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 |
|---|---|---|---|---|---|---|
Submodule.mem_span_mul_finite_of_mem_span_mul | Mathlib/Algebra/Algebra/Operations.lean | theorem mem_span_mul_finite_of_mem_span_mul {R A} [Semiring R] [AddCommMonoid A] [Mul A]
[Module R A] {S : Set A} {S' : Set A} {x : A} (hx : x ∈ span R (S * S')) :
∃ T T' : Finset A, ↑T ⊆ S ∧ ↑T' ⊆ S' ∧ x ∈ span R (T * T' : Set A) | case intro.intro
R : Type u_1
A : Type u_2
inst✝³ : Semiring R
inst✝² : AddCommMonoid A
inst✝¹ : Mul A
inst✝ : Module R A
S S' : Set A
x : A
hx : x ∈ span R (S * S')
U : Finset A
h : ↑U ⊆ S * S'
hU : x ∈ span R ↑U
⊢ ∃ T T', ↑T ⊆ S ∧ ↑T' ⊆ S' ∧ x ∈ span R (↑T * ↑T') | obtain ⟨T, T', hS, hS', h⟩ := Finset.subset_mul h | case intro.intro.intro.intro.intro.intro
R : Type u_1
A : Type u_2
inst✝³ : Semiring R
inst✝² : AddCommMonoid A
inst✝¹ : Mul A
inst✝ : Module R A
S S' : Set A
x : A
hx : x ∈ span R (S * S')
U : Finset A
h✝ : ↑U ⊆ S * S'
hU : x ∈ span R ↑U
T T' : Finset A
hS : ↑T ⊆ S
hS' : ↑T' ⊆ S'
h : U ⊆ T * T'
⊢ ∃ T T', ↑T ⊆ S ∧ ↑T' ... | 5f035c580e6603c0 |
CauSeq.trichotomy | Mathlib/Algebra/Order/CauSeq/Basic.lean | theorem trichotomy (f : CauSeq α abs) : Pos f ∨ LimZero f ∨ Pos (-f) | case inr.intro.intro.intro.refine_1.intro
α : Type u_1
inst✝ : LinearOrderedField α
f : CauSeq α abs
h✝ : ¬f.LimZero
K : α
K0 : K > 0
hK : ∃ i, ∀ j ≥ i, K ≤ |↑f j|
i : ℕ
hi : ∀ j ≥ i, K ≤ |↑f j| ∧ ∀ k ≥ j, |↑f k - ↑f j| < K
h : 0 ≤ ↑f i
j : ℕ
ij : j ≥ i
this : K ≤ |↑f j|
h₁ : K ≤ ↑f i
h₂ : ∀ k ≥ i, |↑f k - ↑f i| < K
⊢ ... | exact
(le_add_iff_nonneg_right _).1
(le_trans h₁ <| neg_le_sub_iff_le_add'.1 <| le_of_lt (abs_lt.1 <| h₂ _ ij).1) | no goals | b071776ee71c7fea |
TopCat.Presheaf.SheafConditionEqualizerProducts.w | Mathlib/Topology/Sheaves/SheafCondition/EqualizerProducts.lean | theorem w : res F U ≫ leftRes F U = res F U ≫ rightRes F U | C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
x✝ : Discrete (ι × ι)
⊢ F.map (leSupr U x✝.as.1).op ≫ F.map ((U x✝.as.1).infLELeft (U x✝.as.2)).op =
F.map (leSupr U x✝.as.2).op ≫ F.map ((U x✝.as.1).infLERight (U x✝.as.2)).op | rw [← F.map_comp] | C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
x✝ : Discrete (ι × ι)
⊢ F.map ((leSupr U x✝.as.1).op ≫ ((U x✝.as.1).infLELeft (U x✝.as.2)).op) =
F.map (leSupr U x✝.as.2).op ≫ F.map ((U x✝.as.1).infLERight (U x✝.as.2)).op | 54ddd6c29dd19829 |
List.filterMap_eq_append_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean | theorem filterMap_eq_append_iff {f : α → Option β} :
filterMap f l = L₁ ++ L₂ ↔ ∃ l₁ l₂, l = l₁ ++ l₂ ∧ filterMap f l₁ = L₁ ∧ filterMap f l₂ = L₂ | case mp.cons.h_2
α : Type u_1
β : Type u_2
L₂ : List β
f : α → Option β
x : α
l : List α
