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 |
|---|---|---|---|---|---|---|
Differential.algHom_deriv | Mathlib/RingTheory/Derivation/MapCoeffs.lean | lemma algHom_deriv (f : R →ₐ[A] R') (hf : Function.Injective f) (x : R) (h : IsSeparable A x) :
f (x′) = (f x)′ | case h
A : Type u_1
inst✝¹¹ : CommRing A
inst✝¹⁰ : Differential A
R : Type u_2
inst✝⁹ : CommRing R
inst✝⁸ : Differential R
inst✝⁷ : Algebra A R
inst✝⁶ : DifferentialAlgebra A R
R' : Type u_3
inst✝⁵ : CommRing R'
inst✝⁴ : Differential R'
inst✝³ : Algebra A R'
inst✝² : DifferentialAlgebra A R'
inst✝¹ : IsDomain R'
inst✝ ... | conv => lhs; rw [Polynomial.aeval_algHom] | case h
A : Type u_1
inst✝¹¹ : CommRing A
inst✝¹⁰ : Differential A
R : Type u_2
inst✝⁹ : CommRing R
inst✝⁸ : Differential R
inst✝⁷ : Algebra A R
inst✝⁶ : DifferentialAlgebra A R
R' : Type u_3
inst✝⁵ : CommRing R'
inst✝⁴ : Differential R'
inst✝³ : Algebra A R'
inst✝² : DifferentialAlgebra A R'
inst✝¹ : IsDomain R'
inst✝ ... | 09e31f39ee7ae7cb |
MvPowerSeries.X_pow_dvd_iff | Mathlib/RingTheory/MvPowerSeries/Basic.lean | theorem X_pow_dvd_iff {s : σ} {n : ℕ} {φ : MvPowerSeries σ R} :
(X s : MvPowerSeries σ R) ^ n ∣ φ ↔ ∀ m : σ →₀ ℕ, m s < n → coeff R m φ = 0 | σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
⊢ X s ^ n ∣ φ ↔ ∀ (m : σ →₀ ℕ), m s < n → (coeff R m) φ = 0 | constructor | case mp
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
⊢ X s ^ n ∣ φ → ∀ (m : σ →₀ ℕ), m s < n → (coeff R m) φ = 0
case mpr
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
⊢ (∀ (m : σ →₀ ℕ), m s < n → (coeff R m) φ = 0) → X s ^ n ∣ φ | c235f2aaea0f4970 |
DyckWord.firstReturn_nest | Mathlib/Combinatorics/Enumerative/DyckWord.lean | @[simp]
lemma firstReturn_nest : p.nest.firstReturn = p.toList.length + 1 | p : DyckWord
u : ↑p.nest = U :: ↑p ++ [D]
⊢ count U (List.take ((↑p).length + 1 + 1) (U :: ↑p ++ [D])) =
count D (List.take ((↑p).length + 1 + 1) (U :: ↑p ++ [D])) ∧
∀ j < (↑p).length + 1,
decide (count U (List.take (j + 1) (U :: ↑p ++ [D])) = count D (List.take (j + 1) (U :: ↑p ++ [D]))) = false | constructor | case left
p : DyckWord
u : ↑p.nest = U :: ↑p ++ [D]
⊢ count U (List.take ((↑p).length + 1 + 1) (U :: ↑p ++ [D])) =
count D (List.take ((↑p).length + 1 + 1) (U :: ↑p ++ [D]))
case right
p : DyckWord
u : ↑p.nest = U :: ↑p ++ [D]
⊢ ∀ j < (↑p).length + 1,
decide (count U (List.take (j + 1) (U :: ↑p ++ [D])) = coun... | 7eee40486e6509d1 |
Nat.ofDigits_add_ofDigits_eq_ofDigits_zipWith_of_length_eq | Mathlib/Data/Nat/Digits.lean | theorem ofDigits_add_ofDigits_eq_ofDigits_zipWith_of_length_eq {b : ℕ} {l1 l2 : List ℕ}
(h : l1.length = l2.length) :
ofDigits b l1 + ofDigits b l2 = ofDigits b (l1.zipWith (· + ·) l2) | case cons.cons
b hd₁ hd₂ : ℕ
tl₂ tl₁ : List ℕ
ih₂ :
∀ (tl₁_1 : List ℕ),
(∀ {l2 : List ℕ},
tl₁_1.length = l2.length →
ofDigits b tl₁_1 + ofDigits b l2 = ofDigits b (List.zipWith (fun x1 x2 => x1 + x2) tl₁_1 l2)) →
tl₁.length = tl₁_1.length + 1 →
hd₁ + b * ofDigits b tl₁_1 + ofDigits... | rw [← ih₁ h.symm, mul_add] | case cons.cons
b hd₁ hd₂ : ℕ
tl₂ tl₁ : List ℕ
ih₂ :
∀ (tl₁_1 : List ℕ),
(∀ {l2 : List ℕ},
tl₁_1.length = l2.length →
ofDigits b tl₁_1 + ofDigits b l2 = ofDigits b (List.zipWith (fun x1 x2 => x1 + x2) tl₁_1 l2)) →
tl₁.length = tl₁_1.length + 1 →
hd₁ + b * ofDigits b tl₁_1 + ofDigits... | 455c84b9a6947b4e |
Polynomial.cauchyBound_zero | Mathlib/Analysis/Polynomial/CauchyBound.lean | @[simp]
lemma cauchyBound_zero : cauchyBound (0 : K[X]) = 1 | K : Type u_1
inst✝ : NormedDivisionRing K
⊢ cauchyBound 0 = 1 | simp [cauchyBound] | no goals | d7384768753ed59c |
Complex.equivRealProd_apply_le' | Mathlib/Analysis/Complex/Basic.lean | theorem equivRealProd_apply_le' (z : ℂ) : ‖equivRealProd z‖ ≤ 1 * ‖z‖ | z : ℂ
⊢ ‖equivRealProd z‖ ≤ 1 * ‖z‖ | simpa using equivRealProd_apply_le z | no goals | 75c9c4dfc1f38349 |
tendsto_sub_mul_tsum_nat_rpow | Mathlib/NumberTheory/LSeries/RiemannZeta.lean | theorem tendsto_sub_mul_tsum_nat_rpow :
Tendsto (fun s : ℝ ↦ (s - 1) * ∑' (n : ℕ), 1 / (n : ℝ) ^ s) (𝓝[>] 1) (𝓝 1) | ⊢ Tendsto (fun x => ↑((x - 1) * ∑' (n : ℕ), 1 / ↑n ^ x)) (𝓝[>] 1) (𝓝 1) | have : Tendsto (fun s : ℝ ↦ (s : ℂ)) (𝓝[>] 1) (𝓝[{s | 1 < re s}] 1) :=
continuous_ofReal.continuousWithinAt.tendsto_nhdsWithin (fun _ _ ↦ by aesop) | this : Tendsto (fun s => ↑s) (𝓝[>] 1) (𝓝[{s | 1 < s.re}] 1)
⊢ Tendsto (fun x => ↑((x - 1) * ∑' (n : ℕ), 1 / ↑n ^ x)) (𝓝[>] 1) (𝓝 1) | 4cff7aa1c0a1ed76 |
BoxIntegral.Integrable.sum_integral_congr | Mathlib/Analysis/BoxIntegral/Basic.lean | theorem sum_integral_congr (h : Integrable I l f vol) {π₁ π₂ : Prepartition I}
(hU : π₁.iUnion = π₂.iUnion) :
∑ J ∈ π₁.boxes, integral J l f vol = ∑ J ∈ π₂.boxes, integral J l f vol | ι : Type u
E : Type v
F : Type w
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
I : Box ι
inst✝¹ : Fintype ι
l : IntegrationParams
f : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
inst✝ : CompleteSpace F
h : Integrable I l f vol
π₁ π₂ : Prepartition I
hU : π₁.iUnio... | rw [l.toFilteriUnion_congr _ hU] | ι : Type u
E : Type v
F : Type w
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
I : Box ι
inst✝¹ : Fintype ι
l : IntegrationParams
f : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
inst✝ : CompleteSpace F
h : Integrable I l f vol
π₁ π₂ : Prepartition I
hU : π₁.iUnio... | 62b055929ceba53b |
Odd.add_odd | Mathlib/Algebra/Ring/Parity.lean | lemma Odd.add_odd : Odd a → Odd b → Even (a + b) | α : Type u_2
inst✝ : Semiring α
a b : α
⊢ Odd a → Odd b → Even (a + b) | rintro ⟨a, rfl⟩ ⟨b, rfl⟩ | case intro.intro
α : Type u_2
inst✝ : Semiring α
a b : α
⊢ Even (2 * a + 1 + (2 * b + 1)) | f6ceec06f69eb743 |
schnirelmannDensity_le_one | Mathlib/Combinatorics/Schnirelmann.lean | lemma schnirelmannDensity_le_one : schnirelmannDensity A ≤ 1 :=
schnirelmannDensity_le_of_le 1 one_ne_zero <|
by rw [Nat.cast_one, div_one, Nat.cast_le_one]; exact card_filter_le _ _
| A : Set ℕ
inst✝ : DecidablePred fun x => x ∈ A
⊢ ↑(#(filter (fun a => a ∈ A) (Ioc 0 1))) / ↑1 ≤ 1 | rw [Nat.cast_one, div_one, Nat.cast_le_one] | A : Set ℕ
inst✝ : DecidablePred fun x => x ∈ A
⊢ #(filter (fun a => a ∈ A) (Ioc 0 1)) ≤ 1 | cd68ce3caa827ed0 |
RootPairing.Base.cartanMatrix_le_zero_of_ne | Mathlib/LinearAlgebra/RootSystem/CartanMatrix.lean | lemma cartanMatrix_le_zero_of_ne [Finite ι] [NoZeroDivisors R]
[NoZeroSMulDivisors R M] [NoZeroSMulDivisors R N]
(i j : b.support) (h : i ≠ j) :
b.cartanMatrix i j ≤ 0 :=
b.pairingIn_le_zero_of_ne (by rwa [ne_eq, ← Subtype.ext_iff]) i.property j.property
| ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : AddCommGroup M
inst✝⁸ : Module R M
inst✝⁷ : AddCommGroup N
inst✝⁶ : Module R N
P : RootPairing ι R M N
b : P.Base
inst✝⁵ : P.IsCrystallographic
inst✝⁴ : CharZero R
inst✝³ : Finite ι
inst✝² : NoZeroDivisors R
inst✝¹ : NoZeroSMulDivisors R ... | rwa [ne_eq, ← Subtype.ext_iff] | no goals | a81e948ee27e6a37 |
map_inv_lt_map_inv_iff | Mathlib/Order/Hom/Basic.lean | theorem map_inv_lt_map_inv_iff (f : F) {a b : β} :
EquivLike.inv f b < EquivLike.inv f a ↔ b < a | F : Type u_1
α : Type u_2
β : Type u_3
inst✝³ : Preorder α
inst✝² : Preorder β
inst✝¹ : EquivLike F α β
inst✝ : OrderIsoClass F α β
f : F
a b : β
⊢ EquivLike.inv f b < EquivLike.inv f a ↔ b < a | simp | no goals | 0df3832abafeccd6 |
Finset.card_le_card_of_surjOn | Mathlib/Data/Finset/Card.lean | lemma card_le_card_of_surjOn (f : α → β) (hf : Set.SurjOn f s t) : #t ≤ #s | α : Type u_1
β : Type u_2
s : Finset α
t : Finset β
f : α → β
hf : Set.SurjOn f ↑s ↑t
⊢ #t ≤ #s | classical unfold Set.SurjOn at hf; exact (card_le_card (mod_cast hf)).trans card_image_le | no goals | 50a60dfea1ce9fd7 |
krullTopology_mem_nhds_one_iff_of_normal | Mathlib/FieldTheory/KrullTopology.lean | lemma krullTopology_mem_nhds_one_iff_of_normal (K L : Type*) [Field K] [Field L] [Algebra K L]
[Normal K L] (s : Set (L ≃ₐ[K] L)) : s ∈ 𝓝 1 ↔ ∃ E : IntermediateField K L,
FiniteDimensional K E ∧ Normal K E ∧ (E.fixingSubgroup : Set (L ≃ₐ[K] L)) ⊆ s | case h
K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : Normal K L
s : Set (L ≃ₐ[K] L)
x✝ : ∃ E, FiniteDimensional K ↥E ∧ ↑E.fixingSubgroup ⊆ s
E : IntermediateField K L
left✝ : FiniteDimensional K ↥E
hE : ↑E.fixingSubgroup ⊆ s
⊢ ↑(IntermediateField.normalClosure K (↥E) L).fixing... | exact le_trans (E.fixingSubgroup_anti E.le_normalClosure) hE | no goals | 7bcf0176217ac3b2 |
AlgebraicGeometry.ValuativeCriterion.Existence.of_specializingMap | Mathlib/AlgebraicGeometry/ValuativeCriterion.lean | lemma of_specializingMap (H : (topologically @SpecializingMap).universally f) :
