = K.find x + 1 | X : Type u_1
inst✝ : TopologicalSpace X
K : CompactExhaustion X
x : X
⊢ K.shiftr.find x = K.find x + 1 | exact Nat.find_comp_succ _ _ (not_mem_empty _) | no goals | 40378a12b786ce5b |
MeasureTheory.IsProjectiveMeasureFamily.congr_cylinder_of_subset | Mathlib/MeasureTheory/Constructions/Projective.lean | lemma congr_cylinder_of_subset (hP : IsProjectiveMeasureFamily P)
{S : Set (∀ i : I, α i)} {T : Set (∀ i : J, α i)} (hT : MeasurableSet T)
(h_eq : cylinder I S = cylinder J T) (hJI : J ⊆ I) :
P I S = P J T | case inl.intro
ι : Type u_1
α : ι → Type u_2
inst✝ : (i : ι) → MeasurableSpace (α i)
P : (J : Finset ι) → Measure ((j : { x // x ∈ J }) → α ↑j)
I✝ J : Finset ι
hP : IsProjectiveMeasureFamily P
S : Set ((i : { x // x ∈ I✝ }) → α ↑i)
T : Set ((i : { x // x ∈ J }) → α ↑i)
hT : MeasurableSet T
h_eq : cylinder I✝ S = cylind... | simp [this] | no goals | 25653748ec5eb76b |
Finset.piAntidiag_zero | Mathlib/Algebra/Order/Antidiag/Pi.lean | @[simp] lemma piAntidiag_zero (s : Finset ι) : piAntidiag s (0 : μ) = {0} | case h
ι : Type u_1
μ : Type u_2
inst✝⁴ : DecidableEq ι
inst✝³ : OrderedAddCommMonoid μ
inst✝² : CanonicallyOrderedAdd μ
inst✝¹ : HasAntidiagonal μ
inst✝ : DecidableEq μ
s : Finset ι
a✝ : ι → μ
⊢ a✝ ∈ s.piAntidiag 0 ↔ a✝ ∈ {0} | simp [Fintype.sum_eq_zero_iff_of_nonneg, funext_iff, not_imp_comm, ← forall_and] | no goals | 9b8c45c99edc8f9b |
Vector.all_eq | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Lemmas.lean | theorem all_eq {xs : Vector α n} {p : α → Bool} : xs.all p = decide (∀ i, (_ : i < n) → p xs[i]) | α : Type u_1
n : Nat
xs : Vector α n
p : α → Bool
⊢ xs.all p = decide (∀ (i : Nat) (x : i < n), p xs[i] = true) | by_cases h : xs.all p | case pos
α : Type u_1
n : Nat
xs : Vector α n
p : α → Bool
h : xs.all p = true
⊢ xs.all p = decide (∀ (i : Nat) (x : i < n), p xs[i] = true)
case neg
α : Type u_1
n : Nat
xs : Vector α n
p : α → Bool
h : ¬xs.all p = true
⊢ xs.all p = decide (∀ (i : Nat) (x : i < n), p xs[i] = true) | 1a2b4e146136e5cc |
ProbabilityTheory.iIndepFun.integrable_exp_mul_sum | Mathlib/Probability/Moments/Basic.lean | theorem iIndepFun.integrable_exp_mul_sum [IsFiniteMeasure μ] {X : ι → Ω → ℝ}
(h_indep : iIndepFun (fun _ => inferInstance) X μ) (h_meas : ∀ i, Measurable (X i))
{s : Finset ι} (h_int : ∀ i ∈ s, Integrable (fun ω => exp (t * X i ω)) μ) :
Integrable (fun ω => exp (t * (∑ i ∈ s, X i) ω)) μ | case insert
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
μ : Measure Ω
t : ℝ
inst✝ : IsFiniteMeasure μ
X : ι → Ω → ℝ
h_indep : iIndepFun (fun x => inferInstance) X μ
h_meas : ∀ (i : ι), Measurable (X i)
i : ι
s : Finset ι
hi_notin_s : i ∉ s
h_int : ∀ i_1 ∈ insert i s, Integrable (fun ω => rexp (t * X i_1 ω)) μ
this ... | refine IndepFun.integrable_exp_mul_add ?_ (h_int i (mem_insert_self _ _)) h_rec | case insert
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
μ : Measure Ω
t : ℝ
inst✝ : IsFiniteMeasure μ
X : ι → Ω → ℝ
h_indep : iIndepFun (fun x => inferInstance) X μ
h_meas : ∀ (i : ι), Measurable (X i)
i : ι
s : Finset ι
hi_notin_s : i ∉ s
h_int : ∀ i_1 ∈ insert i s, Integrable (fun ω => rexp (t * X i_1 ω)) μ
this ... | 97e890d858de8868 |
Array.foldl_toList_eq_map | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean | theorem foldl_toList_eq_map (l : List α) (acc : Array β) (G : α → β) :
(l.foldl (fun acc a => acc.push (G a)) acc).toList = acc.toList ++ l.map G | α : Type u_1
β : Type u_2
l : List α
acc : Array β
G : α → β
⊢ (List.foldl (fun acc a => acc.push (G a)) acc l).toList = acc.toList ++ List.map G l | induction l generalizing acc <;> simp [*] | no goals | 4e33a55be9877b31 |
CategoryTheory.Functor.partialLeftAdjointHomEquiv_map | Mathlib/CategoryTheory/Adjunction/PartialAdjoint.lean | @[simp]
lemma partialLeftAdjointHomEquiv_map {X Y : F.PartialLeftAdjointSource}
(f : X ⟶ Y) :
F.partialLeftAdjointHomEquiv (F.partialLeftAdjointMap f) =
by exact f ≫ F.partialLeftAdjointHomEquiv (𝟙 _) | C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F : D ⥤ C
X Y : F.PartialLeftAdjointSource
f : X ⟶ Y
⊢ F.partialLeftAdjointHomEquiv (F.partialLeftAdjointMap f) =
f ≫ F.partialLeftAdjointHomEquiv (𝟙 (F.partialLeftAdjointObj Y)) | simp [partialLeftAdjointMap] | no goals | c9c9890f4082c996 |
DirichletCharacter.factorsThrough_one_iff | Mathlib/NumberTheory/DirichletCharacter/Basic.lean | lemma factorsThrough_one_iff : FactorsThrough χ 1 ↔ χ = 1 | R : Type u_1
inst✝ : CommMonoidWithZero R
n : ℕ
χ : DirichletCharacter R n
h : χ = 1
⊢ χ = (changeLevel ⋯) 1 | rw [h, changeLevel_one] | no goals | 6ab1863f75a5a581 |
ZMod.natCast_natAbs_valMinAbs | Mathlib/Data/ZMod/ValMinAbs.lean | lemma natCast_natAbs_valMinAbs [NeZero n] (a : ZMod n) :
(a.valMinAbs.natAbs : ZMod n) = if a.val ≤ (n : ℕ) / 2 then a else -a | case neg
n : ℕ
inst✝ : NeZero n
a : ZMod n
this : ↑a.val - ↑n ≤ 0
h✝ : ¬a.val ≤ n / 2
⊢ ↑(↑a.val - ↑n).natAbs = -a | rw [← Int.cast_natCast, Int.ofNat_natAbs_of_nonpos this, Int.cast_neg, Int.cast_sub,
Int.cast_natCast, Int.cast_natCast, natCast_self, sub_zero, natCast_zmod_val] | no goals | c0525c25e68605f4 |
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})) | case hm
μ : Measure ℝ
inst✝ : IsFiniteMeasureOnCompacts μ
b : ℝ
⊢ ∀ (i j : ℝ), 0 < i → i ≤ j → Icc (b - i) (b + i) ⊆ Icc (b - j) (b + j) | intro r s _rpos hrs | case hm
μ : Measure ℝ
inst✝ : IsFiniteMeasureOnCompacts μ
b r s : ℝ
_rpos : 0 < r
hrs : r ≤ s
⊢ Icc (b - r) (b + r) ⊆ Icc (b - s) (b + s) | 234976a60451ee77 |
MeasureTheory.setIntegral_eq_zero_of_ae_eq_zero | Mathlib/MeasureTheory/Integral/SetIntegral.lean | theorem setIntegral_eq_zero_of_ae_eq_zero (ht_eq : ∀ᵐ x ∂μ, x ∈ t → f x = 0) :
∫ x in t, f x ∂μ = 0 | case h
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
t : Set X
μ : Measure X
ht_eq : ∀ᵐ (x : X) ∂μ, x ∈ t → f x = 0
hf : AEStronglyMeasurable f (μ.restrict t)
x : X
hx : x ∈ t → f x = AEStronglyMeasurable.mk f hf x
h'x : x ∈ t → f x = 0
h''x : x ∈ t
⊢ A... | rw [← hx h''x] | case h
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
t : Set X
μ : Measure X
ht_eq : ∀ᵐ (x : X) ∂μ, x ∈ t → f x = 0
hf : AEStronglyMeasurable f (μ.restrict t)
x : X
hx : x ∈ t → f x = AEStronglyMeasurable.mk f hf x
h'x : x ∈ t → f x = 0
h''x : x ∈ t
⊢ f... | ea5d4ba165902531 |
Ideal.Filtration.submodule_eq_span_le_iff_stable_ge | Mathlib/RingTheory/Filtration.lean | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | case neg.intro.intro.refine_2
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F : I.Filtration M
n₀ : ℕ
F' : Submodule (↥(reesAlgebra I)) (PolynomialModule R M) :=
Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(F.N i))
hF : ∀ n ≥ n₀, I •... | exact F'.add_mem hx hy | no goals | fba02d553ffa8877 |
IsNoetherian.injective_of_surjective_of_injective | Mathlib/RingTheory/Noetherian/Orzech.lean | theorem IsNoetherian.injective_of_surjective_of_injective (i f : N →ₗ[R] M)
(hi : Injective i) (hf : Surjective f) : Injective f | R : Type u_1
M : Type u_2
