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
|
|---|---|---|---|---|---|---|
Real.tendsto_eulerMascheroniSeq'
|
Mathlib/NumberTheory/Harmonic/EulerMascheroni.lean
|
lemma tendsto_eulerMascheroniSeq' :
Tendsto eulerMascheroniSeq' atTop (𝓝 eulerMascheroniConstant)
|
⊢ Tendsto (fun n => eulerMascheroniSeq' n - eulerMascheroniSeq n) atTop (𝓝 0)
|
suffices Tendsto (fun x : ℝ ↦ log (x + 1) - log x) atTop (𝓝 0) by
apply (this.comp tendsto_natCast_atTop_atTop).congr'
filter_upwards [eventually_ne_atTop 0] with n hn
simp [eulerMascheroniSeq, eulerMascheroniSeq', eq_false_intro hn]
|
⊢ Tendsto (fun x => log (x + 1) - log x) atTop (𝓝 0)
|
cc6480414e009cf7
|
String.Iterator.ValidFor.remainingToString
|
Mathlib/.lake/packages/batteries/Batteries/Data/String/Lemmas.lean
|
theorem remainingToString {it} (h : ValidFor l r it) : it.remainingToString = ⟨r⟩
|
l r : List Char
it : Iterator
h : ValidFor l r it
⊢ it.remainingToString = { data := r }
|
cases h.out
|
case refl
l r : List Char
h : ValidFor l r { s := { data := l.reverseAux r }, i := { byteIdx := utf8Len l } }
⊢ { s := { data := l.reverseAux r }, i := { byteIdx := utf8Len l } }.remainingToString = { data := r }
|
782b70dd556d5667
|
RootPairing.polarization_apply_eq_zero_iff
|
Mathlib/LinearAlgebra/RootSystem/Finite/CanonicalBilinear.lean
|
lemma polarization_apply_eq_zero_iff (m : M) :
P.Polarization m = 0 ↔ P.RootForm m = 0
|
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : AddCommGroup N
inst✝¹ : Module R N
P : RootPairing ι R M N
inst✝ : Fintype ι
m : M
⊢ P.Polarization m = 0 ↔ P.RootForm m = 0
|
rw [← flip_comp_polarization_eq_rootForm]
|
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : AddCommGroup N
inst✝¹ : Module R N
P : RootPairing ι R M N
inst✝ : Fintype ι
m : M
⊢ P.Polarization m = 0 ↔ (P.flip.toLin ∘ₗ P.Polarization) m = 0
|
8405c462392e53d5
|
BddAbove.continuous_convolution_right_of_integrable
|
Mathlib/Analysis/Convolution.lean
|
theorem _root_.BddAbove.continuous_convolution_right_of_integrable
[FirstCountableTopology G] [SecondCountableTopologyEither G E']
(hbg : BddAbove (range fun x => ‖g x‖)) (hf : Integrable f μ) (hg : Continuous g) :
Continuous (f ⋆[L, μ] g)
|
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedAddCommGroup E'
inst✝¹² : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : NormedSpace 𝕜 E'
inst✝⁸ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝⁷ : MeasurableSpace G
μ : Measure G
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : AddGroup G
inst✝⁴ : TopologicalSpace G
inst✝³ : IsTopologicalAddGroup G
inst✝² : BorelSpace G
inst✝¹ : FirstCountableTopology G
inst✝ : SecondCountableTopologyEither G E'
hbg : BddAbove (range fun x => ‖g x‖)
hf : Integrable f μ
hg : Continuous g
x₀ : G
⊢ ContinuousAt (f ⋆[L, μ] g) x₀
|
have : ∀ᶠ x in 𝓝 x₀, ∀ᵐ t : G ∂μ, ‖L (f t) (g (x - t))‖ ≤ ‖L‖ * ‖f t‖ * ⨆ i, ‖g i‖ := by
filter_upwards with x; filter_upwards with t
apply_rules [L.le_of_opNorm₂_le_of_le, le_rfl, le_ciSup hbg (x - t)]
|
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedAddCommGroup E'
inst✝¹² : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : NormedSpace 𝕜 E'
inst✝⁸ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝⁷ : MeasurableSpace G
μ : Measure G
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : AddGroup G
inst✝⁴ : TopologicalSpace G
inst✝³ : IsTopologicalAddGroup G
inst✝² : BorelSpace G
inst✝¹ : FirstCountableTopology G
inst✝ : SecondCountableTopologyEither G E'
hbg : BddAbove (range fun x => ‖g x‖)
hf : Integrable f μ
hg : Continuous g
x₀ : G
this : ∀ᶠ (x : G) in 𝓝 x₀, ∀ᵐ (t : G) ∂μ, ‖(L (f t)) (g (x - t))‖ ≤ ‖L‖ * ‖f t‖ * ⨆ i, ‖g i‖
⊢ ContinuousAt (f ⋆[L, μ] g) x₀
|
7cef45832d5d6aeb
|
MeasureTheory.Measure.Regular.restrict_of_measure_ne_top
|
Mathlib/MeasureTheory/Measure/Regular.lean
|
theorem restrict_of_measure_ne_top [R1Space α] [BorelSpace α] [Regular μ]
{A : Set α} (h'A : μ A ≠ ∞) : Regular (μ.restrict A)
|
case innerRegular
α : Type u_1
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : TopologicalSpace α
inst✝² : R1Space α
inst✝¹ : BorelSpace α
inst✝ : μ.Regular
A : Set α
h'A : μ A ≠ ⊤
this : (μ.restrict A).WeaklyRegular
⊢ (μ.restrict A).InnerRegularWRT IsCompact IsOpen
|
intro V hV r hr
|
case innerRegular
α : Type u_1
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : TopologicalSpace α
inst✝² : R1Space α
inst✝¹ : BorelSpace α
inst✝ : μ.Regular
A : Set α
h'A : μ A ≠ ⊤
this : (μ.restrict A).WeaklyRegular
V : Set α
hV : IsOpen V
r : ℝ≥0∞
hr : r < (μ.restrict A) V
⊢ ∃ K ⊆ V, IsCompact K ∧ r < (μ.restrict A) K
|
99942bfc62b2d596
|
HolorIndex.drop_drop
|
Mathlib/Data/Holor.lean
|
theorem drop_drop : ∀ t : HolorIndex (ds₁ ++ ds₂ ++ ds₃), t.assocRight.drop.drop = t.drop
| ⟨is, h⟩ => Subtype.eq (by simp [add_comm, assocRight, drop, cast_type, List.drop_drop])
|
ds₁ ds₂ ds₃ is : List ℕ
h : Forall₂ (fun x1 x2 => x1 < x2) is (ds₁ ++ ds₂ ++ ds₃)
⊢ ↑(assocRight ⟨is, h⟩).drop.drop = ↑(drop ⟨is, h⟩)
|
simp [add_comm, assocRight, drop, cast_type, List.drop_drop]
|
no goals
|
82d0a6c432db2ef0
|
smul_finprod'
|
Mathlib/Algebra/BigOperators/GroupWithZero/Action.lean
|
theorem smul_finprod' {ι : Sort*} [Finite ι] {f : ι → β} (r : α) :
r • ∏ᶠ x : ι, f x = ∏ᶠ x : ι, r • (f x)
|
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : CommMonoid β
inst✝¹ : MulDistribMulAction α β
ι : Sort u_4
inst✝ : Finite ι
f : ι → β
r : α
⊢ r • ∏ᶠ (x : ι), f x = ∏ᶠ (x : ι), r • f x
|
cases nonempty_fintype (PLift ι)
|
case intro
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : CommMonoid β
inst✝¹ : MulDistribMulAction α β
ι : Sort u_4
inst✝ : Finite ι
f : ι → β
r : α
val✝ : Fintype (PLift ι)
⊢ r • ∏ᶠ (x : ι), f x = ∏ᶠ (x : ι), r • f x
|
c39a4be152c4f848
|
Cycle.support_formPerm
|
Mathlib/GroupTheory/Perm/Cycle/Concrete.lean
|
theorem support_formPerm [Fintype α] (s : Cycle α) (h : Nodup s) (hn : Nontrivial s) :
support (formPerm s h) = s.toFinset
|
case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : List α
h : Nodup (Quot.mk (⇑(IsRotated.setoid α)) s)
hn : Nontrivial (Quot.mk (⇑(IsRotated.setoid α)) s)
⊢ ∀ (x : α), s ≠ [x]
|
rintro _ rfl
|
case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
x✝ : α
h : Nodup (Quot.mk ⇑(IsRotated.setoid α) [x✝])
hn : Nontrivial (Quot.mk ⇑(IsRotated.setoid α) [x✝])
⊢ False
|
c975caac8b3ab4b4
|
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.insertUnitInvariant_insertUnit
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddResult.lean
|
theorem insertUnitInvariant_insertUnit {n : Nat} (assignments0 : Array Assignment)
(assignments0_size : assignments0.size = n) (units : Array (Literal (PosFin n)))
(assignments : Array Assignment) (assignments_size : assignments.size = n)
(foundContradiction : Bool) (l : Literal (PosFin n)) :
InsertUnitInvariant assignments0 assignments0_size units assignments assignments_size →
let update_res := insertUnit (units, assignments, foundContradiction) l
have update_res_size : update_res.snd.fst.size = n
|
n : Nat
assignments0 : Array Assignment
assignments0_size : assignments0.size = n
units : Array (Literal (PosFin n))
assignments : Array Assignment
assignments_size : assignments.size = n
foundContradiction : Bool
l : Literal (PosFin n)
i : Fin n
i_in_bounds : ↑i < assignments.size
l_in_bounds : l.fst.val < assignments.size
j : Fin units.size
b : Bool
i_gt_zero : ↑i > 0
h4 : ∀ (k : Fin units.size), ¬k = j → ¬units[↑k].fst.val = ↑i
h5 : ¬hasAssignment l.snd assignments[l.fst.val]! = true
i_eq_l : ↑i = l.fst.val
units_size_lt_updatedUnits_size : units.size < (insertUnit (units, assignments, foundContradiction) l).fst.size
mostRecentUnitIdx : Fin (insertUnit (units, assignments, foundContradiction) l).fst.size :=
⟨units.size, units_size_lt_updatedUnits_size⟩
j_lt_updatedUnits_size : ↑j < (insertUnit (units, assignments, foundContradiction) l).fst.size
h1 : units[↑j] = (⟨↑i, ⋯⟩, true)
h3 : hasAssignment true assignments0[↑i] = false
hb : b = true
hl : l.snd = false
h : assignments0[↑i] = neg
h2 : assignments[l.fst.val] = both
⊢ neg = unassigned
|
simp [hasAssignment, hl, getElem!, l_in_bounds, h2, hasNegAssignment, decidableGetElem?] at h5
|
no goals
|
4e9908103f3f6d5f
|
InfiniteGalois.fixingSubgroup_fixedField
|
Mathlib/FieldTheory/Galois/Infinite.lean
|
lemma fixingSubgroup_fixedField (H : ClosedSubgroup (K ≃ₐ[k] K)) [IsGalois k K] :
(IntermediateField.fixedField H).fixingSubgroup = H.1
|
k : Type u_1
K : Type u_2
inst✝³ : Field k
inst✝² : Field K
inst✝¹ : Algebra k K
H : ClosedSubgroup (K ≃ₐ[k] K)
inst✝ : IsGalois k K
σ : K ≃ₐ[k] K
hσ : σ ∈ (fixedField ↑H).fixingSubgroup
⊢ σ ∈ ↑H
|
by_contra h
|
k : Type u_1
K : Type u_2
inst✝³ : Field k
inst✝² : Field K
inst✝¹ : Algebra k K
H : ClosedSubgroup (K ≃ₐ[k] K)
inst✝ : IsGalois k K
σ : K ≃ₐ[k] K
hσ : σ ∈ (fixedField ↑H).fixingSubgroup
h : σ ∉ ↑H
⊢ False
|
4195a2a4cd22a820
|
ProbabilityTheory.sum_variance_truncation_le
|
Mathlib/Probability/StrongLaw.lean
|
theorem sum_variance_truncation_le {X : Ω → ℝ} (hint : Integrable X) (hnonneg : 0 ≤ X) (K : ℕ) :
∑ j ∈ range K, ((j : ℝ) ^ 2)⁻¹ * 𝔼[truncation X j ^ 2] ≤ 2 * 𝔼[X]
|
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : Ω → ℝ
hint : Integrable X ℙ
hnonneg : 0 ≤ X
K : ℕ
Y : ℕ → Ω → ℝ := fun n => truncation X ↑n
ρ : Measure ℝ := Measure.map X ℙ
Y2 : ∀ (n : ℕ), ∫ (a : Ω), (Y n ^ 2) a = ∫ (x : ℝ) in 0 ..↑n, x ^ 2 ∂ρ
k : ℕ
x✝ : k ∈ range K
Ik : ↑k ≤ ↑(k + 1)
x : ℝ
hx : x ∈ Set.Ioc ↑k ↑(k + 1)
⊢ x / (↑k + 1) ≤ 1
|
convert (div_le_one _).2 hx.2
|
case h.e'_3.h.e'_6
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : Ω → ℝ
hint : Integrable X ℙ
hnonneg : 0 ≤ X
K : ℕ
Y : ℕ → Ω → ℝ := fun n => truncation X ↑n
ρ : Measure ℝ := Measure.map X ℙ
Y2 : ∀ (n : ℕ), ∫ (a : Ω), (Y n ^ 2) a = ∫ (x : ℝ) in 0 ..↑n, x ^ 2 ∂ρ
k : ℕ
x✝ : k ∈ range K
Ik : ↑k ≤ ↑(k + 1)
x : ℝ
hx : x ∈ Set.Ioc ↑k ↑(k + 1)
⊢ ↑k + 1 = ↑(k + 1)
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : Ω → ℝ
hint : Integrable X ℙ
hnonneg : 0 ≤ X
K : ℕ
Y : ℕ → Ω → ℝ := fun n => truncation X ↑n
ρ : Measure ℝ := Measure.map X ℙ
Y2 : ∀ (n : ℕ), ∫ (a : Ω), (Y n ^ 2) a = ∫ (x : ℝ) in 0 ..↑n, x ^ 2 ∂ρ
k : ℕ
x✝ : k ∈ range K
Ik : ↑k ≤ ↑(k + 1)
x : ℝ
