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 |
|---|---|---|---|---|---|---|
intervalIntegral.integral_pos_iff_support_of_nonneg_ae' | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | theorem integral_pos_iff_support_of_nonneg_ae' (hf : 0 ≤ᵐ[μ.restrict (Ι a b)] f)
(hfi : IntervalIntegrable f μ a b) :
(0 < ∫ x in a..b, f x ∂μ) ↔ a < b ∧ 0 < μ (support f ∩ Ioc a b) | case inr
f : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
hf : 0 ≤ᶠ[ae (μ.restrict (Ι a b))] f
hfi : IntervalIntegrable f μ a b
hba : b ≤ a
⊢ ∫ (x : ℝ) in a..b, f x ∂μ ≤ 0 | rw [integral_of_ge hba, neg_nonpos] | case inr
f : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
hf : 0 ≤ᶠ[ae (μ.restrict (Ι a b))] f
hfi : IntervalIntegrable f μ a b
hba : b ≤ a
⊢ 0 ≤ ∫ (x : ℝ) in Ioc b a, f x ∂μ | 98c9d1e3c30988ab |
Int.bmod_mul_bmod | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean | theorem bmod_mul_bmod : Int.bmod (Int.bmod x n * y) n = Int.bmod (x * y) n | case isFalse
x : Int
n : Nat
y : Int
h✝ : ¬x % ↑n < (↑n + 1) / 2
⊢ ((x % ↑n - ↑n) * y).bmod n = (x * y).bmod n | next p =>
rw [Int.sub_mul, Int.sub_eq_add_neg, ← Int.mul_neg, bmod_add_mul_cancel, emod_mul_bmod_congr] | no goals | c390cd06841d2283 |
CategoryTheory.isPullback_initial_to_of_cofan_isVanKampen | Mathlib/CategoryTheory/Limits/VanKampen.lean | theorem isPullback_initial_to_of_cofan_isVanKampen [HasInitial C] {ι : Type*} {F : Discrete ι ⥤ C}
{c : Cocone F} (hc : IsVanKampenColimit c) (i j : Discrete ι) (hi : i ≠ j) :
IsPullback (initial.to _) (initial.to _) (c.ι.app i) (c.ι.app j) | C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasInitial C
ι : Type u_3
F : Discrete ι ⥤ C
c : Cocone F
hc : IsVanKampenColimit c
i j : Discrete ι
hi : i ≠ j
⊢ IsPullback (initial.to (F.obj i)) (initial.to (F.obj j)) (c.ι.app i) (c.ι.app j) | classical
let f : ι → C := F.obj ∘ Discrete.mk
have : F = Discrete.functor f :=
Functor.hext (fun i ↦ rfl) (by rintro ⟨i⟩ ⟨j⟩ ⟨⟨rfl : i = j⟩⟩; simp [f])
clear_value f
subst this
have : ∀ i, Subsingleton (⊥_ C ⟶ (Discrete.functor f).obj i) := inferInstance
convert isPullback_of_cofan_isVanKampen hc i.as j.as
exact (if... | no goals | d267d3f87162638e |
PartialHomeomorph.extend_target_mem_nhdsWithin | Mathlib/Geometry/Manifold/IsManifold/ExtChartAt.lean | theorem extend_target_mem_nhdsWithin {y : M} (hy : y ∈ f.source) :
(f.extend I).target ∈ 𝓝[range I] f.extend I y | 𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace M
f : PartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
y : M
hy : y ∈ f.source
⊢ ↑(f.extend I) '' (f.extend I).source ∈ m... | exact image_mem_map (extend_source_mem_nhds _ hy) | no goals | 9f61ee0dd9db10cc |
Function.piCongrLeft'_update | Mathlib/Logic/Equiv/Basic.lean | theorem piCongrLeft'_update [DecidableEq α] [DecidableEq β] (P : α → Sort*) (e : α ≃ β)
(f : ∀ a, P a) (b : β) (x : P (e.symm b)) :
e.piCongrLeft' P (update f (e.symm b) x) = update (e.piCongrLeft' P f) b x | case h.inr.h
α : Sort u_1
β : Sort u_4
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
P : α → Sort u_10
e : α ≃ β
f : (a : α) → P a
b : β
x : P (e.symm b)
b' : β
h : b' ≠ b
h' : e.symm b' = e.symm b
⊢ False | cases e.symm.injective h' |> h | no goals | 7974ca7dfc401d98 |
Metric.Sigma.completeSpace | Mathlib/Topology/MetricSpace/Gluing.lean | theorem completeSpace [∀ i, CompleteSpace (E i)] : CompleteSpace (Σi, E i) | ι : Type u_1
E : ι → Type u_2
inst✝¹ : (i : ι) → MetricSpace (E i)
inst✝ : ∀ (i : ι), CompleteSpace (E i)
s : ι → Set ((i : ι) × E i) := fun i => Sigma.fst ⁻¹' {i}
U : Set (((k : ι) × E k) × (k : ι) × E k) := {p | dist p.1 p.2 < 1}
hc : ∀ (i : ι), IsComplete (s i)
⊢ CompleteSpace ((i : ι) × E i) | have hd : ∀ (i j), ∀ x ∈ s i, ∀ y ∈ s j, (x, y) ∈ U → i = j := fun i j x hx y hy hxy =>
(Eq.symm hx).trans ((fst_eq_of_dist_lt_one _ _ hxy).trans hy) | ι : Type u_1
E : ι → Type u_2
inst✝¹ : (i : ι) → MetricSpace (E i)
inst✝ : ∀ (i : ι), CompleteSpace (E i)
s : ι → Set ((i : ι) × E i) := fun i => Sigma.fst ⁻¹' {i}
U : Set (((k : ι) × E k) × (k : ι) × E k) := {p | dist p.1 p.2 < 1}
hc : ∀ (i : ι), IsComplete (s i)
hd : ∀ (i j : ι), ∀ x ∈ s i, ∀ y ∈ s j, (x, y) ∈ U → i ... | c3518d39e29749a3 |
IsPrimitiveRoot.zmodEquivZPowers_apply_coe_int | Mathlib/RingTheory/RootsOfUnity/PrimitiveRoots.lean | theorem zmodEquivZPowers_apply_coe_int (i : ℤ) :
h.zmodEquivZPowers i = Additive.ofMul (⟨ζ ^ i, i, rfl⟩ : Subgroup.zpowers ζ) | R : Type u_4
k : ℕ
inst✝ : CommRing R
ζ : Rˣ
h : IsPrimitiveRoot ζ k
i : ℤ
⊢ (((Int.castAddHom (ZMod k)).liftOfRightInverse ZMod.cast ⋯)
⟨{ toFun := fun i => Additive.ofMul ⟨(fun x => ζ ^ x) i, ⋯⟩, map_zero' := ⋯, map_add' := ⋯ }, ⋯⟩)
↑i =
Additive.ofMul ⟨ζ ^ i, ⋯⟩ | exact AddMonoidHom.liftOfRightInverse_comp_apply _ _ ZMod.intCast_rightInverse _ _ | no goals | d9a59ad1ebca66bb |
norm_commutator_units_sub_one_le | Mathlib/Analysis/Normed/Field/Basic.lean | lemma norm_commutator_units_sub_one_le (a b : αˣ) :
‖(a * b * a⁻¹ * b⁻¹).val - 1‖ ≤ 2 * ‖a⁻¹.val‖ * ‖b⁻¹.val‖ * ‖a.val - 1‖ * ‖b.val - 1‖ :=
calc
‖(a * b * a⁻¹ * b⁻¹).val - 1‖ = ‖(a * b - b * a) * a⁻¹.val * b⁻¹.val‖ | α : Type u_2
inst✝ : SeminormedRing α
a b : αˣ
⊢ ‖(↑a - 1) * (↑b - 1) - (↑b - 1) * (↑a - 1)‖ * ‖↑a⁻¹‖ * ‖↑b⁻¹‖ ≤
(‖(↑a - 1) * (↑b - 1)‖ + ‖(↑b - 1) * (↑a - 1)‖) * ‖↑a⁻¹‖ * ‖↑b⁻¹‖ | gcongr | case h.h
α : Type u_2
inst✝ : SeminormedRing α
a b : αˣ
⊢ ‖(↑a - 1) * (↑b - 1) - (↑b - 1) * (↑a - 1)‖ ≤ ‖(↑a - 1) * (↑b - 1)‖ + ‖(↑b - 1) * (↑a - 1)‖ | 23e2a090ea0ebe1c |
PFunctor.M.mk_dest | Mathlib/Data/PFunctor/Univariate/M.lean | theorem mk_dest (x : M F) : M.mk (dest x) = x | case H.zero
F : PFunctor.{u}
x : F.M
⊢ Approx.sMk x.dest 0 = x.approx 0 | apply @Subsingleton.elim _ CofixA.instSubsingleton | no goals | 475d78b79340343f |
Function.monotoneOn_of_rightInvOn_of_mapsTo | Mathlib/Data/Set/Monotone.lean | theorem monotoneOn_of_rightInvOn_of_mapsTo {α β : Type*} [PartialOrder α] [LinearOrder β]
{φ : β → α} {ψ : α → β} {t : Set β} {s : Set α} (hφ : MonotoneOn φ t)
(φψs : Set.RightInvOn ψ φ s) (ψts : Set.MapsTo ψ s t) : MonotoneOn ψ s | case inl
α : Type u_4
β : Type u_5
inst✝¹ : PartialOrder α
inst✝ : LinearOrder β
φ : β → α
ψ : α → β
t : Set β
s : Set α
hφ : MonotoneOn φ t
φψs : RightInvOn ψ φ s
ψts : MapsTo ψ s t
x : α
xs : x ∈ s
y : α
ys : y ∈ s
l : x ≤ y
ψxy : ψ x ≤ ψ y
⊢ ψ x ≤ ψ y | exact ψxy | no goals | 9a6af1077e41fc7e |
finRotate_succ_eq_decomposeFin | Mathlib/GroupTheory/Perm/Fin.lean | theorem finRotate_succ_eq_decomposeFin {n : ℕ} :
finRotate n.succ = decomposeFin.symm (1, finRotate n) | case H.h.succ.refine_1
n✝ : ℕ
i : Fin (n✝ + 1).succ
⊢ ↑((finRotate (n✝ + 1).succ) 0) = ↑((decomposeFin.symm (1, finRotate (n✝ + 1))) 0) | simp | no goals | bcacb481255d2ca2 |
MulAction.mem_subgroup_orbit_iff | Mathlib/GroupTheory/GroupAction/Defs.lean | @[to_additive]
lemma mem_subgroup_orbit_iff {H : Subgroup G} {x : α} {a b : orbit G x} :
a ∈ MulAction.orbit H b ↔ (a : α) ∈ MulAction.orbit H (b : α) | case refine_2.intro
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
H : Subgroup G
x : α
a b : ↑(orbit G x)
g : ↥H
h : g • ↑b = ↑a
⊢ a ∈ orbit (↥H) b | erw [← orbit.coe_smul, ← Subtype.ext_iff] at h | case refine_2.intro
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
H : Subgroup G
x : α
a b : ↑(orbit G x)
g : ↥H
h : ↑g • b = a
⊢ a ∈ orbit (↥H) b | 455130db4c4a28b8 |
DedekindDomain.FiniteAdeleRing.submodulesRingBasis | Mathlib/RingTheory/DedekindDomain/FiniteAdeleRing.lean | theorem submodulesRingBasis : SubmodulesRingBasis
(fun (r : R⁰) ↦ Submodule.span (R_hat R K) {((r : R) : FiniteAdeleRing R K)}) where
inter i j := ⟨i * j, by
push_cast
simp only [le_inf_iff, Submodule.span_singleton_le_iff_mem, Submodule.mem_span_singleton]
exact ⟨⟨((j : R) : R_hat R K), by rw [mul_co... | R : Type u_1
K : Type u_2
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
i j : ↥R⁰
⊢ Submodule.span (R_hat R K) {↑↑(i * j)} ≤ Submodule.span (R_hat R K) {↑↑i} ⊓ Submodule.span (R_hat R K) {↑↑j} | push_cast | R : Type u_1
K : Type u_2