N : Type w
inst✝⁵ : Ring R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : IsNoetherian R M
i f : N →ₗ[R] M
hi : Injective ⇑i
hf : Surjective ⇑f
⊢ Injective ⇑f | haveI := isNoetherian_of_injective i hi | R : Type u_1
M : Type u_2
N : Type w
inst✝⁵ : Ring R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : IsNoetherian R M
i f : N →ₗ[R] M
hi : Injective ⇑i
hf : Surjective ⇑f
this : IsNoetherian R N
⊢ Injective ⇑f | 103a63e5bb557d5a |
Odd.strictMono_pow | Mathlib/Algebra/Order/Ring/Basic.lean | lemma Odd.strictMono_pow (hn : Odd n) : StrictMono fun a : R => a ^ n | case inr.inr.intro.intro
R : Type u_3
inst✝¹ : LinearOrderedSemiring R
n : ℕ
inst✝ : ExistsAddOfLE R
hn : Odd n
hn₀ : n ≠ 0
a b : R
hab : a < b
ha : a ≤ 0
hb : b ≤ 0
c : R
hac : 0 = a + c
d : R
hbd : 0 = b + d
hd : 0 ≤ d
⊢ (fun a => a ^ n) a + (c ^ n + d ^ n) < (fun a => a ^ n) b + (c ^ n + d ^ n) | dsimp | case inr.inr.intro.intro
R : Type u_3
inst✝¹ : LinearOrderedSemiring R
n : ℕ
inst✝ : ExistsAddOfLE R
hn : Odd n
hn₀ : n ≠ 0
a b : R
hab : a < b
ha : a ≤ 0
hb : b ≤ 0
c : R
hac : 0 = a + c
d : R
hbd : 0 = b + d
hd : 0 ≤ d
⊢ a ^ n + (c ^ n + d ^ n) < b ^ n + (c ^ n + d ^ n) | 789fc8d7e45ac2d6 |
ENNReal.tendsto_toNNReal | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | theorem tendsto_toNNReal {a : ℝ≥0∞} (ha : a ≠ ∞) :
Tendsto ENNReal.toNNReal (𝓝 a) (𝓝 a.toNNReal) | case intro
a : ℝ≥0
⊢ Tendsto ENNReal.toNNReal (𝓝 ↑a) (𝓝 (↑a).toNNReal) | rw [nhds_coe, tendsto_map'_iff] | case intro
a : ℝ≥0
⊢ Tendsto (ENNReal.toNNReal ∘ ofNNReal) (𝓝 a) (𝓝 (↑a).toNNReal) | 847cc9335ef2aff3 |
AlgebraicGeometry.quasiSeparatedSpace_iff_affine | Mathlib/AlgebraicGeometry/Morphisms/QuasiSeparated.lean | theorem quasiSeparatedSpace_iff_affine (X : Scheme) :
QuasiSeparatedSpace X ↔ ∀ U V : X.affineOpens, IsCompact (U ∩ V : Set X) | case mp
X : Scheme
⊢ (∀ (U V : Set ↑↑X.toPresheafedSpace), IsOpen U → IsCompact U → IsOpen V → IsCompact V → IsCompact (U ∩ V)) →
∀ (U V : ↑X.affineOpens), IsCompact (↑↑U ∩ ↑↑V) | intro H U V | case mp
X : Scheme
H : ∀ (U V : Set ↑↑X.toPresheafedSpace), IsOpen U → IsCompact U → IsOpen V → IsCompact V → IsCompact (U ∩ V)
U V : ↑X.affineOpens
⊢ IsCompact (↑↑U ∩ ↑↑V) | 297e2d46fd541533 |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.confirmRupHint_preserves_invariant_helper | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddResult.lean | theorem confirmRupHint_preserves_invariant_helper {n : Nat} (f : DefaultFormula n)
(f_assignments_size : f.assignments.size = n)
(acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool) (hsize : acc.1.size = n)
(l : Literal (PosFin n)) (ih : DerivedLitsInvariant f f_assignments_size acc.1 hsize acc.2.... | case right.right.right.left
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool
hsize : acc.fst.size = n
l : Literal (PosFin n)
ih : f.DerivedLitsInvariant f_assignments_size acc.fst hsize acc.snd.fst
h : ¬hasAssignment l.snd acc.fst[l.fs... | simp only [hasAssignment, l_eq_true, getElem!, l_eq_i, i_in_bounds,
Array.get_eq_getElem, ↓reduceIte, ↓reduceDIte, h1, addAssignment, l'_eq_false,
hasPos_addNeg, decidableGetElem?, reduceCtorEq] at h | case right.right.right.left
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool
hsize : acc.fst.size = n
l : Literal (PosFin n)
ih : f.DerivedLitsInvariant f_assignments_size acc.fst hsize acc.snd.fst
hsize'✝ : (acc.fst.modify l.fst.val (... | 02ca1706b397b254 |
ENNReal.hasBasis_nhds_of_ne_top | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | theorem hasBasis_nhds_of_ne_top (xt : x ≠ ∞) :
(𝓝 x).HasBasis (0 < ·) (fun ε => Icc (x - ε) (x + ε)) | x : ℝ≥0∞
xt : x ≠ ⊤
⊢ (𝓝 x).HasBasis (fun x => 0 < x) fun ε => Icc (x - ε) (x + ε) | simpa only [pos_iff_ne_zero] using hasBasis_nhds_of_ne_top' xt | no goals | 639e8394832b9678 |
FreeCommRing.isSupported_of | Mathlib/RingTheory/FreeCommRing.lean | theorem isSupported_of {p} {s : Set α} : IsSupported (of p) s ↔ p ∈ s :=
suffices IsSupported (of p) s → p ∈ s from ⟨this, fun hps => Subring.subset_closure ⟨p, hps, rfl⟩⟩
fun hps : IsSupported (of p) s => by
classical
haveI := Classical.decPred s
have : ∀ x, IsSupported x s →
∃ n : ℤ, lift (fun a => ... | case h
α : Type u
p : α
s : Set α
hps : (of p).IsSupported s
this : DecidablePred s
x : FreeCommRing α
hx : x.IsSupported s
⊢ 1 = ↑1 | norm_cast | no goals | 962f6b77428c03f2 |
ProbabilityTheory.meas_ge_le_evariance_div_sq | Mathlib/Probability/Variance.lean | theorem meas_ge_le_evariance_div_sq {X : Ω → ℝ} (hX : AEStronglyMeasurable X μ) {c : ℝ≥0}
(hc : c ≠ 0) : μ {ω | ↑c ≤ |X ω - μ[X]|} ≤ evariance X μ / c ^ 2 | case h.e'_4.e_a
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
hX : AEStronglyMeasurable X μ
c : ℝ≥0
hc : c ≠ 0
A : ↑c ≠ 0
B : AEStronglyMeasurable (fun x => ∫ (x : Ω), X x ∂μ) μ
⊢ eVar[X; μ] = ((∫⁻ (x : Ω), ‖X x - ∫ (x : Ω), X x ∂μ‖ₑ ^ 2 ∂μ) ^ 2⁻¹) ^ 2 | simp_rw [← ENNReal.rpow_mul, inv_mul_cancel₀ (two_ne_zero : (2 : ℝ) ≠ 0), ENNReal.rpow_two,
ENNReal.rpow_one, evariance] | no goals | bfd0c12c76eae2a1 |
ContMDiffWithinAt.prod_mk_space | Mathlib/Geometry/Manifold/ContMDiff/Constructions.lean | theorem ContMDiffWithinAt.prod_mk_space {f : M → E'} {g : M → F'}
(hf : ContMDiffWithinAt I 𝓘(𝕜, E') n f s x) (hg : ContMDiffWithinAt I 𝓘(𝕜, F') n g s x) :
ContMDiffWithinAt I 𝓘(𝕜, E' × F') n (fun x => (f x, g x)) s x | 𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
E' : Type u_5
inst✝³ : NormedAddCommGroup E'
inst✝² : NormedSpace... | rw [contMDiffWithinAt_iff] at * | 𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
E' : Type u_5
inst✝³ : NormedAddCommGroup E'
inst✝² : NormedSpace... | d81bc559f7e409cc |
HurwitzZeta.hurwitzZeta_one_sub_two_mul_nat | Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean | private lemma hurwitzZeta_one_sub_two_mul_nat (hk : k ≠ 0) (hx : x ∈ Icc (0 : ℝ) 1) :
hurwitzZeta x (1 - 2 * k) =
-1 / (2 * k) * ((Polynomial.bernoulli (2 * k)).map (algebraMap ℚ ℂ)).eval (x : ℂ) | k : ℕ
x : ℝ
hk : k ≠ 0
hx : x ∈ Icc 0 1
⊢ hurwitzZeta (↑x) (1 - 2 * ↑k) =
-1 / (2 * ↑k) * Polynomial.eval (↑x) (Polynomial.map (algebraMap ℚ ℂ) (Polynomial.bernoulli (2 * k))) | suffices hurwitzZetaOdd x (1 - 2 * k) = 0 by
rw [hurwitzZeta, this, add_zero, hurwitzZetaEven_one_sub_two_mul_nat hk hx] | k : ℕ
x : ℝ
hk : k ≠ 0
hx : x ∈ Icc 0 1
⊢ hurwitzZetaOdd (↑x) (1 - 2 * ↑k) = 0 | 5028dc0dd1e49d06 |
Finset.map_add_left_Ioc | Mathlib/Algebra/Order/Interval/Finset/Basic.lean | @[simp] lemma map_add_left_Ioc (a b c : α) :
(Ioc a b).map (addLeftEmbedding c) = Ioc (c + a) (c + b) | α : Type u_2
inst✝² : OrderedCancelAddCommMonoid α
inst✝¹ : ExistsAddOfLE α
inst✝ : LocallyFiniteOrder α
a b c : α
⊢ map (addLeftEmbedding c) (Ioc a b) = Ioc (c + a) (c + b) | rw [← coe_inj, coe_map, coe_Ioc, coe_Ioc] | α : Type u_2
inst✝² : OrderedCancelAddCommMonoid α
inst✝¹ : ExistsAddOfLE α
inst✝ : LocallyFiniteOrder α
a b c : α
⊢ ⇑(addLeftEmbedding c) '' Set.Ioc a b = Set.Ioc (c + a) (c + b) | 36fff7f04ca22237 |
