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 |
|---|---|---|---|---|---|---|
PresheafOfModules.surjective_of_epi | Mathlib/Algebra/Category/ModuleCat/Presheaf/EpiMono.lean | lemma surjective_of_epi [Epi f] (X : Cᵒᵖ) :
Function.Surjective (f.app X) | C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
R : Cᵒᵖ ⥤ RingCat
M₁ M₂ : PresheafOfModules R
f : M₁ ⟶ M₂
inst✝ : Epi f
X : Cᵒᵖ
⊢ Function.Surjective ⇑(ConcreteCategory.hom (f.app X)) | rw [← ModuleCat.epi_iff_surjective] | C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
R : Cᵒᵖ ⥤ RingCat
M₁ M₂ : PresheafOfModules R
f : M₁ ⟶ M₂
inst✝ : Epi f
X : Cᵒᵖ
⊢ Epi (f.app X) | 85bb95b9b4768201 |
isNonarchimedean_smoothingFun | Mathlib/Analysis/Normed/Ring/SmoothingSeminorm.lean | theorem isNonarchimedean_smoothingFun (hμ1 : μ 1 ≤ 1) (hna : IsNonarchimedean μ) :
IsNonarchimedean (smoothingFun μ) | R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
hna : IsNonarchimedean ⇑μ
x y : R
hn : ∀ (n : ℕ), ∃ m < n + 1, μ ((x + y) ^ n) ^ (1 / ↑n) ≤ (μ (x ^ m) * μ (y ^ (n - m))) ^ (1 / ↑n)
mu : ℕ → ℕ := fun n => _root_.mu μ hn n
nu : ℕ → ℕ := fun n => n - mu n
hnu : nu = fun n => n - mu n
hmu_le : ∀ (n : ℕ), m... | apply le_trans hxy' (mul_le_mul hx hy (le_limsup_of_frequently_le (Frequently.of_forall
(fun n ↦ rpow_nonneg (apply_nonneg μ _) _)) h_bdd)
(rpow_nonneg (smoothingFun_nonneg μ hμ1 x) _)) | no goals | 003e5c0c3a2046fe |
exists_open_between_and_isCompact_closure | Mathlib/Topology/Separation/Regular.lean | theorem exists_open_between_and_isCompact_closure [LocallyCompactSpace X] [RegularSpace X]
{K U : Set X} (hK : IsCompact K) (hU : IsOpen U) (hKU : K ⊆ U) :
∃ V, IsOpen V ∧ K ⊆ V ∧ closure V ⊆ U ∧ IsCompact (closure V) | case intro.intro.intro.intro
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : LocallyCompactSpace X
inst✝ : RegularSpace X
K U : Set X
hK : IsCompact K
hU : IsOpen U
hKU : K ⊆ U
L : Set X
L_compact : IsCompact L
L_closed : IsClosed L
KL : K ⊆ interior L
LU : L ⊆ U
⊢ ∃ V, IsOpen V ∧ K ⊆ V ∧ closure V ⊆ U ∧ IsCompact (c... | have A : closure (interior L) ⊆ L := by
apply (closure_mono interior_subset).trans (le_of_eq L_closed.closure_eq) | case intro.intro.intro.intro
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : LocallyCompactSpace X
inst✝ : RegularSpace X
K U : Set X
hK : IsCompact K
hU : IsOpen U
hKU : K ⊆ U
L : Set X
L_compact : IsCompact L
L_closed : IsClosed L
KL : K ⊆ interior L
LU : L ⊆ U
A : closure (interior L) ⊆ L
⊢ ∃ V, IsOpen V ∧ K ⊆ V ∧... | 3aa20e5b298ad5e3 |
Submodule.mem_iSup_finset_iff_exists_sum | Mathlib/LinearAlgebra/DFinsupp.lean | theorem mem_iSup_finset_iff_exists_sum {s : Finset ι} (p : ι → Submodule R N) (a : N) :
(a ∈ ⨆ i ∈ s, p i) ↔ ∃ μ : ∀ i, p i, (∑ i ∈ s, (μ i : N)) = a | case mpr.intro.refine_2
ι : Type u_1
R : Type u_2
N : Type u_5
inst✝² : Semiring R
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
s : Finset ι
p : ι → Submodule R N
a : N
μ : (i : ι) → ↥(p i)
hμ : ∑ i ∈ s, ↑(μ i) = a
x : ι
a✝ : x ∈ s
hx : x ∉ (DFinsupp.mk s fun i => ⟨↑(μ ↑i), ⋯⟩).support
⊢ ↑((DFinsupp.mk s fun i => ⟨↑(μ ↑... | rw [mem_support_iff, not_ne_iff] at hx | case mpr.intro.refine_2
ι : Type u_1
R : Type u_2
N : Type u_5
inst✝² : Semiring R
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
s : Finset ι
p : ι → Submodule R N
a : N
μ : (i : ι) → ↥(p i)
hμ : ∑ i ∈ s, ↑(μ i) = a
x : ι
a✝ : x ∈ s
hx : (DFinsupp.mk s fun i => ⟨↑(μ ↑i), ⋯⟩) x = 0
⊢ ↑((DFinsupp.mk s fun i => ⟨↑(μ ↑i), ⋯⟩... | 0146bdbac3178433 |
CategoryTheory.Functor.op_commShiftIso_inv_app | Mathlib/CategoryTheory/Triangulated/Opposite/Functor.lean | @[reassoc]
lemma op_commShiftIso_inv_app (X : Cᵒᵖ) (n m : ℤ) (h : n + m = 0):
(F.op.commShiftIso n).inv.app X =
(shiftFunctorOpIso D n m h).hom.app (op (F.obj X.unop)) ≫
((F.commShiftIso m).hom.app X.unop).op ≫
(F.map ((shiftFunctorOpIso C n m h).inv.app X).unop).op | C : Type u_1
D : Type u_2
inst✝⁴ : Category.{u_4, u_1} C
inst✝³ : Category.{u_3, u_2} D
inst✝² : HasShift C ℤ
inst✝¹ : HasShift D ℤ
F : C ⥤ D
inst✝ : F.CommShift ℤ
X : Cᵒᵖ
n m : ℤ
h : n + m = 0
⊢ 𝟙 ((shiftFunctor Cᵒᵖ n ⋙ F.op).obj X) =
(F.map ((shiftFunctorOpIso C n m h).hom.app X).unop).op ≫
((F.commShiftIs... | simp [← op_comp, ← F.map_comp] | no goals | 645552664dc076c9 |
Cardinal.mk_subset_mk_lt_cof | Mathlib/SetTheory/Cardinal/Cofinality.lean | theorem mk_subset_mk_lt_cof {α : Type*} (h : ∀ x < #α, 2 ^ x < #α) :
#{ s : Set α // #s < cof (#α).ord } = #α | case inr.intro.intro.a
α : Type u_1
h : ∀ x < #α, 2 ^ x < #α
ha : #α ≠ 0
h' : (#α).IsStrongLimit
r : α → α → Prop
wo : IsWellOrder α r
hr : (#α).ord = type r
this : IsWellOrder α r
⊢ #{ s // #↑s < (#α).ord.cof } ≤ #{ s // Bounded r s } | apply mk_le_mk_of_subset | case inr.intro.intro.a.h
α : Type u_1
h : ∀ x < #α, 2 ^ x < #α
ha : #α ≠ 0
h' : (#α).IsStrongLimit
r : α → α → Prop
wo : IsWellOrder α r
hr : (#α).ord = type r
this : IsWellOrder α r
⊢ (fun x =>
Quotient.liftOn₂ (#↑x) (#α).ord.cof (fun α β => Nonempty (α ↪ β)) instLE.proof_1 ∧
¬Quotient.liftOn₂ (#α).ord.c... | 6c44834aaa390522 |
Std.Sat.CNF.Clause.not_mem_nil | Mathlib/.lake/packages/lean4/src/lean/Std/Sat/CNF/Basic.lean | theorem not_mem_nil {v : α} : ¬Mem v ([] : Clause α) | α : Type u_1
v : α
⊢ ¬Mem v [] | simp [Mem] | no goals | c788e15905b5018d |
AlgebraicGeometry.coprodMk_inl | Mathlib/AlgebraicGeometry/Limits.lean | @[simp]
lemma coprodMk_inl (x : X) :
coprodMk X Y (.inl x) = (coprod.inl : X ⟶ X ⨿ Y).base x | case e_a.e_a
X Y : Scheme
x : ↑↑X.toPresheafedSpace
⊢ colimit.ι (pair ↑X.toPresheafedSpace ↑Y.toPresheafedSpace) { as := WalkingPair.left } ≫
(PreservesColimitPair.iso Scheme.forgetToTop X Y).hom =
Scheme.forgetToTop.map coprod.inl | exact coprodComparison_inl Scheme.forgetToTop | no goals | d6974cd9a51dc16b |
CategoryTheory.Limits.prod_rightUnitor_inv_naturality | Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean | theorem prod_rightUnitor_inv_naturality [HasBinaryProducts C] (f : X ⟶ Y) :
(prod.rightUnitor X).inv ≫ prod.map f (𝟙 _) = f ≫ (prod.rightUnitor Y).inv | C : Type u
inst✝² : Category.{v, u} C
X Y : C
inst✝¹ : HasTerminal C
inst✝ : HasBinaryProducts C
f : X ⟶ Y
⊢ (prod.rightUnitor X).inv ≫ prod.map f (𝟙 (⊤_ C)) = f ≫ (prod.rightUnitor Y).inv | rw [Iso.inv_comp_eq, ← Category.assoc, Iso.eq_comp_inv, prod.rightUnitor_hom_naturality] | no goals | acc5bf3fda4b9252 |
CategoryTheory.IsCofiltered.small_fullSubcategory_cofilteredClosure | Mathlib/CategoryTheory/Filtered/Small.lean | theorem small_fullSubcategory_cofilteredClosure :
Small.{max v w} (FullSubcategory (CofilteredClosure f)) | case mk.base
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsCofilteredOrEmpty C
α : Type w
f : α → C
j : C
x : α
⊢ ∃ a,
CategoryTheory.IsCofiltered.CofilteredClosureSmall.abstractCofilteredClosureRealization f a =
{ obj := f x, property := ⋯ }.obj | exact ⟨⟨0, ⟨x⟩⟩, rfl⟩ | no goals | b8c9f1d29d461ed7 |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean | theorem exists_subset_restrict_nonpos (hi : s i < 0) :
∃ j : Set α, MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ s j < 0 | α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬s ≤[i] 0
hn : ∀ (n : ℕ), ¬s ≤[i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
A : Set α := i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ l, MeasureTh... | convert atTop.tendsto_atTop_add_const_right (-1) h₃ | case h.e'_3.h
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬s ≤[i] 0
hn : ∀ (n : ℕ), ¬s ≤[i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
A : Set α := i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ... | 37ee21864bec50c9 |
Multiset.rel_add_right | Mathlib/Data/Multiset/AddSub.lean | theorem rel_add_right {as bs₀ bs₁} :
Rel r as (bs₀ + bs₁) ↔ ∃ as₀ as₁, Rel r as₀ bs₀ ∧ Rel r as₁ bs₁ ∧ as = as₀ + as₁ | α : Type u_1
β : Type v
r : α → β → Prop
as : Multiset α
bs₀ bs₁ : Multiset β
⊢ Rel r as (bs₀ + bs₁) ↔ ∃ as₀ as₁, Rel r as₀ bs₀ ∧ Rel r as₁ bs₁ ∧ as = as₀ + as₁ | rw [← rel_flip, rel_add_left] | α : Type u_1
β : Type v
r : α → β → Prop
as : Multiset α
bs₀ bs₁ : Multiset β
⊢ (∃ bs₀_1 bs₁_1, Rel (flip r) bs₀ bs₀_1 ∧ Rel (flip r) bs₁ bs₁_1 ∧ as = bs₀_1 + bs₁_1) ↔
∃ as₀ as₁, Rel r as₀ bs₀ ∧ Rel r as₁ bs₁ ∧ as = as₀ + as₁ | c17425a7acb7e301 |
CategoryTheory.Presheaf.isLocallyInjective_forget_iff | Mathlib/CategoryTheory/Sites/LocallyInjective.lean | lemma isLocallyInjective_forget_iff :
IsLocallyInjective J (whiskerRight φ (forget D)) ↔ IsLocallyInjective J φ | case mpr