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
i j : ↥R⁰
⊢ Submodule.span (R_hat R K) {↑↑i * ↑↑j} ≤ Submodule.span (R_hat R K) {↑↑i} ⊓ Submodule.span (R_hat R K) {↑↑j} | a7e62a339be55b1a |
MeasureTheory.Measure.mconv_zero | Mathlib/MeasureTheory/Group/Convolution.lean | theorem mconv_zero (μ : Measure M) : (0 : Measure M) ∗ μ = (0 : Measure M) | M : Type u_1
inst✝¹ : Monoid M
inst✝ : MeasurableSpace M
μ : Measure M
⊢ 0 ∗ μ = 0 | unfold mconv | M : Type u_1
inst✝¹ : Monoid M
inst✝ : MeasurableSpace M
μ : Measure M
⊢ map (fun x => x.1 * x.2) (Measure.prod 0 μ) = 0 | 2c52f80d7ac9c3ce |
MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le | Mathlib/Probability/Martingale/Upcrossing.lean | theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k ∈ Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] | case h.e'_6.h
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n : ℕ
ℱ : Filtration ℕ m0
inst✝ : IsFiniteMeasure μ
hf : Submartingale f ℱ μ
h₁ : 0 ≤ ∫ (x : Ω), (∑ k ∈ Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) x ∂μ
x✝ : Ω
⊢ ∑ x ∈ Finset.range n, upcrossingStrat a b f N... | simp | no goals | 4c851012d18abd58 |
Ideal.Quotient.maximal_of_isField | Mathlib/RingTheory/Ideal/Quotient/Basic.lean | theorem maximal_of_isField {R} [CommRing R] (I : Ideal R) (hqf : IsField (R ⧸ I)) :
I.IsMaximal | R : Type u_1
inst✝ : CommRing R
I : Ideal R
hqf : IsField (R ⧸ I)
⊢ 1 ∉ I ∧ ∀ (J : Ideal R) (x : R), I ≤ J → x ∉ I → x ∈ J → 1 ∈ J | constructor | case left
R : Type u_1
inst✝ : CommRing R
I : Ideal R
hqf : IsField (R ⧸ I)
⊢ 1 ∉ I
case right
R : Type u_1
inst✝ : CommRing R
I : Ideal R
hqf : IsField (R ⧸ I)
⊢ ∀ (J : Ideal R) (x : R), I ≤ J → x ∉ I → x ∈ J → 1 ∈ J | 5e093035780dd060 |
Asymptotics.isBigOWith_congr | Mathlib/Analysis/Asymptotics/Defs.lean | theorem isBigOWith_congr (hc : c₁ = c₂) (hf : f₁ =ᶠ[l] f₂) (hg : g₁ =ᶠ[l] g₂) :
IsBigOWith c₁ l f₁ g₁ ↔ IsBigOWith c₂ l f₂ g₂ | α : Type u_1
E : Type u_3
F : Type u_4
inst✝¹ : Norm E
inst✝ : Norm F
c₁ : ℝ
l : Filter α
f₁ f₂ : α → E
g₁ g₂ : α → F
hf : f₁ =ᶠ[l] f₂
hg : g₁ =ᶠ[l] g₂
⊢ (∀ᶠ (x : α) in l, ‖f₁ x‖ ≤ c₁ * ‖g₁ x‖) ↔ ∀ᶠ (x : α) in l, ‖f₂ x‖ ≤ c₁ * ‖g₂ x‖ | apply Filter.eventually_congr | case h
α : Type u_1
E : Type u_3
F : Type u_4
inst✝¹ : Norm E
inst✝ : Norm F
c₁ : ℝ
l : Filter α
f₁ f₂ : α → E
g₁ g₂ : α → F
hf : f₁ =ᶠ[l] f₂
hg : g₁ =ᶠ[l] g₂
⊢ ∀ᶠ (x : α) in l, ‖f₁ x‖ ≤ c₁ * ‖g₁ x‖ ↔ ‖f₂ x‖ ≤ c₁ * ‖g₂ x‖ | ed385ef33338e32f |
MeasureTheory.lintegral_abs_det_fderiv_le_addHaar_image_aux1 | Mathlib/MeasureTheory/Function/Jacobian.lean | theorem lintegral_abs_det_fderiv_le_addHaar_image_aux1 (hs : MeasurableSet s)
(hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : InjOn f s) {ε : ℝ≥0} (εpos : 0 < ε) :
(∫⁻ x in s, ENNReal.ofReal |(f' x).det| ∂μ) ≤ μ (f '' s) + 2 * ε * μ s | E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
ε : ℝ≥0
εp... | conv_lhs => rw [s_eq] | E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
ε : ℝ≥0
εp... | 10b49d97c45d9730 |
functionField_iff | Mathlib/NumberTheory/FunctionField.lean | theorem functionField_iff (Fqt : Type*) [Field Fqt] [Algebra Fq[X] Fqt]
[IsFractionRing Fq[X] Fqt] [Algebra (RatFunc Fq) F] [Algebra Fqt F] [Algebra Fq[X] F]
[IsScalarTower Fq[X] Fqt F] [IsScalarTower Fq[X] (RatFunc Fq) F] :
FunctionField Fq F ↔ FiniteDimensional Fqt F | Fq : Type u_1
F : Type u_2
inst✝⁹ : Field Fq
inst✝⁸ : Field F
Fqt : Type u_3
inst✝⁷ : Field Fqt
inst✝⁶ : Algebra Fq[X] Fqt
inst✝⁵ : IsFractionRing Fq[X] Fqt
inst✝⁴ : Algebra (RatFunc Fq) F
inst✝³ : Algebra Fqt F
inst✝² : Algebra Fq[X] F
inst✝¹ : IsScalarTower Fq[X] Fqt F
inst✝ : IsScalarTower Fq[X] (RatFunc Fq) F
e : R... | rw [Algebra.smul_def, Algebra.smul_def] | Fq : Type u_1
F : Type u_2
inst✝⁹ : Field Fq
inst✝⁸ : Field F
Fqt : Type u_3
inst✝⁷ : Field Fqt
inst✝⁶ : Algebra Fq[X] Fqt
inst✝⁵ : IsFractionRing Fq[X] Fqt
inst✝⁴ : Algebra (RatFunc Fq) F
inst✝³ : Algebra Fqt F
inst✝² : Algebra Fq[X] F
inst✝¹ : IsScalarTower Fq[X] Fqt F
inst✝ : IsScalarTower Fq[X] (RatFunc Fq) F
e : R... | 3d099644c240a495 |
MeasureTheory.condExp_ae_eq_condExpL1 | Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean | theorem condExp_ae_eq_condExpL1 (hm : m ≤ m₀) [hμm : SigmaFinite (μ.trim hm)] (f : α → E) :
μ[f|m] =ᵐ[μ] condExpL1 hm μ f | 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)
f : α → E
hfi : Integrable f μ
⊢ (if StronglyMeasurable f then f else AEStronglyMeasurable.mk ↑↑(condExpL1 hm μ f) ⋯) =ᶠ[ae μ... | by_cases hfm : StronglyMeasurable[m] f | 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)
f : α → E
hfi : Integrable f μ
hfm : StronglyMeasurable f
⊢ (if StronglyMeasurable f then f else AEStronglyMeasurable.mk ↑↑(c... | 1968c39efda9fc0f |
Ordinal.lt_nmul_iff | Mathlib/SetTheory/Ordinal/NaturalOps.lean | theorem lt_nmul_iff : c < a ⨳ b ↔ ∃ a' < a, ∃ b' < b, c ♯ a' ⨳ b' ≤ a' ⨳ b ♯ a ⨳ b' | case refine_1
a b c : Ordinal.{u}
h : c < sInf {c | ∀ a' < a, ∀ b' < b, a' ⨳ b ♯ a ⨳ b' < c ♯ a' ⨳ b'}
⊢ ∃ a' < a, ∃ b' < b, c ♯ a' ⨳ b' ≤ a' ⨳ b ♯ a ⨳ b' | simpa using not_mem_of_lt_csInf h ⟨0, fun _ _ => bot_le⟩ | no goals | 598fd2131c85b39a |
CategoryTheory.Functor.Monoidal.map_associator_inv | Mathlib/CategoryTheory/Monoidal/Functor.lean | theorem map_associator_inv (X Y Z : C) :
F.map (α_ X Y Z).inv =
δ F X (Y ⊗ Z) ≫ F.obj X ◁ δ F Y Z ≫
(α_ (F.obj X) (F.obj Y) (F.obj Z)).inv ≫ μ F X Y ▷ F.obj Z ≫ μ F (X ⊗ Y) Z | C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : MonoidalCategory D
F : C ⥤ D
inst✝ : F.Monoidal
X Y Z : C
⊢ F.map (α_ X Y Z).inv =
δ F X (Y ⊗ Z) ≫ F.obj X ◁ δ F Y Z ≫ (α_ (F.obj X) (F.obj Y) (F.obj Z)).inv ≫ μ F X Y ▷ F.obj Z ≫ μ F (X ⊗ Y) Z | rw [← cancel_epi (F.map (α_ X Y Z).hom), Iso.map_hom_inv_id, map_associator,
assoc, assoc, assoc, assoc, OplaxMonoidal.associativity_inv_assoc,
whiskerRight_δ_μ_assoc, δ_μ, comp_id, LaxMonoidal.associativity_inv,
Iso.hom_inv_id_assoc, whiskerRight_δ_μ_assoc, δ_μ] | no goals | c47ae625bb59e643 |
ProbabilityTheory.eq_condKernel_of_measure_eq_compProd | Mathlib/Probability/Kernel/Disintegration/Unique.lean | theorem eq_condKernel_of_measure_eq_compProd (κ : Kernel α Ω) [IsFiniteKernel κ]
(hκ : ρ = ρ.fst ⊗ₘ κ) :
∀ᵐ x ∂ρ.fst, κ x = ρ.condKernel x | α : Type u_1
Ω : Type u_3
mα : MeasurableSpace α
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
ρ : Measure (α × Ω)
inst✝¹ : IsFiniteMeasure ρ
κ : Kernel α Ω
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
f : Ω → ℝ := embeddingReal Ω
hf : MeasurableEmbedding (embeddingReal Ω)
⊢ ∀ᵐ (x : α) ∂ρ... | set ρ' : Measure (α × ℝ) := ρ.map (Prod.map id f) with hρ'def | α : Type u_1
Ω : Type u_3
mα : MeasurableSpace α
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
ρ : Measure (α × Ω)
inst✝¹ : IsFiniteMeasure ρ
κ : Kernel α Ω
inst✝ : IsFiniteKernel κ
hκ : ρ = ρ.fst ⊗ₘ κ
f : Ω → ℝ := embeddingReal Ω
hf : MeasurableEmbedding (embeddingReal Ω)
ρ' : Measure (α... | 69d76b4765c3d24f |
MvPolynomial.IsWeightedHomogeneous.pderiv | Mathlib/RingTheory/MvPolynomial/EulerIdentity.lean | protected lemma IsWeightedHomogeneous.pderiv [AddCancelCommMonoid M] {w : σ → M} {n n' : M} {i : σ}
(h : φ.IsWeightedHomogeneous w n) (h' : n' + w i = n) :
(pderiv i φ).IsWeightedHomogeneous w n' | case neg
R : Type u_1
σ : Type u_2
M : Type u_3
inst✝¹ : CommSemiring R
φ : MvPolynomial σ R
inst✝ : AddCancelCommMonoid M
w : σ → M
n n' : M
i : σ
h : φ ∈ Submodule.span R ((fun i => single i 1) '' {d | (weight w) d = n})
h' : n' + w i = n
m : σ →₀ ℕ
hm : m ∈ {d | (weight w) d = n}
hi : ¬m i = 0
⊢ IsWeightedHomogeneou... | convert isWeightedHomogeneous_monomial .. | case neg.convert_10
R : Type u_1
σ : Type u_2
M : Type u_3
inst✝¹ : CommSemiring R
φ : MvPolynomial σ R
inst✝ : AddCancelCommMonoid M
w : σ → M
n n' : M
i : σ
h : φ ∈ Submodule.span R ((fun i => single i 1) '' {d | (weight w) d = n})
h' : n' + w i = n
m : σ →₀ ℕ
hm : m ∈ {d | (weight w) d = n}
hi : ¬m i = 0
⊢ (weight w... | 8ec031de5f3c8a44 |
ZetaAsymptotics.term_one | Mathlib/NumberTheory/Harmonic/ZetaAsymp.lean | lemma term_one {n : ℕ} (hn : 0 < n) :
