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 |
|---|---|---|---|---|---|---|
List.mem_eraseIdx_iff_getElem? | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/Basic.lean | theorem mem_eraseIdx_iff_getElem? {x : α} {l} {k} : x ∈ eraseIdx l k ↔ ∃ i ≠ k, l[i]? = some x | α : Type u_1
x : α
l : List α
k : Nat
⊢ x ∈ l.eraseIdx k ↔ ∃ i, i ≠ k ∧ l[i]? = some x | simp only [mem_eraseIdx_iff_getElem, getElem_eq_iff, exists_and_left] | α : Type u_1
x : α
l : List α
k : Nat
⊢ (∃ i, i ≠ k ∧ ∃ x_1, l[i]? = some x) ↔ ∃ i, i ≠ k ∧ l[i]? = some x | b696caa5872e7aba |
Metric.toInductiveLimit_commute | Mathlib/Topology/MetricSpace/Gluing.lean | theorem toInductiveLimit_commute (I : ∀ n, Isometry (f n)) (n : ℕ) :
toInductiveLimit I n.succ ∘ f n = toInductiveLimit I n | X : ℕ → Type u
inst✝ : (n : ℕ) → MetricSpace (X n)
f : (n : ℕ) → X n → X (n + 1)
I : ∀ (n : ℕ), Isometry (f n)
n : ℕ
⊢ toInductiveLimit I n.succ ∘ f n = toInductiveLimit I n | let h := inductivePremetric I | X : ℕ → Type u
inst✝ : (n : ℕ) → MetricSpace (X n)
f : (n : ℕ) → X n → X (n + 1)
I : ∀ (n : ℕ), Isometry (f n)
n : ℕ
h : PseudoMetricSpace ((n : ℕ) × X n) := inductivePremetric I
⊢ toInductiveLimit I n.succ ∘ f n = toInductiveLimit I n | 235b808c7c2a0bb6 |
List.cons_le_cons_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lex.lean | theorem cons_le_cons_iff [DecidableEq α] [LT α] [DecidableLT α]
[i₀ : Std.Irrefl (· < · : α → α → Prop)]
[i₁ : Std.Asymm (· < · : α → α → Prop)]
[i₂ : Std.Antisymm (¬ · < · : α → α → Prop)]
{a b} {l₁ l₂ : List α} :
(a :: l₁) ≤ (b :: l₂) ↔ a < b ∨ a = b ∧ l₁ ≤ l₂ | case mp.inl.intro.h.h
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : LT α
inst✝ : DecidableLT α
i₀ : Std.Irrefl fun x1 x2 => x1 < x2
i₁ : Std.Asymm fun x1 x2 => x1 < x2
i₂ : Std.Antisymm fun x1 x2 => ¬x1 < x2
a b : α
l₁ l₂ : List α
h₁ : ¬b < a
h₂ : ¬b = a
h₃ : ¬a < b
⊢ False | apply h₂ | case mp.inl.intro.h.h
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : LT α
inst✝ : DecidableLT α
i₀ : Std.Irrefl fun x1 x2 => x1 < x2
i₁ : Std.Asymm fun x1 x2 => x1 < x2
i₂ : Std.Antisymm fun x1 x2 => ¬x1 < x2
a b : α
l₁ l₂ : List α
h₁ : ¬b < a
h₂ : ¬b = a
h₃ : ¬a < b
⊢ b = a | 75deba79ccf9ad38 |
Finite.surjective_of_injective | Mathlib/Data/Fintype/Card.lean | theorem surjective_of_injective {f : α → α} (hinj : Injective f) : Surjective f | α : Type u_1
inst✝ : Finite α
f : α → α
hinj : Injective f
x : α
this : (a : Prop) → Decidable a
⊢ ∃ a, f a = x | cases nonempty_fintype α | case intro
α : Type u_1
inst✝ : Finite α
f : α → α
hinj : Injective f
x : α
this : (a : Prop) → Decidable a
val✝ : Fintype α
⊢ ∃ a, f a = x | ed9038464ba3ead0 |
ProbabilityTheory.sum_meas_smul_cond_fiber | Mathlib/Probability/ConditionalProbability.lean | /-- The **law of total probability** for a random variable taking finitely many values: a measure
`μ` can be expressed as a linear combination of its conditional measures `μ[|X ← x]` on fibers of a
random variable `X` valued in a fintype. -/
lemma sum_meas_smul_cond_fiber {X : Ω → α} (hX : Measurable X) (μ : Measure Ω)... | Ω : Type u_1
α : Type u_3
m : MeasurableSpace Ω
inst✝³ : Fintype α
inst✝² : MeasurableSpace α
inst✝¹ : DiscreteMeasurableSpace α
X : Ω → α
hX : Measurable X
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
⊢ ∑ x : α, μ (X ⁻¹' {x}) • μ[|X ⁻¹' {x}] = μ | ext E hE | case h
Ω : Type u_1
α : Type u_3
m : MeasurableSpace Ω
inst✝³ : Fintype α
inst✝² : MeasurableSpace α
inst✝¹ : DiscreteMeasurableSpace α
X : Ω → α
hX : Measurable X
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
E : Set Ω
hE : MeasurableSet E
⊢ (∑ x : α, μ (X ⁻¹' {x}) • μ[|X ⁻¹' {x}]) E = μ E | 184ad315f536fdf8 |
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 | case intro.intro.intro.intro.intro.intro
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
b : Set (K ≃ₐ[k] K)
sub : b ⊆ (fun y => σ * y) ⁻¹' (↑H).carrierᶜ
gp : Subgroup (K... | have := hL.out | case intro.intro.intro.intro.intro.intro
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
b : Set (K ≃ₐ[k] K)
sub : b ⊆ (fun y => σ * y) ⁻¹' (↑H).carrierᶜ
gp : Subgroup (K... | 4195a2a4cd22a820 |
intervalIntegral.intervalIntegrable_log' | Mathlib/Analysis/SpecialFunctions/Integrals.lean | theorem intervalIntegrable_log' : IntervalIntegrable log volume a b | case h₂f.hab.h
a b x : ℝ
hx : 0 < x
⊢ IntervalIntegrable (-log) volume 0 1 | apply intervalIntegrable_deriv_of_nonneg (g := fun x ↦ -(x * log x - x)) | case h₂f.hab.h.hcont
a b x : ℝ
hx : 0 < x
⊢ ContinuousOn (fun x => -(x * log x - x)) [[0, 1]]
case h₂f.hab.h.hderiv
a b x : ℝ
hx : 0 < x
⊢ ∀ x ∈ Set.Ioo (0 ⊓ 1) (0 ⊔ 1), HasDerivAt (fun x => -(x * log x - x)) ((-log) x) x
case h₂f.hab.h.hpos
a b x : ℝ
hx : 0 < x
⊢ ∀ x ∈ Set.Ioo (0 ⊓ 1) (0 ⊔ 1), 0 ≤ (-log) x | d485782e2df5cd2b |
MvPolynomial.rename_eq | Mathlib/Algebra/MvPolynomial/Rename.lean | theorem rename_eq (f : σ → τ) (p : MvPolynomial σ R) :
rename f p = Finsupp.mapDomain (Finsupp.mapDomain f) p | σ : Type u_1
τ : Type u_2
R : Type u_4
inst✝ : CommSemiring R
f : σ → τ
p : MvPolynomial σ R
⊢ (sum p fun s a => (monomial (s.sum fun a => Finsupp.single (f a))) a) =
sum p fun a => Finsupp.single (a.sum fun a => Finsupp.single (f a)) | rfl | no goals | eccdbc558349cdfe |
FormalMultilinearSeries.comp_assoc | Mathlib/Analysis/Analytic/Composition.lean | theorem comp_assoc (r : FormalMultilinearSeries 𝕜 G H) (q : FormalMultilinearSeries 𝕜 F G)
(p : FormalMultilinearSeries 𝕜 E F) : (r.comp q).comp p = r.comp (q.comp p) | case h.H.mk
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : ... | intro j hj1 hj2 | case h.H.mk
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : ... | e6d21021eddd0ef4 |
isIntegralCurve_comp_sub | Mathlib/Geometry/Manifold/IntegralCurve/Transform.lean | lemma isIntegralCurve_comp_sub {dt : ℝ} :
IsIntegralCurve γ v ↔ IsIntegralCurve (γ ∘ (· - dt)) v | E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
H : Type u_2
inst✝² : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
γ : ℝ → M
v : (x : M) → TangentSpace I x
dt : ℝ
⊢ IsIntegralCurve γ v ↔ IsIntegralCurve (γ ∘ fun x => x - dt) v | simpa using isIntegralCurve_comp_add (dt := -dt) | no goals | 32dfe455607413c0 |
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 : i + j < as.length)
(hj : 0 < j) :
(as.set i as[i + j]).set (i + j) as[i] ~ as | α : Type u_1
as : List α
i j : Nat
h₁ : i < as.length
h₂ : i + j < as.length
hj : 0 < j
this : as = take i as ++ as[i] :: drop (i + 1) (take (i + j) as) ++ as[i + j] :: drop (i + j + 1) as
⊢ i - min i as.length = 0 | omega | no goals | 9b51820a9079599a |
Measurable.iSup_Prop | Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean | theorem Measurable.iSup_Prop {α} {mα : MeasurableSpace α} [ConditionallyCompleteLattice α]
(p : Prop) {f : δ → α} (hf : Measurable f) : Measurable fun b => ⨆ _ : p, f b | δ : Type u_4
mδ : MeasurableSpace δ
α : Type u_5
mα : MeasurableSpace α
inst✝ : ConditionallyCompleteLattice α
p : Prop
f : δ → α
hf : Measurable f
⊢ Measurable fun b => if h : p then f b else sSup ∅ | split_ifs with h | case pos
δ : Type u_4
mδ : MeasurableSpace δ
α : Type u_5
mα : MeasurableSpace α
inst✝ : ConditionallyCompleteLattice α
p : Prop
f : δ → α
hf : Measurable f
h : p
⊢ Measurable fun b => f b
case neg
δ : Type u_4
mδ : MeasurableSpace δ
α : Type u_5
mα : MeasurableSpace α
inst✝ : ConditionallyCompleteLattice α
p : Prop
f... | 5181856a76d27a1a |
ZNum.mod_to_int | Mathlib/Data/Num/Lemmas.lean | theorem mod_to_int : ∀ n d, ((n % d : ZNum) : ℤ) = n % d
| 0, _ => (Int.zero_emod _).symm
| pos n, d =>
(Num.cast_toZNum _).trans <| by
rw [← Num.to_nat_to_int, cast_pos, Num.mod_to_nat, ← PosNum.to_nat_to_int, abs_to_nat]
rfl
| neg n, d =>
(Num.cast_sub' _ _).trans <| by
rw [← Num.to_na... | n : PosNum
d : ZNum
⊢ ↑(↑(Num.pos n) % (↑d).natAbs) = ↑↑n % ↑d | rfl | no goals | 7156f7699ce96cc6 |
LinearMap.polyCharpoly_coeff_eq_zero_of_basis | Mathlib/Algebra/Module/LinearMap/Polynomial.lean | lemma polyCharpoly_coeff_eq_zero_of_basis (b : Basis ι R L) (b' : Basis ι' R L) (k : ℕ)
(H : (polyCharpoly φ b).coeff k = 0) :
(polyCharpoly φ b').coeff k = 0 | R : Type u_1
L : Type u_2
M : Type u_3
ι : Type u_5
ι' : Type u_6
inst✝¹⁰ : CommRing R
inst✝⁹ : AddCommGroup L
inst✝⁸ : Module R L
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
φ : L →ₗ[R] End R M
inst✝⁵ : Fintype ι
inst✝⁴ : Fintype ι'
inst✝³ : DecidableEq ι
inst✝² : DecidableEq ι'
inst✝¹ : Free R M
inst✝ : Module.Finite... | apply_fun (MvPolynomial.bind₁ g) at H | R : Type u_1
L : Type u_2
M : Type u_3
ι : Type u_5
ι' : Type u_6
inst✝¹⁰ : CommRing R
inst✝⁹ : AddCommGroup L
inst✝⁸ : Module R L
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
φ : L →ₗ[R] End R M
inst✝⁵ : Fintype ι