GromovHausdorff.totallyBounded | Mathlib/Topology/MetricSpace/GromovHausdorff.lean | theorem totallyBounded {t : Set GHSpace} {C : ℝ} {u : ℕ → ℝ} {K : ℕ → ℕ}
(ulim : Tendsto u atTop (𝓝 0)) (hdiam : ∀ p ∈ t, diam (univ : Set (GHSpace.Rep p)) ≤ C)
(hcov : ∀ p ∈ t, ∀ n : ℕ, ∃ s : Set (GHSpace.Rep p),
(#s) ≤ K n ∧ univ ⊆ ⋃ x ∈ s, ball x (u n)) :
TotallyBounded t | t : Set GHSpace
C : ℝ
u : ℕ → ℝ
K : ℕ → ℕ
ulim : Tendsto u atTop (𝓝 0)
hdiam : ∀ p ∈ t, diam univ ≤ C
hcov : ∀ p ∈ t, ∀ (n : ℕ), ∃ s, #↑s ≤ ↑(K n) ∧ univ ⊆ ⋃ x ∈ s, ball x (u n)
δ : ℝ
δpos : δ > 0
ε : ℝ := 1 / 5 * δ
εpos : 0 < ε
n : ℕ
hn : ∀ n_1 ≥ n, dist (u n_1) 0 < ε
u_le_ε : u n ≤ ε
s : (p : GHSpace) → Set p.Rep
N ... | simp only [i, Ψ, Equiv.apply_symm_apply, Fin.coe_cast] | no goals | c10025d34debaff9 |
List.not_nil_lex_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lex.lean | theorem not_nil_lex_iff : ¬Lex r [] l ↔ l = [] | case mp
α✝ : Type u_1
r : α✝ → α✝ → Prop
l : List α✝
⊢ ¬Lex r [] l → l = [] | rintro h | case mp
α✝ : Type u_1
r : α✝ → α✝ → Prop
l : List α✝
h : ¬Lex r [] l
⊢ l = [] | 1f504df95e7b72cc |
List.filterMapM_loop_eq | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Monadic.lean | theorem filterMapM_loop_eq [Monad m] [LawfulMonad m]
(f : α → m (Option β)) (l : List α) (acc : List β) :
filterMapM.loop f l acc = (acc.reverse ++ ·) <$> filterMapM.loop f l [] | case cons
m : Type u_1 → Type u_2
α β : Type u_1
inst✝¹ : Monad m
inst✝ : LawfulMonad m
f : α → m (Option β)
a : α
l : List α
ih : ∀ (acc : List β), filterMapM.loop f l acc = (fun x => acc.reverse ++ x) <$> filterMapM.loop f l []
acc : List β
⊢ (do
let __do_lift ← f a
match __do_lift with
| none => ... | congr | case cons.e_a
m : Type u_1 → Type u_2
α β : Type u_1
inst✝¹ : Monad m
inst✝ : LawfulMonad m
f : α → m (Option β)
a : α
l : List α
ih : ∀ (acc : List β), filterMapM.loop f l acc = (fun x => acc.reverse ++ x) <$> filterMapM.loop f l []
acc : List β
⊢ (fun __do_lift =>
match __do_lift with
| none => filterMapM... | 3de0a791d3e5ce3f |
Doset.doset_eq_of_mem | Mathlib/GroupTheory/DoubleCoset.lean | theorem doset_eq_of_mem {H K : Subgroup G} {a b : G} (hb : b ∈ doset a H K) :
doset b H K = doset a H K | G : Type u_1
inst✝ : Group G
H K : Subgroup G
a b : G
hb : b ∈ doset a ↑H ↑K
⊢ doset b ↑H ↑K = doset a ↑H ↑K | obtain ⟨h, hh, k, hk, rfl⟩ := mem_doset.1 hb | case intro.intro.intro.intro
G : Type u_1
inst✝ : Group G
H K : Subgroup G
a h : G
hh : h ∈ ↑H
k : G
hk : k ∈ ↑K
hb : h * a * k ∈ doset a ↑H ↑K
⊢ doset (h * a * k) ↑H ↑K = doset a ↑H ↑K | 40616e2eb7ee6747 |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.restoreAssignments_performRupCheck_base_case | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddResult.lean | theorem restoreAssignments_performRupCheck_base_case {n : Nat} (f : DefaultFormula n)
(f_assignments_size : f.assignments.size = n)
(f' : DefaultFormula n) (_f'_def : f' = (performRupCheck f rupHints).1)
(f'_assignments_size : f'.assignments.size = n) (derivedLits : CNF.Clause (PosFin n))
(derivedLits_a... | case inr.inl.intro.intro.intro.intro
rupHints : Array Nat
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
f' : DefaultFormula n
_f'_def : f' = (f.performRupCheck rupHints).fst
f'_assignments_size : f'.assignments.size = n
derivedLits : CNF.Clause (PosFin n)
derivedLits_arr : Array (Literal (Pos... | have i_gt_zero : i.1 > 0 := by rw [← j_eq_i]; exact (List.get derivedLits j).1.2.1 | case inr.inl.intro.intro.intro.intro
rupHints : Array Nat
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
f' : DefaultFormula n
_f'_def : f' = (f.performRupCheck rupHints).fst
f'_assignments_size : f'.assignments.size = n
derivedLits : CNF.Clause (PosFin n)
derivedLits_arr : Array (Literal (Pos... | 92c646ad371b893c |
Turing.TM2to1.tr_respects_aux₂ | Mathlib/Computability/TuringMachine.lean | theorem tr_respects_aux₂ [DecidableEq K] {k : K} {q : TM1.Stmt (Γ' K Γ) (Λ' K Γ Λ σ) σ} {v : σ}
{S : ∀ k, List (Γ k)} {L : ListBlank (∀ k, Option (Γ k))}
(hL : ∀ k, L.map (proj k) = ListBlank.mk ((S k).map some).reverse) (o : StAct K Γ σ k) :
let v' := stVar v (S k) o
let Sk' := stWrite v (S k) o
le... | case neg.inl
K : Type u_1
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
inst✝ : DecidableEq K
k : K
q : TM1.Stmt (Γ' K Γ) (Λ' K Γ Λ σ) σ
v : σ
S : (k : K) → List (Γ k)
L : ListBlank ((k : K) → Option (Γ k))
hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.map some (S k)).reverse
f : σ → Option (Γ k) → σ
hd : ... | rw [List.getI_append] | case neg.inl.h
K : Type u_1
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
inst✝ : DecidableEq K
k : K
q : TM1.Stmt (Γ' K Γ) (Λ' K Γ Λ σ) σ
v : σ
S : (k : K) → List (Γ k)
L : ListBlank ((k : K) → Option (Γ k))
hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.map some (S k)).reverse
f : σ → Option (Γ k) → σ
hd ... | 77a7f2ed036af218 |
sup_himp_self_right | Mathlib/Order/Heyting/Basic.lean | theorem sup_himp_self_right (a b : α) : a ⊔ b ⇨ b = a ⇨ b | α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a b : α
⊢ a ⊔ b ⇨ b = a ⇨ b | rw [sup_himp_distrib, himp_self, inf_top_eq] | no goals | 79c36fca84f242c4 |
Int.lt_zpow_succ_log_self | Mathlib/Data/Int/Log.lean | theorem lt_zpow_succ_log_self {b : ℕ} (hb : 1 < b) (r : R) : r < (b : R) ^ (log b r + 1) | case inr
R : Type u_1
inst✝¹ : LinearOrderedSemifield R
inst✝ : FloorSemiring R
b : ℕ
hb : 1 < b
r : R
hr : 0 < r
⊢ r < ↑b ^ (log b r + 1) | rcases le_or_lt 1 r with hr1 | hr1 | case inr.inl
R : Type u_1
inst✝¹ : LinearOrderedSemifield R
inst✝ : FloorSemiring R
b : ℕ
hb : 1 < b
r : R
hr : 0 < r
hr1 : 1 ≤ r
⊢ r < ↑b ^ (log b r + 1)
case inr.inr
R : Type u_1
inst✝¹ : LinearOrderedSemifield R
inst✝ : FloorSemiring R
b : ℕ
hb : 1 < b
r : R
hr : 0 < r
hr1 : r < 1
⊢ r < ↑b ^ (log b r + 1) | d923e983e06cf99e |
List.getElem_insertIdx | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/InsertIdx.lean | theorem getElem_insertIdx {l : List α} {x : α} {n k : Nat} (h : k < (insertIdx n x l).length) :
(insertIdx n x l)[k] =
if h₁ : k < n then
l[k]'(by simp [length_insertIdx] at h; split at h <;> omega)
else
if h₂ : k = n then
x
else
l[k-1]'(by simp [length_insert... | case isFalse
α : Type u
l : List α
x : α
n k : Nat
h : k < (insertIdx n x l).length
h₁ : ¬k < n
⊢ (insertIdx n x l)[k] = if h₂ : k = n then x else l[k - 1] | split <;> rename_i h₂ | case isFalse.isTrue
α : Type u
l : List α
x : α
n k : Nat
h : k < (insertIdx n x l).length
h₁ : ¬k < n
h₂ : k = n
⊢ (insertIdx n x l)[k] = x
case isFalse.isFalse
α : Type u
l : List α
x : α
n k : Nat
h : k < (insertIdx n x l).length
h₁ : ¬k < n
h₂ : ¬k = n
⊢ (insertIdx n x l)[k] = l[k - 1] | 2303cbe5977d4288 |
Submodule.IsPrincipal.generator_mem | Mathlib/RingTheory/PrincipalIdealDomain.lean | theorem generator_mem (S : Submodule R M) [S.IsPrincipal] : generator S ∈ S | R : Type u
M : Type v
inst✝³ : AddCommMonoid M
inst✝² : Semiring R
inst✝¹ : Module R M
S : Submodule R M
inst✝ : S.IsPrincipal