hx : x ∈ Set.Ioc ↑k ↑(k + 1)
⊢ 0 < ↑(k + 1)
|
4c37cf41985268cf
|
MeasureTheory.VectorMeasure.trim_eq_self
|
Mathlib/MeasureTheory/VectorMeasure/Basic.lean
|
theorem trim_eq_self : v.trim le_rfl = v
|
case h
α : Type u_1
M : Type u_4
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
n : MeasurableSpace α
v : VectorMeasure α M
i : Set α
hi : MeasurableSet i
⊢ ↑(v.trim ⋯) i = ↑v i
|
exact if_pos hi
|
no goals
|
e5d0658cec430037
|
Equiv.Perm.IsCycleOn.pow_apply_eq
|
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
theorem IsCycleOn.pow_apply_eq {s : Finset α} (hf : f.IsCycleOn s) (ha : a ∈ s) {n : ℕ} :
(f ^ n) a = a ↔ #s ∣ n
|
case inr
α : Type u_2
f : Perm α
a : α
s : Finset α
hf : f.IsCycleOn ↑s
ha : a ∈ s
n : ℕ
hs : s.Nontrivial
h : ∀ (x : { x // x ∈ s }), ¬f ↑x = ↑x
this : orderOf (f.subtypePerm ⋯) = #(f.subtypePerm ⋯).support
⊢ (f ^ n) a = a ↔ #s ∣ n
|
simp only [coe_sort_coe, support_subtype_perm, ne_eq, h, not_false_eq_true, univ_eq_attach,
mem_attach, imp_self, implies_true, filter_true_of_mem, card_attach] at this
|
case inr
α : Type u_2
f : Perm α
a : α
s : Finset α
hf : f.IsCycleOn ↑s
ha : a ∈ s
n : ℕ
hs : s.Nontrivial
h : ∀ (x : { x // x ∈ s }), ¬f ↑x = ↑x
this : orderOf (f.subtypePerm ⋯) = #s
⊢ (f ^ n) a = a ↔ #s ∣ n
|
86354cb7edac4dca
|
AntitoneOn.integral_le_sum
|
Mathlib/Analysis/SumIntegralComparisons.lean
|
theorem AntitoneOn.integral_le_sum (hf : AntitoneOn f (Icc x₀ (x₀ + a))) :
(∫ x in x₀..x₀ + a, f x) ≤ ∑ i ∈ Finset.range a, f (x₀ + i)
|
x₀ : ℝ
a : ℕ
f : ℝ → ℝ
hf : AntitoneOn f (Icc x₀ (x₀ + ↑a))
hint : ∀ k < a, IntervalIntegrable f volume (x₀ + ↑k) (x₀ + ↑(k + 1))
i : ℕ
hi : i ∈ Finset.range a
ia : i < a
⊢ IntervalIntegrable (fun x => f (x₀ + ↑i)) volume (x₀ + ↑i) (x₀ + ↑(i + 1))
|
simp
|
no goals
|
709ac30a4a15e5af
|
Real.exists_extension_norm_eq
|
Mathlib/Analysis/NormedSpace/HahnBanach/Extension.lean
|
theorem exists_extension_norm_eq (p : Subspace ℝ E) (f : p →L[ℝ] ℝ) :
∃ g : E →L[ℝ] ℝ, (∀ x : p, g x = f x) ∧ ‖g‖ = ‖f‖
|
case intro.intro
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
p : Subspace ℝ E
f : ↥p →L[ℝ] ℝ
g : E →ₗ[ℝ] ℝ
g_eq : ∀ (x : ↥{ domain := p, toFun := ↑f }.domain), g ↑x = ↑{ domain := p, toFun := ↑f } x
g_le : ∀ (x : E), g x ≤ ‖f‖ * ‖x‖
g' : E →L[ℝ] ℝ := g.mkContinuous ‖f‖ ⋯
⊢ ‖f‖ ≤ ‖g.mkContinuous ‖f‖ ⋯‖
|
refine f.opNorm_le_bound (norm_nonneg _) fun x => ?_
|
case intro.intro
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
p : Subspace ℝ E
f : ↥p →L[ℝ] ℝ
g : E →ₗ[ℝ] ℝ
g_eq : ∀ (x : ↥{ domain := p, toFun := ↑f }.domain), g ↑x = ↑{ domain := p, toFun := ↑f } x
g_le : ∀ (x : E), g x ≤ ‖f‖ * ‖x‖
g' : E →L[ℝ] ℝ := g.mkContinuous ‖f‖ ⋯
x : ↥p
⊢ ‖f x‖ ≤ ‖g.mkContinuous ‖f‖ ⋯‖ * ‖x‖
|
0f306ef1f16e3415
|
Array.anyM_loop_iff_exists
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean
|
theorem anyM_loop_iff_exists {p : α → Bool} {as : Array α} {start stop} (h : stop ≤ as.size) :
anyM.loop (m := Id) p as stop h start = true ↔
∃ (i : Nat) (_ : i < as.size), start ≤ i ∧ i < stop ∧ p as[i] = true
|
α : Type u_1
p : α → Bool
as : Array α
start stop : Nat
h✝ : stop ≤ as.size
h₁ : start < stop
h₂ : ¬p as[start] = true
i : Nat
hi : i < as.size
ge : start + 1 ≤ i
lt : i < stop
h : p as[i] = true
this : start ≠ i
⊢ start ≤ i
|
omega
|
no goals
|
ffc0dbb5549e1171
|
Finset.pi_insert
|
Mathlib/Data/Finset/Pi.lean
|
theorem pi_insert [∀ a, DecidableEq (β a)] {s : Finset α} {t : ∀ a : α, Finset (β a)} {a : α}
(ha : a ∉ s) : pi (insert a s) t = (t a).biUnion fun b => (pi s t).image (Pi.cons s a b)
|
case a
α : Type u_1
β : α → Type u
inst✝¹ : DecidableEq α
inst✝ : (a : α) → DecidableEq (β a)
s : Finset α
t : (a : α) → Finset (β a)
a : α
ha : a ∉ s
⊢ ((insert a s).pi t).val = ((t a).biUnion fun b => image (Pi.cons s a b) (s.pi t)).val
|
rw [← (pi (insert a s) t).2.dedup]
|
case a
α : Type u_1
β : α → Type u
inst✝¹ : DecidableEq α
inst✝ : (a : α) → DecidableEq (β a)
s : Finset α
t : (a : α) → Finset (β a)
a : α
ha : a ∉ s
⊢ ((insert a s).pi t).val.dedup = ((t a).biUnion fun b => image (Pi.cons s a b) (s.pi t)).val
|
6b4ab4a69f59cfdf
|
isComplete_iUnion_separated
|
Mathlib/Topology/UniformSpace/Cauchy.lean
|
theorem isComplete_iUnion_separated {ι : Sort*} {s : ι → Set α} (hs : ∀ i, IsComplete (s i))
{U : Set (α × α)} (hU : U ∈ 𝓤 α) (hd : ∀ (i j : ι), ∀ x ∈ s i, ∀ y ∈ s j, (x, y) ∈ U → i = j) :
IsComplete (⋃ i, s i)
|
case intro.intro
α : Type u
uniformSpace : UniformSpace α
ι : Sort u_1
s : ι → Set α
hs : ∀ (i : ι), IsComplete (s i)
U : Set (α × α)
hU : U ∈ 𝓤 α
hd : ∀ (i j : ι), ∀ x ∈ s i, ∀ y ∈ s j, (x, y) ∈ U → i = j
S : Set α := ⋃ i, s i
l : Filter α
hl : Cauchy l
hls : S ∈ l
hl_ne : l.NeBot
hl' : ∀ s ∈ 𝓤 α, ∃ t ∈ l, t ×ˢ t ⊆ s
t : Set α
htS : t ⊆ S
htl : t ∈ l
htU : t ×ˢ t ⊆ U
x : α
hx : x ∈ t
i : ι
hi : x ∈ s i
y : α
hy : y ∈ t
⊢ y ∈ s i
|
rcases mem_iUnion.1 (htS hy) with ⟨j, hj⟩
|
case intro.intro.intro
α : Type u
uniformSpace : UniformSpace α
ι : Sort u_1
s : ι → Set α
hs : ∀ (i : ι), IsComplete (s i)
U : Set (α × α)
hU : U ∈ 𝓤 α
hd : ∀ (i j : ι), ∀ x ∈ s i, ∀ y ∈ s j, (x, y) ∈ U → i = j
S : Set α := ⋃ i, s i
l : Filter α
hl : Cauchy l
hls : S ∈ l
hl_ne : l.NeBot
hl' : ∀ s ∈ 𝓤 α, ∃ t ∈ l, t ×ˢ t ⊆ s
t : Set α
htS : t ⊆ S
htl : t ∈ l
htU : t ×ˢ t ⊆ U
x : α
hx : x ∈ t
i : ι
hi : x ∈ s i
y : α
hy : y ∈ t
j : ι
hj : y ∈ s j
⊢ y ∈ s i
|
bb147cb415fdbcec
|
inv_le_inv_iff
|
Mathlib/Algebra/Order/Group/Unbundled/Basic.lean
|
theorem inv_le_inv_iff : a⁻¹ ≤ b⁻¹ ↔ b ≤ a
|
α : Type u
inst✝³ : Group α
inst✝² : LE α
inst✝¹ : MulLeftMono α
a b : α
inst✝ : MulRightMono α
⊢ a * a⁻¹ * b ≤ a * b⁻¹ * b ↔ b ≤ a
|
simp
|
no goals
|
85f86bc66c862d56
|
Order.sub_one_covBy
|
Mathlib/Algebra/Order/SuccPred.lean
|
theorem sub_one_covBy [NoMinOrder α] (x : α) : x - 1 ⋖ x
|
α : Type u_1
inst✝⁴ : Preorder α
inst✝³ : Sub α
inst✝² : One α
inst✝¹ : PredSubOrder α
inst✝ : NoMinOrder α
x : α
⊢ x - 1 ⋖ x
|
rw [← pred_eq_sub_one]
|
α : Type u_1
inst✝⁴ : Preorder α
inst✝³ : Sub α
inst✝² : One α
inst✝¹ : PredSubOrder α
inst✝ : NoMinOrder α
x : α
⊢ pred x ⋖ x
|
46680f86c4bcf2a0
|
FDerivMeasurableAux.D_subset_differentiable_set
|
Mathlib/Analysis/Calculus/FDeriv/Measurable.lean
|
theorem D_subset_differentiable_set {K : Set (E →L[𝕜] F)} (hK : IsComplete K) :
D f K ⊆ { x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K }
|
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → E →L[𝕜] F
hn :
∀ (e p q : ℕ),
n e ≤ p →
n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
M :
∀ (e p q e' p' q' : ℕ),
n e ≤ p → n e ≤ q → n e' ≤ p' → n e' ≤ q' → e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
L0 : ℕ → E →L[𝕜] F := fun e => L e (n e) (n e)
this : CauchySeq L0
f' : E →L[𝕜] F
f'K : f' ∈ K
hf' : Tendsto L0 atTop (𝓝 f')
Lf' : ∀ (e p : ℕ), n e ≤ p → ‖L e (n e) p - f'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
ε : ℝ
εpos : 0 < ε
pos : 0 < 4 + 12 * ‖c‖
e : ℕ
he : (1 / 2) ^ e < ε / (4 + 12 * ‖c‖)
y : E
hy : y ∈ ball 0 ((1 / 2) ^ (n e + 1))
y_pos : ¬y = 0
yzero : 0 < ‖y‖
y_lt : ‖y‖ < (1 / 2) ^ (n e + 1)
yone : ‖y‖ ≤ 1
k : ℕ
k_gt : n e < k
m : ℕ := k - 1
h'k : ‖y‖ ≤ (1 / 2) ^ (m + 1)
hk : (1 / 2) ^ (m + 1 + 1) < ‖y‖
m_ge : n e ≤ m
km : k = m + 1
J1 : ‖f (x + y) - f x - (L e (n e) m) (x + y - x)‖ ≤ (1 / 2) ^ e * (1 / 2) ^ m
J2 : ‖f (x + y) - f x - (L e (n e) m) y‖ ≤ 4 * (1 / 2) ^ e * ‖y‖
⊢ (4 + 12 * ‖c‖) * ‖y‖ * ε = ε * ‖y‖ * (4 + 12 * ‖c‖)
|
ring
|
no goals
|
28c357cacce7671e
|
AEMeasurable.sum_measure
|
Mathlib/MeasureTheory/Measure/AEMeasurable.lean
|
theorem sum_measure [Countable ι] {μ : ι → Measure α} (h : ∀ i, AEMeasurable f (μ i)) :
AEMeasurable f (sum μ)
|
case refine_2
ι : Type u_1
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
f : α → β
inst✝ : Countable ι
μ : ι → Measure α
h : ∀ (i : ι), AEMeasurable f (μ i)
a✝ : Nontrivial β
inhabited_h : Inhabited β
s : ι → Set α := fun i => toMeasurable (μ i) {x | f x ≠ mk f ⋯ x}
hsμ : ∀ (i : ι), (μ i) (s i) = 0
hsm : MeasurableSet (⋂ i, s i)
hs : ∀ (i : ι), ∀ x ∉ s i, f x = mk f ⋯ x
g : α → β := (⋂ i, s i).piecewise (const α default) f
⊢ Measurable ((⋃ i, (s i)ᶜ).restrict f)
|
intro t ht
|
case refine_2
ι : Type u_1
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
f : α → β
inst✝ : Countable ι
μ : ι → Measure α
h : ∀ (i : ι), AEMeasurable f (μ i)
a✝ : Nontrivial β
inhabited_h : Inhabited β
s : ι → Set α := fun i => toMeasurable (μ i) {x | f x ≠ mk f ⋯ x}
hsμ : ∀ (i : ι), (μ i) (s i) = 0
hsm : MeasurableSet (⋂ i, s i)
hs : ∀ (i : ι), ∀ x ∉ s i, f x = mk f ⋯ x
g : α → β := (⋂ i, s i).piecewise (const α default) f
t : Set β
ht : MeasurableSet t
⊢ MeasurableSet ((⋃ i, (s i)ᶜ).restrict f ⁻¹' t)
|
df9a789791001abe
|
EuclideanGeometry.tan_oangle_right_mul_dist_of_oangle_eq_pi_div_two
|
Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean
|
theorem tan_oangle_right_mul_dist_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P}
(h : ∡ p₁ p₂ p₃ = ↑(π / 2)) : Real.Angle.tan (∡ p₂ p₃ p₁) * dist p₃ p₂ = dist p₁ p₂
|
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p₁ p₂ p₃ : P
h : ∡ p₁ p₂ p₃ = ↑(π / 2)
⊢ (∡ p₂ p₃ p₁).tan * dist p₃ p₂ = dist p₁ p₂
|
have hs : (∡ p₂ p₃ p₁).sign = 1 := by rw [oangle_rotate_sign, h, Real.Angle.sign_coe_pi_div_two]
|
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p₁ p₂ p₃ : P
h : ∡ p₁ p₂ p₃ = ↑(π / 2)
hs : (∡ p₂ p₃ p₁).sign = 1
⊢ (∡ p₂ p₃ p₁).tan * dist p₃ p₂ = dist p₁ p₂
|
b67c0a9de406c5b9
|
ContinuousMap.induction_on
|
Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean
|
theorem ContinuousMap.induction_on {𝕜 : Type*} [RCLike 𝕜] {s : Set 𝕜}
{p : C(s, 𝕜) → Prop} (const : ∀ r, p (.const s r)) (id : p (.restrict s <| .id 𝕜))
(star_id : p (star (.restrict s <| .id 𝕜)))
(add : ∀ f g, p f → p g → p (f + g)) (mul : ∀ f g, p f → p g → p (f * g))
(closure : (∀ f ∈ (polynomialFunctions s).starClosure, p f) → ∀ f, p f) (f : C(s, 𝕜)) :
p f
|
case mem.inr
𝕜 : Type u_1
inst✝ : RCLike 𝕜
s : Set 𝕜
p : C(↑s, 𝕜) → Prop
const : ∀ (r : 𝕜), p (ContinuousMap.const (↑s) r)
id : p (restrict s (ContinuousMap.id 𝕜))
star_id : p (star (restrict s (ContinuousMap.id 𝕜)))
add : ∀ (f g : C(↑s, 𝕜)), p f → p g → p (f + g)
mul : ∀ (f g : C(↑s, 𝕜)), p f → p g → p (f * g)
closure : (∀ f ∈ (polynomialFunctions s).starClosure, p f) → ∀ (f : C(↑s, 𝕜)), p f