inst✝⁴ : Fintype ι'
inst✝³ : DecidableEq ι
inst✝² : DecidableEq ι'
inst✝¹ : Free R M
inst✝ : Module.Finite... | f34d7291222df9d6 |
MeasureTheory.Measure.haarMeasure_apply | Mathlib/MeasureTheory/Measure/Haar/Basic.lean | theorem haarMeasure_apply {K₀ : PositiveCompacts G} {s : Set G} (hs : MeasurableSet s) :
haarMeasure K₀ s = (haarContent K₀).outerMeasure s / (haarContent K₀).measure K₀ | G : Type u_1
inst✝⁴ : Group G
inst✝³ : TopologicalSpace G
inst✝² : IsTopologicalGroup G
inst✝¹ : MeasurableSpace G
inst✝ : BorelSpace G
K₀ : PositiveCompacts G
s : Set G
hs : MeasurableSet s
⊢ (haarMeasure K₀) s = (haarContent K₀).outerMeasure s / (haarContent K₀).measure ↑K₀ | change ((haarContent K₀).measure K₀)⁻¹ * (haarContent K₀).measure s = _ | G : Type u_1
inst✝⁴ : Group G
inst✝³ : TopologicalSpace G
inst✝² : IsTopologicalGroup G
inst✝¹ : MeasurableSpace G
inst✝ : BorelSpace G
K₀ : PositiveCompacts G
s : Set G
hs : MeasurableSet s
⊢ ((haarContent K₀).measure ↑K₀)⁻¹ * (haarContent K₀).measure s =
(haarContent K₀).outerMeasure s / (haarContent K₀).measure ... | cf45bef96e152173 |
SimpleGraph.isEmpty_of_colorable_zero | Mathlib/Combinatorics/SimpleGraph/Coloring.lean | theorem isEmpty_of_colorable_zero (h : G.Colorable 0) : IsEmpty V | case false
V : Type u
G : SimpleGraph V
h : G.Colorable 0
⊢ V → False | intro v | case false
V : Type u
G : SimpleGraph V
h : G.Colorable 0
v : V
⊢ False | 0a299945b9ed7b1a |
MvPolynomial.mapAlgEquiv_trans | Mathlib/Algebra/MvPolynomial/Equiv.lean | theorem mapAlgEquiv_trans (e : A₁ ≃ₐ[R] A₂) (f : A₂ ≃ₐ[R] A₃) :
(mapAlgEquiv σ e).trans (mapAlgEquiv σ f) = mapAlgEquiv σ (e.trans f) | case h.a
R : Type u
σ : Type u_1
inst✝⁶ : CommSemiring R
A₁ : Type u_2
A₂ : Type u_3
A₃ : Type u_4
inst✝⁵ : CommSemiring A₁
inst✝⁴ : CommSemiring A₂
inst✝³ : CommSemiring A₃
inst✝² : Algebra R A₁
inst✝¹ : Algebra R A₂
inst✝ : Algebra R A₃
e : A₁ ≃ₐ[R] A₂
f : A₂ ≃ₐ[R] A₃
a✝ : MvPolynomial σ A₁
m✝ : σ →₀ ℕ
⊢ coeff m✝ (((... | simp only [AlgEquiv.trans_apply, mapAlgEquiv_apply, map_map] | case h.a
R : Type u
σ : Type u_1
inst✝⁶ : CommSemiring R
A₁ : Type u_2
A₂ : Type u_3
A₃ : Type u_4
inst✝⁵ : CommSemiring A₁
inst✝⁴ : CommSemiring A₂
inst✝³ : CommSemiring A₃
inst✝² : Algebra R A₁
inst✝¹ : Algebra R A₂
inst✝ : Algebra R A₃
e : A₁ ≃ₐ[R] A₂
f : A₂ ≃ₐ[R] A₃
a✝ : MvPolynomial σ A₁
m✝ : σ →₀ ℕ
⊢ coeff m✝ ((m... | 0a9dc017bd662ba7 |
FirstOrder.Language.BoundedFormula.realize_mapTermRel_add_castLe | Mathlib/ModelTheory/Semantics.lean | theorem realize_mapTermRel_add_castLe [L'.Structure M] {k : ℕ}
{ft : ∀ n, L.Term (α ⊕ (Fin n)) → L'.Term (β ⊕ (Fin (k + n)))}
{fr : ∀ n, L.Relations n → L'.Relations n} {n} {φ : L.BoundedFormula α n}
(v : ∀ {n}, (Fin (k + n) → M) → α → M) {v' : β → M} (xs : Fin (k + n) → M)
(h1 :
∀ (n) (t : L.Term... | case imp
L : Language
L' : Language
M : Type w
inst✝¹ : L.Structure M
α : Type u'
β : Type v'
inst✝ : L'.Structure M
k : ℕ
ft : (n : ℕ) → L.Term (α ⊕ Fin n) → L'.Term (β ⊕ Fin (k + n))
fr : (n : ℕ) → L.Relations n → L'.Relations n
n : ℕ
v : {n : ℕ} → (Fin (k + n) → M) → α → M
v' : β → M
h1 :
∀ (n : ℕ) (t : L.Term (α ... | simp [mapTermRel, Realize, ih1, ih2] | no goals | ba6899fcea9af06b |
Pell.pos_generator_iff_fundamental | Mathlib/NumberTheory/Pell.lean | theorem pos_generator_iff_fundamental (a : Solution₁ d) :
(1 < a.x ∧ 0 < a.y ∧ ∀ b : Solution₁ d, ∃ n : ℤ, b = a ^ n ∨ b = -a ^ n) ↔ IsFundamental a | d : ℤ
a : Solution₁ d
h : 1 < a.x ∧ 0 < a.y ∧ ∀ (b : Solution₁ d), ∃ n, b = a ^ n ∨ b = -a ^ n
⊢ IsFundamental a | have h₀ := d_pos_of_one_lt_x h.1 | d : ℤ
a : Solution₁ d
h : 1 < a.x ∧ 0 < a.y ∧ ∀ (b : Solution₁ d), ∃ n, b = a ^ n ∨ b = -a ^ n
h₀ : 0 < d
⊢ IsFundamental a | b8f76271aac4cb87 |
FirstOrder.Language.Theory.imp_sup_left | Mathlib/ModelTheory/Equivalence.lean | lemma imp_sup_left (φ ψ : L.BoundedFormula α n) : φ ⟹[T] φ ⊔ ψ := fun M v xs => by
simp only [BoundedFormula.realize_imp, BoundedFormula.realize_sup]
exact Or.inl
| L : Language
T : L.Theory
α : Type w
n : ℕ
φ ψ : L.BoundedFormula α n
M : T.ModelType
v : α → ↑M
xs : Fin n → ↑M
⊢ φ.Realize v xs → φ.Realize v xs ∨ ψ.Realize v xs | exact Or.inl | no goals | c41c16b12d86778c |
QPF.Wequiv.symm | Mathlib/Data/QPF/Univariate/Basic.lean | theorem Wequiv.symm (x y : q.P.W) : Wequiv x y → Wequiv y x | case abs
F : Type u → Type u
q : QPF F
x y : (P F).W
a : (P F).A
f : (P F).B a → (P F).W
a' : (P F).A
f' : (P F).B a' → (P F).W
h : QPF.abs ⟨a, f⟩ = QPF.abs ⟨a', f'⟩
⊢ Wequiv (WType.mk a' f') (WType.mk a f) | exact Wequiv.abs _ _ _ _ h.symm | no goals | 45cc86f6cee0bf26 |
Std.Tactic.BVDecide.BVExpr.bitblast.blastUmod.denote_go_eq_divRec_r | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/Umod.lean | theorem denote_go_eq_divRec_r (aig : AIG α) (assign : α → Bool) (curr : Nat) (lhs rhs rbv qbv : BitVec w)
(falseRef trueRef : AIG.Ref aig) (n d q r : AIG.RefVec aig w) (wn wr : Nat)
(hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, n.get idx hidx, assign⟧ = lhs.getLsbD idx)
(hright : ∀ (idx : Nat) (hidx : idx ... | case hright.hstart
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
assign : α → Bool
lhs rhs : BitVec w
curr : Nat
ih :
∀ (aig : AIG α) (rbv qbv : BitVec w) (falseRef trueRef : aig.Ref) (n d q r : aig.RefVec w) (wn wr : Nat),
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := n.get idx hidx... | simp [Ref.hgate] | no goals | 123e2e165ae62131 |
MeasureTheory.SignedMeasure.mutuallySingular_iff | Mathlib/MeasureTheory/Decomposition/Jordan.lean | theorem mutuallySingular_iff (s t : SignedMeasure α) :
s ⟂ᵥ t ↔ s.totalVariation ⟂ₘ t.totalVariation | case mp.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1
α : Type u_1
inst✝ : MeasurableSpace α
s t : SignedMeasure α
u : Set α
hmeas : MeasurableSet u
hu₁ : ∀ t ⊆ u, ↑s t = 0
hu₂ : ∀ t_1 ⊆ uᶜ, ↑t t_1 = 0
i : Set α
hi₁ : MeasurableSet i
hi₂ : 0 ≤[i] s
hi₃ : s ≤[iᶜ] 0
hipos : s.toJo... | simp [hu₁ _ Set.inter_subset_right] | no goals | ec18b191248f717a |
IsAlgClosed.ringEquiv_of_equiv_of_charZero | Mathlib/FieldTheory/IsAlgClosed/Classification.lean | theorem ringEquiv_of_equiv_of_charZero [CharZero K] [CharZero L] (hK : ℵ₀ < #K)
(hKL : Nonempty (K ≃ L)) : Nonempty (K ≃+* L) | case intro.intro.intro
K : Type u
L : Type v
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : IsAlgClosed K
inst✝² : IsAlgClosed L
inst✝¹ : CharZero K
inst✝ : CharZero L
hK : ℵ₀ < #K
hKL : Nonempty (K ≃ L)
s : Set K
hs : IsTranscendenceBasis ℤ Subtype.val
t : Set L
ht : IsTranscendenceBasis ℤ Subtype.val
hL : ℵ₀ < #L
this : ... | exact ⟨equivOfTranscendenceBasis _ _ e hs ht⟩ | no goals | 7b4ef9f678e59f88 |
MvPolynomial.leadingCoeff_toLex | Mathlib/Algebra/MvPolynomial/Variables.lean | lemma leadingCoeff_toLex : p.leadingCoeff toLex = p.coeff (ofLex <| p.supDegree toLex) | R : Type u
σ : Type u_1
inst✝¹ : CommSemiring R
p : MvPolynomial σ R
inst✝ : LinearOrder σ
⊢ leadingCoeff (⇑toLex) p = coeff (ofLex (supDegree (⇑toLex) p)) p | rw [leadingCoeff] | R : Type u
σ : Type u_1
inst✝¹ : CommSemiring R
p : MvPolynomial σ R
inst✝ : LinearOrder σ
⊢ p (invFun (⇑toLex) (supDegree (⇑toLex) p)) = coeff (ofLex (supDegree (⇑toLex) p)) p | 8879da913b438f75 |
Nat.nth_comp_of_strictMono | Mathlib/Data/Nat/Nth.lean | lemma nth_comp_of_strictMono {n : ℕ} {f : ℕ → ℕ} (hf : StrictMono f)
(h0 : ∀ k, p k → k ∈ Set.range f) (h : ∀ hfi : (setOf p).Finite, n < hfi.toFinset.card) :
f (nth (fun i ↦ p (f i)) n) = nth p n | case h
p : ℕ → Prop
n : ℕ
f : ℕ → ℕ
hf : StrictMono f
h0 : ∀ (k : ℕ), p k → k ∈ Set.range f
h : ∀ (hfi : (setOf p).Finite), n < #hfi.toFinset
p' : ℕ → Prop
h0p' : ∀ (k : ℕ), p' k → k ∈ Set.range f
i : ℕ
he : i ∈ setOf p'
⊢ i ∈ f '' {i | p' (f i)} | rcases h0p' _ he with ⟨t, rfl⟩ | case h.intro
p : ℕ → Prop
n : ℕ
f : ℕ → ℕ
hf : StrictMono f
h0 : ∀ (k : ℕ), p k → k ∈ Set.range f
h : ∀ (hfi : (setOf p).Finite), n < #hfi.toFinset
p' : ℕ → Prop
h0p' : ∀ (k : ℕ), p' k → k ∈ Set.range f
t : ℕ
he : f t ∈ setOf p'
⊢ f t ∈ f '' {i | p' (f i)} | 03d0809f8aba723f |
Ordnode.Valid'.rotateL | Mathlib/Data/Ordmap/Ordset.lean | theorem Valid'.rotateL {l} {x : α} {r o₁ o₂} (hl : Valid' o₁ l x) (hr : Valid' x r o₂)
(H1 : ¬size l + size r ≤ 1) (H2 : delta * size l < size r)
(H3 : 2 * size r ≤ 9 * size l + 5 ∨ size r ≤ 3) : Valid' o₁ (@rotateL α l x r) o₂ | case neg.inl
α : Type u_1
inst✝ : Preorder α
l : Ordnode α
x : α
o₁ : WithBot α
o₂ : WithTop α
hl : Valid' o₁ l ↑x
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
hr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂
H1 : ¬l.size + (Ordnode.node rs rl rx rr).size ≤ 1
H2 : delta * l.size ≤ rl.size + rr.size
H3 : 2 * (rl.size + rr.... | rw [rl0, not_lt, Nat.le_zero, Nat.mul_eq_zero] at h | case neg.inl
α : Type u_1
inst✝ : Preorder α
l : Ordnode α