ValuativeCriterion.Existence f | case mk.mk.intro.intro
X Y : Scheme
f : X ⟶ Y
R : Type u
commRing✝ : CommRing R
domain✝ : IsDomain R
valuationRing✝ : ValuationRing R
K : Type u
field✝ : Field K
algebra✝ : Algebra R K
isFractionRing✝ : IsFractionRing R K
i₁ : Spec (CommRingCat.of K) ⟶ X
i₂ : Spec (CommRingCat.of R) ⟶ Y
w : i₁ ≫ f = Spec.map (CommRingC... | have hαφ : α ≫ φ = 𝟙 _ := by ext x; exact (RingEquiv.ofBijective _ hbij).symm_apply_apply x | case mk.mk.intro.intro
X Y : Scheme
f : X ⟶ Y
R : Type u
commRing✝ : CommRing R
domain✝ : IsDomain R
valuationRing✝ : ValuationRing R
K : Type u
field✝ : Field K
algebra✝ : Algebra R K
isFractionRing✝ : IsFractionRing R K
i₁ : Spec (CommRingCat.of K) ⟶ X
i₂ : Spec (CommRingCat.of R) ⟶ Y
w : i₁ ≫ f = Spec.map (CommRingC... | 3a778dfa0b125889 |
IsPrimitiveRoot.minpoly_dvd_expand | Mathlib/RingTheory/RootsOfUnity/Minpoly.lean | theorem minpoly_dvd_expand {p : ℕ} (hdiv : ¬p ∣ n) :
minpoly ℤ μ ∣ expand ℤ p (minpoly ℤ (μ ^ p)) | case inr
n : ℕ
K : Type u_1
inst✝² : CommRing K
μ : K
h : IsPrimitiveRoot μ n
inst✝¹ : IsDomain K
inst✝ : CharZero K
p : ℕ
hdiv : ¬p ∣ n
hpos : n > 0
this : IsIntegrallyClosed ℤ := GCDMonoid.toIsIntegrallyClosed
⊢ minpoly ℤ μ ∣ (expand ℤ p) (minpoly ℤ (μ ^ p)) | refine minpoly.isIntegrallyClosed_dvd (h.isIntegral hpos) ?_ | case inr
n : ℕ
K : Type u_1
inst✝² : CommRing K
μ : K
h : IsPrimitiveRoot μ n
inst✝¹ : IsDomain K
inst✝ : CharZero K
p : ℕ
hdiv : ¬p ∣ n
hpos : n > 0
this : IsIntegrallyClosed ℤ := GCDMonoid.toIsIntegrallyClosed
⊢ (Polynomial.aeval μ) ((expand ℤ p) (minpoly ℤ (μ ^ p))) = 0 | 765e9f715d5c1ba0 |
range_mfderiv_coe_sphere | Mathlib/Geometry/Manifold/Instances/Sphere.lean | theorem range_mfderiv_coe_sphere {n : ℕ} [Fact (finrank ℝ E = n + 1)] (v : sphere (0 : E) 1) :
LinearMap.range (mfderiv (𝓡 n) 𝓘(ℝ, E) ((↑) : sphere (0 : E) 1 → E) v :
TangentSpace (𝓡 n) v →L[ℝ] E) = (ℝ ∙ (v : E))ᗮ | case h.e'_3
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
n : ℕ
inst✝ : Fact (finrank ℝ E = n + 1)
v : ↑(sphere 0 1)
U : ↥(Submodule.span ℝ {↑(-v)})ᗮ ≃ₗᵢ[ℝ] EuclideanSpace ℝ (Fin n) :=
(OrthonormalBasis.fromOrthogonalSpanSingleton n ⋯).repr
this : HasFDerivAt (stereoInvFunAux (-↑v) ∘ Subty... | congr 1 | case h.e'_3.e_K
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
n : ℕ
inst✝ : Fact (finrank ℝ E = n + 1)
v : ↑(sphere 0 1)
U : ↥(Submodule.span ℝ {↑(-v)})ᗮ ≃ₗᵢ[ℝ] EuclideanSpace ℝ (Fin n) :=
(OrthonormalBasis.fromOrthogonalSpanSingleton n ⋯).repr
this : HasFDerivAt (stereoInvFunAux (-↑v) ∘ S... | f44b78981a1f8a7d |
Profinite.NobelingProof.succ_mono | Mathlib/Topology/Category/Profinite/Nobeling.lean | theorem succ_mono : CategoryTheory.Mono (ModuleCat.ofHom (πs C o)) | I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
o : Ordinal.{u}
⊢ Mono (ModuleCat.ofHom (πs C o)) | rw [ModuleCat.mono_iff_injective] | I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
o : Ordinal.{u}
⊢ Function.Injective ⇑(ConcreteCategory.hom (ModuleCat.ofHom (πs C o))) | 7056669d7d8761aa |
contMDiff_tangentBundleModelSpaceHomeomorph_symm | Mathlib/Geometry/Manifold/VectorBundle/Tangent.lean | theorem contMDiff_tangentBundleModelSpaceHomeomorph_symm {n : ℕ∞} :
ContMDiff (I.prod 𝓘(𝕜, E)) I.tangent n
((tangentBundleModelSpaceHomeomorph I).symm : ModelProd H E → TangentBundle I H) | 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
H : Type u_4
inst✝ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
n : ℕ∞
x : ModelProd H E
y : TangentBundle I H
⊢ ∀ (a b : E) (x : H),
↑I x = a →
↑I
({ toFun := fun x => (x.pro... | rintro a b x rfl | 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
H : Type u_4
inst✝ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
n : ℕ∞
x✝ : ModelProd H E
y : TangentBundle I H
b : E
x : H
⊢ ↑I
({ toFun := fun x => (x.proj, x.snd), invFun := fun x => { p... | 657294e89406becb |
mul_neg_geom_series | Mathlib/Analysis/SpecificLimits/Normed.lean | theorem mul_neg_geom_series (x : R) (h : ‖x‖ < 1) : (1 - x) * ∑' i : ℕ, x ^ i = 1 | R : Type u_2
inst✝¹ : NormedRing R
inst✝ : HasSummableGeomSeries R
x : R
h : ‖x‖ < 1
this : HasSum (fun i => (1 - x) * x ^ i) ((1 - x) * ∑' (b : ℕ), x ^ b)
⊢ Tendsto (fun n => 1 - x ^ n) atTop (𝓝 1) | simpa using tendsto_const_nhds.sub (tendsto_pow_atTop_nhds_zero_of_norm_lt_one h) | no goals | 9c96f6b2cfbada68 |
Nat.sub_one_mul_sum_log_div_pow_eq_sub_sum_digits | Mathlib/Data/Nat/Digits.lean | theorem sub_one_mul_sum_log_div_pow_eq_sub_sum_digits {p : ℕ} (n : ℕ) :
(p - 1) * ∑ i ∈ range (log p n).succ, n / p ^ i.succ = n - (p.digits n).sum | case h.e'_3.h.e'_5
p n : ℕ
h : 1 < p
hn : n ≠ 0
⊢ n = ofDigits p (p.digits n) | exact (ofDigits_digits p n).symm | no goals | 7f8293a002c12a35 |
Ordinal.IsLimit.iSup_Iio | Mathlib/SetTheory/Ordinal/Arithmetic.lean | theorem IsLimit.iSup_Iio {o : Ordinal} (h : IsLimit o) : ⨆ a : Iio o, a.1 = o | o : Ordinal.{u_4}
h : o.IsLimit
⊢ ⨆ a, ↑a = o | rw [← sSup_eq_iSup', h.sSup_Iio] | no goals | 625bd6c601073215 |
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.mk.mk.intro.mk.mk.intro
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 = ∅
this : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ t }
n : ℕ
u : ℕ → α
hu :... | let w i := if i ≤ n then u i else v (i - (n + 1)) | case neg.mk.mk.intro.mk.mk.intro
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 = ∅
this : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ t }
n : ℕ
u : ℕ → α
hu :... | 4878120f0018b659 |
CategoryTheory.constant_of_preserves_morphisms | Mathlib/CategoryTheory/IsConnected.lean | theorem constant_of_preserves_morphisms [IsPreconnected J] {α : Type u₂} (F : J → α)
(h : ∀ (j₁ j₂ : J) (_ : j₁ ⟶ j₂), F j₁ = F j₂) (j j' : J) : F j = F j' | J : Type u₁
inst✝¹ : Category.{v₁, u₁} J
inst✝ : IsPreconnected J
α : Type u₂
F : J → α
h : ∀ (j₁ j₂ : J), (j₁ ⟶ j₂) → F j₁ = F j₂
j j' : J
⊢ F j = F j' | simpa using
any_functor_const_on_obj
{ obj := Discrete.mk ∘ F
map := fun f => eqToHom (by ext; exact h _ _ f) }
j j' | no goals | bfd10d5e6cac67ef |
SchwartzMap.integrable_of_le_of_pow_mul_le | Mathlib/Analysis/Distribution/SchwartzSpace.lean | /-- Given a function such that `f` and `x ^ (k + l) * f` are bounded for a suitable `l`, then
`x ^ k * f` is integrable. The bounds are not relevant for the integrability conclusion, but they
are relevant for bounding the integral in `integral_pow_mul_le_of_le_of_pow_mul_le`. We formulate
the two lemmas with the same s... | case hf
D : Type u_3
inst✝⁵ : NormedAddCommGroup D
inst✝⁴ : MeasurableSpace D
inst✝³ : BorelSpace D
inst✝² : SecondCountableTopology D
E : Type u_8
inst✝¹ : NormedAddCommGroup E
μ : Measure D
inst✝ : μ.HasTemperateGrowth
f : D → E
C₁ C₂ : ℝ
k : ℕ
hf : ∀ (x : D), ‖f x‖ ≤ C₁
h'f : ∀ (x : D), ‖x‖ ^ (k + μ.integrablePower)... | exact AEStronglyMeasurable.mul (aestronglyMeasurable_id.norm.pow _) h''f.norm | no goals | 30d7f9baab6716a7 |
CategoryTheory.Subgroupoid.isNormal_comap | Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean | theorem isNormal_comap {S : Subgroupoid D} (Sn : IsNormal S) : IsNormal (comap φ S) where
wide c | C : Type u
inst✝¹ : Groupoid C
D : Type u_1
inst✝ : Groupoid D
φ : C ⥤ D
S : Subgroupoid D
Sn : S.IsNormal
c : C
⊢ 𝟙 (φ.obj c) ∈ S.arrows (φ.obj c) (φ.obj c) | apply Sn.wide | no goals | 6cc9be588efac4d5 |
Submodule.mem_smul_iff_inv_mul_mem | Mathlib/Algebra/Algebra/Operations.lean | lemma mem_smul_iff_inv_mul_mem {S} [Field S] [Algebra R S] {x : S} {p : Submodule R S} {y : S}
(hx : x ≠ 0) : y ∈ x • p ↔ x⁻¹ * y ∈ p | case mp
R : Type u
inst✝² : CommSemiring R
S : Type u_1
inst✝¹ : Field S
inst✝ : Algebra R S
x : S
p : Submodule R S
y : S
hx : x ≠ 0
⊢ y ∈ x • p → x⁻¹ * y ∈ p | rintro ⟨a, ha : a ∈ p, rfl⟩ | case mp.intro.intro
R : Type u
inst✝² : CommSemiring R
S : Type u_1
inst✝¹ : Field S
inst✝ : Algebra R S
x : S
p : Submodule R S
hx : x ≠ 0
a : S
ha : a ∈ p
⊢ x⁻¹ * (DistribMulAction.toLinearMap R S x) a ∈ p | 6fc660e1d3d54d45 |
List.forIn'_congr | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Monadic.lean | theorem forIn'_congr [Monad m] {as bs : List α} (w : as = bs)
{b b' : β} (hb : b = b')
{f : (a' : α) → a' ∈ as → β → m (ForInStep β)}
{g : (a' : α) → a' ∈ bs → β → m (ForInStep β)}
(h : ∀ a m b, f a (by simpa [w] using m) b = g a m b) :
forIn' as b f = forIn' bs b' g | case nil
m : Type u_1 → Type u_2
α : Type u_3
β : Type u_1
inst✝ : Monad m
b b' : β
hb : b = b'