C : Type u
inst✝³ : Category.{v, u} C
D : Type u'
inst✝² : Category.{v', u'} D
FD : D → D → Type u_1
CD : D → Type w
inst✝¹ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
inst✝ : ConcreteCategory D FD
J : GrothendieckTopology C
F₁ F₂ : Cᵒᵖ ⥤ D
φ : F₁ ⟶ F₂
a✝ : IsLocallyInjective J φ
⊢ IsLocallyInjective J (whisk... | infer_instance | no goals | 7dfb54e59afbafdb |
not_neZero | Mathlib/Algebra/NeZero.lean | theorem not_neZero {n : R} : ¬NeZero n ↔ n = 0 | R : Type u_1
inst✝ : Zero R
n : R
⊢ ¬NeZero n ↔ n = 0 | simp [neZero_iff] | no goals | 9ed88cd01485d3aa |
Finset.Nat.antidiagonal_succ' | Mathlib/Data/Finset/NatAntidiagonal.lean | theorem antidiagonal_succ' (n : ℕ) :
antidiagonal (n + 1) =
cons (n + 1, 0)
((antidiagonal n).map
(Embedding.prodMap (Embedding.refl _) ⟨Nat.succ, Nat.succ_injective⟩))
(by simp) | case a
n : ℕ
⊢ (antidiagonal (n + 1)).val =
(n + 1, 0) ::ₘ
Multiset.map (⇑((Embedding.refl ℕ).prodMap { toFun := Nat.succ, inj' := Nat.succ_injective }))
(antidiagonal n).val | exact Multiset.Nat.antidiagonal_succ' | no goals | 67edfd3e224a6fab |
SimpleGraph.induce_sUnion_connected_of_pairwise_not_disjoint | Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean | lemma induce_sUnion_connected_of_pairwise_not_disjoint {S : Set (Set V)} (Sn : S.Nonempty)
(Snd : ∀ {s t}, s ∈ S → t ∈ S → (s ∩ t).Nonempty)
(Sc : ∀ {s}, s ∈ S → (G.induce s).Connected) :
(G.induce (⋃₀ S)).Connected | case intro.intro.mk
V : Type u
G : SimpleGraph V
S : Set (Set V)
Snd : ∀ {s t : Set V}, s ∈ S → t ∈ S → (s ∩ t).Nonempty
Sc : ∀ {s : Set V}, s ∈ S → (induce s G).Connected
s : Set V
sS : s ∈ S
v : V
vs : v ∈ s
w : V
hw : ∃ t ∈ S, w ∈ t
⊢ ∃ s' ⊆ ⋃₀ S, ∃ (hu' : v ∈ s') (hv' : w ∈ s'), (induce s' G).Reachable ⟨v, hu'⟩ ⟨w,... | obtain ⟨t, tS, wt⟩ := hw | case intro.intro.mk.intro.intro
V : Type u
G : SimpleGraph V
S : Set (Set V)
Snd : ∀ {s t : Set V}, s ∈ S → t ∈ S → (s ∩ t).Nonempty
Sc : ∀ {s : Set V}, s ∈ S → (induce s G).Connected
s : Set V
sS : s ∈ S
v : V
vs : v ∈ s
w : V
t : Set V
tS : t ∈ S
wt : w ∈ t
⊢ ∃ s' ⊆ ⋃₀ S, ∃ (hu' : v ∈ s') (hv' : w ∈ s'), (induce s' G... | 1c868cecae3e30d1 |
LinearMap.range_dualMap_le_dualAnnihilator_ker | Mathlib/LinearAlgebra/Dual.lean | theorem range_dualMap_le_dualAnnihilator_ker :
LinearMap.range f.dualMap ≤ f.ker.dualAnnihilator | case intro
R : Type uR
inst✝⁴ : CommSemiring R
M₁ : Type uM₁
M₂ : Type uM₂
inst✝³ : AddCommMonoid M₁
inst✝² : Module R M₁
inst✝¹ : AddCommMonoid M₂
inst✝ : Module R M₂
f : M₁ →ₗ[R] M₂
ψ : Dual R M₂
x : M₁
hx : f x = 0
⊢ (f.dualMap ψ) x = 0 | rw [dualMap_apply, hx, map_zero] | no goals | 8fa65959090431db |
HasFPowerSeriesWithinAt.comp | Mathlib/Analysis/Analytic/Composition.lean | theorem HasFPowerSeriesWithinAt.comp {g : F → G} {f : E → F} {q : FormalMultilinearSeries 𝕜 F G}
{p : FormalMultilinearSeries 𝕜 E F} {x : E} {t : Set F} {s : Set E}
(hg : HasFPowerSeriesWithinAt g q t (f x)) (hf : HasFPowerSeriesWithinAt f p s x)
(hs : Set.MapsTo f s t) : HasFPowerSeriesWithinAt (g ∘ f) (... | case hy
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
g : F → G
f : E → F
q : FormalMultilinearSeries 𝕜 F G
p... | simpa [edist_eq_enorm_sub] using fy_mem.2 | no goals | 919c288392263270 |
Std.Sat.AIG.mkBEqCached_le_size | Mathlib/.lake/packages/lean4/src/lean/Std/Sat/AIG/CachedGatesLemmas.lean | theorem mkBEqCached_le_size (aig : AIG α) {input : BinaryInput aig} :
aig.decls.size ≤ (aig.mkBEqCached input).aig.decls.size | α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
input : aig.BinaryInput
⊢ aig.decls.size ≤ (aig.mkBEqCached input).aig.decls.size | simp only [mkBEqCached] | α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
input : aig.BinaryInput
⊢ aig.decls.size ≤
(((aig.mkGateCached (input.asGateInput false true)).aig.mkGateCached
((input.asGateInput true false).cast ⋯)).aig.mkGateCached
{ lhs := { ref := (aig.mkGateCached (input.asGateInpu... | e1438273e879dabe |
Module.Relations.toQuotient_relation | Mathlib/Algebra/Module/Presentation/Basic.lean | @[simp]
lemma toQuotient_relation (r : relations.R) :
relations.toQuotient (relations.relation r) = 0 | A : Type u
inst✝ : Ring A
relations : Relations A
r : relations.R
⊢ relations.relation r ∈ Set.range relations.relation | simp | no goals | b2c7e74b19353cb0 |
Zsqrtd.norm_zero | Mathlib/NumberTheory/Zsqrtd/Basic.lean | theorem norm_zero : norm (0 : ℤ√d) = 0 | d : ℤ
⊢ norm 0 = 0 | simp [norm] | no goals | fc5be22acd0f7508 |
Std.DHashMap.Internal.List.perm_insertList | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/List/Associative.lean | theorem perm_insertList [BEq α] [EquivBEq α]
{l toInsert : List ((a : α) × β a)}
(distinct_l : DistinctKeys l)
(distinct_toInsert : toInsert.Pairwise (fun a b => (a.1 == b.1) = false))
(distinct_both : ∀ (a : α), containsKey a l → (toInsert.map Sigma.fst).contains a = false) :
Perm (insertList l toI... | case nil
α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : EquivBEq α
l : List ((a : α) × β a)
distinct_l : DistinctKeys l
distinct_toInsert : DistinctKeys []
distinct_both : ∀ (a : α), containsKey a l = true → (List.map Sigma.fst []).contains a = false
⊢ (insertList l []).Perm (l ++ []) | simp only [insertList, List.append_nil, Perm.refl] | no goals | 92b666bd097e806e |
ContinuousLinearMap.sSup_unit_ball_eq_nnnorm | Mathlib/Analysis/NormedSpace/OperatorNorm/NNNorm.lean | theorem sSup_unit_ball_eq_nnnorm {𝕜 𝕜₂ E F : Type*} [NormedAddCommGroup E]
[SeminormedAddCommGroup F] [DenselyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] {σ₁₂ : 𝕜 →+* 𝕜₂}
[NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F) :
sSup ((fun x => ‖f x‖₊) '' ball 0 1) = ‖f‖₊ | case refine_1
𝕜 : Type u_11
𝕜₂ : Type u_12
E : Type u_13
F : Type u_14
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : SeminormedAddCommGroup F
inst✝⁴ : DenselyNormedField 𝕜
inst✝³ : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜₂ F
inst✝ : RingHomIsometric σ₁₂
f : E →SL[σ₁₂] F... | rintro - ⟨x, hx, rfl⟩ | case refine_1.intro.intro
𝕜 : Type u_11
𝕜₂ : Type u_12
E : Type u_13
F : Type u_14
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : SeminormedAddCommGroup F
inst✝⁴ : DenselyNormedField 𝕜
inst✝³ : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜₂ F
inst✝ : RingHomIsometric σ₁₂
f : ... | fc882bf3a0401bee |
Polynomial.eval₂_reflect_eq_zero_iff | Mathlib/Algebra/Polynomial/Reverse.lean | theorem eval₂_reflect_eq_zero_iff (i : R →+* S) (x : S) [Invertible x] (N : ℕ) (f : R[X])
(hf : f.natDegree ≤ N) : eval₂ i (⅟ x) (reflect N f) = 0 ↔ eval₂ i x f = 0 | case mpr
R : Type u_1
inst✝² : Semiring R
S : Type u_2
inst✝¹ : CommSemiring S
i : R →+* S
x : S
inst✝ : Invertible x
N : ℕ
f : R[X]
hf : f.natDegree ≤ N
h : eval₂ i (⅟x) (reflect N f) * x ^ N = 0
⊢ eval₂ i (⅟x) (reflect N f) = 0 | rw [← mul_one (eval₂ i (⅟ x) _), ← one_pow N, ← mul_invOf_self x, mul_pow, ← mul_assoc, h,
zero_mul] | no goals | 9fac0fc899d0f16f |
CFC.posPart_mul_negPart | Mathlib/Analysis/SpecialFunctions/ContinuousFunctionalCalculus/PosPart.lean | @[simp]
lemma posPart_mul_negPart (a : A) : a⁺ * a⁻ = 0 | case neg
A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : Module ℝ A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : IsScalarTower ℝ A A
inst✝² : StarRing A
inst✝¹ : TopologicalSpace A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ IsSelfAdjoint
a : A
ha : ¬IsSelfAdjoint a
⊢ cfcₙ (fun x => x⁺) a * cfcₙ (fun x => x⁻) a = 0 | simp [cfcₙ_apply_of_not_predicate a ha] | no goals | 649801fe2e219314 |
IsLocalRing.finrank_quotient_map | Mathlib/RingTheory/LocalRing/Quotient.lean | theorem finrank_quotient_map :
finrank (R ⧸ p) (S ⧸ pS) = finrank R S | R : Type u_1
S : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : IsLocalRing R
inst✝¹ : Module.Finite R S
inst✝ : Free R S
this : Module.Finite R (S ⧸ pS)
⊢ finrank (R ⧸ p) (S ⧸ pS) = finrank R S | have : Module.Finite (R ⧸ p) (S ⧸ pS) := Module.Finite.of_restrictScalars_finite R _ _ | R : Type u_1
S : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : IsLocalRing R
inst✝¹ : Module.Finite R S
inst✝ : Free R S
this✝ : Module.Finite R (S ⧸ pS)
this : Module.Finite (R ⧸ p) (S ⧸ pS)
⊢ finrank (R ⧸ p) (S ⧸ pS) = finrank R S | 1f22b61b5755b1e2 |
ENNReal.lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_ne_top | Mathlib/MeasureTheory/Integral/MeanInequalities.lean | theorem lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_ne_top {p q : ℝ} (hpq : p.IsConjExponent q)