term n 1 = (log (n + 1) - log n) - 1 / (n + 1) | n : ℕ
hn : 0 < n
hv : ∀ x ∈ uIcc (↑n) (↑n + 1), 0 < x
x : ℝ
hx : x ∈ uIcc (↑n) (↑n + 1)
⊢ (x - ↑n) * (x * x ^ 2) = (x ^ 2 - x * ↑n) * x ^ 2 | ring | no goals | b352dfbb4af79ad9 |
Polynomial.leadingCoeff_multiset_prod' | Mathlib/Algebra/Polynomial/BigOperators.lean | theorem leadingCoeff_multiset_prod' (h : (t.map leadingCoeff).prod ≠ 0) :
t.prod.leadingCoeff = (t.map leadingCoeff).prod | case cons
R : Type u
inst✝ : CommSemiring R
t✝ : Multiset R[X]
a : R[X]
t : Multiset R[X]
ih : (Multiset.map leadingCoeff t).prod ≠ 0 → t.prod.leadingCoeff = (Multiset.map leadingCoeff t).prod
h : (Multiset.map leadingCoeff (a ::ₘ t)).prod ≠ 0
⊢ (a ::ₘ t).prod.leadingCoeff = (Multiset.map leadingCoeff (a ::ₘ t)).prod | simp only [Multiset.map_cons, Multiset.prod_cons] at h ⊢ | case cons
R : Type u
inst✝ : CommSemiring R
t✝ : Multiset R[X]
a : R[X]
t : Multiset R[X]
ih : (Multiset.map leadingCoeff t).prod ≠ 0 → t.prod.leadingCoeff = (Multiset.map leadingCoeff t).prod
h : a.leadingCoeff * (Multiset.map leadingCoeff t).prod ≠ 0
⊢ (a * t.prod).leadingCoeff = a.leadingCoeff * (Multiset.map leadin... | 08d45fcd3d906d49 |
le_iff_exists_one_le_mul | Mathlib/Algebra/Order/Monoid/Unbundled/ExistsOfLE.lean | @[to_additive] lemma le_iff_exists_one_le_mul [MulLeftMono α]
[MulLeftReflectLE α] : a ≤ b ↔ ∃ c, 1 ≤ c ∧ a * c = b :=
⟨exists_one_le_mul_of_le, by rintro ⟨c, hc, rfl⟩; exact le_mul_of_one_le_right' hc⟩
| case intro.intro
α : Type u
inst✝⁴ : MulOneClass α
inst✝³ : Preorder α
inst✝² : ExistsMulOfLE α
a : α
inst✝¹ : MulLeftMono α
inst✝ : MulLeftReflectLE α
c : α
hc : 1 ≤ c
⊢ a ≤ a * c | exact le_mul_of_one_le_right' hc | no goals | e9fc7aa0b63b7ad1 |
MeasureTheory.condExp_mul_of_stronglyMeasurable_left | Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean | theorem condExp_mul_of_stronglyMeasurable_left {f g : α → ℝ} (hf : StronglyMeasurable[m] f)
(hfg : Integrable (f * g) μ) (hg : Integrable g μ) : μ[f * g|m] =ᵐ[μ] f * μ[g|m] | case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f g : α → ℝ
hf : StronglyMeasurable f
hfg : Integrable (f * g) μ
hg : Integrable g μ
hm : ¬m ≤ m0
⊢ 0 =ᶠ[ae μ] f * 0 | rw [mul_zero] | no goals | baffe7f7f5a39936 |
disjointed_succ | Mathlib/Order/Disjointed.lean | lemma disjointed_succ (f : ι → α) {i : ι} (hi : ¬IsMax i) :
disjointed f (succ i) = f (succ i) \ partialSups f i | α : Type u_1
ι : Type u_2
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : SuccOrder ι
f : ι → α
i : ι
hi : ¬IsMax i
⊢ f (succ i) \ (Iio (succ i)).sup f = f (succ i) \ (Iic i).sup f | congr 2 with m | case e_a.e_s.h
α : Type u_1
ι : Type u_2
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : SuccOrder ι
f : ι → α
i : ι
hi : ¬IsMax i
m : ι
⊢ m ∈ Iio (succ i) ↔ m ∈ Iic i | b9cdec018ed9bfb6 |
Vitali.exists_disjoint_subfamily_covering_enlargement | Mathlib/MeasureTheory/Covering/Vitali.lean | theorem exists_disjoint_subfamily_covering_enlargement (B : ι → Set α) (t : Set ι) (δ : ι → ℝ)
(τ : ℝ) (hτ : 1 < τ) (δnonneg : ∀ a ∈ t, 0 ≤ δ a) (R : ℝ) (δle : ∀ a ∈ t, δ a ≤ R)
(hne : ∀ a ∈ t, (B a).Nonempty) :
∃ u ⊆ t,
u.PairwiseDisjoint B ∧ ∀ a ∈ t, ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ δ a ≤ τ * δ b | case inr.intro.intro
α : Type u_1
ι : Type u_2
B : ι → Set α
t : Set ι
δ : ι → ℝ
τ : ℝ
hτ : 1 < τ
δnonneg : ∀ a ∈ t, 0 ≤ δ a
R : ℝ
δle : ∀ a ∈ t, δ a ≤ R
hne : ∀ a ∈ t, (B a).Nonempty
T : Set (Set ι) :=
{u |
u ⊆ t ∧
u.PairwiseDisjoint B ∧ ∀ a ∈ t, ∀ b ∈ u, (B a ∩ B b).Nonempty → ∃ c ∈ u, (B a ∩ B c).Nonempt... | rcases (mem_image _ _ _).1 xA with ⟨a', ha', rfl⟩ | case inr.intro.intro.intro.intro
α : Type u_1
ι : Type u_2
B : ι → Set α
t : Set ι
δ : ι → ℝ
τ : ℝ
hτ : 1 < τ
δnonneg : ∀ a ∈ t, 0 ≤ δ a
R : ℝ
δle : ∀ a ∈ t, δ a ≤ R
hne : ∀ a ∈ t, (B a).Nonempty
T : Set (Set ι) :=
{u |
u ⊆ t ∧
u.PairwiseDisjoint B ∧ ∀ a ∈ t, ∀ b ∈ u, (B a ∩ B b).Nonempty → ∃ c ∈ u, (B a ∩ ... | e38a150de4970f9c |
List.set_set_perm | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/Perm.lean | theorem set_set_perm {as : List α} {i j : Nat} (h₁ : i < as.length) (h₂ : j < as.length) :
(as.set i as[j]).set j as[i] ~ as | α : Type u_1
as : List α
i j : Nat
h₁ : i < as.length
h₂ : j < as.length
h₃ : i = j
⊢ (as.set i as[j]).set j as[i] ~ as | simp [h₃] | no goals | 93c9fe7546c9ab4b |
CategoryTheory.Limits.Sigma.map_comp_map' | Mathlib/CategoryTheory/Limits/Shapes/Products.lean | lemma Sigma.map_comp_map' {f g : α → C} {h : β → C} [HasCoproduct f] [HasCoproduct g]
[HasCoproduct h] (p : α → β) (q : ∀ (a : α), f a ⟶ g a) (q' : ∀ (a : α), g a ⟶ h (p a)) :
Sigma.map q ≫ Sigma.map' p q' = Sigma.map' p (fun a => q a ≫ q' a) | β : Type w
α : Type w₂
C : Type u
inst✝³ : Category.{v, u} C
f g : α → C
h : β → C
inst✝² : HasCoproduct f
inst✝¹ : HasCoproduct g
inst✝ : HasCoproduct h
p : α → β
q : (a : α) → f a ⟶ g a
q' : (a : α) → g a ⟶ h (p a)
⊢ map q ≫ map' p q' = map' p fun a => q a ≫ q' a | ext | case h
β : Type w
α : Type w₂
C : Type u
inst✝³ : Category.{v, u} C
f g : α → C
h : β → C
inst✝² : HasCoproduct f
inst✝¹ : HasCoproduct g
inst✝ : HasCoproduct h
p : α → β
q : (a : α) → f a ⟶ g a
q' : (a : α) → g a ⟶ h (p a)
b✝ : α
⊢ ι f b✝ ≫ map q ≫ map' p q' = ι f b✝ ≫ map' p fun a => q a ≫ q' a | d3b8f8d83e547df6 |
CategoryTheory.initiallySmall_of_small_weakly_initial_set | Mathlib/CategoryTheory/Limits/FinallySmall.lean | theorem initiallySmall_of_small_weakly_initial_set [IsCofilteredOrEmpty J] (s : Set J) [Small.{v} s]
(hs : ∀ i, ∃ j ∈ s, Nonempty (j ⟶ i)) : InitiallySmall.{v} J | J : Type u
inst✝² : Category.{v, u} J
inst✝¹ : IsCofilteredOrEmpty J
s : Set J
inst✝ : Small.{v, u} ↑s
hs : ∀ (i : J), ∃ j ∈ s, Nonempty (j ⟶ i)
i : J
⊢ ∃ c, Nonempty ((fullSubcategoryInclusion fun x => x ∈ s).obj c ⟶ i) | obtain ⟨j, hj₁, hj₂⟩ := hs i | case intro.intro
J : Type u
inst✝² : Category.{v, u} J
inst✝¹ : IsCofilteredOrEmpty J
s : Set J
inst✝ : Small.{v, u} ↑s
hs : ∀ (i : J), ∃ j ∈ s, Nonempty (j ⟶ i)
i j : J
hj₁ : j ∈ s
hj₂ : Nonempty (j ⟶ i)
⊢ ∃ c, Nonempty ((fullSubcategoryInclusion fun x => x ∈ s).obj c ⟶ i) | 5146c16b291c6e05 |
SemiNormedGrp.explicitCokernel_hom_ext | Mathlib/Analysis/Normed/Group/SemiNormedGrp/Kernels.lean | theorem explicitCokernel_hom_ext {X Y Z : SemiNormedGrp.{u}} {f : X ⟶ Y}
(e₁ e₂ : explicitCokernel f ⟶ Z) (h : explicitCokernelπ f ≫ e₁ = explicitCokernelπ f ≫ e₂) :
e₁ = e₂ | X Y Z : SemiNormedGrp
f : X ⟶ Y
e₁ e₂ : explicitCokernel f ⟶ Z
h : explicitCokernelπ f ≫ e₁ = explicitCokernelπ f ≫ e₂
g : Y ⟶ Z := explicitCokernelπ f ≫ e₂
w : f ≫ g = 0
⊢ e₁ = e₂ | have : e₂ = explicitCokernelDesc w := by apply explicitCokernelDesc_unique; rfl | X Y Z : SemiNormedGrp
f : X ⟶ Y
e₁ e₂ : explicitCokernel f ⟶ Z
h : explicitCokernelπ f ≫ e₁ = explicitCokernelπ f ≫ e₂
g : Y ⟶ Z := explicitCokernelπ f ≫ e₂
w : f ≫ g = 0
this : e₂ = explicitCokernelDesc w
⊢ e₁ = e₂ | 1608ec920b7bd483 |
CategoryTheory.IsFiltered.sup_objs_exists | Mathlib/CategoryTheory/Filtered/Basic.lean | theorem sup_objs_exists (O : Finset C) : ∃ S : C, ∀ {X}, X ∈ O → Nonempty (X ⟶ S) | case h.inl
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFiltered C
O' : Finset C
S' : C
w' : ∀ {X : C}, X ∈ O' → Nonempty (X ⟶ S')
Y : C
nm : Y ∉ O'
mY : Y ∈ insert Y O'
⊢ Nonempty (Y ⟶ max Y S') | exact ⟨leftToMax _ _⟩ | no goals | 6340905fdf09a34b |
CategoryTheory.Limits.Types.unique_of_type_equalizer | Mathlib/CategoryTheory/Limits/Shapes/Types.lean | theorem unique_of_type_equalizer (t : IsLimit (Fork.ofι _ w)) (y : Y) (hy : g y = h y) :
∃! x : X, f x = y | X Y Z : Type u
f : X ⟶ Y
g h : Y ⟶ Z
w : f ≫ g = f ≫ h
t : IsLimit (Fork.ofι f w)
y : Y
hy : g y = h y
⊢ ∃! x, f x = y | let y' : PUnit ⟶ Y := fun _ => y | X Y Z : Type u
f : X ⟶ Y
g h : Y ⟶ Z
w : f ≫ g = f ≫ h
t : IsLimit (Fork.ofι f w)
y : Y
hy : g y = h y
y' : PUnit.{u + 1} ⟶ Y := fun x => y
⊢ ∃! x, f x = y | 5853f29b8c3636cd |
SimpleGraph.adjMatrix_mulVec_const_apply_of_regular | Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | theorem adjMatrix_mulVec_const_apply_of_regular [NonAssocSemiring α] {d : ℕ} {a : α}