this : generator S ∈ span R {generator S}
⊢ generator S ∈ S | convert this | case h.e'_4
R : Type u
M : Type v
inst✝³ : AddCommMonoid M
inst✝² : Semiring R
inst✝¹ : Module R M
S : Submodule R M
inst✝ : S.IsPrincipal
this : generator S ∈ span R {generator S}
⊢ S = span R {generator S} | cd3340efd272d171 |
subsingleton_of_disjoint_isClopen | Mathlib/Topology/Connected/Clopen.lean | /-- In a preconnected space, any disjoint family of non-empty clopen subsets has at most one
element. -/
lemma subsingleton_of_disjoint_isClopen
(h_clopen : ∀ i, IsClopen (s i)) :
Subsingleton ι | α : Type u
ι : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : PreconnectedSpace α
s : ι → Set α
h_disj : Pairwise (Disjoint on s)
h_clopen : ∀ (i : ι), IsClopen (s i)
h_nonempty : ∀ (i : ι), s i ≠ ∅
⊢ Subsingleton ι | rw [← not_nontrivial_iff_subsingleton] | α : Type u
ι : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : PreconnectedSpace α
s : ι → Set α
h_disj : Pairwise (Disjoint on s)
h_clopen : ∀ (i : ι), IsClopen (s i)
h_nonempty : ∀ (i : ι), s i ≠ ∅
⊢ ¬Nontrivial ι | 6e05e6622254f214 |
Pell.eq_pow_of_pell | Mathlib/NumberTheory/PellMatiyasevic.lean | theorem eq_pow_of_pell {m n k} :
n ^ k = m ↔ k = 0 ∧ m = 1 ∨0 < k ∧ (n = 0 ∧ m = 0 ∨
0 < n ∧ ∃ (w a t z : ℕ) (a1 : 1 < a), xn a1 k ≡ yn a1 k * (a - n) + m [MOD t] ∧
2 * a * n = t + (n * n + 1) ∧ m < t ∧
n ≤ w ∧ k ≤ w ∧ a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1) | case mpr.inr.intro.inr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
m n k : ℕ
hk0 : 0 < k
hn0 : 0 < n
w t z : ℕ
mt : m < t
nw : n ≤ w
kw : k ≤ w
hw0 : 0 < w
hw1 : 1 < w + 1
j : ℕ
yj : w * z = yn hw1 j
a1 : 1 < xn hw1 j
tm : xn a1 k ≡ yn a1 k * (xn hw1 j - n) + m [MOD t]
ta : 2 * xn hw1 ... | have : xn a1 k ≡ yn a1 k * (xn hw1 j - n) + n ^ k [MOD t] := by
apply modEq_of_dvd
rw [te, Nat.cast_add, Nat.cast_mul, Int.ofNat_sub na]
exact x_sub_y_dvd_pow a1 n k | case mpr.inr.intro.inr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
m n k : ℕ
hk0 : 0 < k
hn0 : 0 < n
w t z : ℕ
mt : m < t
nw : n ≤ w
kw : k ≤ w
hw0 : 0 < w
hw1 : 1 < w + 1
j : ℕ
yj : w * z = yn hw1 j
a1 : 1 < xn hw1 j
tm : xn a1 k ≡ yn a1 k * (xn hw1 j - n) + m [MOD t]
ta : 2 * xn hw1 ... | a363d7dbe39afd6c |
Turing.Tape.move_right_mk' | Mathlib/Computability/Tape.lean | theorem Tape.move_right_mk' {Γ} [Inhabited Γ] (L R : ListBlank Γ) :
(Tape.mk' L R).move Dir.right = Tape.mk' (L.cons R.head) R.tail | Γ : Type u_1
inst✝ : Inhabited Γ
L R : ListBlank Γ
⊢ move Dir.right (mk' L R) = mk' (ListBlank.cons R.head L) R.tail | simp only [Tape.move, Tape.mk', ListBlank.head_cons, eq_self_iff_true, ListBlank.cons_head_tail,
and_self_iff, ListBlank.tail_cons] | no goals | a02cdca4d4db7aec |
Padic.padicNormE_lim_le | Mathlib/NumberTheory/Padics/PadicNumbers.lean | theorem padicNormE_lim_le {f : CauSeq ℚ_[p] norm} {a : ℝ} (ha : 0 < a) (hf : ∀ i, ‖f i‖ ≤ a) :
‖f.lim‖ ≤ a | p : ℕ
hp : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] norm
a : ℝ
ha : 0 < a
hf : ∀ (i : ℕ), ‖↑f i‖ ≤ a
N : ℕ
hN : ∀ j ≥ N, ‖↑(const norm f.lim - f) j‖ < a
⊢ ‖f.lim‖ = ‖f.lim - ↑f N + ↑f N‖ | simp | no goals | 1cb646383844b024 |
ENNReal.liminf_toReal_eq | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | /-- If `xs : ι → ℝ≥0∞` is bounded, then we have `liminf (toReal ∘ xs) = toReal (liminf xs)`. -/
lemma liminf_toReal_eq {ι : Type*} {F : Filter ι} [NeBot F] {b : ℝ≥0∞} (b_ne_top : b ≠ ∞)
{xs : ι → ℝ≥0∞} (le_b : ∀ᶠ i in F, xs i ≤ b) :
F.liminf (fun i ↦ (xs i).toReal) = (F.liminf xs).toReal | ι : Type u_5
F : Filter ι
inst✝ : F.NeBot
b : ℝ≥0∞
b_ne_top : b ≠ ⊤
xs : ι → ℝ≥0∞
le_b : ∀ᶠ (i : ι) in F, xs i ≤ b
⊢ ∀ (b_1 : ℝ≥0∞), (∀ᶠ (n : ι) in F, b_1 ≤ xs n) → b_1 ≤ b | intro y h | ι : Type u_5
F : Filter ι
inst✝ : F.NeBot
b : ℝ≥0∞
b_ne_top : b ≠ ⊤
xs : ι → ℝ≥0∞
le_b : ∀ᶠ (i : ι) in F, xs i ≤ b
y : ℝ≥0∞
h : ∀ᶠ (n : ι) in F, y ≤ xs n
⊢ y ≤ b | d76541420cf17a1a |
AlgebraicTopology.NormalizedMooreComplex.d_squared | Mathlib/AlgebraicTopology/MooreComplex.lean | theorem d_squared (n : ℕ) : objD X (n + 1) ≫ objD X n = 0 | C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
X : SimplicialObject C
n : ℕ
⊢ 0 ∈ Finset.univ | simp | no goals | eb0da68b3e20fac8 |
ONote.scale_eq_mul | Mathlib/SetTheory/Ordinal/Notation.lean | theorem scale_eq_mul (x) [NF x] : ∀ (o) [NF o], scale x o = oadd x 1 0 * o
| 0, _ => rfl
| oadd e n a, h => by
simp only [HMul.hMul]; simp only [scale]
haveI := h.snd
by_cases e0 : e = 0
· simp_rw [scale_eq_mul]
simp [Mul.mul, mul, scale_eq_mul, e0, h.zero_of_zero,
show x + 0 = x from ... | case pos
x : ONote
inst✝ : x.NF
e : ONote
n : ℕ+
a : ONote
h : (e.oadd n a).NF
this : a.NF
e0 : e = 0
⊢ (x + e).oadd n (x.scale a) = Mul.mul (x.oadd 1 0) (e.oadd n a) | simp_rw [scale_eq_mul] | case pos
x : ONote
inst✝ : x.NF
e : ONote
n : ℕ+
a : ONote
h : (e.oadd n a).NF
this : a.NF
e0 : e = 0
⊢ (x + e).oadd n (x.oadd 1 0 * a) = Mul.mul (x.oadd 1 0) (e.oadd n a) | 0690b6dca477b6fa |
PNat.mod_add_div | Mathlib/Data/PNat/Basic.lean | theorem mod_add_div (m k : ℕ+) : (mod m k + k * div m k : ℕ) = m | m k : ℕ+
h₀ : ↑m % ↑k + ↑k * (↑m / ↑k) = ↑m := Nat.mod_add_div ↑m ↑k
this✝ : ¬(↑m % ↑k = 0 ∧ ↑m / ↑k = 0)
this : ↑(k.modDivAux (↑m % ↑k) (↑m / ↑k)).1 + ↑k * (k.modDivAux (↑m % ↑k) (↑m / ↑k)).2 = ↑m % ↑k + ↑k * (↑m / ↑k)
⊢ ↑(m.mod k) + ↑k * m.div k = ↑m | exact this.trans h₀ | no goals | 7284aa8586f5f0ff |
HasProd.sigma | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | theorem HasProd.sigma {γ : β → Type*} {f : (Σ b : β, γ b) → α} {g : β → α} {a : α}
(ha : HasProd f a) (hf : ∀ b, HasProd (fun c ↦ f ⟨b, c⟩) (g b)) : HasProd g a | case right
α : Type u_1
β : Type u_2
inst✝³ : CommMonoid α
inst✝² : TopologicalSpace α
inst✝¹ : ContinuousMul α
inst✝ : RegularSpace α
γ : β → Type u_4
f : (b : β) × γ b → α
g : β → α
a : α
ha : HasProd f a
hf : ∀ (b : β), HasProd (fun c => f ⟨b, c⟩) (g b)
s : Set α
hs : s ∈ 𝓝 a
hsc : IsClosed s
u : Finset ((b : β) × ... | intro bs hbs | case right
α : Type u_1
β : Type u_2
inst✝³ : CommMonoid α
inst✝² : TopologicalSpace α
inst✝¹ : ContinuousMul α
inst✝ : RegularSpace α
γ : β → Type u_4
f : (b : β) × γ b → α
g : β → α
a : α
ha : HasProd f a
hf : ∀ (b : β), HasProd (fun c => f ⟨b, c⟩) (g b)
s : Set α
hs : s ∈ 𝓝 a
hsc : IsClosed s
u : Finset ((b : β) × ... | d606f86187f6c747 |
Complex.arg_mul_cos_add_sin_mul_I_sub | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) :
arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ | r : ℝ
hr : 0 < r
θ : ℝ
⊢ ↑(- -⌊(-π + 2 * π - θ) / (2 * π)⌋) * (2 * π) = 2 * π * ↑⌊(π - θ) / (2 * π)⌋ | ring_nf | no goals | 6bc458bee1a50231 |
BooleanAlgebra.le_iff_atom_le_imp | Mathlib/Order/Atoms.lean | theorem le_iff_atom_le_imp {α} [BooleanAlgebra α] [IsAtomic α] {x y : α} :
x ≤ y ↔ ∀ a, IsAtom a → a ≤ x → a ≤ y | α : Type u_4
inst✝¹ : BooleanAlgebra α
inst✝ : IsAtomic α
x y : α
h : ∀ (a : α), IsAtom a → a ≤ x → a ≤ y
⊢ x ≤ y | have : x ⊓ yᶜ = ⊥ := of_not_not fun hbot =>
have ⟨a, ha, hle⟩ := (eq_bot_or_exists_atom_le _).resolve_left hbot