g f : (a' : α) → a' ∈ [] → β → m (ForInStep β)
h : ∀ (a : α) (m_1 : a ∈ []) (b : β), f a ⋯ b = g a m_1 b
⊢ forIn' [] b f = forIn' [] b' g | simp [hb, forIn'_nil] | no goals | b21e9d91722268c4 |
Std.DHashMap.Raw.getThenInsertIfNew?_fst | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/RawLemmas.lean | theorem getThenInsertIfNew?_fst [LawfulBEq α] (h : m.WF) {k : α} {v : β k} :
(m.getThenInsertIfNew? k v).1 = m.get? k | α : Type u
β : α → Type v
m : Raw α β
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : LawfulBEq α
h : m.WF
k : α
v : β k
⊢ (m.getThenInsertIfNew? k v).fst = m.get? k | simp_to_raw using Raw₀.getThenInsertIfNew?_fst | no goals | a1beb5ed8111f091 |
PadicInt.norm_sub_modPart | Mathlib/NumberTheory/Padics/RingHoms.lean | theorem norm_sub_modPart (h : ‖(r : ℚ_[p])‖ ≤ 1) : ‖(⟨r, h⟩ - modPart p r : ℤ_[p])‖ < 1 | p : ℕ
hp_prime : Fact (Nat.Prime p)
r : ℚ
h : ‖↑r‖ ≤ 1
n : ℤ := modPart p r
⊢ ↑p ∣ (⟨↑r, h⟩ - ↑(modPart p r)) * ↑r.den | suffices ↑p ∣ r.num - n * r.den by
convert (Int.castRingHom ℤ_[p]).map_dvd this
simp only [n, sub_mul, Int.cast_natCast, eq_intCast, Int.cast_mul, sub_left_inj,
Int.cast_sub]
apply Subtype.coe_injective
simp only [coe_mul, Subtype.coe_mk, coe_natCast]
rw_mod_cast [@Rat.mul_den_eq_num r]
rfl | p : ℕ
hp_prime : Fact (Nat.Prime p)
r : ℚ
h : ‖↑r‖ ≤ 1
n : ℤ := modPart p r
⊢ ↑p ∣ r.num - n * ↑r.den | 7311994057bd6f68 |
CategoryTheory.MorphismProperty.map_eq_iff_postcomp | Mathlib/CategoryTheory/Localization/CalculusOfFractions.lean | lemma MorphismProperty.map_eq_iff_postcomp {X Y : C} (f₁ f₂ : X ⟶ Y) :
L.map f₁ = L.map f₂ ↔ ∃ (Z : C) (s : Y ⟶ Z) (_ : W s), f₁ ≫ s = f₂ ≫ s | case mpr.intro.intro.intro
C : Type u_1
D : Type u_2
inst✝³ : Category.{u_3, u_1} C
inst✝² : Category.{u_4, u_2} D
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
f₁ f₂ : X ⟶ Y
Z : C
s : Y ⟶ Z
hs : W s
fac : f₁ ≫ s = f₂ ≫ s
⊢ L.map f₁ = L.map f₂ | simp only [← cancel_mono (Localization.isoOfHom L W s hs).hom,
Localization.isoOfHom_hom, ← L.map_comp, fac] | no goals | 078433f6a97f8f9b |
Batteries.HashMap.empty_find? | Mathlib/.lake/packages/batteries/Batteries/Data/HashMap/Lemmas.lean | theorem empty_find? [BEq α] [Hashable α] {a : α} :
(∅ : HashMap α β).find? a = none | α : Type u_1
β : Type u_2
inst✝¹ : BEq α
inst✝ : Hashable α
a : α
⊢ ∅.find? a = none | simp [find?, Imp.find?] | no goals | 4995f96ba5aa9ae7 |
GromovHausdorff.ghDist_le_of_approx_subsets | Mathlib/Topology/MetricSpace/GromovHausdorff.lean | theorem ghDist_le_of_approx_subsets {s : Set X} (Φ : s → Y) {ε₁ ε₂ ε₃ : ℝ}
(hs : ∀ x : X, ∃ y ∈ s, dist x y ≤ ε₁) (hs' : ∀ x : Y, ∃ y : s, dist x (Φ y) ≤ ε₃)
(H : ∀ x y : s, |dist x y - dist (Φ x) (Φ y)| ≤ ε₂) : ghDist X Y ≤ ε₁ + ε₂ / 2 + ε₃ | X : Type u
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
Y : Type v
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
s : Set X
Φ : ↑s → Y
ε₁ ε₂ ε₃ : ℝ
hs : ∀ (x : X), ∃ y ∈ s, dist x y ≤ ε₁
hs' : ∀ (x : Y), ∃ y, dist x (Φ y) ≤ ε₃
H : ∀ (x y : ↑s), |dist x y - dist (Φ x) (Φ y)| ≤ ε₂... | refine hausdorffDist_le_of_mem_dist (by linarith) ?_ ?_ | case refine_1
X : Type u
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
Y : Type v
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
s : Set X
Φ : ↑s → Y
ε₁ ε₂ ε₃ : ℝ
hs : ∀ (x : X), ∃ y ∈ s, dist x y ≤ ε₁
hs' : ∀ (x : Y), ∃ y, dist x (Φ y) ≤ ε₃
H : ∀ (x y : ↑s), |dist x y - dist (Φ ... | ca73aa8d17706ff3 |
MeasureTheory.L1.SimpleFunc.integral_eq_norm_posPart_sub | Mathlib/MeasureTheory/Integral/BochnerL1.lean | theorem integral_eq_norm_posPart_sub (f : α →₁ₛ[μ] ℝ) : integral f = ‖posPart f‖ - ‖negPart f‖ | α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ↥(simpleFunc ℝ 1 μ)
⊢ integral f = ‖posPart f‖ - ‖negPart f‖ | have ae_eq₁ : (toSimpleFunc f).posPart =ᵐ[μ] (toSimpleFunc (posPart f)).map norm := by
filter_upwards [posPart_toSimpleFunc f] with _ h
rw [SimpleFunc.map_apply, h]
conv_lhs => rw [← SimpleFunc.posPart_map_norm, SimpleFunc.map_apply] | α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ↥(simpleFunc ℝ 1 μ)
ae_eq₁ : ⇑(toSimpleFunc f).posPart =ᶠ[ae μ] ⇑(SimpleFunc.map norm (toSimpleFunc (posPart f)))
⊢ integral f = ‖posPart f‖ - ‖negPart f‖ | 464e4ba9ec3511ad |
MvQPF.supp_eq | Mathlib/Data/QPF/Multivariate/Basic.lean | theorem supp_eq {α : TypeVec n} {i} (x : F α) :
supp x i = { u | ∀ a f, abs ⟨a, f⟩ = x → u ∈ f i '' univ } | n : ℕ
F : TypeVec.{u} n → Type u_1
q : MvQPF F
α : TypeVec.{u} n
i : Fin2 n
x : F α
⊢ supp x i = {u | ∀ (a : (P F).A) (f : (P F).B a ⟹ α), abs ⟨a, f⟩ = x → u ∈ f i '' univ} | ext | case h
n : ℕ
F : TypeVec.{u} n → Type u_1
q : MvQPF F
α : TypeVec.{u} n
i : Fin2 n
x : F α
x✝ : α i
⊢ x✝ ∈ supp x i ↔ x✝ ∈ {u | ∀ (a : (P F).A) (f : (P F).B a ⟹ α), abs ⟨a, f⟩ = x → u ∈ f i '' univ} | f337d415d55c7932 |
MeasureTheory.q | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean | theorem q {hs : MeasurableSet s} {hμs : μ s ≠ ∞} {x : G} :
MemLp (condExpIndSMul hm hs hμs x) 1 μ | α : Type u_1
G : Type u_4
inst✝² : NormedAddCommGroup G
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
hs : MeasurableSet s
hμs : μ s ≠ ⊤
x : G
⊢ MemLp (↑↑(condExpIndSMul hm hs hμs x)) 1 μ | rw [memLp_one_iff_integrable] | α : Type u_1
G : Type u_4
inst✝² : NormedAddCommGroup G
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
hs : MeasurableSet s
hμs : μ s ≠ ⊤
x : G
⊢ Integrable (↑↑(condExpIndSMul hm hs hμs x)) μ | 04475e394d69e1bd |
Batteries.HashMap.Imp.expand_size | Mathlib/.lake/packages/batteries/Batteries/Data/HashMap/WF.lean | theorem expand_size [Hashable α] {buckets : Buckets α β} :
(expand sz buckets).buckets.size = buckets.size | case e_a.e_a
α : Type u_1
β : Type u_2
sz : Nat
inst✝ : Hashable α
buckets : Buckets α β
i : Nat
source : Array (AssocList α β)
target : Buckets α β
hs : ∀ (j : Nat), j < i → source.toList[j]?.getD AssocList.nil = AssocList.nil
H : i < source.size
l₁ l₂ : List (AssocList α β)
h₁ : source.toList = l₁ ++ source.toList[i]... | simp [Buckets.size] at this | case e_a.e_a
α : Type u_1
β : Type u_2
sz : Nat
inst✝ : Hashable α
buckets : Buckets α β
i : Nat
source : Array (AssocList α β)
target : Buckets α β
hs : ∀ (j : Nat), j < i → source.toList[j]?.getD AssocList.nil = AssocList.nil
H : i < source.size
l₁ l₂ : List (AssocList α β)
h₁ : source.toList = l₁ ++ source.toList[i]... | 5e56d234c0d941c0 |
orderOf_eq_iff | Mathlib/GroupTheory/OrderOfElement.lean | theorem orderOf_eq_iff {n} (h : 0 < n) :
orderOf x = n ↔ x ^ n = 1 ∧ ∀ m, m < n → 0 < m → x ^ m ≠ 1 | case pos
G : Type u_1
inst✝ : Monoid G
x : G
n : ℕ
h : 0 < n
h1 : 1 ∈ periodicPts fun x_1 => x * x_1
⊢ Nat.find h1 = n ↔ IsPeriodicPt (fun x_1 => x * x_1) n 1 ∧ ∀ m < n, 0 < m → ¬IsPeriodicPt (fun x_1 => x * x_1) m 1 | rw [find_eq_iff] | case pos
G : Type u_1
inst✝ : Monoid G
x : G
n : ℕ
h : 0 < n
h1 : 1 ∈ periodicPts fun x_1 => x * x_1
⊢ ((n > 0 ∧ IsPeriodicPt (fun x_1 => x * x_1) n 1) ∧ ∀ n_1 < n, ¬(n_1 > 0 ∧ IsPeriodicPt (fun x_1 => x * x_1) n_1 1)) ↔
IsPeriodicPt (fun x_1 => x * x_1) n 1 ∧ ∀ m < n, 0 < m → ¬IsPeriodicPt (fun x_1 => x * x_1) m 1 | 8667a5de67464649 |
AddMonoid.isTorsionFree_iff_noZeroSMulDivisors_int | Mathlib/GroupTheory/Torsion.lean | lemma isTorsionFree_iff_noZeroSMulDivisors_int [AddGroup G] :
IsTorsionFree G ↔ NoZeroSMulDivisors ℤ G | G : Type u_1
inst✝ : AddGroup G
⊢ IsTorsionFree G ↔ NoZeroSMulDivisors ℤ G | simp_rw [AddMonoid.IsTorsionFree, isOfFinAddOrder_iff_zsmul_eq_zero, not_exists, not_and,
noZeroSMulDivisors_iff, forall_swap (β := ℤ)] | G : Type u_1
inst✝ : AddGroup G
⊢ (∀ (y : ℤ) (x : G), x ≠ 0 → y ≠ 0 → ¬y • x = 0) ↔ ∀ {c : ℤ} {x : G}, c • x = 0 → c = 0 ∨ x = 0 | cb14b7e659ef08f2 |
MeasureTheory.tendsto_setToFun_approxOn_of_measurable_of_range_subset | Mathlib/MeasureTheory/Integral/SetToL1.lean | theorem tendsto_setToFun_approxOn_of_measurable_of_range_subset
(hT : DominatedFinMeasAdditive μ T C) [MeasurableSpace E] [BorelSpace E] {f : α → E}
(fmeas : Measurable f) (hf : Integrable f μ) (s : Set E) [SeparableSpace s]
(hs : range f ∪ {0} ⊆ s) :
Tendsto (fun n => setToFun μ T hT (SimpleFunc.approx... | α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝³ : CompleteSpace F
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace... | refine tendsto_setToFun_approxOn_of_measurable hT hf fmeas ?_ _ (integrable_zero _ _ _) | α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝³ : CompleteSpace F
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace... | 129fa37e00f8db6a |