ih : ∀ {L₁ : List β}, filterMap f l = L₁ ++ L₂ → ∃ l₁ l₂, l = l₁ ++ l₂ ∧ filterMap f l₁ = L₁ ∧ filterMap f l₂ = L₂
L₁ : List β
x✝ : Option β
b✝ : β
heq✝ : f x = some b✝
⊢ b✝ :: filterMap f l = L₁ ++ L₂ → ∃ l₁ l₂, x :: l = l₁ ++ l₂ ... | rename_i b w | case mp.cons.h_2
α : Type u_1
β : Type u_2
L₂ : List β
f : α → Option β
x : α
l : List α
ih : ∀ {L₁ : List β}, filterMap f l = L₁ ++ L₂ → ∃ l₁ l₂, l = l₁ ++ l₂ ∧ filterMap f l₁ = L₁ ∧ filterMap f l₂ = L₂
L₁ : List β
x✝ : Option β
b : β
w : f x = some b
⊢ b :: filterMap f l = L₁ ++ L₂ → ∃ l₁ l₂, x :: l = l₁ ++ l₂ ∧ filt... | e698eece5419b8d8 |
Matroid.IsCircuit.inter_isCocircuit_ne_singleton | Mathlib/Data/Matroid/Circuit.lean | /-- A cocircuit and a circuit cannot meet in exactly one element. -/
lemma IsCircuit.inter_isCocircuit_ne_singleton (hC : M.IsCircuit C) (hK : M.IsCocircuit K) :
C ∩ K ≠ {e} | α : Type u_1
M : Matroid α
C : Set α
e : α
K : Set α
hC : M.IsCircuit C
he : C ∩ K = {e}
heC : e ∈ C
hK : ¬M.Spanning (M.E \ K) ∧ ∀ ⦃t : Set α⦄, t ⊂ K → M.Spanning (M.E \ t)
hKe : M.Spanning (M.E \ (K \ {e}))
⊢ M.closure (M.E \ K ∪ {e}) ⊆ M.closure (M.E \ K) | refine (M.closure_subset_closure (subset_union_left (t := C))).trans ?_ | α : Type u_1
M : Matroid α
C : Set α
e : α
K : Set α
hC : M.IsCircuit C
he : C ∩ K = {e}
heC : e ∈ C
hK : ¬M.Spanning (M.E \ K) ∧ ∀ ⦃t : Set α⦄, t ⊂ K → M.Spanning (M.E \ t)
hKe : M.Spanning (M.E \ (K \ {e}))
⊢ M.closure (M.E \ K ∪ {e} ∪ C) ⊆ M.closure (M.E \ K) | 9b34fb8eaea54357 |
isClosedMap_inr | Mathlib/Topology/Constructions.lean | theorem isClosedMap_inr : IsClosedMap (@inr X Y) := fun u hu ↦ by
simpa [isClosed_sum_iff, preimage_image_eq u Sum.inr_injective]
| X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
u : Set Y
hu : IsClosed u
⊢ IsClosed (inr '' u) | simpa [isClosed_sum_iff, preimage_image_eq u Sum.inr_injective] | no goals | 279e7b6759dca07a |
MeasureTheory.Measure.MutuallySingular.compProd_of_right | Mathlib/Probability/Kernel/Composition/AbsolutelyContinuous.lean | lemma MutuallySingular.compProd_of_right (μ ν : Measure α) (hκη : ∀ᵐ a ∂μ, κ a ⟂ₘ η a) :
μ ⊗ₘ κ ⟂ₘ ν ⊗ₘ η | case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ η : Kernel α β
inst✝² : IsFiniteKernel κ
inst✝¹ : IsFiniteKernel η
inst✝ : MeasurableSpace.CountableOrCountablyGenerated α β
μ ν : Measure α
hκη : ∀ᵐ (a : α) ∂μ, κ a ⟂ₘ η a
hμ : SFinite μ
hν : SFinite ν
s : Set (α × β) := κ.mutuallySingu... | have h1 a : η a (Prod.mk a ⁻¹' s) = 0 := by rw [h_eq, Kernel.measure_mutuallySingularSetSlice] | case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ η : Kernel α β
inst✝² : IsFiniteKernel κ
inst✝¹ : IsFiniteKernel η
inst✝ : MeasurableSpace.CountableOrCountablyGenerated α β