f✝ f : C(↑s, 𝕜)
⊢ p (star ((toContinuousMapOnAlgHom s) X))
|
simpa only [toContinuousMapOnAlgHom_apply, toContinuousMapOn_X_eq_restrict_id]
|
no goals
|
01b202580110cbda
|
Polynomial.degree_pow_le
|
Mathlib/Algebra/Polynomial/Degree/Definitions.lean
|
theorem degree_pow_le (p : R[X]) : ∀ n : ℕ, degree (p ^ n) ≤ n • degree p
| 0 => by rw [pow_zero, zero_nsmul]; exact degree_one_le
| n + 1 =>
calc
degree (p ^ (n + 1)) ≤ degree (p ^ n) + degree p
|
R : Type u
inst✝ : Semiring R
p : R[X]
n : ℕ
⊢ (p ^ (n + 1)).degree ≤ (p ^ n).degree + p.degree
|
rw [pow_succ]
|
R : Type u
inst✝ : Semiring R
p : R[X]
n : ℕ
⊢ (p ^ n * p).degree ≤ (p ^ n).degree + p.degree
|
1a032331b3bf632a
|
sUnion_memPartition
|
Mathlib/Data/Set/MemPartition.lean
|
@[simp]
lemma sUnion_memPartition (f : ℕ → Set α) (n : ℕ) : ⋃₀ memPartition f n = univ
|
case succ.h
α : Type u_1
f : ℕ → Set α
n : ℕ
ih : ⋃₀ memPartition f n = univ
x : α
this : ∃ t ∈ memPartition f n, x ∈ t
⊢ ∃ t ∈ {s | ∃ u ∈ memPartition f n, s = u ∩ f n ∨ s = u \ f n}, x ∈ t
|
obtain ⟨t, ht, hxt⟩ := this
|
case succ.h.intro.intro
α : Type u_1
f : ℕ → Set α
n : ℕ
ih : ⋃₀ memPartition f n = univ
x : α
t : Set α
ht : t ∈ memPartition f n
hxt : x ∈ t
⊢ ∃ t ∈ {s | ∃ u ∈ memPartition f n, s = u ∩ f n ∨ s = u \ f n}, x ∈ t
|
92b61a089b6c0707
|
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.assignmentsInvariant_insertRatUnits
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RatAddSound.lean
|
theorem assignmentsInvariant_insertRatUnits {n : Nat} (f : DefaultFormula n)
(hf : f.ratUnits = #[] ∧ AssignmentsInvariant f) (units : CNF.Clause (PosFin n)) :
AssignmentsInvariant (insertRatUnits f units).1
|
case left
n : Nat
f : DefaultFormula n
hf : f.ratUnits = #[] ∧ f.AssignmentsInvariant
units : CNF.Clause (PosFin n)
h :
let assignments := (f.insertRatUnits units).fst.assignments;
let_fun hsize := ⋯;
let ratUnits := (f.insertRatUnits units).fst.ratUnits;
InsertUnitInvariant f.assignments ⋯ ratUnits assignments hsize
hsize : (f.insertRatUnits units).fst.assignments.size = n
i : PosFin n
b : Bool
hb : hasAssignment b (f.insertRatUnits units).fst.assignments[i.val] = true
p : PosFin n → Bool
hp :
∀ (x : DefaultClause n),
(some x ∈ (f.insertRatUnits units).fst.clauses.toList ∨
(∃ a,
(a, false) ∈ (f.insertRatUnits units).fst.rupUnits.toList ∧ unit (a, false) = x ∨
(a, true) ∈ (f.insertRatUnits units).fst.rupUnits.toList ∧ unit (a, true) = x) ∨
∃ a,
(a, false) ∈ (f.insertRatUnits units).fst.ratUnits.toList ∧ unit (a, false) = x ∨
(a, true) ∈ (f.insertRatUnits units).fst.ratUnits.toList ∧ unit (a, true) = x) →
∃ a, (a, false) ∈ Clause.toList x ∧ decide (p a = false) = true ∨ (a, true) ∈ Clause.toList x ∧ p a = true
pf : p ⊨ f
j1 j2 : Fin (f.insertRatUnits units).fst.ratUnits.size
i_gt_zero : ↑⟨i.val, ⋯⟩ > 0
h1 : (f.insertRatUnits units).fst.ratUnits[j1] = (⟨↑⟨i.val, ⋯⟩, ⋯⟩, true)
h2 : (f.insertRatUnits units).fst.ratUnits[j2] = (⟨↑⟨i.val, ⋯⟩, ⋯⟩, false)
left✝¹ : (f.insertRatUnits units).fst.assignments[↑⟨i.val, ⋯⟩] = both
left✝ : f.assignments[↑⟨i.val, ⋯⟩] = unassigned
right✝ :
∀ (k : Fin (f.insertRatUnits units).fst.ratUnits.size),
k ≠ j1 → k ≠ j2 → (f.insertRatUnits units).fst.ratUnits[k].fst.val ≠ ↑⟨i.val, ⋯⟩
j1_unit : DefaultClause n := unit (f.insertRatUnits units).fst.ratUnits[j1]
j1_unit_def : j1_unit = unit (f.insertRatUnits units).fst.ratUnits[j1]
j1_unit_in_insertRatUnits_res :
∃ i,
(i, false) ∈ (f.insertRatUnits units).fst.ratUnits.toList ∧ unit (i, false) = j1_unit ∨
(i, true) ∈ (f.insertRatUnits units).fst.ratUnits.toList ∧ unit (i, true) = j1_unit
j2_unit : DefaultClause n := unit (f.insertRatUnits units).fst.ratUnits[j2]
j2_unit_def : j2_unit = unit (f.insertRatUnits units).fst.ratUnits[j2]
⊢ (i, false) ∈ (f.insertRatUnits units).fst.ratUnits.toList
|
have h2 : (insertRatUnits f units).fst.ratUnits[j2] = (i, false) := by
rw [h2]
simp only [Prod.mk.injEq, and_true]
rfl
|
case left
n : Nat
f : DefaultFormula n
hf : f.ratUnits = #[] ∧ f.AssignmentsInvariant
units : CNF.Clause (PosFin n)
h :
let assignments := (f.insertRatUnits units).fst.assignments;
let_fun hsize := ⋯;
let ratUnits := (f.insertRatUnits units).fst.ratUnits;
InsertUnitInvariant f.assignments ⋯ ratUnits assignments hsize
hsize : (f.insertRatUnits units).fst.assignments.size = n
i : PosFin n
b : Bool
hb : hasAssignment b (f.insertRatUnits units).fst.assignments[i.val] = true
p : PosFin n → Bool
hp :
∀ (x : DefaultClause n),
(some x ∈ (f.insertRatUnits units).fst.clauses.toList ∨
(∃ a,
(a, false) ∈ (f.insertRatUnits units).fst.rupUnits.toList ∧ unit (a, false) = x ∨
(a, true) ∈ (f.insertRatUnits units).fst.rupUnits.toList ∧ unit (a, true) = x) ∨
∃ a,
(a, false) ∈ (f.insertRatUnits units).fst.ratUnits.toList ∧ unit (a, false) = x ∨
(a, true) ∈ (f.insertRatUnits units).fst.ratUnits.toList ∧ unit (a, true) = x) →
∃ a, (a, false) ∈ Clause.toList x ∧ decide (p a = false) = true ∨ (a, true) ∈ Clause.toList x ∧ p a = true
pf : p ⊨ f
j1 j2 : Fin (f.insertRatUnits units).fst.ratUnits.size
i_gt_zero : ↑⟨i.val, ⋯⟩ > 0
h1 : (f.insertRatUnits units).fst.ratUnits[j1] = (⟨↑⟨i.val, ⋯⟩, ⋯⟩, true)
h2✝ : (f.insertRatUnits units).fst.ratUnits[j2] = (⟨↑⟨i.val, ⋯⟩, ⋯⟩, false)
left✝¹ : (f.insertRatUnits units).fst.assignments[↑⟨i.val, ⋯⟩] = both
left✝ : f.assignments[↑⟨i.val, ⋯⟩] = unassigned
right✝ :
∀ (k : Fin (f.insertRatUnits units).fst.ratUnits.size),
k ≠ j1 → k ≠ j2 → (f.insertRatUnits units).fst.ratUnits[k].fst.val ≠ ↑⟨i.val, ⋯⟩
j1_unit : DefaultClause n := unit (f.insertRatUnits units).fst.ratUnits[j1]
j1_unit_def : j1_unit = unit (f.insertRatUnits units).fst.ratUnits[j1]
j1_unit_in_insertRatUnits_res :
∃ i,
(i, false) ∈ (f.insertRatUnits units).fst.ratUnits.toList ∧ unit (i, false) = j1_unit ∨
(i, true) ∈ (f.insertRatUnits units).fst.ratUnits.toList ∧ unit (i, true) = j1_unit
j2_unit : DefaultClause n := unit (f.insertRatUnits units).fst.ratUnits[j2]
j2_unit_def : j2_unit = unit (f.insertRatUnits units).fst.ratUnits[j2]
h2 : (f.insertRatUnits units).fst.ratUnits[j2] = (i, false)
⊢ (i, false) ∈ (f.insertRatUnits units).fst.ratUnits.toList
|
b6d274376b14ee0b
|
ProbabilityTheory.Kernel.iIndepSet.indep_generateFrom_of_disjoint
|
Mathlib/Probability/Independence/Kernel.lean
|
theorem iIndepSet.indep_generateFrom_of_disjoint {s : ι → Set Ω}
(hsm : ∀ n, MeasurableSet (s n)) (hs : iIndepSet s κ μ) (S T : Set ι) (hST : Disjoint S T) :
Indep (generateFrom { t | ∃ n ∈ S, s n = t }) (generateFrom { t | ∃ k ∈ T, s k = t }) κ μ
|
case inr
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
s : ι → Set Ω
hsm : ∀ (n : ι), MeasurableSet (s n)
hs : iIndepSet s κ μ
S T : Set ι
hST : Disjoint S T
hμ : μ ≠ 0
⊢ Indep (generateFrom {t | ∃ n ∈ S, s n = t}) (generateFrom {t | ∃ k ∈ T, s k = t}) κ μ
|
obtain ⟨η, η_eq, hη⟩ : ∃ (η : Kernel α Ω), κ =ᵐ[μ] η ∧ IsMarkovKernel η :=
exists_ae_eq_isMarkovKernel hs.ae_isProbabilityMeasure hμ
|
case inr.intro.intro
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
s : ι → Set Ω
hsm : ∀ (n : ι), MeasurableSet (s n)
hs : iIndepSet s κ μ
S T : Set ι
hST : Disjoint S T
hμ : μ ≠ 0
η : Kernel α Ω
η_eq : ⇑κ =ᶠ[ae μ] ⇑η
hη : IsMarkovKernel η
⊢ Indep (generateFrom {t | ∃ n ∈ S, s n = t}) (generateFrom {t | ∃ k ∈ T, s k = t}) κ μ
|
a2cf118cd07d37b1
|
Filter.eventually_lt_of_lt_liminf
|
Mathlib/Order/LiminfLimsup.lean
|
theorem eventually_lt_of_lt_liminf {f : Filter α} [ConditionallyCompleteLinearOrder β] {u : α → β}
{b : β} (h : b < liminf u f)
(hu : f.IsBoundedUnder (· ≥ ·) u
|
α : Type u_1
β : Type u_2
f : Filter α
inst✝ : ConditionallyCompleteLinearOrder β
u : α → β
b : β
h : b < liminf u f
hu : autoParam (IsBoundedUnder (fun x1 x2 => x1 ≥ x2) f u) _auto✝
⊢ ∃ c, ∃ (_ : c ∈ {c | ∀ᶠ (n : α) in f, c ≤ u n}), b < c
|
simp_rw [exists_prop]
|
α : Type u_1
β : Type u_2
f : Filter α
inst✝ : ConditionallyCompleteLinearOrder β
u : α → β
b : β
h : b < liminf u f
hu : autoParam (IsBoundedUnder (fun x1 x2 => x1 ≥ x2) f u) _auto✝
⊢ ∃ c ∈ {c | ∀ᶠ (n : α) in f, c ≤ u n}, b < c
|
82ef5ffe328c509b
|
MonomialOrder.leadingCoeff_monomial
|
Mathlib/RingTheory/MvPolynomial/MonomialOrder.lean
|
theorem leadingCoeff_monomial {d : σ →₀ ℕ} (c : R) :
m.leadingCoeff (monomial d c) = c
|
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommSemiring R
d : σ →₀ ℕ
c : R
⊢ m.leadingCoeff ((monomial d) c) = c
|
classical
simp only [leadingCoeff, degree_monomial]
split_ifs with hc <;> simp [hc]
|
no goals
|
f8562f63559ed44e
|
StrictMono.not_bddAbove_range_of_wellFoundedLT
|
Mathlib/Order/WellFounded.lean
|
theorem StrictMono.not_bddAbove_range_of_wellFoundedLT {f : β → β} [WellFoundedLT β] [NoMaxOrder β]
(hf : StrictMono f) : ¬ BddAbove (Set.range f)
|
case intro.intro
β : Type u_2
inst✝² : LinearOrder β
f : β → β
inst✝¹ : WellFoundedLT β
inst✝ : NoMaxOrder β
hf : StrictMono f
a : β
ha : a ∈ upperBounds (Set.range f)
b : β
hb : a < b
⊢ False
|
exact ((hf.le_apply.trans_lt (hf hb)).trans_le <| ha (Set.mem_range_self _)).false
|
no goals
|
5c12033566e5c122
|
MeasureTheory.SimpleFunc.measure_preimage_lt_top_of_memLp
|
Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean
|
theorem measure_preimage_lt_top_of_memLp (hp_pos : p ≠ 0) (hp_ne_top : p ≠ ∞) (f : α →ₛ E)
(hf : MemLp f p μ) (y : E) (hy_ne : y ≠ 0) : μ (f ⁻¹' {y}) < ∞
|
case pos
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
hp_pos : p ≠ 0
hp_ne_top : p ≠ ⊤
f : α →ₛ E
hf : MemLp (⇑f) p μ
y : E
hy_ne : y ≠ 0
hp_pos_real : 0 < p.toReal
hf_eLpNorm : ∀ a ∈ f.range, ‖a‖ₑ ^ p.toReal * μ (⇑f ⁻¹' {a}) < ⊤
hyf : y ∈ f.range
⊢ μ (⇑f ⁻¹' {y}) < ⊤
|
specialize hf_eLpNorm y hyf
|
case pos
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
hp_pos : p ≠ 0
hp_ne_top : p ≠ ⊤
f : α →ₛ E
hf : MemLp (⇑f) p μ
y : E
hy_ne : y ≠ 0
hp_pos_real : 0 < p.toReal
hyf : y ∈ f.range
hf_eLpNorm : ‖y‖ₑ ^ p.toReal * μ (⇑f ⁻¹' {y}) < ⊤
⊢ μ (⇑f ⁻¹' {y}) < ⊤
|
33a6699fa8a77822
|
Matrix.PosSemidef.conjTranspose_mul_mul_same
|
Mathlib/LinearAlgebra/Matrix/PosDef.lean
|
lemma conjTranspose_mul_mul_same {A : Matrix n n R} (hA : PosSemidef A)
{m : Type*} [Fintype m] (B : Matrix n m R) :
PosSemidef (Bᴴ * A * B)
|
case right
n : Type u_2
R : Type u_3
inst✝⁴ : Fintype n