{f g : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hf_nontop : (∫⁻ a, f a ^ p ∂μ) ≠ ⊤)
(hg_nontop : (∫⁻ a, g a ^ q ∂μ) ≠ ⊤) (hf_nonzero : (∫⁻ a, f a ^ p ∂μ) ≠ 0)
(hg_nonzero : (∫⁻ a, g a ^ q ∂μ) ≠ 0) :
(∫⁻ a, (f * g) a ∂... | α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p q : ℝ
hpq : p.IsConjExponent q
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hf_nontop : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤
hg_nontop : ∫⁻ (a : α), g a ^ q ∂μ ≠ ⊤
hf_nonzero : ∫⁻ (a : α), f a ^ p ∂μ ≠ 0
hg_nonzero : ∫⁻ (a : α), g a ^ q ∂μ ≠ 0
npf : ℝ≥0∞ := (∫⁻ (c : α), f c ^ p ∂μ) ... | calc
(∫⁻ a : α, (f * g) a ∂μ) =
∫⁻ a : α, (funMulInvSnorm f p μ * funMulInvSnorm g q μ) a * (npf * nqg) ∂μ := by
refine lintegral_congr fun a => ?_
rw [Pi.mul_apply, fun_eq_funMulInvSnorm_mul_eLpNorm f hf_nonzero hf_nontop,
fun_eq_funMulInvSnorm_mul_eLpNorm g hg_nonzero hg_nontop, Pi.mul_apply]
... | no goals | fad475627c8ad9fe |
FirstOrder.Language.BoundedFormula.induction_on_all_ex | Mathlib/ModelTheory/Complexity.lean | theorem induction_on_all_ex {P : ∀ {m}, L.BoundedFormula α m → Prop} (φ : L.BoundedFormula α n)
(hqf : ∀ {m} {ψ : L.BoundedFormula α m}, IsQF ψ → P ψ)
(hall : ∀ {m} {ψ : L.BoundedFormula α (m + 1)}, P ψ → P ψ.all)
(hex : ∀ {m} {φ : L.BoundedFormula α (m + 1)}, P φ → P φ.ex)
(hse : ∀ {m} {φ₁ φ₂ : L.Bound... | L : Language
α : Type u'
n : ℕ
P : {m : ℕ} → L.BoundedFormula α m → Prop
φ : L.BoundedFormula α n
hqf : ∀ {m : ℕ} {ψ : L.BoundedFormula α m}, ψ.IsQF → P ψ
hall : ∀ {m : ℕ} {ψ : L.BoundedFormula α (m + 1)}, P ψ → P (∀'ψ)
hex : ∀ {m : ℕ} {φ : L.BoundedFormula α (m + 1)}, P φ → P (∃'φ)
hse : ∀ {m : ℕ} {φ₁ φ₂ : L.BoundedFo... | suffices h' : ∀ {m} {φ : L.BoundedFormula α m}, φ.IsPrenex → P φ from
(hse φ.iff_toPrenex).2 (h' φ.toPrenex_isPrenex) | L : Language
α : Type u'
n : ℕ
P : {m : ℕ} → L.BoundedFormula α m → Prop
φ : L.BoundedFormula α n
hqf : ∀ {m : ℕ} {ψ : L.BoundedFormula α m}, ψ.IsQF → P ψ
hall : ∀ {m : ℕ} {ψ : L.BoundedFormula α (m + 1)}, P ψ → P (∀'ψ)
hex : ∀ {m : ℕ} {φ : L.BoundedFormula α (m + 1)}, P φ → P (∃'φ)
hse : ∀ {m : ℕ} {φ₁ φ₂ : L.BoundedFo... | 46d6537499c676b9 |
ApproximatesLinearOn.norm_fderiv_sub_le | Mathlib/MeasureTheory/Function/Jacobian.lean | theorem _root_.ApproximatesLinearOn.norm_fderiv_sub_le {A : E →L[ℝ] E} {δ : ℝ≥0}
(hf : ApproximatesLinearOn f A s δ) (hs : MeasurableSet s) (f' : E → E →L[ℝ] E)
(hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) : ∀ᵐ x ∂μ.restrict s, ‖f' x - A‖₊ ≤ δ | E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
A : E →L[ℝ] E
δ : ℝ≥0
hf : ApproximatesLinearOn f A s δ
hs : MeasurableSet s
f' : E → E →L[ℝ] E
hf' : ∀ x ∈ s, ... | simp only [ContinuousLinearMap.map_smul, norm_smul, Real.norm_eq_abs, abs_of_nonneg rpos.le] | no goals | 99ae4fc7a76b96c9 |
AkraBazziRecurrence.smoothingFn_mul_asympBound_isBigO_T | Mathlib/Computability/AkraBazzi/AkraBazzi.lean | /-- The main proof of the lower bound part of the Akra-Bazzi theorem. The factor
`1 + ε n` does not change the asymptotic order, but is needed for the induction step to go
through. -/
lemma smoothingFn_mul_asympBound_isBigO_T :
(fun (n : ℕ) => (1 + ε n) * asympBound g a b n) =O[atTop] T | α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
b' : ℝ := b (min_bi b) / 2
hb_pos : 0 < b'
c₁ : ℝ
hc₁ : c₁ > 0
h_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n
n₀ : ℕ
n₀_ge_Rn₀ : R.n₀ ≤ n₀... | exact hm_mem.1 | no goals | 82bb51116c3f899e |
Nimber.add_assoc | Mathlib/SetTheory/Nimber/Basic.lean | theorem add_assoc (a b c : Nimber) : a + b + c = a + (b + c) | case a.h₁.inl.intro.intro
a b c y : Nimber
hy : y < a
hx : y + b < a + b
⊢ y ≠ a
case a.h₁.inr.intro.intro
a b c y : Nimber
hy : y < b
hx : a + y < a + b
⊢ y ≠ b
case a.h₂
a b c x : Nimber
hx : x < c
⊢ a + b + x ≠ a + (b + c)
case a.h₁
a b c x : Nimber
hx : x < a
⊢ x + (b + c) ≠ a + b + c
case a.h₂.inl.intro.intro
... | on_goal 3 => rw [Nimber.add_assoc _ _ x, add_ne_add_right, add_ne_add_right] | case a.h₁.inl.intro.intro
a b c y : Nimber
hy : y < a
hx : y + b < a + b
⊢ y ≠ a
case a.h₁.inr.intro.intro
a b c y : Nimber
hy : y < b
hx : a + y < a + b
⊢ y ≠ b
case a.h₂
a b c x : Nimber
hx : x < c
⊢ x ≠ c
case a.h₁
a b c x : Nimber
hx : x < a
⊢ x + (b + c) ≠ a + b + c
case a.h₂.inl.intro.intro
a b c y : Nimber
h... | 4588e1e396a7b689 |
Polynomial.coeff_le_of_roots_le | Mathlib/Topology/Algebra/Polynomial.lean | theorem coeff_le_of_roots_le {p : F[X]} {f : F →+* K} {B : ℝ} (i : ℕ) (h1 : p.Monic)
(h2 : Splits f p) (h3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B) :
‖(map f p).coeff i‖ ≤ B ^ (p.natDegree - i) * p.natDegree.choose i | case inl
F : Type u_3
K : Type u_4
inst✝¹ : CommRing F
inst✝ : NormedField K
p : F[X]
f : F →+* K
B : ℝ
i : ℕ
h1 : p.Monic
h2 : Splits f p
h3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B
hB : B < 0
⊢ ‖if i = 0 then 1 else 0‖ ≤ ↑(Nat.choose 0 i) | split_ifs with h <;> simp [h] | no goals | 15177c00698e7d89 |
div_sq_cancel | Mathlib/Algebra/GroupWithZero/Basic.lean | lemma div_sq_cancel (a b : G₀) : a ^ 2 * b / a = a * b | case inr
G₀ : Type u_2
inst✝ : CommGroupWithZero G₀
a b : G₀
ha : a ≠ 0
⊢ a ^ 2 * b / a = a * b | rw [sq, mul_assoc, mul_div_cancel_left₀ _ ha] | no goals | 1f5ff9ebeacb3a51 |
mfderivWithin_of_mem_nhds | Mathlib/Geometry/Manifold/MFDeriv/Basic.lean | theorem mfderivWithin_of_mem_nhds (h : s ∈ 𝓝 x) : mfderivWithin I I' f s x = mfderiv I I' f x | 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpac... | rw [← mfderivWithin_univ, ← univ_inter s, mfderivWithin_inter h] | no goals | 9dc3f5423af6e347 |
Module.Dual.range_eq_top_of_ne_zero | Mathlib/LinearAlgebra/Dual.lean | lemma range_eq_top_of_ne_zero :
LinearMap.range f = ⊤ | K : Type uK
inst✝² : Field K
V₁ : Type uV₁
inst✝¹ : AddCommGroup V₁
inst✝ : Module K V₁
f : Dual K V₁
hf : f ≠ 0
v : V₁
hv : f v ≠ 0
x : K
x✝ : x ∈ ⊤
⊢ f (x • (f v)⁻¹ • v) = x | simp [inv_mul_cancel₀ hv] | no goals | 63c5fb9b9c37d7e1 |
Subalgebra.mem_of_finset_sum_eq_one_of_pow_smul_mem | Mathlib/Algebra/Algebra/Subalgebra/Operations.lean | theorem mem_of_finset_sum_eq_one_of_pow_smul_mem
{ι : Type*} (ι' : Finset ι) (s : ι → S) (l : ι → S)
(e : ∑ i ∈ ι', l i * s i = 1) (hs : ∀ i, s i ∈ S') (hl : ∀ i, l i ∈ S') (x : S)
(H : ∀ i, ∃ n : ℕ, (s i ^ n : S) • x ∈ S') : x ∈ S' | R : Type u_1
S : Type u_2
inst✝² : CommSemiring R
inst✝¹ : CommRing S
inst✝ : Algebra R S
S' : Subalgebra R S
ι : Type u_3
ι' : Finset ι
s l : ι → S
e : ∑ i ∈ ι', l i * s i = 1
hs : ∀ (i : ι), s i ∈ S'
hl : ∀ (i : ι), l i ∈ S'
x : S
n : ι → ℕ
hn : ∀ (i : ι), s i ^ n i • x ∈ S'
s' : ι → ↥S' := fun x => ⟨s x, ⋯⟩
⊢ x ∈ to... | let l' : ι → S' := fun x => ⟨l x, hl x⟩ | R : Type u_1
S : Type u_2
inst✝² : CommSemiring R
inst✝¹ : CommRing S
inst✝ : Algebra R S
S' : Subalgebra R S
ι : Type u_3
ι' : Finset ι
s l : ι → S
e : ∑ i ∈ ι', l i * s i = 1
hs : ∀ (i : ι), s i ∈ S'
hl : ∀ (i : ι), l i ∈ S'
x : S
n : ι → ℕ
hn : ∀ (i : ι), s i ^ n i • x ∈ S'
s' : ι → ↥S' := fun x => ⟨s x, ⋯⟩
l' : ι →... | 911706ddaafddc82 |
Configuration.ofField.crossProduct_eq_zero_of_dotProduct_eq_zero | Mathlib/Combinatorics/Configuration.lean | lemma crossProduct_eq_zero_of_dotProduct_eq_zero {a b c d : Fin 3 → K} (hac : dotProduct a c = 0)
(hbc : dotProduct b c = 0) (had : dotProduct a d = 0) (hbd : dotProduct b d = 0) :
crossProduct a b = 0 ∨ crossProduct c d = 0 | K : Type u_3
inst✝ : Field K
a b c d : Fin 3 → K
hac : a ⬝ᵥ c = 0
hbc : b ⬝ᵥ c = 0
had : a ⬝ᵥ d = 0
hbd : b ⬝ᵥ d = 0
h : LinearIndependent K ![a, b] ∧ LinearIndependent K ![c, d]
⊢ False | let A : Matrix (Fin 2) (Fin 3) K := ![a, b] | K : Type u_3
inst✝ : Field K
a b c d : Fin 3 → K
hac : a ⬝ᵥ c = 0
hbc : b ⬝ᵥ c = 0
had : a ⬝ᵥ d = 0
hbd : b ⬝ᵥ d = 0
h : LinearIndependent K ![a, b] ∧ LinearIndependent K ![c, d]
A : Matrix (Fin 2) (Fin 3) K := ![a, b]
⊢ False | b06607b2f5468528 |
CircleDeg1Lift.translationNumber_lt_of_forall_lt_add | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | theorem translationNumber_lt_of_forall_lt_add (hf : Continuous f) {z : ℝ} (hz : ∀ x, f x < x + z) :
τ f < z | case intro.intro
f : CircleDeg1Lift
hf : Continuous ⇑f
z : ℝ
hz : ∀ (x : ℝ), f x < x + z
x : ℝ
hx : ∀ y ∈ Icc 0 1, f y - y ≤ f x - x
⊢ τ f ≤ f x - x | apply translationNumber_le_of_le_add | case intro.intro.hz
f : CircleDeg1Lift
hf : Continuous ⇑f
z : ℝ
hz : ∀ (x : ℝ), f x < x + z
x : ℝ
hx : ∀ y ∈ Icc 0 1, f y - y ≤ f x - x
⊢ ∀ (x_1 : ℝ), f x_1 ≤ x_1 + (f x - x) | e891823a8179b7be |