have ⟨hx, hy'⟩ := le_inf_iff.1 hle
have hy := h a ha hx
have : a ≤ y ⊓ yᶜ := le_inf_iff.2 ⟨hy, hy'⟩
ha.1 (by simpa using this) | α : Type u_4
inst✝¹ : BooleanAlgebra α
inst✝ : IsAtomic α
x y : α
h : ∀ (a : α), IsAtom a → a ≤ x → a ≤ y
this : x ⊓ yᶜ = ⊥
⊢ x ≤ y | 59053ce0d08205f9 |
MeasureTheory.integral_Iic_of_hasDerivAt_of_tendsto | Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | theorem integral_Iic_of_hasDerivAt_of_tendsto (hcont : ContinuousWithinAt f (Iic a) a)
(hderiv : ∀ x ∈ Iio a, HasDerivAt f (f' x) x) (f'int : IntegrableOn f' (Iic a))
(hf : Tendsto f atBot (𝓝 m)) : ∫ x in Iic a, f' x = f a - m | case h
E : Type u_1
f f' : ℝ → E
a : ℝ
m : E
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hcont✝ : ContinuousWithinAt f (Iic a) a
hderiv : ∀ x ∈ Iio a, HasDerivAt f (f' x) x
f'int : IntegrableOn f' (Iic a) volume
hf : Tendsto f atBot (𝓝 m)
hcont : ContinuousOn f (Iic a)
x : ℝ
hx : x ∈... | symm | case h
E : Type u_1
f f' : ℝ → E
a : ℝ
m : E
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hcont✝ : ContinuousWithinAt f (Iic a) a
hderiv : ∀ x ∈ Iio a, HasDerivAt f (f' x) x
f'int : IntegrableOn f' (Iic a) volume
hf : Tendsto f atBot (𝓝 m)
hcont : ContinuousOn f (Iic a)
x : ℝ
hx : x ∈... | ce873e66b888f054 |
Std.DHashMap.Internal.Raw.containsThenInsertIfNew_fst_eq | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/Raw.lean | theorem containsThenInsertIfNew_fst_eq [BEq α] [Hashable α] {m : Raw α β} (h : m.WF) {a : α}
{b : β a} : (m.containsThenInsertIfNew a b).1 =
(Raw₀.containsThenInsertIfNew ⟨m, h.size_buckets_pos⟩ a b).1 | α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : Hashable α
m : Raw α β
h : m.WF
a : α
b : β a
⊢ (m.containsThenInsertIfNew a b).fst = (Raw₀.containsThenInsertIfNew ⟨m, ⋯⟩ a b).fst | simp [Raw.containsThenInsertIfNew, h.size_buckets_pos] | no goals | 1e61bb2dc9bfcf21 |
EuclideanGeometry.existsUnique_dist_eq_of_insert | Mathlib/Geometry/Euclidean/Circumcenter.lean | theorem existsUnique_dist_eq_of_insert {s : AffineSubspace ℝ P}
[HasOrthogonalProjection s.direction] {ps : Set P} (hnps : ps.Nonempty) {p : P} (hps : ps ⊆ s)
(hp : p ∉ s) (hu : ∃! cs : Sphere P, cs.center ∈ s ∧ ps ⊆ (cs : Set P)) :
∃! cs₂ : Sphere P,
cs₂.center ∈ affineSpan ℝ (insert p (s : Set P)) ∧... | case h.right.mk.intro.intro.intro.intro.intro.refl
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
s : AffineSubspace ℝ P
inst✝ : HasOrthogonalProjection s.direction
ps : Set P
hnps : ps.Nonempty
p : P
hps : ps ⊆ ↑s
hp : p ∉ s
th... | have hcr₃val : cr₃ = √(cr * cr + t₃ * y * (t₃ * y)) := by
obtain ⟨p0, hp0⟩ := hnps
have h' : ↑(⟨cc, hcc₃'⟩ : s) = cc := rfl
rw [← dist_of_mem_subset_mk_sphere (Set.mem_insert_of_mem _ hp0) hcr₃, hcc₃'', ←
mul_self_inj_of_nonneg dist_nonneg (Real.sqrt_nonneg _),
Real.mul_self_sqrt (add_nonneg (mul_self_non... | case h.right.mk.intro.intro.intro.intro.intro.refl
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
s : AffineSubspace ℝ P
inst✝ : HasOrthogonalProjection s.direction
ps : Set P
hnps : ps.Nonempty
p : P
hps : ps ⊆ ↑s
hp : p ∉ s
th... | 6d31314b3548c9e7 |
BitVec.zero_umod | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem zero_umod {x : BitVec w} : (0#w) % x = 0#w | w : Nat
x : BitVec w
⊢ 0#w % x = 0#w | simp [bv_toNat] | no goals | 98867339fd702458 |
AlgebraicGeometry.is_localization_basicOpen_of_qcqs | Mathlib/AlgebraicGeometry/Morphisms/QuasiSeparated.lean | theorem is_localization_basicOpen_of_qcqs {X : Scheme} {U : X.Opens} (hU : IsCompact U.1)
(hU' : IsQuasiSeparated U.1) (f : Γ(X, U)) :
IsLocalization.Away f (Γ(X, X.basicOpen f)) | case exists_of_eq.intro
X : Scheme
U : X.Opens
hU : IsCompact U.carrier
hU' : IsQuasiSeparated U.carrier
f x y z : ↑Γ(X, U)
H : (CommRingCat.Hom.hom (X.presheaf.map (homOfLE ⋯).op)) z = 0
n : ℕ
e : f ^ n * z = 0
⊢ ↑⟨(fun x => f ^ x) n, ⋯⟩ * z = 0 | simpa [mul_comm z] using e | no goals | f0552aeee5d1d94b |
Lean.Order.Array.monotone_anyM | Mathlib/.lake/packages/lean4/src/lean/Init/Internal/Order/Lemmas.lean | theorem monotone_anyM
{m : Type → Type v} [Monad m] [∀ α, PartialOrder (m α)] [MonoBind m] {α : Type u}
(f : γ → α → m Bool) (xs : Array α) (start stop : Nat) (hmono : monotone f) :
monotone (fun x => xs.anyM (f x) start stop) | case isFalse
γ : Type w
inst✝³ : PartialOrder γ
m : Type → Type v
inst✝² : Monad m
inst✝¹ : (α : Type) → PartialOrder (m α)
inst✝ : MonoBind m
α : Type u
f : γ → α → m Bool
xs : Array α
start stop : Nat
hmono : monotone f
h✝ : ¬stop ≤ xs.size
⊢ monotone fun x =>
let any := fun stop h => Array.anyM.loop (f x) xs sto... | apply monotone_anyM_loop | case isFalse.hmono
γ : Type w
inst✝³ : PartialOrder γ
m : Type → Type v
inst✝² : Monad m
inst✝¹ : (α : Type) → PartialOrder (m α)
inst✝ : MonoBind m
α : Type u
f : γ → α → m Bool
xs : Array α
start stop : Nat
hmono : monotone f
h✝ : ¬stop ≤ xs.size
⊢ monotone f | 6f37728987761f9e |
meas_essSup_lt | Mathlib/MeasureTheory/Function/EssSup.lean | theorem meas_essSup_lt
(hf : IsBoundedUnder (· ≤ ·) (ae μ) f | α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝³ : ConditionallyCompleteLinearOrder β
f : α → β
inst✝² : TopologicalSpace β
inst✝¹ : FirstCountableTopology β
inst✝ : OrderTopology β
hf : autoParam (IsBoundedUnder (fun x1 x2 => x1 ≤ x2) (ae μ) f) _auto✝
⊢ μ {y | ¬f y ≤ essSup f μ} = 0 | exact ae_le_essSup hf | no goals | 456c526abae644e2 |
Module.exists_nontrivial_relation_sum_zero_of_finrank_succ_lt_card | Mathlib/LinearAlgebra/Dimension/Finite.lean | theorem Module.exists_nontrivial_relation_sum_zero_of_finrank_succ_lt_card
{t : Finset M} (h : finrank R M + 1 < t.card) :
∃ f : M → R, ∑ e ∈ t, f e • e = 0 ∧ ∑ e ∈ t, f e = 0 ∧ ∃ x ∈ t, f x ≠ 0 | case intro.intro.intro.intro.intro.refine_3
R : Type u
M : Type v
inst✝⁴ : Ring R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Finite R M
inst✝ : StrongRankCondition R
t : Finset M
h : finrank R M + 1 < #t
x₀ : M
x₀_mem : x₀ ∈ t
shift : M ↪ M := { toFun := fun x => x - x₀, inj' := ⋯ }
t' : Finset M := Fi... | obtain ⟨x₁, x₁_mem', rfl⟩ := Finset.mem_map.mp x₁_mem | case intro.intro.intro.intro.intro.refine_3.intro.intro
R : Type u
M : Type v
inst✝⁴ : Ring R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Finite R M
inst✝ : StrongRankCondition R
t : Finset M
h : finrank R M + 1 < #t
x₀ : M
x₀_mem : x₀ ∈ t
shift : M ↪ M := { toFun := fun x => x - x₀, inj' := ⋯ }
t' : Fi... | 9ccb45a24efab47f |
sum_div_nat_floor_pow_sq_le_div_sq | Mathlib/Analysis/SpecificLimits/FloorPow.lean | theorem sum_div_nat_floor_pow_sq_le_div_sq (N : ℕ) {j : ℝ} (hj : 0 < j) {c : ℝ} (hc : 1 < c) :
(∑ i ∈ range N with j < ⌊c ^ i⌋₊, (1 : ℝ) / (⌊c ^ i⌋₊ : ℝ) ^ 2) ≤
c ^ 5 * (c - 1)⁻¹ ^ 3 / j ^ 2 | case e_a
N : ℕ
j : ℝ
hj : 0 < j
c : ℝ
hc : 1 < c
cpos : 0 < c
A : 0 < 1 - c⁻¹
⊢ c ^ 2 * c ^ 3 * (c - 1) ^ 3 = c ^ 5 * ((c - 1) ^ 2 * (c - 1)) | ring! | no goals | a23322a47b698b23 |
Finset.range_image_pred_top_sub | Mathlib/Order/Interval/Finset/Nat.lean | theorem range_image_pred_top_sub (n : ℕ) :
