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 |
|---|---|---|---|---|---|---|
CategoryTheory.shiftFunctorCompIsoId_add'_inv_app | Mathlib/CategoryTheory/Shift/Basic.lean | lemma shiftFunctorCompIsoId_add'_inv_app :
(shiftFunctorCompIsoId C p' p hp).inv.app X =
(shiftFunctorCompIsoId C n' n hn).inv.app X ≫
(shiftFunctorCompIsoId C m' m hm).inv.app (X⟦n'⟧)⟦n⟧' ≫
(shiftFunctorAdd' C m n p h).inv.app (X⟦n'⟧⟦m'⟧) ≫
((shiftFunctorAdd' C n' m' p'
(by rw [← ad... | C : Type u
A : Type u_1
inst✝² : Category.{v, u} C
inst✝¹ : AddGroup A
inst✝ : HasShift C A
X : C
m n p m' n' p' : A
hm : m' + m = 0
hn : n' + n = 0
hp : p' + p = 0
h : m + n = p
⊢ p' + m + n = 0 | rw [add_assoc, h, hp] | no goals | 1740b8167511ce6d |
CliffordAlgebra.evenOdd_induction | Mathlib/LinearAlgebra/CliffordAlgebra/Grading.lean | theorem evenOdd_induction (n : ZMod 2) {motive : ∀ x, x ∈ evenOdd Q n → Prop}
(range_ι_pow : ∀ (v) (h : v ∈ LinearMap.range (ι Q) ^ n.val),
motive v (Submodule.mem_iSup_of_mem ⟨n.val, n.natCast_zmod_val⟩ h))
(add : ∀ x y hx hy, motive x hx → motive y hy → motive (x + y) (Submodule.add_mem _ hx hy))
... | R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
n : ZMod 2
motive : (x : CliffordAlgebra Q) → x ∈ evenOdd Q n → Prop
range_ι_pow : ∀ (v : CliffordAlgebra Q) (h : v ∈ LinearMap.range (ι Q) ^ n.val), motive v ⋯
add :
∀ (x y : CliffordAlgebra Q) (hx : x ∈ ev... | apply Submodule.iSup_induction' (C := motive) _ _ (range_ι_pow 0 (Submodule.zero_mem _)) add | R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
n : ZMod 2
motive : (x : CliffordAlgebra Q) → x ∈ evenOdd Q n → Prop
range_ι_pow : ∀ (v : CliffordAlgebra Q) (h : v ∈ LinearMap.range (ι Q) ^ n.val), motive v ⋯
add :
∀ (x y : CliffordAlgebra Q) (hx : x ∈ ev... | 3b3d6ba6e257fdcc |
List.sublist_of_orderEmbedding_getElem?_eq | Mathlib/Data/List/NodupEquivFin.lean | theorem sublist_of_orderEmbedding_getElem?_eq {l l' : List α} (f : ℕ ↪o ℕ)
(hf : ∀ ix : ℕ, l[ix]? = l'[f ix]?) : l <+ l' | α : Type u_1
hd : α
tl : List α
IH : ∀ {l' : List α} (f : ℕ ↪o ℕ), (∀ (ix : ℕ), tl[ix]? = l'[f ix]?) → tl <+ l'
l' : List α
f : ℕ ↪o ℕ
hf : ∀ (ix : ℕ), (hd :: tl)[ix]? = l'[f ix]?
w : f 0 < l'.length
h : l'[f 0] = hd
a b : ℕ
⊢ 0 < b + 1 | exact b.succ_pos | no goals | 1ef7e5e229e83d6a |
Real.volume_pi_ball | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | theorem volume_pi_ball (a : ι → ℝ) {r : ℝ} (hr : 0 < r) :
volume (Metric.ball a r) = ENNReal.ofReal ((2 * r) ^ Fintype.card ι) | ι : Type u_1
inst✝ : Fintype ι
a : ι → ℝ
r : ℝ
hr : 0 < r
⊢ volume (Metric.ball a r) = ofReal ((2 * r) ^ Fintype.card ι) | simp only [MeasureTheory.volume_pi_ball a hr, volume_ball, Finset.prod_const] | ι : Type u_1
inst✝ : Fintype ι
a : ι → ℝ
r : ℝ
hr : 0 < r
⊢ ofReal (2 * r) ^ Finset.univ.card = ofReal ((2 * r) ^ Fintype.card ι) | 6926f174b772b926 |
Submodule.goursat | Mathlib/LinearAlgebra/Goursat.lean | /-- **Goursat's lemma** for an arbitrary submodule of a product.
If `L` is a submodule of `M × N`, then there exist submodules `M'' ≤ M' ≤ M` and `N'' ≤ N' ≤ N` such
that `L ≤ M' × N'`, and `L` is (the image in `M × N` of) the preimage of the graph of an `R`-linear
isomorphism `M' ⧸ M'' ≃ N' ⧸ N''`. -/
lemma goursat :... | case h.h.mk.mp
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : Ring R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
L : Submodule R (M × N)
M' : Submodule R M := map (LinearMap.fst R M N) L
N' : Submodule R N := map (LinearMap.snd R M N) L
P : ↥L →ₗ[R] ↥M' := (LinearMap.fst R M ... | simp only [mem_map, LinearMap.mem_range, prod_apply, Subtype.exists, Prod.exists, coe_prodMap,
coe_subtype, Prod.map_apply, Prod.mk.injEq, exists_and_right, exists_eq_right_right,
exists_eq_right, M', N', fst_apply, snd_apply] | case h.h.mk.mp
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : Ring R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
L : Submodule R (M × N)
M' : Submodule R M := map (LinearMap.fst R M N) L
N' : Submodule R N := map (LinearMap.snd R M N) L
P : ↥L →ₗ[R] ↥M' := (LinearMap.fst R M ... | 8bafbf13c72f3711 |
Finset.support_sum_eq | Mathlib/Data/Finsupp/BigOperators.lean | theorem Finset.support_sum_eq [AddCommMonoid M] (s : Finset (ι →₀ M))
(hs : (s : Set (ι →₀ M)).PairwiseDisjoint Finsupp.support) :
(s.sum id).support = Finset.sup s Finsupp.support | case intro.intro.hr
ι : Type u_1
M : Type u_2
inst✝¹ : DecidableEq ι
inst✝ : AddCommMonoid M
l : List (ι →₀ M)
hn : l.Nodup
hs : (↑l.toFinset).PairwiseDisjoint Finsupp.support
⊢ Symmetric (Disjoint on Finsupp.support) | intro x y hxy | case intro.intro.hr
ι : Type u_1
M : Type u_2
inst✝¹ : DecidableEq ι
inst✝ : AddCommMonoid M
l : List (ι →₀ M)
hn : l.Nodup
hs : (↑l.toFinset).PairwiseDisjoint Finsupp.support
x y : ι →₀ M
hxy : (Disjoint on Finsupp.support) x y
⊢ (Disjoint on Finsupp.support) y x | 67f7aca84a3dab37 |
FirstOrder.Language.Theory.isSatisfiable_iff_isFinitelySatisfiable | Mathlib/ModelTheory/Satisfiability.lean | theorem isSatisfiable_iff_isFinitelySatisfiable {T : L.Theory} :
T.IsSatisfiable ↔ T.IsFinitelySatisfiable :=
⟨Theory.IsSatisfiable.isFinitelySatisfiable, fun h => by
classical
set M : Finset T → Type max u v := fun T0 : Finset T =>
(h (T0.map (Function.Embedding.subtype fun x => x ∈ T)) T0.map_... | L : Language
T : L.Theory
h : T.IsFinitelySatisfiable
M : Finset ↑T → Type (max u v) := fun T0 => ↑(Nonempty.some ⋯)
M' : Type (max u v) := (↑(Ultrafilter.of Filter.atTop)).Product M
⊢ M' ⊨ T | refine ⟨fun φ hφ => ?_⟩ | L : Language
T : L.Theory
h : T.IsFinitelySatisfiable
M : Finset ↑T → Type (max u v) := fun T0 => ↑(Nonempty.some ⋯)
M' : Type (max u v) := (↑(Ultrafilter.of Filter.atTop)).Product M
φ : L.Sentence
hφ : φ ∈ T
⊢ M' ⊨ φ | 360797a9dc61769a |
PartialHomeomorph.contDiffAt_symm | Mathlib/Analysis/Calculus/ContDiff/Operations.lean | theorem PartialHomeomorph.contDiffAt_symm [CompleteSpace E] (f : PartialHomeomorph E F)
{f₀' : E ≃L[𝕜] F} {a : F} (ha : a ∈ f.target)
(hf₀' : HasFDerivAt f (f₀' : E →L[𝕜] F) (f.symm a)) (hf : ContDiffAt 𝕜 n f (f.symm a)) :
ContDiffAt 𝕜 n f.symm a | case hsuc.intro.intro.intro.intro.refine_2
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type uF
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
n✝ : WithTop ℕ∞
inst✝ : CompleteSpace E
f : PartialHomeomorph E F
f₀' : E ≃L[𝕜] F
a : F
ha... | have h_deriv₂ : ContDiffAt 𝕜 n f.symm a := by
refine IH (hf.of_le ?_)
norm_cast
exact Nat.le_succ n | case hsuc.intro.intro.intro.intro.refine_2
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type uF
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
n✝ : WithTop ℕ∞
inst✝ : CompleteSpace E
f : PartialHomeomorph E F
f₀' : E ≃L[𝕜] F
a : F
ha... | db52077be919f1af |
krullTopology_mem_nhds_one_iff | Mathlib/FieldTheory/KrullTopology.lean | lemma krullTopology_mem_nhds_one_iff (K L : Type*) [Field K] [Field L] [Algebra K L]
(s : Set (L ≃ₐ[K] L)) : s ∈ 𝓝 1 ↔ ∃ E : IntermediateField K L,
FiniteDimensional K E ∧ (E.fixingSubgroup : Set (L ≃ₐ[K] L)) ⊆ s | case mp.intro.intro.intro.intro.intro.intro
K : Type u_1
L : Type u_2
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
s : Set (L ≃ₐ[K] L)
E : IntermediateField K L
fin : E ∈ finiteExts K L
hE : (fun g => g.carrier) E.fixingSubgroup ⊆ s
⊢ ∃ E, FiniteDimensional K ↥E ∧ ↑E.fixingSubgroup ⊆ s | exact ⟨E, fin, hE⟩ | no goals | f860557d167209ce |
inr_comp_cfcₙHom_eq_cfcₙAux | Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Basic.lean | lemma inr_comp_cfcₙHom_eq_cfcₙAux {A : Type*} [NonUnitalCStarAlgebra A] (a : A)
[ha : IsStarNormal a] : (inrNonUnitalStarAlgHom ℂ A).comp (cfcₙHom ha) =
cfcₙAux (isStarNormal_inr (R := ℂ) (A := A)) a ha | A : Type u_2
inst✝ : NonUnitalCStarAlgebra A
a : A
ha : IsStarNormal a
h : ∀ (a : A), IsStarNormal ↑a ↔ IsStarNormal a
⊢ ((inrNonUnitalStarAlgHom ℂ A).comp (cfcₙHom ha))
{ toContinuousMap := ContinuousMap.restrict (σₙ ℂ a) (ContinuousMap.id ℂ), map_zero' := ⋯ } =
↑a | congrm(inr $(cfcₙHom_id ha)) | no goals | 6489793b55e07998 |
LocalSubring.exists_le_valuationSubring | Mathlib/RingTheory/Valuation/LocalSubring.lean | @[stacks 00IA]
lemma LocalSubring.exists_le_valuationSubring (A : LocalSubring K) :
∃ B : ValuationSubring K, A ≤ B.toLocalSubring | case refine_2.mk.intro.mk.intro
K : Type u_3
inst✝ : Field K
A✝ : LocalSubring K
s : Set (LocalSubring K)
hs : s ⊆ Set.Ici A✝
H : IsChain (fun x1 x2 => x1 ≤ x2) s
y : LocalSubring K
hys : y ∈ s
inst : Nonempty ↑s
hdir : Directed LE.le (toSubring ∘ fun x => ↑x)
A : LocalSubring K
hA : A ∈ s
a : K
haA : a ∈ A.toSubring
h... | obtain ⟨C, hCA, hCB⟩ := H.directed ⟨A, hA⟩ B | case refine_2.mk.intro.mk.intro.intro.intro
K : Type u_3
inst✝ : Field K
A✝ : LocalSubring K
s : Set (LocalSubring K)
hs : s ⊆ Set.Ici A✝
H : IsChain (fun x1 x2 => x1 ≤ x2) s
y : LocalSubring K
hys : y ∈ s
inst : Nonempty ↑s
hdir : Directed LE.le (toSubring ∘ fun x => ↑x)
A : LocalSubring K
hA : A ∈ s
a : K
haA : a ∈ A... | 926576b935296dd2 |
smul_singleton_mem_nhds_of_sigmaCompact | Mathlib/Topology/Algebra/Group/OpenMapping.lean | theorem smul_singleton_mem_nhds_of_sigmaCompact
{U : Set G} (hU : U ∈ 𝓝 1) (x : X) : U • {x} ∈ 𝓝 x | G : Type u_1
X : Type u_2
inst✝⁹ : TopologicalSpace G
inst✝⁸ : TopologicalSpace X
inst✝⁷ : Group G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : MulAction G X
inst✝⁴ : SigmaCompactSpace G
inst✝³ : BaireSpace X
inst✝² : T2Space X
inst✝¹ : ContinuousSMul G X
inst✝ : IsPretransitive G X
U : Set G
hU : U ∈ 𝓝 1
x : X
V : Set G
V_... | simpa only [F, smul_singleton] using H | no goals | 09fff4470c6f8515 |
List.mapFinIdx_append | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/MapIdx.lean | theorem mapFinIdx_append {K L : List α} {f : (i : Nat) → α → (h : i < (K ++ L).length) → β} :
(K ++ L).mapFinIdx f =
K.mapFinIdx (fun i a h => f i a (by simp; omega)) ++
L.mapFinIdx (fun i a h => f (i + K.length) a (by simp; omega)) | case h
α : Type u_1
β : Type u_2
K L : List α
f : (i : Nat) → α → i < (K ++ L).length → β
i : Nat
h₁ : i < ((K ++ L).mapFinIdx f).length
h₂ : i < ((K.mapFinIdx fun i a h => f i a ⋯) ++ L.mapFinIdx fun i a h => f (i + K.length) a ⋯).length