x : α
o₁ : WithBot α
o₂ : WithTop α
hl : Valid' o₁ l ↑x
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
hr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂
H1 : ¬l.size + (Ordnode.node rs rl rx rr).size ≤ 1
H2 : delta * l.size ≤ rl.size + rr.size
H3 : 2 * (rl.size + rr.... | 0ebe7132c42c2866 |
AffineSubspace.isPreconnected_setOf_wOppSide | Mathlib/Analysis/Convex/Side.lean | theorem isPreconnected_setOf_wOppSide (s : AffineSubspace ℝ P) (x : P) :
IsPreconnected { y | s.WOppSide x y } | V : Type u_2
P : Type u_4
inst✝³ : SeminormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
s : AffineSubspace ℝ P
x : P
⊢ IsPreconnected {y | s.WOppSide x y} | rcases Set.eq_empty_or_nonempty (s : Set P) with (h | h) | case inl
V : Type u_2
P : Type u_4
inst✝³ : SeminormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
s : AffineSubspace ℝ P
x : P
h : ↑s = ∅
⊢ IsPreconnected {y | s.WOppSide x y}
case inr
V : Type u_2
P : Type u_4
inst✝³ : SeminormedAddCommGroup V
inst✝² : NormedSpace ... | 838ed4940b3b67ed |
YoungDiagram.transpose_le_iff | Mathlib/Combinatorics/Young/YoungDiagram.lean | theorem transpose_le_iff {μ ν : YoungDiagram} : μ.transpose ≤ ν.transpose ↔ μ ≤ ν :=
⟨fun h => by
convert YoungDiagram.le_of_transpose_le h
simp, fun h => by
rw [← transpose_transpose μ] at h
exact YoungDiagram.le_of_transpose_le h ⟩
| μ ν : YoungDiagram
h : μ.transpose.transpose ≤ ν
⊢ μ.transpose ≤ ν.transpose | exact YoungDiagram.le_of_transpose_le h | no goals | 06ad74b9927fc7bc |
Matrix.det_fromBlocks_zero₂₁ | Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean | theorem det_fromBlocks_zero₂₁ (A : Matrix m m R) (B : Matrix m n R) (D : Matrix n n R) :
(Matrix.fromBlocks A B 0 D).det = A.det * D.det | case h.e'_3.refine_4
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A : Matrix m m R
B : Matrix m n R
D : Matrix n n R
σ₁ : Perm m
σ₂ : Perm n
⊢ ↑↑(sign σ₁) * ↑↑(sign σ₂) * ((∏ x : m, A (σ₁ x) x) * ∏ x : n, D (σ₂ x) x) =
↑↑... | congr | case h.e'_3.refine_4.e_a
m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A : Matrix m m R
B : Matrix m n R
D : Matrix n n R
σ₁ : Perm m
σ₂ : Perm n
⊢ ↑↑(sign σ₁) * ↑↑(sign σ₂) = ↑↑(sign (σ₁.sumCongr σ₂)) | cabab2834e390e10 |
EuclideanDomain.mod_eq_zero | Mathlib/Algebra/EuclideanDomain/Basic.lean | theorem mod_eq_zero {a b : R} : a % b = 0 ↔ b ∣ a :=
⟨fun h => by
rw [← div_add_mod a b, h, add_zero]
exact dvd_mul_right _ _, fun ⟨c, e⟩ => by
rw [e, ← add_left_cancel_iff, div_add_mod, add_zero]
haveI := Classical.dec
by_cases b0 : b = 0
· simp only [b0, zero_mul]
· rw [mul_div_cancel_le... | case pos
R : Type u
inst✝ : EuclideanDomain R
a b : R
x✝ : b ∣ a
c : R
e : a = b * c
this : (p : Prop) → Decidable p
b0 : b = 0
⊢ b * c = b * (b * c / b) | simp only [b0, zero_mul] | no goals | b42d0fe6e6a8faaf |
PEquiv.toMatrix_injective | Mathlib/Data/Matrix/PEquiv.lean | theorem toMatrix_injective [DecidableEq n] [MonoidWithZero α] [Nontrivial α] :
Function.Injective (@toMatrix m n α _ _ _) | case h.some
m : Type u_3
n : Type u_4
α : Type u_5
inst✝² : DecidableEq n
inst✝¹ : MonoidWithZero α
inst✝ : Nontrivial α
f g : m ≃. n
i : m
hi : ¬f i = g i
fi : n
hf : f i = some fi
⊢ ∃ x, ¬(if x ∈ some fi then 1 else 0) = if x ∈ g i then 1 else 0 | use fi | case h
m : Type u_3
n : Type u_4
α : Type u_5
inst✝² : DecidableEq n
inst✝¹ : MonoidWithZero α
inst✝ : Nontrivial α
f g : m ≃. n
i : m
hi : ¬f i = g i
fi : n
hf : f i = some fi
⊢ ¬(if fi ∈ some fi then 1 else 0) = if fi ∈ g i then 1 else 0 | b54d765579e0eac3 |
CategoryTheory.Equivalence.unit_app_inverse | Mathlib/CategoryTheory/Equivalence.lean | theorem unit_app_inverse (e : C ≌ D) (Y : D) :
e.unit.app (e.inverse.obj Y) = e.inverse.map (e.counitInv.app Y) | C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
e : C ≌ D
Y : D
⊢ 𝟙 (e.inverse.obj Y) = 𝟙 (e.inverse.obj ((𝟭 D).obj Y)) | dsimp | no goals | 556855ed2fb68b57 |
HomologicalComplex.HomologySequence.quasiIso_τ₃ | Mathlib/Algebra/Homology/HomologySequenceLemmas.lean | lemma quasiIso_τ₃ (h₁ : QuasiIso φ.τ₁) (h₂ : QuasiIso φ.τ₂) :
QuasiIso φ.τ₃ | case h₄
C : Type u_1
ι : Type u_2
inst✝¹ : Category.{u_3, u_1} C
inst✝ : Abelian C
c : ComplexShape ι
S₁ S₂ : ShortComplex (HomologicalComplex C c)
φ : S₁ ⟶ S₂
hS₁ : S₁.ShortExact
hS₂ : S₂.ShortExact
h₁ : QuasiIso φ.τ₁
h₂ : QuasiIso φ.τ₂
i : ι
⊢ ∀ (j : ι), c.Rel i j → Mono (homologyMap φ.τ₂ j) | infer_instance | no goals | e88f0b981fbf6b92 |
biInf_congr' | Mathlib/Order/CompleteLattice.lean | theorem biInf_congr' {p : ι → Prop} {f g : (i : ι) → p i → α}
(h : ∀ i (hi : p i), f i hi = g i hi) :
⨅ i, ⨅ (hi : p i), f i hi = ⨅ i, ⨅ (hi : p i), g i hi | case e_s
α : Type u_1
ι : Sort u_4
inst✝ : InfSet α
p : ι → Prop
f g : (i : ι) → p i → α
h : ∀ (i : ι) (hi : p i), f i hi = g i hi
⊢ (fun i => ⨅ (hi : p i), f i hi) = fun i => ⨅ (hi : p i), g i hi | ext i | case e_s.h
α : Type u_1
ι : Sort u_4
inst✝ : InfSet α
p : ι → Prop
f g : (i : ι) → p i → α
h : ∀ (i : ι) (hi : p i), f i hi = g i hi
i : ι
⊢ ⨅ (hi : p i), f i hi = ⨅ (hi : p i), g i hi | d964f936d2f38d10 |
Stream'.WSeq.liftRel_join.lem | Mathlib/Data/Seq/WSeq.lean | theorem liftRel_join.lem (R : α → β → Prop) {S T} {U : WSeq α → WSeq β → Prop}
(ST : LiftRel (LiftRel R) S T)
(HU :
∀ s1 s2,
(∃ s t S T,
s1 = append s (join S) ∧
s2 = append t (join T) ∧ LiftRel R s t ∧ LiftRel (LiftRel R) S T) →
U s1 s2)
{a} (ma : a ∈ destr... | α : Type u
β : Type v
R : α → β → Prop
U : WSeq α → WSeq β → Prop
HU :
∀ (s1 : WSeq α) (s2 : WSeq β),
(∃ s t S T, s1 = s.append S.join ∧ s2 = t.append T.join ∧ LiftRel R s t ∧ LiftRel (LiftRel R) S T) → U s1 s2
n : ℕ
IH :
∀ (m : ℕ),
m < n →
∀ {S : WSeq (WSeq α)} {T : WSeq (WSeq β)},
LiftRel (L... | apply think_mem | case a
α : Type u
β : Type v
R : α → β → Prop
U : WSeq α → WSeq β → Prop
HU :
∀ (s1 : WSeq α) (s2 : WSeq β),
(∃ s t S T, s1 = s.append S.join ∧ s2 = t.append T.join ∧ LiftRel R s t ∧ LiftRel (LiftRel R) S T) → U s1 s2
n : ℕ
IH :
∀ (m : ℕ),
m < n →
∀ {S : WSeq (WSeq α)} {T : WSeq (WSeq β)},
Lif... | d6cf069ebb5bc575 |
List.tendsto_insertIdx' | Mathlib/Topology/List.lean | theorem tendsto_insertIdx' {a : α} :
∀ {n : ℕ} {l : List α},
Tendsto (fun p : α × List α => insertIdx n p.1 p.2) (𝓝 a ×ˢ 𝓝 l) (𝓝 (insertIdx n a l))
| 0, _ => tendsto_cons
| n + 1, [] => by simp
| n + 1, a'::l => by
have : 𝓝 a ×ˢ 𝓝 (a'::l) =
(𝓝 a ×ˢ (𝓝 a' ×ˢ 𝓝 l)).map fun p : α × α × ... | α : Type u_1
inst✝ : TopologicalSpace α
a : α
n : ℕ
a' : α
l : List α
⊢ Prod.mk <$> 𝓝 a <*> (cons <$> 𝓝 a' <*> 𝓝 l) =
(fun p => (p.1, p.2.1 :: p.2.2)) <$> (Prod.mk <$> 𝓝 a <*> (Prod.mk <$> 𝓝 a' <*> 𝓝 l)) | simp [-Filter.map_def, Function.comp_def, functor_norm] | no goals | a72e8370bcaa424a |
FractionalIdeal.mem_canonicalEquiv_apply | Mathlib/RingTheory/FractionalIdeal/Operations.lean | theorem mem_canonicalEquiv_apply {I : FractionalIdeal S P} {x : P'} :
x ∈ canonicalEquiv S P P' I ↔
∃ y ∈ I,
IsLocalization.map P' (RingHom.id R) (fun y (hy : y ∈ S) => show RingHom.id R y ∈ S from hy)
(y : P) =
x | R : Type u_1
inst✝⁶ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁵ : CommRing P
inst✝⁴ : Algebra R P
P' : Type u_3
inst✝³ : CommRing P'
inst✝² : Algebra R P'
inst✝¹ : IsLocalization S P
inst✝ : IsLocalization S P'
I : FractionalIdeal S P
x : P'
⊢ x ∈ (canonicalEquiv S P P') I ↔ ∃ y ∈ I, (IsLocalization.map P' (RingHo... | rw [canonicalEquiv, mapEquiv_apply, mem_map] | R : Type u_1
inst✝⁶ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁵ : CommRing P
inst✝⁴ : Algebra R P
P' : Type u_3
inst✝³ : CommRing P'
inst✝² : Algebra R P'
inst✝¹ : IsLocalization S P
inst✝ : IsLocalization S P'
I : FractionalIdeal S P
x : P'
⊢ (∃ x_1 ∈ I,
↑(let __src := ringEquivOfRingEquiv P P' (RingEquiv.r... | a125d8dc5607ebe5 |
Function.Injective.dite | Mathlib/Logic/Function/Basic.lean | lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (fun x ↦ if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) := fun x₁... | α : Sort u_1
β : Sort u_2
p : α → Prop
inst✝ : DecidablePred p
f : { a // p a } → β
f' : { a // ¬p a } → β
hf : Injective f
hf' : Injective f'
im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩
x₁ x₂ : α
h : (if h : p x₁ then f ⟨x₁, h⟩ else f' ⟨x₁, h⟩) = if h : p x₂ then f ⟨x₂, h⟩ else f' ⟨x₂, h⟩... | by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂ | case pos
α : Sort u_1
β : Sort u_2
p : α → Prop
inst✝ : DecidablePred p
f : { a // p a } → β
f' : { a // ¬p a } → β
hf : Injective f
hf' : Injective f'
im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩
x₁ x₂ : α
h : (if h : p x₁ then f ⟨x₁, h⟩ else f' ⟨x₁, h⟩) = if h : p x₂ then f ⟨x₂, h⟩ else f... | 6cdf4aadfc45899a |
IsMeagre.of_isSigmaCompact_null | Mathlib/MeasureTheory/Measure/OpenPos.lean | /-- A σ-compact measure zero subset is meagre.