μ ν : Measure α
hκη : ∀ᵐ (a : α) ∂μ, κ a ⟂ₘ η a
hμ : SFinite μ
hν : SFinite ν
s : Set (α × β) := κ.mutuallySingu... | 6b225dc6dedbf7b5 |
CategoryTheory.Limits.Types.surjective_π_app_zero_of_surjective_map_aux | Mathlib/CategoryTheory/Limits/Types.lean | /-- Auxiliary lemma. Use `limit_of_surjections_surjective` instead. -/
lemma surjective_π_app_zero_of_surjective_map_aux :
Function.Surjective ((limitCone F).π.app ⟨0⟩) | F : ℕᵒᵖ ⥤ Type u
hF : ∀ (n : ℕ), Function.Surjective (F.map (homOfLE ⋯).op)
a : F.obj (Opposite.op 0)
n m : ℕ
h : m ≤ n
⊢ F.map (Opposite.op { down := { down := h } })
((fun x =>
match x with
| Opposite.op n => CategoryTheory.Limits.Types.limitOfSurjectionsSurjective.preimage hF a n)
(... | induction h with
| refl =>
erw [CategoryTheory.Functor.map_id, types_id_apply]
| @step p h ih =>
rw [← ih]
have h' : m ≤ p := h
erw [CategoryTheory.Functor.map_comp (f := (homOfLE (Nat.le_succ p)).op) (g := (homOfLE h').op),
types_comp_apply, (hF p _).choose_spec]
rfl | no goals | 18a8c352cca791cc |
MeasureTheory.exists_subordinate_pairwise_disjoint | Mathlib/MeasureTheory/Measure/NullMeasurable.lean | theorem exists_subordinate_pairwise_disjoint [Countable ι] {s : ι → Set α}
(h : ∀ i, NullMeasurableSet (s i) μ) (hd : Pairwise (AEDisjoint μ on s)) :
∃ t : ι → Set α,
(∀ i, t i ⊆ s i) ∧
(∀ i, s i =ᵐ[μ] t i) ∧ (∀ i, MeasurableSet (t i)) ∧ Pairwise (Disjoint on t) | case intro.intro.intro
ι : Type u_1
α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst✝ : Countable ι
s : ι → Set α
h : ∀ (i : ι), NullMeasurableSet (s i) μ
hd : Pairwise (AEDisjoint μ on s)
t : ι → Set α
ht_sub : ∀ (i : ι), t i ⊆ s i
htm : ∀ (i : ι), MeasurableSet (t i)
ht_eq : ∀ (i : ι), t i =ᶠ[ae μ] s i
u : ι → ... | exact
⟨fun i => t i \ u i, fun i => diff_subset.trans (ht_sub _), fun i =>
(ht_eq _).symm.trans (diff_null_ae_eq_self (hu₀ i)).symm, fun i => (htm i).diff (hum i),
hud.mono fun i j h =>
h.mono (diff_subset_diff_left (ht_sub i)) (diff_subset_diff_left (ht_sub j))⟩ | no goals | 6122c5c5890a3735 |
Complex.cos_ne_zero_of_arctan_bounds | Mathlib/Analysis/SpecialFunctions/Complex/Arctan.lean | /-- `cos z` is nonzero when the bounds in `arctan_tan` are met (`z` lies in the vertical strip
`-π / 2 < z.re < π / 2` and `z ≠ π / 2`). -/
lemma cos_ne_zero_of_arctan_bounds {z : ℂ} (h₀ : z ≠ π / 2) (h₁ : -(π / 2) < z.re)
(h₂ : z.re ≤ π / 2) : cos z ≠ 0 | case inl.h
z : ℂ
h₁ : -(π / 2) < z.re
h₂ : z.re ≤ π / 2
k : ℤ
nr : ¬z.re = π / 2
⊢ ¬z.re = ↑(2 * k + 1) * π / 2 | contrapose! nr | case inl.h
z : ℂ
h₁ : -(π / 2) < z.re
h₂ : z.re ≤ π / 2
k : ℤ
nr : z.re = ↑(2 * k + 1) * π / 2
⊢ z.re = π / 2 | b518e846241a8df1 |