List.getElem_eraseIdx | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/Erase.lean | theorem getElem_eraseIdx (l : List α) (i : Nat) (j : Nat) (h : j < (l.eraseIdx i).length) :
(l.eraseIdx i)[j] = if h' : j < i then
l[j]'(by have := length_eraseIdx_le l i; omega)
else
l[j + 1]'(by rw [length_eraseIdx] at h; split at h <;> omega) | case x
α : Type u_1
l : List α
i j : Nat
h : j < (l.eraseIdx i).length
⊢ some (l.eraseIdx i)[j] = some (if h' : j < i then l[j] else l[j + 1]) | rw [← getElem?_eq_getElem, getElem?_eraseIdx] | case x
α : Type u_1
l : List α
i j : Nat
h : j < (l.eraseIdx i).length
⊢ (if j < i then l[j]? else l[j + 1]?) = some (if h' : j < i then l[j] else l[j + 1]) | 29c9d8a2fe1d6782 |
Matroid.Indep.union_indep_iff_forall_not_mem_closure_left | Mathlib/Data/Matroid/Closure.lean | lemma Indep.union_indep_iff_forall_not_mem_closure_left (hI : M.Indep I) (hJ : M.Indep J) :
M.Indep (I ∪ J) ↔ ∀ e ∈ I \ J, e ∉ M.closure ((I \ {e}) ∪ J) | α : Type u_2
M : Matroid α
I J : Set α
hI : M.Indep I
hJ : M.Indep J
⊢ M.Indep (I ∪ J) ↔ ∀ e ∈ I \ J, e ∉ M.closure (I \ {e} ∪ J) | simp_rw [union_comm I J, hJ.union_indep_iff_forall_not_mem_closure_right hI, union_comm] | no goals | c60d2940e605eed8 |
AlgebraicTopology.DoldKan.MorphComponents.id_φ | Mathlib/AlgebraicTopology/DoldKan/Decomposition.lean | theorem id_φ : (id X n).φ = 𝟙 _ | C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
n : ℕ
⊢ PInfty.f (n + 1) ≫ (id X n).a + ∑ i : Fin (n + 1), (P ↑i).f (n + 1) ≫ X.δ i.rev.succ ≫ (id X n).b i.rev =
(P (n + 1)).f (n + 1) + (Q (n + 1)).f (n + 1) | congr 1 | case e_a
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
n : ℕ
⊢ PInfty.f (n + 1) ≫ (id X n).a = (P (n + 1)).f (n + 1)
case e_a
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
n : ℕ
⊢ ∑ i : Fin (n + 1), (P ↑i).f (n + 1) ≫ X.δ i.rev.succ ... | 61503b4830313759 |
swap_apply_apply | Mathlib/Algebra/Group/End.lean | theorem swap_apply_apply (f : Perm α) (x y : α) : swap (f x) (f y) = f * swap x y * f⁻¹ | α : Type u_4
inst✝ : DecidableEq α
f : Perm α
x y : α
⊢ swap (f x) (f y) = f * swap x y * f⁻¹ | rw [mul_swap_eq_swap_mul, mul_inv_cancel_right] | no goals | 5e80ee108090a134 |
sum_range_pow | Mathlib/NumberTheory/Bernoulli.lean | theorem sum_range_pow (n p : ℕ) :
(∑ k ∈ range n, (k : ℚ) ^ p) =
∑ i ∈ range (p + 1), bernoulli i * ((p + 1).choose i) * (n : ℚ) ^ (p + 1 - i) / (p + 1) | case h
n p : ℕ
hne : ∀ (m : ℕ), ↑m ! ≠ 0
q : ℕ
f : ℕ → ℕ → ℚ := fun a b => bernoulli a / ↑a ! * (coeff ℚ (b + 1)) (exp ℚ ^ n)
m : ℕ
h : m < q + 1
⊢ bernoulli m * ↑((q + 1)! / (m ! * (q + 1 - m)!)) * ↑n ^ (q + 1 - m) =
bernoulli m * ↑q.succ ! / ↑m ! *
(coeff ℚ (q - m + 1))
({ toFun := fun f => PowerSer... | simp only [add_eq, add_zero, IsUnit.mul_iff, Nat.isUnit_iff, succ.injEq, cast_mul,
cast_succ, MonoidHom.coe_mk, OneHom.coe_mk, coeff_exp, Algebra.id.map_eq_id, one_div,
map_inv₀, map_natCast, coeff_mk, mul_inv_rev] | case h
n p : ℕ
hne : ∀ (m : ℕ), ↑m ! ≠ 0
q : ℕ
f : ℕ → ℕ → ℚ := fun a b => bernoulli a / ↑a ! * (coeff ℚ (b + 1)) (exp ℚ ^ n)
m : ℕ
h : m < q + 1
⊢ bernoulli m * ↑((q + 1)! / (m ! * (q + 1 - m)!)) * ↑n ^ (q + 1 - m) =
bernoulli m * ↑q.succ ! / ↑m ! * (↑n ^ (q - m + 1) * (↑(q - m + 1)!)⁻¹) | fc82cef36f6524aa |
Specializes.joinedIn | Mathlib/Topology/Connected/PathConnected.lean | theorem Specializes.joinedIn (h : x ⤳ y) (hx : x ∈ F) (hy : y ∈ F) : JoinedIn F x y | case refine_1
X : Type u_1
inst✝ : TopologicalSpace X
x y : X
F : Set X
h : x ⤳ y
hx : x ∈ F
hy : y ∈ F
⊢ Continuous ({1}.piecewise (const (↑I) y) (const (↑I) x)) | exact isClosed_singleton.continuous_piecewise_of_specializes continuous_const continuous_const
fun _ ↦ h | no goals | c2e84f08712afacd |
Real.exists_isLUB | Mathlib/Data/Real/Archimedean.lean | theorem exists_isLUB (hne : s.Nonempty) (hbdd : BddAbove s) : ∃ x, IsLUB s x | case intro.intro
s : Set ℝ
L : ℝ
hL : L ∈ s
U : ℝ
hU : U ∈ upperBounds s
this : ∀ (d : ℕ), BddAbove {m | ∃ y ∈ s, ↑m ≤ y * ↑d}
f : ℕ → ℤ
hf : ∀ (d : ℕ), f d ∈ {m | ∃ y ∈ s, ↑m ≤ y * ↑d} ∧ ∀ z ∈ {m | ∃ y ∈ s, ↑m ≤ y * ↑d}, z ≤ f d
hf₁ : ∀ n > 0, ∃ y ∈ s, ↑(↑(f n) / ↑n) ≤ y
hf₂ : ∀ n > 0, ∀ y ∈ s, y - (↑n)⁻¹ < ↑(↑(f n) /... | refine ⟨mk g, ⟨fun x xS => ?_, fun y h => ?_⟩⟩ | case intro.intro.refine_1
s : Set ℝ
L : ℝ
hL : L ∈ s
U : ℝ
hU : U ∈ upperBounds s
this : ∀ (d : ℕ), BddAbove {m | ∃ y ∈ s, ↑m ≤ y * ↑d}
f : ℕ → ℤ
hf : ∀ (d : ℕ), f d ∈ {m | ∃ y ∈ s, ↑m ≤ y * ↑d} ∧ ∀ z ∈ {m | ∃ y ∈ s, ↑m ≤ y * ↑d}, z ≤ f d
hf₁ : ∀ n > 0, ∃ y ∈ s, ↑(↑(f n) / ↑n) ≤ y
hf₂ : ∀ n > 0, ∀ y ∈ s, y - (↑n)⁻¹ < ↑... | a49996b1e72b07b6 |
HasLineDerivAt.congr_of_eventuallyEq | Mathlib/Analysis/Calculus/LineDeriv/Basic.lean | theorem HasLineDerivAt.congr_of_eventuallyEq (h : HasLineDerivAt 𝕜 f f' x v) (h₁ : f₁ =ᶠ[𝓝 x] f) :
HasLineDerivAt 𝕜 f₁ f' x v | 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
F✝ : Type u_2
inst✝³ : NormedAddCommGroup F✝
inst✝² : NormedSpace 𝕜 F✝
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f f₁ : E → F✝
f' : F✝
x v : E
h : HasLineDerivAt 𝕜 f f' x v
F : 𝕜 → E := fun t => x + t • v
h₁ : f₁ =ᶠ[𝓝 (F 0)] f
⊢ (fun t => f... | exact (Continuous.continuousAt (by fun_prop)).preimage_mem_nhds h₁ | no goals | ce049ab776c3e79a |
Batteries.UnionFind.root_link | Mathlib/.lake/packages/batteries/Batteries/Data/UnionFind/Lemmas.lean | theorem root_link {self : UnionFind} {x y : Fin self.size}
(xroot : self.parent x = x) (yroot : self.parent y = y) :
∃ r, (r = x ∨ r = y) ∧ ∀ i,
(link self x y yroot).rootD i =
if self.rootD i = x ∨ self.rootD i = y then r.1 else self.rootD i | self : UnionFind
x✝¹ y✝ : Fin self.size
xroot✝ : self.parent ↑x✝¹ = ↑x✝¹
yroot✝ : self.parent ↑y✝ = ↑y✝
h✝ : ¬↑x✝¹ = ↑y✝
x y : Fin self.size
xroot : self.parent ↑x = ↑x
yroot : self.parent ↑y = ↑y
m : UnionFind
hm : ∀ (i : Nat), m.parent i = if ↑y = i then ↑x else self.parent i
i : Nat
h : ¬m.parent i = i
x✝ : m.rankMa... | split <;> [subst i; rw [rootD_parent]] | case isTrue
self : UnionFind
x✝¹ y✝ : Fin self.size
xroot✝ : self.parent ↑x✝¹ = ↑x✝¹
yroot✝ : self.parent ↑y✝ = ↑y✝
h✝ : ¬↑x✝¹ = ↑y✝
x y : Fin self.size
xroot : self.parent ↑x = ↑x
yroot : self.parent ↑y = ↑y
m : UnionFind
hm : ∀ (i : Nat), m.parent i = if ↑y = i then ↑x else self.parent i
h : ¬m.parent ↑y = ↑y
x✝ : m.... | d17902f7f8b46530 |
Ordinal.accPt_subtype | Mathlib/SetTheory/Ordinal/Topology.lean | theorem accPt_subtype {p o : Ordinal} (S : Set Ordinal) (hpo : p < o) :
AccPt p (𝓟 S) ↔ AccPt ⟨p, hpo⟩ (𝓟 (Iio o ↓∩ S)) | case h.right
p o : Ordinal.{u_1}
S : Set Ordinal.{u_1}
hpo : p < o
u : Set Ordinal.{u_1}
hu : u ∈ 𝓝 p
ho : p + 1 < o
ppos : p ≠ 0
h : ¬p.IsLimit
q : Ordinal.{u_1}
hq : p = succ q
y✝ : ↑(Iio o)
mem : y✝ ∈ Ioo ⟨q, ⋯⟩ ⟨p + 1, ho⟩ ∩ Iio o ↓∩ S
aux1 : q < ↑y✝
aux2 : ↑y✝ < p + 1
⊢ ↑y✝ = p | rw [hq] at aux2 ⊢ | case h.right
p o : Ordinal.{u_1}
S : Set Ordinal.{u_1}
hpo : p < o
u : Set Ordinal.{u_1}
hu : u ∈ 𝓝 p
ho : p + 1 < o
ppos : p ≠ 0
h : ¬p.IsLimit
q : Ordinal.{u_1}
hq : p = succ q
y✝ : ↑(Iio o)
mem : y✝ ∈ Ioo ⟨q, ⋯⟩ ⟨p + 1, ho⟩ ∩ Iio o ↓∩ S
aux1 : q < ↑y✝
aux2 : ↑y✝ < succ q + 1
⊢ ↑y✝ = succ q | 830d905e8e28c141 |
Real.Angle.abs_sin_eq_of_two_nsmul_eq | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | theorem abs_sin_eq_of_two_nsmul_eq {θ ψ : Angle} (h : (2 : ℕ) • θ = (2 : ℕ) • ψ) :
|sin θ| = |sin ψ| | θ ψ : Angle
h : θ = ψ ∨ θ = ψ + ↑π
⊢ |θ.sin| = |ψ.sin| | rcases h with (rfl | rfl) | case inl
θ : Angle
⊢ |θ.sin| = |θ.sin|
case inr
ψ : Angle
⊢ |(ψ + ↑π).sin| = |ψ.sin| | 163937340a56a11c |
hasFDerivAt_of_tendstoLocallyUniformlyOn | Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean | theorem hasFDerivAt_of_tendstoLocallyUniformlyOn [NeBot l] {s : Set E} (hs : IsOpen s)
(hf' : TendstoLocallyUniformlyOn f' g' l s) (hf : ∀ n, ∀ x ∈ s, HasFDerivAt (f n) (f' n x) x)
(hfg : ∀ x ∈ s, Tendsto (fun n => f n x) l (𝓝 (g x))) (hx : x ∈ s) :
HasFDerivAt g (g' x) x | ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : IsRCLikeNormedField 𝕜
inst✝³ : NormedSpace 𝕜 E
G : Type u_4
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
f : ι → E → G
g : E → G
f' : ι → E → E →L[𝕜] G
g' : E → E →L[𝕜] G
x : E
... | refine hasFDerivAt_of_tendstoUniformlyOnFilter ?_ h4 (eventually_of_mem h1 hfg) | ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : IsRCLikeNormedField 𝕜
inst✝³ : NormedSpace 𝕜 E
G : Type u_4
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
f : ι → E → G
g : E → G
f' : ι → E → E →L[𝕜] G
g' : E → E →L[𝕜] G
x : E
... | c808385b50a11572 |