MeasureTheory.levyProkhorovEDist_le_of_forall_le | Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean | /-- A simple sufficient condition for bounding `levyProkhorovEDist` between probability measures
from above. The condition involves only one of two natural bounds, the other bound is for free. -/
lemma levyProkhorovEDist_le_of_forall_le
(μ ν : Measure Ω) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] (δ : ℝ≥0∞)
... | Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoEMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : IsProbabilityMeasure ν
δ ε : ℝ≥0∞
B : Set Ω
ε_gt : δ < ε
ε_lt_top : ε < ⊤
B_mble : MeasurableSet B
B_subset : B ⊆ (thickening ε.toReal (thickening ε.toReal B)ᶜ)ᶜ
... | apply (tsub_le_tsub_right almost _).trans | Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : PseudoEMetricSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ ν : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
inst✝ : IsProbabilityMeasure ν
δ ε : ℝ≥0∞
B : Set Ω
ε_gt : δ < ε
ε_lt_top : ε < ⊤
B_mble : MeasurableSet B
B_subset : B ⊆ (thickening ε.toReal (thickening ε.toReal B)ᶜ)ᶜ
... | 44fd10ff870a09b2 |
ZNum.div_to_int | Mathlib/Data/Num/Lemmas.lean | theorem div_to_int : ∀ n d, ((n / d : ZNum) : ℤ) = n / d
| 0, 0 => by simp [Int.ediv_zero]
| 0, pos _ => (Int.zero_ediv _).symm
| 0, neg _ => (Int.zero_ediv _).symm
| pos _, 0 => (Int.ediv_zero _).symm
| neg _, 0 => (Int.ediv_zero _).symm
| pos n, pos d => (Num.cast_toZNum _).trans <| by rw [← Num.to_nat_to... | n d : PosNum
⊢ ↑(n.pred' / Num.pos d).succ' = -↑n / -↑d | rw [n.to_int_eq_succ_pred, d.to_int_eq_succ_pred, ← PosNum.to_nat_to_int, Num.succ'_to_nat,
Num.div_to_nat] | n d : PosNum
⊢ ↑(↑n.pred' / ↑(Num.pos d) + 1) = -(↑↑n.pred' + 1) / -(↑↑d.pred' + 1) | c03febc878a6cda7 |
RootPairing.coxeterWeight_nonneg | Mathlib/LinearAlgebra/RootSystem/RootPositive.lean | lemma coxeterWeight_nonneg : 0 ≤ P.coxeterWeightIn S i j | ι : Type u_1
R : Type u_2
S : Type u_3
M : Type u_4
N : Type u_5
inst✝¹⁰ : LinearOrderedCommRing S
inst✝⁹ : CommRing R
inst✝⁸ : Algebra S R
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
P : RootPairing ι R M N
inst✝³ : P.IsValuedIn S
B : RootPositiveForm S P
i j : ι
inst✝² : Fa... | dsimp [coxeterWeightIn] | ι : Type u_1
R : Type u_2
S : Type u_3
M : Type u_4
N : Type u_5
inst✝¹⁰ : LinearOrderedCommRing S
inst✝⁹ : CommRing R
inst✝⁸ : Algebra S R
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
P : RootPairing ι R M N
inst✝³ : P.IsValuedIn S
B : RootPositiveForm S P
i j : ι
inst✝² : Fa... | 18f7e4e37928fa1e |
Equiv.Perm.cycleFactorsFinset_mul_inv_mem_eq_sdiff | Mathlib/GroupTheory/Perm/Cycle/Factors.lean | theorem cycleFactorsFinset_mul_inv_mem_eq_sdiff [DecidableEq α] [Fintype α] {f g : Perm α}
(h : f ∈ cycleFactorsFinset g) : cycleFactorsFinset (g * f⁻¹) = cycleFactorsFinset g \ {f} | case pos
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g f✝ σ τ : Perm α
hd : σ.Disjoint τ
a✝ : σ.IsCycle
hσ : ∀ {f : Perm α}, f ∈ σ.cycleFactorsFinset → (σ * f⁻¹).cycleFactorsFinset = σ.cycleFactorsFinset \ {f}
hτ : ∀ {f : Perm α}, f ∈ τ.cycleFactorsFinset → (τ * f⁻¹).cycleFactorsFinset = τ.cycleFactorsFinset ... | simpa [hx] using hfx.symm | no goals | 3d77fd0de031f082 |
AkraBazziRecurrence.rpow_p_mul_one_add_smoothingFn_ge | Mathlib/Computability/AkraBazzi/AkraBazzi.lean | lemma rpow_p_mul_one_add_smoothingFn_ge :
∀ᶠ (n : ℕ) in atTop, ∀ i, (b i) ^ (p a b) * n ^ (p a b) * (1 + ε n)
≤ (r i n) ^ (p a b) * (1 + ε (r i n)) | case h
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
i : α
q : ℝ → ℝ := fun x => x ^ p a b * (1 + ε x)
h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1)
h_deriv_q : deriv q =O[atTop] fun x => x ^ (p a b - 1)
h_main_norm :
(fun n => ‖q ↑(r ... | show (b i) ^ (p a b) * n ^ (p a b) * (1 + ε n) ≤ q (r i n) | case h
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
i : α
q : ℝ → ℝ := fun x => x ^ p a b * (1 + ε x)
h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1)
h_deriv_q : deriv q =O[atTop] fun x => x ^ (p a b - 1)
h_main_norm :
(fun n => ‖q ↑(r ... | 6b09e1ce84116b26 |
BoundedContinuousFunction.dist_extend_extend | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | theorem dist_extend_extend (f : α ↪ δ) (g₁ g₂ : α →ᵇ β) (h₁ h₂ : δ →ᵇ β) :
dist (g₁.extend f h₁) (g₂.extend f h₂) =
max (dist g₁ g₂) (dist (h₁.restrict (range f)ᶜ) (h₂.restrict (range f)ᶜ)) | case refine_3
α : Type u
β : Type v
inst✝³ : TopologicalSpace α
inst✝² : PseudoMetricSpace β
δ : Type u_2
inst✝¹ : TopologicalSpace δ
inst✝ : DiscreteTopology δ
f : α ↪ δ
g₁ g₂ : α →ᵇ β
h₁ h₂ : δ →ᵇ β
⊢ dist (h₁.restrict (range ⇑f)ᶜ) (h₂.restrict (range ⇑f)ᶜ) ≤ dist (extend f g₁ h₁) (extend f g₂ h₂) | refine (dist_le dist_nonneg).2 fun x => ?_ | case refine_3
α : Type u
β : Type v
inst✝³ : TopologicalSpace α
inst✝² : PseudoMetricSpace β
δ : Type u_2
inst✝¹ : TopologicalSpace δ
inst✝ : DiscreteTopology δ
f : α ↪ δ
g₁ g₂ : α →ᵇ β
h₁ h₂ : δ →ᵇ β
x : ↑(range ⇑f)ᶜ
⊢ dist ((h₁.restrict (range ⇑f)ᶜ) x) ((h₂.restrict (range ⇑f)ᶜ) x) ≤ dist (extend f g₁ h₁) (extend f g... | 45cb1096070ba961 |
BitVec.toFin_and | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem toFin_and (x y : BitVec v) :
BitVec.toFin (x &&& y) = BitVec.toFin x &&& BitVec.toFin y | v : Nat
x y : BitVec v
⊢ (x &&& y).toFin = x.toFin &&& y.toFin | apply Fin.eq_of_val_eq | case a
v : Nat
x y : BitVec v
⊢ ↑(x &&& y).toFin = ↑(x.toFin &&& y.toFin) | 20fae5ed112ca1d4 |
Submodule.mkQ_surjective | Mathlib/LinearAlgebra/Quotient/Defs.lean | theorem mkQ_surjective : Function.Surjective p.mkQ | case mk
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
p : Submodule R M
b✝ : M ⧸ p
x : M
⊢ ∃ a, p.mkQ a = Quot.mk (⇑p.quotientRel) x | exact ⟨x, rfl⟩ | no goals | bc318a0d1befeea0 |
PEquiv.transpose_toMatrix_toPEquiv_apply | Mathlib/Data/Matrix/PEquiv.lean | theorem transpose_toMatrix_toPEquiv_apply
[DecidableEq m] [DecidableEq n] [Zero α] [One α] (f : m ≃ n) (j) :
f.toPEquiv.toMatrixᵀ j = Pi.single (f.symm j) (1 : α) | case h
m : Type u_3
n : Type u_4
α : Type u_5
inst✝³ : DecidableEq m
inst✝² : DecidableEq n
inst✝¹ : Zero α
inst✝ : One α
f : m ≃ n
j : n
x✝ : m
⊢ f.toPEquiv.toMatrixᵀ j x✝ = Pi.single (f.symm j) 1 x✝ | simp [toMatrix_apply, Pi.single_apply, eq_comm, ← Equiv.apply_eq_iff_eq_symm_apply] | no goals | b2ca71f63f5ed545 |
List.splitByLoop_eq_append | Mathlib/Data/List/SplitBy.lean | theorem splitByLoop_eq_append {r : α → α → Bool} {l : List α} {a : α} {g : List α}
(gs : List (List α)) : splitBy.loop r l a g gs = gs.reverse ++ splitBy.loop r l a g [] | case cons.h_2
α : Type u_1
r : α → α → Bool
b : α
l : List α
IH : ∀ {a : α} {g : List α} (gs : List (List α)), splitBy.loop r l a g gs = gs.reverse ++ splitBy.loop r l a g []
a : α
g : List α
gs : List (List α)
x✝ : Bool
heq✝ : r a b = false
⊢ ((a :: g).reverse :: gs).reverse ++ splitBy.loop r l b [] [] =
gs.revers... | simp | no goals | 593e90637e5bb968 |
ZMod.sum_mul_div_add_sum_mul_div_eq_mul | Mathlib/NumberTheory/LegendreSymbol/GaussEisensteinLemmas.lean | theorem sum_mul_div_add_sum_mul_div_eq_mul (p q : ℕ) [hp : Fact p.Prime] (hq0 : (q : ZMod p) ≠ 0) :
∑ a ∈ Ico 1 (p / 2).succ, a * q / p + ∑ a ∈ Ico 1 (q / 2).succ, a * p / q =
p / 2 * (q / 2) | p q : ℕ
hp : Fact (Nat.Prime p)
hq0 : ↑q ≠ 0
hswap :
#(filter (fun x => x.2 * q ≤ x.1 * p) (Ico 1 (q / 2).succ ×ˢ Ico 1 (p / 2).succ)) =
#(filter (fun x => x.1 * q ≤ x.2 * p) (Ico 1 (p / 2).succ ×ˢ Ico 1 (q / 2).succ))
x : ℕ × ℕ
hx : x ∈ Ico 1 (p / 2).succ ×ˢ Ico 1 (q / 2).succ
hpq : x.2 * p ≤ x.1 * q
hqp : x.1 *... | rw [val_cast_of_lt hxp, val_zero] at this | p q : ℕ
hp : Fact (Nat.Prime p)
hq0 : ↑q ≠ 0
hswap :
#(filter (fun x => x.2 * q ≤ x.1 * p) (Ico 1 (q / 2).succ ×ˢ Ico 1 (p / 2).succ)) =
#(filter (fun x => x.1 * q ≤ x.2 * p) (Ico 1 (p / 2).succ ×ˢ Ico 1 (q / 2).succ))
x : ℕ × ℕ
hx : x ∈ Ico 1 (p / 2).succ ×ˢ Ico 1 (q / 2).succ
hpq : x.2 * p ≤ x.1 * q
hqp : x.1 *... | 1909d4eead5b0e9e |
tendsto_factorial_div_pow_self_atTop | Mathlib/Analysis/SpecificLimits/Basic.lean | theorem tendsto_factorial_div_pow_self_atTop :
Tendsto (fun n ↦ n ! / (n : ℝ) ^ n : ℕ → ℝ) atTop (𝓝 0) :=
tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds
(tendsto_const_div_atTop_nhds_zero_nat 1)
(Eventually.of_forall fun n ↦
div_nonneg (mod_cast n.factorial_pos.le)
(pow_nonne... | case intro.refine_2
k : ℕ
hn : 0 < k.succ
x : ℕ
hx : x < k
⊢ (↑x + 1 + 1) * (↑k + 1)⁻¹ ≤ 1 | refine (div_le_one <| mod_cast hn).mpr ?_ | case intro.refine_2
k : ℕ