⊢ f i (K ++ L)[i] ⋯ = if h : i < K.length then f i K[i] ⋯ else f (i - K.length + ... | split <;> rename_i h | case h.isTrue
α : Type u_1
β : Type u_2
K L : List α
f : (i : Nat) → α → i < (K ++ L).length → β
i : Nat
h₁ : i < ((K ++ L).mapFinIdx f).length
h₂ : i < ((K.mapFinIdx fun i a h => f i a ⋯) ++ L.mapFinIdx fun i a h => f (i + K.length) a ⋯).length
h : i < K.length
⊢ f i (K ++ L)[i] ⋯ = f i K[i] ⋯
case h.isFalse
α : Type... | 3176fb50755ce896 |
Measurable.liminf' | Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean | theorem Measurable.liminf' {ι ι'} {f : ι → δ → α} {v : Filter ι} (hf : ∀ i, Measurable (f i))
{p : ι' → Prop} {s : ι' → Set ι} (hv : v.HasCountableBasis p s) (hs : ∀ j, (s j).Countable) :
Measurable fun x => liminf (f · x) v | α : Type u_1
δ : Type u_4
inst✝⁴ : TopologicalSpace α
mα : MeasurableSpace α
inst✝³ : BorelSpace α
mδ : MeasurableSpace δ
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_5
ι' : Type u_6
f : ι → δ → α
v : Filter ι
hf : ∀ (i : ι), Measurable (f i)
p : ι' →... | by_cases H : ∃ k, x ∈ m k | case pos
α : Type u_1
δ : Type u_4
inst✝⁴ : TopologicalSpace α
mα : MeasurableSpace α
inst✝³ : BorelSpace α
mδ : MeasurableSpace δ
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_5
ι' : Type u_6
f : ι → δ → α
v : Filter ι
hf : ∀ (i : ι), Measurable (f i)... | 92aa2e449ca3c5d1 |
Lat.id_apply | Mathlib/Order/Category/Lat.lean | lemma id_apply (X : Lat) (x : X) :
(𝟙 X : X ⟶ X) x = x | X : Lat
x : ↑X
⊢ (ConcreteCategory.hom (𝟙 X)) x = x | simp | no goals | 32c5b178eb0b3749 |
MeasureTheory.condExp_bot_ae_eq | Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean | theorem condExp_bot_ae_eq (f : α → E) :
μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ | case inl
α : Type u_1
E : Type u_3
m₀ : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : α → E
⊢ 0[f|⊥] =ᶠ[ae 0] fun x => (0 Set.univ).toReal⁻¹ • ∫ (x : α), f x ∂0 | rw [ae_zero] | case inl
α : Type u_1
E : Type u_3
m₀ : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : α → E
⊢ 0[f|⊥] =ᶠ[⊥] fun x => (0 Set.univ).toReal⁻¹ • ∫ (x : α), f x ∂0 | 767e3c75448f429e |
AlgebraicTopology.AlternatingFaceMapComplex.d_squared | Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean | theorem d_squared (n : ℕ) : objD X (n + 1) ≫ objD X n = 0 | case h.mk.e_a.H
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
n : ℕ
P : Type := Fin (n + 2) × Fin (n + 3)
S : Finset P := Finset.filter (fun ij => ↑ij.2 ≤ ↑ij.1) Finset.univ
φ : (ij : P) → ij ∈ S → P := fun ij hij => (ij.2.castLT ⋯, ij.1.succ)
i : Fin (n + 2)
j : Fin (n + 3)
h... | simpa [S] using hij | no goals | 795ab48b2006c7af |
AlgebraicGeometry.ProjectiveSpectrum.Proj.isLocalization_atPrime | Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean | /--
If `x` is a point in the basic open set `D(f)` where `f` is a homogeneous element of positive
degree, then the homogeneously localized ring `A⁰ₓ` has the universal property of the localization
of `A⁰_f` at `φ(x)` where `φ : Proj|D(f) ⟶ Spec A⁰_f` is the morphism of locally ringed space
constructed as above.
-/
lemm... | case exists_of_eq.intro.intro.intro.intro.intro.intro
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
x : ↥(pbo f)
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
this : Algebra (A⁰_ f) (AtPrime 𝒜 (↑x).asHomogeneousIdeal.toIdeal) := (mapId ... | simp only [val_mul, val_mk, mk_eq_mk', ← IsLocalization.mk'_mul, Submonoid.mk_mul_mk,
IsLocalization.mk'_eq_iff_eq, mul_assoc] | case exists_of_eq.intro.intro.intro.intro.intro.intro
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
x : ↥(pbo f)
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
this : Algebra (A⁰_ f) (AtPrime 𝒜 (↑x).asHomogeneousIdeal.toIdeal) := (mapId ... | afc745eaf2bf3f81 |
Polynomial.add_scaleRoots_of_natDegree_eq | Mathlib/RingTheory/Polynomial/ScaleRoots.lean | lemma add_scaleRoots_of_natDegree_eq (p q : R[X]) (r : R) (h : natDegree p = natDegree q) :
r ^ (natDegree p - natDegree (p + q)) • (p + q).scaleRoots r =
p.scaleRoots r + q.scaleRoots r | case a
R : Type u_1
inst✝ : CommSemiring R
p q : R[X]
r : R
h : p.natDegree = q.natDegree
n : ℕ
⊢ (r ^ (p.natDegree - (p + q).natDegree) • (p + q).scaleRoots r).coeff n = (p.scaleRoots r + q.scaleRoots r).coeff n | simp only [coeff_smul, coeff_scaleRoots, coeff_add, smul_eq_mul,
mul_comm (r ^ _), ← pow_add, ← h, ← add_mul, add_comm (_ - n)] | case a
R : Type u_1
inst✝ : CommSemiring R
p q : R[X]
r : R
h : p.natDegree = q.natDegree
n : ℕ
⊢ (p.coeff n + q.coeff n) * r ^ ((p + q).natDegree - n) * r ^ (p.natDegree - (p + q).natDegree) =
(p.coeff n + q.coeff n) * r ^ (p.natDegree - n) | 5945b7da60f24c45 |
Subgroup.Normal.commutator_le_of_self_sup_commutative_eq_top | Mathlib/GroupTheory/Abelianization.lean | theorem Subgroup.Normal.commutator_le_of_self_sup_commutative_eq_top {N : Subgroup G} [N.Normal]
{H : Subgroup G} (hHN : N ⊔ H = ⊤) (hH : Subgroup.IsCommutative H) :
_root_.commutator G ≤ N | G : Type u
inst✝¹ : Group G
N : Subgroup G
inst✝ : N.Normal
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : H.IsCommutative
φ : ↥H →ₙ* G ⧸ N := ↑((QuotientGroup.mk' N).comp H.subtype)
⊢ Subgroup.map (QuotientGroup.mk' N) ⊤ = ⊤ | rw [← MonoidHom.range_eq_map, MonoidHom.range_eq_top] | G : Type u
inst✝¹ : Group G
N : Subgroup G
inst✝ : N.Normal
H : Subgroup G
hHN : N ⊔ H = ⊤
hH : H.IsCommutative
φ : ↥H →ₙ* G ⧸ N := ↑((QuotientGroup.mk' N).comp H.subtype)
⊢ Function.Surjective ⇑(QuotientGroup.mk' N) | 4f1c7225fbfaf09a |
ModP.mul_ne_zero_of_pow_p_ne_zero | Mathlib/RingTheory/Perfection.lean | theorem mul_ne_zero_of_pow_p_ne_zero {x y : ModP O p} (hx : x ^ p ≠ 0) (hy : y ^ p ≠ 0) :
x * y ≠ 0 | case pos
K : Type u₁
inst✝² : Field K
v : Valuation K ℝ≥0
O : Type u₂
inst✝¹ : CommRing O
inst✝ : Algebra O K
hv : v.Integers O
p : ℕ
hp : Fact (Nat.Prime p)
r : O
hx : v ↑p ^ (1 / ↑p) < v ((algebraMap O K) r)
s : O
hy : v ↑p ^ (1 / ↑p) < v ((algebraMap O K) s)
h1p : 0 < 1 / ↑p
hvp : v ↑p = 0
⊢ v ↑p ≤ v ↑p ^ (1 / ↑p) *... | rw [hvp] | case pos
K : Type u₁
inst✝² : Field K
v : Valuation K ℝ≥0
O : Type u₂
inst✝¹ : CommRing O
inst✝ : Algebra O K
hv : v.Integers O
p : ℕ
hp : Fact (Nat.Prime p)
r : O
hx : v ↑p ^ (1 / ↑p) < v ((algebraMap O K) r)
s : O
hy : v ↑p ^ (1 / ↑p) < v ((algebraMap O K) s)
h1p : 0 < 1 / ↑p
hvp : v ↑p = 0
⊢ 0 ≤ 0 ^ (1 / ↑p) * 0 ^ (... | fcaa6e56fffb7efd |
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‖ | E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
p : Subspace ℝ E
f : ↥p →L[ℝ] ℝ
c : ℝ
hc : 0 < c
x : E
⊢ (fun x => ‖f‖ * ‖x‖) (c • x) = c * (fun x => ‖f‖ * ‖x‖) x | simp only [norm_smul c x, Real.norm_eq_abs, abs_of_pos hc, mul_left_comm] | no goals | 0f306ef1f16e3415 |
CategoryTheory.Limits.preservesBinaryBiproduct_of_preservesBinaryProduct | Mathlib/CategoryTheory/Preadditive/Biproducts.lean | /-- A functor between preadditive categories that preserves (zero morphisms and) binary products
preserves binary biproducts. -/
lemma preservesBinaryBiproduct_of_preservesBinaryProduct {X Y : C} [PreservesLimit (pair X Y) F] :
PreservesBinaryBiproduct X Y F where
preserves {b} hb := ⟨isBinaryBilimitOfIsLimit... | C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
D : Type u'
inst✝³ : Category.{v', u'} D
inst✝² : Preadditive D
F : C ⥤ D
inst✝¹ : F.PreservesZeroMorphisms
X Y : C
inst✝ : PreservesLimit (pair X Y) F
b : BinaryBicone X Y
hb : b.IsBilimit
⊢ F.obj b.pt ≅ F.obj b.pt | rfl | no goals | 07a7c7d65f60322c |
PMF.toOuterMeasure_bindOnSupport_apply | Mathlib/Probability/ProbabilityMassFunction/Monad.lean | theorem toOuterMeasure_bindOnSupport_apply :
(p.bindOnSupport f).toOuterMeasure s =
∑' a, p a * if h : p a = 0 then 0 else (f a h).toOuterMeasure s | α : Type u_1
β : Type u_2
p : PMF α
f : (a : α) → a ∈ p.support → PMF β
s : Set β
⊢ (p.bindOnSupport f).toOuterMeasure s = ∑' (a : α), p a * if h : p a = 0 then 0 else (f a h).toOuterMeasure s | simp only [toOuterMeasure_apply, Set.indicator_apply, bindOnSupport_apply] | α : Type u_1
β : Type u_2
p : PMF α
f : (a : α) → a ∈ p.support → PMF β
s : Set β
⊢ ∑' (x : β), s.indicator (⇑(p.bindOnSupport f)) x =
∑' (a : α), p a * if h : p a = 0 then 0 else ∑' (x : β), s.indicator (⇑(f a ⋯)) x | 48552b8cdb86bb57 |
MeasureTheory.withDensity_inv_same₀ | Mathlib/MeasureTheory/Measure/WithDensity.lean | lemma withDensity_inv_same₀ {μ : Measure α} {f : α → ℝ≥0∞}
(hf : AEMeasurable f μ) (hf_ne_zero : ∀ᵐ x ∂μ, f x ≠ 0) (hf_ne_top : ∀ᵐ x ∂μ, f x ≠ ∞) :
(μ.withDensity f).withDensity (fun x ↦ (f x)⁻¹) = μ | case h
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
hf_ne_zero✝ : ∀ᵐ (x : α) ∂μ, f x ≠ 0
hf_ne_top✝ : ∀ᵐ (x : α) ∂μ, f x ≠ ⊤
x : α
hf_ne_zero : f x ≠ 0
hf_ne_top : f x ≠ ⊤
⊢ f x * (f x)⁻¹ = 1 x | rw [ENNReal.mul_inv_cancel hf_ne_zero hf_ne_top, Pi.one_apply] | no goals | afe625e716585194 |
MeasureTheory.absolutelyContinuous_of_isMulLeftInvariant | Mathlib/MeasureTheory/Group/Prod.lean | theorem absolutelyContinuous_of_isMulLeftInvariant [IsMulLeftInvariant ν] (hν : ν ≠ 0) : μ ≪ ν | G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
μ ν : Measure G
inst✝⁴ : SFinite ν
inst✝³ : SFinite μ
inst✝² : MeasurableInv G
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : ν.IsMulLeftInvariant
hν : ν ≠ 0
s : Set G
sm : MeasurableSet s
hνs : ν s = 0
h1 : μ s = 0
⊢ μ s = 0 | exact h1 | no goals | 3b30d34d996bb593 |
Array.toList_fst_unzip | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean | theorem toList_fst_unzip (as : Array (α × β)) :
as.unzip.1.toList = as.toList.unzip.1 | α : Type u_1
β : Type u_2
as : Array (α × β)
⊢ as.unzip.fst.toList = as.toList.unzip.fst | simp | no goals | 5d0be5483eada628 |
tprod_empty | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | theorem tprod_empty [IsEmpty β] : ∏' b, f b = 1 | α : Type u_1
β : Type u_2
inst✝² : CommMonoid α
inst✝¹ : TopologicalSpace α
f : β → α
inst✝ : IsEmpty β
⊢ ∏' (b : β), f b = 1 | rw [tprod_eq_prod (s := (∅ : Finset β))] <;> simp | no goals | 38ca6830aa7e9b06 |
ZMod.LFunction_stdAddChar_eq_expZeta | Mathlib/NumberTheory/LSeries/ZMod.lean | /--
The `LFunction` of the function `x ↦ e (j * x)`, where `e : ZMod N → ℂ` is the standard additive
character, is `expZeta (j / N)`.