CategoryTheory.Square.IsPullback.iff_of_equiv | Mathlib/CategoryTheory/Limits/Preserves/Shapes/Square.lean | lemma IsPullback.iff_of_equiv : sq₁.IsPullback ↔ sq₂.IsPullback | case refine_4.h
sq₁ : Square (Type v)
sq₂ : Square (Type u)
e₁ : sq₁.X₁ ≃ sq₂.X₁
e₂ : sq₁.X₂ ≃ sq₂.X₂
e₃ : sq₁.X₃ ≃ sq₂.X₃
e₄ : sq₁.X₄ ≃ sq₂.X₄
comm₁₂ : ⇑e₂ ∘ sq₁.f₁₂ = sq₂.f₁₂ ∘ ⇑e₁
comm₁₃ : ⇑e₃ ∘ sq₁.f₁₃ = sq₂.f₁₃ ∘ ⇑e₁
comm₂₄ : ⇑e₄ ∘ sq₁.f₂₄ = sq₂.f₂₄ ∘ ⇑e₂
comm₃₄ : ⇑e₄ ∘ sq₁.f₃₄ = sq₂.f₃₄ ∘ ⇑e₃
a✝ : (sq₁.map uliftF... | apply ULift.down_injective | case refine_4.h.a
sq₁ : Square (Type v)
sq₂ : Square (Type u)
e₁ : sq₁.X₁ ≃ sq₂.X₁
e₂ : sq₁.X₂ ≃ sq₂.X₂
e₃ : sq₁.X₃ ≃ sq₂.X₃
e₄ : sq₁.X₄ ≃ sq₂.X₄
comm₁₂ : ⇑e₂ ∘ sq₁.f₁₂ = sq₂.f₁₂ ∘ ⇑e₁
comm₁₃ : ⇑e₃ ∘ sq₁.f₁₃ = sq₂.f₁₃ ∘ ⇑e₁
comm₂₄ : ⇑e₄ ∘ sq₁.f₂₄ = sq₂.f₂₄ ∘ ⇑e₂
comm₃₄ : ⇑e₄ ∘ sq₁.f₃₄ = sq₂.f₃₄ ∘ ⇑e₃
a✝ : (sq₁.map ulif... | 9c84a54ae8a3d82b |
MeasureTheory.Measure.IsEverywherePos.IsGdelta_of_isMulLeftInvariant | Mathlib/MeasureTheory/Measure/EverywherePos.lean | /-- If a compact closed set is everywhere positive with respect to a left-invariant measure on a
topological group, then it is a Gδ set. This is nontrivial, as there is no second-countability or
metrizability assumption in the statement, so a general compact closed set has no reason to be
a countable intersection of op... | case intro.intro
G : Type u_2
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : LocallyCompactSpace G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : μ.InnerRegularCompactLTTop
k : Set G
h : μ.IsE... | have I : V i ⊆ W n := iInter₂_subset n (by simp [ni]) | case intro.intro
G : Type u_2
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : LocallyCompactSpace G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : μ.InnerRegularCompactLTTop
k : Set G
h : μ.IsE... | 5dba91f75a485a07 |
cfc_zero | Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Unital.lean | @[simp]
lemma cfc_zero : cfc (0 : R → R) a = 0 | case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : MetricSpace R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : ContinuousStar R
inst✝³ : TopologicalSpace A
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R p
a : A
ha : p a
⊢ cf... | exact cfc_apply (0 : R → R) a ▸ map_zero (cfcHom ha) | no goals | 3185221de0d73ef9 |
Stream'.Seq.coinduction2 | Mathlib/Data/Seq/Seq.lean | theorem coinduction2 (s) (f g : Seq α → Seq β)
(H :
∀ s,
BisimO (fun s1 s2 : Seq β => ∃ s : Seq α, s1 = f s ∧ s2 = g s) (destruct (f s))
(destruct (g s))) :
f s = g s | case intro.intro
α : Type u
β : Type v
s✝ : Seq α
f g : Seq α → Seq β
H : ∀ (s : Seq α), BisimO (fun s1 s2 => ∃ s, s1 = f s ∧ s2 = g s) (f s).destruct (g s).destruct
s1 s2 : Seq β
s : Seq α
h1 : s1 = f s
h2 : s2 = g s