MeasureTheory.integral_countable | Mathlib/MeasureTheory/Integral/Bochner.lean | theorem integral_countable [MeasurableSingletonClass α] (f : α → E) {s : Set α} (hs : s.Countable)
(hf : IntegrableOn f s μ) :
∫ a in s, f a ∂μ = ∑' a : s, (μ {(a : α)}).toReal • f a | α : Type u_1
E : Type u_2
inst✝² : NormedAddCommGroup E
hE : CompleteSpace E
inst✝¹ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
inst✝ : MeasurableSingletonClass α
f : α → E
s : Set α
hs : s.Countable
hf : Integrable (f ∘ Subtype.val) (Measure.comap Subtype.val μ)
hi : Countable { x // x ∈ s }
⊢ Integrable (fu... | apply hf | no goals | b6c93f6d1ed571eb |
Polynomial.derivative_map | Mathlib/Algebra/Polynomial/Derivative.lean | theorem derivative_map [Semiring S] (p : R[X]) (f : R →+* S) :
derivative (p.map f) = p.derivative.map f | R : Type u
S : Type v
inst✝¹ : Semiring R
inst✝ : Semiring S
p : R[X]
f : R →+* S
⊢ derivative (map f p) = map f (derivative p) | let n := max p.natDegree (map f p).natDegree | R : Type u
S : Type v
inst✝¹ : Semiring R
inst✝ : Semiring S
p : R[X]
f : R →+* S
n : ℕ := p.natDegree ⊔ (map f p).natDegree
⊢ derivative (map f p) = map f (derivative p) | a52cfaee492122be |
le_iff_transGen_wcovBy | Mathlib/Order/Interval/Finset/Basic.lean | /-- In a locally finite preorder, `≤` is the transitive closure of `⩿`. -/
lemma le_iff_transGen_wcovBy [Preorder α] [LocallyFiniteOrder α] {x y : α} :
x ≤ y ↔ TransGen (· ⩿ ·) x y | case single
α : Type u_2
inst✝¹ : Preorder α
inst✝ : LocallyFiniteOrder α
x y b✝ : α
h : x ⩿ b✝
⊢ x ≤ b✝ | exact h.le | no goals | dac2077edce8014c |
Set.Countable.exists_pos_hasSum_le | Mathlib/Analysis/SpecificLimits/Basic.lean | theorem Set.Countable.exists_pos_hasSum_le {ι : Type*} {s : Set ι} (hs : s.Countable) {ε : ℝ}
(hε : 0 < ε) : ∃ ε' : ι → ℝ, (∀ i, 0 < ε' i) ∧ ∃ c, HasSum (fun i : s ↦ ε' i) c ∧ c ≤ ε | ι : Type u_4
s : Set ι
hs : s.Countable
ε : ℝ
hε : 0 < ε
⊢ ∃ ε', (∀ (i : ι), 0 < ε' i) ∧ ∃ c, HasSum (fun i => ε' ↑i) c ∧ c ≤ ε | haveI := hs.toEncodable | ι : Type u_4
s : Set ι
hs : s.Countable
ε : ℝ
hε : 0 < ε
this : Encodable ↑s
⊢ ∃ ε', (∀ (i : ι), 0 < ε' i) ∧ ∃ c, HasSum (fun i => ε' ↑i) c ∧ c ≤ ε | 3ba9ae4a7acadf22 |
Subgroup.rank_le_index_mul_rank | Mathlib/GroupTheory/Schreier.lean | theorem rank_le_index_mul_rank [hG : Group.FG G] [FiniteIndex H] :
Group.rank H ≤ H.index * Group.rank G | G : Type u_1
inst✝¹ : Group G
H : Subgroup G
hG : Group.FG G
inst✝ : H.FiniteIndex
this : Group.FG ↥H
⊢ Group.rank ↥H ≤ H.index * Group.rank G | obtain ⟨S, hS₀, hS⟩ := Group.rank_spec G | case intro.intro
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
hG : Group.FG G
inst✝ : H.FiniteIndex
this : Group.FG ↥H
S : Finset G
hS₀ : S.card = Group.rank G
hS : closure ↑S = ⊤
⊢ Group.rank ↥H ≤ H.index * Group.rank G | 901611d39b0c0868 |
Polynomial.content_X_mul | Mathlib/RingTheory/Polynomial/Content.lean | theorem content_X_mul {p : R[X]} : content (X * p) = content p | R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
⊢ (X * p).content = p.content | rw [content, content, Finset.gcd_def, Finset.gcd_def] | R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
⊢ (Multiset.map (X * p).coeff (X * p).support.val).gcd = (Multiset.map p.coeff p.support.val).gcd | ac071a5a6c3a3683 |
SimpleGraph.triple_eq_triple_of_mem | Mathlib/Combinatorics/SimpleGraph/Triangle/Counting.lean | private lemma triple_eq_triple_of_mem (hst : Disjoint s t) (hsu : Disjoint s u) (htu : Disjoint t u)
{x₁ x₂ y₁ y₂ z₁ z₂ : α} (h : ({x₁, y₁, z₁} : Finset α) = {x₂, y₂, z₂})
(hx₁ : x₁ ∈ s) (hx₂ : x₂ ∈ s) (hy₁ : y₁ ∈ t) (hy₂ : y₂ ∈ t) (hz₁ : z₁ ∈ u) (hz₂ : z₂ ∈ u) :
(x₁, y₁, z₁) = (x₂, y₂, z₂) | α : Type u_1
s t u : Finset α
inst✝ : DecidableEq α
hst : Disjoint s t
hsu : Disjoint s u
htu : Disjoint t u
x₁ x₂ y₁ y₂ z₁ z₂ : α
h : {x₁, y₁, z₁} = {x₂, y₂, z₂}
hx₁ : x₁ ∈ s
hx₂ : x₂ ∈ s
hy₁ : y₁ ∈ t
hy₂ : y₂ ∈ t
hz₁ : z₁ ∈ u
hz₂ : z₂ ∈ u
⊢ (x₁, y₁, z₁) = (x₂, y₂, z₂) | simp only [Finset.Subset.antisymm_iff, subset_iff, mem_insert, mem_singleton, forall_eq_or_imp,
forall_eq] at h | α : Type u_1
s t u : Finset α
inst✝ : DecidableEq α
hst : Disjoint s t
hsu : Disjoint s u
htu : Disjoint t u
x₁ x₂ y₁ y₂ z₁ z₂ : α
hx₁ : x₁ ∈ s
hx₂ : x₂ ∈ s
hy₁ : y₁ ∈ t
hy₂ : y₂ ∈ t
hz₁ : z₁ ∈ u
hz₂ : z₂ ∈ u
h :
((x₁ = x₂ ∨ x₁ = y₂ ∨ x₁ = z₂) ∧ (y₁ = x₂ ∨ y₁ = y₂ ∨ y₁ = z₂) ∧ (z₁ = x₂ ∨ z₁ = y₂ ∨ z₁ = z₂)) ∧
(x₂... | 6861db9e331e05e9 |
Std.DHashMap.Const.getD_insertManyIfNewUnit_list | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Lemmas.lean | theorem getD_insertManyIfNewUnit_list
{l : List α} {k : α} {fallback : Unit} :
getD (insertManyIfNewUnit m l) k fallback = () | α : Type u
x✝¹ : BEq α
x✝ : Hashable α
m : DHashMap α fun x => Unit
l : List α
k : α
fallback : Unit
⊢ getD (insertManyIfNewUnit m l) k fallback = () | simp | no goals | 63d6f2153aea646f |
Valued.closure_coe_completion_v_lt | Mathlib/Topology/Algebra/Valued/ValuedField.lean | theorem closure_coe_completion_v_lt {γ : Γ₀ˣ} :
closure ((↑) '' { x : K | v x < (γ : Γ₀) }) =
{ x : hat K | extensionValuation x < (γ : Γ₀) } | case h.refine_1
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
γ : Γ₀ˣ
x : hat K
γ₀ : Γ₀ := extensionValuation x
h : γ₀ ≠ 0
hγ₀ : extension ⁻¹' {γ₀} ∈ 𝓝 x
hx : ∀ t ∈ 𝓝 x, ∃ y, ↑y ∈ t
⊢ γ₀ < ↑γ | obtain ⟨⟨-, y, hy₁ : v y < (γ : Γ₀), rfl⟩, hy₂⟩ := hx _ hγ₀ | case h.refine_1.intro.mk.intro.intro
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
γ : Γ₀ˣ
x : hat K
γ₀ : Γ₀ := extensionValuation x
h : γ₀ ≠ 0
hγ₀ : extension ⁻¹' {γ₀} ∈ 𝓝 x
hx : ∀ t ∈ 𝓝 x, ∃ y, ↑y ∈ t
y : K
hy₁ : v y < ↑γ
hy₂ : ↑⟨↑y, ⋯⟩ ∈ extension ⁻¹' {γ₀}
⊢... | 543e7e76040e703b |
LieModule.coe_lowerCentralSeries_eq_int_aux | Mathlib/Algebra/Lie/Nilpotent.lean | theorem coe_lowerCentralSeries_eq_int_aux (R₁ R₂ L M : Type*)
[CommRing R₁] [CommRing R₂] [AddCommGroup M]
[LieRing L] [LieAlgebra R₁ L] [LieAlgebra R₂ L] [Module R₁ M] [Module R₂ M] [LieRingModule L M]
[LieModule R₁ L M] (k : ℕ) :
let I := lowerCentralSeries R₂ L M k; let S : Set M := {⁅a, b⁆ | (a : L)... | R₁ : Type u_1
R₂ : Type u_2
L : Type u_3
M : Type u_4
inst✝⁹ : CommRing R₁
inst✝⁸ : CommRing R₂
inst✝⁷ : AddCommGroup M
inst✝⁶ : LieRing L
inst✝⁵ : LieAlgebra R₁ L
inst✝⁴ : LieAlgebra R₂ L
inst✝³ : Module R₁ M
inst✝² : Module R₂ M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R₁ L M
k : ℕ
I : LieSubmodule R₂ L M := lowe... | simp only [SetLike.mem_coe] at hx ⊢ | R₁ : Type u_1
R₂ : Type u_2
L : Type u_3
M : Type u_4
inst✝⁹ : CommRing R₁
inst✝⁸ : CommRing R₂
inst✝⁷ : AddCommGroup M
inst✝⁶ : LieRing L
inst✝⁵ : LieAlgebra R₁ L
inst✝⁴ : LieAlgebra R₂ L
inst✝³ : Module R₁ M
inst✝² : Module R₂ M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R₁ L M
k : ℕ
I : LieSubmodule R₂ L M := lowe... | a9c4e1be58181123 |
Algebra.FormallySmooth.exists_lift | Mathlib/RingTheory/Smooth/Basic.lean | theorem exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g | case h₁
R : Type u
inst✝⁴ : CommSemiring R
A : Type u
inst✝³ : Semiring A
inst✝² : Algebra R A
B : Type u
inst✝¹ : CommRing B
inst✝ : FormallySmooth R A
I : Ideal B
hI : IsNilpotent I
⊢ ∀ ⦃S : Type u⦄ [inst : CommRing S] (I : Ideal S),
I ^ 2 = ⊥ → ∀ [_RB : Algebra R S], Function.Surjective (Ideal.Quotient.mkₐ R I).... | intro B _ I hI _ | case h₁
R : Type u
inst✝⁵ : CommSemiring R
A : Type u
inst✝⁴ : Semiring A
inst✝³ : Algebra R A
B✝ : Type u
inst✝² : CommRing B✝
inst✝¹ : FormallySmooth R A
I✝ : Ideal B✝
hI✝ : IsNilpotent I✝
B : Type u
inst✝ : CommRing B
I : Ideal B
hI : I ^ 2 = ⊥
_RB✝ : Algebra R B
⊢ Function.Surjective (Ideal.Quotient.mkₐ R I).comp | 76eb1b2fc562dd62 |
MeasureTheory.Integrable.measure_ge_lt_top | Mathlib/MeasureTheory/Function/L1Space/Integrable.lean | /-- If `f` is `ℝ`-valued and integrable, then for any `c > 0` the set `{x | f x ≥ c}` has finite
measure. -/
lemma Integrable.measure_ge_lt_top {f : α → ℝ} (hf : Integrable f μ) {ε : ℝ} (ε_pos : 0 < ε) :
μ {a : α | ε ≤ f a} < ∞ | α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f μ
ε : ℝ
ε_pos : 0 < ε
⊢ μ {a | ε ≤ f a} < ⊤ | refine lt_of_le_of_lt (measure_mono ?_) (hf.measure_norm_ge_lt_top ε_pos) | α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f μ
ε : ℝ
ε_pos : 0 < ε
⊢ {a | ε ≤ f a} ⊆ {x | ε ≤ ‖f x‖} | b0e27a7abbd12e4e |
List.monotone_filter_right | Mathlib/Data/List/Basic.lean | theorem monotone_filter_right (l : List α) ⦃p q : α → Bool⦄
(h : ∀ a, p a → q a) : l.filter p <+ l.filter q | case nil
α : Type u
p q : α → Bool
h : ∀ (a : α), p a = true → q a = true