(hd : G.IsRegularOfDegree d) {v : V} : (G.adjMatrix α *ᵥ Function.const _ a) v = d * a | V : Type u_1
α : Type u_2
G : SimpleGraph V
inst✝² : DecidableRel G.Adj
inst✝¹ : Fintype V
inst✝ : NonAssocSemiring α
d : ℕ
a : α
hd : G.IsRegularOfDegree d
v : V
⊢ (adjMatrix α G *ᵥ Function.const V a) v = ↑d * a | simp [hd v] | no goals | b014a0784e862b2d |
spectrum.zero_mem_resolventSet_of_unit | Mathlib/Algebra/Algebra/Spectrum.lean | theorem zero_mem_resolventSet_of_unit (a : Aˣ) : 0 ∈ resolventSet R (a : A) | R : Type u
A : Type v
inst✝² : CommSemiring R
inst✝¹ : Ring A
inst✝ : Algebra R A
a : Aˣ
⊢ 0 ∈ resolventSet R ↑a | simpa only [mem_resolventSet_iff, ← not_mem_iff, zero_not_mem_iff] using a.isUnit | no goals | 81e4510fac416c38 |
StrictConvexSpace.of_norm_add_ne_two | Mathlib/Analysis/Convex/StrictConvexSpace.lean | theorem StrictConvexSpace.of_norm_add_ne_two
(h : ∀ ⦃x y : E⦄, ‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ‖x + y‖ ≠ 2) : StrictConvexSpace ℝ E | E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
h : ∀ ⦃x y : E⦄, ‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ‖x + y‖ ≠ 2
x y : E
hx : ‖x‖ = 1
hy : ‖y‖ = 1
hne : x ≠ y
⊢ ‖(1 / 2) • x + (1 / 2) • y‖ ≠ 1 | rw [← smul_add, norm_smul, Real.norm_of_nonneg one_half_pos.le, one_div, ← div_eq_inv_mul, Ne,
div_eq_one_iff_eq (two_ne_zero' ℝ)] | E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
h : ∀ ⦃x y : E⦄, ‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ‖x + y‖ ≠ 2
x y : E
hx : ‖x‖ = 1
hy : ‖y‖ = 1
hne : x ≠ y
⊢ ¬‖x + y‖ = 2 | 5f3613c8c199beaa |
Filter.mem_coprod_iff | Mathlib/Order/Filter/Prod.lean | theorem mem_coprod_iff {s : Set (α × β)} {f : Filter α} {g : Filter β} :
s ∈ f.coprod g ↔ (∃ t₁ ∈ f, Prod.fst ⁻¹' t₁ ⊆ s) ∧ ∃ t₂ ∈ g, Prod.snd ⁻¹' t₂ ⊆ s | α : Type u_1
β : Type u_2
s : Set (α × β)
f : Filter α
g : Filter β
⊢ s ∈ f.coprod g ↔ (∃ t₁ ∈ f, Prod.fst ⁻¹' t₁ ⊆ s) ∧ ∃ t₂ ∈ g, Prod.snd ⁻¹' t₂ ⊆ s | simp [Filter.coprod] | no goals | e1e0d4e150e2fb13 |
equicontinuousWithinAt_univ | Mathlib/Topology/UniformSpace/Equicontinuity.lean | @[simp] lemma equicontinuousWithinAt_univ (F : ι → X → α) (x₀ : X) :
EquicontinuousWithinAt F univ x₀ ↔ EquicontinuousAt F x₀ | ι : Type u_1
X : Type u_3
α : Type u_6
tX : TopologicalSpace X
uα : UniformSpace α
F : ι → X → α
x₀ : X
⊢ EquicontinuousWithinAt F univ x₀ ↔ EquicontinuousAt F x₀ | rw [EquicontinuousWithinAt, EquicontinuousAt, nhdsWithin_univ] | no goals | 0577120a7718dbac |
smul_sphere' | Mathlib/Analysis/NormedSpace/Pointwise.lean | theorem smul_sphere' {c : 𝕜} (hc : c ≠ 0) (x : E) (r : ℝ) :
c • sphere x r = sphere (c • x) (‖c‖ * r) | 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
hc : c ≠ 0
x : E
r : ℝ
⊢ c • sphere x r = sphere (c • x) (‖c‖ * r) | ext y | case h
𝕜 : Type u_1
E : Type u_2
inst✝² : NormedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
hc : c ≠ 0
x : E
r : ℝ
y : E
⊢ y ∈ c • sphere x r ↔ y ∈ sphere (c • x) (‖c‖ * r) | 38b245f7ebbdc477 |
Decidable.iff_congr_right | Mathlib/.lake/packages/lean4/src/lean/Init/PropLemmas.lean | theorem Decidable.iff_congr_right {P Q R : Prop} [Decidable P] [Decidable Q] [Decidable R] :
((P ↔ Q) ↔ (P ↔ R)) ↔ (Q ↔ R) :=
if h : P then by simp_all [Decidable.not_iff_not] else by simp_all [Decidable.not_iff_not]
| P Q R : Prop
inst✝² : Decidable P
inst✝¹ : Decidable Q
inst✝ : Decidable R
h : P
⊢ ((P ↔ Q) ↔ (P ↔ R)) ↔ (Q ↔ R) | simp_all [Decidable.not_iff_not] | no goals | 5dd66a3e1d09dd56 |
MeasureTheory.measure_mul_laverage | Mathlib/MeasureTheory/Integral/Average.lean | theorem measure_mul_laverage [IsFiniteMeasure μ] (f : α → ℝ≥0∞) :
μ univ * ⨍⁻ x, f x ∂μ = ∫⁻ x, f x ∂μ | case inr
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
f : α → ℝ≥0∞
hμ : μ ≠ 0
⊢ μ univ * ⨍⁻ (x : α), f x ∂μ = ∫⁻ (x : α), f x ∂μ | rw [laverage_eq, ENNReal.mul_div_cancel (measure_univ_ne_zero.2 hμ) (measure_ne_top _ _)] | no goals | 640c4e9d5430e11d |
WeierstrassCurve.Jacobian.nonsingular_neg | Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean | lemma nonsingular_neg {P : Fin 3 → F} (hP : W.Nonsingular P) : W.Nonsingular <| W.neg P | case pos
F : Type u
inst✝ : Field F
W : Jacobian F
P : Fin 3 → F
hP : W.Nonsingular P
hPz : P z = 0
⊢ W.Nonsingular (W.neg P) | simp only [neg_of_Z_eq_zero hP hPz, nonsingular_smul _
((isUnit_Y_of_Z_eq_zero hP hPz).div <| isUnit_X_of_Z_eq_zero hP hPz).neg, nonsingular_zero] | no goals | 554a9cfe06518499 |
Lean.Order.Array.monotone_foldlM_loop | Mathlib/.lake/packages/lean4/src/lean/Init/Internal/Order/Lemmas.lean | theorem monotone_foldlM_loop
(f : γ → β → α → m β) (xs : Array α) (stop : Nat) (h : stop ≤ xs.size) (i j : Nat) (b : β)
(hmono : monotone f) : monotone (fun x => Array.foldlM.loop (f x) xs stop h i j b) | case case2.hmono₁
m : Type u → Type v
inst✝³ : Monad m
inst✝² : (α : Type u) → PartialOrder (m α)
inst✝¹ : MonoBind m
α β : Type u
γ : Type w
inst✝ : PartialOrder γ
f : γ → β → α → m β
xs : Array α
stop : Nat
h : stop ≤ xs.size
hmono : monotone f
j✝ : Nat
b✝ : β
h✝ : j✝ < stop
i'✝ : Nat
this✝ : j✝ < xs.size
ih : ∀ (__d... | apply monotone_apply | case case2.hmono₁.h
m : Type u → Type v
inst✝³ : Monad m
inst✝² : (α : Type u) → PartialOrder (m α)
inst✝¹ : MonoBind m
α β : Type u
γ : Type w
inst✝ : PartialOrder γ
f : γ → β → α → m β
xs : Array α
stop : Nat
h : stop ≤ xs.size
hmono : monotone f
j✝ : Nat
b✝ : β
h✝ : j✝ < stop
i'✝ : Nat
this✝ : j✝ < xs.size
ih : ∀ (_... | 644dcc71261616e9 |
four_functions_theorem | Mathlib/Combinatorics/SetFamily/FourFunctions.lean | /-- The **Four Functions Theorem**, aka **Ahlswede-Daykin Inequality**. -/
lemma four_functions_theorem [DecidableEq α] (h₁ : 0 ≤ f₁) (h₂ : 0 ≤ f₂) (h₃ : 0 ≤ f₃) (h₄ : 0 ≤ f₄)
(h : ∀ a b, f₁ a * f₂ b ≤ f₃ (a ⊓ b) * f₄ (a ⊔ b)) (s t : Finset α) :
(∑ a ∈ s, f₁ a) * ∑ a ∈ t, f₂ a ≤ (∑ a ∈ s ⊼ t, f₃ a) * ∑ a ∈ s ⊻ ... | case intro.intro.intro.intro.refine_1.inr
α : Type u_1
β✝ : Type u_2
inst✝³ : DistribLattice α
inst✝² : LinearOrderedCommSemiring β✝
inst✝¹ : ExistsAddOfLE β✝
f₁ f₂ f₃ f₄ : α → β✝
inst✝ : DecidableEq α
h₁ : 0 ≤ f₁
h₂ : 0 ≤ f₂
h₃ : 0 ≤ f₃
h₄ : 0 ≤ f₄
h : ∀ (a b : α), f₁ a * f₂ b ≤ f₃ (a ⊓ b) * f₄ (a ⊔ b)
s✝ t✝ : Finset ... | simpa [extend_apply' _ _ _ hs] using mul_nonneg
(extend_nonneg (fun a : L ↦ h₃ a) le_rfl _) (extend_nonneg (fun a : L ↦ h₄ a) le_rfl _) | no goals | bda3cc5a4c2e31b0 |
Nat.ordCompl_dvd_ordCompl_of_dvd | Mathlib/Data/Nat/Factorization/Basic.lean | theorem ordCompl_dvd_ordCompl_of_dvd {a b : ℕ} (hab : a ∣ b) (p : ℕ) :
ordCompl[p] a ∣ ordCompl[p] b | case inr.inl
a p : ℕ
pp : Prime p
hab : a ∣ 0
⊢ a / p ^ a.factorization p ∣ 0 / p ^ (factorization 0) p | simp | no goals | c02459c7c27ede19 |
Complex.integral_exp_neg_rpow | Mathlib/MeasureTheory/Integral/Gamma.lean | theorem Complex.integral_exp_neg_rpow {p : ℝ} (hp : 1 ≤ p) :
∫ x : ℂ, rexp (- ‖x‖ ^ p) = π * Real.Gamma (2 / p + 1) | p : ℝ
hp : 1 ≤ p
⊢ p ≠ 0 | linarith | no goals | 05de202e9a0c853c |
List.sum_map_count_dedup_filter_eq_countP | Mathlib/Algebra/BigOperators/Group/List/Lemmas.lean | theorem sum_map_count_dedup_filter_eq_countP (p : α → Bool) (l : List α) :
((l.dedup.filter p).map fun x => l.count x).sum = l.countP p | case neg.intro
α : Type u_2
inst✝ : DecidableEq α
p : α → Bool
a : α
as : List α
h : (map (fun x => count x as) (filter p as.dedup)).sum = countP p as
hp : ¬p a = true
n : ℕ
hn : n ∈ map (fun i => if a = i then 1 else 0) (filter p (a :: as).dedup)
a' : α
ha' : a' ∈ filter p (a :: as).dedup ∧ (if a = a' then 1 else 0) =... | split_ifs at ha' with ha | case pos
α : Type u_2
inst✝ : DecidableEq α
p : α → Bool
a : α
as : List α
h : (map (fun x => count x as) (filter p as.dedup)).sum = countP p as
hp : ¬p a = true
n : ℕ
hn : n ∈ map (fun i => if a = i then 1 else 0) (filter p (a :: as).dedup)
a' : α
ha : a = a'
ha' : a' ∈ filter p (a :: as).dedup ∧ 1 = n
⊢ n = 0
case n... | 26eedee1a0b217fa |