hn : 0 < k.succ
x : ℕ
hx : x < k
⊢ ↑x + 1 + 1 ≤ ↑k + 1 | 74d4f8c80b3eb104 |
Submodule.biSup_comap_eq_top_of_surjective | Mathlib/LinearAlgebra/Span/Basic.lean | lemma biSup_comap_eq_top_of_surjective {ι : Type*} (s : Set ι) (hs : s.Nonempty)
(p : ι → Submodule R₂ M₂) (hp : ⨆ i ∈ s, p i = ⊤)
(f : M →ₛₗ[τ₁₂] M₂) (hf : Surjective f) :
⨆ i ∈ s, (p i).comap f = ⊤ | R : Type u_1
R₂ : Type u_2
M : Type u_4
M₂ : Type u_5
inst✝⁶ : Semiring R
inst✝⁵ : Semiring R₂
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : AddCommGroup M₂
inst✝¹ : Module R₂ M₂
τ₁₂ : R →+* R₂
inst✝ : RingHomSurjective τ₁₂
ι : Type u_9
s : Set ι
p : ι → Submodule R₂ M₂
hp : ⨆ i ∈ s, p i = ⊤
f : M →ₛₗ[τ₁₂] M₂
hf... | exact le_trans f.ker_le_comap (le_biSup (fun i ↦ (p i).comap f) hk) | no goals | e3cd39b5141fc4e5 |
IsDedekindDomain.HeightOneSpectrum.intValuation.map_mul' | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | theorem intValuation.map_mul' (x y : R) :
v.intValuationDef (x * y) = v.intValuationDef x * v.intValuationDef y | R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
x y : R
hx : ¬x = 0
hy : ¬y = 0
⊢ Associates.mk (Ideal.span {x}) ≠ 0 | apply Associates.mk_ne_zero'.mpr hx | no goals | d0a356c2f02f103d |
IsTopologicalGroup.exists_antitone_basis_nhds_one | Mathlib/Topology/Algebra/Group/Basic.lean | theorem IsTopologicalGroup.exists_antitone_basis_nhds_one [FirstCountableTopology G] :
∃ u : ℕ → Set G, (𝓝 1).HasAntitoneBasis u ∧ ∀ n, u (n + 1) * u (n + 1) ⊆ u n | case intro.mk
G : Type w
inst✝³ : TopologicalSpace G
inst✝² : Group G
inst✝¹ : IsTopologicalGroup G
inst✝ : FirstCountableTopology G
u : ℕ → Set G
hu : (𝓝 1).HasBasis (fun x => True) u
u_anti : Antitone u
⊢ ∃ u, (𝓝 1).HasAntitoneBasis u ∧ ∀ (n : ℕ), u (n + 1) * u (n + 1) ⊆ u n | have :=
((hu.prod_nhds hu).tendsto_iff hu).mp
(by simpa only [mul_one] using continuous_mul.tendsto ((1, 1) : G × G)) | case intro.mk
G : Type w
inst✝³ : TopologicalSpace G
inst✝² : Group G
inst✝¹ : IsTopologicalGroup G
inst✝ : FirstCountableTopology G
u : ℕ → Set G
hu : (𝓝 1).HasBasis (fun x => True) u
u_anti : Antitone u
this : ∀ (ib : ℕ), True → ∃ ia, (True ∧ True) ∧ ∀ x ∈ u ia.1 ×ˢ u ia.2, x.1 * x.2 ∈ u ib
⊢ ∃ u, (𝓝 1).HasAntitone... | b18d56050ad7f478 |
MonoidAlgebra.mul_apply_mul_eq_mul_of_uniqueMul | Mathlib/Algebra/MonoidAlgebra/NoZeroDivisors.lean | theorem mul_apply_mul_eq_mul_of_uniqueMul [Mul A] {f g : MonoidAlgebra R A} {a0 b0 : A}
(h : UniqueMul f.support g.support a0 b0) :
(f * g) (a0 * b0) = f a0 * g b0 | case refine_2.inl
R : Type u_1
A : Type u_2
inst✝¹ : Semiring R
inst✝ : Mul A
f g : MonoidAlgebra R A
a0 b0 : A
h : UniqueMul f.support g.support a0 b0
hnmem : ¬(a0 ∈ f.support ∧ b0 ∈ g.support)
x✝ : True
af : a0 ∉ f.support
⊢ f a0 * g b0 = 0 | rw [not_mem_support_iff.mp af, zero_mul] | no goals | dfd38470e4ef2425 |
Std.DHashMap.Internal.List.getValueCast?_eq_getEntry? | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/List/Associative.lean | theorem getValueCast?_eq_getEntry? [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {a : α} :
getValueCast? a l = Option.dmap (getEntry? a l)
(fun p h => cast (congrArg β (eq_of_beq (getEntry?_eq_some h))) p.2) | α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : LawfulBEq α
a k : α
v : β k
t : List ((a : α) × β a)
ih : getValueCast? a t = Std.DHashMap.Internal.List.Option.dmap (getEntry? a t) fun p h => cast ⋯ p.snd
⊢ getValueCast? a (⟨k, v⟩ :: t) =
Std.DHashMap.Internal.List.Option.dmap (getEntry? a (⟨k, v⟩ :: t)) fun p h =... | cases h : k == a | case false
α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : LawfulBEq α
a k : α
v : β k
t : List ((a : α) × β a)
ih : getValueCast? a t = Std.DHashMap.Internal.List.Option.dmap (getEntry? a t) fun p h => cast ⋯ p.snd
h : (k == a) = false
⊢ getValueCast? a (⟨k, v⟩ :: t) =
Std.DHashMap.Internal.List.Option.dmap (getE... | 28f6f201c63cc747 |
Finset.sum_range_by_parts | Mathlib/Algebra/BigOperators/Module.lean | theorem sum_range_by_parts :
∑ i ∈ range n, f i • g i =
f (n - 1) • G n - ∑ i ∈ range (n - 1), (f (i + 1) - f i) • G (i + 1) | case neg
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : ℕ → R
g : ℕ → M
n : ℕ
hn : ¬n = 0
⊢ ∑ i ∈ range n, f i • g i =
f (n - 1) • ∑ i ∈ range n, g i - ∑ i ∈ range (n - 1), (f (i + 1) - f i) • ∑ i ∈ range (i + 1), g i | rw [range_eq_Ico, sum_Ico_by_parts f g (Nat.pos_of_ne_zero hn), sum_range_zero, smul_zero,
sub_zero, range_eq_Ico] | no goals | 2414517ad053dda7 |
CategoryTheory.HomOrthogonal.matrixDecomposition_comp | Mathlib/CategoryTheory/Preadditive/HomOrthogonal.lean | theorem matrixDecomposition_comp (o : HomOrthogonal s) {α β γ : Type} [Finite α] [Fintype β]
[Finite γ] {f : α → ι} {g : β → ι} {h : γ → ι} (z : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b))
(w : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)) (i : ι) :
o.matrixDecomposition (z ≫ w) i = o.matrixDecomposition w i * ... | case a.mk.refl.mk
C : Type u
inst✝⁵ : Category.{v, u} C
ι : Type u_1
s : ι → C
inst✝⁴ : Preadditive C
inst✝³ : HasFiniteBiproducts C
o : HomOrthogonal s
α β γ : Type
inst✝² : Finite α
inst✝¹ : Fintype β
inst✝ : Finite γ
f : α → ι
g : β → ι
h : γ → ι
z : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)
w : (⨁ fun b => s (g b))... | apply Finset.sum_congr_set | case a.mk.refl.mk.w
C : Type u
inst✝⁵ : Category.{v, u} C
ι : Type u_1
s : ι → C
inst✝⁴ : Preadditive C
inst✝³ : HasFiniteBiproducts C
o : HomOrthogonal s
α β γ : Type
inst✝² : Finite α
inst✝¹ : Fintype β
inst✝ : Finite γ
f : α → ι
g : β → ι
h : γ → ι
z : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)
w : (⨁ fun b => s (g b... | 0b929e3994fac12e |
Fin.univ_castSuccEmb | Mathlib/Data/Fintype/Basic.lean | theorem Fin.univ_castSuccEmb (n : ℕ) :
(univ : Finset (Fin (n + 1))) =
Finset.cons (Fin.last n) (univ.map Fin.castSuccEmb) (by simp [map_eq_image]) | α : Type u_1
β : Type u_2
γ : Type u_3
n : ℕ
⊢ last n ∉ map castSuccEmb univ | simp [map_eq_image] | no goals | 2df9ff6debd7e37d |
MeasureTheory.setIntegral_tilted | Mathlib/MeasureTheory/Measure/Tilted.lean | lemma setIntegral_tilted [SFinite μ] (f : α → ℝ) (g : α → E) (s : Set α) :
∫ x in s, g x ∂(μ.tilted f) = ∫ x in s, (exp (f x) / ∫ x, exp (f x) ∂μ) • (g x) ∂μ | α : Type u_1
mα : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : SFinite μ
f : α → ℝ
g : α → E
s : Set α
hf : AEMeasurable f μ
this : AEMeasurable (fun x => rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) μ
⊢ AEMeasurable (fun x => ↑⟨rexp (f x) / ∫ (x : α), rexp (f x)... | refine AEMeasurable.restrict ?_ | α : Type u_1
mα : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : SFinite μ
f : α → ℝ
g : α → E
s : Set α
hf : AEMeasurable f μ
this : AEMeasurable (fun x => rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) μ
⊢ AEMeasurable (fun x => ↑⟨rexp (f x) / ∫ (x : α), rexp (f x)... | 9479368488db4a2b |
List.nextOr_eq_nextOr_of_mem_of_ne | Mathlib/Data/List/Cycle.lean | theorem nextOr_eq_nextOr_of_mem_of_ne (xs : List α) (x d d' : α) (x_mem : x ∈ xs)
(x_ne : x ≠ xs.getLast (ne_nil_of_mem x_mem)) : nextOr xs x d = nextOr xs x d' | case neg
α : Type u_1
inst✝ : DecidableEq α
x d d' y z : α
zs : List α
IH : ∀ (x_mem : x ∈ z :: zs), x ≠ (z :: zs).getLast ⋯ → (z :: zs).nextOr x d = (z :: zs).nextOr x d'
x_mem : x ∈ y :: z :: zs
x_ne : x ≠ (y :: z :: zs).getLast ⋯
h : ¬x = y
⊢ (y :: z :: zs).nextOr x d = (y :: z :: zs).nextOr x d' | rw [nextOr, nextOr, IH] | case neg.x_mem
α : Type u_1
inst✝ : DecidableEq α
x d d' y z : α
zs : List α
IH : ∀ (x_mem : x ∈ z :: zs), x ≠ (z :: zs).getLast ⋯ → (z :: zs).nextOr x d = (z :: zs).nextOr x d'
x_mem : x ∈ y :: z :: zs
x_ne : x ≠ (y :: z :: zs).getLast ⋯
h : ¬x = y
⊢ x ∈ z :: zs
case neg.x_ne
α : Type u_1
inst✝ : DecidableEq α
x d d'... | 2ab17110c5253750 |
preNormEDS'_even | Mathlib/NumberTheory/EllipticDivisibilitySequence.lean | lemma preNormEDS'_even (m : ℕ) : preNormEDS' b c d (2 * (m + 3)) =
preNormEDS' b c d (m + 2) ^ 2 * preNormEDS' b c d (m + 3) * preNormEDS' b c d (m + 5) -
preNormEDS' b c d (m + 1) * preNormEDS' b c d (m + 3) * preNormEDS' b c d (m + 4) ^ 2 | R : Type u
inst✝ : CommRing R
b c d : R
m : ℕ
⊢ 2 * (m + 3) = 2 * m + 1 + 5 | rfl | no goals | ebac1282960d77ba |
memℓp_gen_iff | Mathlib/Analysis/Normed/Lp/lpSpace.lean | theorem memℓp_gen_iff (hp : 0 < p.toReal) {f : ∀ i, E i} :
Memℓp f p ↔ Summable fun i => ‖f i‖ ^ p.toReal | α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
hp : 0 < p ∧ p < ⊤
f : (i : α) → E i