Note this is not at all obvious from the definitions, and we prove it by analytic continuation
from the convergence range.
-/
lemma LFunction_stdAddChar_eq_expZeta (j : ZMod N) (s : ℂ) ... | N : ℕ
inst✝ : NeZero N
j : ZMod N
s : ℂ
hjs : j ≠ 0 ∨ s ≠ 1
U : Set ℂ := if j = 0 then {z | z ≠ 1} else Set.univ
V : Set ℂ := {z | 1 < z.re}
hUo : IsOpen U
f : ℂ → ℂ := LFunction fun k => 𝕖 (j * k)
g : ℂ → ℂ := expZeta (toAddCircle j)
hU : ∀ {u : ℂ}, u ∈ U ↔ u ≠ 1 ∨ j ≠ 0
hf : AnalyticOnNhd ℂ f U
hg : AnalyticOnNhd ℂ ... | filter_upwards [hV] with z using LFunction_stdAddChar_eq_expZeta_of_one_lt_re _ | no goals | 11c978f7c59083e2 |
CategoryTheory.Adjunction.full_L_of_isSplitEpi_unit_app | Mathlib/CategoryTheory/Adjunction/FullyFaithful.lean | /-- If each component of the unit is a split epimorphism, then the left adjoint is full. -/
lemma full_L_of_isSplitEpi_unit_app [∀ X, IsSplitEpi (h.unit.app X)] : L.Full where
map_surjective {X Y} f | case h
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
L : C ⥤ D
R : D ⥤ C
h : L ⊣ R
inst✝ : ∀ (X : C), IsSplitEpi (h.unit.app X)
X Y : C
f : L.obj X ⟶ L.obj Y
⊢ L.map (section_ (h.unit.app Y)) = h.counit.app (L.obj Y) | rw [← comp_id (L.map (section_ (h.unit.app Y)))] | case h
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
L : C ⥤ D
R : D ⥤ C
h : L ⊣ R
inst✝ : ∀ (X : C), IsSplitEpi (h.unit.app X)
X Y : C
f : L.obj X ⟶ L.obj Y
⊢ L.map (section_ (h.unit.app Y)) ≫ 𝟙 (L.obj ((𝟭 C).obj Y)) = h.counit.app (L.obj Y) | 99582905f384847b |
List.splitLengths_length_getElem | Mathlib/Data/List/SplitLengths.lean | theorem splitLengths_length_getElem {α : Type*} (l : List α) (sz : List ℕ)
(h : sz.sum ≤ l.length) (i : ℕ) (hi : i < (sz.splitLengths l).length) :
(sz.splitLengths l)[i].length = sz[i]'(by simpa using hi) | α : Type u_2
l : List α
sz : List ℕ
h : sz.sum ≤ l.length
i : ℕ
hi : i < (sz.splitLengths l).length
⊢ (sz.splitLengths l)[i].length = sz[i] | have := map_splitLengths_length l sz h | α : Type u_2
l : List α
sz : List ℕ
h : sz.sum ≤ l.length
i : ℕ
hi : i < (sz.splitLengths l).length
this : map length (sz.splitLengths l) = sz
⊢ (sz.splitLengths l)[i].length = sz[i] | 4e55a93adad03450 |
MeasureTheory.exists_measure_iInter_lt | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | theorem exists_measure_iInter_lt {α ι : Type*} {_ : MeasurableSpace α} {μ : Measure α}
[SemilatticeSup ι] [Countable ι] {f : ι → Set α}
(hm : ∀ i, NullMeasurableSet (f i) μ) {ε : ℝ≥0∞} (hε : 0 < ε) (hfin : ∃ i, μ (f i) ≠ ∞)
(hfem : ⋂ n, f n = ∅) : ∃ m, μ (⋂ n ≤ m, f n) < ε | α : Type u_8
ι : Type u_9
x✝ : MeasurableSpace α
μ : Measure α
inst✝¹ : SemilatticeSup ι
inst✝ : Countable ι
f : ι → Set α
hm : ∀ (i : ι), NullMeasurableSet (f i) μ
ε : ℝ≥0∞
hε : 0 < ε
hfin : ∃ i, μ (f i) ≠ ⊤
hfem : ⋂ n, f n = ∅
⊢ ∃ m, μ (⋂ n, ⋂ (_ : n ≤ m), f n) < ε | let F m := μ (⋂ n ≤ m, f n) | α : Type u_8
ι : Type u_9
x✝ : MeasurableSpace α
μ : Measure α
inst✝¹ : SemilatticeSup ι
inst✝ : Countable ι
f : ι → Set α
hm : ∀ (i : ι), NullMeasurableSet (f i) μ
ε : ℝ≥0∞
hε : 0 < ε
hfin : ∃ i, μ (f i) ≠ ⊤
hfem : ⋂ n, f n = ∅
F : ι → ℝ≥0∞ := fun m => μ (⋂ n, ⋂ (_ : n ≤ m), f n)
⊢ ∃ m, μ (⋂ n, ⋂ (_ : n ≤ m), f n) < ε | 1f737bad22d86841 |
PhragmenLindelof.horizontal_strip | Mathlib/Analysis/Complex/PhragmenLindelof.lean | theorem horizontal_strip (hfd : DiffContOnCl ℂ f (im ⁻¹' Ioo a b))
(hB : ∃ c < π / (b - a), ∃ B, f =O[comap (_root_.abs ∘ re) atTop ⊓ 𝓟 (im ⁻¹' Ioo a b)]
fun z ↦ expR (B * expR (c * |z.re|)))
(hle_a : ∀ z : ℂ, im z = a → ‖f z‖ ≤ C) (hle_b : ∀ z, im z = b → ‖f z‖ ≤ C) (hza : a ≤ im z)
(hzb : im z ≤ b)... | case neg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C✝ : ℝ
f : ℂ → E
z : ℂ
C : ℝ
hC₀ : 0 < C
a b : ℝ
hza : a - b < z.im
hle_a : ∀ (z : ℂ), z.im = a - b → ‖f z‖ ≤ C
hzb : z.im < a + b
hle_b : ∀ (z : ℂ), z.im = a + b → ‖f z‖ ≤ C
hfd : DiffContOnCl ℂ f (im ⁻¹' Ioo (a - b) (a + b))
hab : a - b < a +... | have hw' := eq_endpoints_or_mem_Ioo_of_mem_Icc hw.2 | case neg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C✝ : ℝ
f : ℂ → E
z : ℂ
C : ℝ
hC₀ : 0 < C
a b : ℝ
hza : a - b < z.im
hle_a : ∀ (z : ℂ), z.im = a - b → ‖f z‖ ≤ C
hzb : z.im < a + b
hle_b : ∀ (z : ℂ), z.im = a + b → ‖f z‖ ≤ C
hfd : DiffContOnCl ℂ f (im ⁻¹' Ioo (a - b) (a + b))
hab : a - b < a +... | 96530d0dcac40a71 |
Finset.antidiagonal.snd_le | Mathlib/Algebra/Order/Antidiag/Prod.lean | theorem antidiagonal.snd_le {n : A} {kl : A × A} (hlk : kl ∈ antidiagonal n) : kl.2 ≤ n | A : Type u_1
inst✝² : OrderedAddCommMonoid A
inst✝¹ : CanonicallyOrderedAdd A
inst✝ : HasAntidiagonal A
n : A
kl : A × A
hlk : kl ∈ antidiagonal n
⊢ kl.2 ≤ n | rw [le_iff_exists_add] | A : Type u_1
inst✝² : OrderedAddCommMonoid A
inst✝¹ : CanonicallyOrderedAdd A
inst✝ : HasAntidiagonal A
n : A
kl : A × A
hlk : kl ∈ antidiagonal n
⊢ ∃ c, n = kl.2 + c | a346e570b0197919 |
EReal.add_pos | Mathlib/Data/Real/EReal.lean | theorem add_pos {a b : EReal} (ha : 0 < a) (hb : 0 < b) : 0 < a + b | case h_real
b : EReal
hb : 0 < b
a✝ : ℝ
ha : 0 < ↑a✝
⊢ 0 < ↑a✝ + b | induction b | case h_real.h_bot
a✝ : ℝ
ha : 0 < ↑a✝
hb : 0 < ⊥
⊢ 0 < ↑a✝ + ⊥
case h_real.h_real
a✝¹ : ℝ
ha : 0 < ↑a✝¹
a✝ : ℝ
hb : 0 < ↑a✝
⊢ 0 < ↑a✝¹ + ↑a✝
case h_real.h_top
a✝ : ℝ
ha : 0 < ↑a✝
hb : 0 < ⊤
⊢ 0 < ↑a✝ + ⊤ | ea146a9f917f5977 |
CategoryTheory.Grothendieck.fiber_eqToHom | Mathlib/CategoryTheory/Grothendieck.lean | theorem fiber_eqToHom {X Y : Grothendieck F} (h : X = Y) :
(eqToHom h).fiber = eqToHom (by subst h; simp) | C : Type u
inst✝¹ : Category.{v, u} C
D : Type u₁
inst✝ : Category.{v₁, u₁} D
F : C ⥤ Cat
X : Grothendieck F
⊢ (F.map (eqToHom ⋯).base).obj X.fiber = X.fiber | simp | no goals | 6408252bccdc4edc |
Array.toList_filterMap' | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean | theorem toList_filterMap' (f : α → Option β) (l : Array α) (w : stop = l.size) :
(l.filterMap f 0 stop).toList = l.toList.filterMap f | α : Type u_1
β : Type u_2
f : α → Option β
l : Array α
⊢ (filterMap f l).toList = List.filterMap f l.toList | dsimp only [filterMap, filterMapM] | α : Type u_1
β : Type u_2
f : α → Option β
l : Array α
⊢ (foldlM
(fun bs a => do
let __do_lift ← f a
match __do_lift with
| some b => pure (bs.push b)
| none => pure bs)
#[] l).run.toList =
List.filterMap f l.toList | 0bf06d250b5cdb33 |
gramSchmidt_orthogonal | Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean | theorem gramSchmidt_orthogonal (f : ι → E) {a b : ι} (h₀ : a ≠ b) :
⟪gramSchmidt 𝕜 f a, gramSchmidt 𝕜 f b⟫ = 0 | case neg.h
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
b✝ b : ι
ih : ∀ y < b, ∀ a < y, inner (gramSchmidt 𝕜 f a) (gramSchmidt 𝕜 f y) = 0
a : ι
h₀ : a <... | rwa [inner_self_ne_zero] | no goals | a5e33da8a7158e66 |
ProbabilityTheory.cond_iInter | Mathlib/Probability/Independence/Basic.lean | /-- The probability of an intersection of preimages conditioning on another intersection factors
into a product. -/
lemma cond_iInter [Finite ι] (hY : ∀ i, Measurable (Y i))
(hindep : iIndepFun (fun _ ↦ mα.prod mβ) (fun i ω ↦ (X i ω, Y i ω)) μ)
(hf : ∀ i ∈ s, MeasurableSet[mα.comap (X i)] (f i))
(hy : ∀ i ∉... | ι : Type u_6
Ω : Type u_7
α : Type u_8
β : Type u_9
mΩ : MeasurableSpace Ω
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure Ω
X : ι → Ω → α
Y : ι → Ω → β
f : ι → Set Ω
t : ι → Set β
s : Finset ι
inst✝ : Finite ι
hY : ∀ (i : ι), Measurable (Y i)
hindep : iIndepFun (fun x => mα.prod mβ) (fun i ω => (X i ω, Y i ω... | by_cases hi : i ∈ s <;> simp only [hi, ↓reduceIte, g] | case pos
ι : Type u_6
Ω : Type u_7
α : Type u_8
β : Type u_9
mΩ : MeasurableSpace Ω
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure Ω
X : ι → Ω → α
Y : ι → Ω → β
f : ι → Set Ω
t : ι → Set β
s : Finset ι
inst✝ : Finite ι
hY : ∀ (i : ι), Measurable (Y i)
hindep : iIndepFun (fun x => mα.prod mβ) (fun i ω => (X i... | 299952e2d8d690c9 |
Order.Ideal.isProper_of_not_mem | Mathlib/Order/Ideal.lean | theorem isProper_of_not_mem {I : Ideal P} {p : P} (nmem : p ∉ I) : IsProper I :=
⟨fun hp ↦ by
have := mem_univ p
rw [← hp] at this
exact nmem this⟩
| P : Type u_1
inst✝ : LE P
I : Ideal P
p : P
nmem : p ∉ I
hp : ↑I = univ
⊢ False | have := mem_univ p | P : Type u_1
inst✝ : LE P
I : Ideal P
p : P
nmem : p ∉ I
hp : ↑I = univ
this : p ∈ univ
⊢ False | eedecb497d07f70f |