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 | 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 : ¬FiniteDimensional k ↥s.direction
h : FiniteDimensional k ↥(vectorSpan k (insert p ↑s))
h' : s.direction ≤ vectorSpan k (insert p ↑s)
⊢ False | exact hf (Submodule.finiteDimensional_of_le h') | no goals | de5f4f906d2ae1ab |
PFun.fix_fwd | Mathlib/Data/PFun.lean | theorem fix_fwd {f : α →. β ⊕ α} {b : β} {a a' : α} (hb : b ∈ f.fix a) (ha' : Sum.inr a' ∈ f a) :
b ∈ f.fix a' | α : Type u_1
β : Type u_2
f : α →. β ⊕ α
b : β
a a' : α
hb : b ∈ f.fix a
ha' : Sum.inr a' ∈ f a
⊢ b ∈ f.fix a' | rwa [← fix_fwd_eq ha'] | no goals | 881693509a429c45 |
NumberField.InfinitePlace.card_filter_mk_eq | Mathlib/NumberTheory/NumberField/Embeddings.lean | theorem card_filter_mk_eq [NumberField K] (w : InfinitePlace K) : #{φ | mk φ = w} = mult w | case neg
K : Type u_2
inst✝¹ : Field K
inst✝ : NumberField K
w : InfinitePlace K
hw : ¬w.IsReal
⊢ #({w.embedding} ∪ {ComplexEmbedding.conjugate w.embedding}) = 2 | refine Finset.card_pair ?_ | case neg
K : Type u_2
inst✝¹ : Field K
inst✝ : NumberField K
w : InfinitePlace K
hw : ¬w.IsReal
⊢ w.embedding ≠ ComplexEmbedding.conjugate w.embedding | 87d68d4562228a68 |
CategoryTheory.InjectiveResolution.rightDerivedToHomotopyCategory_app_eq | Mathlib/CategoryTheory/Abelian/RightDerived.lean | lemma InjectiveResolution.rightDerivedToHomotopyCategory_app_eq
{F G : C ⥤ D} [F.Additive] [G.Additive] (α : F ⟶ G) {X : C} (P : InjectiveResolution X) :
(NatTrans.rightDerivedToHomotopyCategory α).app X =
(P.isoRightDerivedToHomotopyCategoryObj F).hom ≫
(HomotopyCategory.quotient _ _).map
... | case intro
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{u_2, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasInjectiveResolutions C
inst✝² : Abelian D
F G : C ⥤ D
inst✝¹ : F.Additive
inst✝ : G.Additive
α : F ⟶ G
X : C
P : InjectiveResolution X
β : (injectiveResolution X).cocomplex ⟶ P.cocomplex
hβ : (Hom... | rw [← hβ] | case intro
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{u_2, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasInjectiveResolutions C
inst✝² : Abelian D
F G : C ⥤ D
inst✝¹ : F.Additive
inst✝ : G.Additive
α : F ⟶ G
X : C
P : InjectiveResolution X
β : (injectiveResolution X).cocomplex ⟶ P.cocomplex
hβ : (Hom... | fb3a9807ff9e85f1 |
AnalyticOnNhd.eqOn_of_preconnected_of_frequently_eq | Mathlib/Analysis/Analytic/IsolatedZeros.lean | theorem eqOn_of_preconnected_of_frequently_eq (hf : AnalyticOnNhd 𝕜 f U) (hg : AnalyticOnNhd 𝕜 g U)
(hU : IsPreconnected U) (h₀ : z₀ ∈ U) (hfg : ∃ᶠ z in 𝓝[≠] z₀, f z = g z) : EqOn f g U | 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f g : 𝕜 → E
z₀ : 𝕜
U : Set 𝕜
hf : AnalyticOnNhd 𝕜 f U
hg : AnalyticOnNhd 𝕜 g U
hU : IsPreconnected U
h₀ : z₀ ∈ U
hfg : ∃ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z = g z
z : 𝕜
h : f z = g z
⊢ (f - g) z = 0 | rw [Pi.sub_apply, h, sub_self] | no goals | 56e6a713a07d326c |
PadicSeq.norm_neg | Mathlib/NumberTheory/Padics/PadicNumbers.lean | theorem norm_neg (a : PadicSeq p) : (-a).norm = a.norm :=
norm_eq <| by simp
| p : ℕ
hp : Fact (Nat.Prime p)
a : PadicSeq p
⊢ ∀ (k : ℕ), padicNorm p (↑(-a) k) = padicNorm p (↑a k) | simp | no goals | e221d9e6d0e403dd |
nilpotencyClass_le_of_ker_le_center | Mathlib/GroupTheory/Nilpotent.lean | theorem nilpotencyClass_le_of_ker_le_center {H : Type*} [Group H] (f : G →* H)
(hf1 : f.ker ≤ center G) (hH : IsNilpotent H) :
Group.nilpotencyClass (hG := isNilpotent_of_ker_le_center f hf1 hH) ≤
Group.nilpotencyClass H + 1 | case h
G : Type u_1
inst✝¹ : Group G
H : Type u_2
inst✝ : Group H
f : G →* H
hf1 : f.ker ≤ center G
hH : IsNilpotent H
this : IsNilpotent G
⊢ lowerCentralSeries G (nilpotencyClass H + 1) = ⊥ | refine lowerCentralSeries_succ_eq_bot (le_trans ((Subgroup.map_eq_bot_iff _).mp ?_) hf1) | case h
G : Type u_1
inst✝¹ : Group G
H : Type u_2
inst✝ : Group H
f : G →* H
hf1 : f.ker ≤ center G
hH : IsNilpotent H
this : IsNilpotent G
⊢ Subgroup.map f (lowerCentralSeries G (nilpotencyClass H)) = ⊥ | 74aedaa0ee74fba5 |
Real.mulExpNegMulSq_one_le_one | Mathlib/Analysis/SpecialFunctions/MulExpNegMulSq.lean | theorem mulExpNegMulSq_one_le_one (x : ℝ) : mulExpNegMulSq 1 x ≤ 1 | x : ℝ
⊢ mulExpNegMulSq 1 x ≤ 1 | simp [mulExpNegMulSq] | x : ℝ
⊢ x * rexp (-(x * x)) ≤ 1 | b6227aba2a3e92f1 |
MeasureTheory.upcrossingsBefore_eq_sum | Mathlib/Probability/Martingale/Upcrossing.lean | theorem upcrossingsBefore_eq_sum (hab : a < b) : upcrossingsBefore a b f N ω =
∑ i ∈ Finset.Ico 1 (N + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 i | Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
N : ℕ
ω : Ω
hab : a < b
hN : ¬N = 0
h₁ : ∀ k ∈ Finset.Ico 1 (upcrossingsBefore a b f N ω + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 k = 1
k : ℕ
hk : upcrossingsBefore a b f N ω < k ∧ k < N + 1
⊢ {n | upperCrossingTime a b f N n ω < N}.indicator 1 k = 0 | rw [Set.indicator_of_not_mem] | case h
Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
N : ℕ
ω : Ω
hab : a < b
hN : ¬N = 0
h₁ : ∀ k ∈ Finset.Ico 1 (upcrossingsBefore a b f N ω + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 k = 1
k : ℕ
hk : upcrossingsBefore a b f N ω < k ∧ k < N + 1
⊢ k ∉ {n | upperCrossingTime a b f N n ω < N} | 50c14e6e447fb83a |
MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul_of_measurable_of_sigmaFinite | Mathlib/MeasureTheory/Integral/Layercake.lean | theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable_of_sigmaFinite
(μ : Measure α) [SFinite μ]
(f_nn : 0 ≤ f) (f_mble : Measurable f)
(g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t) (g_mble : Measurable g)
(g_nn : ∀ t > 0, 0 ≤ g t) :
∫⁻ ω, ENNReal.ofReal (∫ t in (0)..f ω, g t) ∂μ =
... | case h.h
α : Type u_1
inst✝¹ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
inst✝ : SFinite μ
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
g_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t
integrand_eq : ∀ (ω : α... | rw [lintegral_indicator₀] | case h.h
α : Type u_1
inst✝¹ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
inst✝ : SFinite μ
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
g_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t
integrand_eq : ∀ (ω : α... | 028c71a79fc18349 |
Complex.uniformContinuous_ringHom_eq_id_or_conj | Mathlib/Topology/Instances/Complex.lean | theorem Complex.uniformContinuous_ringHom_eq_id_or_conj (K : Subfield ℂ) {ψ : K →+* ℂ}
(hc : UniformContinuous ψ) : ψ.toFun = K.subtype ∨ ψ.toFun = conj ∘ K.subtype | case h.e'_2
K : Subfield ℂ
ψ : ↥K →+* ℂ
hc : UniformContinuous ⇑ψ
this✝ : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk
this : IsTopologicalRing ↥K.topologicalClosure := Subring.instIsTopologicalRing K.topologicalClosure.toSubring
ι : ↥K → ↥K.topologicalClosure := ⇑(Subfield.inclusion ⋯)
ui : IsUniformInducin... | exact (IsDenseInducing.extend_eq di hc.continuous z).symm | no goals | f78bcdf9d891573b |
PowerSeries.trunc_coe_eq_self | Mathlib/RingTheory/PowerSeries/Trunc.lean | theorem trunc_coe_eq_self {n} {f : R[X]} (hn : natDegree f < n) : trunc n (f : R⟦X⟧) = f | R : Type u_2
inst✝ : CommSemiring R
n : ℕ
f : R[X]
hn : f.natDegree < n
⊢ trunc n ↑f = f | rw [← Polynomial.coe_inj] | R : Type u_2
inst✝ : CommSemiring R
n : ℕ
f : R[X]
hn : f.natDegree < n
⊢ ↑(trunc n ↑f) = ↑f | 5bf3fb6492face06 |
ProbabilityTheory.Kernel.withDensity_one_sub_rnDerivAux | Mathlib/Probability/Kernel/RadonNikodym.lean | lemma withDensity_one_sub_rnDerivAux (κ η : Kernel α γ) [IsFiniteKernel κ] [IsFiniteKernel η] :
withDensity (κ + η) (fun a x ↦ Real.toNNReal (1 - rnDerivAux κ (κ + η) a x)) = η | case hfg
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : Kernel α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
h_le : κ ≤ κ + η
this : ∀ (b : ℝ), ↑b.toNNReal = ENNReal.ofReal b
⊢ ∀ (a : α), (fun x => ENNReal.ofReal (κ.rnDerivA... | intro a | case hfg
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : Kernel α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
h_le : κ ≤ κ + η