⊢ Memℓp f p ↔ Summable fun i => ‖f i‖ ^ p.toReal | dsimp [Memℓp] | α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
hp : 0 < p ∧ p < ⊤
f : (i : α) → E i
⊢ (if p = 0 then {i | ¬f i = 0}.Finite
else if p = ⊤ then BddAbove (Set.range fun i => ‖f i‖) else Summable fun i => ‖f i‖ ^ p.toReal) ↔
Summable fun i => ‖f i‖ ^ p.toReal | d46858059545142b |
BitVec.getMsbD_neg | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Bitblast.lean | theorem getMsbD_neg {i : Nat} {x : BitVec w} :
getMsbD (-x) i =
(getMsbD x i ^^ decide (∃ j < w, i < j ∧ getMsbD x j = true)) | case mp
w i : Nat
x : BitVec w
hi : i < w
h₁ : w - 1 - i < w
⊢ (∃ j, j < w - 1 - i ∧ x.getLsbD j = true) → ∃ j, j < w ∧ i < j ∧ j < w ∧ x.getLsbD (w - 1 - j) = true | rintro ⟨j, hj, h⟩ | case mp.intro.intro
w i : Nat
x : BitVec w
hi : i < w
h₁ : w - 1 - i < w
j : Nat
hj : j < w - 1 - i
h : x.getLsbD j = true
⊢ ∃ j, j < w ∧ i < j ∧ j < w ∧ x.getLsbD (w - 1 - j) = true | a66804a595b3aedc |
MeasureTheory.JordanDecomposition.toSignedMeasure_zero | Mathlib/MeasureTheory/Decomposition/Jordan.lean | theorem toSignedMeasure_zero : (0 : JordanDecomposition α).toSignedMeasure = 0 | α : Type u_1
inst✝ : MeasurableSpace α
⊢ toSignedMeasure 0 = 0 | ext1 i hi | case h
α : Type u_1
inst✝ : MeasurableSpace α
i : Set α
hi : MeasurableSet i
⊢ ↑(toSignedMeasure 0) i = ↑0 i | f6b6ab034ca177b1 |
FilterBasis.generate | Mathlib/Order/Filter/Bases.lean | theorem generate (B : FilterBasis α) : generate B.sets = B.filter | case a.intro.intro
α : Type u_1
B : FilterBasis α
U : Set α
U_in : U ∈ B.filter
V : Set α
V_in : V ∈ B
h : V ⊆ U
⊢ U ∈ generate B.sets | exact GenerateSets.superset (GenerateSets.basic V_in) h | no goals | 4f5ec27baff95828 |
Polynomial.leadingCoeff_prod' | Mathlib/Algebra/Polynomial/BigOperators.lean | theorem leadingCoeff_prod' (h : (∏ i ∈ s, (f i).leadingCoeff) ≠ 0) :
(∏ i ∈ s, f i).leadingCoeff = ∏ i ∈ s, (f i).leadingCoeff | R : Type u
ι : Type w
s : Finset ι
inst✝ : CommSemiring R
f : ι → R[X]
h : ∏ i ∈ s, (f i).leadingCoeff ≠ 0
⊢ (Multiset.map leadingCoeff (Multiset.map f s.val)).prod ≠ 0 | simpa using h | no goals | fbe3ca3182d428c2 |
MeasureTheory.Measure.rnDeriv_smul_right | Mathlib/MeasureTheory/Decomposition/Lebesgue.lean | theorem rnDeriv_smul_right (ν μ : Measure α) [IsFiniteMeasure ν]
[ν.HaveLebesgueDecomposition μ] {r : ℝ≥0} (hr : r ≠ 0) :
ν.rnDeriv (r • μ) =ᵐ[μ] r⁻¹ • ν.rnDeriv μ | case hf
α : Type u_1
m : MeasurableSpace α
ν μ : Measure α
inst✝¹ : IsFiniteMeasure ν
inst✝ : ν.HaveLebesgueDecomposition μ
r : ℝ≥0
hr : r ≠ 0
⊢ AEMeasurable (ν.rnDeriv (r • μ)) (r • μ) | exact (measurable_rnDeriv _ _).aemeasurable | no goals | 4ecbf63aa5ac0667 |
eq_finite_iUnion_of_isTopologicalBasis_of_isCompact_open | Mathlib/Topology/Compactness/Compact.lean | lemma eq_finite_iUnion_of_isTopologicalBasis_of_isCompact_open (b : ι → Set X)
(hb : IsTopologicalBasis (Set.range b)) (U : Set X) (hUc : IsCompact U) (hUo : IsOpen U) :
∃ s : Set ι, s.Finite ∧ U = ⋃ i ∈ s, b i | case intro.intro.intro.intro.refine_2
X : Type u
ι : Type u_1
inst✝ : TopologicalSpace X
b : ι → Set X
hb : IsTopologicalBasis (range b)
U : Set X
hUc : IsCompact U
hUo : IsOpen U
Y : Type u
f' : Y → ι
e : U = ⋃ i, (b ∘ f') i
hf' : ∀ (i : Y), b (f' i) = (b ∘ f') i
t : Finset Y
ht : U ⊆ ⋃ i ∈ t, (b ∘ f') i
⊢ ⋃ i ∈ ↑(Fin... | apply Set.iUnion₂_subset | case intro.intro.intro.intro.refine_2.h
X : Type u
ι : Type u_1
inst✝ : TopologicalSpace X
b : ι → Set X
hb : IsTopologicalBasis (range b)
U : Set X
hUc : IsCompact U
hUo : IsOpen U
Y : Type u
f' : Y → ι
e : U = ⋃ i, (b ∘ f') i
hf' : ∀ (i : Y), b (f' i) = (b ∘ f') i
t : Finset Y
ht : U ⊆ ⋃ i ∈ t, (b ∘ f') i
⊢ ∀ i ∈ ↑(F... | 0a3b3cc3a1e26713 |
bergelson' | Mathlib/MeasureTheory/Function/Intersectivity.lean | /-- **Bergelson Intersectivity Lemma**: In a finite measure space, a sequence of events that have
measure at least `r` has an infinite subset whose finite intersections all have positive volume.
TODO: The infinity of `t` should be strengthened to `t` having positive natural density. -/
lemma bergelson' {s : ℕ → Set α}... | case intro.intro.refine_2.intro
α : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
r : ℝ≥0∞
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hr₀ : r ≠ 0
hr : ∀ (n : ℕ), r ≤ μ (s n)
M : (α → ℝ) → Set α := fun f => {x | eLpNormEssSup f μ < ↑‖f x‖₊}
N : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicat... | rw [laverage_eq] at hn | case intro.intro.refine_2.intro
α : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
r : ℝ≥0∞
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hr₀ : r ≠ 0
hr : ∀ (n : ℕ), r ≤ μ (s n)
M : (α → ℝ) → Set α := fun f => {x | eLpNormEssSup f μ < ↑‖f x‖₊}
N : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicat... | 8eca484049265c21 |
Finset.divConst_le_mulConst_sq | Mathlib/Combinatorics/Additive/DoublingConst.lean | /-- If `A` has small doubling, then it has small difference, with the constant squared.
This is a consequence of the Ruzsa triangle inequality. -/
@[to_additive
"If `A` has small doubling, then it has small difference, with the constant squared.
This is a consequence of the Ruzsa triangle inequality."]
lemma divConst... | case inr
G : Type u_1
inst✝¹ : CommGroup G
inst✝ : DecidableEq G
A : Finset G
hA' : A.Nonempty
⊢ δₘ[A] * (↑(#A) * ↑(#A)) ≤ σₘ[A] ^ 2 * (↑(#A) * ↑(#A)) | calc
_ = #(A / A) * (#A : ℚ≥0) := by rw [← mul_assoc, divConst_mul_card]
_ ≤ #(A * A) * #(A * A) := by norm_cast; exact ruzsa_triangle_inequality_div_mul_mul ..
_ = _ := by rw [← mulConst_mul_card]; ring | no goals | a3c2025e31fd48a2 |
MeasureTheory.condExp_nonpos | Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean | lemma condExp_nonpos (hf : f ≤ᵐ[μ] 0) : μ[f|m] ≤ᵐ[μ] 0 | case pos
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
f : α → E
inst✝³ : NormedLatticeAddCommGroup E
inst✝² : CompleteSpace E
inst✝¹ : NormedSpace ℝ E
inst✝ : OrderedSMul ℝ E
hf : f ≤ᶠ[ae μ] 0
hfint : Integrable f μ
⊢ μ[f|m] ≤ᶠ[ae μ] μ[0|m] | exact condExp_mono hfint (integrable_zero _ _ _) hf | no goals | 33f8ebe2647b18ac |
WeierstrassCurve.Jacobian.neg_smul | Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean | lemma neg_smul (P : Fin 3 → R) (u : R) : W'.neg (u • P) = u • W'.neg P | R : Type r
inst✝ : CommRing R
W' : Jacobian R
P : Fin 3 → R
u : R
⊢ ![(u • P) x, u ^ 3 * W'.negY P, (u • P) z] = u • W'.neg P | rfl | no goals | b4ab73b934926950 |
le_limsup_mul | Mathlib/Topology/Algebra/Order/LiminfLimsup.lean | lemma le_limsup_mul (h₁ : 0 ≤ᶠ[f] u) (h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u)
(h₃ : 0 ≤ᶠ[f] v) (h₄ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f v) :
(limsup u f) * liminf v f ≤ limsup (u * v) f | ι : Type u_1
f : Filter ι
inst✝ : f.NeBot
u v : ι → ℝ
h₁ : 0 ≤ᶠ[f] u
h₂ : IsBoundedUnder (fun x1 x2 => x1 ≤ x2) f u
h₃ : 0 ≤ᶠ[f] v
h₄ : IsBoundedUnder (fun x1 x2 => x1 ≤ x2) f v
h : IsCoboundedUnder (fun x1 x2 => x1 ≤ x2) f fun x => u x * v x
h' : IsBoundedUnder (fun x1 x2 => x1 ≤ x2) f (u * v)
u0 : 0 ≤ limsup u f
⊢ li... | have uv : 0 ≤ limsup (u * v) f :=
le_limsup_of_frequently_le ((h₁.and h₃).mono fun _ ⟨hu, hv⟩ ↦ mul_nonneg hu hv).frequently h' | ι : Type u_1
f : Filter ι
inst✝ : f.NeBot
u v : ι → ℝ
h₁ : 0 ≤ᶠ[f] u
h₂ : IsBoundedUnder (fun x1 x2 => x1 ≤ x2) f u
h₃ : 0 ≤ᶠ[f] v
h₄ : IsBoundedUnder (fun x1 x2 => x1 ≤ x2) f v
h : IsCoboundedUnder (fun x1 x2 => x1 ≤ x2) f fun x => u x * v x
h' : IsBoundedUnder (fun x1 x2 => x1 ≤ x2) f (u * v)
u0 : 0 ≤ limsup u f
uv :... | 0975ba9cb811681a |
Batteries.RBNode.Balanced.append | Mathlib/.lake/packages/batteries/Batteries/Data/RBMap/WF.lean | theorem Balanced.append {l r : RBNode α}
(hl : l.Balanced c₁ n) (hr : r.Balanced c₂ n) :
(l.append r).RedRed (c₁ = black → c₂ ≠ black) n | α : Type u_1
c₁ : RBColor
n : Nat
c₂ : RBColor
x✝² x✝¹ a✝ : RBNode α
x✝ : α
b c : RBNode α
y✝ : α
d✝ : RBNode α
hl : (node black a✝ x✝ b).Balanced c₁ n
c₁✝ : RBColor
n✝ : Nat
c₂✝ : RBColor
ha : a✝.Balanced c₁✝ n✝
hb : b.Balanced c₂✝ n✝
hr : (node black c y✝ d✝).Balanced c₂ (n✝ + 1)
⊢ RedRed (black = black → c₂ ≠ black)... | have .black hc hd := hr | α : Type u_1
c₁ : RBColor
n : Nat
c₂ : RBColor
x✝² x✝¹ a✝ : RBNode α
x✝ : α
b c : RBNode α
y✝ : α
d✝ : RBNode α
hl : (node black a✝ x✝ b).Balanced c₁ n
c₁✝¹ : RBColor
n✝ : Nat
c₂✝¹ : RBColor
ha : a✝.Balanced c₁✝¹ n✝
hb : b.Balanced c₂✝¹ n✝
hr : (node black c y✝ d✝).Balanced c₂ (n✝ + 1)
c₁✝ c₂✝ : RBColor
hc : c.Balanced... | b3ecc8a07b6c9ec6 |
