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
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|---|---|---|---|---|---|---|
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_neg (mt Discrete.ext hi.symm)).symm
|
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 ∈ map (↑(f.extend I)) (𝓝 y)
|
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 = j
⊢ CompleteSpace ((i : ι) × E 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_comm]; rfl⟩, ⟨((i : R) : R_hat R K), rfl⟩⟩⟩
leftMul a r
|
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 x x✝ * (f (x + 1) x✝ - f x x✝) =
(∑ k ∈ Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)) x✝
|
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
εpos : 0 < ε
δ : (E →L[ℝ] E) → ℝ≥0
hδ :
∀ (A : E →L[ℝ] E),
0 < δ A ∧
(∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |B.det - A.det| ≤ ↑ε) ∧
∀ (t : Set E) (g : E → E),
ApproximatesLinearOn g A t (δ A) → ENNReal.ofReal |A.det| * μ t ≤ μ (g '' t) + ↑ε * μ t
t : ℕ → Set E
A : ℕ → E →L[ℝ] E
t_disj : Pairwise (Disjoint on t)
t_meas : ∀ (n : ℕ), MeasurableSet (t n)
t_cover : s ⊆ ⋃ n, t n
ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n))
s_eq : s = ⋃ n, s ∩ t n
⊢ ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| ∂μ = ∑' (n : ℕ), ∫⁻ (x : E) in s ∩ t n, ENNReal.ofReal |(f' x).det| ∂μ
|
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
εpos : 0 < ε
δ : (E →L[ℝ] E) → ℝ≥0
hδ :
∀ (A : E →L[ℝ] E),
0 < δ A ∧
(∀ (B : E →L[ℝ] E), ‖B - A‖ ≤ ↑(δ A) → |B.det - A.det| ≤ ↑ε) ∧
∀ (t : Set E) (g : E → E),
ApproximatesLinearOn g A t (δ A) → ENNReal.ofReal |A.det| * μ t ≤ μ (g '' t) + ↑ε * μ t
t : ℕ → Set E
A : ℕ → E →L[ℝ] E
t_disj : Pairwise (Disjoint on t)
t_meas : ∀ (n : ℕ), MeasurableSet (t n)
t_cover : s ⊆ ⋃ n, t n
ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) (δ (A n))
s_eq : s = ⋃ n, s ∩ t n
⊢ ∫⁻ (x : E) in ⋃ n, s ∩ t n, ENNReal.ofReal |(f' x).det| ∂μ =
∑' (n : ℕ), ∫⁻ (x : E) in s ∩ t n, ENNReal.ofReal |(f' x).det| ∂μ
|
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 : RatFunc Fq ≃ₐ[Fq[X]] Fqt := IsLocalization.algEquiv Fq[X]⁰ (RatFunc Fq) Fqt
c : RatFunc Fq
x : F
⊢ e c • x = c • x
|
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 : RatFunc Fq ≃ₐ[Fq[X]] Fqt := IsLocalization.algEquiv Fq[X]⁰ (RatFunc Fq) Fqt
c : RatFunc Fq
x : F
⊢ (algebraMap Fqt F) (e c) * x = (algebraMap (RatFunc Fq) F) c * x
|
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] ↑↑(condExpL1 hm μ f)
|
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 ↑↑(condExpL1 hm μ f) ⋯) =ᶠ[ae μ] ↑↑(condExpL1 hm μ f)
case neg
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 ↑↑(condExpL1 hm μ f) ⋯) =ᶠ[ae μ] ↑↑(condExpL1 hm μ f)
|
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.fst, κ x = ρ.condKernel x
|
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 × ℝ) := Measure.map (Prod.map id f) ρ
hρ'def : ρ' = Measure.map (Prod.map id f) ρ
⊢ ∀ᵐ (x : α) ∂ρ.fst, κ x = ρ.condKernel x
|
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
⊢ IsWeightedHomogeneous w ((monomial (m - single i 1)) ↑(m i)) n'
|