(More generally, every Fσ set of measure zero is meagre.) -/
lemma IsMeagre.of_isSigmaCompact_null [T2Space X] (h₁s : IsSigmaCompact s) (h₂s : μ s = 0) :
IsMeagre s | case intro.intro
X : Type u_1
inst✝³ : TopologicalSpace X
inst✝² : MeasurableSpace X
s : Set X
μ : Measure X
inst✝¹ : μ.IsOpenPosMeasure
inst✝ : T2Space X
h₂s : μ s = 0
K : ℕ → Set X
hcompact : ∀ (n : ℕ), IsCompact (K n)
hcover : ⋃ n, K n = s
h : ∀ (n : ℕ), IsNowhereDense (K n)
⊢ IsMeagre s | rw [isMeagre_iff_countable_union_isNowhereDense] | case intro.intro
X : Type u_1
inst✝³ : TopologicalSpace X
inst✝² : MeasurableSpace X
s : Set X
μ : Measure X
inst✝¹ : μ.IsOpenPosMeasure
inst✝ : T2Space X
h₂s : μ s = 0
K : ℕ → Set X
hcompact : ∀ (n : ℕ), IsCompact (K n)
hcover : ⋃ n, K n = s
h : ∀ (n : ℕ), IsNowhereDense (K n)
⊢ ∃ S, (∀ t ∈ S, IsNowhereDense t) ∧ S.Co... | d808e5fac8c86d7c |
AlgebraicTopology.DoldKan.Compatibility.equivalenceCounitIso_eq | Mathlib/AlgebraicTopology/DoldKan/Compatibility.lean | theorem equivalenceCounitIso_eq (hη : τ₀ = τ₁ hF hG η) :
(equivalence hF hG).counitIso = equivalenceCounitIso η | A : Type u_1
A' : Type u_2
B : Type u_3
B' : Type u_4
inst✝³ : Category.{u_8, u_1} A
inst✝² : Category.{u_7, u_2} A'
inst✝¹ : Category.{u_6, u_3} B
inst✝ : Category.{u_5, u_4} B'
eA : A ≌ A'
eB : B ≌ B'
e' : A' ≌ B'
F : A ⥤ B'
hF : eA.functor ⋙ e'.functor ≅ F
G : B ⥤ A
hG : eB.functor ⋙ e'.inverse ≅ G ⋙ eA.functor
η : ... | ext1 | case w
A : Type u_1
A' : Type u_2
B : Type u_3
B' : Type u_4
inst✝³ : Category.{u_8, u_1} A
inst✝² : Category.{u_7, u_2} A'
inst✝¹ : Category.{u_6, u_3} B
inst✝ : Category.{u_5, u_4} B'
eA : A ≌ A'
eB : B ≌ B'
e' : A' ≌ B'
F : A ⥤ B'
hF : eA.functor ⋙ e'.functor ≅ F
G : B ⥤ A
hG : eB.functor ⋙ e'.inverse ≅ G ⋙ eA.funct... | 176b71307e064a16 |
List.toArray_inj | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/ToArray.lean | theorem toArray_inj {a b : List α} (h : a.toArray = b.toArray) : a = b | case nil
α : Type u_1
b : List α
h : #[] = b.toArray
⊢ [] = b | simpa using h | no goals | 08276af1c4b419d4 |
NumberField.house.house_le_bound | Mathlib/NumberTheory/NumberField/House.lean | theorem house_le_bound : ∀ l, house (ξ K x l).1 ≤ (c₁ K) *
((c₁ K * q * A)^((p : ℝ) / (q - p))) | case calc_1
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
α : Type u_2
β : Type u_3
a : Matrix α β (𝓞 K)
p q : ℕ
hpq : p < q
x : β × (K →+* ℂ) → ℤ
inst✝² : Fintype β
A : ℝ
habs : ∀ (k : α) (l : β), house ((algebraMap (𝓞 K) K) (a k l)) ≤ A
inst✝¹ : DecidableEq (K →+* ℂ)
inst✝ : Fintype α
Apos : 0 ≤ A
hxbound : ... | simp_rw [← _root_.map_mul, map_sum] | case calc_1
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
α : Type u_2
β : Type u_3
a : Matrix α β (𝓞 K)
p q : ℕ
hpq : p < q
x : β × (K →+* ℂ) → ℤ
inst✝² : Fintype β
A : ℝ
habs : ∀ (k : α) (l : β), house ((algebraMap (𝓞 K) K) (a k l)) ≤ A
inst✝¹ : DecidableEq (K →+* ℂ)
inst✝ : Fintype α
Apos : 0 ≤ A
hxbound : ... | fdc35d0bd9de650e |
Nat.fold_eq_foldTR | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Fold.lean | theorem fold_eq_foldTR : @fold = @foldTR :=
funext fun α => funext fun n => funext fun f => funext fun init =>
let rec go : ∀ m n f, fold (m + n) f init = foldTR.loop (m + n) f m (by omega) (fold n (fun i h => f i (by omega)) init)
| 0, n, f => by
simp only [foldTR.loop]
have t : 0 + n = n | α : Type u_1
n✝ : Nat
f✝ : (i : Nat) → i < n✝ → α → α
init : α
m n : Nat
f : (i : Nat) → i < m.succ + n → α → α
t : m + 1 + n = m + (n + 1)
⊢ (m.succ + n).fold f init = foldTR.loop (m.succ + n) f m.succ ⋯ (n.fold (fun i h => f i ⋯) init) | rw [foldTR.loop] | α : Type u_1
n✝ : Nat
f✝ : (i : Nat) → i < n✝ → α → α
init : α
m n : Nat
f : (i : Nat) → i < m.succ + n → α → α
t : m + 1 + n = m + (n + 1)
⊢ (m.succ + n).fold f init = foldTR.loop (m.succ + n) f m ⋯ (f (m.succ + n - m.succ) ⋯ (n.fold (fun i h => f i ⋯) init)) | 5cd6e0cb48120b26 |
RingHom.locally_isStableUnderBaseChange | Mathlib/RingTheory/RingHom/Locally.lean | /-- If `P` is stable under base change, then so is `Locally P`. -/
lemma locally_isStableUnderBaseChange (hPi : RespectsIso P) (hPb : IsStableUnderBaseChange P) :
IsStableUnderBaseChange (Locally P) | case intro.intro.refine_2
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hPi : RespectsIso fun {R S} [CommRing R] [CommRing S] => P
hPb : IsStableUnderBaseChange fun {R S} [CommRing R] [CommRing S] => P
R S T : Type u
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝... | rw [← isLocalizedModule_iff_isLocalization, isLocalizedModule_iff_isBaseChange
(S := Submonoid.powers a.val) (A := Localization.Away a.val)] | case intro.intro.refine_2
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hPi : RespectsIso fun {R S} [CommRing R] [CommRing S] => P
hPb : IsStableUnderBaseChange fun {R S} [CommRing R] [CommRing S] => P
R S T : Type u
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝... | b116e3feacfa87ad |
countable_setOf_covBy_right | Mathlib/Topology/Order/Basic.lean | theorem countable_setOf_covBy_right [OrderTopology α] [SecondCountableTopology α] :
Set.Countable { x : α | ∃ y, x ⋖ y } | α : Type u
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
a✝ : Nontrivial α
s : Set α := {x | ∃ y, x ⋖ y}
y : α → α
hy : ∀ x ∈ s, x ⋖ y x
Hy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x
⊢ {x | ∃ y, x ⋖ y}.Countable | suffices H : ∀ a : Set α, IsOpen a → Set.Countable { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a } by
have : s ⊆ ⋃ a ∈ countableBasis α, { x | x ∈ s ∧ x ∈ a ∧ y x ∉ a } := fun x hx => by
rcases (isBasis_countableBasis α).exists_mem_of_ne (hy x hx).ne with ⟨a, ab, xa, ya⟩
exact mem_iUnion₂.2 ⟨a, ab, hx, xa, ya⟩
refine Set.C... | α : Type u
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
a✝ : Nontrivial α
s : Set α := {x | ∃ y, x ⋖ y}
y : α → α
hy : ∀ x ∈ s, x ⋖ y x
Hy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x
⊢ ∀ (a : Set α), IsOpen a → {x | x ∈ s ∧ x ∈ a ∧ y x ∉ a}.Countable | 7a450f28627ecdc7 |
MeasureTheory.Measure.haveLebesgueDecomposition_spec | Mathlib/MeasureTheory/Decomposition/Lebesgue.lean | theorem haveLebesgueDecomposition_spec (μ ν : Measure α) [h : HaveLebesgueDecomposition μ ν] :
Measurable (μ.rnDeriv ν) ∧
μ.singularPart ν ⟂ₘ ν ∧ μ = μ.singularPart ν + ν.withDensity (μ.rnDeriv ν) | α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
h : μ.HaveLebesgueDecomposition ν
⊢ Measurable (Classical.choose ⋯).2 ∧
(Classical.choose ⋯).1 ⟂ₘ ν ∧ μ = (Classical.choose ⋯).1 + ν.withDensity (Classical.choose ⋯).2 | exact Classical.choose_spec h.lebesgue_decomposition | no goals | d00ff8c00e79ec10 |
PiNat.exists_lipschitz_retraction_of_isClosed | Mathlib/Topology/MetricSpace/PiNat.lean | theorem exists_lipschitz_retraction_of_isClosed {s : Set (∀ n, E n)} (hs : IsClosed s)
(hne : s.Nonempty) :
∃ f : (∀ n, E n) → ∀ n, E n, (∀ x ∈ s, f x = x) ∧ range f = s ∧ LipschitzWith 1 f | case neg
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed s
hne : s.Nonempty
f : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x => if x ∈ s then x else ⋯.some
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
hfxfy : f x ≠ f y
I2 : ... | have fy : f y = Ay.some := by simp_rw [f, if_neg ys] | case neg
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed s
hne : s.Nonempty
f : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x => if x ∈ s then x else ⋯.some
fs : ∀ x ∈ s, f x = x
x y : (n : ℕ) → E n
hxy : x ≠ y
hfxfy : f x ≠ f y
I2 : ... | bd92edd65dec8c2a |
mul_mul_inv_cancel'_right | Mathlib/Algebra/Group/Basic.lean | theorem mul_mul_inv_cancel'_right (a b : G) : a * (b * a⁻¹) = b | G : Type u_3
inst✝ : CommGroup G
a b : G
⊢ a * (b * a⁻¹) = b | rw [← div_eq_mul_inv, mul_div_cancel a b] | no goals | 7aee51ceea45ed27 |
Std.DHashMap.Internal.apply_bucket | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/Model.lean | theorem apply_bucket [BEq α] [Hashable α] [PartialEquivBEq α] [LawfulHashable α] {m : Raw₀ α β}
(hm : Raw.WFImp m.1) {a : α} {f : AssocList α β → γ} {g : List ((a : α) × β a) → γ}
(hfg : ∀ {l}, f l = g l.toList) (hg₁ : ∀ {l l'}, DistinctKeys l → Perm l l' → g l = g l')
(hg₂ : ∀ {l l'}, containsKey a l' = fa... | case intro.intro
α : Type u
β : α → Type v
γ : Type w
inst✝³ : BEq α
inst✝² : Hashable α
inst✝¹ : PartialEquivBEq α
inst✝ : LawfulHashable α
m : Raw₀ α β
hm : Raw.WFImp m.val
a : α
f : AssocList α β → γ
g : List ((a : α) × β a) → γ
hfg : ∀ {l : AssocList α β}, f l = g l.toList
hg₁ : ∀ {l l' : List ((a : α) × β a)}, Dis... | refine Eq.trans ?_ (hg₁ (hm.distinct.perm hl.symm) hl.symm) | case intro.intro
α : Type u
β : α → Type v
γ : Type w
inst✝³ : BEq α
inst✝² : Hashable α
inst✝¹ : PartialEquivBEq α
inst✝ : LawfulHashable α
m : Raw₀ α β
hm : Raw.WFImp m.val