Std.Sat.AIG.RefVec.zip.go_get_aux | Mathlib/.lake/packages/lean4/src/lean/Std/Sat/AIG/RefVecOperator/Zip.lean | theorem go_get_aux {aig : AIG α} (curr : Nat) (hcurr : curr ≤ len) (s : RefVec aig curr)
(lhs rhs : RefVec aig len) (f : (aig : AIG α) → BinaryInput aig → Entrypoint α)
[LawfulOperator α BinaryInput f] [chainable : LawfulZipOperator α f] :
-- The hfoo here is a trick to make the dependent type gods happy
... | case isFalse
α : Type
inst✝² : Hashable α
inst✝¹ : DecidableEq α
len : Nat
aig : AIG α
curr : Nat
hcurr : curr ≤ len
s : aig.RefVec curr
lhs rhs : aig.RefVec len
f : (aig : AIG α) → aig.BinaryInput → Entrypoint α
inst✝ : LawfulOperator α BinaryInput f
chainable : LawfulZipOperator α f
idx : Nat
hidx : idx < curr
res : ... | rw [← hgo] | case isFalse
α : Type
inst✝² : Hashable α
inst✝¹ : DecidableEq α
len : Nat
aig : AIG α
curr : Nat
hcurr : curr ≤ len
s : aig.RefVec curr
lhs rhs : aig.RefVec len
f : (aig : AIG α) → aig.BinaryInput → Entrypoint α
inst✝ : LawfulOperator α BinaryInput f
chainable : LawfulZipOperator α f
idx : Nat
hidx : idx < curr
res : ... | 90c056b58a9b7a80 |
orderOf_pow_dvd | Mathlib/GroupTheory/OrderOfElement.lean | theorem orderOf_pow_dvd (n : ℕ) : orderOf (x ^ n) ∣ orderOf x | G : Type u_1
inst✝ : Monoid G
x : G
n : ℕ
⊢ orderOf (x ^ n) ∣ orderOf x | rw [orderOf_dvd_iff_pow_eq_one, pow_right_comm, pow_orderOf_eq_one, one_pow] | no goals | 68c1a84e882fb930 |
FreeGroup.Red.red_iff_irreducible | Mathlib/GroupTheory/FreeGroup/Basic.lean | theorem red_iff_irreducible {x1 b1 x2 b2} (h : (x1, b1) ≠ (x2, b2)) :
Red [(x1, !b1), (x2, b2)] L ↔ L = [(x1, !b1), (x2, b2)] | α : Type u
L : List (α × Bool)
x1 : α
b1 : Bool
x2 : α
b2 : Bool
h : (x1, b1) ≠ (x2, b2)
⊢ Red [(x1, !b1), (x2, b2)] L ↔ L = [(x1, !b1), (x2, b2)] | apply reflTransGen_iff_eq | case h
α : Type u
L : List (α × Bool)
x1 : α
b1 : Bool
x2 : α
b2 : Bool
h : (x1, b1) ≠ (x2, b2)
⊢ ∀ (b : List (α × Bool)), ¬Step [(x1, !b1), (x2, b2)] b | f87886f761db64c8 |
Array.mapM_map_eq_foldl | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean | theorem mapM_map_eq_foldl (as : Array α) (f : α → β) (i) :
mapM.map (m := Id) f as i b = as.foldl (start := i) (fun r a => r.push (f a)) b | α : Type u_1
β : Type u_2
b : Array β
as : Array α
f : α → β
i : Nat
⊢ (if hlt : i < as.size then do
let __do_lift ← f as[i]
mapM.map f as (i + 1) (b.push __do_lift)
else pure b) =
foldl (fun r a => r.push (f a)) b as i | split <;> rename_i h | case isTrue
α : Type u_1
β : Type u_2
b : Array β
as : Array α
f : α → β
i : Nat
h : i < as.size
⊢ (do
let __do_lift ← f as[i]
mapM.map f as (i + 1) (b.push __do_lift)) =
foldl (fun r a => r.push (f a)) b as i
case isFalse
α : Type u_1
β : Type u_2
b : Array β
as : Array α
f : α → β
i : Nat
h : ¬i < as... | a2d615417b2bd811 |
MonoidHom.noncommCoprod_range | Mathlib/GroupTheory/NoncommCoprod.lean | lemma noncommCoprod_range {M N P : Type*} [Group M] [Group N] [Group P]
(f : M →* P) (g : N →* P) (comm : ∀ (m : M) (n : N), Commute (f m) (g n)) :
(noncommCoprod f g comm).range = f.range ⊔ g.range | case a.left.intro
M : Type u_4
N : Type u_5
P : Type u_6
inst✝² : Group M
inst✝¹ : Group N
inst✝ : Group P
f : M →* P
g : N →* P
comm : ∀ (m : M) (n : N), Commute (f m) (g n)
a : M
⊢ f a ∈ (f.noncommCoprod g comm).range | exact ⟨(a, 1), by rw [noncommCoprod_apply, map_one, mul_one]⟩ | no goals | a43981614b1c4850 |
SimpleGraph.Walk.takeUntil_takeUntil | Mathlib/Combinatorics/SimpleGraph/Connectivity/WalkDecomp.lean | lemma takeUntil_takeUntil {w x : V} (p : G.Walk u v) (hw : w ∈ p.support)
(hx : x ∈ (p.takeUntil w hw).support) :
(p.takeUntil w hw).takeUntil x hx = p.takeUntil x (p.support_takeUntil_subset hw hx) | case case3
V : Type u
G : SimpleGraph V
v u : V
inst✝ : DecidableEq V
w x : V
p : G.Walk u v
a w' v' : V
hadj : G.Adj a v'
q : G.Walk v' w'
u' : V
hu'✝ : u' ∈ (cons hadj q).support
hau'✝ : ¬a = u'
hau' : a ≠ u'
x✝ : u' ∈ (cons hadj q).support
ih : ∀ (hx : x ∈ (q.takeUntil u' ⋯).support), (q.takeUntil u' ⋯).takeUntil x ... | by_cases hx' : x = a | case pos
V : Type u
G : SimpleGraph V
v u : V
inst✝ : DecidableEq V
w x : V
p : G.Walk u v
a w' v' : V
hadj : G.Adj a v'
q : G.Walk v' w'
u' : V
hu'✝ : u' ∈ (cons hadj q).support
hau'✝ : ¬a = u'
hau' : a ≠ u'
x✝ : u' ∈ (cons hadj q).support
ih : ∀ (hx : x ∈ (q.takeUntil u' ⋯).support), (q.takeUntil u' ⋯).takeUntil x hx... | c3b1f330685fe6f0 |
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 | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
s : Cycle α
h : s.Nodup
hn : s.Nontrivial
⊢ (s.formPerm h).support = s.toFinset | induction' s using Quot.inductionOn with s | 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)
⊢ (formPerm (Quot.mk (⇑(IsRotated.setoid α)) s) h).support = toFinset (Quot.mk (⇑(IsRotated.setoid α)) s) | c975caac8b3ab4b4 |
Vector.toArray_mapM_go | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Lemmas.lean | theorem toArray_mapM_go [Monad m] [LawfulMonad m] (f : α → m β) (v : Vector α n) (i h r) :
toArray <$> mapM.go f v i h r = Array.mapM.map f v.toArray i r.toArray | m : Type u_1 → Type u_2
α : Type u_3
β : Type u_1
n : Nat
inst✝¹ : Monad m
inst✝ : LawfulMonad m
f : α → m β
v : Vector α n
i : Nat
h : i ≤ n
r : Vector β i
⊢ (toArray <$>
if h' : i < n then do
let __do_lift ← f v[i]
mapM.go f v (i + 1) ⋯ (r.push __do_lift)
else pure (Vector.cast ⋯ r)) =
... | unfold Array.mapM.map | m : Type u_1 → Type u_2
α : Type u_3
β : Type u_1
n : Nat
inst✝¹ : Monad m
inst✝ : LawfulMonad m
f : α → m β
v : Vector α n
i : Nat
h : i ≤ n
r : Vector β i
⊢ (toArray <$>
if h' : i < n then do
let __do_lift ← f v[i]
mapM.go f v (i + 1) ⋯ (r.push __do_lift)
else pure (Vector.cast ⋯ r)) =
... | b8bb1e3d17c268d2 |
IsAntichain.volume_eq_zero | Mathlib/MeasureTheory/Order/UpperLower.lean | theorem IsAntichain.volume_eq_zero [Nonempty ι] (hs : IsAntichain (· ≤ ·) s) : volume s = 0 | ι : Type u_1
inst✝¹ : Fintype ι
s : Set (ι → ℝ)
inst✝ : Nonempty ι
hs : IsAntichain (fun x1 x2 => x1 ≤ x2) s
⊢ s ⊆ frontier s | rw [← closure_diff_interior, hs.interior_eq_empty, diff_empty] | ι : Type u_1
inst✝¹ : Fintype ι
s : Set (ι → ℝ)
inst✝ : Nonempty ι
hs : IsAntichain (fun x1 x2 => x1 ≤ x2) s
⊢ s ⊆ closure s | e697af406be31b26 |
PFunctor.M.ext | Mathlib/Data/PFunctor/Univariate/M.lean | theorem ext [Inhabited (M F)] (x y : M F) (H : ∀ ps : Path F, iselect ps x = iselect ps y) :
x = y | case H.succ.hrec
F : PFunctor.{u}
inst✝ : Inhabited F.M
x y : F.M
H : ∀ (ps : Path F), (isubtree ps x).head = (isubtree ps y).head
i : ℕ
i_ih : x.approx i = y.approx i
ps : Path F
H' : i = length ps
⊢ iselect ps x = iselect ps y | cases H' | case H.succ.hrec.refl
F : PFunctor.{u}
inst✝ : Inhabited F.M
x y : F.M
H : ∀ (ps : Path F), (isubtree ps x).head = (isubtree ps y).head
ps : Path F
i_ih : x.approx (length ps) = y.approx (length ps)
⊢ iselect ps x = iselect ps y | d42c81fe4cdd416b |
Submodule.span_preimage_eq | Mathlib/LinearAlgebra/Quotient/Basic.lean | theorem span_preimage_eq [RingHomSurjective τ₁₂] {f : M →ₛₗ[τ₁₂] M₂} {s : Set M₂} (h₀ : s.Nonempty)
(h₁ : s ⊆ range f) : span R (f ⁻¹' s) = (span R₂ s).comap f | R : Type u_1
M : Type u_2
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
R₂ : Type u_3
M₂ : Type u_4
inst✝³ : Ring R₂
inst✝² : AddCommGroup M₂
inst✝¹ : Module R₂ M₂
τ₁₂ : R →+* R₂
inst✝ : RingHomSurjective τ₁₂
f : M →ₛₗ[τ₁₂] M₂
s : Set M₂
h₀ : s.Nonempty
h₁ : s ⊆ ↑(range f)
y : M₂ := Classical.choose h₀
hy... | rw [ker_le_iff] | R : Type u_1
M : Type u_2
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
R₂ : Type u_3
M₂ : Type u_4
inst✝³ : Ring R₂
inst✝² : AddCommGroup M₂
inst✝¹ : Module R₂ M₂
τ₁₂ : R →+* R₂
inst✝ : RingHomSurjective τ₁₂
f : M →ₛₗ[τ₁₂] M₂
s : Set M₂
h₀ : s.Nonempty
h₁ : s ⊆ ↑(range f)
y : M₂ := Classical.choose h₀
hy... | 5560c859997de634 |
Matrix.det_eq_of_forall_row_eq_smul_add_const_aux | Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean | theorem det_eq_of_forall_row_eq_smul_add_const_aux {A B : Matrix n n R} {s : Finset n} :
∀ (c : n → R) (_ : ∀ i, i ∉ s → c i = 0) (k : n) (_ : k ∉ s)
(_ : ∀ i j, A i j = B i j + c i * B k j), det A = det B | case empty
n : Type u_2
inst✝² : DecidableEq n
inst✝¹ : Fintype n
R : Type v
inst✝ : CommRing R
A B : Matrix n n R
⊢ ∀ (c : n → R), (∀ i ∉ ∅, c i = 0) → ∀ k ∉ ∅, (∀ (i j : n), A i j = B i j + c i * B k j) → A.det = B.det | rintro c hs k - A_eq | case empty