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) (m - single i 1) = n'
|
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 leadingCoeff t).prod
|
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).Nonempty ∧ δ a ≤ τ * δ c}
u : Set ι
hu : Maximal (fun x => x ∈ T) u
a : ι
hat : a ∈ t
hcon : ∀ b ∈ u, (B a ∩ B b).Nonempty → τ * δ b < δ a
a_disj : ∀ c ∈ u, Disjoint (B a) (B c)
A : Set ι := {a' | a' ∈ t ∧ ∀ c ∈ u, Disjoint (B a') (B c)}
Anonempty : A.Nonempty
m : ℝ := sSup (δ '' A)
bddA : BddAbove (δ '' A)
this : 0 ≤ m
mpos : 0 < m
I : m / τ < m
x : ℝ
xA : x ∈ δ '' A
hx : m / τ < x
⊢ ∃ a' ∈ A, m / τ ≤ δ a'
|
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 ∩ B c).Nonempty ∧ δ a ≤ τ * δ c}
u : Set ι
hu : Maximal (fun x => x ∈ T) u
a : ι
hat : a ∈ t
hcon : ∀ b ∈ u, (B a ∩ B b).Nonempty → τ * δ b < δ a
a_disj : ∀ c ∈ u, Disjoint (B a) (B c)
A : Set ι := {a' | a' ∈ t ∧ ∀ c ∈ u, Disjoint (B a') (B c)}
Anonempty : A.Nonempty
m : ℝ := sSup (δ '' A)
bddA : BddAbove (δ '' A)
this : 0 ≤ m
mpos : 0 < m
I : m / τ < m
a' : ι
ha' : a' ∈ A
xA : δ a' ∈ δ '' A
hx : m / τ < δ a'
⊢ ∃ a' ∈ A, m / τ ≤ δ 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 : ∀ (__do_lift : β), monotone fun x => Array.foldlM.loop (f x) xs stop h i'✝ (j✝ + 1) __do_lift
⊢ monotone fun x => f x b✝ xs[j✝]
|
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 : ∀ (__do_lift : β), monotone fun x => Array.foldlM.loop (f x) xs stop h i'✝ (j✝ + 1) __do_lift
⊢ monotone fun x => f x b✝
|
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 ⊻ t, f₄ a
|
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 α
L : Sublattice α := { carrier := latticeClosure (↑s✝ ∪ ↑t✝), supClosed' := ⋯, infClosed' := ⋯ }
this : Finite ↥L
t' : Finset ↥L
ht' : map { toFun := ⇑L.subtype, inj' := ⋯ } t' = t✝
s' : Finset ↥L
hs' : map { toFun := ⇑L.subtype, inj' := ⋯ } s' = s✝
β : Type u_1
w✝¹ : DecidableEq β
w✝ : Fintype β
g : LatticeHom (↥L) (Finset β)
hg : Injective ⇑g
s t : Finset β
hs : ¬∃ a, g a = s
⊢ extend (⇑g) (f₁ ∘ Subtype.val) 0 s * extend (⇑g) (f₂ ∘ Subtype.val) 0 t ≤
extend (⇑g) (f₃ ∘ Subtype.val) 0 (s ∩ t) * extend (⇑g) (f₄ ∘ Subtype.val) 0 (s ∪ t)
|
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) = n
⊢ n = 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 neg
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 ∧ 0 = n
⊢ n = 0
|
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
((NatTrans.mapHomologicalComplex α _).app P.cocomplex) ≫
(P.isoRightDerivedToHomotopyCategoryObj G).inv
|
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 : (HomotopyCategory.quotient C (ComplexShape.up ℕ)).map β = P.iso.hom
⊢ (HomotopyCategory.quotient D (ComplexShape.up ℕ)).map
((NatTrans.mapHomologicalComplex α (ComplexShape.up ℕ)).app ((injectiveResolutions C).obj X).as) ≫
(G.mapHomotopyCategory (ComplexShape.up ℕ)).map P.iso.hom =
(F.mapHomotopyCategory (ComplexShape.up ℕ)).map P.iso.hom ≫
(HomotopyCategory.quotient D (ComplexShape.up ℕ)).map
((NatTrans.mapHomologicalComplex α (ComplexShape.up ℕ)).app P.cocomplex)
|