⊢ BisimO (fun s1 s2 => ∃ s, s1 = f s ∧ s2 = g s) (f s).destruct (g s).destruct | apply H | no goals | b52202dbb190e447 |
Fin.ofNat'_self | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean | theorem ofNat'_self {n : Nat} [NeZero n] : Fin.ofNat' n n = 0 | case h
n : Nat
inst✝ : NeZero n
⊢ 0 = ↑0 | congr | no goals | 9b75a3e9808a3817 |
CategoryTheory.Limits.PreservesPullback.iso_hom_snd | Mathlib/CategoryTheory/Limits/Preserves/Shapes/Pullbacks.lean | theorem PreservesPullback.iso_hom_snd :
(PreservesPullback.iso G f g).hom ≫ pullback.snd _ _ = G.map (pullback.snd f g) | C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
G : C ⥤ D
X Y Z : C
f : X ⟶ Z
g : Y ⟶ Z
inst✝² : PreservesLimit (cospan f g) G
inst✝¹ : HasPullback f g
inst✝ : HasPullback (G.map f) (G.map g)
⊢ (iso G f g).hom ≫ pullback.snd (G.map f) (G.map g) = G.map (pullback.snd f g) | simp [PreservesPullback.iso] | no goals | 6f55da981739a3d7 |
AlgHom.IsArithFrobAt.isArithFrobAt_localize | Mathlib/RingTheory/Frobenius.lean | lemma isArithFrobAt_localize [Q.IsPrime] : H.localize.IsArithFrobAt (maximalIdeal _) | R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
φ : S →ₐ[R] S
Q : Ideal S
H : φ.IsArithFrobAt Q
inst✝ : Q.IsPrime
h :
Nat.card (R ⧸ Ideal.comap (algebraMap R (Localization.AtPrime Q)) (maximalIdeal (Localization.AtPrime Q))) =
Nat.card (R ⧸ Ideal.under R Q)
⊢ H.localize.IsAr... | intro x | R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
φ : S →ₐ[R] S
Q : Ideal S
H : φ.IsArithFrobAt Q
inst✝ : Q.IsPrime
h :
Nat.card (R ⧸ Ideal.comap (algebraMap R (Localization.AtPrime Q)) (maximalIdeal (Localization.AtPrime Q))) =
Nat.card (R ⧸ Ideal.under R Q)
x : Localization.... | 3affdd8fbfc529cd |
CategoryTheory.Limits.Concrete.widePushout_exists_rep | Mathlib/CategoryTheory/Limits/Shapes/ConcreteCategory.lean | theorem widePushout_exists_rep {B : C} {α : Type _} {X : α → C} (f : ∀ j : α, B ⟶ X j)
[HasWidePushout.{v} B X f] [PreservesColimit (wideSpan B X f) (forget C)]
(x : ToType (widePushout B X f)) :
(∃ y : ToType B, head f y = x) ∨ ∃ (i : α) (y : ToType (X i)), ι f i y = x | C : Type u
inst✝⁴ : Category.{v, u} C
FC : C → C → Type u_1
CC : C → Type v
inst✝³ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝² : ConcreteCategory C FC
B : C
α : Type v
X : α → C
f : (j : α) → B ⟶ X j
inst✝¹ : HasWidePushout B X f
inst✝ : PreservesColimit (wideSpan B X f) (forget C)
x : ToType (widePushout B X f... | obtain ⟨_ | j, y, rfl⟩ := Concrete.colimit_exists_rep _ x | case intro.none.intro
C : Type u
inst✝⁴ : Category.{v, u} C
FC : C → C → Type u_1
CC : C → Type v
inst✝³ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝² : ConcreteCategory C FC
B : C
α : Type v
X : α → C