⊢ filter p [] <+ filter q [] | rfl | no goals | ec84442b80b4abf0 |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.clear_insert_inductive_case | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddResult.lean | theorem clear_insert_inductive_case {n : Nat} (f : DefaultFormula n) (f_assignments_size : f.assignments.size = n)
(units : Array (Literal (PosFin n))) (units_nodup : ∀ i : Fin units.size, ∀ j : Fin units.size, i ≠ j → units[i] ≠ units[j])
(idx : Fin units.size) (assignments : Array Assignment)
(ih : ClearI... | case right.right.right.right.left
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
units : Array (Literal (PosFin n))
units_nodup : ∀ (i j : Fin units.size), i ≠ j → units[i] ≠ units[j]
idx : Fin units.size
assignments : Array Assignment
hsize : assignments.size = n
hsize' : (clearUnit assignmen... | exact ih3 | no goals | c90be518eb885cb1 |
UV.compress_self | Mathlib/Combinatorics/SetFamily/Compression/UV.lean | theorem compress_self (u a : α) : compress u u a = a | α : Type u_1
inst✝² : GeneralizedBooleanAlgebra α
inst✝¹ : DecidableRel Disjoint
inst✝ : DecidableRel fun x1 x2 => x1 ≤ x2
u a : α
⊢ (if Disjoint u a ∧ u ≤ a then (a ⊔ u) \ u else a) = a | split_ifs with h | case pos
α : Type u_1
inst✝² : GeneralizedBooleanAlgebra α
inst✝¹ : DecidableRel Disjoint
inst✝ : DecidableRel fun x1 x2 => x1 ≤ x2
u a : α
h : Disjoint u a ∧ u ≤ a
⊢ (a ⊔ u) \ u = a
case neg
α : Type u_1
inst✝² : GeneralizedBooleanAlgebra α
inst✝¹ : DecidableRel Disjoint
inst✝ : DecidableRel fun x1 x2 => x1 ≤ x2
u a ... | f5149a55bc843279 |
MultilinearMap.map_piecewise_smul | Mathlib/LinearAlgebra/Multilinear/Basic.lean | theorem map_piecewise_smul [DecidableEq ι] (c : ι → R) (m : ∀ i, M₁ i) (s : Finset ι) :
f (s.piecewise (fun i => c i • m i) m) = (∏ i ∈ s, c i) • f m | R : Type uR
ι : Type uι
M₁ : ι → Type v₁
M₂ : Type v₂
inst✝⁵ : CommSemiring R
inst✝⁴ : (i : ι) → AddCommMonoid (M₁ i)
inst✝³ : AddCommMonoid M₂
inst✝² : (i : ι) → Module R (M₁ i)
inst✝¹ : Module R M₂
f : MultilinearMap R M₁ M₂
inst✝ : DecidableEq ι
c : ι → R
m : (i : ι) → M₁ i
s : Finset ι
⊢ f (s.piecewise (fun i => c ... | refine s.induction_on (by simp) ?_ | R : Type uR
ι : Type uι
M₁ : ι → Type v₁
M₂ : Type v₂
inst✝⁵ : CommSemiring R
inst✝⁴ : (i : ι) → AddCommMonoid (M₁ i)
inst✝³ : AddCommMonoid M₂
inst✝² : (i : ι) → Module R (M₁ i)
inst✝¹ : Module R M₂
f : MultilinearMap R M₁ M₂
inst✝ : DecidableEq ι
c : ι → R
m : (i : ι) → M₁ i
s : Finset ι
⊢ ∀ ⦃a : ι⦄ {s : Finset ι},
... | 964f1fabb64024ea |
MeasureTheory.Measure.hausdorffMeasure_le_one_of_subsingleton | Mathlib/MeasureTheory/Measure/Hausdorff.lean | theorem hausdorffMeasure_le_one_of_subsingleton {s : Set X} (hs : s.Subsingleton) {d : ℝ}
(hd : 0 ≤ d) : μH[d] s ≤ 1 | case inr.intro.inr
X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
s : Set X
hs : s.Subsingleton
d : ℝ
hd : 0 ≤ d
x : X
hx : x ∈ s
dpos : 0 < d
this : NoAtoms μH[d]
⊢ μH[d] {x} ≤ 1 | simp only [zero_le, measure_singleton] | no goals | ecec90d28f4c1dc1 |
affineIndependent_iff_indicator_eq_of_affineCombination_eq | Mathlib/LinearAlgebra/AffineSpace/Independent.lean | theorem affineIndependent_iff_indicator_eq_of_affineCombination_eq (p : ι → P) :
AffineIndependent k p ↔
∀ (s1 s2 : Finset ι) (w1 w2 : ι → k),
∑ i ∈ s1, w1 i = 1 →
∑ i ∈ s2, w2 i = 1 →
s1.affineCombination k p w1 = s2.affineCombination k p w2 →
Set.indicator (↑s1) w... | k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
ι : Type u_4
p : ι → P
ha : AffineIndependent k p
s1 s2 : Finset ι
w1 w2 : ι → k
hw1 : ∑ i ∈ s1, w1 i = 1
hw2 : ∑ i ∈ s2, w2 i = 1
heq : (affineCombination k s1 p) w1 = (affineCombination k s2 p) w... | exact (mt (@Set.mem_union_right _ i ↑s2 ↑s1) hi) h | no goals | 4f9d6693c0403c52 |
Order.krullDim_of_isSimpleOrder | Mathlib/Order/KrullDimension.lean | @[simp] lemma krullDim_of_isSimpleOrder {α : Type*} [PartialOrder α] [BoundedOrder α]
[IsSimpleOrder α] : krullDim α = 1 | α : Type u_1
inst✝² : PartialOrder α
inst✝¹ : BoundedOrder α
inst✝ : IsSimpleOrder α
q : LTSeries α := { length := 1, toFun := fun n => if (n == 0) = true then ⊥ else ⊤, step := ⋯ }
p : LTSeries α
h : 1 < p.length
h2 : 2 < p.length + 1
h0' : 0 < p.length
h1 : 1 < p.length + 1
this✝ : p.toFun ⟨1, h1⟩ = ⊤
this : p.toFun ... | simp_all | no goals | 8e220ae7485cc733 |
Ordinal.le_of_dvd | Mathlib/SetTheory/Ordinal/Arithmetic.lean | theorem le_of_dvd : ∀ {a b : Ordinal}, b ≠ 0 → a ∣ b → a ≤ b
-- Porting note: `⟨b, rfl⟩ => by` → `⟨b, e⟩ => by subst e`
| a, _, b0, ⟨b, e⟩ => by
subst e
-- Porting note: `Ne` is required.
simpa only [mul_one] using
mul_le_mul_left'
(one_le_iff_ne_zero.2 fun h : b = 0 => by
simp o... | a x✝ : Ordinal.{u_4}
b0 : x✝ ≠ 0
b : Ordinal.{u_4}
e : x✝ = a * b
⊢ a ≤ x✝ | subst e | a b : Ordinal.{u_4}
b0 : a * b ≠ 0
⊢ a ≤ a * b | 0fe35c6d81df8019 |
MvPolynomial.IsHomogeneous.totalDegree | Mathlib/RingTheory/MvPolynomial/Homogeneous.lean | theorem totalDegree (hφ : IsHomogeneous φ n) (h : φ ≠ 0) : totalDegree φ = n | case intro
σ : Type u_1
R : Type u_3
inst✝ : CommSemiring R
φ : MvPolynomial σ R
n : ℕ
hφ : φ.IsHomogeneous n
h : φ ≠ 0
d : σ →₀ ℕ
hd : d ∈ φ.support
⊢ (weight 1) d ≤ φ.support.sup fun s => ∑ x ∈ s.support, s x | simp only [weight_apply, Pi.one_apply, smul_eq_mul, mul_one] | case intro
σ : Type u_1
R : Type u_3
inst✝ : CommSemiring R
φ : MvPolynomial σ R
n : ℕ
hφ : φ.IsHomogeneous n
h : φ ≠ 0
d : σ →₀ ℕ
hd : d ∈ φ.support
⊢ (d.sum fun i c => c) ≤ φ.support.sup fun s => ∑ x ∈ s.support, s x | 50bc7e2b28ea88e4 |
CategoryTheory.isTriangulated_of_essSurj_mapComposableArrows_two | Mathlib/CategoryTheory/Triangulated/Functor.lean | /-- If `F : C ⥤ D` is a triangulated functor from a triangulated category, then `D`
is also triangulated if tuples of composables arrows in `D` can be lifted to `C`. -/
lemma isTriangulated_of_essSurj_mapComposableArrows_two
(F : C ⥤ D) [F.CommShift ℤ] [F.IsTriangulated]
[(F.mapComposableArrows 2).EssSurj] [IsT... | case octahedron_axiom.intro.intro
C : Type u_1
D : Type u_2
inst✝¹⁵ : Category.{u_3, u_1} C
inst✝¹⁴ : Category.{u_4, u_2} D
inst✝¹³ : HasShift C ℤ
inst✝¹² : HasShift D ℤ
inst✝¹¹ : HasZeroObject C
inst✝¹⁰ : HasZeroObject D
inst✝⁹ : Preadditive C
inst✝⁸ : Preadditive D
inst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst... | obtain ⟨X₁, X₂, X₃, f, g, rfl⟩ := ComposableArrows.mk₂_surjective α | case octahedron_axiom.intro.intro.intro.intro.intro.intro.intro
C : Type u_1
D : Type u_2
inst✝¹⁵ : Category.{u_3, u_1} C
inst✝¹⁴ : Category.{u_4, u_2} D
inst✝¹³ : HasShift C ℤ
inst✝¹² : HasShift D ℤ
inst✝¹¹ : HasZeroObject C
inst✝¹⁰ : HasZeroObject D
inst✝⁹ : Preadditive C
inst✝⁸ : Preadditive D
inst✝⁷ : ∀ (n : ℤ), (s... | b514da98fd8dcf0f |
CategoryTheory.Functor.isRightKanExtension_iff_postcomp₁ | Mathlib/CategoryTheory/Functor/KanExtension/Basic.lean | lemma isRightKanExtension_iff_postcomp₁ (α : L' ⋙ F' ⟶ F) :
F'.IsRightKanExtension α ↔ (G ⋙ F').IsRightKanExtension
((Functor.associator _ _ _).inv ≫ whiskerRight e.hom F' ≫ α) | case mp
C : Type u_1
H : Type u_3
D : Type u_4
D' : Type u_5
inst✝⁴ : Category.{u_7, u_1} C
inst✝³ : Category.{u_6, u_3} H
inst✝² : Category.{u_9, u_4} D
inst✝¹ : Category.{u_8, u_5} D'
L : C ⥤ D
L' : C ⥤ D'
G : D ⥤ D'
inst✝ : G.IsEquivalence
e : L ⋙ G ≅ L'
F : C ⥤ H
F' : D' ⥤ H
α : L' ⋙ F' ⟶ F
eq : CostructuredArrow.I... | exact fun _ => ⟨⟨eq (isUniversalOfIsRightKanExtension _ _)⟩⟩ | no goals | 1f78713265ce7333 |
CategoryTheory.Functor.final_of_final_costructuredArrowToOver | Mathlib/CategoryTheory/Comma/StructuredArrow/Final.lean | theorem final_of_final_costructuredArrowToOver (L : A ⥤ T) (R : B ⥤ T) [Final R]
[hB : ∀ b : B, Final (CostructuredArrow.toOver L (R.obj b))] : Final L | A : Type u₁
inst✝³ : Category.{v₁, u₁} A
B : Type u₂
inst✝² : Category.{v₂, u₂} B
T : Type u₃
inst✝¹ : Category.{v₃, u₃} T
L : A ⥤ T
R : B ⥤ T
inst✝ : R.Final
hB : ∀ (b : B), (pre L (𝟭 T) (R.obj b)).Final
sA : A ≌ AsSmall A := AsSmall.equiv
sB : B ≌ AsSmall B := AsSmall.equiv
sT : T ≌ AsSmall T := AsSmall.equiv
L' : A... | simp [F'', hsT] | no goals | a234934b59747f70 |
HasCompactSupport.enorm_le_lintegral_Ici_deriv | Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | lemma _root_.HasCompactSupport.enorm_le_lintegral_Ici_deriv
{F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
{f : ℝ → F} (hf : ContDiff ℝ 1 f) (h'f : HasCompactSupport f) (x : ℝ) :
‖f x‖ₑ ≤ ∫⁻ y in Iic x, ‖deriv f y‖ₑ | case h.e'_4.h.e'_4.h
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
hf : ContDiff ℝ 1 f
h'f : HasCompactSupport f
x : ℝ