Metric.hausdorffDist_le_of_mem_dist | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | theorem hausdorffDist_le_of_mem_dist {r : ℝ} (hr : 0 ≤ r) (H1 : ∀ x ∈ s, ∃ y ∈ t, dist x y ≤ r)
(H2 : ∀ x ∈ t, ∃ y ∈ s, dist x y ≤ r) : hausdorffDist s t ≤ r | α : Type u
inst✝ : PseudoMetricSpace α
s t : Set α
r : ℝ
hr : 0 ≤ r
H1 : ∀ x ∈ s, ∃ y ∈ t, dist x y ≤ r
H2 : ∀ x ∈ t, ∃ y ∈ s, dist x y ≤ r
⊢ hausdorffDist s t ≤ r | apply hausdorffDist_le_of_infDist hr | case H1
α : Type u
inst✝ : PseudoMetricSpace α
s t : Set α
r : ℝ
hr : 0 ≤ r
H1 : ∀ x ∈ s, ∃ y ∈ t, dist x y ≤ r
H2 : ∀ x ∈ t, ∃ y ∈ s, dist x y ≤ r
⊢ ∀ x ∈ s, infDist x t ≤ r
case H2
α : Type u
inst✝ : PseudoMetricSpace α
s t : Set α
r : ℝ
hr : 0 ≤ r
H1 : ∀ x ∈ s, ∃ y ∈ t, dist x y ≤ r
H2 : ∀ x ∈ t, ∃ y ∈ s, dist x y ... | 628fd108fcb99ee2 |
AffineSubspace.setOf_wSameSide_eq_image2 | Mathlib/Analysis/Convex/Side.lean | theorem setOf_wSameSide_eq_image2 {s : AffineSubspace R P} {x p : P} (hx : x ∉ s) (hp : p ∈ s) :
{ y | s.WSameSide x y } = Set.image2 (fun (t : R) q => t • (x -ᵥ p) +ᵥ q) (Set.Ici 0) s | case h.mp.intro.intro.inr.inl
R : Type u_1
V : Type u_2
P : Type u_4
inst✝³ : LinearOrderedField R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x p : P
hx : x ∉ s
hp : p ∈ s
y p₂ : P
hp₂ : p₂ ∈ s
h : y = p₂
⊢ ∃ a, 0 ≤ a ∧ ∃ b ∈ ↑s, a • (x -ᵥ p) +ᵥ b = y | refine ⟨0, le_rfl, p₂, hp₂, ?_⟩ | case h.mp.intro.intro.inr.inl
R : Type u_1
V : Type u_2
P : Type u_4
inst✝³ : LinearOrderedField R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x p : P
hx : x ∉ s
hp : p ∈ s
y p₂ : P
hp₂ : p₂ ∈ s
h : y = p₂
⊢ 0 • (x -ᵥ p) +ᵥ p₂ = y | 0ddcbc2292b9cdb2 |
Module.finitePresentation_of_ker | Mathlib/Algebra/Module/FinitePresentation.lean | lemma Module.finitePresentation_of_ker [Module.FinitePresentation R N]
(l : M →ₗ[R] N) (hl : Function.Surjective l) [Module.FinitePresentation R (LinearMap.ker l)] :
Module.FinitePresentation R M | R : Type u_1
M : Type u_3
N : Type u_2
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : FinitePresentation R N
l : M →ₗ[R] N
hl : Function.Surjective ⇑l
inst✝ : FinitePresentation R ↥(LinearMap.ker l)
s : Finset M
hs : Submodule.span R ↑s = ⊤
π : ({ x // x... | rfl | no goals | 888cd3f368652bf6 |
Std.Tactic.BVDecide.BVExpr.bitblast.blastRotateLeft.go_get | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/RotateLeft.lean | theorem go_get (aig : AIG α) (distance : Nat) (input : AIG.RefVec aig w)
(curr : Nat) (hcurr : curr ≤ w) (s : AIG.RefVec aig curr) :
∀ (idx : Nat) (hidx1 : idx < w),
curr ≤ idx
→
(go input distance curr hcurr s).get idx hidx1
=
if hidx3 : idx < distance % w then
... | case isTrue.inl.isTrue.isTrue
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
distance : Nat
input : aig.RefVec w
curr : Nat
hcurr : curr ≤ w
s : aig.RefVec curr
idx : Nat
hidx1 : idx < w
hidx2 : curr ≤ idx
h✝² : curr < w
heq : curr = idx
h✝¹ : curr < distance % w
h✝ : idx < distance % w
⊢ input.... | simp [heq] | no goals | 51e04392aa60443b |
List.Duplicate.of_duplicate_cons | Mathlib/Data/List/Duplicate.lean | theorem Duplicate.of_duplicate_cons {y : α} (h : x ∈+ y :: l) (hx : x ≠ y) : x ∈+ l | α : Type u_1
l : List α
x y : α
h : x ∈+ y :: l
hx : x ≠ y
⊢ x ∈+ l | simpa [duplicate_cons_iff, hx.symm] using h | no goals | 34e682ff54b9e8af |
FDerivMeasurableAux.norm_sub_le_of_mem_A | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | theorem norm_sub_le_of_mem_A {c : 𝕜} (hc : 1 < ‖c‖) {r ε : ℝ} (hε : 0 < ε) (hr : 0 < r) {x : E}
{L₁ L₂ : E →L[𝕜] F} (h₁ : x ∈ A f L₁ r ε) (h₂ : x ∈ A f L₂ r ε) : ‖L₁ - L₂‖ ≤ 4 * ‖c‖ * ε | 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
c : 𝕜
hc : 1 < ‖c‖
r ε : ℝ
hε : 0 < ε
hr : 0 < r
x : E
L₁ L₂ : E →L[𝕜] F
h₁ : x ∈ A f L₁ r ε
h₂ : x ∈ A f L₂ r ε
y : E
le... | rw [div_div, div_le_iff₀' (mul_pos (by norm_num : (0 : ℝ) < 2) (zero_lt_one.trans hc))] at ley | 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
c : 𝕜
hc : 1 < ‖c‖
r ε : ℝ
hε : 0 < ε
hr : 0 < r
x : E
L₁ L₂ : E →L[𝕜] F
h₁ : x ∈ A f L₁ r ε
h₂ : x ∈ A f L₂ r ε
y : E
le... | 2ee71d508cf2305e |
smul_algebraMap | Mathlib/Algebra/Algebra/Defs.lean | theorem _root_.smul_algebraMap {α : Type*} [Monoid α] [MulDistribMulAction α A]
[SMulCommClass α R A] (a : α) (r : R) : a • algebraMap R A r = algebraMap R A r | R : Type u
A : Type w
inst✝⁵ : CommSemiring R
inst✝⁴ : Semiring A
inst✝³ : Algebra R A
α : Type u_2
inst✝² : Monoid α
inst✝¹ : MulDistribMulAction α A
inst✝ : SMulCommClass α R A
a : α
r : R
⊢ a • (algebraMap R A) r = (algebraMap R A) r | rw [algebraMap_eq_smul_one, smul_comm a r (1 : A), smul_one] | no goals | 4986d6937124a688 |
CategoryTheory.Presheaf.isLocallySurjective_comp_iff | Mathlib/CategoryTheory/Sites/LocallySurjective.lean | lemma isLocallySurjective_comp_iff
{F₁ F₂ F₃ : Cᵒᵖ ⥤ A} (f₁ : F₁ ⟶ F₂) (f₂ : F₂ ⟶ F₃)
[IsLocallyInjective J f₂] [IsLocallySurjective J f₂] :
IsLocallySurjective J (f₁ ≫ f₂) ↔ IsLocallySurjective J f₁ | C : Type u
inst✝⁵ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁴ : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w'
inst✝³ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝² : ConcreteCategory A FA
F₁ F₂ F₃ : Cᵒᵖ ⥤ A
f₁ : F₁ ⟶ F₂
f₂ : F₂ ⟶ F₃
inst✝¹ : IsLocallyInjective J f₂
inst✝ : IsLoca... | constructor | case mp
C : Type u
inst✝⁵ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁴ : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w'
inst✝³ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝² : ConcreteCategory A FA
F₁ F₂ F₃ : Cᵒᵖ ⥤ A
f₁ : F₁ ⟶ F₂
f₂ : F₂ ⟶ F₃
inst✝¹ : IsLocallyInjective J f₂
inst✝ ... | 00b0636da0c707ef |
DFinsupp.single_smul | Mathlib/Data/DFinsupp/Module.lean | theorem single_smul {i : ι} (c : γ) (x : β i) : single i (c • x) = c • single i x :=
ext fun i => by
simp only [smul_apply, single_apply]
split_ifs with h
· cases h; rfl
· rw [smul_zero]
| case pos.refl
ι : Type u
γ : Type w
β : ι → Type v
inst✝³ : DecidableEq ι
inst✝² : Monoid γ
inst✝¹ : (i : ι) → AddMonoid (β i)
inst✝ : (i : ι) → DistribMulAction γ (β i)
i : ι
c : γ
x : β i
⊢ ⋯ ▸ (c • x) = c • ⋯ ▸ x | rfl | no goals | bd1a7af5b70948b5 |
Nat.mul_add_mod | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean | theorem mul_add_mod (m x y : Nat) : (m * x + y) % m = y % m | m x✝ y x : Nat
⊢ (m * (x + 1) + y) % m = y % m | simp [Nat.mul_succ, Nat.add_assoc _ m, mul_add_mod _ x] | no goals | 05002e8ccd98db69 |
GaussianFourier.integrable_cexp_neg_mul_sq_norm_add_of_euclideanSpace | Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean | theorem integrable_cexp_neg_mul_sq_norm_add_of_euclideanSpace
{ι : Type*} [Fintype ι] (hb : 0 < b.re) (c : ℂ) (w : EuclideanSpace ℝ ι) :
Integrable (fun (v : EuclideanSpace ℝ ι) ↦ cexp (- b * ‖v‖^2 + c * ⟪w, v⟫)) | b : ℂ
ι : Type u_2
inst✝ : Fintype ι
hb : 0 < b.re
c : ℂ
w : EuclideanSpace ℝ ι
this : MeasurePreserving (⇑(EuclideanSpace.measurableEquiv ι)) volume volume
⊢ Integrable (fun v => cexp (-b * ↑‖v‖ ^ 2 + c * ↑(inner w v))) volume | rw [← MeasurePreserving.integrable_comp_emb this.symm (MeasurableEquiv.measurableEmbedding _)] | b : ℂ
ι : Type u_2
inst✝ : Fintype ι
hb : 0 < b.re
c : ℂ
w : EuclideanSpace ℝ ι
this : MeasurePreserving (⇑(EuclideanSpace.measurableEquiv ι)) volume volume
⊢ Integrable ((fun v => cexp (-b * ↑‖v‖ ^ 2 + c * ↑(inner w v))) ∘ ⇑(EuclideanSpace.measurableEquiv ι).symm) volume | 6a1e7e95e621a0ac |
List.nodup_append | Mathlib/Data/List/Nodup.lean | theorem nodup_append {l₁ l₂ : List α} :
Nodup (l₁ ++ l₂) ↔ Nodup l₁ ∧ Nodup l₂ ∧ Disjoint l₁ l₂ | α : Type u
l₁ l₂ : List α
⊢ (l₁ ++ l₂).Nodup ↔ l₁.Nodup ∧ l₂.Nodup ∧ l₁.Disjoint l₂ | simp only [Nodup, pairwise_append, disjoint_iff_ne] | no goals | 763ad2aefa23a0b8 |
Nat.totient_gcd_mul_totient_mul | Mathlib/Data/Nat/Totient.lean | theorem totient_gcd_mul_totient_mul (a b : ℕ) : φ (a.gcd b) * φ (a * b) = φ a * φ b * a.gcd b | a b : ℕ
⊢ ∀ (a1 a2 b1 b2 c1 c2 : ℕ), b1 ∣ a1 → b2 ∣ a2 → a1 / b1 * c1 * (a2 / b2 * c2) = a1 * a2 / (b1 * b2) * (c1 * c2) | intro a1 a2 b1 b2 c1 c2 h1 h2 | a b a1 a2 b1 b2 c1 c2 : ℕ
h1 : b1 ∣ a1
h2 : b2 ∣ a2
⊢ a1 / b1 * c1 * (a2 / b2 * c2) = a1 * a2 / (b1 * b2) * (c1 * c2) | 35f1994c784b46d9 |
ContinuousLinearMap.exists_approx_preimage_norm_le | Mathlib/Analysis/Normed/Operator/Banach.lean | theorem exists_approx_preimage_norm_le (surj : Surjective f) :
∃ C ≥ 0, ∀ y, ∃ x, dist (f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ C * ‖y‖ | 𝕜 : Type u_1
𝕜' : Type u_2
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NontriviallyNormedField 𝕜'