this : ∀ (b : ℝ), ↑b.toNNReal = ENNReal.ofReal b
a : α
⊢ (fun x => ENNReal.ofReal (κ.rnDerivAux (κ... | 33aa60e00817c2c6 |
Polynomial.eval₂_comp' | Mathlib/Algebra/Polynomial/Eval/Algebra.lean | theorem eval₂_comp' : eval₂ (algebraMap R S) x (p.comp q) =
eval₂ (algebraMap R S) (eval₂ (algebraMap R S) x q) p | R : Type u
S : Type v
inst✝² : CommSemiring R
inst✝¹ : Semiring S
inst✝ : Algebra R S
x : S
p q : R[X]
⊢ eval₂ (algebraMap R S) x (p.comp q) = eval₂ (algebraMap R S) (eval₂ (algebraMap R S) x q) p | induction p using Polynomial.induction_on' with
| h_add r s hr hs => simp only [add_comp, eval₂_add, hr, hs]
| h_monomial n a => simp only [monomial_comp, eval₂_mul', eval₂_C, eval₂_monomial, eval₂_pow'] | no goals | 069f9445c1f8dd0e |
MvPolynomial.degreeOf_mul_X_self | Mathlib/Algebra/MvPolynomial/Degrees.lean | theorem degreeOf_mul_X_self (j : σ) (f : MvPolynomial σ R) :
degreeOf j (f * X j) ≤ degreeOf j f + 1 | case h.e'_4
R : Type u
σ : Type u_1
inst✝ : CommSemiring R
j : σ
f : MvPolynomial σ R
⊢ 1 = Multiset.count j {j} | rw [Multiset.count_singleton_self] | no goals | 102a8d529ad3906a |
PartialHomeomorph.MDifferentiable.trans | Mathlib/Geometry/Manifold/MFDeriv/Atlas.lean | theorem trans (he' : e'.MDifferentiable I' I'') : (e.trans e').MDifferentiable I I'' | case right
𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace H M
E' : Type u_5
inst✝⁹ : NormedAddCommGroup E'
ins... | simp only [mfld_simps] at hx | case right
𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace H M
E' : Type u_5
inst✝⁹ : NormedAddCommGroup E'
ins... | 822be95d99e5c4f0 |
Algebra.baseChange_lmul | Mathlib/RingTheory/TensorProduct/Basic.lean | lemma Algebra.baseChange_lmul {R B : Type*} [CommSemiring R] [Semiring B] [Algebra R B]
{A : Type*} [CommSemiring A] [Algebra R A] (f : B) :
(Algebra.lmul R B f).baseChange A = Algebra.lmul A (A ⊗[R] B) (1 ⊗ₜ f) | R : Type u_1
B : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : Semiring B
inst✝² : Algebra R B
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f : B
⊢ LinearMap.baseChange A ((lmul R B) f) = (lmul A (A ⊗[R] B)) (1 ⊗ₜ[R] f) | ext i | case a.h.h
R : Type u_1
B : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : Semiring B
inst✝² : Algebra R B
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f i : B
⊢ ((AlgebraTensorModule.curry (LinearMap.baseChange A ((lmul R B) f))) 1) i =
((AlgebraTensorModule.curry ((lmul A (A ⊗[R] B)) (1 ⊗ₜ[R] f))) 1) i | 3e3d1363ac8d60bc |
TopCat.Presheaf.map_restrict | Mathlib/Topology/Sheaves/Presheaf.lean | theorem map_restrict
{F G : X.Presheaf C} (e : F ⟶ G) {U V : Opens X} (h : U ≤ V) (x : ToType (F.obj (op V))) :
e.app _ (x |_ U) = e.app _ x |_ U | X : TopCat
C : Type u_1
inst✝² : Category.{u_5, u_1} C
FC : C → C → Type u_2
CC : C → Type u_3
inst✝¹ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝ : ConcreteCategory C FC
F G : Presheaf C X
e : F ⟶ G
U V : Opens ↑X
h : U ≤ V
x : ToType (F.obj (op V))
⊢ (ConcreteCategory.hom (e.app (op U))) ((ConcreteCategory.hom ... | rw [← ConcreteCategory.comp_apply, NatTrans.naturality, ConcreteCategory.comp_apply] | no goals | dc647c24b4e1e955 |
Topology.IsClosedEmbedding.polishSpace | Mathlib/Topology/MetricSpace/Polish.lean | theorem _root_.Topology.IsClosedEmbedding.polishSpace [TopologicalSpace α] [TopologicalSpace β]
[PolishSpace β] {f : α → β} (hf : IsClosedEmbedding f) : PolishSpace α | α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : PolishSpace β
f : α → β
hf : IsClosedEmbedding f
this✝¹ : UpgradedPolishSpace β := upgradePolishSpace β
this✝ : MetricSpace α := IsEmbedding.comapMetricSpace f ⋯
this : SecondCountableTopology α
⊢ CompleteSpace α | rw [completeSpace_iff_isComplete_range hf.isEmbedding.to_isometry.isUniformInducing] | α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : PolishSpace β
f : α → β
hf : IsClosedEmbedding f
this✝¹ : UpgradedPolishSpace β := upgradePolishSpace β
this✝ : MetricSpace α := IsEmbedding.comapMetricSpace f ⋯
this : SecondCountableTopology α
⊢ IsComplete (range f) | 4a7835d35855f550 |
Nat.SOM.Expr.toPoly_denote | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/SOM.lean | theorem Expr.toPoly_denote (ctx : Context) (e : Expr) : e.toPoly.denote ctx = e.denote ctx | case num
ctx : Context
k : Nat
⊢ Poly.denote ctx (bif k == 0 then [] else [(k, [])]) = k | by_cases h : k == 0 <;> simp! [*] | case pos
ctx : Context
k : Nat
h : (k == 0) = true
⊢ 0 = k | 7a9fb10409122baf |
HomotopyCategory.mappingConeCompTriangleh_distinguished | Mathlib/Algebra/Homology/HomotopyCategory/Triangulated.lean | lemma mappingConeCompTriangleh_distinguished :
(mappingConeCompTriangleh f g) ∈
distTriang (HomotopyCategory C (ComplexShape.up ℤ)) | C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
inst✝¹ : HasBinaryBiproducts C
X₁ X₂ X₃ : CochainComplex C ℤ
f : X₁ ⟶ X₂
g : X₂ ⟶ X₃
inst✝ : HasZeroObject C
⊢ mappingConeCompTriangleh f g ∈ distinguishedTriangles | refine ⟨_, _, (mappingConeCompTriangle f g).mor₁, ⟨?_⟩⟩ | C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
inst✝¹ : HasBinaryBiproducts C
X₁ X₂ X₃ : CochainComplex C ℤ
f : X₁ ⟶ X₂
g : X₂ ⟶ X₃
inst✝ : HasZeroObject C
⊢ mappingConeCompTriangleh f g ≅ mappingCone.triangleh (mappingConeCompTriangle f g).mor₁ | 32c9ce5e781b428a |
algebraMap_monotone | Mathlib/Algebra/Order/Algebra.lean | theorem algebraMap_monotone : Monotone (algebraMap R A) := fun a b h => by
rw [Algebra.algebraMap_eq_smul_one, Algebra.algebraMap_eq_smul_one, ← sub_nonneg, ← sub_smul]
trans (b - a) • (0 : A)
· simp
· exact smul_le_smul_of_nonneg_left zero_le_one (sub_nonneg.mpr h)
| R : Type u_1
A : Type u_2
inst✝³ : OrderedCommRing R
inst✝² : OrderedRing A
inst✝¹ : Algebra R A
inst✝ : OrderedSMul R A
a b : R
h : a ≤ b
⊢ (b - a) • 0 ≤ (b - a) • 1 | exact smul_le_smul_of_nonneg_left zero_le_one (sub_nonneg.mpr h) | no goals | 662eaaf11dd4ba11 |
List.nodup_finRange | Mathlib/Data/List/FinRange.lean | theorem nodup_finRange (n : ℕ) : (finRange n).Nodup | n : ℕ
⊢ (finRange n).Nodup | rw [finRange_eq_pmap_range] | n : ℕ
⊢ (pmap Fin.mk (range n) ⋯).Nodup | 82aa54221459055c |
AlgebraicGeometry.Scheme.Spec_map_stalkMap_fromSpecStalk | Mathlib/AlgebraicGeometry/Stalk.lean | @[reassoc (attr := simp)]
lemma Spec_map_stalkMap_fromSpecStalk {x} :
Spec.map (f.stalkMap x) ≫ Y.fromSpecStalk _ = X.fromSpecStalk x ≫ f | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
X Y : Scheme
f : X ⟶ Y
x : ↑↑X.toPresheafedSpace
U : TopologicalSpace.Opens ↑↑Y.toPresheafedSpace
hU : U ∈ Y.affineOpens
hxU : (ConcreteCategory.hom f.base) x ∈ ↑U
V : TopologicalSpace.Opens ↑↑X.toPresheafedSpace
hV : V ∈ X.affineOpens
hxV : x ∈ ↑V
hVU : ... | rw [← hU.fromSpecStalk_eq_fromSpecStalk hxU, ← hV.fromSpecStalk_eq_fromSpecStalk hxV,
IsAffineOpen.fromSpecStalk, ← Spec.map_comp_assoc, Scheme.stalkMap_germ f _ x hxU,
IsAffineOpen.fromSpecStalk, Spec.map_comp_assoc, ← X.presheaf.germ_res (homOfLE hVU) x hxV,
Spec.map_comp_assoc, Category.assoc, ← Spec.map_comp_... | no goals | f19f7c7123ab26ca |
NumberField.mixedEmbedding.iUnion_negAt_plusPart_union | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | theorem iUnion_negAt_plusPart_union :
(⋃ s, negAt s '' (plusPart A)) ∪ (A ∩ (⋃ w, {x | x.1 w = 0})) = A | case h
K : Type u_1
inst✝ : Field K
A : Set (mixedSpace K)
hA : ∀ (x : mixedSpace K), x ∈ A ↔ (fun w => |x.1 w|, x.2) ∈ A
x : mixedSpace K
⊢ ((∃ i, x ∈ ⇑(negAt i) '' plusPart A) ∨ x ∈ A ∧ ∃ i, x ∈ {x | x.1 i = 0}) ↔ x ∈ A | refine ⟨?_, fun h ↦ ?_⟩ | case h.refine_1
K : Type u_1
inst✝ : Field K
A : Set (mixedSpace K)
hA : ∀ (x : mixedSpace K), x ∈ A ↔ (fun w => |x.1 w|, x.2) ∈ A
x : mixedSpace K
⊢ ((∃ i, x ∈ ⇑(negAt i) '' plusPart A) ∨ x ∈ A ∧ ∃ i, x ∈ {x | x.1 i = 0}) → x ∈ A
case h.refine_2
K : Type u_1
inst✝ : Field K
A : Set (mixedSpace K)
hA : ∀ (x : mixedSpa... | 618707fcf4bf60a6 |
MulAction.stabilizer_orbit_eq | Mathlib/GroupTheory/GroupAction/Blocks.lean | theorem stabilizer_orbit_eq {a : X} {H : Subgroup G} (hH : stabilizer G a ≤ H) :