inst✝³ : CommRing R
inst✝² : PartialOrder R
inst✝¹ : StarRing R
A : Matrix n n R
hA : A.PosSemidef
m : Type u_5
inst✝ : Fintype m
B : Matrix n m R
⊢ ∀ (x : m → R), 0 ≤ star x ⬝ᵥ (Bᴴ * A * B) *ᵥ x
|
intro x
|
case right
n : Type u_2
R : Type u_3
inst✝⁴ : Fintype n
inst✝³ : CommRing R
inst✝² : PartialOrder R
inst✝¹ : StarRing R
A : Matrix n n R
hA : A.PosSemidef
m : Type u_5
inst✝ : Fintype m
B : Matrix n m R
x : m → R
⊢ 0 ≤ star x ⬝ᵥ (Bᴴ * A * B) *ᵥ x
|
6cfd56aa47280d8c
|
gramSchmidt_ne_zero_coe
|
Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean
|
theorem gramSchmidt_ne_zero_coe {f : ι → E} (n : ι)
(h₀ : LinearIndependent 𝕜 (f ∘ ((↑) : Set.Iic n → ι))) : gramSchmidt 𝕜 f n ≠ 0
|
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : WellFoundedLT ι
f : ι → E
n : ι
h₀ : LinearIndependent 𝕜 (f ∘ Subtype.val)
h : gramSchmidt 𝕜 f n = 0
⊢ ∑ i ∈ Finset.Iio n, ↑((orthogonalProjection (span 𝕜 {gramSchmidt 𝕜 f i})) (f n)) ∈
span 𝕜 (gramSchmidt 𝕜 f '' Set.Iio n)
|
apply Submodule.sum_mem _ _
|
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : WellFoundedLT ι
f : ι → E
n : ι
h₀ : LinearIndependent 𝕜 (f ∘ Subtype.val)
h : gramSchmidt 𝕜 f n = 0
⊢ ∀ c ∈ Finset.Iio n,
↑((orthogonalProjection (span 𝕜 {gramSchmidt 𝕜 f c})) (f n)) ∈ span 𝕜 (gramSchmidt 𝕜 f '' Set.Iio n)
|
69b931e205d84775
|
Nat.log_eq_iff
|
Mathlib/Data/Nat/Log.lean
|
theorem log_eq_iff {b m n : ℕ} (h : m ≠ 0 ∨ 1 < b ∧ n ≠ 0) :
log b n = m ↔ b ^ m ≤ n ∧ n < b ^ (m + 1)
|
case inr
b m n : ℕ
h : m ≠ 0 ∨ 1 < b ∧ n ≠ 0
hbn : ¬(1 < b ∧ n ≠ 0)
hm : m ≠ 0
⊢ log b n = m ↔ b ^ m ≤ n ∧ n < b ^ (m + 1)
|
rw [not_and_or, not_lt, Ne, not_not] at hbn
|
case inr
b m n : ℕ
h : m ≠ 0 ∨ 1 < b ∧ n ≠ 0
hbn : b ≤ 1 ∨ n = 0
hm : m ≠ 0
⊢ log b n = m ↔ b ^ m ≤ n ∧ n < b ^ (m + 1)
|
d0a2456541327695
|
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.safe_insert_of_performRupCheck_insertRat
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RatAddSound.lean
|
theorem safe_insert_of_performRupCheck_insertRat {n : Nat} (f : DefaultFormula n)
(hf : f.ratUnits = #[] ∧ AssignmentsInvariant f) (c : DefaultClause n) (rupHints : Array Nat) :
(performRupCheck (insertRatUnits f (negate c)).1 rupHints).2.2.1 = true
→
Limplies (PosFin n) f (f.insert c)
|
case inr
n : Nat
f : DefaultFormula n
hf : f.ratUnits = #[] ∧ f.AssignmentsInvariant
c : DefaultClause n
rupHints : Array Nat
performRupCheck_success :
(Array.foldl (confirmRupHint (f.insertRatUnits c.negate).1.clauses)
((f.insertRatUnits c.negate).1.assignments, [], false, false) rupHints).2.2.fst =
true
p : PosFin n → Bool
c' : DefaultClause n
c'_in_f : c' ∈ f.toList
pf : ∀ (x : DefaultClause n), x ∈ f.toList → p ⊨ x
⊢ p ⊨ c'
|
exact pf c' c'_in_f
|
no goals
|
7b923cb96f92e5d9
|
Associates.prod_le_prod_iff_le
|
Mathlib/RingTheory/UniqueFactorizationDomain/Basic.lean
|
theorem prod_le_prod_iff_le [Nontrivial α] {p q : Multiset (Associates α)}
(hp : ∀ a ∈ p, Irreducible a) (hq : ∀ a ∈ q, Irreducible a) : p.prod ≤ q.prod ↔ p ≤ q
|
case intro.refine_2.e_a
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
p q : Multiset (Associates α)
hp : ∀ a ∈ p, Irreducible a
hq : ∀ a ∈ q, Irreducible a
c : Associates α
eqc : q.prod = p.prod * c
hc : c = 0
⊢ 0 ∈ q
|
rw [← prod_eq_zero_iff, eqc, hc, mul_zero]
|
no goals
|
9197e5d1051792cf
|
MeasureTheory.measure_univ_of_isMulLeftInvariant
|
Mathlib/MeasureTheory/Group/Measure.lean
|
theorem measure_univ_of_isMulLeftInvariant [WeaklyLocallyCompactSpace G] [NoncompactSpace G]
(μ : Measure G) [IsOpenPosMeasure μ] [μ.IsMulLeftInvariant] : μ univ = ∞
|
G : Type u_1
inst✝⁸ : MeasurableSpace G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : BorelSpace G
inst✝⁵ : Group G
inst✝⁴ : IsTopologicalGroup G
inst✝³ : WeaklyLocallyCompactSpace G
inst✝² : NoncompactSpace G
μ : Measure G
inst✝¹ : μ.IsOpenPosMeasure
inst✝ : μ.IsMulLeftInvariant
K : Set G
K1 : K ∈ 𝓝 1
hK : IsCompact K
Kclosed : IsClosed K
K_pos : 0 < μ K
g : Set G → G
hg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)
L : ℕ → Set G := fun n => (fun T => T ∪ g T • K)^[n] K
Lcompact : ∀ (n : ℕ), IsCompact (L n)
Lclosed : ∀ (n : ℕ), IsClosed (L n)
M : ∀ (n : ℕ), μ (L n) = ↑(n + 1) * μ K
⊢ Tendsto (fun x => ↑(x + 1)) atTop (𝓝 ⊤)
|
exact ENNReal.tendsto_nat_nhds_top.comp (tendsto_add_atTop_nat _)
|
no goals
|
d58a29d70ac786c1
|
Multiset.bind_bind
|
Mathlib/Data/Multiset/Bind.lean
|
theorem bind_bind (m : Multiset α) (n : Multiset β) {f : α → β → Multiset γ} :
((bind m) fun a => (bind n) fun b => f a b) = (bind n) fun b => (bind m) fun a => f a b :=
Multiset.induction_on m (by simp) (by simp +contextual)
|
α : Type u_1
β : Type v
γ : Type u_2
m : Multiset α
n : Multiset β
f : α → β → Multiset γ
⊢ ∀ (a : α) (s : Multiset α),
((s.bind fun a => n.bind fun b => f a b) = n.bind fun b => s.bind fun a => f a b) →
((a ::ₘ s).bind fun a => n.bind fun b => f a b) = n.bind fun b => (a ::ₘ s).bind fun a => f a b
|
simp +contextual
|
no goals
|
abdc90d1a3f89e14
|
HomologicalComplex.pOpcycles_extendOpcyclesIso_inv
|
Mathlib/Algebra/Homology/Embedding/ExtendHomology.lean
|
@[reassoc (attr := simp)]
lemma pOpcycles_extendOpcyclesIso_inv :
K.pOpcycles j ≫ (K.extendOpcyclesIso e hj').inv =
(K.extendXIso e hj').inv ≫ (K.extend e).pOpcycles j'
|
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁴ : Category.{u_4, u_3} C
inst✝³ : HasZeroMorphisms C
inst✝² : HasZeroObject C
K : HomologicalComplex C c
e : c.Embedding c'
j : ι
j' : ι'
hj' : e.f j = j'
inst✝¹ : K.HasHomology j
inst✝ : (K.extend e).HasHomology j'
⊢ (K.sc j).pOpcycles =
(K.extendXIso e hj').inv ≫
((K.extend e).sc j').pOpcycles ≫
(extend.homologyData' K e hj' ⋯ ⋯ (K.sc j).homologyData).right.opcyclesIso.hom ≫
(K.sc j).homologyData.right.opcyclesIso.inv
|
rw [ShortComplex.RightHomologyData.pOpcycles_comp_opcyclesIso_hom_assoc]
|
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁴ : Category.{u_4, u_3} C
inst✝³ : HasZeroMorphisms C
inst✝² : HasZeroObject C
K : HomologicalComplex C c
e : c.Embedding c'
j : ι
j' : ι'
hj' : e.f j = j'
inst✝¹ : K.HasHomology j
inst✝ : (K.extend e).HasHomology j'
⊢ (K.sc j).pOpcycles =
(K.extendXIso e hj').inv ≫
(extend.homologyData' K e hj' ⋯ ⋯ (K.sc j).homologyData).right.p ≫ (K.sc j).homologyData.right.opcyclesIso.inv
|
d598f615c159cc2f
|
PrimeSpectrum.iSup_basicOpen_eq_top_iff'
|
Mathlib/RingTheory/Spectrum/Prime/Topology.lean
|
lemma iSup_basicOpen_eq_top_iff' {s : Set R} :
(⨆ i ∈ s, PrimeSpectrum.basicOpen i) = ⊤ ↔ Ideal.span s = ⊤
|
R : Type u
inst✝ : CommSemiring R
s : Set R
⊢ ⨆ i ∈ s, basicOpen i = ⊤ ↔ ⨆ i, basicOpen ↑i = ⊤
|
simp
|
no goals
|
c483e22b2c8fc038
|
HasFiniteFPowerSeriesOnBall.fderiv
|
Mathlib/Analysis/Calculus/FDeriv/Analytic.lean
|
theorem HasFiniteFPowerSeriesOnBall.fderiv
(h : HasFiniteFPowerSeriesOnBall f p x (n + 1) r) :
HasFiniteFPowerSeriesOnBall (fderiv 𝕜 f) p.derivSeries x n r
|
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
n : ℕ
f : E → F
x : E
h : HasFiniteFPowerSeriesOnBall f p x (n + 1) r
⊢ HasFiniteFPowerSeriesOnBall (fderiv 𝕜 f) p.derivSeries x n r
|
refine .congr (f := fun z ↦ continuousMultilinearCurryFin1 𝕜 E F (p.changeOrigin (z - x) 1)) ?_
fun z hz ↦ ?_
|
case refine_1
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
n : ℕ
f : E → F
x : E
h : HasFiniteFPowerSeriesOnBall f p x (n + 1) r
⊢ HasFiniteFPowerSeriesOnBall (fun z => (continuousMultilinearCurryFin1 𝕜 E F) (p.changeOrigin (z - x) 1)) p.derivSeries
x n r
case refine_2
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
n : ℕ
f : E → F
x : E
h : HasFiniteFPowerSeriesOnBall f p x (n + 1) r
z : E
hz : z ∈ EMetric.ball x r
⊢ (fun z => (continuousMultilinearCurryFin1 𝕜 E F) (p.changeOrigin (z - x) 1)) z = fderiv 𝕜 f z
|
b335a221088906cf
|
List.take_of_length_le
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/TakeDrop.lean
|
theorem take_of_length_le {l : List α} (h : l.length ≤ i) : take i l = l
|
α : Type u_1
i : Nat
l : List α
h : l.length ≤ i
this : take i l ++ drop i l = l
⊢ take i l = l
|
rw [drop_of_length_le h, append_nil] at this
|
α : Type u_1
i : Nat
l : List α
h : l.length ≤ i
this : take i l = l
⊢ take i l = l
|
6c0eac07c538a813
|
ComplexShape.π_symm
|
Mathlib/Algebra/Homology/ComplexShapeSigns.lean
|
lemma π_symm (i₁ : I₁) (i₂ : I₂) :
π c₂ c₁ c₁₂ ⟨i₂, i₁⟩ = π c₁ c₂ c₁₂ ⟨i₁, i₂⟩
|
I₁ : Type u_1
I₂ : Type u_2
I₁₂ : Type u_4
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
c₁₂ : ComplexShape I₁₂
inst✝² : TotalComplexShape c₁ c₂ c₁₂
inst✝¹ : TotalComplexShape c₂ c₁ c₁₂
inst✝ : TotalComplexShapeSymmetry c₁ c₂ c₁₂
i₁ : I₁
i₂ : I₂
⊢ c₂.π c₁ c₁₂ (i₂, i₁) = c₁.π c₂ c₁₂ (i₁, i₂)
|
apply TotalComplexShapeSymmetry.symm
|
no goals
|
aefa0261e4cfe0e7
|
finiteMultiplicity_mul_aux
|
Mathlib/RingTheory/Multiplicity.lean
|
theorem finiteMultiplicity_mul_aux {p : α} (hp : Prime p) {a b : α} :
∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b
| n, m => fun ha hb ⟨s, hs⟩ =>
have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩
(hp.2.2 a b this).elim
(fun ⟨x, hx⟩ =>
have hn0 : 0 < n :=
Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha
have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ =>
ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by
rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy
simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩)
have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m)
finiteMultiplicity_mul_aux hp hpx hb
⟨s, mul_right_cancel₀ hp.1 (by
rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this]
simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩)
fun ⟨x, hx⟩ =>
have hm0 : 0 < m :=
Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb
have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ =>
hb
(hx.symm ▸
⟨y,
mul_right_cancel₀ hp.1 <| by
rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy
simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩)
finiteMultiplicity_mul_aux hp ha hpx
⟨s, mul_right_cancel₀ hp.1 (by
rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)]
simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩
|
α : Type u_1
inst✝ : CancelCommMonoidWithZero α
p : α
hp : Prime p
a b : α
n m : ℕ
ha : ¬p ^ (n + 1) ∣ a
hb : ¬p ^ (m + 1) ∣ b