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 : (HomotopyCategory.quotient C (ComplexShape.up ℕ)).map β = P.iso.hom
⊢ (HomotopyCategory.quotient D (ComplexShape.up ℕ)).map
((NatTrans.mapHomologicalComplex α (ComplexShape.up ℕ)).app ((injectiveResolutions C).obj X).as) ≫
(G.mapHomotopyCategory (ComplexShape.up ℕ)).map ((HomotopyCategory.quotient C (ComplexShape.up ℕ)).map β) =
(F.mapHomotopyCategory (ComplexShape.up ℕ)).map ((HomotopyCategory.quotient C (ComplexShape.up ℕ)).map β) ≫
(HomotopyCategory.quotient D (ComplexShape.up ℕ)).map
((NatTrans.mapHomologicalComplex α (ComplexShape.up ℕ)).app P.cocomplex)
|
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 =
∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (g t)
|
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), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)
s : ℝ
aux₁ :
(fun x => (Ioc 0 (f x)).indicator (fun t => ENNReal.ofReal (g t)) s) = fun x =>
ENNReal.ofReal (g s) * (Ioi 0).indicator (fun x => 1) s * (Ici s).indicator (fun x => 1) (f x)
⊢ ENNReal.ofReal (g s) * (Ioi 0).indicator (fun x => 1) s * ∫⁻ (a : α), {a | s ≤ f a}.indicator (fun x => 1) a ∂μ =
(Ioi 0).indicator (fun t => μ {a | t ≤ f a} * ENNReal.ofReal (g t)) s
|
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), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)
s : ℝ
aux₁ :
(fun x => (Ioc 0 (f x)).indicator (fun t => ENNReal.ofReal (g t)) s) = fun x =>
ENNReal.ofReal (g s) * (Ioi 0).indicator (fun x => 1) s * (Ici s).indicator (fun x => 1) (f x)
⊢ ENNReal.ofReal (g s) * (Ioi 0).indicator (fun x => 1) s * ∫⁻ (a : α) in {a | s ≤ f a}, 1 ∂μ =
(Ioi 0).indicator (fun t => μ {a | t ≤ f a} * ENNReal.ofReal (g t)) s
case h.h.hs
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), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)
s : ℝ
aux₁ :
(fun x => (Ioc 0 (f x)).indicator (fun t => ENNReal.ofReal (g t)) s) = fun x =>
ENNReal.ofReal (g s) * (Ioi 0).indicator (fun x => 1) s * (Ici s).indicator (fun x => 1) (f x)
⊢ NullMeasurableSet {a | s ≤ f a} μ
|
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 : IsUniformInducing ι
di : IsDenseInducing ι := IsUniformInducing.isDenseInducing ui ?refine_1
extψ : ↥K.topologicalClosure →+* ℂ := IsDenseInducing.extendRingHom ui ⋯ hc
hψ : Continuous (⋯.extend ⇑ψ)
h✝ : K.topologicalClosure = ⊤
ψ₁ : ℂ →+* ℂ := extψ.comp ((RingEquiv.subfieldCongr h✝).symm.toRingHom.comp Subfield.topEquiv.symm.toRingHom)
hψ₁ : Continuous ⇑ψ₁
h : ψ₁ = RingHom.id ℂ
z : ↥K
⊢ (↑↑ψ).toFun z = ψ₁ ↑z
|
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.rnDerivAux (κ + η) a x)) ≤ᶠ[ae ((κ + η) a)] fun x => 1
|
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 + η) a x)) ≤ᶠ[ae ((κ + η) a)] fun x => 1
|
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'
inst✝⁸ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁷ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁶ : TopologicalSpace M'
inst✝⁵ : ChartedSpace H' M'
E'' : Type u_8
inst✝⁴ : NormedAddCommGroup E''
inst✝³ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝² : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝¹ : TopologicalSpace M''