n : Type u_2
inst✝² : DecidableEq n
inst✝¹ : Fintype n
R : Type v
inst✝ : CommRing R
A B : Matrix n n R
c : n → R
hs : ∀ i ∉ ∅, c i = 0
k : n
A_eq : ∀ (i j : n), A i j = B i j + c i * B k j
⊢ A.det = B.det | 7739d96b149cd534 |
Equiv.Perm.cycle_zpow_mem_support_iff | Mathlib/GroupTheory/Perm/Cycle/Basic.lean | theorem cycle_zpow_mem_support_iff {g : Perm α}
(hg : g.IsCycle) {n : ℤ} {x : α} (hx : g x ≠ x) :
(g ^ n) x = x ↔ n % #g.support = 0 | case mp
α : Type u_2
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g : Perm α
hg : g.IsCycle
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(#g.support)
r : ℤ := n % ↑(#g.support)
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
⊢ (g ^ m) x = x → g ^ m = 1 | intro hgm | case mp
α : Type u_2
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g : Perm α
hg : g.IsCycle
n : ℤ
x : α
hx : g x ≠ x
q : ℤ := n / ↑(#g.support)
r : ℤ := n % ↑(#g.support)
m : ℕ
hm : r = ↑m
div_euc : ↑m + ↑(orderOf g) * q = n ∧ m < orderOf g
hgm : (g ^ m) x = x
⊢ g ^ m = 1 | 53814599c33ac8eb |
HomologicalComplex.extendCyclesIso_inv_iCycles | Mathlib/Algebra/Homology/Embedding/ExtendHomology.lean | @[reassoc (attr := simp)]
lemma extendCyclesIso_inv_iCycles :
(K.extendCyclesIso e hj').inv ≫ (K.extend e).iCycles j' =
K.iCycles j ≫ (K.extendXIso e hj').inv | 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.extendCyclesI... | simp only [← cancel_epi (K.extendCyclesIso e hj').hom, Iso.hom_inv_id_assoc,
extendCyclesIso_hom_iCycles_assoc, Iso.hom_inv_id, comp_id] | no goals | e2549df1b0ab6814 |
SeminormFamily.basisSets_intersect | Mathlib/Analysis/LocallyConvex/WithSeminorms.lean | theorem basisSets_intersect (U V : Set E) (hU : U ∈ p.basisSets) (hV : V ∈ p.basisSets) :
∃ z ∈ p.basisSets, z ⊆ U ∩ V | case h
𝕜 : Type u_1
E : Type u_5
ι : Type u_8
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : SeminormFamily 𝕜 E ι
U V : Set E
hU✝ : U ∈ p.basisSets
hV✝ : V ∈ p.basisSets
s : Finset ι
r₁ : ℝ
hr₁ : 0 < r₁
hU : U = (s.sup p).ball 0 r₁
t : Finset ι
r₂ : ℝ
hr₂ : 0 < r₂
hV : V = (t.sup p).ball 0 r₂... | refine ⟨p.basisSets_mem (s ∪ t) (lt_min_iff.mpr ⟨hr₁, hr₂⟩), ?_⟩ | case h
𝕜 : Type u_1
E : Type u_5
ι : Type u_8
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : SeminormFamily 𝕜 E ι
U V : Set E
hU✝ : U ∈ p.basisSets
hV✝ : V ∈ p.basisSets
s : Finset ι
r₁ : ℝ
hr₁ : 0 < r₁
hU : U = (s.sup p).ball 0 r₁
t : Finset ι
r₂ : ℝ
hr₂ : 0 < r₂
hV : V = (t.sup p).ball 0 r₂... | b495b149a3cd9a84 |
Matrix.mulVec_surjective_iff_exists_right_inverse | Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | theorem mulVec_surjective_iff_exists_right_inverse
[DecidableEq m] [Finite m] [Fintype n] {A : Matrix m n R} :
Function.Surjective A.mulVec ↔ ∃ B : Matrix n m R, A * B = 1 | m : Type u
n : Type u'
R : Type u_2
inst✝³ : Semiring R
inst✝² : DecidableEq m
inst✝¹ : Finite m
inst✝ : Fintype n
A : Matrix m n R
⊢ Function.Surjective A.mulVec ↔ ∃ B, A * B = 1 | cases nonempty_fintype m | case intro
m : Type u
n : Type u'
R : Type u_2
inst✝³ : Semiring R
inst✝² : DecidableEq m
inst✝¹ : Finite m
inst✝ : Fintype n
A : Matrix m n R
val✝ : Fintype m
⊢ Function.Surjective A.mulVec ↔ ∃ B, A * B = 1 | e2574278ca6dc051 |
Vector.any_eq_false | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Lemmas.lean | theorem any_eq_false {p : α → Bool} {xs : Vector α n} :
xs.any p = false ↔ ∀ (i : Nat) (_ : i < n), ¬p xs[i] | α : Type u_1
n : Nat
p : α → Bool
xs : Vector α n
⊢ (¬∃ i x, p xs[i] = true) ↔ ∀ (i : Nat) (x : i < n), ¬p xs[i] = true | simp | no goals | f691f53d621bf9a1 |
MeasureTheory.integral_simpleFunc_larger_space | Mathlib/MeasureTheory/Integral/Bochner.lean | theorem integral_simpleFunc_larger_space (hm : m ≤ m0) (f : @SimpleFunc β m F)
(hf_int : Integrable f μ) :
∫ x, f x ∂μ = ∑ x ∈ @SimpleFunc.range β F m f, ENNReal.toReal (μ (f ⁻¹' {x})) • x | F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
β : Type u_6
m m0 : MeasurableSpace β
μ : Measure β
hm : m ≤ m0
f : β →ₛ F
hf_int : Integrable (⇑f) μ
⊢ Integrable (⇑(SimpleFunc.toLargerSpace hm f)) μ | rwa [SimpleFunc.coe_toLargerSpace_eq] | no goals | f561880c405f28b4 |
Fintype.nonempty_field_iff | Mathlib/FieldTheory/Cardinality.lean | theorem Fintype.nonempty_field_iff {α} [Fintype α] : Nonempty (Field α) ↔ IsPrimePow ‖α‖ | α : Type u_1
inst✝ : Fintype α
⊢ IsPrimePow ‖α‖ → Nonempty (Field α) | rintro ⟨p, n, hp, hn, hα⟩ | case intro.intro.intro.intro
α : Type u_1
inst✝ : Fintype α
p n : ℕ
hp : Prime p
hn : 0 < n
hα : p ^ n = ‖α‖
⊢ Nonempty (Field α) | 3fa2a395eda1ffe4 |
MeasureTheory.not_frequently_of_upcrossings_lt_top | Mathlib/Probability/Martingale/Convergence.lean | theorem not_frequently_of_upcrossings_lt_top (hab : a < b) (hω : upcrossings a b f ω ≠ ∞) :
¬((∃ᶠ n in atTop, f n ω < a) ∧ ∃ᶠ n in atTop, b < f n ω) | case intro
Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
ω : Ω
hab : a < b
hω : ∃ k, ∀ (N : ℕ), upcrossingsBefore a b f N ω < k
h₁ : ∀ (a_1 : ℕ), ∃ b ≥ a_1, f b ω < a
h₂ : ∀ (a : ℕ), ∃ b_1 ≥ a, b < f b_1 ω
⊢ False | refine Classical.not_not.2 hω ?_ | case intro
Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
ω : Ω
hab : a < b
hω : ∃ k, ∀ (N : ℕ), upcrossingsBefore a b f N ω < k
h₁ : ∀ (a_1 : ℕ), ∃ b ≥ a_1, f b ω < a
h₂ : ∀ (a : ℕ), ∃ b_1 ≥ a, b < f b_1 ω
⊢ ¬∃ k, ∀ (N : ℕ), upcrossingsBefore a b f N ω < k | 123352cc62fc6065 |
FractionalIdeal.mul_one_div_le_one | Mathlib/RingTheory/FractionalIdeal/Operations.lean | theorem mul_one_div_le_one {I : FractionalIdeal R₁⁰ K} : I * (1 / I) ≤ 1 | case neg
R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
I : FractionalIdeal R₁⁰ K
hI : ¬I = 0
⊢ I * (1 / I) ≤ 1 | rw [← coe_le_coe, coe_mul, coe_div hI, coe_one] | case neg
R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
I : FractionalIdeal R₁⁰ K
hI : ¬I = 0
⊢ ↑I * (1 / ↑I) ≤ 1 | 7b63428ed5392dc6 |
ProbabilityTheory.strong_law_Lp | Mathlib/Probability/StrongLaw.lean | theorem strong_law_Lp {p : ℝ≥0∞} (hp : 1 ≤ p) (hp' : p ≠ ∞) (X : ℕ → Ω → E)
(hℒp : MemLp (X 0) p μ) (hindep : Pairwise ((IndepFun · · μ) on X))
(hident : ∀ i, IdentDistrib (X i) (X 0) μ μ) :
Tendsto (fun (n : ℕ) => eLpNorm (fun ω => (n : ℝ) ⁻¹ • (∑ i ∈ range n, X i ω) - μ[X 0]) p μ)
atTop (𝓝 0) | case neg
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ⊤
X : ℕ → Ω → E
hℒp : MemLp (X 0) p μ
hindep : Pairwise ((fun x1 x2 => IndepFun x1 x2 μ) o... | apply UniformIntegrable.unifIntegrable | case neg.hf
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ⊤
X : ℕ → Ω → E
hℒp : MemLp (X 0) p μ
hindep : Pairwise ((fun x1 x2 => IndepFun x1 x2 μ... | 8d69987f63824be4 |
MvPowerSeries.coeff_mul_right_one_sub_of_lt_order | Mathlib/RingTheory/MvPowerSeries/Order.lean | theorem coeff_mul_right_one_sub_of_lt_order (d : σ →₀ ℕ) (h : degree d < g.order) :
coeff R d ((1 - g) * f) = coeff R d f | σ : Type u_1
R : Type u_3
inst✝ : Ring R
f g : MvPowerSeries σ R
d : σ →₀ ℕ
h : ↑((weight fun x => 1) d) < g.order
⊢ (coeff R d) ((1 - g) * f) = (coeff R d) f | exact coeff_mul_right_one_sub_of_lt_weightedOrder _ h | no goals | 9ca22027bb019392 |
Monotone.tendsto_le_alternating_series | Mathlib/Analysis/SpecificLimits/Normed.lean | theorem Monotone.tendsto_le_alternating_series
(hfl : Tendsto (fun n ↦ ∑ i ∈ range n, (-1) ^ i * f i) atTop (𝓝 l))
(hfm : Monotone f) (k : ℕ) : l ≤ ∑ i ∈ range (2 * k), (-1) ^ i * f i | E : Type u_2
inst✝² : OrderedRing E
inst✝¹ : TopologicalSpace E
inst✝ : OrderClosedTopology E
l : E
f : ℕ → E
hfl : Tendsto (fun n => ∑ i ∈ Finset.range n, (-1) ^ i * f i) atTop (𝓝 l)
hfm : Monotone f
k : ℕ
ha : Antitone fun n => ∑ i ∈ Finset.range (2 * n), (-1) ^ i * f i
n : ℕ
⊢ n ≤ 2 * n | omega | no goals | 8204a47cf4df7c65 |
Matrix.Nondegenerate.toBilin' | Mathlib/LinearAlgebra/Matrix/BilinearForm.lean | theorem _root_.Matrix.Nondegenerate.toBilin' {M : Matrix ι ι R₂} (h : M.Nondegenerate) :
M.toBilin'.Nondegenerate := fun x hx =>
h.eq_zero_of_ortho fun y => by simpa only [toBilin'_apply'] using hx y
| R₂ : Type u_3
inst✝² : CommRing R₂
ι : Type u_6
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
M : Matrix ι ι R₂
h : M.Nondegenerate
x : ι → R₂
hx : ∀ (n : ι → R₂), ((Matrix.toBilin' M) x) n = 0
y : ι → R₂
⊢ x ⬝ᵥ M *ᵥ y = 0 | simpa only [toBilin'_apply'] using hx y | no goals | 1afb909a5fd4bc4d |