a : α
f : AssocList α β → γ
g : List ((a : α) × β a) → γ
hfg : ∀ {l : AssocList α β}, f l = g l.toList
hg₁ : ∀ {l l' : List ((a : α) × β a)}, Dis... | a11d4a82e8ac83df |
exists_rat_btwn | Mathlib/Algebra/Order/Archimedean/Basic.lean | theorem exists_rat_btwn {x y : α} (h : x < y) : ∃ q : ℚ, x < q ∧ (q : α) < y | case intro.intro.hp
α : Type u_1
inst✝¹ : LinearOrderedField α
inst✝ : Archimedean α
x y : α
h : x < y
n : ℕ
nh : (y - x)⁻¹ < ↑n
z : ℤ
zh : ∀ (z_1 : ℤ), z_1 ≤ z ↔ ↑z_1 ≤ x * ↑n
n0' : 0 < ↑n
n0 : 0 < n
⊢ 1 ≠ 0 | exact one_ne_zero | no goals | 876a28cf777622dc |
MeasureTheory.measure_unitBall_eq_integral_div_gamma | Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean | theorem MeasureTheory.measure_unitBall_eq_integral_div_gamma {E : Type*} {p : ℝ}
[NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E]
[BorelSpace E] (μ : Measure E) [IsAddHaarMeasure μ] (hp : 0 < p) :
μ (Metric.ball 0 1) =
.ofReal ((∫ (x : E), Real.exp (- ‖x‖ ^ p) ∂μ) /... | case inl
E : Type u_1
p : ℝ
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hp : 0 < p
hE : Subsingleton E
⊢ μ (Metric.ball 0 1) = ENNReal.ofReal ((∫ (x : E), Real.exp (-‖x‖ ^ p) ∂μ) / Real.Ga... | rw [(Metric.nonempty_ball.mpr zero_lt_one).eq_zero, ← setIntegral_univ,
Set.univ_nonempty.eq_zero, integral_singleton, finrank_zero_of_subsingleton, Nat.cast_zero,
zero_div, zero_add, Real.Gamma_one, div_one, norm_zero, Real.zero_rpow hp.ne', neg_zero,
Real.exp_zero, smul_eq_mul, mul_one, ofReal_toReal (measure_n... | no goals | b730568993c5e381 |
Complex.cos_three_mul | Mathlib/Data/Complex/Trigonometric.lean | theorem cos_three_mul : cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x | x : ℂ
h1 : x + 2 * x = 3 * x
h2 : 4 * cos x ^ 3 = 2 * cos x * cos x * cos x + 2 * cos x * cos x ^ 2
⊢ cos x * (2 * (cos x * cos x)) - cos x - sin x * (2 * sin x * cos x) =
2 * cos x * cos x * cos x + 2 * cos x * (1 - sin x ^ 2) - 3 * cos x | ring | no goals | 5c33ec6495ac8ed8 |
untrop_sum_eq_sInf_image | Mathlib/Algebra/Tropical/BigOperators.lean | theorem untrop_sum_eq_sInf_image [ConditionallyCompleteLinearOrder R] (s : Finset S)
(f : S → Tropical (WithTop R)) : untrop (∑ i ∈ s, f i) = sInf (untrop ∘ f '' s) | R : Type u_1
S : Type u_2
inst✝ : ConditionallyCompleteLinearOrder R
s : Finset S
f : S → Tropical (WithTop R)
⊢ untrop (∑ i ∈ s, f i) = sInf (untrop ∘ f '' ↑s) | rcases s.eq_empty_or_nonempty with (rfl | h) | case inl
R : Type u_1
S : Type u_2
inst✝ : ConditionallyCompleteLinearOrder R
f : S → Tropical (WithTop R)
⊢ untrop (∑ i ∈ ∅, f i) = sInf (untrop ∘ f '' ↑∅)
case inr
R : Type u_1
S : Type u_2
inst✝ : ConditionallyCompleteLinearOrder R
s : Finset S
f : S → Tropical (WithTop R)
h : s.Nonempty
⊢ untrop (∑ i ∈ s, f i) = s... | 180a6bb362964f2c |
LipschitzWith.hasFDerivAt_of_hasLineDerivAt_of_closure | Mathlib/Analysis/Calculus/Rademacher.lean | theorem hasFDerivAt_of_hasLineDerivAt_of_closure
{f : E → F} (hf : LipschitzWith C f) {s : Set E} (hs : sphere 0 1 ⊆ closure s)
{L : E →L[ℝ] F} {x : E} (hL : ∀ v ∈ s, HasLineDerivAt ℝ f (L v) x v) :
HasFDerivAt f L x | E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
F : Type u_2
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
C : ℝ≥0
inst✝ : FiniteDimensional ℝ E
f : E → F
hf : LipschitzWith C f
s : Set E
hs : sphere 0 1 ⊆ closure s
L : E →L[ℝ] F
x : E
hL : ∀ v ∈ s, HasLineDerivAt ℝ f (L v) x v
ε : ℝ
εpos :... | apply (Finite.eventually_all q_fin).2 (fun v hv ↦ ?_) | E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
F : Type u_2
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
C : ℝ≥0
inst✝ : FiniteDimensional ℝ E
f : E → F
hf : LipschitzWith C f
s : Set E
hs : sphere 0 1 ⊆ closure s
L : E →L[ℝ] F
x : E
hL : ∀ v ∈ s, HasLineDerivAt ℝ f (L v) x v
ε : ℝ
εpos :... | b201788e5c940afd |
Orientation.oangle_eq_two_zsmul_oangle_sub_of_norm_eq | Mathlib/Geometry/Euclidean/Angle/Sphere.lean | theorem oangle_eq_two_zsmul_oangle_sub_of_norm_eq {x y z : V} (hxyne : x ≠ y) (hxzne : x ≠ z)
(hxy : ‖x‖ = ‖y‖) (hxz : ‖x‖ = ‖z‖) : o.oangle y z = (2 : ℤ) • o.oangle (y - x) (z - x) | V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y z : V
hxyne : x ≠ y
hxzne : x ≠ z
hxy : ‖x‖ = ‖y‖
hxz : ‖x‖ = ‖z‖
hy : y ≠ 0
hx : x ≠ 0
hz : z ≠ 0
⊢ ↑π - 2 • o.oangle (x - z) x - (↑π - 2 • o.oangle (x - y) x) = 2 • (o.oangle (x - y... | abel | no goals | c803d605d4f215bf |
lt_of_mul_lt_mul_of_le₀ | Mathlib/Algebra/Order/GroupWithZero/Canonical.lean | theorem lt_of_mul_lt_mul_of_le₀ (h : a * b < c * d) (hc : 0 < c) (hh : c ≤ a) : b < d | α : Type u_1
inst✝ : LinearOrderedCommGroupWithZero α
a b c d : α
h : a * b < c * d
hc : 0 < c
hh : c ≤ a
ha : a ≠ 0
⊢ b < d | rw [← inv_le_inv₀ (zero_lt_iff.2 ha) hc] at hh | α : Type u_1
inst✝ : LinearOrderedCommGroupWithZero α
a b c d : α
h : a * b < c * d
hc : 0 < c
hh : a⁻¹ ≤ c⁻¹
ha : a ≠ 0
⊢ b < d | 1ce58fbde2d14549 |
Finsupp.single_multiset_sum | Mathlib/Algebra/BigOperators/Finsupp.lean | theorem single_multiset_sum [AddCommMonoid M] (s : Multiset M) (a : α) :
single a s.sum = (s.map (single a)).sum :=
Multiset.induction_on s (single_zero _) fun a s ih => by
rw [Multiset.sum_cons, single_add, ih, Multiset.map_cons, Multiset.sum_cons]
| α : Type u_1
M : Type u_8
inst✝ : AddCommMonoid M
s✝ : Multiset M
a✝ : α
a : M
s : Multiset M
ih : single a✝ s.sum = (Multiset.map (single a✝) s).sum
⊢ single a✝ (a ::ₘ s).sum = (Multiset.map (single a✝) (a ::ₘ s)).sum | rw [Multiset.sum_cons, single_add, ih, Multiset.map_cons, Multiset.sum_cons] | no goals | a9e9c3a2261ec553 |
MeasureTheory.det_one_smulRight | Mathlib/MeasureTheory/Function/Jacobian.lean | theorem det_one_smulRight {𝕜 : Type*} [CommRing 𝕜] [TopologicalSpace 𝕜] [ContinuousMul 𝕜] (v : 𝕜) :
((1 : 𝕜 →L[𝕜] 𝕜).smulRight v).det = v | 𝕜 : Type u_3
inst✝² : CommRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : ContinuousMul 𝕜
v : 𝕜
a✝ : Nontrivial 𝕜
this : ContinuousLinearMap.smulRight 1 v = v • 1
⊢ (ContinuousLinearMap.smulRight 1 v).det = v | rw [this, ContinuousLinearMap.det, ContinuousLinearMap.coe_smul,
ContinuousLinearMap.one_def, ContinuousLinearMap.coe_id, LinearMap.det_smul,
Module.finrank_self, LinearMap.det_id, pow_one, mul_one] | no goals | 60cf35004d7f50e7 |
CategoryTheory.PreGaloisCategory.has_decomp_quotients | Mathlib/CategoryTheory/Galois/EssSurj.lean | /-- If `X` is a finite discrete `G`-set, it can be written as the finite disjoint union
of quotients of the form `G ⧸ Uᵢ` for open subgroups `(Uᵢ)`. Note that this
is simply the decomposition into orbits. -/
lemma has_decomp_quotients (X : Action FintypeCat G)
[TopologicalSpace X.V] [DiscreteTopology X.V] [Continuo... | case intro.intro.intro.intro
G : Type u_1
inst✝⁶ : Group G
inst✝⁵ : TopologicalSpace G
inst✝⁴ : IsTopologicalGroup G
inst✝³ : CompactSpace G
X : Action FintypeCat G
inst✝² : TopologicalSpace X.V.carrier
inst✝¹ : DiscreteTopology X.V.carrier
inst✝ : ContinuousSMul G X.V.carrier
ι : Type
hf : Finite ι
f : ι → Action Fint... | haveI (i : ι) : DiscreteTopology (f i).V := ⟨rfl⟩ | case intro.intro.intro.intro
G : Type u_1
inst✝⁶ : Group G
inst✝⁵ : TopologicalSpace G
inst✝⁴ : IsTopologicalGroup G
inst✝³ : CompactSpace G
X : Action FintypeCat G
inst✝² : TopologicalSpace X.V.carrier
inst✝¹ : DiscreteTopology X.V.carrier
inst✝ : ContinuousSMul G X.V.carrier
ι : Type
hf : Finite ι
f : ι → Action Fint... | 19a19434ac8aa279 |
List.lt_length_drop | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/TakeDrop.lean | theorem lt_length_drop (L : List α) {i j : Nat} (h : i + j < L.length) : j < (L.drop i).length | α : Type u_1
L : List α
i j : Nat
h : i + j < (take i L ++ drop i L).length
A : i < L.length
⊢ j < (drop i L).length | simpa only [Nat.le_of_lt A, Nat.min_eq_left, Nat.add_lt_add_iff_left, length_take,
length_append] using h | no goals | 157020fb94c4ecbd |
lp.hasSum_single | Mathlib/Analysis/Normed/Lp/lpSpace.lean | theorem hasSum_single [Fact (1 ≤ p)] (hp : p ≠ ⊤) (f : lp E p) :
HasSum (fun i : α => lp.single p i (f i : E i)) f | α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝² : (i : α) → NormedAddCommGroup (E i)
inst✝¹ : DecidableEq α
inst✝ : Fact (1 ≤ p)
hp : p ≠ ⊤
f : ↥(lp E p)
hp₀ : 0 < p
hp' : 0 < p.toReal
this : ∀ ε > 0, ∀ᶠ (x : Finset α) in Filter.atTop, dist (∑ b ∈ x, ‖↑f b‖ ^ p.toReal) (‖f‖ ^ p.toReal) < ε
ε : ℝ
hε : ε > 0
⊢ ∀ᶠ (x : Fins... | refine (this _ (Real.rpow_pos_of_pos hε p.toReal)).mono ?_ | α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝² : (i : α) → NormedAddCommGroup (E i)
inst✝¹ : DecidableEq α
inst✝ : Fact (1 ≤ p)
hp : p ≠ ⊤
f : ↥(lp E p)
hp₀ : 0 < p
hp' : 0 < p.toReal
this : ∀ ε > 0, ∀ᶠ (x : Finset α) in Filter.atTop, dist (∑ b ∈ x, ‖↑f b‖ ^ p.toReal) (‖f‖ ^ p.toReal) < ε
ε : ℝ
hε : ε > 0