inst✝ : ChartedSpace H'' M''
e : PartialHomeomorph M M'
he : MDifferentiable I I' e
e' : PartialHomeomorph M' M''
he' : MDifferentiable I' I'' e'
x : M''
hx : x ∈ (e ≫ₕ e').target
⊢ MDifferentiableWithinAt I'' I (↑(e ≫ₕ e').symm) (e ≫ₕ e').target x
|
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'
inst✝⁸ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁷ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁶ : TopologicalSpace M'
inst✝⁵ : ChartedSpace H' M'
E'' : Type u_8
inst✝⁴ : NormedAddCommGroup E''
inst✝³ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝² : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝¹ : TopologicalSpace M''
inst✝ : ChartedSpace H'' M''
e : PartialHomeomorph M M'
he : MDifferentiable I I' e
e' : PartialHomeomorph M' M''
he' : MDifferentiable I' I'' e'
x : M''
hx : x ∈ e'.target ∧ ↑e'.symm x ∈ e.target
⊢ MDifferentiableWithinAt I'' I (↑(e ≫ₕ e').symm) (e ≫ₕ e').target x
|
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 (F.map (homOfLE h).op)) x) =
(ConcreteCategory.hom (G.map (homOfLE h).op)) ((ConcreteCategory.hom (e.app (op V))) x)
|
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 : ↑V ⊆ (f ⁻¹ᵁ U).carrier
⊢ Spec.map (Hom.stalkMap f x) ≫ Y.fromSpecStalk ((ConcreteCategory.hom f.base) x) = X.fromSpecStalk x ≫ f
|
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_assoc (f.app _),
Hom.app_eq_appLE, Hom.appLE_map, IsAffineOpen.Spec_map_appLE_fromSpec]
|
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 : mixedSpace K), x ∈ A ↔ (fun w => |x.1 w|, x.2) ∈ A
x : mixedSpace K
h : x ∈ A
⊢ (∃ i, x ∈ ⇑(negAt i) '' plusPart A) ∨ x ∈ A ∧ ∃ i, x ∈ {x | x.1 i = 0}
|
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.condKernel a)) ∧
Integrable (fun a => ∫ (ω : Ω), ‖f (a, ω)‖ ∂ρ.condKernel a) ρ.fst ↔
Integrable f ρ
|
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 => f (a, ω)) (ρ.condKernel a)) ∧
Integrable (fun a => ∫ (ω : Ω), ‖f (a, ω)‖ ∂ρ.condKernel a) ρ.fst ↔
Integrable f ρ
|
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 (n + 1)} ∈ 𝓤 E
|
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 : Bimod X Y
P : Bimod Y Z
⊢ (tensorLeft M.X).map (coequalizer.π (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft)) ≫
f.hom ▷ coequalizer (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) ≫
colimit.ι
(parallelPair (M'.actRight ▷ coequalizer (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft))
((α_ M'.X X.X (coequalizer (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft))).hom ≫
M'.X ◁ TensorBimod.actLeft N P))
WalkingParallelPair.one =
(tensorLeft M.X).map (coequalizer.π (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft)) ≫
(((PreservesCoequalizer.iso (tensorLeft M.X) (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X P.X).inv ≫
coequalizer.π (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) ▷ P.X ≫
coequalizer.π (TensorBimod.actRight M N ▷ P.X)
((α_ (TensorBimod.X M N) Y.X P.X).hom ≫ TensorBimod.X M N ◁ P.actLeft))
⋯) ≫
colimMap
(parallelPairHom (TensorBimod.actRight M N ▷ P.X)