spectrum_diagonal | Mathlib/LinearAlgebra/Eigenspace/Matrix.lean | /-- The spectrum of the diagonal operator is the range of the diagonal viewed as a function. -/
lemma spectrum_diagonal [Field R] (d : n → R) :
spectrum R (diagonal d) = Set.range d | case h
R : Type u_1
n : Type u_2
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : Field R
d : n → R
μ : R
⊢ μ ∈ spectrum R (diagonal d) ↔ μ ∈ Set.range d | rw [← AlgEquiv.spectrum_eq (toLinAlgEquiv <| Pi.basisFun R n), ← hasEigenvalue_iff_mem_spectrum] | case h
R : Type u_1
n : Type u_2
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : Field R
d : n → R
μ : R
⊢ HasEigenvalue ((toLinAlgEquiv (Pi.basisFun R n)) (diagonal d)) μ ↔ μ ∈ Set.range d | bc4a66fe7e0f27f7 |
MonomialOrder.div | Mathlib/RingTheory/MvPolynomial/Groebner.lean | theorem div {ι : Type*} {b : ι → MvPolynomial σ R}
(hb : ∀ i, IsUnit (m.leadingCoeff (b i))) (f : MvPolynomial σ R) :
∃ (g : ι →₀ (MvPolynomial σ R)) (r : MvPolynomial σ R),
f = Finsupp.linearCombination _ b g + r ∧
(∀ i, m.degree (b i * (g i)) ≼[m] m.degree f) ∧
(∀ c ∈ r.support, ∀ i, ¬ (... | case neg
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommRing R
ι : Type u_3
b : ι → MvPolynomial σ R
hb : ∀ (i : ι), IsUnit (m.leadingCoeff (b i))
f : MvPolynomial σ R
hb' : ∀ (i : ι), m.degree (b i) ≠ 0
hf0 : ¬f = 0
hf : ∀ (i : ι), ¬m.degree (b i) ≤ m.degree f
g' : ι →₀ MvPolynomial σ R
r' : MvPolynomial σ ... | convert hf i | case h.e'_1.h.e'_4
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommRing R
ι : Type u_3
b : ι → MvPolynomial σ R
hb : ∀ (i : ι), IsUnit (m.leadingCoeff (b i))
f : MvPolynomial σ R
hb' : ∀ (i : ι), m.degree (b i) ≠ 0
hf0 : ¬f = 0
hf : ∀ (i : ι), ¬m.degree (b i) ≤ m.degree f
g' : ι →₀ MvPolynomial σ R
r' : MvPol... | f995acdac7645677 |
EReal.limsup_add_bot_of_ne_top | Mathlib/Topology/Instances/EReal/Lemmas.lean | lemma limsup_add_bot_of_ne_top (h : limsup u f = ⊥) (h' : limsup v f ≠ ⊤) :
limsup (u + v) f = ⊥ | α : Type u_3
f : Filter α
u v : α → EReal
h : limsup u f = ⊥
h' : limsup v f ≠ ⊤
⊢ ⊥ ≠ ⊤ ∨ limsup v f ≠ ⊥ | exact .inl bot_ne_top | no goals | 41c306a0a8f0f954 |
HallMarriageTheorem.hall_hard_inductive_step_A | Mathlib/Combinatorics/Hall/Finite.lean | theorem hall_hard_inductive_step_A {n : ℕ} (hn : Fintype.card ι = n + 1)
(ht : ∀ s : Finset ι, #s ≤ #(s.biUnion t))
(ih :
∀ {ι' : Type u} [Fintype ι'] (t' : ι' → Finset α),
Fintype.card ι' ≤ n →
(∀ s' : Finset ι', #s' ≤ #(s'.biUnion t')) →
∃ f : ι' → α, Function.Injective f ∧... | ι : Type u
α : Type v
inst✝¹ : DecidableEq α
t : ι → Finset α
inst✝ : Fintype ι
n : ℕ
hn : Fintype.card ι = n + 1
ht : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
ih :
∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),
Fintype.card ι' ≤ n →
(∀ (s' : Finset ι'), #s' ≤ #(s'.biUnion t')) → ∃ f, Function.Injec... | intro x h | ι : Type u
α : Type v
inst✝¹ : DecidableEq α
t : ι → Finset α
inst✝ : Fintype ι
n : ℕ
hn : Fintype.card ι = n + 1
ht : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
ih :
∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),
Fintype.card ι' ≤ n →
(∀ (s' : Finset ι'), #s' ≤ #(s'.biUnion t')) → ∃ f, Function.Injec... | 258330b4009c5e5d |
IsPrimitiveRoot.minpoly_dvd_pow_mod | Mathlib/RingTheory/RootsOfUnity/Minpoly.lean | theorem minpoly_dvd_pow_mod {p : ℕ} [hprime : Fact p.Prime] (hdiv : ¬p ∣ n) :
map (Int.castRingHom (ZMod p)) (minpoly ℤ μ) ∣
map (Int.castRingHom (ZMod p)) (minpoly ℤ (μ ^ p)) ^ p | n : ℕ
K : Type u_1
inst✝² : CommRing K
μ : K
h : IsPrimitiveRoot μ n
inst✝¹ : IsDomain K
inst✝ : CharZero K
p : ℕ
hprime : Fact (Nat.Prime p)
hdiv : ¬p ∣ n
Q : ℤ[X] := minpoly ℤ (μ ^ p)
hfrob : map (Int.castRingHom (ZMod p)) Q ^ p = map (Int.castRingHom (ZMod p)) ((expand ℤ p) Q)
⊢ map (Int.castRingHom (ZMod p)) (minpo... | apply RingHom.map_dvd (mapRingHom (Int.castRingHom (ZMod p))) | n : ℕ
K : Type u_1
inst✝² : CommRing K
μ : K
h : IsPrimitiveRoot μ n
inst✝¹ : IsDomain K
inst✝ : CharZero K
p : ℕ
hprime : Fact (Nat.Prime p)
hdiv : ¬p ∣ n
Q : ℤ[X] := minpoly ℤ (μ ^ p)
hfrob : map (Int.castRingHom (ZMod p)) Q ^ p = map (Int.castRingHom (ZMod p)) ((expand ℤ p) Q)
⊢ minpoly ℤ μ ∣ (expand ℤ p) Q | f3a17101a3651aa0 |
MeasureTheory.continuousOn_convolution_right_with_param | Mathlib/Analysis/Convolution.lean | theorem continuousOn_convolution_right_with_param {g : P → G → E'} {s : Set P} {k : Set G}
(hk : IsCompact k) (hgs : ∀ p, ∀ x, p ∈ s → x ∉ k → g p x = 0)
(hf : LocallyIntegrable f μ) (hg : ContinuousOn (↿g) (s ×ˢ univ)) :
ContinuousOn (fun q : P × G => (f ⋆[L, μ] g q.1) q.2) (s ×ˢ univ) | 𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
P : Type uP
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedAddCommGroup E'
inst✝¹¹ : NormedAddCommGroup F
f : G → E
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : NormedSpace 𝕜 E'
inst✝⁷ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F... | rw [this] | 𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
P : Type uP
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedAddCommGroup E'
inst✝¹¹ : NormedAddCommGroup F
f : G → E
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : NormedSpace 𝕜 E'
inst✝⁷ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F... | a579c7593509cfe8 |
MeasurableEmbedding.rnDeriv_map | Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean | lemma _root_.MeasurableEmbedding.rnDeriv_map (hf : MeasurableEmbedding f)
(μ ν : Measure α) [SigmaFinite μ] [SigmaFinite ν] :
(fun x ↦ (μ.map f).rnDeriv (ν.map f) (f x)) =ᵐ[ν] μ.rnDeriv ν | case refine_1
α : Type u_1
β : Type u_2
m : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
this✝¹ : SigmaFinite (map f ν)
this✝ : SigmaFinite (map f (μ.singularPart ν))
this : SigmaFinite (map f (ν.withDensity (μ.rnDeriv ν)))
h_... | refine Measure.rnDeriv_eq_zero_of_mutuallySingular ?_ Measure.AbsolutelyContinuous.rfl | case refine_1
α : Type u_1
β : Type u_2
m : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
hf : MeasurableEmbedding f
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
this✝¹ : SigmaFinite (map f ν)
this✝ : SigmaFinite (map f (μ.singularPart ν))
this : SigmaFinite (map f (ν.withDensity (μ.rnDeriv ν)))
h_... | ea80b5c7d85148f7 |
MvPolynomial.totalDegree_add_eq_left_of_totalDegree_lt | Mathlib/Algebra/MvPolynomial/Degrees.lean | theorem totalDegree_add_eq_left_of_totalDegree_lt {p q : MvPolynomial σ R}
(h : q.totalDegree < p.totalDegree) : (p + q).totalDegree = p.totalDegree | R : Type u
σ : Type u_1
inst✝ : CommSemiring R
p q : MvPolynomial σ R
h : q.totalDegree < p.totalDegree
hp : ¬p = 0
b : σ →₀ ℕ
hb₁ : b ∈ p.support
hb₂ : (p.support.sup fun m => (toMultiset m).card) = (toMultiset b).card
⊢ b ∉ q.support | contrapose! h | R : Type u
σ : Type u_1
inst✝ : CommSemiring R
p q : MvPolynomial σ R
hp : ¬p = 0
b : σ →₀ ℕ
hb₁ : b ∈ p.support
hb₂ : (p.support.sup fun m => (toMultiset m).card) = (toMultiset b).card
h : b ∈ q.support
⊢ p.totalDegree ≤ q.totalDegree | 554f88c9f1d39d60 |
Set.Definable.image_comp_embedding | Mathlib/ModelTheory/Definability.lean | theorem Definable.image_comp_embedding {s : Set (β → M)} (h : A.Definable L s) (f : α ↪ β)
[Finite β] : A.Definable L ((fun g : β → M => g ∘ f) '' s) | case intro
M : Type w
A : Set M
L : Language
inst✝¹ : L.Structure M
α : Type u₁
β : Type u_1
s : Set (β → M)
h : A.Definable L s
f : α ↪ β
inst✝ : Finite β
val✝ : Fintype β
x : α → M
⊢ x ∈
(fun g => g ∘ Sum.inl) ''
((fun g => g ∘ ⇑((Equiv.ofInjective ⇑f ⋯).sumCongr (Fintype.equivFin ↑(range ⇑f)ᶜ).symm)) '... | simp only [mem_preimage, mem_image, exists_exists_and_eq_and] | case intro
M : Type w
A : Set M
L : Language
inst✝¹ : L.Structure M
α : Type u₁
β : Type u_1
s : Set (β → M)
h : A.Definable L s
f : α ↪ β
inst✝ : Finite β
val✝ : Fintype β
x : α → M
⊢ (∃ a ∈ s,
((a ∘ ⇑(Equiv.Set.sumCompl (range ⇑f))) ∘
⇑((Equiv.ofInjective ⇑f ⋯).sumCongr (Fintype.equivFin ↑(range ⇑f)... | 7b3adb8c33237ade |
CategoryTheory.Arrow.functor_ext | Mathlib/CategoryTheory/Comma/Arrow.lean | /-- Extensionality lemma for functors `C ⥤ D` which uses as an assumption
that the induced maps `Arrow C → Arrow D` coincide. -/
lemma Arrow.functor_ext {F G : C ⥤ D} (h : ∀ ⦃X Y : C⦄ (f : X ⟶ Y),
F.mapArrow.obj (Arrow.mk f) = G.mapArrow.obj (Arrow.mk f)) :