((Finset.range n).image fun j => n - 1 - j) = Finset.range n | case zero
⊢ image (fun j => 0 - 1 - j) (range 0) = range 0 | rw [range_zero, image_empty] | no goals | 82d632012d230477 |
Finset.card_le_mul_card_image_of_maps_to | Mathlib/Algebra/Order/BigOperators/Group/Finset.lean | theorem card_le_mul_card_image_of_maps_to {f : α → β} {s : Finset α} {t : Finset β}
(Hf : ∀ a ∈ s, f a ∈ t) (n : ℕ) (hn : ∀ b ∈ t, #{a ∈ s | f a = b} ≤ n) : #s ≤ n * #t :=
calc
#s = ∑ b ∈ t, #{a ∈ s | f a = b} := card_eq_sum_card_fiberwise Hf
_ ≤ ∑ _b ∈ t, n := sum_le_sum hn
_ = _ | α : Type u_2
β : Type u_3
inst✝ : DecidableEq β
f : α → β
s : Finset α
t : Finset β
Hf : ∀ a ∈ s, f a ∈ t
n : ℕ
hn : ∀ b ∈ t, #(filter (fun a => f a = b) s) ≤ n
⊢ ∑ _b ∈ t, n = n * #t | simp [mul_comm] | no goals | b1bf76b420221628 |
finrank_vectorSpan_insert_le | Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean | theorem finrank_vectorSpan_insert_le (s : AffineSubspace k P) (p : P) :
finrank k (vectorSpan k (insert p (s : Set P))) ≤ finrank k s.direction + 1 | case h.e'_3
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : AffineSubspace k P
p : P
hf this : FiniteDimensional k ↥s.direction
hs : s = ⊥
⊢ finrank k ↥(vectorSpan k (insert p ∅)) = finrank k ↥⊥ | convert rfl <;> simp | no goals | de5f4f906d2ae1ab |
count_span_normalizedFactors_eq | Mathlib/RingTheory/DedekindDomain/Ideal.lean | theorem count_span_normalizedFactors_eq {r X : R} (hr : r ≠ 0) (hX : Prime X) :
Multiset.count (Ideal.span {X} : Ideal R) (normalizedFactors (Ideal.span {r})) =
Multiset.count (normalize X) (normalizedFactors r) | R : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
inst✝² : NormalizationMonoid R
inst✝¹ : DecidableEq R
inst✝ : DecidableEq (Ideal R)
r X : R
hr : r ≠ 0
hX : Prime X
this : emultiplicity (span {X}) (span {r}) = ↑(Multiset.count (normalize X) (normalizedFactors r))
⊢ Prime (span {X}) | simpa only [prime_span_singleton_iff] | no goals | 920e101bccbbf1ec |
ProbabilityTheory.eqOn_complexMGF_of_mgf' | Mathlib/Probability/Moments/ComplexMGF.lean | /-- If two random variables have the same moment generating function then they have
the same `complexMGF` on the vertical strip `{z | z.re ∈ interior (integrableExpSet X μ)}`.
TODO: once we know that equal `mgf` implies equal distributions, we will be able to show that
the `complexMGF` are equal everywhere, not only o... | case neg.intro
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
Ω' : Type u_3
mΩ' : MeasurableSpace Ω'
Y : Ω' → ℝ
μ' : Measure Ω'
hXY : mgf X μ = mgf Y μ'
hμμ' : μ = 0 ↔ μ' = 0
t : ℝ
ht : t ∈ interior (integrableExpSet Y μ')
hX : AnalyticOnNhd ℂ (complexMGF X μ) {z | z.re ∈ interior (integrableExpSet Y μ')}
h... | obtain ⟨xs, hx_tendsto, hx_eq⟩ := h_real | case neg.intro.intro.intro
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
Ω' : Type u_3
mΩ' : MeasurableSpace Ω'
Y : Ω' → ℝ
μ' : Measure Ω'
hXY : mgf X μ = mgf Y μ'
hμμ' : μ = 0 ↔ μ' = 0
t : ℝ
ht : t ∈ interior (integrableExpSet Y μ')
hX : AnalyticOnNhd ℂ (complexMGF X μ) {z | z.re ∈ interior (integrableExp... | a1be17aeb598938f |
dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_isEisensteinAt | Mathlib/RingTheory/Polynomial/Eisenstein/IsIntegral.lean | theorem dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_isEisensteinAt {B : PowerBasis K L}
(hp : Prime p) (hBint : IsIntegral R B.gen) {z : L} {Q : R[X]} (hQ : aeval B.gen Q = p • z)
(hzint : IsIntegral R z) (hei : (minpoly R B.gen).IsEisensteinAt 𝓟) : p ∣ Q.coeff 0 | case intro
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 : PowerBasi... | have finrank_K_L : Module.finrank K L = B.dim := B.finrank | case intro
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 : PowerBasi... | d2e1ed24dbb826cf |
extremePoints_pi | Mathlib/Analysis/Convex/Extreme.lean | theorem extremePoints_pi (s : ∀ i, Set (π i)) :
(univ.pi s).extremePoints 𝕜 = univ.pi fun i ↦ (s i).extremePoints 𝕜 | case h.refine_1.refine_2.inl
𝕜 : Type u_1
ι : Type u_4
π : ι → Type u_5
inst✝² : OrderedSemiring 𝕜
inst✝¹ : (i : ι) → AddCommGroup (π i)
inst✝ : (i : ι) → Module 𝕜 (π i)
s : (i : ι) → Set (π i)
x : (i : ι) → π i
hx : ∀ (i : ι), x i ∈ s i
h :
∀ (x₁ : (i : ι) → π i),
(∀ (i : ι), x₁ i ∈ s i) →
∀ (x₂ : (i : ... | rwa [update_self] | no goals | dae1f2fb7600454d |
AntitoneOn.integral_le_sum_Ico | Mathlib/Analysis/SumIntegralComparisons.lean | theorem AntitoneOn.integral_le_sum_Ico (hab : a ≤ b) (hf : AntitoneOn f (Set.Icc a b)) :
(∫ x in a..b, f x) ≤ ∑ x ∈ Finset.Ico a b, f x | a b : ℕ
f : ℝ → ℝ
hab : a ≤ b
hf : AntitoneOn f (Icc ↑a ↑b)
⊢ ∫ (x : ℝ) in ↑a..↑a + ↑(b - a), f x ≤ ∑ x ∈ Finset.range (b - a), f (↑a + ↑x) | apply AntitoneOn.integral_le_sum | case hf
a b : ℕ
f : ℝ → ℝ
hab : a ≤ b
hf : AntitoneOn f (Icc ↑a ↑b)
⊢ AntitoneOn f (Icc (↑a) (↑a + ↑(b - a))) | cc24ebd63bbbebff |
CompositionSeries.exists_last_eq_snoc_equivalent | Mathlib/Order/JordanHolder.lean | theorem exists_last_eq_snoc_equivalent (s : CompositionSeries X) (x : X) (hm : IsMaximal x s.last)
(hb : s.head ≤ x) :
∃ t : CompositionSeries X,
t.head = s.head ∧ t.length + 1 = s.length ∧
∃ htx : t.last = x,
Equivalent s (snoc t s.last (show IsMaximal t.last _ from htx.symm ▸ hm)) | case h
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
n : ℕ
ih :
∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (last s)),
head s ≤ x →
s.length = n → ∃ t, head t = head s ∧ t.length + 1 = n ∧ ∃ (htx : last t = x), s.Equivalent (snoc t (last s) ⋯)
s : CompositionSeries X
x : X
hm : IsMax... | have : s.Equivalent ((snoc t s.eraseLast.last <| show IsMaximal t.last _ from
htt.symm ▸ imxs).snoc s.last
(by simpa using isMaximal_eraseLast_last h0s)) := by
conv_lhs => rw [eq_snoc_eraseLast h0s]
exact Equivalent.snoc hteqv (by simpa using (isMaximal_eraseLast_last h0s).iso_refl) | case h
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
n : ℕ
ih :
∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (last s)),
head s ≤ x →
s.length = n → ∃ t, head t = head s ∧ t.length + 1 = n ∧ ∃ (htx : last t = x), s.Equivalent (snoc t (last s) ⋯)
s : CompositionSeries X
x : X
hm : IsMax... | 738321de520b9239 |
EuclideanSpace.volume_closedBall_fin_three | Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean | @[simp]
lemma volume_closedBall_fin_three (x : EuclideanSpace ℝ (Fin 3)) (r : ℝ) :
volume (closedBall x r) = .ofReal r ^ 3 * .ofReal (π * 4 / 3) | x : EuclideanSpace ℝ (Fin 3)
r : ℝ
⊢ volume (closedBall x r) = ENNReal.ofReal r ^ 3 * ENNReal.ofReal (π * 4 / 3) | rw [addHaar_closedBall_eq_addHaar_ball, volume_ball_fin_three x] | no goals | 4cbe6d6781524bbf |
Real.rpow_lt_one_iff | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | theorem rpow_lt_one_iff (hx : 0 ≤ x) :
x ^ y < 1 ↔ x = 0 ∧ y ≠ 0 ∨ 1 < x ∧ y < 0 ∨ x < 1 ∧ 0 < y | case inr
x y : ℝ
hx✝ : 0 ≤ x
hx : 0 < x
⊢ x ^ y < 1 ↔ x = 0 ∧ y ≠ 0 ∨ 1 < x ∧ y < 0 ∨ x < 1 ∧ 0 < y | simp [rpow_lt_one_iff_of_pos hx, hx.ne.symm] | no goals | 03f9bcae4a503688 |