⊢ ∀ (x : Finse... | ee4b5cdb4a55cc5d |
Affine.Simplex.regular_reindex_iff | Mathlib/Analysis/Normed/Affine/Simplex.lean | @[simp] lemma regular_reindex_iff {s : Simplex R P m} (e : Fin (m + 1) ≃ Fin (n + 1)) :
(s.reindex e).Regular ↔ s.Regular | R : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : Ring R
inst✝³ : SeminormedAddCommGroup V
inst✝² : PseudoMetricSpace P
inst✝¹ : Module R V
inst✝ : NormedAddTorsor V P
m n : ℕ
s : Simplex R P m
e : Fin (m + 1) ≃ Fin (n + 1)
⊢ (s.reindex e).Regular ↔ s.Regular | refine ⟨fun h σ ↦ ?_, fun h σ ↦ ?_⟩ | case refine_1
R : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : Ring R
inst✝³ : SeminormedAddCommGroup V
inst✝² : PseudoMetricSpace P
inst✝¹ : Module R V
inst✝ : NormedAddTorsor V P
m n : ℕ
s : Simplex R P m
e : Fin (m + 1) ≃ Fin (n + 1)
h : (s.reindex e).Regular
σ : Equiv.Perm (Fin (m + 1))
⊢ ∃ x, s.points ∘ ⇑σ = ⇑x ∘ s... | b54b7b2655405c05 |
AkraBazziRecurrence.isLittleO_deriv_smoothingFn | Mathlib/Computability/AkraBazzi/AkraBazzi.lean | lemma isLittleO_deriv_smoothingFn : deriv ε =o[atTop] fun x => x⁻¹ := calc
deriv ε =ᶠ[atTop] fun x => -x⁻¹ / (log x ^ 2) | ⊢ deriv ε =ᶠ[atTop] fun x => -x⁻¹ / log x ^ 2 | filter_upwards [eventually_gt_atTop 1] with x hx | case h
x : ℝ
hx : 1 < x
⊢ deriv ε x = -x⁻¹ / log x ^ 2 | 40d277aad09d54d8 |
MeasureTheory.eLpNorm_const_smul | Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | theorem eLpNorm_const_smul (c : 𝕜) (f : α → F) (p : ℝ≥0∞) (μ : Measure α):
eLpNorm (c • f) p μ = ‖c‖ₑ * eLpNorm f p μ | case inl
α : Type u_1
F : Type u_4
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup F
𝕜 : Type u_6
inst✝² : NormedDivisionRing 𝕜
inst✝¹ : Module 𝕜 F
inst✝ : BoundedSMul 𝕜 F
f : α → F
p : ℝ≥0∞
μ : Measure α
⊢ eLpNorm (0 • f) p μ = ‖0‖ₑ * eLpNorm f p μ | simp | no goals | 3fa26b7e3c41d869 |
CategoryTheory.Localization.Construction.natTrans_hcomp_injective | Mathlib/CategoryTheory/Localization/Construction.lean | theorem natTrans_hcomp_injective {F G : W.Localization ⥤ D} {τ₁ τ₂ : F ⟶ G}
(h : 𝟙 W.Q ◫ τ₁ = 𝟙 W.Q ◫ τ₂) : τ₁ = τ₂ | case w.h
C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
D : Type uD
inst✝ : Category.{uD', uD} D
F G : W.Localization ⥤ D
τ₁ τ₂ : F ⟶ G
h : 𝟙 W.Q ◫ τ₁ = 𝟙 W.Q ◫ τ₂
X : W.Localization
⊢ τ₁.app X = τ₂.app X | have eq := (objEquiv W).right_inv X | case w.h
C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
D : Type uD
inst✝ : Category.{uD', uD} D
F G : W.Localization ⥤ D
τ₁ τ₂ : F ⟶ G
h : 𝟙 W.Q ◫ τ₁ = 𝟙 W.Q ◫ τ₂
X : W.Localization
eq : (objEquiv W).toFun ((objEquiv W).invFun X) = X
⊢ τ₁.app X = τ₂.app X | 4cab32a892bffbaf |
CoxeterSystem.IsReflection.conj | Mathlib/GroupTheory/Coxeter/Inversion.lean | theorem conj (w : W) : cs.IsReflection (w * t * w⁻¹) | B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
t : W
ht : cs.IsReflection t
w : W
⊢ cs.IsReflection (w * t * w⁻¹) | obtain ⟨u, i, rfl⟩ := ht | case intro.intro
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
w u : W
i : B
⊢ cs.IsReflection (w * (u * cs.simple i * u⁻¹) * w⁻¹) | a4d00e0380504ca9 |
Turing.PartrecToTM2.pred_ok | Mathlib/Computability/TMToPartrec.lean | theorem pred_ok (q₁ q₂ s v) (c d : List Γ') : ∃ s',
Reaches₁ (TM2.step tr) ⟨some (Λ'.pred q₁ q₂), s, K'.elim (trList v) [] c d⟩
(v.headI.rec ⟨some q₁, s', K'.elim (trList v.tail) [] c d⟩ fun n _ =>
⟨some q₂, s', K'.elim (trList (n::v.tail)) [] c d⟩) | case cons.succ.zero
q₁ q₂ : Λ'
s : Option Γ'
c d : List Γ'
v : List ℕ
n : ℕ
⊢ Reaches₁ (TM2.step tr) { l := some (q₁.pred q₂), var := s, stk := elim (Γ'.bit1 :: Γ'.cons :: trList v) [] c d }
{ l := some q₂, var := none, stk := elim (Γ'.cons :: trList v) [] c d } | refine TransGen.head rfl ?_ | case cons.succ.zero
q₁ q₂ : Λ'
s : Option Γ'
c d : List Γ'
v : List ℕ
n : ℕ
⊢ TransGen (fun a b => b ∈ TM2.step tr a)
(TM2.stepAux (tr (q₁.pred q₂)) s (elim (Γ'.bit1 :: Γ'.cons :: trList v) [] c d))
{ l := some q₂, var := none, stk := elim (Γ'.cons :: trList v) [] c d } | b57c75f21d3c3b50 |
DFinsupp.comul_comp_lsingle | Mathlib/RingTheory/Coalgebra/Basic.lean | theorem comul_comp_lsingle (i : ι) :
comul ∘ₗ (lsingle i : A i →ₗ[R] _) = TensorProduct.map (lsingle i) (lsingle i) ∘ₗ comul | R : Type u
ι : Type v
A : ι → Type w
inst✝⁴ : DecidableEq ι
inst✝³ : CommSemiring R
inst✝² : (i : ι) → AddCommMonoid (A i)
inst✝¹ : (i : ι) → Module R (A i)
inst✝ : (i : ι) → Coalgebra R (A i)
i : ι
⊢ comul ∘ₗ lsingle i = TensorProduct.map (lsingle i) (lsingle i) ∘ₗ comul | ext | case h
R : Type u
ι : Type v
A : ι → Type w
inst✝⁴ : DecidableEq ι
inst✝³ : CommSemiring R
inst✝² : (i : ι) → AddCommMonoid (A i)
inst✝¹ : (i : ι) → Module R (A i)
inst✝ : (i : ι) → Coalgebra R (A i)
i : ι
x✝ : A i
⊢ (comul ∘ₗ lsingle i) x✝ = (TensorProduct.map (lsingle i) (lsingle i) ∘ₗ comul) x✝ | 5df540cc2e0244c8 |
Fin.tail_update_zero | Mathlib/Data/Fin/Tuple/Basic.lean | theorem tail_update_zero : tail (update q 0 z) = tail q | n : ℕ
α : Fin (n + 1) → Sort u
q : (i : Fin (n + 1)) → α i
z : α 0
⊢ tail (update q 0 z) = tail q | ext j | case h
n : ℕ
α : Fin (n + 1) → Sort u
q : (i : Fin (n + 1)) → α i
z : α 0
j : Fin n
⊢ tail (update q 0 z) j = tail q j | d2d2eccf320429b2 |
sum_Ico_pow | Mathlib/NumberTheory/Bernoulli.lean | theorem sum_Ico_pow (n p : ℕ) :
(∑ k ∈ Ico 1 (n + 1), (k : ℚ) ^ p) =
∑ i ∈ range (p + 1), bernoulli' i * (p + 1).choose i * (n : ℚ) ^ (p + 1 - i) / (p + 1) | case succ
n p : ℕ
f : ℕ → ℚ := fun i => bernoulli i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ
f' : ℕ → ℚ := fun i => bernoulli' i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ
hle : 1 ≤ n + 1
hne : ↑p + 1 + 1 ≠ 0
h1 : ∀ (r : ℚ), r * (↑p + 1 + 1) * ↑n ^ p.succ / (↑p + 1 + 1) =... | calc
(∑ k ∈ Ico 1 n.succ, (k : ℚ) ^ p.succ)
_ = ∑ k ∈ range n.succ, (k : ℚ) ^ p.succ := by simp [sum_Ico_eq_sub _ hle, succ_ne_zero]
_ = (∑ k ∈ range n, (k : ℚ) ^ p.succ) + (n : ℚ) ^ p.succ := by rw [sum_range_succ]
_ = (∑ i ∈ range p.succ.succ, f i) + (n : ℚ) ^ p.succ := by simp [f, sum_range_pow]
_ = (∑ i ∈... | no goals | 0112bf36c164f141 |
ContinuousMultilinearMap.norm_compContinuousLinearMap_le | Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean | theorem norm_compContinuousLinearMap_le (g : ContinuousMultilinearMap 𝕜 E₁ G)
(f : ∀ i, E i →L[𝕜] E₁ i) : ‖g.compContinuousLinearMap f‖ ≤ ‖g‖ * ∏ i, ‖f i‖ :=
opNorm_le_bound (by positivity) fun m =>
calc
‖g fun i => f i (m i)‖ ≤ ‖g‖ * ∏ i, ‖f i (m i)‖ := g.le_opNorm _
_ ≤ ‖g‖ * ∏ i, ‖f i‖ * ‖m i... | 𝕜 : Type u
ι : Type v
E : ι → Type wE
E₁ : ι → Type wE₁
G : Type wG
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁵ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝² : SeminormedAddCommGroup G
ins... | rw [prod_mul_distrib, mul_assoc] | no goals | 9b2619295592db8a |
Complex.abs_arg_lt_pi_div_two_iff | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | theorem abs_arg_lt_pi_div_two_iff {z : ℂ} : |arg z| < π / 2 ↔ 0 < re z ∨ z = 0 | case inl
z : ℂ
hz : z = 0
⊢ 0 < z.re ∨ 0 ≤ z.im ∧ (z.im < 0 ∨ z = 0) ↔ 0 < z.re ∨ z = 0 | simp [hz] | no goals | 86a383eb96fa5adb |
mul_inv_cancel | Mathlib/Algebra/Group/Defs.lean | theorem mul_inv_cancel (a : G) : a * a⁻¹ = 1 | G : Type u_1
inst✝ : Group G
a : G
⊢ a * a⁻¹ = 1 | rw [← inv_mul_cancel a⁻¹, inv_eq_of_mul (inv_mul_cancel a)] | no goals | 24a54758dc0c035b |
Projectivization.mk_eq_mk_iff' | Mathlib/LinearAlgebra/Projectivization/Basic.lean | theorem mk_eq_mk_iff' (v w : V) (hv : v ≠ 0) (hw : w ≠ 0) :
mk K v hv = mk K w hw ↔ ∃ a : K, a • w = v | case mpr
K : Type u_1
V : Type u_2
inst✝² : DivisionRing K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
v w : V
hv : v ≠ 0
hw : w ≠ 0
⊢ (∃ a, a • w = v) → ∃ a, a • w = v | rintro ⟨a, ha⟩ | case mpr.intro
K : Type u_1
V : Type u_2
inst✝² : DivisionRing K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
v w : V
hv : v ≠ 0
hw : w ≠ 0
a : K
ha : a • w = v
⊢ ∃ a, a • w = v | 4ffdffc595d7cf7b |
Ideal.Quotient.torsionBy_eq_span_singleton | Mathlib/Algebra/Module/Torsion.lean | theorem torsionBy_eq_span_singleton {R : Type w} [CommRing R] (a b : R) (ha : a ∈ R⁰) :
torsionBy R (R ⧸ R ∙ a * b) a = R ∙ mk (R ∙ a * b) b | case h
R : Type w
inst✝ : CommRing R
a b : R
ha : a ∈ R⁰
x : R ⧸ Submodule.span R {a * b}
⊢ x ∈ torsionBy R (R ⧸ Submodule.span R {a * b}) a ↔ x ∈ Submodule.span R {(mk (Submodule.span R {a * b})) b} | rw [mem_torsionBy_iff, Submodule.mem_span_singleton] | case h
R : Type w
inst✝ : CommRing R
a b : R
ha : a ∈ R⁰
x : R ⧸ Submodule.span R {a * b}
⊢ a • x = 0 ↔ ∃ a_1, a_1 • (mk (Submodule.span R {a * b})) b = x | e9ef7b1dd7f59d1b |
Subgroup.eq_one_of_noncommProd_eq_one_of_iSupIndep | Mathlib/GroupTheory/NoncommPiCoprod.lean | theorem eq_one_of_noncommProd_eq_one_of_iSupIndep {ι : Type*} (s : Finset ι) (f : ι → G) (comm)