I : F →L[ℝ] Completion F := Completion.toComplL
f' : ℝ → Completion F := ⇑I ∘ f
hf' : ContDiff ℝ 1 f'
h'f' : HasCompactSupport f'
this : ‖f' x‖ₑ ≤ ∫⁻ (y : ℝ) in Iic x, ‖d... | simp only [ContinuousLinearMap.fderiv] | case h.e'_4.h.e'_4.h
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
hf : ContDiff ℝ 1 f
h'f : HasCompactSupport f
x : ℝ
I : F →L[ℝ] Completion F := Completion.toComplL
f' : ℝ → Completion F := ⇑I ∘ f
hf' : ContDiff ℝ 1 f'
h'f' : HasCompactSupport f'
this : ‖f' x‖ₑ ≤ ∫⁻ (y : ℝ) in Iic x, ‖d... | 43d9795b86d3ea3e |
Bimod.whiskerRight_comp_bimod | Mathlib/CategoryTheory/Monoidal/Bimod.lean | theorem whiskerRight_comp_bimod {W X Y Z : Mon_ C} {M M' : Bimod W X} (f : M ⟶ M') (N : Bimod X Y)
(P : Bimod Y Z) :
whiskerRight f (N.tensorBimod P) =
(associatorBimod M N P).inv ≫
whiskerRight (whiskerRight f N) P ≫ (associatorBimod M' N P).hom | case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
W X Y Z : Mon_ C
M M' : Bimod W X
f : M ⟶ M'
N : Bi... | slice_rhs 2 3 => rw [← comp_whiskerRight, ι_colimMap, parallelPairHom_app_one] | case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
W X Y Z : Mon_ C
M M' : Bimod W X
f : M ⟶ M'
N : Bi... | 2725ee4aa075b079 |
HasFTaylorSeriesUpToOn.shift_of_succ | Mathlib/Analysis/Calculus/ContDiff/FTaylorSeries.lean | theorem HasFTaylorSeriesUpToOn.shift_of_succ
{n : ℕ} (H : HasFTaylorSeriesUpToOn (n + 1 : ℕ) f p s) :
(HasFTaylorSeriesUpToOn n (fun x => continuousMultilinearCurryFin1 𝕜 E F (p x 1))
(fun x => (p x).shift)) s | 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
H : HasFTaylorSeriesUpToOn (↑(n + 1)) f p s
m : ℕ
hm : m < n
x : E
hx : ... | exact Nat.succ_lt_succ hm | no goals | cf26e196073ba975 |
enorm_indicator_le_enorm_self | Mathlib/Analysis/NormedSpace/IndicatorFunction.lean | theorem enorm_indicator_le_enorm_self : ‖indicator s f a‖ₑ ≤ ‖f a‖ₑ | α : Type u_1
E : Type u_2
inst✝ : SeminormedAddCommGroup E
s : Set α
f : α → E
a : α
⊢ s.indicator (fun a => ‖f a‖ₑ) a ≤ ‖f a‖ₑ | apply indicator_enorm_le_enorm_self | no goals | 91e8d779faf01ab9 |
IsLinearMap.isLinearMap_sub | Mathlib/Algebra/Module/LinearMap/Prod.lean | theorem isLinearMap_sub [AddCommGroup M] [Module R M] :
IsLinearMap R fun x : M × M => x.1 - x.2 | case map_smul
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
x : R
y : M × M
⊢ (x • y).1 - (x • y).2 = x • (y.1 - y.2) | simp [smul_sub] | no goals | 6c17256583bdbbb4 |
Matrix.aeval_self_charpoly | Mathlib/LinearAlgebra/Matrix/Charpoly/Basic.lean | theorem aeval_self_charpoly (M : Matrix n n R) : aeval M M.charpoly = 0 | R : Type u_1
inst✝² : CommRing R
n : Type u_4
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
h : M.charpoly • 1 = M.charmatrix.adjugate * M.charmatrix
⊢ (aeval M) M.charpoly = 0 | apply_fun matPolyEquiv at h | R : Type u_1
inst✝² : CommRing R
n : Type u_4
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
h : matPolyEquiv (M.charpoly • 1) = matPolyEquiv (M.charmatrix.adjugate * M.charmatrix)
⊢ (aeval M) M.charpoly = 0 | 650c4547af86a3b8 |
CategoryTheory.shift_shiftFunctorCompIsoId_hom_app | Mathlib/CategoryTheory/Shift/Basic.lean | theorem shift_shiftFunctorCompIsoId_hom_app (n m : A) (h : n + m = 0) (X : C) :
((shiftFunctorCompIsoId C n m h).hom.app X)⟦n⟧' =
(shiftFunctorCompIsoId C m n
(by rw [← neg_eq_of_add_eq_zero_left h, add_neg_cancel])).hom.app (X⟦n⟧) | C : Type u
A : Type u_1
inst✝² : Category.{v, u} C
inst✝¹ : AddGroup A
inst✝ : HasShift C A
n m : A
h : n + m = 0
X : C
⊢ (shiftFunctor C n).map ((shiftFunctorCompIsoId C n m h).hom.app X) =
(shiftFunctorCompIsoId C m n ⋯).hom.app ((shiftFunctor C n).obj X) | dsimp [shiftFunctorCompIsoId] | C : Type u
A : Type u_1
inst✝² : Category.{v, u} C
inst✝¹ : AddGroup A
inst✝ : HasShift C A
n m : A
h : n + m = 0
X : C
⊢ (shiftFunctor C n).map ((shiftFunctorAdd' C n m 0 h).inv.app X ≫ (shiftFunctorZero C A).hom.app X) =
(shiftFunctorAdd' C m n 0 ⋯).inv.app ((shiftFunctor C n).obj X) ≫
(shiftFunctorZero C A... | c1724fd6871d1253 |
VectorField.mlieBracketWithin_add_right | Mathlib/Geometry/Manifold/VectorField.lean | lemma mlieBracketWithin_add_right
(hW : MDifferentiableWithinAt I I.tangent (fun x ↦ (W x : TangentBundle I M)) s x)
(hW₁ : MDifferentiableWithinAt I I.tangent (fun x ↦ (W₁ x : TangentBundle I M)) s x)
(hs : UniqueMDiffWithinAt I s x) :
mlieBracketWithin I V (W + W₁) s x =
mlieBracketWithin I V W ... | 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁶ : TopologicalSpace H
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
s : Set M
x : M
V W W₁ : (x : M) → TangentSpace I x
inst✝¹ : IsMa... | rw [mlieBracketWithin_swap, Pi.neg_apply, mlieBracketWithin_add_left hW hW₁ hs,
mlieBracketWithin_swap (V := V), mlieBracketWithin_swap (V := V), Pi.neg_apply, Pi.neg_apply] | 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁶ : TopologicalSpace H
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
s : Set M
x : M
V W W₁ : (x : M) → TangentSpace I x
inst✝¹ : IsMa... | 5a0aa9619be433c2 |
Submodule.LinearDisjoint.not_linearIndependent_pair_of_commute_of_flat_left | Mathlib/LinearAlgebra/LinearDisjoint.lean | theorem not_linearIndependent_pair_of_commute_of_flat_left [Module.Flat R M]
(a b : ↥(M ⊓ N)) (hc : Commute a.1 b.1) : ¬LinearIndependent R ![a, b] := fun h ↦ by
let n : Fin 2 → N := (inclusion inf_le_right) ∘ ![a, b]
have hn : LinearIndependent R n := h.map' _ (ker_inclusion _ _ _)
-- need this instance othe... | R : Type u
S : Type v
inst✝⁴ : CommRing R
inst✝³ : Ring S
inst✝² : Algebra R S
M N : Submodule R S
H : M.LinearDisjoint N
inst✝¹ : Nontrivial R
inst✝ : Module.Flat R ↥M
a b : ↥(M ⊓ N)
hc : Commute ↑a ↑b
h : LinearIndependent R ![a, b]
n : Fin 2 → ↥N := ⇑(inclusion ⋯) ∘ ![a, b]
hn : LinearIndependent R n
this : AddCommG... | rw [AddSubmonoid.mk_eq_zero, ZeroMemClass.coe_eq_zero] at hm | R : Type u
S : Type v
inst✝⁴ : CommRing R
inst✝³ : Ring S
inst✝² : Algebra R S
M N : Submodule R S
H : M.LinearDisjoint N
inst✝¹ : Nontrivial R
inst✝ : Module.Flat R ↥M
a b : ↥(M ⊓ N)
hc : Commute ↑a ↑b
h : LinearIndependent R ![a, b]
n : Fin 2 → ↥N := ⇑(inclusion ⋯) ∘ ![a, b]
hn : LinearIndependent R n
this : AddCommG... | 518b4123cdd75b80 |
Batteries.RBNode.Path.Ordered.del | Mathlib/.lake/packages/batteries/Batteries/Data/RBMap/Alter.lean | theorem Ordered.del : ∀ {path : Path α} {t : RBNode α} {c},
t.Ordered cmp → path.Ordered cmp → t.All (path.RootOrdered cmp) → (path.del t c).Ordered cmp
| .root, _, _, ht, _, _ => Ordered.setBlack.2 ht
| .left _ parent x b, a, red, ha, ⟨hp, xb, xp, bp, hb⟩, H => by
unfold Path.del
have ⟨ax, ap⟩ := All_a... | α : Type u_1
cmp : α → α → Ordering
c✝ : RBColor
parent : Path α
x : α
b a : RBNode α
ha : RBNode.Ordered cmp a
hp : Ordered cmp parent
xb : All (fun x_1 => cmpLT cmp x x_1) b
xp : RootOrdered cmp parent x
bp : All (RootOrdered cmp parent) b
hb : RBNode.Ordered cmp b
H : All (RootOrdered cmp (left c✝ parent x b)) a
⊢ R... | have ⟨ax, ap⟩ := All_and.1 H | α : Type u_1
cmp : α → α → Ordering
c✝ : RBColor
parent : Path α
x : α
b a : RBNode α
ha : RBNode.Ordered cmp a
hp : Ordered cmp parent
xb : All (fun x_1 => cmpLT cmp x x_1) b
xp : RootOrdered cmp parent x
bp : All (RootOrdered cmp parent) b
hb : RBNode.Ordered cmp b
H : All (RootOrdered cmp (left c✝ parent x b)) a
ax ... | 75992fc423b90d31 |
Concept.ext | Mathlib/Order/Concept.lean | theorem ext (h : c.fst = d.fst) : c = d | case mk.mk.mk.mk
α : Type u_2
β : Type u_3
r : α → β → Prop
s₁ : Set α
t₁ : Set β
h₁ : intentClosure r (s₁, t₁).1 = (s₁, t₁).2
closure_snd✝¹ : extentClosure r (s₁, t₁).2 = (s₁, t₁).1
s₂ : Set α
t₂ : Set β
h₂ : intentClosure r (s₂, t₂).1 = (s₂, t₂).2
closure_snd✝ : extentClosure r (s₂, t₂).2 = (s₂, t₂).1
h :
{ toProd ... | dsimp at h₁ h₂ h | case mk.mk.mk.mk
α : Type u_2
β : Type u_3
r : α → β → Prop
s₁ : Set α
t₁ : Set β
h₁ : intentClosure r s₁ = t₁
closure_snd✝¹ : extentClosure r (s₁, t₁).2 = (s₁, t₁).1
s₂ : Set α
t₂ : Set β
h₂ : intentClosure r s₂ = t₂
closure_snd✝ : extentClosure r (s₂, t₂).2 = (s₂, t₂).1
h : s₁ = s₂
⊢ { toProd := (s₁, t₁), closure_fst... | dd6b7dc0117169a9 |
ArithmeticFunction.IsMultiplicative.mul | Mathlib/NumberTheory/ArithmeticFunction.lean | theorem mul [CommSemiring R] {f g : ArithmeticFunction R} (hf : f.IsMultiplicative)
(hg : g.IsMultiplicative) : IsMultiplicative (f * g) | R : Type u_1
inst✝ : CommSemiring R
f g : ArithmeticFunction R
hf : f.IsMultiplicative
hg : g.IsMultiplicative
a1 a2 b1 b2 : ℕ
ha : ¬((a1, a2), b1, b2).1.1 * ((a1, a2), b1, b2).1.2 = 0