F = G :=
Functor.ext (fun X ↦ congr_arg Comma.left (... | C : Type u_1
D : Type u_2
inst✝¹ : Category.{u_3, u_1} C
inst✝ : Category.{u_4, u_2} D
F G : C ⥤ D
h : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), F.mapArrow.obj (mk f) = G.mapArrow.obj (mk f)
X Y : C
f : X ⟶ Y
this : ∃ hX hY, F.map (mk f).hom = eqToHom hX ≫ G.map (mk f).hom ≫ eqToHom ⋯
⊢ F.map f = eqToHom ⋯ ≫ G.map f ≫ eqToHom ⋯ | tauto | no goals | 8fbf458ec71e16c4 |
Polynomial.Monic.sub_of_right | Mathlib/Algebra/Polynomial/Monic.lean | theorem Monic.sub_of_right {p q : R[X]} (hq : q.leadingCoeff = -1) (hpq : degree p < degree q) :
Monic (p - q) | R : Type u
inst✝ : Ring R
p q : R[X]
hq : q.leadingCoeff = -1
hpq : p.degree < q.degree
this : (-q).coeff (-q).natDegree = 1
⊢ (p - q).Monic | rw [sub_eq_add_neg] | R : Type u
inst✝ : Ring R
p q : R[X]
hq : q.leadingCoeff = -1
hpq : p.degree < q.degree
this : (-q).coeff (-q).natDegree = 1
⊢ (p + -q).Monic | 0f87bd252fa796db |
Matrix.stdBasisMatrix_eq_of_single_single | Mathlib/Data/Matrix/Basis.lean | theorem stdBasisMatrix_eq_of_single_single (i : m) (j : n) (a : α) :
stdBasisMatrix i j a = Matrix.of (Pi.single i (Pi.single j a)) | m : Type u_2
n : Type u_3
α : Type u_5
inst✝² : DecidableEq m
inst✝¹ : DecidableEq n
inst✝ : Zero α
i : m
j : n
a : α
⊢ stdBasisMatrix i j a = of (Pi.single i (Pi.single j a)) | ext a b | case a
m : Type u_2
n : Type u_3
α : Type u_5
inst✝² : DecidableEq m
inst✝¹ : DecidableEq n
inst✝ : Zero α
i : m
j : n
a✝ : α
a : m
b : n
⊢ stdBasisMatrix i j a✝ a b = of (Pi.single i (Pi.single j a✝)) a b | 4ae8f56c1d529f84 |
ProbabilityTheory.Kernel.IndepSet.measure_inter_eq_mul | Mathlib/Probability/Independence/Kernel.lean | theorem IndepSet.measure_inter_eq_mul {_m0 : MeasurableSpace Ω} (κ : Kernel α Ω) (μ : Measure α)
(h : IndepSet s t κ μ) : ∀ᵐ a ∂μ, κ a (s ∩ t) = κ a s * κ a t :=
Indep.indepSets h _ _ (by simp) (by simp)
| α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
s t : Set Ω
_m0 : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
h : IndepSet s t κ μ
⊢ s ∈ {s} | simp | no goals | 04d47f0b74fc340f |
AlgebraicGeometry.SurjectiveOnStalks.isEmbedding_pullback | Mathlib/AlgebraicGeometry/Morphisms/SurjectiveOnStalks.lean | /-- If `Y ⟶ S` is surjective on stalks, then for every `X ⟶ S`, `X ×ₛ Y` is a subset of
`X × Y` (cartesian product as topological spaces) with the induced topology. -/
lemma isEmbedding_pullback {X Y S : Scheme.{u}} (f : X ⟶ S) (g : Y ⟶ S) [SurjectiveOnStalks g] :
IsEmbedding (fun x ↦ ((pullback.fst f g).base x, (p... | case refine_2.intro.intro.intro.intro.intro.intro.refine_3
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
inst✝ : SurjectiveOnStalks g
L : ↑↑(pullback f g).toPresheafedSpace → ↑↑X.toPresheafedSpace × ↑↑Y.toPresheafedSpace :=
fun x => ((ConcreteCategory.hom (pullback.fst f g).base) x, (ConcreteCategory.hom (pullback.snd f g).base... | congr 5 | case refine_2.intro.intro.intro.intro.intro.intro.refine_3.e_a.e_a.e_self.e_self.e_self
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
inst✝ : SurjectiveOnStalks g
L : ↑↑(pullback f g).toPresheafedSpace → ↑↑X.toPresheafedSpace × ↑↑Y.toPresheafedSpace :=
fun x => ((ConcreteCategory.hom (pullback.fst f g).base) x, (ConcreteCategor... | 9509c1734d72b9a6 |
RingHom.finiteType_holdsForLocalizationAway | Mathlib/RingTheory/RingHom/FiniteType.lean | theorem finiteType_holdsForLocalizationAway : HoldsForLocalizationAway @FiniteType | case h.e'_5.h
R S : Type u_1
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
r : R
inst✝ : IsLocalization.Away r S
this : Algebra.FiniteType R S
r✝ : R
x✝ : S
⊢ (let_fun I := (algebraMap R S).toAlgebra;
r✝ • x✝) =
r✝ • x✝ | rw [Algebra.smul_def] | case h.e'_5.h
R S : Type u_1
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
r : R
inst✝ : IsLocalization.Away r S
this : Algebra.FiniteType R S
r✝ : R
x✝ : S
⊢ (let_fun I := (algebraMap R S).toAlgebra;
r✝ • x✝) =
(algebraMap R S) r✝ * x✝ | 97309f76efa63e45 |
Int.le_ceil_iff | Mathlib/Algebra/Order/Floor.lean | lemma le_ceil_iff : z ≤ ⌈a⌉ ↔ z - 1 < a | α : Type u_2
inst✝¹ : LinearOrderedRing α
inst✝ : FloorRing α
z : ℤ
a : α
⊢ z ≤ ⌈a⌉ ↔ ↑z - 1 < a | rw [← sub_one_lt_iff, lt_ceil] | α : Type u_2
inst✝¹ : LinearOrderedRing α
inst✝ : FloorRing α
z : ℤ
a : α
⊢ ↑(z - 1) < a ↔ ↑z - 1 < a | a0620efd241d7464 |
MeasureTheory.aemeasurable_withDensity_ennreal_iff' | Mathlib/MeasureTheory/Measure/WithDensity.lean | theorem aemeasurable_withDensity_ennreal_iff' {f : α → ℝ≥0}
(hf : AEMeasurable f μ) {g : α → ℝ≥0∞} :
AEMeasurable g (μ.withDensity fun x => (f x : ℝ≥0∞)) ↔
AEMeasurable (fun x => (f x : ℝ≥0∞) * g x) μ | case h
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0
hf : AEMeasurable f μ
g : α → ℝ≥0∞
f' : α → ℝ≥0
hf'_m : Measurable f'
hf'_ae : f =ᶠ[ae μ] f'
g' : α → ℝ≥0∞
g'meas : Measurable g'
hg' : ∀ᵐ (x : α) ∂μ, ↑(f x) ≠ 0 → g x = g' x
A : MeasurableSet {x | f' x ≠ 0}
a : α
ha : ↑(f a) ≠ 0 → g a = g' a
h'a : f ... | have : (f' a : ℝ≥0∞) ≠ 0 := by simpa only [Ne, ENNReal.coe_eq_zero] using h_a_nonneg | case h
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0
hf : AEMeasurable f μ
g : α → ℝ≥0∞
f' : α → ℝ≥0
hf'_m : Measurable f'
hf'_ae : f =ᶠ[ae μ] f'
g' : α → ℝ≥0∞
g'meas : Measurable g'
hg' : ∀ᵐ (x : α) ∂μ, ↑(f x) ≠ 0 → g x = g' x
A : MeasurableSet {x | f' x ≠ 0}
a : α
ha : ↑(f a) ≠ 0 → g a = g' a
h'a : f ... | 8b1f28ad3ee0d17a |
bernsteinPolynomial.iterate_derivative_at_0 | Mathlib/RingTheory/Polynomial/Bernstein.lean | theorem iterate_derivative_at_0 (n ν : ℕ) :
(Polynomial.derivative^[ν] (bernsteinPolynomial R n ν)).eval 0 =
(ascPochhammer R ν).eval ((n - (ν - 1) : ℕ) : R) | case pos.succ.inr
R : Type u_1
inst✝ : CommRing R
ν : ℕ
ih :
∀ (n : ℕ), ν ≤ n → eval 0 ((⇑derivative)^[ν] (bernsteinPolynomial R n ν)) = eval (↑(n - (ν - 1))) (ascPochhammer R ν)
n : ℕ
h : ν + 1 ≤ n
h' : ν ≤ n - 1
h'' : ν > 0
⊢ ↑n * eval (↑(n - 1 - (ν - 1))) (ascPochhammer R ν) = ↑(n - ν) * eval (↑(n - ν) + 1) (ascPo... | have : n - 1 - (ν - 1) = n - ν := by omega | case pos.succ.inr
R : Type u_1
inst✝ : CommRing R
ν : ℕ
ih :
∀ (n : ℕ), ν ≤ n → eval 0 ((⇑derivative)^[ν] (bernsteinPolynomial R n ν)) = eval (↑(n - (ν - 1))) (ascPochhammer R ν)
n : ℕ
h : ν + 1 ≤ n
h' : ν ≤ n - 1
h'' : ν > 0
this : n - 1 - (ν - 1) = n - ν
⊢ ↑n * eval (↑(n - 1 - (ν - 1))) (ascPochhammer R ν) = ↑(n - ... | 2869d2a115228592 |
Ordinal.principal_mul_iff_mul_left_eq | Mathlib/SetTheory/Ordinal/Principal.lean | theorem principal_mul_iff_mul_left_eq : Principal (· * ·) o ↔ ∀ a, 0 < a → a < o → a * o = o | case h.e'_2.h.e'_5.a
o : Ordinal.{u}
h : Principal (fun x1 x2 => x1 * x2) o
a : Ordinal.{u}
ha₀ : 0 < a
hao : a < o
ho : o ≤ 2
⊢ a < 2 | exact hao.trans_le ho | no goals | 5ae376ccba37fa39 |
MeasurableSet.iUnion_of_monotone_of_frequently | Mathlib/MeasureTheory/MeasurableSpace/Basic.lean | theorem iUnion_of_monotone_of_frequently
{ι : Type*} [Preorder ι] [(atTop : Filter ι).IsCountablyGenerated] {s : ι → Set α}
(hsm : Monotone s) (hs : ∃ᶠ i in atTop, MeasurableSet (s i)) : MeasurableSet (⋃ i, s i) | case intro.intro
α : Type u_1
inst✝² : MeasurableSpace α
ι : Type u_6
inst✝¹ : Preorder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set α
hsm : Monotone s
hs : ∃ᶠ (i : ι) in atTop, MeasurableSet (s i)
x : ℕ → ι
hx : Tendsto x atTop atTop
hxm : ∀ (n : ℕ), MeasurableSet (s (x n))
⊢ MeasurableSet (⋃ i, s i) | rw [← hsm.iUnion_comp_tendsto_atTop hx] | case intro.intro
α : Type u_1
inst✝² : MeasurableSpace α
ι : Type u_6
inst✝¹ : Preorder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set α
hsm : Monotone s
hs : ∃ᶠ (i : ι) in atTop, MeasurableSet (s i)
x : ℕ → ι
hx : Tendsto x atTop atTop
hxm : ∀ (n : ℕ), MeasurableSet (s (x n))
⊢ MeasurableSet (⋃ a, s (x a)) | eaf9c9d5846defea |
AdicCompletion.ofTensorProduct_iso | Mathlib/RingTheory/AdicCompletion/AsTensorProduct.lean | private lemma ofTensorProduct_iso [Fintype ι] [IsNoetherianRing R] :
IsIso (ModuleCat.ofHom (ofTensorProduct I M)) | case refine_4
R : Type u
inst✝⁴ : CommRing R
I : Ideal R
M : Type u
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type
f : (ι → R) →ₗ[R] M
hf : Function.Surjective ⇑f
inst✝¹ : Fintype ι
inst✝ : IsNoetherianRing R
⊢ Mono (ComposableArrows.app' (AdicCompletion.firstRowToSecondRow I M f) 4 ⋯) | apply ConcreteCategory.mono_of_injective | case refine_4.i