σ : 𝕜 →+* 𝕜'
E : Type u_3
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜' F
f : E →SL[σ] F
σ' : 𝕜' →+* 𝕜
inst✝³ : RingHomInvPair σ σ'
inst... | gcongr | no goals | a856ccf4b69cb2d9 |
UniqueMul.iff_card_le_one | Mathlib/Algebra/Group/UniqueProds/Basic.lean | theorem iff_card_le_one [DecidableEq G] (ha0 : a0 ∈ A) (hb0 : b0 ∈ B) :
UniqueMul A B a0 b0 ↔ #{p ∈ A ×ˢ B | p.1 * p.2 = a0 * b0} ≤ 1 | case refine_1
G : Type u_1
inst✝¹ : Mul G
A B : Finset G
a0 b0 : G
inst✝ : DecidableEq G
ha0 : a0 ∈ A
hb0 : b0 ∈ B
h : UniqueMul A B a0 b0
p1 p2 : G × G
x✝¹ : (p1.1 ∈ A ∧ p1.2 ∈ B) ∧ p1.1 * p1.2 = a0 * b0
x✝ : (p2.1 ∈ A ∧ p2.2 ∈ B) ∧ p2.1 * p2.2 = a0 * b0
ha1 : p1.1 ∈ A
hb1 : p1.2 ∈ B
he1 : p1.1 * p1.2 = a0 * b0
ha2 : ... | have h1 := h ha1 hb1 he1 | case refine_1
G : Type u_1
inst✝¹ : Mul G
A B : Finset G
a0 b0 : G
inst✝ : DecidableEq G
ha0 : a0 ∈ A
hb0 : b0 ∈ B
h : UniqueMul A B a0 b0
p1 p2 : G × G
x✝¹ : (p1.1 ∈ A ∧ p1.2 ∈ B) ∧ p1.1 * p1.2 = a0 * b0
x✝ : (p2.1 ∈ A ∧ p2.2 ∈ B) ∧ p2.1 * p2.2 = a0 * b0
ha1 : p1.1 ∈ A
hb1 : p1.2 ∈ B
he1 : p1.1 * p1.2 = a0 * b0
ha2 : ... | 1934bc605be2ea62 |
AlgebraicGeometry.IsProper.eq_valuativeCriterion | Mathlib/AlgebraicGeometry/ValuativeCriterion.lean | /-- The **valuative criterion** for proper morphisms. -/
@[stacks 0BX5]
lemma IsProper.eq_valuativeCriterion :
@IsProper = ValuativeCriterion ⊓ @QuasiCompact ⊓ @QuasiSeparated ⊓ @LocallyOfFiniteType | ⊢ ValuativeCriterion.Uniqueness ⊓
(@QuasiSeparated ⊓ (ValuativeCriterion.Existence ⊓ (@QuasiCompact ⊓ @LocallyOfFiniteType))) =
ValuativeCriterion.Existence ⊓
(ValuativeCriterion.Uniqueness ⊓ (@QuasiCompact ⊓ (@QuasiSeparated ⊓ @LocallyOfFiniteType))) | ext X Y f | case h.h.h.a
X Y : Scheme
f : X ⟶ Y
⊢ (ValuativeCriterion.Uniqueness ⊓
(@QuasiSeparated ⊓ (ValuativeCriterion.Existence ⊓ (@QuasiCompact ⊓ @LocallyOfFiniteType))))
f ↔
(ValuativeCriterion.Existence ⊓
(ValuativeCriterion.Uniqueness ⊓ (@QuasiCompact ⊓ (@QuasiSeparated ⊓ @LocallyOfFiniteType))))
... | 4b305d897fe3df6a |
Submodule.map_unop_one | Mathlib/Algebra/Algebra/Operations.lean | theorem map_unop_one :
map (↑(opLinearEquiv R : A ≃ₗ[R] Aᵐᵒᵖ).symm : Aᵐᵒᵖ →ₗ[R] A) (1 : Submodule R Aᵐᵒᵖ) = 1 | R : Type u
inst✝² : CommSemiring R
A : Type v
inst✝¹ : Semiring A
inst✝ : Algebra R A
⊢ map (↑(opLinearEquiv R).symm) 1 = 1 | rw [← comap_equiv_eq_map_symm, comap_op_one] | no goals | cb5f0f564660db12 |
LinearMap.exists_map_addHaar_eq_smul_addHaar' | Mathlib/MeasureTheory/Measure/Haar/Disintegration.lean | theorem LinearMap.exists_map_addHaar_eq_smul_addHaar' (h : Function.Surjective L) :
∃ (c : ℝ≥0∞), 0 < c ∧ c < ∞ ∧ μ.map L = (c * addHaar (univ : Set (LinearMap.ker L))) • ν | case intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace... | let μS : Measure S := addHaar | case intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : CompleteSpace 𝕜
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : MeasurableSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : BorelSpace F
inst✝³ : NormedSpace... | 2e191355893f8fd9 |
StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart_source_aux | Mathlib/Geometry/Manifold/LocalInvariantProperties.lean | theorem liftPropWithinAt_indep_chart_source_aux (g : M → H') (he : e ∈ G.maximalAtlas M)
(xe : x ∈ e.source) (he' : e' ∈ G.maximalAtlas M) (xe' : x ∈ e'.source) :
P (g ∘ e.symm) (e.symm ⁻¹' s) (e x) ↔ P (g ∘ e'.symm) (e'.symm ⁻¹' s) (e' x) | case refine_1
H : Type u_1
M : Type u_2
H' : Type u_3
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : TopologicalSpace H'
G : StructureGroupoid H
G' : StructureGroupoid H'
e e' : PartialHomeomorph M H
P : (H → H') → Set H → H → Prop
s : Set M
x : M
hG : G.LocalInvariantProp G' ... | simp_rw [e'.left_inv xe', xe] | no goals | 545faefef09162d9 |
AlgebraicGeometry.LocallyRingedSpace.isLocalHom_stalkMap_congr | Mathlib/Geometry/RingedSpace/LocallyRingedSpace/HasColimits.lean | theorem isLocalHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalHom (f.stalkMap x).hom) :
IsLocalHom (g.stalkMap x).hom | X Y : RingedSpace
f g : X ⟶ Y
H : f = g
x : ↑↑X.toPresheafedSpace
h : IsLocalHom (CommRingCat.Hom.hom (PresheafedSpace.Hom.stalkMap f x))
⊢ IsLocalHom (CommRingCat.Hom.hom (eqToHom ⋯ ≫ PresheafedSpace.Hom.stalkMap f x)) | infer_instance | no goals | 6280e2a39847d6a8 |
MeasureTheory.Measure.setIntegral_toReal_rnDeriv_le | Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean | lemma setIntegral_toReal_rnDeriv_le [SigmaFinite μ] {s : Set α} (hμs : μ s ≠ ∞) :
∫ x in s, (μ.rnDeriv ν x).toReal ∂ν ≤ (μ s).toReal | α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝ : SigmaFinite μ
s : Set α
hμs : μ s ≠ ⊤
t : Set α := toMeasurable μ s
ht : t = toMeasurable μ s
ht_m : MeasurableSet t
hμt : μ t ≠ ⊤
⊢ ((ν.withDensity (μ.rnDeriv ν)) t).toReal ≤ (μ t).toReal | gcongr | case hb
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝ : SigmaFinite μ
s : Set α
hμs : μ s ≠ ⊤
t : Set α := toMeasurable μ s
ht : t = toMeasurable μ s
ht_m : MeasurableSet t
hμt : μ t ≠ ⊤
⊢ μ t ≠ ⊤
case h
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝ : SigmaFinite μ
s : Set α
hμs : μ s ≠ ⊤
t : Set... | c48b568e2fc34da6 |
Real.fourierIntegral_deriv | Mathlib/Analysis/Fourier/FourierTransformDeriv.lean | theorem fourierIntegral_deriv
{f : ℝ → E} (hf : Integrable f) (h'f : Differentiable ℝ f) (hf' : Integrable (deriv f)) :
𝓕 (deriv f) = fun (x : ℝ) ↦ (2 * π * I * x) • (𝓕 f x) | E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
hf : Integrable f volume
h'f : Differentiable ℝ f
hf' : Integrable (deriv f) volume
⊢ 𝓕 (deriv f) = fun x => (2 * ↑π * I * ↑x) • 𝓕 f x | ext x | case h
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
hf : Integrable f volume
h'f : Differentiable ℝ f
hf' : Integrable (deriv f) volume
x : ℝ
⊢ 𝓕 (deriv f) x = (2 * ↑π * I * ↑x) • 𝓕 f x | 2342c63a4643b700 |
TopCat.isTopologicalBasis_cofiltered_limit | Mathlib/Topology/Category/TopCat/Limits/Cofiltered.lean | theorem isTopologicalBasis_cofiltered_limit (hC : IsLimit C) (T : ∀ j, Set (Set (F.obj j)))
(hT : ∀ j, IsTopologicalBasis (T j)) (univ : ∀ i : J, Set.univ ∈ T i)
(inter : ∀ (i) (U1 U2 : Set (F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i)
(compat : ∀ (i j : J) (f : i ⟶ j) (V : Set (F.obj j)) (_hV : V ∈ T j)... | case empty
J : Type v
inst✝¹ : Category.{w, v} J
inst✝ : IsCofiltered J
F : J ⥤ TopCat
C : Cone F
hC : IsLimit C
T : (j : J) → Set (Set ↑(F.obj j))
hT : ∀ (j : J), IsTopologicalBasis (T j)
univ : ∀ (i : J), Set.univ ∈ T i
inter : ∀ (i : J) (U1 U2 : Set ↑(F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i
compat : ∀ (i j : ... | simpa | no goals | 8a9de6a458f61cf0 |
inf_symmDiff_distrib_right | Mathlib/Order/SymmDiff.lean | theorem inf_symmDiff_distrib_right : a ∆ b ⊓ c = (a ⊓ c) ∆ (b ⊓ c) | α : Type u_2
inst✝ : GeneralizedBooleanAlgebra α
a b c : α
⊢ a ∆ b ⊓ c = (a ⊓ c) ∆ (b ⊓ c) | simp_rw [inf_comm _ c, inf_symmDiff_distrib_left] | no goals | 9ed666c0810beb95 |
Finset.card_biUnion_le_of_intersecting | Mathlib/Combinatorics/SetFamily/Kleitman.lean | theorem Finset.card_biUnion_le_of_intersecting (s : Finset ι) (f : ι → Finset (Finset α))
(hf : ∀ i ∈ s, (f i : Set (Finset α)).Intersecting) :
#(s.biUnion f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - #s) | case inr.cons.refine_1
ι : Type u_1
α : Type u_2
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : Nonempty α
this : DecidableEq ι
i : ι
s : Finset ι
hi : i ∉ s
ih :
∀ (f : ι → Finset (Finset α)),
(∀ i ∈ s, (↑(f i)).Intersecting) →
#s ≤ Fintype.card α → #(s.biUnion f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.car... | exact isUpperSet_iUnion₂ fun i hi ↦ hf₂ _ <| subset_cons _ hi | no goals | e2f32d8e4f6db312 |
FreeGroup.Red.Step.cons_left_iff | Mathlib/GroupTheory/FreeGroup/Basic.lean | theorem Step.cons_left_iff {a : α} {b : Bool} :
Step ((a, b) :: L₁) L₂ ↔ (∃ L, Step L₁ L ∧ L₂ = (a, b) :: L) ∨ L₁ = (a, ! b) :: L₂ | case mp.not.cons.intro
α : Type u
a : α
b : Bool
e : List (α × Bool)
a' : α
b' : Bool
s' : List (α × Bool)
⊢ (∃ L, Step (s' ++ (a', b') :: (a', !b') :: e) L ∧ (a, b) :: s' ++ e = (a, b) :: L) ∨
s' ++ (a', b') :: (a', !b') :: e = (a, !b) :: ((a, b) :: s' ++ e) | refine Or.inl ⟨s' ++ e, Step.not, ?_⟩ | case mp.not.cons.intro
α : Type u
a : α
b : Bool
e : List (α × Bool)
a' : α
b' : Bool
s' : List (α × Bool)
⊢ (a, b) :: s' ++ e = (a, b) :: (s' ++ e) | c671293c37f8dfaf |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.