((α_ (coequalizer (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft)) Y.X P.X).hom ≫
coequalizer (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) ◁ P.actLeft)
(TensorBimod.actRight M' N ▷ P.X)
((α_ (coequalizer (M'.actRight ▷ N.X) ((α_ M'.X X.X N.X).hom ≫ M'.X ◁ N.actLeft)) Y.X P.X).hom ≫
coequalizer (M'.actRight ▷ N.X) ((α_ M'.X X.X N.X).hom ≫ M'.X ◁ N.actLeft) ◁ P.actLeft)
(colimMap
(parallelPairHom (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) (M'.actRight ▷ N.X)
((α_ M'.X X.X N.X).hom ≫ M'.X ◁ N.actLeft) (f.hom ▷ X.X ▷ N.X) (f.hom ▷ N.X) ⋯ ⋯) ▷
Y.X ▷
P.X)
(colimMap
(parallelPairHom (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) (M'.actRight ▷ N.X)
((α_ M'.X X.X N.X).hom ≫ M'.X ◁ N.actLeft) (f.hom ▷ X.X ▷ N.X) (f.hom ▷ N.X) ⋯ ⋯) ▷
P.X)
⋯ ⋯)) ≫
coequalizer.desc (AssociatorBimod.homAux M' N P) ⋯
|
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 : Bimod X Y
P : Bimod Y Z
⊢ M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) ≫
f.hom ▷ coequalizer (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) ≫
colimit.ι
(parallelPair (M'.actRight ▷ coequalizer (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft))
((α_ M'.X X.X (coequalizer (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft))).hom ≫
M'.X ◁ TensorBimod.actLeft N P))
WalkingParallelPair.one =
M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) ≫
(((PreservesCoequalizer.iso (tensorLeft M.X) (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
coequalizer.desc
((α_ M.X N.X P.X).inv ≫
coequalizer.π (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) ▷ P.X ≫
coequalizer.π (TensorBimod.actRight M N ▷ P.X)
((α_ (TensorBimod.X M N) Y.X P.X).hom ≫ TensorBimod.X M N ◁ P.actLeft))
⋯) ≫
colimMap
(parallelPairHom (TensorBimod.actRight M N ▷ P.X)
((α_ (coequalizer (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft)) Y.X P.X).hom ≫
coequalizer (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) ◁ P.actLeft)
(TensorBimod.actRight M' N ▷ P.X)
((α_ (coequalizer (M'.actRight ▷ N.X) ((α_ M'.X X.X N.X).hom ≫ M'.X ◁ N.actLeft)) Y.X P.X).hom ≫
coequalizer (M'.actRight ▷ N.X) ((α_ M'.X X.X N.X).hom ≫ M'.X ◁ N.actLeft) ◁ P.actLeft)
(colimMap
(parallelPairHom (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) (M'.actRight ▷ N.X)
((α_ M'.X X.X N.X).hom ≫ M'.X ◁ N.actLeft) (f.hom ▷ X.X ▷ N.X) (f.hom ▷ N.X) ⋯ ⋯) ▷
Y.X ▷
P.X)
(colimMap
(parallelPairHom (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) (M'.actRight ▷ N.X)
((α_ M'.X X.X N.X).hom ≫ M'.X ◁ N.actLeft) (f.hom ▷ X.X ▷ N.X) (f.hom ▷ N.X) ⋯ ⋯) ▷
P.X)
⋯ ⋯)) ≫
coequalizer.desc (AssociatorBimod.homAux M' N P) ⋯
|
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_ruzsaSzemerediNumber _
_ = ruzsaSzemerediNumberNat (6 * n + 3) := by
simp_rw [← ruzsaSzemerediNumberNat_card, Fintype.card_sum, α, Fintype.card_fin]
ring_nf
|
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), 1 ∈ U → IsOpen U → ∃ N, ∀ (n : ℕ), N ≤ n → μ x ^ (↑(s (ψ n)) * (1 / ↑(ψ n))) ∈ U
⊢ ∀ (ε : ℝ), 0 < ε → c ≤ 1 + ε
|