(K : ι → Subgroup G) (hind : iSupIndep K) (hmem : ∀ x ∈ s, f x ∈ K x)
(heq1 : s.noncommProd f comm = 1) : ∀ i ∈ s, f i = 1 | G : Type u_1
inst✝ : Group G
ι : Type u_2
s : Finset ι
f : ι → G
comm : (↑s).Pairwise (Function.onFun Commute f)
K : ι → Subgroup G
hind : iSupIndep K
hmem : ∀ x ∈ s, f x ∈ K x
heq1 : s.noncommProd f comm = 1
⊢ ∀ i ∈ s, f i = 1 | revert heq1 | G : Type u_1
inst✝ : Group G
ι : Type u_2
s : Finset ι
f : ι → G
comm : (↑s).Pairwise (Function.onFun Commute f)
K : ι → Subgroup G
hind : iSupIndep K
hmem : ∀ x ∈ s, f x ∈ K x
⊢ s.noncommProd f comm = 1 → ∀ i ∈ s, f i = 1 | 658142e21f9fd208 |
FractionalIdeal.exists_ne_zero_mem_isInteger | Mathlib/RingTheory/FractionalIdeal/Operations.lean | theorem exists_ne_zero_mem_isInteger [Nontrivial R] (hI : I ≠ 0) :
∃ x, x ≠ 0 ∧ algebraMap R K x ∈ I | R : Type u_1
inst✝⁴ : CommRing R
K : Type u_3
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
I : FractionalIdeal R⁰ K
inst✝ : Nontrivial R
hI : I ≠ 0
⊢ ?m.160379 < ?m.160380 | simpa only using bot_lt_iff_ne_bot.mpr hI | no goals | 40c281908b66bc3c |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.confirmRupHint_preserves_invariant_helper | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddResult.lean | theorem confirmRupHint_preserves_invariant_helper {n : Nat} (f : DefaultFormula n)
(f_assignments_size : f.assignments.size = n)
(acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool) (hsize : acc.1.size = n)
(l : Literal (PosFin n)) (ih : DerivedLitsInvariant f f_assignments_size acc.1 hsize acc.2.... | n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool
hsize : acc.fst.size = n
l : Literal (PosFin n)
ih : f.DerivedLitsInvariant f_assignments_size acc.fst hsize acc.snd.fst
h : ¬hasAssignment l.snd acc.fst[l.fst.val]! = true
hsize'✝ : (ac... | have l_in_bounds : l.1.1 < acc.1.size := by rw [hsize]; exact l.1.2.2 | n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool
hsize : acc.fst.size = n
l : Literal (PosFin n)
ih : f.DerivedLitsInvariant f_assignments_size acc.fst hsize acc.snd.fst
h : ¬hasAssignment l.snd acc.fst[l.fst.val]! = true
hsize'✝ : (ac... | 02ca1706b397b254 |
Finset.uIcc_injective_left | Mathlib/Order/Interval/Finset/Basic.lean | theorem uIcc_injective_left (a : α) : Injective (uIcc a) | α : Type u_2
inst✝¹ : DistribLattice α
inst✝ : LocallyFiniteOrder α
a : α
⊢ Injective (uIcc a) | simpa only [uIcc_comm] using uIcc_injective_right a | no goals | 96b1545a99bc13cf |
Fin.addCases_right | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean | theorem addCases_right {m n : Nat} {motive : Fin (m + n) → Sort _} {left right} (i : Fin n) :
addCases (motive := motive) left right (natAdd m i) = right i | m n : Nat
motive : Fin (m + n) → Sort u_1
left : (i : Fin m) → motive (castAdd n i)
right : (i : Fin n) → motive (natAdd m i)
i : Fin n
this : ¬↑(natAdd m i) < m
⊢ HEq (right (subNat m (Fin.cast ⋯ (natAdd m i)) ⋯)) (right i) | congr 1 | case e_1
m n : Nat
motive : Fin (m + n) → Sort u_1
left : (i : Fin m) → motive (castAdd n i)
right : (i : Fin n) → motive (natAdd m i)
i : Fin n
this : ¬↑(natAdd m i) < m
⊢ subNat m (Fin.cast ⋯ (natAdd m i)) ⋯ = i | b9a986d08cdf67ef |
stableUnderSpecialization_iff_exists_sUnion_eq | Mathlib/Topology/Inseparable.lean | /-- A set is stable under specialization iff it is a union of closed sets. -/
lemma stableUnderSpecialization_iff_exists_sUnion_eq {s : Set X} :
StableUnderSpecialization s ↔ ∃ (S : Set (Set X)), (∀ s ∈ S, IsClosed s) ∧ ⋃₀ S = s | case refine_2
X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
H : StableUnderSpecialization s
⊢ ⋃₀ ((fun x => closure {x}) '' s) = ⋃ x ∈ s, closure {x} | simp | no goals | 865566a191c555ed |
Finsupp.range_linearCombination | Mathlib/LinearAlgebra/Finsupp/LinearCombination.lean | theorem range_linearCombination : LinearMap.range (linearCombination R v) = span R (range v) | case h.mp
α : Type u_1
M : Type u_2
R : Type u_5
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
v : α → M
x : M
⊢ x ∈ LinearMap.range (linearCombination R v) → x ∈ span R (Set.range v) | intro hx | case h.mp
α : Type u_1
M : Type u_2
R : Type u_5
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
v : α → M
x : M
hx : x ∈ LinearMap.range (linearCombination R v)
⊢ x ∈ span R (Set.range v) | 28c3b4ab3746ce9c |
EReal.coe_toENNReal | Mathlib/Data/Real/EReal.lean | @[simp]
lemma coe_toENNReal {x : EReal} (hx : 0 ≤ x) : (x.toENNReal : EReal) = x | x : EReal
hx : 0 ≤ x
⊢ ↑x.toENNReal = x | rw [toENNReal] | x : EReal
hx : 0 ≤ x
⊢ ↑(if x = ⊤ then ⊤ else ENNReal.ofReal x.toReal) = x | cea0103de28d8072 |
Ordinal.bsup_not_succ_of_ne_bsup | Mathlib/SetTheory/Ordinal/Arithmetic.lean | theorem bsup_not_succ_of_ne_bsup {o : Ordinal.{u}} {f : ∀ a < o, Ordinal.{max u v}}
(hf : ∀ {i : Ordinal} (h : i < o), f i h ≠ bsup.{_, v} o f) (a) :
a < bsup.{_, v} o f → succ a < bsup.{_, v} o f | o : Ordinal.{u}
f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v}
hf : ∀ {i : Ordinal.{u}} (h : i < o), f i h ≠ o.bsup f
a : Ordinal.{max u v}
⊢ a < o.bsup f → succ a < o.bsup f | rw [← sup_eq_bsup] at * | o : Ordinal.{u}
f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v}
hf : ∀ {i : Ordinal.{u}} (h : i < o), f i h ≠ sup (o.familyOfBFamily f)
a : Ordinal.{max u v}
⊢ a < sup (o.familyOfBFamily f) → succ a < sup (o.familyOfBFamily f) | 61c5063cab2968ad |
Function.Exact.exact_mapQ_iff | Mathlib/Algebra/Exact.lean | /-- A necessary and sufficient condition for an exact sequence to descend to a quotient. -/
lemma Exact.exact_mapQ_iff
(hfg : Exact f g) {p q r} (hpq : p ≤ comap f q) (hqr : q ≤ comap g r) :
Exact (mapQ p q f hpq) (mapQ q r g hqr) ↔ range g ⊓ r ≤ map g q | R : Type u_1
M : Type u_2
N : Type u_4
P : Type u_6
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup N
inst✝³ : AddCommGroup P
inst✝² : Module R M
inst✝¹ : Module R N
inst✝ : Module R P
f : M →ₗ[R] N
g : N →ₗ[R] P
hfg : Exact ⇑f ⇑g
p : Submodule R M
q : Submodule R N
r : Submodule R P
hpq : p ≤ comap f q
h... | dsimp only [mapQ] | R : Type u_1
M : Type u_2
N : Type u_4
P : Type u_6
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup N
inst✝³ : AddCommGroup P
inst✝² : Module R M
inst✝¹ : Module R N
inst✝ : Module R P
f : M →ₗ[R] N
g : N →ₗ[R] P
hfg : Exact ⇑f ⇑g
p : Submodule R M
q : Submodule R N
r : Submodule R P
hpq : p ≤ comap f q
h... | ea373fb6168a15ef |
geom_series_mul_neg | Mathlib/Analysis/SpecificLimits/Normed.lean | theorem geom_series_mul_neg (x : R) (h : ‖x‖ < 1) : (∑' i : ℕ, x ^ i) * (1 - x) = 1 | R : Type u_2
inst✝¹ : NormedRing R
inst✝ : HasSummableGeomSeries R
x : R
h : ‖x‖ < 1
this✝ : HasSum (fun i => x ^ i * (1 - x)) ((∑' (b : ℕ), x ^ b) * (1 - x))
this : Tendsto (fun n => 1 - x ^ n) atTop (𝓝 1)
⊢ Tendsto (fun n => ∑ i ∈ Finset.range n, x ^ i * (1 - x)) atTop (𝓝 1) | convert← this | case h.e'_3.h
R : Type u_2
inst✝¹ : NormedRing R
inst✝ : HasSummableGeomSeries R
x : R
h : ‖x‖ < 1
this✝ : HasSum (fun i => x ^ i * (1 - x)) ((∑' (b : ℕ), x ^ b) * (1 - x))
this : Tendsto (fun n => 1 - x ^ n) atTop (𝓝 1)
x✝ : ℕ
⊢ 1 - x ^ x✝ = ∑ i ∈ Finset.range x✝, x ^ i * (1 - x) | acaf2eb12ca8d928 |
Polynomial.taylor_zero' | Mathlib/Algebra/Polynomial/Taylor.lean | theorem taylor_zero' : taylor (0 : R) = LinearMap.id | case h.h.a
R : Type u_1
inst✝ : Semiring R
n✝¹ n✝ : ℕ
⊢ ((taylor 0 ∘ₗ monomial n✝¹) 1).coeff n✝ = ((LinearMap.id ∘ₗ monomial n✝¹) 1).coeff n✝ | simp only [taylor_apply, add_zero, comp_X, _root_.map_zero, LinearMap.id_comp,
Function.comp_apply, LinearMap.coe_comp] | no goals | cb158bfeb4d2d35d |
MeasureTheory.IntegrableOn.hasBoxIntegral | Mathlib/Analysis/BoxIntegral/Integrability.lean | theorem IntegrableOn.hasBoxIntegral [CompleteSpace E] {f : (ι → ℝ) → E} {μ : Measure (ι → ℝ)}
[IsLocallyFiniteMeasure μ] {I : Box ι} (hf : IntegrableOn f I μ) (l : IntegrationParams)
(hl : l.bRiemann = false) :
HasIntegral.{u, v, v} I l f μ.toBoxAdditive.toSMul (∫ x in I, f x ∂μ) | case intro.intro.intro.intro.intro.intro.intro.intro
ι : Type u
E : Type v
inst✝⁴ : Fintype ι
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
I : Box ι
l : IntegrationParams
hl : l.bRiemann = false
this✝¹ : MeasurableSpace E := borel E... | set r : ℝ≥0 → (ι → ℝ) → Ioi (0 : ℝ) := fun c x => (hfi' <| Nx x).convergenceR (δ <| Nx x) c x | case intro.intro.intro.intro.intro.intro.intro.intro
ι : Type u
E : Type v
inst✝⁴ : Fintype ι
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
I : Box ι
l : IntegrationParams
hl : l.bRiemann = false
this✝¹ : MeasurableSpace E := borel E... | 1cd867d174f2adc3 |
ZMod.eisenstein_lemma | Mathlib/NumberTheory/LegendreSymbol/GaussEisensteinLemmas.lean | theorem eisenstein_lemma {p : ℕ} [Fact p.Prime] (hp : p ≠ 2) {a : ℕ} (ha1 : a % 2 = 1)