stabilizer G (orbit H a) = H | case h.mp
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
⊢ g ∈ stabilizer G (orbit (↥H) a) → g ∈ H | intro hg | case h.mp
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g ∈ stabilizer G (orbit (↥H) a)
⊢ g ∈ H | 71a4cf416a3754d6 |
Real.invariant | Mathlib/NumberTheory/DiophantineApproximation/Basic.lean | theorem invariant : ContfracLegendre.Ass (fract ξ)⁻¹ v (u - ⌊ξ⌋ * v) | ξ : ℝ
u v : ℤ
hv : 2 ≤ v
h : ContfracLegendre.Ass ξ u v
huv : u - ⌊ξ⌋ * v = 1
hv₀' : 0 < 2 * ↑v - 1
⊢ (↑v * (2 * ↑v - 1))⁻¹ + (↑v)⁻¹ = 2 / (2 * ↑v - 1) | field_simp | ξ : ℝ
u v : ℤ
hv : 2 ≤ v
h : ContfracLegendre.Ass ξ u v
huv : u - ⌊ξ⌋ * v = 1
hv₀' : 0 < 2 * ↑v - 1
⊢ (↑v + ↑v * (2 * ↑v - 1)) * (2 * ↑v - 1) = 2 * (↑v * (2 * ↑v - 1) * ↑v) | 8eb0aea68cc1bde9 |
MeasureTheory.AEStronglyMeasurable.ae_integrable_condKernel_iff | Mathlib/Probability/Kernel/Disintegration/Integral.lean | theorem AEStronglyMeasurable.ae_integrable_condKernel_iff {f : α × Ω → F}
(hf : AEStronglyMeasurable f ρ) :
(∀ᵐ a ∂ρ.fst, Integrable (fun ω ↦ f (a, ω)) (ρ.condKernel a)) ∧
Integrable (fun a ↦ ∫ ω, ‖f (a, ω)‖ ∂ρ.condKernel a) ρ.fst ↔ Integrable f ρ | α : Type u_1
Ω : Type u_2
F : Type u_4
mα : MeasurableSpace α
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
inst✝¹ : NormedAddCommGroup F
ρ : Measure (α × Ω)
inst✝ : IsFiniteMeasure ρ
f : α × Ω → F
hf : AEStronglyMeasurable f ρ
⊢ (∀ᵐ (a : α) ∂ρ.fst, Integrable (fun ω => f (a, ω)) (ρ.condK... | rw [← ρ.disintegrate ρ.condKernel] at hf | α : Type u_1
Ω : Type u_2
F : Type u_4
mα : MeasurableSpace α
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
inst✝¹ : NormedAddCommGroup F
ρ : Measure (α × Ω)
inst✝ : IsFiniteMeasure ρ
f : α × Ω → F
hf : AEStronglyMeasurable f (ρ.fst ⊗ₘ ρ.condKernel)
⊢ (∀ᵐ (a : α) ∂ρ.fst, Integrable (fun ω... | 7bdc19fddcc9d6a0 |
CochainComplex.shiftFunctorZero_hom_app_f | Mathlib/Algebra/Homology/HomotopyCategory/Shift.lean | lemma shiftFunctorZero_hom_app_f (K : CochainComplex C ℤ) (n : ℤ) :
((CategoryTheory.shiftFunctorZero (CochainComplex C ℤ) ℤ).hom.app K).f n =
(K.XIsoOfEq (by dsimp; rw [add_zero])).hom | C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K : CochainComplex C ℤ
n : ℤ
⊢ IsIso (((shiftFunctorZero (CochainComplex C ℤ) ℤ).inv.app K).f n) | rw [shiftFunctorZero_inv_app_f] | C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K : CochainComplex C ℤ
n : ℤ
⊢ IsIso (XIsoOfEq K ⋯).hom | 3d3f0579f35bf4a8 |
CategoryTheory.OverPresheafAux.MakesOverArrow.of_yoneda_arrow | Mathlib/CategoryTheory/Comma/Presheaf/Basic.lean | lemma of_yoneda_arrow {Y : C} {η : yoneda.obj Y ⟶ A} {X : C} {s : yoneda.obj X ⟶ A} {f : X ⟶ Y}
(hf : yoneda.map f ≫ η = s) : MakesOverArrow η s f | C : Type u
inst✝ : Category.{v, u} C
A : Cᵒᵖ ⥤ Type v
Y : C
η : yoneda.obj Y ⟶ A
X : C
s : yoneda.obj X ⟶ A
f : X ⟶ Y
hf : yoneda.map f ≫ η = s
⊢ MakesOverArrow η s f | simpa only [yonedaEquiv_yoneda_map f] using of_arrow hf | no goals | 4a9c2a75a34c347c |
IsDiscreteValuationRing.addVal_eq_top_iff | Mathlib/RingTheory/DiscreteValuationRing/Basic.lean | theorem addVal_eq_top_iff {a : R} : addVal R a = ⊤ ↔ a = 0 | case mp.intro.intro
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsDiscreteValuationRing R
hi : Irreducible (Classical.choose ⋯)
n : ℕ
u : Rˣ
h : ¬Classical.choose ⋯ ^ n * ↑u = 0
ha : Associated (Classical.choose ⋯ ^ n * ↑u) (Classical.choose ⋯ ^ n)
⊢ ¬(addVal R) (Classical.choose ⋯ ^ n * ↑u) = ⊤ | rw [mul_comm, addVal_def' u hi n] | case mp.intro.intro
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsDiscreteValuationRing R
hi : Irreducible (Classical.choose ⋯)
n : ℕ
u : Rˣ
h : ¬Classical.choose ⋯ ^ n * ↑u = 0
ha : Associated (Classical.choose ⋯ ^ n * ↑u) (Classical.choose ⋯ ^ n)
⊢ ¬↑n = ⊤ | 0f5c0cf481b15329 |
controlled_prod_of_mem_closure | Mathlib/Analysis/Normed/Group/Continuity.lean | theorem controlled_prod_of_mem_closure {s : Subgroup E} (hg : a ∈ closure (s : Set E)) {b : ℕ → ℝ}
(b_pos : ∀ n, 0 < b n) :
∃ v : ℕ → E,
Tendsto (fun n => ∏ i ∈ range (n + 1), v i) atTop (𝓝 a) ∧
(∀ n, v n ∈ s) ∧ ‖v 0 / a‖ < b 0 ∧ ∀ n, 0 < n → ‖v n‖ < b n | E : Type u_5
inst✝ : SeminormedCommGroup E
a : E
s : Subgroup E
hg : a ∈ closure ↑s
b : ℕ → ℝ
b_pos : ∀ (n : ℕ), 0 < b n
u : ℕ → E
u_in : ∀ (n : ℕ), u n ∈ s
lim_u : Tendsto u atTop (𝓝 a)
n₀ : ℕ
hn₀ : ∀ n ≥ n₀, ‖u n / a‖ < b 0
z : ℕ → E := fun n => u (n + n₀)
lim_z : Tendsto z atTop (𝓝 a)
n : ℕ
⊢ {p | ‖p.1 / p.2‖ < b ... | simpa [← dist_eq_norm_div] using Metric.dist_mem_uniformity (b_pos <| n + 1) | no goals | 647a783958e68b9a |
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... | rw [tensorLeft_map] | 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 |
rothNumberNat_le_ruzsaSzemerediNumberNat | Mathlib/Combinatorics/Extremal/RuzsaSzemeredi.lean | lemma rothNumberNat_le_ruzsaSzemerediNumberNat (n : ℕ) :
(2 * n + 1) * rothNumberNat n ≤ ruzsaSzemerediNumberNat (6 * n + 3) | n : ℕ
α : Type := Fin (2 * n + 1)
this✝ : Coprime 2 (2 * n + 1)
this : Fact (IsUnit 2)
⊢ (2 * n + 1) * rothNumberNat n ≤ ruzsaSzemerediNumberNat (6 * n + 3) | calc
(2 * n + 1) * rothNumberNat n
_ = Fintype.card α * addRothNumber (Iio (n : α)) := by
rw [Fin.addRothNumber_eq_rothNumberNat le_rfl, Fintype.card_fin]
_ ≤ Fintype.card α * addRothNumber (univ : Finset α) := by
gcongr; exact subset_univ _
_ ≤ ruzsaSzemerediNumber (Sum α (Sum α α)) := addRothNumber_le... | no goals | f10db59dd468e568 |
μ_limsup_le_one | Mathlib/Analysis/Normed/Ring/SmoothingSeminorm.lean | theorem μ_limsup_le_one {s : ℕ → ℕ} (hs_le : ∀ n : ℕ, s n ≤ n) {x : R} {ψ : ℕ → ℕ}
(hψ_lim : Tendsto ((fun n : ℕ => ↑(s n) / (n : ℝ)) ∘ ψ) atTop (𝓝 0)) :
limsup (fun n : ℕ => μ x ^ ((s (ψ n) : ℝ) * (1 / (ψ n : ℝ)))) atTop ≤ 1 | case neg.h
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
s : ℕ → ℕ
hs_le : ∀ (n : ℕ), s n ≤ n
x : R
ψ : ℕ → ℕ
hψ_lim : Tendsto ((fun n => ↑(s n) / ↑n) ∘ ψ) atTop (𝓝 0)
c : ℝ
hc_bd : ∀ (x_1 : ℝ) (x_2 : ℕ), (∀ (b : ℕ), x_2 ≤ b → μ x ^ (↑(s (ψ b)) * (1 / ↑(ψ b))) ≤ x_1) → c ≤ x_1
hμx : ¬μ x < 1
hμ_lim : ∀ (U : Set ℝ... | intro ε hε | case neg.h
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
s : ℕ → ℕ
hs_le : ∀ (n : ℕ), s n ≤ n
x : R
ψ : ℕ → ℕ
hψ_lim : Tendsto ((fun n => ↑(s n) / ↑n) ∘ ψ) atTop (𝓝 0)
c : ℝ
hc_bd : ∀ (x_1 : ℝ) (x_2 : ℕ), (∀ (b : ℕ), x_2 ≤ b → μ x ^ (↑(s (ψ b)) * (1 / ↑(ψ b))) ≤ x_1) → c ≤ x_1
hμx : ¬μ x < 1
hμ_lim : ∀ (U : Set ℝ... | 56e579c9d7dd8cf5 |
Quiver.homOfEq_injective | Mathlib/Combinatorics/Quiver/Basic.lean | lemma homOfEq_injective {X X' Y Y' : V} (hX : X = X') (hY : Y = Y')
{f g : X ⟶ Y} (h : Quiver.homOfEq f hX hY = Quiver.homOfEq g hX hY) : f = g | V : Type u_1
inst✝ : Quiver V
X X' Y Y' : V
hX : X = X'
hY : Y = Y'
f g : X ⟶ Y
h : homOfEq f hX hY = homOfEq g hX hY
⊢ f = g | subst hX hY | V : Type u_1
inst✝ : Quiver V
X Y : V
f g : X ⟶ Y
h : homOfEq f ⋯ ⋯ = homOfEq g ⋯ ⋯
⊢ f = g | 20bc36261cec5cd4 |
DedekindDomain.ProdAdicCompletions.IsFiniteAdele.add | Mathlib/RingTheory/DedekindDomain/FiniteAdeleRing.lean | theorem add {x y : K_hat R K} (hx : x.IsFiniteAdele) (hy : y.IsFiniteAdele) :
(x + y).IsFiniteAdele | R : Type u_1
K : Type u_2
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
x y : K_hat R K
hx : {x_1 | x x_1 ∉ adicCompletionIntegers K x_1}.Finite
hy : {x | y x ∉ adicCompletionIntegers K x}.Finite
v : HeightOneSpectrum R
hv : Valued.v (x v) ⊔ Valued.v (y... | rw [mem_adicCompletionIntegers, Pi.add_apply] | R : Type u_1
K : Type u_2
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
x y : K_hat R K
hx : {x_1 | x x_1 ∉ adicCompletionIntegers K x_1}.Finite
hy : {x | y x ∉ adicCompletionIntegers K x}.Finite
v : HeightOneSpectrum R
hv : Valued.v (x v) ⊔ Valued.v (y... | 82c2f2870b8e321c |
Std.Tactic.BVDecide.BVExpr.bitblast.blastUdiv.denote_blastDivSubtractShift_q | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/Udiv.lean | theorem denote_blastDivSubtractShift_q (aig : AIG α) (assign : α → Bool) (lhs rhs : BitVec w)
(falseRef trueRef : AIG.Ref aig) (n d : AIG.RefVec aig w) (wn wr : Nat)