x✝¹ : p ^ (n + m + 1) ∣ a * b
s : α
hs : a * b = p ^ (n + m + 1) * s
this✝ : p ∣ a * b
x✝ : p ∣ a
x : α
hx : a = p * x
hn0 : 0 < n
hpx : ¬p ^ (n - 1 + 1) ∣ x
this : 1 ≤ n + m
⊢ x * b * p = p ^ (n + m) * s * p
|
simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add]
|
no goals
|
e3775fe83a9cfc31
|
MeasureTheory.integrableOn_univ
|
Mathlib/MeasureTheory/Integral/IntegrableOn.lean
|
theorem integrableOn_univ : IntegrableOn f univ μ ↔ Integrable f μ
|
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
f : α → E
μ : Measure α
⊢ IntegrableOn f univ μ ↔ Integrable f μ
|
rw [IntegrableOn, Measure.restrict_univ]
|
no goals
|
dc45be82ebcf739a
|
Equiv.Perm.ofSign_disjoint
|
Mathlib/GroupTheory/Perm/Sign.lean
|
lemma ofSign_disjoint : _root_.Disjoint (ofSign 1 : Finset (Perm α)) (ofSign (-1))
|
α : Type u
inst✝¹ : DecidableEq α
inst✝ : Fintype α
σ : Perm α
hσ : σ ∈ ofSign 1
hτ : σ ∈ ofSign (-1)
⊢ False
|
rw [mem_ofSign] at hσ hτ
|
α : Type u
inst✝¹ : DecidableEq α
inst✝ : Fintype α
σ : Perm α
hσ : sign σ = 1
hτ : sign σ = -1
⊢ False
|
e53c5c58b6af3a43
|
ProbabilityTheory.integrable_exp_mul_of_abs_le
|
Mathlib/Probability/Moments/IntegrableExpMul.lean
|
/-- If `ω ↦ exp (u * X ω)` is integrable at `u` and `-u`, then it is integrable on `[-u, u]`. -/
lemma integrable_exp_mul_of_abs_le
(hu_int_pos : Integrable (fun ω ↦ exp (u * X ω)) μ)
(hu_int_neg : Integrable (fun ω ↦ exp (- u * X ω)) μ)
(htu : |t| ≤ |u|) :
Integrable (fun ω ↦ exp (t * X ω)) μ
|
case refine_2
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t u : ℝ
hu_int_pos : Integrable (fun ω => rexp (u * X ω)) μ
hu_int_neg : Integrable (fun ω => rexp (-u * X ω)) μ
htu : |t| ≤ |u|
⊢ Integrable (fun ω => rexp (|u| * X ω)) μ
|
rcases le_total 0 u with hu | hu
|
case refine_2.inl
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t u : ℝ
hu_int_pos : Integrable (fun ω => rexp (u * X ω)) μ
hu_int_neg : Integrable (fun ω => rexp (-u * X ω)) μ
htu : |t| ≤ |u|
hu : 0 ≤ u
⊢ Integrable (fun ω => rexp (|u| * X ω)) μ
case refine_2.inr
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t u : ℝ
hu_int_pos : Integrable (fun ω => rexp (u * X ω)) μ
hu_int_neg : Integrable (fun ω => rexp (-u * X ω)) μ
htu : |t| ≤ |u|
hu : u ≤ 0
⊢ Integrable (fun ω => rexp (|u| * X ω)) μ
|
c59caf165b150fde
|
BitVec.zero_sshiftRight
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean
|
theorem zero_sshiftRight {n : Nat} : (0#w).sshiftRight n = 0#w
|
w n : Nat
⊢ (0#w).sshiftRight n = 0#w
|
ext i h
|
case pred
w n i : Nat
h : i < w
⊢ ((0#w).sshiftRight n).getLsbD i = (0#w).getLsbD i
|
d91b7635f218cbff
|
Trivialization.apply_eq_prod_continuousLinearEquivAt
|
Mathlib/Topology/VectorBundle/Basic.lean
|
theorem apply_eq_prod_continuousLinearEquivAt (e : Trivialization F (π F E)) [e.IsLinear R] (b : B)
(hb : b ∈ e.baseSet) (z : E b) : e ⟨b, z⟩ = (b, e.continuousLinearEquivAt R b hb z)
|
case fst
R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁹ : NontriviallyNormedField R
inst✝⁸ : (x : B) → AddCommMonoid (E x)
inst✝⁷ : (x : B) → Module R (E x)
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace R F
inst✝⁴ : TopologicalSpace B
inst✝³ : TopologicalSpace (TotalSpace F E)
inst✝² : (x : B) → TopologicalSpace (E x)
inst✝¹ : FiberBundle F E
e : Trivialization F TotalSpace.proj
inst✝ : Trivialization.IsLinear R e
b : B
hb : b ∈ e.baseSet
z : E b
⊢ { proj := b, snd := z } ∈ TotalSpace.proj ⁻¹' e.baseSet
|
exact hb
|
no goals
|
5868769150d7cbc0
|
PythagoreanTriple.even_odd_of_coprime
|
Mathlib/NumberTheory/PythagoreanTriples.lean
|
theorem even_odd_of_coprime (hc : Int.gcd x y = 1) :
x % 2 = 0 ∧ y % 2 = 1 ∨ x % 2 = 1 ∧ y % 2 = 0
|
case inr.inr
x y z : ℤ
h : PythagoreanTriple x y z
hc : x.gcd y = 1
hx : x % 2 = 1
hy : y % 2 = 1
⊢ False
|
obtain ⟨x0, y0, rfl, rfl⟩ : ∃ x0 y0, x = x0 * 2 + 1 ∧ y = y0 * 2 + 1 := by
obtain ⟨x0, hx2⟩ := exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hx)
obtain ⟨y0, hy2⟩ := exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hy)
rw [sub_eq_iff_eq_add] at hx2 hy2
exact ⟨x0, y0, hx2, hy2⟩
|
case inr.inr.intro.intro.intro
z x0 y0 : ℤ
hx : (x0 * 2 + 1) % 2 = 1
hy : (y0 * 2 + 1) % 2 = 1
h : PythagoreanTriple (x0 * 2 + 1) (y0 * 2 + 1) z
hc : (x0 * 2 + 1).gcd (y0 * 2 + 1) = 1
⊢ False
|
48f64ea05953c676
|
Batteries.RBNode.foldr_reverse
|
Mathlib/.lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean
|
theorem foldr_reverse {α β : Type _} {t : RBNode α} {f : α → β → β} {init : β} :
t.reverse.foldr f init = t.foldl (flip f) init :=
foldl_reverse.symm.trans (by simp; rfl)
|
α : Type u_1
β : Type u_2
t : RBNode α
f : α → β → β
init : β
⊢ foldl (fun a b => f b a) init t.reverse.reverse = foldl (flip f) init t
|
simp
|
α : Type u_1
β : Type u_2
t : RBNode α
f : α → β → β
init : β
⊢ foldl (fun a b => f b a) init t = foldl (flip f) init t
|
50b680d84eae0c3b
|
PartENat.lt_find
|
Mathlib/Data/Nat/PartENat.lean
|
theorem lt_find (n : ℕ) (h : ∀ m ≤ n, ¬P m) : (n : PartENat) < find P
|
P : ℕ → Prop
inst✝ : DecidablePred P
n : ℕ
h : ∀ m ≤ n, ¬P m
⊢ ∀ (h : (find P).Dom), n < (find P).get h
|
intro h₁
|
P : ℕ → Prop
inst✝ : DecidablePred P
n : ℕ
h : ∀ m ≤ n, ¬P m
h₁ : (find P).Dom
⊢ n < (find P).get h₁
|
33f16598e16dc969
|
one_div_pow_le_one_div_pow_of_le
|
Mathlib/Algebra/Order/Field/Basic.lean
|
theorem one_div_pow_le_one_div_pow_of_le (a1 : 1 ≤ a) {m n : ℕ} (mn : m ≤ n) :
1 / a ^ n ≤ 1 / a ^ m
|
α : Type u_2
inst✝ : LinearOrderedSemifield α
a : α
a1 : 1 ≤ a
m n : ℕ
mn : m ≤ n
⊢ 1 / a ^ n ≤ 1 / a ^ m
|
refine (one_div_le_one_div ?_ ?_).mpr (pow_right_mono₀ a1 mn) <;>
exact pow_pos (zero_lt_one.trans_le a1) _
|
no goals
|
30efe5e517cf3a6d
|
Subgroup.fg_iff_submonoid_fg
|
Mathlib/GroupTheory/Finiteness.lean
|
theorem Subgroup.fg_iff_submonoid_fg (P : Subgroup G) : P.FG ↔ P.toSubmonoid.FG
|
case mpr.intro.refine_1
G : Type u_3
inst✝ : Group G
P : Subgroup G
S : Finset G
hS : Submonoid.closure ↑S = P.toSubmonoid
⊢ closure ↑S ≤ P
|
rw [Subgroup.closure_le, ← Subgroup.coe_toSubmonoid, ← hS]
|
case mpr.intro.refine_1
G : Type u_3
inst✝ : Group G
P : Subgroup G
S : Finset G
hS : Submonoid.closure ↑S = P.toSubmonoid
⊢ ↑S ⊆ ↑(Submonoid.closure ↑S)
|
11ccabe6f8ec3164
|
MvPolynomial.vars_sub_subset
|
Mathlib/Algebra/MvPolynomial/CommRing.lean
|
theorem vars_sub_subset [DecidableEq σ] : (p - q).vars ⊆ p.vars ∪ q.vars
|
R : Type u
σ : Type u_1
inst✝¹ : CommRing R
p q : MvPolynomial σ R
inst✝ : DecidableEq σ
⊢ (p - q).vars ⊆ p.vars ∪ q.vars
|
convert vars_add_subset p (-q) using 2 <;> simp [sub_eq_add_neg]
|
no goals
|
2d80017a45e04164
|
IsCompact.exists_forall_le'
|
Mathlib/Topology/Order/Compact.lean
|
theorem IsCompact.exists_forall_le' [ClosedIicTopology α] [NoMaxOrder α] {f : β → α}
{s : Set β} (hs : IsCompact s) (hf : ContinuousOn f s) {a : α} (hf' : ∀ b ∈ s, a < f b) :
∃ a', a < a' ∧ ∀ b ∈ s, a' ≤ f b
|
case inr.intro.intro
α : Type u_2
β : Type u_3
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : ClosedIicTopology α
inst✝ : NoMaxOrder α
f : β → α
s : Set β
hs : IsCompact s
hf : ContinuousOn f s
a : α
hf' : ∀ b ∈ s, a < f b
hs' : s.Nonempty
x : β
hx : x ∈ s
hx' : IsMinOn f s x
⊢ ∃ a', a < a' ∧ ∀ b ∈ s, a' ≤ f b
|
exact ⟨f x, hf' x hx, hx'⟩
|
no goals
|
0928e6d3df8c75af
|
HahnSeries.addOppositeEquiv_orderTop
|
Mathlib/RingTheory/HahnSeries/Addition.lean
|
@[simp]
lemma addOppositeEquiv_orderTop (x : HahnSeries Γ (Rᵃᵒᵖ)) :
(addOppositeEquiv x).unop.orderTop = x.orderTop
|
Γ : Type u_1
R : Type u_3
inst✝¹ : PartialOrder Γ
inst✝ : AddMonoid R
x : HahnSeries Γ Rᵃᵒᵖ
⊢ (AddOpposite.unop (addOppositeEquiv x)).orderTop = x.orderTop
|
classical
simp only [orderTop, AddOpposite.unop_op, mk_eq_zero, EmbeddingLike.map_eq_zero_iff,
addOppositeEquiv_support, ne_eq]
simp only [addOppositeEquiv_apply, AddOpposite.unop_op, mk_eq_zero, coeff_zero]
simp_rw [HahnSeries.ext_iff, funext_iff]
simp only [Pi.zero_apply, AddOpposite.unop_eq_zero_iff, coeff_zero]
|
no goals
|
66cf7b891a048377
|
ContDiffWithinAt.fderivWithin''
|
Mathlib/Analysis/Calculus/ContDiff/Basic.lean
|
theorem ContDiffWithinAt.fderivWithin'' {f : E → F → G} {g : E → F} {t : Set F}
(hf : ContDiffWithinAt 𝕜 n (Function.uncurry f) (insert x₀ s ×ˢ t) (x₀, g x₀))
(hg : ContDiffWithinAt 𝕜 m g s x₀)
(ht : ∀ᶠ x in 𝓝[insert x₀ s] x₀, UniqueDiffWithinAt 𝕜 t (g x)) (hmn : m + 1 ≤ n)
(hgt : t ∈ 𝓝[g '' s] g x₀) :
ContDiffWithinAt 𝕜 m (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀
|
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
x₀ : E
m : WithTop ℕ∞
f : E → F → G
g : E → F
t : Set F
ht : ∀ᶠ (x : E) in 𝓝[insert x₀ s] x₀, UniqueDiffWithinAt 𝕜 t (g x)
hgt : t ∈ 𝓝[g '' s] g x₀
hg : ContDiffWithinAt 𝕜 ω g s x₀
this : ∀ (k : ℕ), ↑k ≤ ω → ContDiffWithinAt 𝕜 (↑k) (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀
hf : ContDiffWithinAt 𝕜 ω (uncurry f) (insert x₀ s ×ˢ t) (x₀, g x₀)
hmn : ω + 1 ≤ ω
⊢ ContDiffWithinAt 𝕜 ω (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀
|
obtain ⟨v, hv, -, f', hvf', hf'⟩ := hf.hasFDerivWithinAt_nhds (by simp) hg hgt
|
case intro.intro.intro.intro.intro
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
x₀ : E
m : WithTop ℕ∞
f : E → F → G
g : E → F
t : Set F
ht : ∀ᶠ (x : E) in 𝓝[insert x₀ s] x₀, UniqueDiffWithinAt 𝕜 t (g x)
hgt : t ∈ 𝓝[g '' s] g x₀
hg : ContDiffWithinAt 𝕜 ω g s x₀
this : ∀ (k : ℕ), ↑k ≤ ω → ContDiffWithinAt 𝕜 (↑k) (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀
hf : ContDiffWithinAt 𝕜 ω (uncurry f) (insert x₀ s ×ˢ t) (x₀, g x₀)
hmn : ω + 1 ≤ ω
v : Set E
hv : v ∈ 𝓝[insert x₀ s] x₀
f' : E → F →L[𝕜] G
hvf' : ∀ x ∈ v, HasFDerivWithinAt (f x) (f' x) t (g x)
hf' : ContDiffWithinAt 𝕜 ω (fun x => f' x) s x₀
⊢ ContDiffWithinAt 𝕜 ω (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀
|
ce8ad388b9bbe46b
|
Set.MulAntidiagonal.fst_eq_fst_iff_snd_eq_snd
|
Mathlib/Data/Set/MulAntidiagonal.lean
|
theorem fst_eq_fst_iff_snd_eq_snd : (x : α × α).1 = (y : α × α).1 ↔ (x : α × α).2 = (y : α × α).2 :=
⟨fun h =>
mul_left_cancel
(y.2.2.2.trans <| by
rw [← h]
exact x.2.2.2.symm).symm,
fun h =>
mul_right_cancel
(y.2.2.2.trans <| by
rw [← h]
exact x.2.2.2.symm).symm⟩
|
α : Type u_1
inst✝ : CancelCommMonoid α
s t : Set α
a : α
x y : ↑(s.mulAntidiagonal t a)