(ha0 : (a : ZMod p) ≠ 0) : legendreSym p a = (-1) ^ ∑ x ∈ Ico 1 (p / 2).succ, x * a / p | p : ℕ
inst✝ : Fact (Nat.Prime p)
hp : p ≠ 2
a : ℕ
ha1 : a % 2 = 1
ha0 : ↑a ≠ 0
hp' : Fact (p % 2 = 1)
ha0' : ↑↑a ≠ 0
⊢ ↑↑a = ↑a | norm_cast | no goals | dddada4c44e9a9e0 |
summable_dirichletSummand | Mathlib/NumberTheory/EulerProduct/DirichletLSeries.lean | /-- When `s.re > 1`, the map `n ↦ χ(n) * n^(-s)` is norm-summable. -/
lemma summable_dirichletSummand {N : ℕ} (χ : DirichletCharacter ℂ N) (hs : 1 < s.re) :
Summable (fun n ↦ ‖dirichletSummandHom χ (ne_zero_of_one_lt_re hs) n‖) | s : ℂ
N : ℕ
χ : DirichletCharacter ℂ N
hs : 1 < s.re
⊢ Summable fun n => ‖(dirichletSummandHom χ ⋯) n‖ | simp only [dirichletSummandHom, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk, norm_mul] | s : ℂ
N : ℕ
χ : DirichletCharacter ℂ N
hs : 1 < s.re
⊢ Summable fun n => ‖χ ↑n‖ * ‖↑n ^ (-s)‖ | 97d5025580211112 |
CategoryTheory.NonemptyParallelPairPresentationAux.hg | Mathlib/CategoryTheory/Limits/Indization/ParallelPair.lean | theorem hg : g = IsColimit.map (isColimit₁ f g P₁ P₂)
(Cocone.mk B (ι₂ f g P₁ P₂)) (whiskerRight (ψ f g P₁ P₂) yoneda) | C : Type u₁
inst✝ : Category.{v₁, u₁} C
A B : Cᵒᵖ ⥤ Type v₁
f g : A ⟶ B
P₁ : IndObjectPresentation A
P₂ : IndObjectPresentation B
⊢ g = (isColimit₁ f g P₁ P₂).map { pt := B, ι := ι₂ f g P₁ P₂ } (whiskerRight (ψ f g P₁ P₂) yoneda) | refine (isColimit₁ f g P₁ P₂).hom_ext (fun i => ?_) | C : Type u₁
inst✝ : Category.{v₁, u₁} C
A B : Cᵒᵖ ⥤ Type v₁
f g : A ⟶ B
P₁ : IndObjectPresentation A
P₂ : IndObjectPresentation B
i : K f g P₁ P₂
⊢ { pt := A, ι := ι₁ f g P₁ P₂ }.ι.app i ≫ g =
{ pt := A, ι := ι₁ f g P₁ P₂ }.ι.app i ≫
(isColimit₁ f g P₁ P₂).map { pt := B, ι := ι₂ f g P₁ P₂ } (whiskerRight (ψ f... | 4aad8c08030044df |
RegularExpression.mul_rmatch_iff | Mathlib/Computability/RegularExpressions.lean | theorem mul_rmatch_iff (P Q : RegularExpression α) (x : List α) :
(P * Q).rmatch x ↔ ∃ t u : List α, x = t ++ u ∧ P.rmatch t ∧ Q.rmatch u | case pos.mpr.intro.intro.intro.intro.nil.h
α : Type u_1
inst✝ : DecidableEq α
a : α
x : List α
ih : ∀ (P Q : RegularExpression α), (P * Q).rmatch x = true ↔ ∃ t u, x = t ++ u ∧ P.rmatch t = true ∧ Q.rmatch u = true
P Q : RegularExpression α
hepsilon : P.matchEpsilon = true
u : List α
hQ : Q.rmatch u = true
h : a :: x =... | rw [← h] at hQ | case pos.mpr.intro.intro.intro.intro.nil.h
α : Type u_1
inst✝ : DecidableEq α
a : α
x : List α
ih : ∀ (P Q : RegularExpression α), (P * Q).rmatch x = true ↔ ∃ t u, x = t ++ u ∧ P.rmatch t = true ∧ Q.rmatch u = true
P Q : RegularExpression α
hepsilon : P.matchEpsilon = true
u : List α
hQ : Q.rmatch (a :: x) = true
h : a... | eb4ea08dacfd0649 |
ContinuousMap.hasSum_of_hasSum_Lp | Mathlib/MeasureTheory/Function/LpSpace/ContinuousFunctions.lean | theorem hasSum_of_hasSum_Lp {β : Type*} [μ.IsOpenPosMeasure] [NormedField 𝕜] [NormedSpace 𝕜 E]
{g : β → C(α, E)} {f : C(α, E)} (hg : Summable g)
(hg2 : HasSum (toLp (E := E) p μ 𝕜 ∘ g) (toLp (E := E) p μ 𝕜 f)) : HasSum g f | case h.e'_6
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : BorelSpace α
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : SecondCountableTopologyEither α E
inst✝⁵ : CompactSpace α
inst✝⁴ : IsFiniteMeasure μ
𝕜 : Type u_3
inst✝³ : Fact (1 ≤ p)
β : Type u_4
inst✝² : μ.Is... | exact toLp_injective μ (hg2.unique ((toLp p μ 𝕜).hasSum <| Summable.hasSum hg)) | no goals | 2415a8dd1726fecd |
NumberField.InfinitePlace.card_isUnramified_compl | Mathlib/NumberTheory/NumberField/Embeddings.lean | lemma card_isUnramified_compl [NumberField k] [IsGalois k K] :
#({w : InfinitePlace K | w.IsUnramified k} : Finset _)ᶜ =
#({w : InfinitePlace k | w.IsUnramifiedIn K} : Finset _)ᶜ * (finrank k K / 2) | case e_s.h
k : Type u_1
inst✝⁵ : Field k
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra k K
inst✝² : NumberField K
inst✝¹ : NumberField k
inst✝ : IsGalois k K
this : Module.Finite k K := Finite.of_restrictScalars_finite ℚ k K
w : InfinitePlace K
hw : ¬IsUnramifiedIn K (w.comap (algebraMap k K))
w' : InfinitePlace K
⊢ w... | simp only [compl_filter, filter_congr_decidable, mem_filter, mem_univ, true_and,
@eq_comm _ (comap w' _), Set.mem_toFinset, mem_orbit_iff, and_iff_right_iff_imp] | case e_s.h
k : Type u_1
inst✝⁵ : Field k
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra k K
inst✝² : NumberField K
inst✝¹ : NumberField k
inst✝ : IsGalois k K
this : Module.Finite k K := Finite.of_restrictScalars_finite ℚ k K
w : InfinitePlace K
hw : ¬IsUnramifiedIn K (w.comap (algebraMap k K))
w' : InfinitePlace K
⊢ w... | d92790b65480f1b5 |
csSup_mem_of_not_isSuccPrelimit | Mathlib/Order/SuccPred/CompleteLinearOrder.lean | lemma csSup_mem_of_not_isSuccPrelimit
(hne : s.Nonempty) (hbdd : BddAbove s) (hlim : ¬ IsSuccPrelimit (sSup s)) : sSup s ∈ s | case intro
α : Type u_2
inst✝ : ConditionallyCompleteLinearOrder α
s : Set α
hne : s.Nonempty
hbdd : BddAbove s
hlim : ¬IsSuccPrelimit (sSup s)
y : α
hy : y ⋖ sSup s
⊢ sSup s ∈ s | obtain ⟨i, his, hi⟩ := exists_lt_of_lt_csSup hne hy.lt | case intro.intro.intro
α : Type u_2
inst✝ : ConditionallyCompleteLinearOrder α
s : Set α
hne : s.Nonempty
hbdd : BddAbove s
hlim : ¬IsSuccPrelimit (sSup s)
y : α
hy : y ⋖ sSup s
i : α
his : i ∈ s
hi : y < i
⊢ sSup s ∈ s | f419e4f7b2c0f097 |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.size_assignemnts_confirmRupHint | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddResult.lean | theorem size_assignemnts_confirmRupHint {n : Nat} (clauses : Array (Option (DefaultClause n)))
(assignments : Array Assignment) (derivedLits : CNF.Clause (PosFin n)) (b1 b2 : Bool) (id : Nat) :
(confirmRupHint clauses (assignments, derivedLits, b1, b2) id).1.size = assignments.size | n : Nat
clauses : Array (Option (DefaultClause n))
assignments : Array Assignment
derivedLits : CNF.Clause (PosFin n)
b1 b2 : Bool
id : Nat
⊢ (if (b2 || b1) = true then (assignments, derivedLits, b1, b2)
else
match clauses[id]? with
| some (some c) =>
match c.reduce assignments w... | repeat first
| rfl
| simp only [Array.size_modify]
| split | no goals | 73de79a1bc76d583 |
MeasureTheory.ae_of_forall_measure_lt_top_ae_restrict' | Mathlib/MeasureTheory/Measure/Typeclasses.lean | theorem ae_of_forall_measure_lt_top_ae_restrict' {μ : Measure α} (ν : Measure α) [SigmaFinite μ]
[SigmaFinite ν] (P : α → Prop)
(h : ∀ s, MeasurableSet s → μ s < ∞ → ν s < ∞ → ∀ᵐ x ∂μ.restrict s, P x) : ∀ᵐ x ∂μ, P x | α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
P : α → Prop
h : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ν s < ⊤ → ∀ᵐ (x : α) ∂μ.restrict s, P x
n : ℕ
this : ∀ᵐ (x : α) ∂μ.restrict (spanningSets (μ + ν) n), P x
⊢ ∀ᵐ (x : α) ∂μ, x ∈ spanningSets (μ + ν) n → P x | exact (ae_restrict_iff' (measurableSet_spanningSets _ _)).mp this | no goals | 1efc730cfe534765 |
MeasureTheory.locallyIntegrableOn_iff_locallyIntegrable_restrict | Mathlib/MeasureTheory/Function/LocallyIntegrable.lean | theorem locallyIntegrableOn_iff_locallyIntegrable_restrict [OpensMeasurableSpace X]
(hs : IsClosed s) : LocallyIntegrableOn f s μ ↔ LocallyIntegrable f (μ.restrict s) | case pos.intro.intro.intro.intro.intro
X : Type u_1
E : Type u_3
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : OpensMeasurableSpace X
hs : IsClosed s
hf : LocallyIntegrableOn f s μ
x : X
h : x ∈ s
t : Set X
ht_nhds : t ∈ 𝓝[s] x
ht_int : I... | rw [IntegrableOn, restrict_restrict hu_o.measurableSet] | case pos.intro.intro.intro.intro.intro
X : Type u_1
E : Type u_3
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : OpensMeasurableSpace X
hs : IsClosed s
hf : LocallyIntegrableOn f s μ
x : X
h : x ∈ s
t : Set X
ht_nhds : t ∈ 𝓝[s] x
ht_int : I... | fa93e0216a501ac1 |
OreLocalization.add_smul | Mathlib/RingTheory/OreLocalization/Ring.lean | theorem add_smul (y z : R[S⁻¹]) (x : X[S⁻¹]) :
(y + z) • x = y • x + z • x | case c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro
R : Type u_1
inst✝³ : Semiring R
S : Submonoid R
inst✝² : OreSet S
X : Type u_2
inst✝¹ : AddCommMonoid X
inst✝ : Module R X
r₁ : X
s₁ : ↥S
r₂ : R
s₂ : ↥S
r₃ : R
s₃ : ↥S
ra : R
sa : ↥S
ha : ↑sa * ↑s₂ = ra * ↑s₃
rb : R
sb : ↥S
hb : ↑sb * sa • r₂ = rb * ↑s₁
hs₃rasb : ↑sb * ra... | rw [OreLocalization.expand' (sa • r₂ + ra • r₃) (sa * s₂) (sc * sb)] | case c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro
R : Type u_1
inst✝³ : Semiring R
S : Submonoid R
inst✝² : OreSet S
X : Type u_2
inst✝¹ : AddCommMonoid X
inst✝ : Module R X
r₁ : X
s₁ : ↥S
r₂ : R
s₂ : ↥S
r₃ : R
s₃ : ↥S
ra : R
sa : ↥S
ha : ↑sa * ↑s₂ = ra * ↑s₃
rb : R
sb : ↥S
hb : ↑sb * sa • r₂ = rb * ↑s₁
hs₃rasb : ↑sb * ra... | 79dc4eb2119c8abc |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.