(q r : AIG.RefVec aig w) (qbv rbv : BitVec w)
(hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, n.get idx hidx, assign⟧ = lhs.getLsbD idx)
(h... | case hleft.hx
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
assign : α → Bool
lhs rhs : BitVec w
falseRef trueRef : aig.Ref
n d : aig.RefVec w
wn wr : Nat
q r : aig.RefVec w
qbv rbv : BitVec w
hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := n.get idx hidx }⟧ = lhs.getLsbD ... | simp [hr] | no goals | 11112d18501f0fd9 |
MeasurableSpace.generateMeasurableRec_omega1 | Mathlib/MeasureTheory/MeasurableSpace/Card.lean | theorem generateMeasurableRec_omega1 (s : Set (Set α)) :
generateMeasurableRec s (ω₁ : Ordinal.{v}) =
⋃ i < (ω₁ : Ordinal.{v}), generateMeasurableRec s i | α : Type u
s : Set (Set α)
t : Set α
ht : t ∈ generateMeasurableRec s (ω_ 1)
⊢ t ∈ ⋃ i, ⋃ (_ : i < ω_ 1), generateMeasurableRec s i | rw [mem_iUnion₂] | α : Type u
s : Set (Set α)
t : Set α
ht : t ∈ generateMeasurableRec s (ω_ 1)
⊢ ∃ i, ∃ (_ : i < ω_ 1), t ∈ generateMeasurableRec s i | 648bd35d385d7978 |
Polynomial.rootMultiplicity_C | Mathlib/Algebra/Polynomial/Div.lean | theorem rootMultiplicity_C (r a : R) : rootMultiplicity a (C r) = 0 | case inr
R : Type u
inst✝ : Ring R
r a : R
h✝ : Nontrivial R
⊢ rootMultiplicity a (C r) = 0 | rw [rootMultiplicity_eq_multiplicity] | case inr
R : Type u
inst✝ : Ring R
r a : R
h✝ : Nontrivial R
⊢ (if C r = 0 then 0 else multiplicity (X - C a) (C r)) = 0 | 409ebcc1f57e5060 |
inner_self_re_eq_norm | Mathlib/Analysis/InnerProductSpace/Basic.lean | theorem inner_self_re_eq_norm (x : E) : re ⟪x, x⟫ = ‖⟪x, x⟫‖ | 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x : E
⊢ re ⟪x, x⟫_𝕜 = ‖⟪x, x⟫_𝕜‖ | conv_rhs => rw [← inner_self_ofReal_re] | 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x : E
⊢ re ⟪x, x⟫_𝕜 = ‖↑(re ⟪x, x⟫_𝕜)‖ | a43961020d75372d |
MvPowerSeries.WithPiTopology.tendsto_pow_of_constantCoeff_nilpotent_iff | Mathlib/RingTheory/MvPowerSeries/PiTopology.lean | theorem tendsto_pow_of_constantCoeff_nilpotent_iff [CommRing R] [DiscreteTopology R] (f) :
Tendsto (fun n : ℕ => f ^ n) atTop (nhds 0) ↔
IsNilpotent (constantCoeff σ R f) | case intro
σ : Type u_1
R : Type u_2
inst✝² : TopologicalSpace R
inst✝¹ : CommRing R
inst✝ : DiscreteTopology R
f : MvPowerSeries σ R
h : Tendsto (fun n => f ^ n) atTop (nhds 0)
m : ℕ
hm : ∀ (b : ℕ), m ≤ b → (constantCoeff σ R) f ^ b = 0
⊢ IsNilpotent ((constantCoeff σ R) f) | use m | case h
σ : Type u_1
R : Type u_2
inst✝² : TopologicalSpace R
inst✝¹ : CommRing R
inst✝ : DiscreteTopology R
f : MvPowerSeries σ R
h : Tendsto (fun n => f ^ n) atTop (nhds 0)
m : ℕ
hm : ∀ (b : ℕ), m ≤ b → (constantCoeff σ R) f ^ b = 0
⊢ (constantCoeff σ R) f ^ m = 0 | 75b216f0c0004b82 |
LSeriesHasSum_congr | Mathlib/NumberTheory/LSeries/Basic.lean | lemma LSeriesHasSum_congr {f g : ℕ → ℂ} (s a : ℂ) (h : ∀ {n}, n ≠ 0 → f n = g n) :
LSeriesHasSum f s a ↔ LSeriesHasSum g s a | f g : ℕ → ℂ
s a : ℂ
h : ∀ {n : ℕ}, n ≠ 0 → f n = g n
⊢ LSeriesHasSum f s a ↔ LSeriesHasSum g s a | simp [LSeriesHasSum_iff, LSeriesSummable_congr s h, LSeries_congr s h] | no goals | 45512968302499ea |
cauchySeq_tendsto_of_isComplete | Mathlib/Topology/UniformSpace/Cauchy.lean | theorem cauchySeq_tendsto_of_isComplete [Preorder β] {K : Set α} (h₁ : IsComplete K)
{u : β → α} (h₂ : ∀ n, u n ∈ K) (h₃ : CauchySeq u) : ∃ v ∈ K, Tendsto u atTop (𝓝 v) :=
h₁ _ h₃ <| le_principal_iff.2 <| mem_map_iff_exists_image.2
⟨univ, univ_mem, by rwa [image_univ, range_subset_iff]⟩
| α : Type u
β : Type v
uniformSpace : UniformSpace α
inst✝ : Preorder β
K : Set α
h₁ : IsComplete K
u : β → α
h₂ : ∀ (n : β), u n ∈ K
h₃ : CauchySeq u
⊢ u '' univ ⊆ K | rwa [image_univ, range_subset_iff] | no goals | 7093a9f0c8d7815c |
MeasureTheory.ae_eq_of_forall_setIntegral_eq_of_sigmaFinite' | Mathlib/MeasureTheory/Function/ConditionalExpectation/Unique.lean | theorem ae_eq_of_forall_setIntegral_eq_of_sigmaFinite' (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn f s μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hfg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ ... | α : Type u_1
F' : Type u_3
m m0 : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
f g : α → F'
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn f s μ
hg_int_finite : ∀ (s : Set α), Measu... | rw [trim_measurableSet_eq hm hs] at hμs | α : Type u_1
F' : Type u_3
m m0 : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
f g : α → F'
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn f s μ
hg_int_finite : ∀ (s : Set α), Measu... | f2ee55d026c11cd8 |
SimpleGraph.Subgraph.image_coe_edgeSet_coe | Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | lemma image_coe_edgeSet_coe (G' : G.Subgraph) : Sym2.map (↑) '' G'.coe.edgeSet = G'.edgeSet | V : Type u
G : SimpleGraph V
G' : G.Subgraph
⊢ Sym2.map Subtype.val '' G'.coe.edgeSet = G'.edgeSet | rw [edgeSet_coe, Set.image_preimage_eq_iff] | V : Type u
G : SimpleGraph V
G' : G.Subgraph
⊢ G'.edgeSet ⊆ Set.range (Sym2.map Subtype.val) | 3392d1b2483292bb |
List.append_eq_appendTR | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Basic.lean | theorem append_eq_appendTR : @List.append = @appendTR | case h.h.h
α : Type u_1
as : List α
⊢ ∀ (x : List α), as.append x = as.appendTR x | intro bs | case h.h.h
α : Type u_1
as bs : List α
⊢ as.append bs = as.appendTR bs | aae20de0c8888126 |
norm_eq_iInf_iff_real_inner_le_zero | Mathlib/Analysis/InnerProductSpace/Projection.lean | theorem norm_eq_iInf_iff_real_inner_le_zero {K : Set F} (h : Convex ℝ K) {u : F} {v : F}
(hv : v ∈ K) : (‖u - v‖ = ⨅ w : K, ‖u - w‖) ↔ ∀ w ∈ K, ⟪u - v, w - v⟫_ℝ ≤ 0 | F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Set F
h : Convex ℝ K
u v : F
hv : v ∈ K
this✝ : Nonempty ↑K := Nonempty.intro ⟨v, hv⟩
eq : ‖u - v‖ = ⨅ w, ‖u - ↑w‖
w : F
hw : w ∈ K
δ : ℝ := ⨅ w, ‖u - ↑w‖
p : ℝ := ⟪u - v, w - v⟫_ℝ
q : ℝ := ‖w - v‖ ^ 2
δ_le : ∀ (w : ↑K), δ ≤ ‖u - ↑w‖
δ_le' : ∀... | have := this (1 : ℝ) (by norm_num) (by norm_num) | F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Set F
h : Convex ℝ K
u v : F
hv : v ∈ K
this✝¹ : Nonempty ↑K := Nonempty.intro ⟨v, hv⟩
eq : ‖u - v‖ = ⨅ w, ‖u - ↑w‖
w : F
hw : w ∈ K
δ : ℝ := ⨅ w, ‖u - ↑w‖
p : ℝ := ⟪u - v, w - v⟫_ℝ
q : ℝ := ‖w - v‖ ^ 2
δ_le : ∀ (w : ↑K), δ ≤ ‖u - ↑w‖
δ_le' : ... | 2a92713ac13db1c9 |
TopCat.pullback_snd_range | Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean | theorem pullback_snd_range {X Y S : TopCat} (f : X ⟶ S) (g : Y ⟶ S) :
Set.range (pullback.snd f g) = { y : Y | ∃ x : X, f x = g y } | case h.mpr.intro
X Y S : TopCat
f : X ⟶ S
g : Y ⟶ S
y : ↑Y
x : ↑X
eq : (ConcreteCategory.hom f) x = (ConcreteCategory.hom g) y
⊢ y ∈ Set.range ⇑(ConcreteCategory.hom (pullback.snd f g)) | use (TopCat.pullbackIsoProdSubtype f g).inv ⟨⟨x, y⟩, eq⟩ | case h
X Y S : TopCat
f : X ⟶ S
g : Y ⟶ S
y : ↑Y
x : ↑X
eq : (ConcreteCategory.hom f) x = (ConcreteCategory.hom g) y
⊢ (ConcreteCategory.hom (pullback.snd f g)) ((ConcreteCategory.hom (pullbackIsoProdSubtype f g).inv) ⟨(x, y), eq⟩) = y | fbd3a97b78944df8 |
TopologicalSpace.NoetherianSpace.exists_open_ne_empty_le_irreducibleComponent | Mathlib/Topology/NoetherianSpace.lean | theorem NoetherianSpace.exists_open_ne_empty_le_irreducibleComponent [NoetherianSpace α]
(Z : Set α) (H : Z ∈ irreducibleComponents α) :
∃ o : Set α, IsOpen o ∧ o ≠ ∅ ∧ o ≤ Z | α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : NoetherianSpace α
Z : Set α
H : Z ∈ irreducibleComponents α
ι : Set (Set α) := irreducibleComponents α \ {Z}
hι : ι.Finite
hι' : Finite ↑ι
U : Set α := Z \ ⋃ x, ↑x
r : U = ∅
⊢ Z ⊆ ⋃₀ ↑hι.toFinset | rw [Set.Finite.coe_toFinset, Set.sUnion_eq_iUnion] | α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : NoetherianSpace α
Z : Set α
H : Z ∈ irreducibleComponents α
ι : Set (Set α) := irreducibleComponents α \ {Z}
hι : ι.Finite
hι' : Finite ↑ι
U : Set α := Z \ ⋃ x, ↑x
r : U = ∅
⊢ Z ⊆ ⋃ i, ↑i | 2cef9854bab7d94c |
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