R : Type u
inst✝⁴ : CommRing R
I : Ideal R
M : Type u
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type
f : (ι → R) →ₗ[R] M
hf : Function.Surjective ⇑f
inst✝¹ : Fintype ι
inst✝ : IsNoetherianRing R
⊢ Function.Injective ⇑(ConcreteCategory.hom (ComposableArrows.app' (AdicCompletion.firstRowToSecondRow ... | 76224af4d5a35f57 |
PreTilt.valAux_eq | Mathlib/RingTheory/Perfection.lean | theorem valAux_eq {f : PreTilt O p} {n : ℕ} (hfn : coeff _ _ n f ≠ 0) :
valAux K v O p f = ModP.preVal K v O p (coeff _ _ n f) ^ p ^ n | case intro.succ.intro
K : Type u₁
inst✝⁴ : Field K
v : Valuation K ℝ≥0
O : Type u₂
inst✝³ : CommRing O
inst✝² : Algebra O K
hv : v.Integers O
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fact ¬IsUnit ↑p
f : PreTilt O p
h : ∃ n, (coeff (ModP O p) p n) f ≠ 0
k : ℕ
ih :
(coeff (ModP O p) p (Nat.find h + k)) f ≠ 0 →
Mod... | have h2 : (Ideal.Quotient.mk _ (x ^ p) : ModP O p) ≠ 0 := by
erw [RingHom.map_pow, hx, ← RingHom.map_pow, coeff_pow_p]
exact coeff_nat_find_add_ne_zero k | case intro.succ.intro
K : Type u₁
inst✝⁴ : Field K
v : Valuation K ℝ≥0
O : Type u₂
inst✝³ : CommRing O
inst✝² : Algebra O K
hv : v.Integers O
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fact ¬IsUnit ↑p
f : PreTilt O p
h : ∃ n, (coeff (ModP O p) p n) f ≠ 0
k : ℕ
ih :
(coeff (ModP O p) p (Nat.find h + k)) f ≠ 0 →
Mod... | 821b0935f5925da0 |
TrivSqZeroExt.norm_inl | Mathlib/Analysis/Normed/Algebra/TrivSqZeroExt.lean | theorem norm_inl (r : R) : ‖(inl r : tsze R M)‖ = ‖r‖ | R : Type u_3
M : Type u_4
inst✝¹ : SeminormedRing R
inst✝ : SeminormedAddCommGroup M
r : R
⊢ ‖inl r‖ = ‖r‖ | simp [norm_def] | no goals | 47694de979f8454e |
PrimeSpectrum.isCompact_basicOpen | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | theorem isCompact_basicOpen (f : R) : IsCompact (basicOpen f : Set (PrimeSpectrum R)) | R : Type u
inst✝ : CommSemiring R
f : R
⊢ IsCompact ↑(basicOpen f) | rw [← localization_away_comap_range (Localization (Submonoid.powers f))] | R : Type u
inst✝ : CommSemiring R
f : R
⊢ IsCompact (Set.range ⇑(comap (algebraMap R (Localization (Submonoid.powers f))))) | 5fb4b47b3c5996fe |
UpperSet.mem_iInf_iff | Mathlib/Order/UpperLower/Basic.lean | theorem mem_iInf_iff {f : ι → UpperSet α} : (a ∈ ⨅ i, f i) ↔ ∃ i, a ∈ f i | α : Type u_1
ι : Sort u_4
inst✝ : LE α
a : α
f : ι → UpperSet α
⊢ a ∈ ⨅ i, f i ↔ ∃ i, a ∈ f i | rw [← SetLike.mem_coe, coe_iInf] | α : Type u_1
ι : Sort u_4
inst✝ : LE α
a : α
f : ι → UpperSet α
⊢ a ∈ ⋃ i, ↑(f i) ↔ ∃ i, a ∈ f i | 6c69d7bdb2398f7e |
AffineIndependent.convexHull_inter | Mathlib/Analysis/Convex/Combination.lean | /-- Two simplices glue nicely if the union of their vertices is affine independent. -/
lemma AffineIndependent.convexHull_inter (hs : AffineIndependent R ((↑) : s → E))
(ht₁ : t₁ ⊆ s) (ht₂ : t₂ ⊆ s) :
convexHull R (t₁ ∩ t₂ : Set E) = convexHull R t₁ ∩ convexHull R t₂ | R : Type u_1
E : Type u_3
inst✝² : LinearOrderedField R
inst✝¹ : AddCommGroup E
inst✝ : Module R E
s t₁ t₂ : Finset E
hs : AffineIndependent R Subtype.val
ht₁ : t₁ ⊆ s
ht₂ : t₂ ⊆ s
x : E
w₁ : E → R
h₁w₁ : ∀ y ∈ t₁, 0 ≤ w₁ y
h₂w₁ : ∑ y ∈ t₁, w₁ y = 1
h₃w₁ : ∑ y ∈ t₁, w₁ y • y = x
w₂ : E → R
h₂w₂ : ∑ y ∈ t₂, w₂ y = 1
h₃w... | simp only [w, sub_smul, zero_smul, ite_smul, Finset.sum_sub_distrib, ← Finset.sum_filter, h₃w₁,
Finset.filter_mem_eq_inter, Finset.inter_eq_right.2 ht₁, Finset.inter_eq_right.2 ht₂, h₃w₂,
sub_self] | no goals | c5414b8b46240fda |
ascending_central_series_le_upper | Mathlib/GroupTheory/Nilpotent.lean | theorem ascending_central_series_le_upper (H : ℕ → Subgroup G) (hH : IsAscendingCentralSeries H) :
∀ n : ℕ, H n ≤ upperCentralSeries G n
| 0 => hH.1.symm ▸ le_refl ⊥
| n + 1 => by
intro x hx
rw [mem_upperCentralSeries_succ_iff]
exact fun y => ascending_central_series_le_upper H hH n (hH.2 x n hx y)
| G : Type u_1
inst✝ : Group G
H : ℕ → Subgroup G
hH : IsAscendingCentralSeries H
n : ℕ
x : G
hx : x ∈ H (n + 1)
⊢ x ∈ upperCentralSeries G (n + 1) | rw [mem_upperCentralSeries_succ_iff] | G : Type u_1
inst✝ : Group G
H : ℕ → Subgroup G
hH : IsAscendingCentralSeries H
n : ℕ
x : G
hx : x ∈ H (n + 1)
⊢ ∀ (y : G), x * y * x⁻¹ * y⁻¹ ∈ upperCentralSeries G n | 67368f32a22ec98a |
integral_mul_rpow_one_add_sq | Mathlib/Analysis/SpecialFunctions/Integrals.lean | theorem integral_mul_rpow_one_add_sq {t : ℝ} (ht : t ≠ -1) :
(∫ x : ℝ in a..b, x * (↑1 + x ^ 2) ^ t) =
(↑1 + b ^ 2) ^ (t + 1) / (↑2 * (t + ↑1)) - (↑1 + a ^ 2) ^ (t + 1) / (↑2 * (t + ↑1)) | case h.e'_2
a b t : ℝ
ht : t ≠ -1
this : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (1 + ↑x ^ 2) ^ ↑s
⊢ ∫ (x : ℝ) in a..b, ↑(x * (1 + x ^ 2) ^ t) = ∫ (x : ℝ) in ?convert_1..?convert_2, ↑x * (1 + ↑x ^ 2) ^ ↑t | congr with x : 1 | case h.e'_2.e_f.h
a b t : ℝ
ht : t ≠ -1
this : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (1 + ↑x ^ 2) ^ ↑s
x : ℝ
⊢ ↑(x * (1 + x ^ 2) ^ t) = ↑x * (1 + ↑x ^ 2) ^ ↑t | 06efdbf5b78963b9 |
Real.tan_zero | Mathlib/Data/Complex/Trigonometric.lean | theorem tan_zero : tan 0 = 0 | ⊢ tan 0 = 0 | simp [tan] | no goals | 6b0d08623a116e2f |
Real.pi_lt_sqrtTwoAddSeries | Mathlib/Data/Real/Pi/Bounds.lean | theorem pi_lt_sqrtTwoAddSeries (n : ℕ) :
π < 2 ^ (n + 1) * √(2 - sqrtTwoAddSeries 0 n) + 1 / 4 ^ n | n : ℕ
⊢ 0 ≤ 2 ^ (n + 2) | positivity | no goals | 02d82b689d50a592 |
Pell.Solution₁.eq_zero_of_d_neg | Mathlib/NumberTheory/Pell.lean | theorem eq_zero_of_d_neg (h₀ : d < 0) (a : Solution₁ d) : a.x = 0 ∨ a.y = 0 | d : ℤ
h₀ : d < 0
a : Solution₁ d
h : a.x ≠ 0 ∧ a.y ≠ 0
h1 : 0 < a.x ^ 2
⊢ a.x ^ 2 - d * a.y ^ 2 ≠ 1 | have h2 := sq_pos_of_ne_zero h.2 | d : ℤ
h₀ : d < 0
a : Solution₁ d
h : a.x ≠ 0 ∧ a.y ≠ 0
h1 : 0 < a.x ^ 2
h2 : 0 < a.y ^ 2
⊢ a.x ^ 2 - d * a.y ^ 2 ≠ 1 | 50a8ed2852157cdb |
ComplexShape.not_mem_range_embeddingUpIntGE_iff | Mathlib/Algebra/Homology/Embedding/Basic.lean | lemma not_mem_range_embeddingUpIntGE_iff (n : ℤ) :
(∀ (i : ℕ), (embeddingUpIntGE p).f i ≠ n) ↔ n < p | p n : ℤ
⊢ (∀ (i : ℕ), (embeddingUpIntGE p).f i ≠ n) ↔ n < p | constructor | case mp
p n : ℤ
⊢ (∀ (i : ℕ), (embeddingUpIntGE p).f i ≠ n) → n < p
case mpr
p n : ℤ
⊢ n < p → ∀ (i : ℕ), (embeddingUpIntGE p).f i ≠ n | 7d00f879499fc2b7 |
Prod.map_surjective | Mathlib/Data/Prod/Basic.lean | theorem map_surjective [Nonempty γ] [Nonempty δ] {f : α → γ} {g : β → δ} :
Surjective (map f g) ↔ Surjective f ∧ Surjective g :=
⟨fun h =>
⟨fun c => by
inhabit δ
obtain ⟨⟨a, b⟩, h⟩ := h (c, default)
exact ⟨a, congr_arg Prod.fst h⟩,
fun d => by
inhabit γ
obtain ⟨⟨a, b⟩, h⟩ :... | α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝¹ : Nonempty γ
inst✝ : Nonempty δ
f : α → γ
g : β → δ
h : Surjective (map f g)
d : δ
⊢ ∃ a, g a = d | inhabit γ | α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝¹ : Nonempty γ
inst✝ : Nonempty δ
f : α → γ
g : β → δ
h : Surjective (map f g)
d : δ
inhabited_h : Inhabited γ
⊢ ∃ a, g a = d | 7bc55de06597f56e |
Surreal.Multiplication.P3.trans | Mathlib/SetTheory/Surreal/Multiplication.lean | lemma P3.trans (h₁ : P3 x₁ x₂ y₁ y₂) (h₂ : P3 x₂ x₃ y₁ y₂) : P3 x₁ x₃ y₁ y₂ | x₁ x₂ x₃ y₁ y₂ : PGame
h₁ : P3 x₁ x₂ y₁ y₂
h₂ : P3 x₂ x₃ y₁ y₂
⊢ P3 x₁ x₃ y₁ y₂ | rw [P3] at h₁ h₂ | x₁ x₂ x₃ y₁ y₂ : PGame
h₁ : ⟦x₁ * y₂⟧ + ⟦x₂ * y₁⟧ < ⟦x₁ * y₁⟧ + ⟦x₂ * y₂⟧
h₂ : ⟦x₂ * y₂⟧ + ⟦x₃ * y₁⟧ < ⟦x₂ * y₁⟧ + ⟦x₃ * y₂⟧
⊢ P3 x₁ x₃ y₁ y₂ | 86ed7f5d4e0f46c8 |
Finset.sq_sum_div_le_sum_sq_div | Mathlib/Algebra/Order/BigOperators/Ring/Finset.lean | theorem sq_sum_div_le_sum_sq_div [LinearOrderedSemifield R] [ExistsAddOfLE R] (s : Finset ι)
(f : ι → R) {g : ι → R} (hg : ∀ i ∈ s, 0 < g i) :
(∑ i ∈ s, f i) ^ 2 / ∑ i ∈ s, g i ≤ ∑ i ∈ s, f i ^ 2 / g i | ι : Type u_1
R : Type u_2
inst✝¹ : LinearOrderedSemifield R
inst✝ : ExistsAddOfLE R
s : Finset ι
f g : ι → R
hg : ∀ i ∈ s, 0 < g i
hg' : ∀ i ∈ s, 0 ≤ g i
⊢ (∑ i ∈ s, f i) ^ 2 / ∑ i ∈ s, g i ≤ ∑ i ∈ s, f i ^ 2 / g i | have H : ∀ i ∈ s, 0 ≤ f i ^ 2 / g i := fun i hi ↦ div_nonneg (sq_nonneg _) (hg' i hi) | ι : Type u_1
R : Type u_2
inst✝¹ : LinearOrderedSemifield R
inst✝ : ExistsAddOfLE R
s : Finset ι
f g : ι → R
hg : ∀ i ∈ s, 0 < g i
hg' : ∀ i ∈ s, 0 ≤ g i
H : ∀ i ∈ s, 0 ≤ f i ^ 2 / g i
⊢ (∑ i ∈ s, f i) ^ 2 / ∑ i ∈ s, g i ≤ ∑ i ∈ s, f i ^ 2 / g i | b364a97d5e265cff |
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