hb : ¬((a1, a2), b1, b2).2.1 * ((a1, a2), b1, b2).2.2 = 0
c1 c2 d1 d2 : ℕ
cop : (((c1, c2), d1, d2).1.1 * ((c1, c2), d1, d2).1.2).Copri... | rw [← hcd.2.1, h.1, mul_comm c1 d1, Nat.gcd_mul_left,
cop.coprime_mul_right.coprime_mul_left_right.symm.gcd_eq_one, mul_one] | no goals | 70b3aa260764f2d8 |
CategoryTheory.Simple.of_iso | Mathlib/CategoryTheory/Simple.lean | theorem Simple.of_iso {X Y : C} [Simple Y] (i : X ≅ Y) : Simple X :=
{ mono_isIso_iff_nonzero := fun f m => by
constructor
· intro h w
have j : IsIso (f ≫ i.hom) | case mp
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
X Y : C
inst✝ : Simple Y
i : X ≅ Y
Y✝ : C
f : Y✝ ⟶ X
m : Mono f
h : IsIso f
w : f = 0
j : IsIso (f ≫ i.hom)
⊢ False | rw [Simple.mono_isIso_iff_nonzero] at j | case mp
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
X Y : C
inst✝ : Simple Y
i : X ≅ Y
Y✝ : C
f : Y✝ ⟶ X
m : Mono f
h : IsIso f
w : f = 0
j : f ≫ i.hom ≠ 0
⊢ False | d9ef7ae0da1428b6 |
Polynomial.quotient_mk_comp_C_isIntegral_of_jacobson' | Mathlib/RingTheory/Jacobson/Ring.lean | theorem quotient_mk_comp_C_isIntegral_of_jacobson' [Nontrivial R] (hR : IsJacobsonRing R)
(hP' : ∀ x : R, C x ∈ P → x = 0) :
((Ideal.Quotient.mk P).comp C : R →+* R[X] ⧸ P).IsIntegral | case refine_2
R : Type u_1
inst✝¹ : CommRing R
P : Ideal R[X]
hP : P.IsMaximal
inst✝ : Nontrivial R
hR : IsJacobsonRing R
hP' : ∀ (x : R), C x ∈ P → x = 0
P' : Ideal R := comap C P
pX : R[X]
hpX : pX ∈ P
hp0 : map (Ideal.Quotient.mk (comap C P)) pX ≠ 0
a : R ⧸ P' := (map (Ideal.Quotient.mk P') pX).leadingCoeff
M : Subm... | have isloc : IsLocalization M' (Localization M') := by infer_instance | case refine_2
R : Type u_1
inst✝¹ : CommRing R
P : Ideal R[X]
hP : P.IsMaximal
inst✝ : Nontrivial R
hR : IsJacobsonRing R
hP' : ∀ (x : R), C x ∈ P → x = 0
P' : Ideal R := comap C P
pX : R[X]
hpX : pX ∈ P
hp0 : map (Ideal.Quotient.mk (comap C P)) pX ≠ 0
a : R ⧸ P' := (map (Ideal.Quotient.mk P') pX).leadingCoeff
M : Subm... | fe75ad9300033bac |
Matrix.reverse_charpoly | Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean | lemma reverse_charpoly (M : Matrix n n R) :
M.charpoly.reverse = M.charpolyRev | R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
a✝ : Nontrivial R
t : R[T;T⁻¹] := T 1
t_inv : R[T;T⁻¹] := T (-1)
p : R[T;T⁻¹] := ((scalar n) t - M.map ⇑LaurentPolynomial.C).det
q : R[T;T⁻¹] := (1 - (scalar n) t * M.map ⇑LaurentPolynomial.C).det
⊢ t_inv * t = 1 | rw [← T_add, neg_add_cancel, T_zero] | no goals | da883ccbbd8b47b4 |
Submodule.exists_sub_one_mem_and_smul_le_of_fg_of_le_sup | Mathlib/RingTheory/Nakayama.lean | @[stacks 00DV "(3) see `Submodule.localized₀_le_localized₀_of_smul_le` for the second conclusion."]
lemma exists_sub_one_mem_and_smul_le_of_fg_of_le_sup {I : Ideal R}
{N N' P : Submodule R M} (hN' : N'.FG) (hN'le : N' ≤ P) (hNN' : P ≤ N ⊔ I • N') :
∃ r : R, r - 1 ∈ I ∧ r • P ≤ N | case intro.intro.smul₁
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
N N' P : Submodule R M
hN' : N'.FG
hN'le : N' ≤ P
hNN' : P ≤ N ⊔ I • N'
hNN'' : P ≤ N ⊔ N'
h1 : map N.mkQ P = map N.mkQ N'
h2 : map N.mkQ P = map N.mkQ (I • N')
hle : map N.mkQ P ≤ I • map N.mkQ P... | exact N.smul_mem _ h | no goals | 991a203cf8881570 |
MvRatFunc.rank_eq_max_lift | Mathlib/FieldTheory/MvRatFunc/Rank.lean | theorem MvRatFunc.rank_eq_max_lift
{σ : Type u} {F : Type v} [Field F] [Nonempty σ] :
Module.rank F (FractionRing (MvPolynomial σ F)) = lift.{u} #F ⊔ lift.{v} #σ ⊔ ℵ₀ | case refine_2.intro
σ : Type u
F : Type v
inst✝¹ : Field F
inst✝ : Nonempty σ
R : Type (max u v) := MvPolynomial σ F
K : Type (max u v) := FractionRing R
hinj : Function.Injective ⇑(algebraMap R K)
h1 : lift.{v, u} #σ ≤ Module.rank F K ∧ ℵ₀ ≤ Module.rank F K
i : σ
hx : Transcendental F ((algebraMap R K) (MvPolynomial.X... | have h2 := hx.linearIndependent_sub_inv.cardinal_lift_le_rank | case refine_2.intro
σ : Type u
F : Type v
inst✝¹ : Field F
inst✝ : Nonempty σ
R : Type (max u v) := MvPolynomial σ F
K : Type (max u v) := FractionRing R
hinj : Function.Injective ⇑(algebraMap R K)
h1 : lift.{v, u} #σ ≤ Module.rank F K ∧ ℵ₀ ≤ Module.rank F K
i : σ
hx : Transcendental F ((algebraMap R K) (MvPolynomial.X... | 4b8f9efec9d5cd0c |
Fin.cons_injective_iff | Mathlib/Data/Fin/Tuple/Basic.lean | theorem cons_injective_iff {α} {x₀ : α} {x : Fin n → α} :
Function.Injective (cons x₀ x : Fin n.succ → α) ↔ x₀ ∉ Set.range x ∧ Function.Injective x | case refine_1.intro
n : ℕ
α : Type u_1
x₀ : α
x : Fin n → α
h : Injective (cons x₀ x)
i : Fin n
hi : x i = x₀
⊢ False | replace h := @h i.succ 0 | case refine_1.intro
n : ℕ
α : Type u_1
x₀ : α
x : Fin n → α
i : Fin n
hi : x i = x₀
h : cons x₀ x i.succ = cons x₀ x 0 → i.succ = 0
⊢ False | 772a7aa54dcd93ab |
EuclideanGeometry.dist_center_eq_dist_center_of_mem_sphere' | Mathlib/Geometry/Euclidean/Sphere/Basic.lean | theorem dist_center_eq_dist_center_of_mem_sphere' {p₁ p₂ : P} {s : Sphere P} (hp₁ : p₁ ∈ s)
(hp₂ : p₂ ∈ s) : dist s.center p₁ = dist s.center p₂ | P : Type u_2
inst✝ : MetricSpace P
p₁ p₂ : P
s : Sphere P
hp₁ : p₁ ∈ s
hp₂ : p₂ ∈ s
⊢ dist s.center p₁ = dist s.center p₂ | rw [mem_sphere'.1 hp₁, mem_sphere'.1 hp₂] | no goals | 0c893550974970b9 |
MeasureTheory.tendsto_Lp_finite_of_tendstoInMeasure | Mathlib/MeasureTheory/Function/UniformIntegrable.lean | theorem tendsto_Lp_finite_of_tendstoInMeasure [IsFiniteMeasure μ] (hp : 1 ≤ p) (hp' : p ≠ ∞)
(hf : ∀ n, AEStronglyMeasurable (f n) μ) (hg : MemLp g p μ) (hui : UnifIntegrable f p μ)
(hfg : TendstoInMeasure μ f atTop g) : Tendsto (fun n ↦ eLpNorm (f n - g) p μ) atTop (𝓝 0) | case intro.intro
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ℕ → α → β
g : α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hg : MemLp g p μ
hui : UnifIntegrable f p μ
hfg : TendstoInMeasure μ f atTop g
ns : ... | exact ⟨ms,
tendsto_Lp_finite_of_tendsto_ae hp hp' (fun _ => hf _) hg (fun ε hε =>
let ⟨δ, hδ, hδ'⟩ := hui hε
⟨δ, hδ, fun i s hs hμs => hδ' _ s hs hμs⟩)
hms'⟩ | no goals | 6a9a7bb3da79aee9 |
Order.height_top_eq_krullDim | Mathlib/Order/KrullDimension.lean | @[simp] -- not as useful as a simp lemma as it looks, due to the coe on the left
lemma height_top_eq_krullDim [OrderTop α] : height (⊤ : α) = krullDim α | α : Type u_1
inst✝¹ : Preorder α
inst✝ : OrderTop α
⊢ height ⊤ = ⨆ p, ↑p.length | apply le_antisymm | case a
α : Type u_1
inst✝¹ : Preorder α
inst✝ : OrderTop α
⊢ height ⊤ ≤ ⨆ p, ↑p.length
case a
α : Type u_1
inst✝¹ : Preorder α
inst✝ : OrderTop α
⊢ ⨆ p, ↑p.length ≤ height ⊤ | 12195e7d9158d145 |
SchwartzMap.one_add_le_sup_seminorm_apply | Mathlib/Analysis/Distribution/SchwartzSpace.lean | theorem one_add_le_sup_seminorm_apply {m : ℕ × ℕ} {k n : ℕ} (hk : k ≤ m.1) (hn : n ≤ m.2)
(f : 𝓢(E, F)) (x : E) :
(1 + ‖x‖) ^ k * ‖iteratedFDeriv ℝ n f x‖ ≤
2 ^ m.1 * (Finset.Iic m).sup (fun m => SchwartzMap.seminorm 𝕜 m.1 m.2) f | 𝕜 : Type u_1
E : Type u_4
F : Type u_5
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 F
inst✝ : SMulCommClass ℝ 𝕜 F
m : ℕ × ℕ
k n : ℕ
hk : k ≤ m.1
hn : n ≤ m.2
f : 𝓢(E, F)
x : E
⊢ (∑ m ∈ Finset.range (k + 1... | simp only [one_pow, mul_one, Finset.sum_congr, Finset.sum_mul] | 𝕜 : Type u_1
E : Type u_4
F : Type u_5
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 F
inst✝ : SMulCommClass ℝ 𝕜 F
m : ℕ × ℕ
k n : ℕ
hk : k ≤ m.1
hn : n ≤ m.2
f : 𝓢(E, F)
x : E
⊢ ∑ i ∈ Finset.range (k + 1)... | 61f60cd7052782c3 |
Ideal.KerLift.map_smul | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | theorem KerLift.map_smul (f : A →ₐ[R₁] B) (r : R₁) (x : A ⧸ (RingHom.ker f)) :
f.kerLift (r • x) = r • f.kerLift x | case intro
R₁ : Type u_1
A : Type u_3
B : Type u_4
inst✝⁴ : CommSemiring R₁
inst✝³ : Ring A
inst✝² : Algebra R₁ A
inst✝¹ : Semiring B
inst✝ : Algebra R₁ B
f : A →ₐ[R₁] B
r : R₁
a : A
⊢ f.kerLift (r • (Quotient.mkₐ R₁ (ker f)) a) = r • f.kerLift ((Quotient.mkₐ R₁ (ker f)) a) | exact _root_.map_smul f _ _ | no goals | b09374e175507dbb |
Submodule.sup_orthogonal_of_completeSpace | Mathlib/Analysis/InnerProductSpace/Projection.lean | theorem Submodule.sup_orthogonal_of_completeSpace [HasOrthogonalProjection K] : K ⊔ Kᗮ = ⊤ | 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
inst✝ : HasOrthogonalProjection K
⊢ K ⊔ Kᗮ = ⊤ | convert Submodule.sup_orthogonal_inf_of_completeSpace (le_top : K ≤ ⊤) using 2 | case h.e'_2.h.e'_4
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
inst✝ : HasOrthogonalProjection K
⊢ Kᗮ = Kᗮ ⊓ ⊤ | 597e065ae0604bd3 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.