h : (↑x).1 = (↑y).1
⊢ a = (↑y).1 * (↑x).2
|
rw [← h]
|
α : Type u_1
inst✝ : CancelCommMonoid α
s t : Set α
a : α
x y : ↑(s.mulAntidiagonal t a)
h : (↑x).1 = (↑y).1
⊢ a = (↑x).1 * (↑x).2
|
c2b68b98fd393ff0
|
contMDiff_of_contMDiff_inl
|
Mathlib/Geometry/Manifold/ContMDiff/Constructions.lean
|
lemma contMDiff_of_contMDiff_inl {f : M → N}
(h : ContMDiff I J n ((@Sum.inl N N') ∘ f)) : ContMDiff I J n f
|
𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
n : WithTop ℕ∞
E' : Type u_17
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
H' : Type u_18
inst✝⁴ : TopologicalSpace H'
J : ModelWithCorners 𝕜 E' H'
N : Type u_20
N' : Type u_21
inst✝³ : TopologicalSpace N
inst✝² : TopologicalSpace N'
inst✝¹ : ChartedSpace H' N
inst✝ : ChartedSpace H' N'
f : M → N
h : ContMDiffOn I J n (Sum.inl ∘ f) univ
a✝ : Nontrivial N
inhabited_h : Inhabited N
aux : N ⊕ N' → N := Sum.elim id fun x => default
this : aux ∘ Sum.inl ∘ f = f
⊢ ContMDiffOn I J n (aux ∘ Sum.inl ∘ f) univ
|
apply (contMDiff_id.sumElim contMDiff_const).contMDiffOn (s := @Sum.inl N N' '' univ).comp h
|
𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
n : WithTop ℕ∞
E' : Type u_17
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
H' : Type u_18
inst✝⁴ : TopologicalSpace H'
J : ModelWithCorners 𝕜 E' H'
N : Type u_20
N' : Type u_21
inst✝³ : TopologicalSpace N
inst✝² : TopologicalSpace N'
inst✝¹ : ChartedSpace H' N
inst✝ : ChartedSpace H' N'
f : M → N
h : ContMDiffOn I J n (Sum.inl ∘ f) univ
a✝ : Nontrivial N
inhabited_h : Inhabited N
aux : N ⊕ N' → N := Sum.elim id fun x => default
this : aux ∘ Sum.inl ∘ f = f
⊢ univ ⊆ Sum.inl ∘ f ⁻¹' (Sum.inl '' univ)
|
9dafec4728754145
|
IsPGroup.iff_card
|
Mathlib/GroupTheory/PGroup.lean
|
theorem iff_card [Fact p.Prime] [Finite G] : IsPGroup p G ↔ ∃ n : ℕ, Nat.card G = p ^ n
|
case intro.intro
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite G
hG : Nat.card G ≠ 0
h : IsPGroup p G
q : ℕ
hq : q ∈ (Nat.card G).primeFactorsList
hq1 : Nat.Prime q
hq2 : q ∣ Nat.card G
this : Fact (Nat.Prime q)
g : G
hg : orderOf g = q
⊢ q = p
|
obtain ⟨k, hk⟩ := (iff_orderOf.mp h) g
|
case intro.intro.intro
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite G
hG : Nat.card G ≠ 0
h : IsPGroup p G
q : ℕ
hq : q ∈ (Nat.card G).primeFactorsList
hq1 : Nat.Prime q
hq2 : q ∣ Nat.card G
this : Fact (Nat.Prime q)
g : G
hg : orderOf g = q
k : ℕ
hk : orderOf g = p ^ k
⊢ q = p
|
cdf7a3b03afadbaa
|
Cardinal.lt_wf
|
Mathlib/SetTheory/Cardinal/Basic.lean
|
theorem lt_wf : @WellFounded Cardinal.{u} (· < ·) :=
⟨fun a =>
by_contradiction fun h => by
let ι := { c : Cardinal // ¬Acc (· < ·) c }
let f : ι → Cardinal := Subtype.val
haveI hι : Nonempty ι := ⟨⟨_, h⟩⟩
obtain ⟨⟨c : Cardinal, hc : ¬Acc (· < ·) c⟩, ⟨h_1 : ∀ j, (f ⟨c, hc⟩).out ↪ (f j).out⟩⟩ :=
Embedding.min_injective fun i => (f i).out
refine hc (Acc.intro _ fun j h' => by_contradiction fun hj => h'.2 ?_)
have : #_ ≤ #_ := ⟨h_1 ⟨j, hj⟩⟩
simpa only [mk_out] using this⟩
|
a : Cardinal.{u}
h : ¬Acc (fun x1 x2 => x1 < x2) a
ι : Type (max 0 (?u.56390 + 1)) := { c // ¬Acc (fun x1 x2 => x1 < x2) c }
⊢ False
|
let f : ι → Cardinal := Subtype.val
|
a : Cardinal.{u}
h : ¬Acc (fun x1 x2 => x1 < x2) a
ι : Type (max 0 (?u.56390 + 1)) := { c // ¬Acc (fun x1 x2 => x1 < x2) c }
f : ι → Cardinal.{?u.56390} := Subtype.val
⊢ False
|
2083f4a91b2b3d48
|
Complex.HadamardThreeLines.interpStrip_eq_of_pos
|
Mathlib/Analysis/Complex/Hadamard.lean
|
/-- Rewrite for `InterpStrip` when `0 < sSupNormIm f 0` and `0 < sSupNormIm f 1`. -/
lemma interpStrip_eq_of_pos (z : ℂ) (h0 : 0 < sSupNormIm f 0) (h1 : 0 < sSupNormIm f 1) :
interpStrip f z = sSupNormIm f 0 ^ (1 - z) * sSupNormIm f 1 ^ z
|
E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
z : ℂ
h0 : 0 < sSupNormIm f 0
h1 : 0 < sSupNormIm f 1
⊢ interpStrip f z = ↑(sSupNormIm f 0) ^ (1 - z) * ↑(sSupNormIm f 1) ^ z
|
simp only [ne_of_gt h0, ne_of_gt h1, interpStrip, if_false, or_false]
|
no goals
|
8db93de84433c1a8
|
VitaliFamily.exists_measurable_supersets_limRatio
|
Mathlib/MeasureTheory/Covering/Differentiation.lean
|
theorem exists_measurable_supersets_limRatio {p q : ℝ≥0} (hpq : p < q) :
∃ a b, MeasurableSet a ∧ MeasurableSet b ∧
{x | v.limRatio ρ x < p} ⊆ a ∧ {x | (q : ℝ≥0∞) < v.limRatio ρ x} ⊆ b ∧ μ (a ∩ b) = 0
|
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α := {x | ∃ c, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : ℕ → Set α := fun n => s ∩ {x | v.limRatio ρ x < ↑p} ∩ o n
w : ℕ → Set α := fun n => s ∩ {x | ↑q < v.limRatio ρ x} ∩ o n
H : ∀ (m n : ℕ), μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0
A :
(toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (u n)) ∩ (toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (w n)) ⊆
toMeasurable μ sᶜ ∪ ⋃ m, ⋃ n, toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)
⊢ μ ((toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (u n)) ∩ (toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (w n))) = 0
|
refine le_antisymm ((measure_mono A).trans ?_) bot_le
|
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α := {x | ∃ c, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : ℕ → Set α := fun n => s ∩ {x | v.limRatio ρ x < ↑p} ∩ o n
w : ℕ → Set α := fun n => s ∩ {x | ↑q < v.limRatio ρ x} ∩ o n
H : ∀ (m n : ℕ), μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0
A :
(toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (u n)) ∩ (toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (w n)) ⊆
toMeasurable μ sᶜ ∪ ⋃ m, ⋃ n, toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)
⊢ μ (toMeasurable μ sᶜ ∪ ⋃ m, ⋃ n, toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤ 0
|
f2acb3dc8c3a0a55
|
SimpleGraph.card_commonNeighbors_top
|
Mathlib/Combinatorics/SimpleGraph/Finite.lean
|
theorem card_commonNeighbors_top [DecidableEq V] {v w : V} (h : v ≠ w) :
Fintype.card ((⊤ : SimpleGraph V).commonNeighbors v w) = Fintype.card V - 2
|
V : Type u_1
inst✝¹ : Fintype V
inst✝ : DecidableEq V
v w : V
h : v ≠ w
⊢ #(Set.univ.toFinset \ {v, w}.toFinset) = Fintype.card V - 2
|
rw [Finset.card_sdiff]
|
V : Type u_1
inst✝¹ : Fintype V
inst✝ : DecidableEq V
v w : V
h : v ≠ w
⊢ #Set.univ.toFinset - #{v, w}.toFinset = Fintype.card V - 2
V : Type u_1
inst✝¹ : Fintype V
inst✝ : DecidableEq V
v w : V
h : v ≠ w
⊢ {v, w}.toFinset ⊆ Set.univ.toFinset
|
3c759119a4b352a3
|
Array.getElem_swap'
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean
|
theorem getElem_swap' (a : Array α) (i j : Nat) {hi hj} (k : Nat) (hk : k < a.size) :
(a.swap i j hi hj)[k]'(by simp_all) = if k = i then a[j] else if k = j then a[i] else a[k]
|
case isTrue
α : Type u_1
a : Array α
i j : Nat
hi : i < a.size
hj : j < a.size
k : Nat
hk : k < a.size
h✝ : k = i
⊢ (a.swap i j hi hj)[k] = a[j]
|
simp_all only [getElem_swap_left]
|
no goals
|
57b3b7cba232ccbc
|
Ideal.subset_union_prime'
|
Mathlib/RingTheory/Ideal/Operations.lean
|
theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
|
ι : Type u_1
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, (f i).IsPrime) →
s.card = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
a b i j : ι
hfji : f j ≤ f i
u : Finset ι
hju : j ∉ u
hit : i ∉ insert j u
hn : (insert j u).card = n
h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i_1 ∈ insert i (insert j ↑u), ↑(f i_1)
hp : (f i).IsPrime ∧ ∀ x ∈ insert j u, (f x).IsPrime
hjt : j ∈ insert j u
hp' : ∀ k ∈ insert i u, (f k).IsPrime
hiu : i ∉ u
hn' : (insert i u).card = n
⊢ ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ k ∈ insert i ↑u, ↑(f k)
|
simp only [Set.biUnion_insert] at h ⊢
|
ι : Type u_1
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, (f i).IsPrime) →
s.card = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
a b i j : ι
hfji : f j ≤ f i
u : Finset ι
hju : j ∉ u
hit : i ∉ insert j u
hn : (insert j u).card = n
hp : (f i).IsPrime ∧ ∀ x ∈ insert j u, (f x).IsPrime
hjt : j ∈ insert j u
hp' : ∀ k ∈ insert i u, (f k).IsPrime
hiu : i ∉ u
hn' : (insert i u).card = n
h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ (↑(f i) ∪ (↑(f j) ∪ ⋃ x ∈ ↑u, ↑(f x)))
⊢ ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ (↑(f i) ∪ ⋃ x ∈ ↑u, ↑(f x))
|
fa41770a7d2d9fd9
|
AlgebraicGeometry.Scheme.GlueData.isOpen_iff
|
Mathlib/AlgebraicGeometry/Gluing.lean
|
theorem isOpen_iff (U : Set D.glued.carrier) : IsOpen U ↔ ∀ i, IsOpen ((D.ι i).base ⁻¹' U)
|
D : GlueData
U : Set ↑↑D.glued.toPresheafedSpace
⊢ (∀ (i : D.toLocallyRingedSpaceGlueData.toSheafedSpaceGlueData.toPresheafedSpaceGlueData.toTopGlueData.J),
IsOpen
(⇑(ConcreteCategory.hom
(D.toLocallyRingedSpaceGlueData.toSheafedSpaceGlueData.toPresheafedSpaceGlueData.toTopGlueData.ι i)) ⁻¹'
(⇑(TopCat.homeoOfIso D.isoCarrier.symm) ⁻¹' U))) ↔
∀ (i : D.J), IsOpen (⇑(ConcreteCategory.hom (D.ι i).base) ⁻¹' U)
|
apply forall_congr'
|
case h
D : GlueData
U : Set ↑↑D.glued.toPresheafedSpace
⊢ ∀ (a : D.toLocallyRingedSpaceGlueData.toSheafedSpaceGlueData.toPresheafedSpaceGlueData.toTopGlueData.J),
IsOpen
(⇑(ConcreteCategory.hom
(D.toLocallyRingedSpaceGlueData.toSheafedSpaceGlueData.toPresheafedSpaceGlueData.toTopGlueData.ι a)) ⁻¹'
(⇑(TopCat.homeoOfIso D.isoCarrier.symm) ⁻¹' U)) ↔
IsOpen (⇑(ConcreteCategory.hom (D.ι a).base) ⁻¹' U)
|
a33f798209b44a35
|
RingHom.isStandardSmoothOfRelativeDimension_isStableUnderBaseChange
|
Mathlib/RingTheory/RingHom/StandardSmooth.lean
|
lemma isStandardSmoothOfRelativeDimension_isStableUnderBaseChange :
IsStableUnderBaseChange (@IsStandardSmoothOfRelativeDimension.{t, w} n)
|
case h.e'_6
n : ℕ
R S T : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
h : Algebra.IsStandardSmoothOfRelativeDimension n R T
this : Algebra.IsStandardSmoothOfRelativeDimension n S (S ⊗[R] T)
⊢ Algebra.TensorProduct.includeLeftRingHom.toAlgebra = Algebra.TensorProduct.leftAlgebra
|
ext
|
case h.e'_6.h
n : ℕ
R S T : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
h : Algebra.IsStandardSmoothOfRelativeDimension n R T
this : Algebra.IsStandardSmoothOfRelativeDimension n S (S ⊗[R] T)
r✝ : S
x✝ : S ⊗[R] T
⊢ (let_fun I := Algebra.TensorProduct.includeLeftRingHom.toAlgebra;
r✝ • x✝) =
r✝ • x✝
|
14172c066b23aad2
|
Batteries.RBNode.Ordered.memP_iff_upperBound?