Measurable.iSup | Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean | theorem Measurable.iSup {ι} [Countable ι] {f : ι → δ → α} (hf : ∀ i, Measurable (f i)) :
Measurable (fun b ↦ ⨆ i, f i b) | α : Type u_1
δ : Type u_4
inst✝⁵ : TopologicalSpace α
mα : MeasurableSpace α
inst✝⁴ : BorelSpace α
mδ : MeasurableSpace δ
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
inst✝ : Countable ι
f : ι → δ → α
hf : ∀ (i : ι), Measurable (f i)
⊢ Measurable f... | rcases isEmpty_or_nonempty ι with hι|hι | case inl
α : Type u_1
δ : Type u_4
inst✝⁵ : TopologicalSpace α
mα : MeasurableSpace α
inst✝⁴ : BorelSpace α
mδ : MeasurableSpace δ
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
inst✝ : Countable ι
f : ι → δ → α
hf : ∀ (i : ι), Measurable (f i)
hι : ... | 85759f720558b7e9 |
MeasureTheory.Content.innerContent_iSup_nat | Mathlib/MeasureTheory/Measure/Content.lean | theorem innerContent_iSup_nat [R1Space G] (U : ℕ → Opens G) :
μ.innerContent (⨆ i : ℕ, U i) ≤ ∑' i : ℕ, μ.innerContent (U i) | case h.e'_3.h.e'_6.h
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), μ (t.sup K) ≤ ∑ i ∈ t, μ (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i... | rw [Compacts.coe_finset_sup, Finset.sup_eq_iSup] | case h.e'_3.h.e'_6.h
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), μ (t.sup K) ≤ ∑ i ∈ t, μ (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i... | 47868df373390eca |
List.eraseIdx_modify_of_gt | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/Modify.lean | theorem eraseIdx_modify_of_gt (f : α → α) (i j) (l : List α) (h : j > i) :
(modify f i l).eraseIdx j = (l.eraseIdx j).modify f i | α : Type u_1
f : α → α
i j : Nat
l : List α
h : j > i
⊢ (modify f i l).eraseIdx j = modify f i (l.eraseIdx j) | apply ext_getElem | case hl
α : Type u_1
f : α → α
i j : Nat
l : List α
h : j > i
⊢ ((modify f i l).eraseIdx j).length = (modify f i (l.eraseIdx j)).length
case h
α : Type u_1
f : α → α
i j : Nat
l : List α
h : j > i
⊢ ∀ (i_1 : Nat) (h₁ : i_1 < ((modify f i l).eraseIdx j).length) (h₂ : i_1 < (modify f i (l.eraseIdx j)).length),
((mod... | 26512ea5cfe6dba0 |
ax_grothendieck_of_locally_finite | Mathlib/FieldTheory/AxGrothendieck.lean | theorem ax_grothendieck_of_locally_finite {ι K R : Type*} [Field K] [Finite K] [CommRing R]
[Finite ι] [Algebra K R] [alg : Algebra.IsAlgebraic K R] (ps : ι → MvPolynomial ι R)
(S : Set (ι → R))
(hm : S.MapsTo (fun v i => eval v (ps i)) S)
(hinj : S.InjOn (fun v i => eval v (ps i))) :
S.SurjOn (fun ... | case a.h.a
ι : Type u_1
K : Type u_2
R : Type u_3
inst✝⁴ : Field K
inst✝³ : Finite K
inst✝² : CommRing R
inst✝¹ : Finite ι
inst✝ : Algebra K R
alg : Algebra.IsAlgebraic K R
ps : ι → MvPolynomial ι R
S : Set (ι → R)
hm : Set.MapsTo (fun v i => (eval v) (ps i)) S S
hinj : Set.InjOn (fun v i => (eval v) (ps i)) S
is_int :... | exact congr_fun (hinj x.2 y.2 (funext hxy)) i | no goals | 6b85e3de256b7145 |
FractionalIdeal.count_mono | Mathlib/RingTheory/DedekindDomain/Factorization.lean | theorem count_mono {I J} (hI : I ≠ 0) (h : I ≤ J) : count K v J ≤ count K v I | case neg
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I J : FractionalIdeal R⁰ K
hI : I ≠ 0
h : I ≤ J
hJ : ¬J = 0
this : ∃ I_1, ↑I_1 = J⁻¹ * I
⊢ count K v J ≤ count K v I | obtain ⟨J', hJ'⟩ := this | case neg.intro
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I J : FractionalIdeal R⁰ K
hI : I ≠ 0
h : I ≤ J
hJ : ¬J = 0
J' : Ideal R
hJ' : ↑J' = J⁻¹ * I
⊢ count K v J ≤ count K v I | c30acf486d479174 |
le_symmDiff_iff_left | Mathlib/Order/SymmDiff.lean | theorem le_symmDiff_iff_left : a ≤ a ∆ b ↔ Disjoint a b | α : Type u_2
inst✝ : GeneralizedBooleanAlgebra α
a b : α
h : a ≤ (a ⊔ b) \ (a ⊓ b)
⊢ Disjoint a b | exact disjoint_iff_inf_le.mpr (le_sdiff_right.1 <| inf_le_of_left_le h).le | no goals | ffc5b042143df3fc |
AlgebraicGeometry.HasAffineProperty.diagonal_of_diagonal_of_isPullback | Mathlib/AlgebraicGeometry/Morphisms/Constructors.lean | theorem HasAffineProperty.diagonal_of_diagonal_of_isPullback
(P) {Q} [HasAffineProperty P Q]
{X Y U V : Scheme.{u}} {f : X ⟶ Y} {g : U ⟶ Y}
[IsAffine U] [IsOpenImmersion g]
{iV : V ⟶ X} {f' : V ⟶ U} (h : IsPullback iV f' f g) (H : P.diagonal f) :
Q.diagonal f' | P : MorphismProperty Scheme
Q : AffineTargetMorphismProperty
inst✝² : HasAffineProperty P Q
X Y U V : Scheme
f : X ⟶ Y
g : U ⟶ Y
inst✝¹ : IsAffine U
inst✝ : IsOpenImmersion g
iV : V ⟶ X
f' : V ⟶ U
h : IsPullback iV f' f g
H : P.diagonal f
this : Q.IsLocal := isLocal_affineProperty P
⊢ Q.diagonal (pullback.snd f g) | rintro U V f₁ f₂ hU hV hf₁ hf₂ | P : MorphismProperty Scheme
Q : AffineTargetMorphismProperty
inst✝² : HasAffineProperty P Q
X Y U✝ V✝ : Scheme
f : X ⟶ Y
g : U✝ ⟶ Y
inst✝¹ : IsAffine U✝
inst✝ : IsOpenImmersion g
iV : V✝ ⟶ X
f' : V✝ ⟶ U✝
h : IsPullback iV f' f g
H : P.diagonal f
this : Q.IsLocal := isLocal_affineProperty P
U V : Scheme
f₁ : U ⟶ pullbac... | 6247e87fffb2135c |
MeasureTheory.integral_mul_norm_le_Lp_mul_Lq | Mathlib/MeasureTheory/Integral/Bochner.lean | theorem integral_mul_norm_le_Lp_mul_Lq {E} [NormedAddCommGroup E] {f g : α → E} {p q : ℝ}
(hpq : p.IsConjExponent q) (hf : MemLp f (ENNReal.ofReal p) μ)
(hg : MemLp g (ENNReal.ofReal q) μ) :
∫ a, ‖f a‖ * ‖g a‖ ∂μ ≤ (∫ a, ‖f a‖ ^ p ∂μ) ^ (1 / p) * (∫ a, ‖g a‖ ^ q ∂μ) ^ (1 / q) | α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_7
inst✝ : NormedAddCommGroup E
f g : α → E
p q : ℝ
hpq : p.IsConjExponent q
hf : MemLp f (ENNReal.ofReal p) μ
hg : MemLp g (ENNReal.ofReal q) μ
h_left : ∫⁻ (a : α), ENNReal.ofReal (‖f a‖ * ‖g a‖) ∂μ = ∫⁻ (a : α), ((fun x => ‖f x‖ₑ) * fun x => ‖g x‖ₑ) a ∂μ
⊢ (∫... | have h_right_f : ∫⁻ a, .ofReal (‖f a‖ ^ p) ∂μ = ∫⁻ a, ‖f a‖ₑ ^ p ∂μ := by
refine lintegral_congr fun x => ?_
rw [← ofReal_norm_eq_enorm, ENNReal.ofReal_rpow_of_nonneg (norm_nonneg _) hpq.nonneg] | α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_7
inst✝ : NormedAddCommGroup E
f g : α → E
p q : ℝ
hpq : p.IsConjExponent q
hf : MemLp f (ENNReal.ofReal p) μ
hg : MemLp g (ENNReal.ofReal q) μ
h_left : ∫⁻ (a : α), ENNReal.ofReal (‖f a‖ * ‖g a‖) ∂μ = ∫⁻ (a : α), ((fun x => ‖f x‖ₑ) * fun x => ‖g x‖ₑ) a ∂μ
h_ri... | 539c08192e88a140 |
Turing.ToPartrec.cont_eval_fix | Mathlib/Computability/TMConfig.lean | theorem cont_eval_fix {f k v} (fok : Code.Ok f) :
Turing.eval step (stepNormal f (Cont.fix f k) v) =
f.fix.eval v >>= fun v => Turing.eval step (Cfg.ret k v) | case mpr.intro.intro
f : Code
k : Cont
v : List ℕ
fok : f.Ok
x : Cfg
v' : List ℕ
he : v' ∈ f.fix.eval v
hr : x ∈ eval step (Cfg.ret k v')
⊢ Reaches step (Cfg.ret k v') ?m.138575 | exact ReflTransGen.single rfl | no goals | 4a2318a05b9f1866 |
IsExposed.isExtreme | Mathlib/Analysis/Convex/Exposed.lean | theorem isExtreme (hAB : IsExposed 𝕜 A B) : IsExtreme 𝕜 A B | 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : TopologicalSpace 𝕜
inst✝³ : LinearOrderedRing 𝕜
inst✝² : AddCommMonoid E
inst✝¹ : TopologicalSpace E
inst✝ : Module 𝕜 E
A B : Set E
hAB : IsExposed 𝕜 A B