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), 1 ∈ U → IsOpen U → ∃ N, ∀ (n : ℕ), N ≤ n → μ x ^ (↑(s (ψ n)) * (1 / ↑(ψ n))) ∈ U
ε : ℝ
hε : 0 < ε
⊢ c ≤ 1 + ε
|
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 v) ≤ 1
⊢ (x + y) v ∈ adicCompletionIntegers K v
|
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 v) ≤ 1
⊢ Valued.v (x v + y v) ≤ 1
|
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)
(hright : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, d.get idx hidx, assign⟧ = rhs.getLsbD idx)
(hq : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, q.get idx hidx, assign⟧ = qbv.getLsbD idx)
(hr : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, r.get idx hidx, assign⟧ = rbv.getLsbD idx)
(hfalse : ⟦aig, falseRef, assign⟧ = false)
(htrue : ⟦aig, trueRef, assign⟧ = true)
:
∀ (idx : Nat) (hidx : idx < w),
⟦
(blastDivSubtractShift aig falseRef trueRef n d wn wr q r).aig,
(blastDivSubtractShift aig falseRef trueRef n d wn wr q r).q.get idx hidx,
assign
⟧
=
(BitVec.divSubtractShift { n := lhs, d := rhs } { wn := wn, wr := wr, q := qbv, r := rbv }).q.getLsbD idx
|
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 idx
hright : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := d.get idx hidx }⟧ = rhs.getLsbD idx
hq : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := q.get idx hidx }⟧ = qbv.getLsbD idx
hr : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := r.get idx hidx }⟧ = rbv.getLsbD idx
hfalse : ⟦assign, { aig := aig, ref := falseRef }⟧ = false
htrue : ⟦assign, { aig := aig, ref := trueRef }⟧ = true
idx✝ : Nat
hidx✝ : idx✝ < w
idx : Nat
hidx : idx < w
⊢ ∀ (idx : Nat) (hidx : idx < w),
⟦assign, { aig := aig, ref := { lhs := r, bit := n.getD (wn - 1) falseRef }.lhs.get idx hidx }⟧ = rbv.getLsbD idx
|
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 < ∞ → ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ)
(hfm : AEStronglyMeasurable[m] f μ) (hgm : AEStronglyMeasurable[m] g μ) : f =ᵐ[μ] g
|
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), MeasurableSet s → μ s < ⊤ → IntegrableOn g s μ
hfg_eq : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in s, g x ∂μ
hfm : AEStronglyMeasurable f μ
hgm : AEStronglyMeasurable g μ
s : Set α
hs : MeasurableSet s
hμs : (μ.trim hm) s < ⊤
⊢ IntegrableOn (AEStronglyMeasurable.mk f hfm) s (μ.trim hm)
|
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), MeasurableSet s → μ s < ⊤ → IntegrableOn g s μ
hfg_eq : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in s, g x ∂μ
hfm : AEStronglyMeasurable f μ
hgm : AEStronglyMeasurable g μ
s : Set α
hs : MeasurableSet s
hμs : μ s < ⊤
⊢ IntegrableOn (AEStronglyMeasurable.mk f hfm) s (μ.trim hm)
|
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' : ∀ w ∈ K, δ ≤ ‖u - w‖
this : ∀ (θ : ℝ), 0 < θ → θ ≤ 1 → 2 * p ≤ θ * 0
hq : q = 0
⊢ p ≤ 0
|
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' : ∀ w ∈ K, δ ≤ ‖u - w‖
this✝ : ∀ (θ : ℝ), 0 < θ → θ ≤ 1 → 2 * p ≤ θ * 0
hq : q = 0
this : 2 * p ≤ 1 * 0
⊢ p ≤ 0
|
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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