|
Mathlib/.lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean
|
theorem Ordered.memP_iff_upperBound? [@TransCmp α cmp] [IsCut cmp cut] (ht : Ordered cmp t) :
t.MemP cut ↔ ∃ x, t.upperBound? cut = some x ∧ cut x = .eq
|
α : Type u_1
cmp : α → α → Ordering
cut : α → Ordering
t : RBNode α
inst✝¹ : TransCmp cmp
inst✝ : IsCut cmp cut
ht : Ordered cmp t
x✝ : ∃ x, x ∈ t ∧ cut x = Ordering.eq
y : α
hy : y ∈ t
ey : cut y = Ordering.eq
x : α
hx : upperBound? cut t = some x
⊢ cut x = Ordering.eq
|
cases ex : cut x
|
case lt
α : Type u_1
cmp : α → α → Ordering
cut : α → Ordering
t : RBNode α
inst✝¹ : TransCmp cmp
inst✝ : IsCut cmp cut
ht : Ordered cmp t
x✝ : ∃ x, x ∈ t ∧ cut x = Ordering.eq
y : α
hy : y ∈ t
ey : cut y = Ordering.eq
x : α
hx : upperBound? cut t = some x
ex : cut x = Ordering.lt
⊢ Ordering.lt = Ordering.eq
case eq
α : Type u_1
cmp : α → α → Ordering
cut : α → Ordering
t : RBNode α
inst✝¹ : TransCmp cmp
inst✝ : IsCut cmp cut
ht : Ordered cmp t
x✝ : ∃ x, x ∈ t ∧ cut x = Ordering.eq
y : α
hy : y ∈ t
ey : cut y = Ordering.eq
x : α
hx : upperBound? cut t = some x
ex : cut x = Ordering.eq
⊢ Ordering.eq = Ordering.eq
case gt
α : Type u_1
cmp : α → α → Ordering
cut : α → Ordering
t : RBNode α
inst✝¹ : TransCmp cmp
inst✝ : IsCut cmp cut
ht : Ordered cmp t
x✝ : ∃ x, x ∈ t ∧ cut x = Ordering.eq
y : α
hy : y ∈ t
ey : cut y = Ordering.eq
x : α
hx : upperBound? cut t = some x
ex : cut x = Ordering.gt
⊢ Ordering.gt = Ordering.eq
|
34c79185e82ce46b
|
Std.Tactic.BVDecide.LRAT.Internal.CNF.convertLRAT_readyForRatAdd
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Convert.lean
|
theorem CNF.convertLRAT_readyForRatAdd (cnf : CNF Nat) :
DefaultFormula.ReadyForRatAdd (CNF.convertLRAT cnf)
|
cnf : CNF Nat
⊢ (convertLRAT cnf).ReadyForRatAdd
|
unfold CNF.convertLRAT
|
cnf : CNF Nat
⊢ (let lifted := lift cnf;
let lratCnf := convertLRAT' lifted;
DefaultFormula.ofArray (none :: lratCnf).toArray).ReadyForRatAdd
|
6721871e1b681c10
|
Continuous.continuousOn
|
Mathlib/Topology/ContinuousOn.lean
|
theorem Continuous.continuousOn (h : Continuous f) : ContinuousOn f s
|
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
s : Set α
h : Continuous f
⊢ ContinuousOn f s
|
rw [continuous_iff_continuousOn_univ] at h
|
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
s : Set α
h : ContinuousOn f univ
⊢ ContinuousOn f s
|
f5a067e05d3992af
|
CategoryTheory.Subgroupoid.mul_mem_cancel_left
|
Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean
|
theorem mul_mem_cancel_left {c d e : C} {f : c ⟶ d} {g : d ⟶ e} (hf : f ∈ S.arrows c d) :
f ≫ g ∈ S.arrows c e ↔ g ∈ S.arrows d e
|
C : Type u
inst✝ : Groupoid C
S : Subgroupoid C
c d e : C
f : c ⟶ d
g : d ⟶ e
hf : f ∈ S.arrows c d
⊢ f ≫ g ∈ S.arrows c e ↔ g ∈ S.arrows d e
|
constructor
|
case mp
C : Type u
inst✝ : Groupoid C
S : Subgroupoid C
c d e : C
f : c ⟶ d
g : d ⟶ e
hf : f ∈ S.arrows c d
⊢ f ≫ g ∈ S.arrows c e → g ∈ S.arrows d e
case mpr
C : Type u
inst✝ : Groupoid C
S : Subgroupoid C
c d e : C
f : c ⟶ d
g : d ⟶ e
hf : f ∈ S.arrows c d
⊢ g ∈ S.arrows d e → f ≫ g ∈ S.arrows c e
|
1dcdf4857cfeffc1
|
Complex.isConnected_of_lowerHalfPlane
|
Mathlib/Analysis/Complex/Convex.lean
|
lemma Complex.isConnected_of_lowerHalfPlane {r} {s : Set ℂ} (hs₁ : {z | z.im < r} ⊆ s)
(hs₂ : s ⊆ {z | z.im ≤ r}) : IsConnected s
|
r : ℝ
s : Set ℂ
hs₁ : {z | z.im < r} ⊆ s
hs₂ : s ⊆ {z | z.im ≤ r}
⊢ IsConnected {z | z.im < r}
|
exact (convex_halfSpace_im_lt r).isConnected ⟨(r - 1) * I, by simp⟩
|
no goals
|
f2eeaafd90e9656f
|
Group.ext
|
Mathlib/Algebra/Group/Ext.lean
|
theorem Group.ext {G : Type*} ⦃g₁ g₂ : Group G⦄ (h_mul : g₁.mul = g₂.mul) : g₁ = g₂
|
G : Type u_1
g₁ g₂ : Group G
h_mul : Mul.mul = Mul.mul
h₁ : One.one = One.one
⊢ g₁ = g₂
|
let f : @MonoidHom G G g₁.toMulOneClass g₂.toMulOneClass :=
@MonoidHom.mk _ _ (_) _ (@OneHom.mk _ _ (_) _ id h₁)
(fun x y => congr_fun (congr_fun h_mul x) y)
|
G : Type u_1
g₁ g₂ : Group G
h_mul : Mul.mul = Mul.mul
h₁ : One.one = One.one
f : G →* G := { toFun := id, map_one' := h₁, map_mul' := ⋯ }
⊢ g₁ = g₂
|
40365b7f868018d4
|
Filter.HasBasis.equicontinuousAt_iff_right
|
Mathlib/Topology/UniformSpace/Equicontinuity.lean
|
theorem Filter.HasBasis.equicontinuousAt_iff_right {p : κ → Prop} {s : κ → Set (α × α)}
{F : ι → X → α} {x₀ : X} (hα : (𝓤 α).HasBasis p s) :
EquicontinuousAt F x₀ ↔ ∀ k, p k → ∀ᶠ x in 𝓝 x₀, ∀ i, (F i x₀, F i x) ∈ s k
|
ι : Type u_1
κ : Type u_2
X : Type u_3
α : Type u_6
tX : TopologicalSpace X
uα : UniformSpace α
p : κ → Prop
s : κ → Set (α × α)
F : ι → X → α
x₀ : X
hα : (𝓤 α).HasBasis p s
⊢ EquicontinuousAt F x₀ ↔ ∀ (k : κ), p k → ∀ᶠ (x : X) in 𝓝 x₀, ∀ (i : ι), (F i x₀, F i x) ∈ s k
|
rw [equicontinuousAt_iff_continuousAt, ContinuousAt,
(UniformFun.hasBasis_nhds_of_basis ι α _ hα).tendsto_right_iff]
|
ι : Type u_1
κ : Type u_2
X : Type u_3
α : Type u_6
tX : TopologicalSpace X
uα : UniformSpace α
p : κ → Prop
s : κ → Set (α × α)
F : ι → X → α
x₀ : X
hα : (𝓤 α).HasBasis p s
⊢ (∀ (i : κ),
p i → ∀ᶠ (x : X) in 𝓝 x₀, (⇑ofFun ∘ swap F) x ∈ {g | ((⇑ofFun ∘ swap F) x₀, g) ∈ UniformFun.gen ι α (s i)}) ↔
∀ (k : κ), p k → ∀ᶠ (x : X) in 𝓝 x₀, ∀ (i : ι), (F i x₀, F i x) ∈ s k
|
bb2ea194c9e25e81
|
Real.Angle.arg_toCircle
|
Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean
|
@[simp] lemma arg_toCircle (θ : Real.Angle) : (arg θ.toCircle : Angle) = θ
|
θ : Angle
⊢ ↑(↑θ.toCircle).arg = θ
|
induction θ using Real.Angle.induction_on
|
case h
x✝ : ℝ
⊢ ↑(↑(↑x✝).toCircle).arg = ↑x✝
|
f3f2bde3cb20d916
|
sign_finRotate
|
Mathlib/GroupTheory/Perm/Fin.lean
|
theorem sign_finRotate (n : ℕ) : Perm.sign (finRotate (n + 1)) = (-1) ^ n
|
case succ
n : ℕ
ih : sign (finRotate (n + 1)) = (-1) ^ n
⊢ sign (finRotate (n + 1 + 1)) = (-1) ^ (n + 1)
|
rw [finRotate_succ_eq_decomposeFin]
|
case succ
n : ℕ
ih : sign (finRotate (n + 1)) = (-1) ^ n
⊢ sign (decomposeFin.symm (1, finRotate (n + 1))) = (-1) ^ (n + 1)
|
f9f2177ec4cba59d
|
zeta_nat_eq_tsum_of_gt_one
|
Mathlib/NumberTheory/LSeries/RiemannZeta.lean
|
theorem zeta_nat_eq_tsum_of_gt_one {k : ℕ} (hk : 1 < k) :
riemannZeta k = ∑' n : ℕ, 1 / (n : ℂ) ^ k
|
k : ℕ
hk : 1 < k
⊢ 1 < (↑k).re
|
rwa [← ofReal_natCast, ofReal_re, ← Nat.cast_one, Nat.cast_lt]
|
no goals
|
a3b6be3587306bf5
|
List.not_lex_antisymm
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/BasicAux.lean
|
theorem not_lex_antisymm [DecidableEq α] {r : α → α → Prop} [DecidableRel r]
(antisymm : ∀ x y : α, ¬ r x y → ¬ r y x → x = y)
{as bs : List α} (h₁ : ¬ Lex r bs as) (h₂ : ¬ Lex r as bs) : as = bs :=
match as, bs with
| [], [] => rfl
| [], _::_ => False.elim <| h₂ (List.Lex.nil ..)
| _::_, [] => False.elim <| h₁ (List.Lex.nil ..)
| a::as, b::bs => by
by_cases hab : r a b
· exact False.elim <| h₂ (List.Lex.rel hab)
· by_cases eq : a = b
· subst eq
have h₁ : ¬ Lex r bs as := fun h => h₁ (List.Lex.cons h)
have h₂ : ¬ Lex r as bs := fun h => h₂ (List.Lex.cons h)
simp [not_lex_antisymm antisymm h₁ h₂]
· exfalso
by_cases hba : r b a
· exact h₁ (Lex.rel hba)
· exact eq (antisymm _ _ hab hba)
|
case pos
α : Type u_1
inst✝¹ : DecidableEq α
r : α → α → Prop
inst✝ : DecidableRel r
antisymm : ∀ (x y : α), ¬r x y → ¬r y x → x = y
as✝ bs✝ : List α
a : α
as : List α
b : α
bs : List α
h₁ : ¬Lex r (b :: bs) (a :: as)
h₂ : ¬Lex r (a :: as) (b :: bs)
hab : ¬r a b
eq : ¬a = b
hba : r b a
⊢ False
|
exact h₁ (Lex.rel hba)
|
no goals
|
bc20d604ba52bc59
|
Set.Iic_mul_Iio_subset'
|
Mathlib/Algebra/Order/Group/Pointwise/Interval.lean
|
theorem Iic_mul_Iio_subset' (a b : α) : Iic a * Iio b ⊆ Iio (a * b)
|
case intro.intro.intro.intro
α : Type u_1
inst✝³ : Mul α
inst✝² : PartialOrder α
inst✝¹ : MulLeftStrictMono α
inst✝ : MulRightStrictMono α
a b : α
this : MulRightMono α
y : α
hya : y ∈ Iic a
z : α
hzb : z ∈ Iio b
⊢ (fun x1 x2 => x1 * x2) y z ∈ Iio (a * b)
|
exact mul_lt_mul_of_le_of_lt hya hzb
|
no goals
|
d29ae3cc2f5221d0
|
SimpleGraph.IsAcyclic.path_unique
|
Mathlib/Combinatorics/SimpleGraph/Acyclic.lean
|
theorem IsAcyclic.path_unique {G : SimpleGraph V} (h : G.IsAcyclic) {v w : V} (p q : G.Path v w) :
p = q
|
case mk.mk.cons.inl.cons.inl.inl.refl
V : Type u
G : SimpleGraph V
v w u✝ v✝ w✝ : V
ph : G.Adj u✝ v✝
p : G.Walk v✝ w✝
ih : p.IsPath → ∀ (q : G.Walk v✝ w✝), q.IsPath → p = q
hp : p.IsPath ∧ u✝ ∉ p.support
h✝ : G.Adj u✝ v✝
q : G.Walk v✝ w✝
hq : q.IsPath ∧ u✝ ∉ q.support
⊢ cons ph p = cons h✝ q
|
cases ih hp.1 q hq.1
|
case mk.mk.cons.inl.cons.inl.inl.refl.refl
V : Type u
G : SimpleGraph V
v w u✝ v✝ w✝ : V
ph : G.Adj u✝ v✝
p : G.Walk v✝ w✝
ih : p.IsPath → ∀ (q : G.Walk v✝ w✝), q.IsPath → p = q
hp : p.IsPath ∧ u✝ ∉ p.support
h✝ : G.Adj u✝ v✝
hq : p.IsPath ∧ u✝ ∉ p.support
⊢ cons ph p = cons h✝ p
|
c6bceaf8dd6ef474
|
Ordinal.card_iSup_Iio_le_card_mul_iSup
|
Mathlib/SetTheory/Cardinal/Arithmetic.lean
|
theorem card_iSup_Iio_le_card_mul_iSup {o : Ordinal.{u}} (f : Iio o → Ordinal.{max u v}) :
(⨆ a : Iio o, f a).card ≤ Cardinal.lift.{v} o.card * ⨆ a : Iio o, (f a).card
|
case h.e'_4.h.e'_5.h.e'_1
o : Ordinal.{u}
f : ↑(Iio o) → Ordinal.{max u v}
⊢ #o.toType = o.card
|
exact mk_toType o
|
no goals
|
c3d702eab5714f39
|
RingHom.finiteType_ofLocalizationSpanTarget
|
Mathlib/RingTheory/RingHom/FiniteType.lean
|
theorem finiteType_ofLocalizationSpanTarget : OfLocalizationSpanTarget @FiniteType
|
case h
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset S
hs : Ideal.span ↑s = ⊤
this : Algebra R S := f.toAlgebra
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.linearCombination S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x : S
⊢ x ∈ Algebra.adjoin R ↑(s.attach.biUnion sf ∪ s ∪ Finset.image (⇑l) l.support)
|
apply Subalgebra.mem_of_span_eq_top_of_smul_pow_mem _ (s : Set S) l hl _ _ x _
|
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset S
hs : Ideal.span ↑s = ⊤
this : Algebra R S := f.toAlgebra
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.linearCombination S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x : S