⊢ IsExtreme 𝕜 A B | refine ⟨hAB.subset, fun x₁ hx₁A x₂ hx₂A x hxB hx => ?_⟩ | 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : TopologicalSpace 𝕜
inst✝³ : LinearOrderedRing 𝕜
inst✝² : AddCommMonoid E
inst✝¹ : TopologicalSpace E
inst✝ : Module 𝕜 E
A B : Set E
hAB : IsExposed 𝕜 A B
x₁ : E
hx₁A : x₁ ∈ A
x₂ : E
hx₂A : x₂ ∈ A
x : E
hxB : x ∈ B
hx : x ∈ openSegment 𝕜 x₁ x₂
⊢ x₁ ∈ B ∧ x₂ ∈ B | fc814994a84ea1e1 |
Algebra.smulTower_leftMulMatrix_algebraMap_ne | Mathlib/LinearAlgebra/Matrix/ToLin.lean | theorem smulTower_leftMulMatrix_algebraMap_ne (x : S) (i j) {k k'} (h : k ≠ k') :
leftMulMatrix (b.smulTower c) (algebraMap _ _ x) (i, k) (j, k') = 0 | R : Type u_1
S : Type u_2
T : Type u_3
inst✝¹⁰ : CommSemiring R
inst✝⁹ : CommSemiring S
inst✝⁸ : Semiring T
inst✝⁷ : Algebra R S
inst✝⁶ : Algebra S T
inst✝⁵ : Algebra R T
inst✝⁴ : IsScalarTower R S T
m : Type u_4
n : Type u_5
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : DecidableEq m
inst✝ : DecidableEq n
b : Basis m... | rw [smulTower_leftMulMatrix_algebraMap, blockDiagonal_apply_ne _ _ _ h] | no goals | 8b34da3cec2a1ad3 |
not_disjoint_segment_convexHull_triple | Mathlib/Analysis/Convex/StoneSeparation.lean | theorem not_disjoint_segment_convexHull_triple {p q u v x y z : E} (hz : z ∈ segment 𝕜 x y)
(hu : u ∈ segment 𝕜 x p) (hv : v ∈ segment 𝕜 y q) :
¬Disjoint (segment 𝕜 u v) (convexHull 𝕜 {p, q, z}) | case «2»
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q x y : E
az bz : 𝕜
haz : 0 ≤ az
hbz : 0 ≤ bz
habz : az + bz = 1
haz' : 0 < az
av bv : 𝕜
hav : 0 ≤ av
hbv : 0 ≤ bv
habv : av + bv = 1
hav' : 0 < av
au bu : 𝕜
hau : 0 ≤ au
hbu : 0 ≤ bu
habu : au + bu = 1
h... | exact mul_nonneg hau hav | no goals | b7449eac49163f66 |
compact_covered_by_mul_left_translates | Mathlib/Topology/Algebra/Group/Basic.lean | theorem compact_covered_by_mul_left_translates {K V : Set G} (hK : IsCompact K)
(hV : (interior V).Nonempty) : ∃ t : Finset G, K ⊆ ⋃ g ∈ t, (g * ·) ⁻¹' V | G : Type w
inst✝² : TopologicalSpace G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
K V : Set G
hK : IsCompact K
hV : (interior V).Nonempty
⊢ K ⊆ ⋃ i, (fun x => interior ((fun x_1 => x * x_1) ⁻¹' V)) i | obtain ⟨g₀, hg₀⟩ := hV | case intro
G : Type w
inst✝² : TopologicalSpace G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
K V : Set G
hK : IsCompact K
g₀ : G
hg₀ : g₀ ∈ interior V
⊢ K ⊆ ⋃ i, (fun x => interior ((fun x_1 => x * x_1) ⁻¹' V)) i | c6b84b90d5740fe1 |
List.exists_perm_sublist | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Perm.lean | theorem exists_perm_sublist {l₁ l₂ l₂' : List α} (s : l₁ <+ l₂) (p : l₂ ~ l₂') :
∃ l₁', l₁' ~ l₁ ∧ l₁' <+ l₂' | α : Type u_1
l₂ l₂' : List α
x y : α
l' l₁ : List α
s✝ : l₁ <+ y :: x :: l'
l₁✝ : List α
s : l₁✝ <+ l'
⊢ ∃ l₁', l₁' ~ y :: l₁✝ ∧ l₁' <+ x :: y :: l' | exact ⟨y :: _, .rfl, (s.cons₂ _).cons _⟩ | no goals | b87d2baff04d68a4 |
ArithmeticFunction.vonMangoldt.summable_residueClass_non_primes_div | Mathlib/NumberTheory/LSeries/PrimesInAP.lean | /-- The function `n ↦ Λ n / n`, restricted to non-primes in a residue class, is summable.
This is used to convert results on `ArithmeticFunction.vonMangoldt.residueClass` to results
on primes in an arithmetic progression. -/
lemma summable_residueClass_non_primes_div :
Summable fun n : ℕ ↦ (if n.Prime then 0 else r... | q : ℕ
a : ZMod q
n : ℕ
this : 0 ≤ residueClass a n
⊢ 0 ≤ (if Nat.Prime n then 0 else residueClass a n) / ↑n | positivity | no goals | 6c8eee8a1fa8d5a9 |
iSup_succ | Mathlib/Order/SuccPred/CompleteLinearOrder.lean | theorem iSup_succ [SuccOrder α] (x : α) : ⨆ a : Iio x, succ a.1 = x | α : Type u_2
inst✝¹ : ConditionallyCompleteLinearOrderBot α
inst✝ : SuccOrder α
x : α
H : BddAbove (range fun a => succ ↑a)
⊢ ⨆ a, succ ↑a ≤ x | rw [ciSup_le_iff' H] | α : Type u_2
inst✝¹ : ConditionallyCompleteLinearOrderBot α
inst✝ : SuccOrder α
x : α
H : BddAbove (range fun a => succ ↑a)
⊢ ∀ (i : ↑(Iio x)), succ ↑i ≤ x | 1ac61462e616078c |
LieIdeal.comap_bracket_eq | Mathlib/Algebra/Lie/IdealOperations.lean | theorem comap_bracket_eq {J₁ J₂ : LieIdeal R L'} (h : f.IsIdealMorphism) :
comap f ⁅f.idealRange ⊓ J₁, f.idealRange ⊓ J₂⁆ = ⁅comap f J₁, comap f J₂⁆ ⊔ f.ker | case e_p.e_s.h.mp.intro.mk.intro.mk.intro.intro.intro.intro
R : Type u
L : Type v
L' : Type w₂
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieAlgebra R L
inst✝¹ : LieRing L'
inst✝ : LieAlgebra R L'
f : L →ₗ⁅R⁆ L'
J₁ J₂ : LieIdeal R L'
h : f.IsIdealMorphism
y x₁ : L'
hx₁ : x₁ ∈ f.idealRange ⊓ J₁
x₂ : L'
hx₂ : x₂ ∈ f... | rw [← hz₂] at hz₂' | case e_p.e_s.h.mp.intro.mk.intro.mk.intro.intro.intro.intro
R : Type u
L : Type v
L' : Type w₂
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieAlgebra R L
inst✝¹ : LieRing L'
inst✝ : LieAlgebra R L'
f : L →ₗ⁅R⁆ L'
J₁ J₂ : LieIdeal R L'
h : f.IsIdealMorphism
y x₁ : L'
hx₁ : x₁ ∈ f.idealRange ⊓ J₁
x₂ : L'
hx₂ : x₂ ∈ f... | 6c6e3d9aa2a88b30 |
CompactT2.Projective.extremallyDisconnected | Mathlib/Topology/ExtremallyDisconnected.lean | theorem CompactT2.Projective.extremallyDisconnected [CompactSpace X] [T2Space X]
(h : CompactT2.Projective X) : ExtremallyDisconnected X | case intro.intro
X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
hZ : IsClosed... | have hφ₁ : ∀ x, (φ x).1 = x := congr_fun g_sec | case intro.intro
X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
h : Projective X
U : Set X
hU : IsOpen U
Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}
Z₂ : Set (X × Bool) := closure U ×ˢ {false}
Z : Set (X × Bool) := Z₁ ∪ Z₂
hZ₁₂ : Disjoint Z₁ Z₂
hZ₁ : IsClosed Z₁
hZ₂ : IsClosed Z₂
hZ : IsClosed... | 541ccb6618c06452 |
reflection_sub | Mathlib/Analysis/InnerProductSpace/Projection.lean | theorem reflection_sub {v w : F} (h : ‖v‖ = ‖w‖) : reflection (ℝ ∙ (v - w))ᗮ v = w | F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
v w : F
h : ‖v‖ = ‖w‖
R : F ≃ₗᵢ[ℝ] F := reflection (Submodule.span ℝ {v - w})ᗮ
⊢ R v = w | suffices R v + R v = w + w by
apply smul_right_injective F (by norm_num : (2 : ℝ) ≠ 0)
simpa [two_smul] using this | F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
v w : F
h : ‖v‖ = ‖w‖
R : F ≃ₗᵢ[ℝ] F := reflection (Submodule.span ℝ {v - w})ᗮ
⊢ R v + R v = w + w | 0665d6b9d7b08a87 |
Real.self_le_sinh_iff | Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean | theorem self_le_sinh_iff : x ≤ sinh x ↔ 0 ≤ x :=
calc
x ≤ sinh x ↔ sinh 0 - 0 ≤ sinh x - x | x : ℝ
⊢ x ≤ sinh x ↔ sinh 0 - 0 ≤ sinh x - x | simp | no goals | c6fbf03a8cd11f83 |
MeasureTheory.setLIntegral_smul_measure | Mathlib/MeasureTheory/Integral/Lebesgue.lean | lemma setLIntegral_smul_measure (c : ℝ≥0∞) (f : α → ℝ≥0∞) (s : Set α) :
∫⁻ a in s, f a ∂(c • μ) = c * ∫⁻ a in s, f a ∂μ | α : Type u_1
m : MeasurableSpace α
μ : Measure α
c : ℝ≥0∞
f : α → ℝ≥0∞
s : Set α
⊢ ∫⁻ (a : α) in s, f a ∂c • μ = c * ∫⁻ (a : α) in s, f a ∂μ | rw [Measure.restrict_smul, lintegral_smul_measure] | no goals | dd61d4b90fab1464 |
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