⊢ ↑s ⊆ ↑(Algebra.adjoin R ↑(s.attach.biUnion sf ∪ s ∪ Finset.image (⇑l) l.support))
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset S
hs : Ideal.span ↑s = ⊤
this : Algebra R S := f.toAlgebra
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.linearCombination S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x : S
⊢ ∀ (i : ↑↑s), l i ∈ Algebra.adjoin R ↑(s.attach.biUnion sf ∪ s ∪ Finset.image (⇑l) l.support)
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset S
hs : Ideal.span ↑s = ⊤
this : Algebra R S := f.toAlgebra
t : (r : { x // x ∈ s }) → Finset (Localization.Away ↑r)
ht : ∀ (r : { x // x ∈ s }), Algebra.adjoin R ↑(t r) = ⊤
l : ↑↑s →₀ S
hl : (Finsupp.linearCombination S Subtype.val) l = 1
sf : { x // x ∈ s } → Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers ↑x) (t x)
x : S
⊢ ∀ (r : ↑↑s), ∃ n, ↑r ^ n • x ∈ Algebra.adjoin R ↑(s.attach.biUnion sf ∪ s ∪ Finset.image (⇑l) l.support)
|
2dc9a560bb7745cb
|
Normal.of_isSplittingField
|
Mathlib/FieldTheory/Normal/Basic.lean
|
theorem Normal.of_isSplittingField (p : F[X]) [hFEp : IsSplittingField F E p] : Normal F E
|
case inr.refine_1
F : Type u_1
inst✝² : Field F
E : Type u_3
inst✝¹ : Field E
inst✝ : Algebra F E
p : F[X]
hFEp : IsSplittingField F E p
hp : p ≠ 0
x : E
this : FiniteDimensional F E
hx : IsIntegral F x
L : Type u_1 := (p * minpoly F x).SplittingField
⊢ p ≠ 0 ∧ minpoly F x ≠ 0
|
exact ⟨hp, minpoly.ne_zero hx⟩
|
no goals
|
0aed749abc5ecaab
|
Commute.geom_sum₂
|
Mathlib/Algebra/GeomSum.lean
|
theorem Commute.geom_sum₂ [DivisionRing α] {x y : α} (h' : Commute x y) (h : x ≠ y)
(n : ℕ) : ∑ i ∈ range n, x ^ i * y ^ (n - 1 - i) = (x ^ n - y ^ n) / (x - y)
|
α : Type u
inst✝ : DivisionRing α
x y : α
h' : Commute x y
h : x ≠ y
n : ℕ
⊢ ∑ i ∈ range n, x ^ i * y ^ (n - 1 - i) = (x ^ n - y ^ n) / (x - y)
|
have : x - y ≠ 0 := by simp_all [sub_eq_iff_eq_add]
|
α : Type u
inst✝ : DivisionRing α
x y : α
h' : Commute x y
h : x ≠ y
n : ℕ
this : x - y ≠ 0
⊢ ∑ i ∈ range n, x ^ i * y ^ (n - 1 - i) = (x ^ n - y ^ n) / (x - y)
|
3c2f97cccd2e98b3
|
Polynomial.contract_mul_expand
|
Mathlib/Algebra/Polynomial/Expand.lean
|
theorem contract_mul_expand {p : ℕ} (hp : p ≠ 0) (f g : R[X]) :
contract p (f * expand R p g) = contract p f * g
|
case a.hf
R : Type u
inst✝ : CommSemiring R
p : ℕ
hp : p ≠ 0
f g : R[X]
n : ℕ
⊢ ∀ (x : ℕ × ℕ),
x.1 + x.2 = n * p → (¬∃ a, a.1 + a.2 = n ∧ (a.1 * p, a.2 * p) = x) → f.coeff x.1 * ((expand R p) g).coeff x.2 = 0
|
intro ⟨x, y⟩ eq nex
|
case a.hf
R : Type u
inst✝ : CommSemiring R
p : ℕ
hp : p ≠ 0
f g : R[X]
n x y : ℕ
eq : (x, y).1 + (x, y).2 = n * p
nex : ¬∃ a, a.1 + a.2 = n ∧ (a.1 * p, a.2 * p) = (x, y)
⊢ f.coeff (x, y).1 * ((expand R p) g).coeff (x, y).2 = 0
|
c12f546c03101c96
|
MulAction.mk'
|
Mathlib/GroupTheory/GroupAction/Primitive.lean
|
theorem mk' (Hnt : fixedPoints G X ≠ ⊤)
(H : ∀ {B : Set X} (_ : IsBlock G B), IsTrivialBlock B) :
IsPreprimitive G X
|
G : Type u_1
X : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G X
Hnt : fixedPoints G X ≠ ⊤
H : ∀ {B : Set X}, IsBlock G B → IsTrivialBlock B
⊢ IsPreprimitive G X
|
simp only [Set.top_eq_univ, Set.ne_univ_iff_exists_not_mem] at Hnt
|
G : Type u_1
X : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G X
H : ∀ {B : Set X}, IsBlock G B → IsTrivialBlock B
Hnt : ∃ a, a ∉ fixedPoints G X
⊢ IsPreprimitive G X
|
93a1b4b478f7903a
|
Polynomial.sum_bernoulli
|
Mathlib/NumberTheory/BernoulliPolynomials.lean
|
theorem sum_bernoulli (n : ℕ) :
(∑ k ∈ range (n + 1), ((n + 1).choose k : ℚ) • bernoulli k) = monomial n (n + 1 : ℚ)
|
n : ℕ
⊢ ∑ x ∈ range (n + 1),
∑ x_1 ∈ range (n + 1 - x), (↑((n + 1 - x).choose x_1) * _root_.bernoulli x_1) • (monomial x) ↑((n + 1).choose x) =
(monomial n) ↑n + (monomial n) 1
|
simp_rw [← sum_smul]
|
n : ℕ
⊢ ∑ x ∈ range (n + 1),
(∑ i ∈ range (n + 1 - x), ↑((n + 1 - x).choose i) * _root_.bernoulli i) • (monomial x) ↑((n + 1).choose x) =
(monomial n) ↑n + (monomial n) 1
|
b01e377955c96550
|
SimpleGraph.Subgraph.map_mono
|
Mathlib/Combinatorics/SimpleGraph/Subgraph.lean
|
@[gcongr] lemma map_mono {H₁ H₂ : G.Subgraph} (hH : H₁ ≤ H₂) : H₁.map f ≤ H₂.map f
|
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
f : G →g G'
H₁ H₂ : G.Subgraph
hH : H₁ ≤ H₂
⊢ Subgraph.map f H₁ ≤ Subgraph.map f H₂
|
constructor
|
case left
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
f : G →g G'
H₁ H₂ : G.Subgraph
hH : H₁ ≤ H₂
⊢ (Subgraph.map f H₁).verts ⊆ (Subgraph.map f H₂).verts
case right
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
f : G →g G'
H₁ H₂ : G.Subgraph
hH : H₁ ≤ H₂
⊢ ∀ ⦃v w : W⦄, (Subgraph.map f H₁).Adj v w → (Subgraph.map f H₂).Adj v w
|
45e01797f0297c07
|
LSeriesSummable.congr'
|
Mathlib/NumberTheory/LSeries/Basic.lean
|
/-- If `f` and `g` agree on large `n : ℕ` and the `LSeries` of `f` converges at `s`,
then so does that of `g`. -/
lemma LSeriesSummable.congr' {f g : ℕ → ℂ} (s : ℂ) (h : f =ᶠ[atTop] g) (hf : LSeriesSummable f s) :
LSeriesSummable g s
|
case intro.intro
f g : ℕ → ℂ
s : ℂ
hf : LSeriesSummable f s
S : Set ℕ
hS : S ∈ cofinite
hS' : Set.EqOn f g S
n : ℕ
hn : n ∈ S ∧ ¬n = 0
⊢ term f s n = term g s n
|
simp [hn.2, hS' hn.1]
|
no goals
|
549909a247ffe2bd
|
ProbabilityTheory.Kernel.measurable_kernel_prod_mk_left_of_finite
|
Mathlib/Probability/Kernel/MeasurableLIntegral.lean
|
theorem measurable_kernel_prod_mk_left_of_finite {t : Set (α × β)} (ht : MeasurableSet t)
(hκs : ∀ a, IsFiniteMeasure (κ a)) : Measurable fun a => κ a (Prod.mk a ⁻¹' t)
|
case compl
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
t✝ : Set (α × β)
hκs : ∀ (a : α), IsFiniteMeasure (κ a)
t : Set (α × β)
htm : MeasurableSet t
iht : Measurable fun a => (κ a) (Prod.mk a ⁻¹' t)
h_eq_sdiff : ∀ (a : α), Prod.mk a ⁻¹' tᶜ = univ \ Prod.mk a ⁻¹' t
this : (fun a => (κ a) (univ \ Prod.mk a ⁻¹' t)) = fun a => (κ a) univ - (κ a) (Prod.mk a ⁻¹' t)
⊢ Measurable fun a => (κ a) (univ \ Prod.mk a ⁻¹' t)
|
rw [this]
|
case compl
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
t✝ : Set (α × β)
hκs : ∀ (a : α), IsFiniteMeasure (κ a)
t : Set (α × β)
htm : MeasurableSet t
iht : Measurable fun a => (κ a) (Prod.mk a ⁻¹' t)
h_eq_sdiff : ∀ (a : α), Prod.mk a ⁻¹' tᶜ = univ \ Prod.mk a ⁻¹' t
this : (fun a => (κ a) (univ \ Prod.mk a ⁻¹' t)) = fun a => (κ a) univ - (κ a) (Prod.mk a ⁻¹' t)
⊢ Measurable fun a => (κ a) univ - (κ a) (Prod.mk a ⁻¹' t)
|
4c14e8ca6e5d1721
|
MeasureTheory.setToFun_congr_measure_of_integrable
|
Mathlib/MeasureTheory/Integral/SetToL1.lean
|
theorem setToFun_congr_measure_of_integrable {μ' : Measure α} (c' : ℝ≥0∞) (hc' : c' ≠ ∞)
(hμ'_le : μ' ≤ c' • μ) (hT : DominatedFinMeasAdditive μ T C)
(hT' : DominatedFinMeasAdditive μ' T C') (f : α → E) (hfμ : Integrable f μ) :
setToFun μ T hT f = setToFun μ' T hT' f
|
case h_closed
α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T : Set α → E →L[ℝ] F
C C' : ℝ
μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ⊤
hμ'_le : μ' ≤ c' • μ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ' T C'
f : α → E
hfμ : Integrable f μ
h_int : ∀ (g : α → E), Integrable g μ → Integrable g μ'
this : (fun f => setToFun μ' T hT' ↑↑f) = fun f => setToFun μ' T hT' ↑↑(Integrable.toL1 ↑↑f ⋯)
⊢ Continuous fun f => setToFun μ' T hT' ↑↑(Integrable.toL1 ↑↑f ⋯)
|
exact (continuous_setToFun hT').comp (continuous_L1_toL1 c' hc' hμ'_le)
|
no goals
|
65e35cd00eda3e5b
|
PartENat.withTopEquiv_symm_coe
|
Mathlib/Data/Nat/PartENat.lean
|
theorem withTopEquiv_symm_coe (n : Nat) : withTopEquiv.symm n = n
|
n : ℕ
⊢ withTopEquiv.symm ↑n = ↑n
|
simp
|
no goals
|
645a799cca9575eb
|
MeromorphicAt.order_smul
|
Mathlib/Analysis/Meromorphic/Order.lean
|
theorem order_smul {f : 𝕜 → 𝕜} {g : 𝕜 → E} {x : 𝕜}
(hf : MeromorphicAt f x) (hg : MeromorphicAt g x) :
(hf.smul hg).order = hf.order + hg.order
|
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → 𝕜
g : 𝕜 → E
x : 𝕜
hf : MeromorphicAt f x
hg : MeromorphicAt g x
m : ℤ
h₂f : hf.order = ↑m
h₂g : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, g z = 0
z : 𝕜
hz : g z = 0
⊢ (f • g) z = 0
|
simp [hz]
|
no goals
|
888980cd950630aa
|
CategoryTheory.Iso.refl_conj
|
Mathlib/CategoryTheory/Conj.lean
|
theorem refl_conj (f : End X) : (Iso.refl X).conj f = f
|
C : Type u
inst✝ : Category.{v, u} C
X : C
f : End X
⊢ (refl X).conj f = f
|
rw [conj_apply, Iso.refl_inv, Iso.refl_hom, Category.id_comp, Category.comp_id]
|
no goals
|
9620a46c53a58c1f
|
Module.finite_dual_iff
|
Mathlib/LinearAlgebra/Dual.lean
|
theorem finite_dual_iff [Free K V] : Module.Finite K (Dual K V) ↔ Module.Finite K V
|
case mk.intro.refine_1
K : Type uK
V : Type uV
inst✝³ : CommRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : Free K V
ι : Type uV
b : Basis ι K V
a✝ : Nontrivial K
s : Finset (Dual K V)
span_s : span K ↑s = ⊤
⊢ Set.range (⇑b.toDual ∘ ⇑b) ≤ ↑(span K ↑s)
|
rw [span_s]
|
case mk.intro.refine_1
K : Type uK
V : Type uV
inst✝³ : CommRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : Free K V
ι : Type uV
b : Basis ι K V
a✝ : Nontrivial K
s : Finset (Dual K V)
span_s : span K ↑s = ⊤
⊢ Set.range (⇑b.toDual ∘ ⇑b) ≤ ↑⊤
|
1a833564690fbd59
|
LiouvilleWith.add_rat
|
Mathlib/NumberTheory/Transcendental/Liouville/LiouvilleWith.lean
|
theorem add_rat (h : LiouvilleWith p x) (r : ℚ) : LiouvilleWith p (x + r)
|
p x : ℝ
h : LiouvilleWith p x
r : ℚ
⊢ LiouvilleWith p (x + ↑r)
|
rcases h.exists_pos with ⟨C, _hC₀, hC⟩
|
case intro.intro
p x : ℝ
h : LiouvilleWith p x
r : ℚ
C : ℝ
_hC₀ : 0 < C
hC : ∃ᶠ (n : ℕ) in atTop, 1 ≤ n ∧ ∃ m, x ≠ ↑m / ↑n ∧ |x - ↑m / ↑n| < C / ↑n ^ p
⊢ LiouvilleWith p (x + ↑r)
|
7652465b354d5b90
|
List.sym2_eq_sym_two
|
Mathlib/Data/List/Sym.lean
|
theorem sym2_eq_sym_two : xs.sym2.map (Sym2.equivSym α) = xs.sym 2
|
case nil
α : Type u_1
xs : List α
⊢ map ⇑(Sym2.equivSym α) [].sym2 = List.sym 2 []
|
simp only [List.sym, map_eq_nil_iff, sym2_eq_nil_iff]
|
no goals
|
02f4fa6d82cb0101
|
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