f : (j : α) → B ⟶ X j
inst✝¹ : HasWidePushout B X f
inst✝ : PreservesColimit (wideSpan B X f) (forget C)
y : ToT... | f096228c4e17e89e |
Nat.count_modEq_card | Mathlib/Data/Int/CardIntervalMod.lean | theorem count_modEq_card (v : ℕ) :
b.count (· ≡ v [MOD r]) = b / r + if v % r < b % r then 1 else 0 | b r : ℕ
hr : 0 < r
v : ℕ
⊢ count (fun x => x ≡ v [MOD r]) b = b / r + if v % r < b % r then 1 else 0 | have hr' : 0 < (r : ℚ) := by positivity | b r : ℕ
hr : 0 < r
v : ℕ
hr' : 0 < ↑r
⊢ count (fun x => x ≡ v [MOD r]) b = b / r + if v % r < b % r then 1 else 0 | 09545fdb161cf7c0 |
Int.dvd_sub | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean | theorem dvd_sub : ∀ {a b c : Int}, a ∣ b → a ∣ c → a ∣ b - c
| _, _, _, ⟨d, rfl⟩, ⟨e, rfl⟩ => ⟨d - e, by rw [Int.mul_sub]⟩
| a✝ d e : Int
⊢ a✝ * d - a✝ * e = a✝ * (d - e) | rw [Int.mul_sub] | no goals | 3da12fba4237b17e |
SimpleGraph.minDegree_of_isEmpty | Mathlib/Combinatorics/SimpleGraph/Finite.lean | /-- If there are no vertices then the `minDegree` is zero. -/
@[simp]
lemma minDegree_of_isEmpty [DecidableRel G.Adj] [IsEmpty V] : G.minDegree = 0 | V : Type u_1
G : SimpleGraph V
inst✝² : Fintype V
inst✝¹ : DecidableRel G.Adj
inst✝ : IsEmpty V
⊢ G.minDegree = 0 | rw [minDegree, WithTop.untopD_eq_self_iff] | V : Type u_1
G : SimpleGraph V
inst✝² : Fintype V
inst✝¹ : DecidableRel G.Adj
inst✝ : IsEmpty V
⊢ (image (fun v => G.degree v) univ).min = ↑0 ∨ (image (fun v => G.degree v) univ).min = ⊤ | 5b625bae44347ba6 |
rank_eq_zero_iff | Mathlib/LinearAlgebra/Dimension/Finite.lean | /-- See `rank_zero_iff` for a stronger version with `NoZeroSMulDivisor R M`. -/
lemma rank_eq_zero_iff :
Module.rank R M = 0 ↔ ∀ x : M, ∃ a : R, a ≠ 0 ∧ a • x = 0 | R : Type u
M : Type v
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
a✝ : Nontrivial R
h : ∀ (x : M), ∃ a, a ≠ 0 ∧ a • x = 0
s : Set M
hs : LinearIndepOn R id s
i : ↑s
a : R
ha : a ≠ 0
ha' : a • ↑i = 0
⊢ (Finsupp.linearCombination R fun x => id ↑x) (Finsupp.single i a) = 0 | simpa | no goals | 0af4e256b38880a1 |
MeasureTheory.OuterMeasure.exists_measurable_superset_eq_trim | Mathlib/MeasureTheory/OuterMeasure/Induced.lean | theorem exists_measurable_superset_eq_trim (m : OuterMeasure α) (s : Set α) :
∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t = m.trim s | case neg
α : Type u_1
inst✝ : MeasurableSpace α
m : OuterMeasure α
s : Set α
ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), m t
hs : ¬ms = ⊤
this : ∀ r > ms, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t < r
t : ℕ → Set α
hsub : ∀ (n : ℕ), s ⊆ t n
hm : ∀ (n : ℕ), MeasurableSet (t n)
hm' : ∀ (n : ℕ), m (t n) < ms + (↑n)... | have : Tendsto (fun n : ℕ => ms + (n : ℝ≥0∞)⁻¹) atTop (𝓝 (ms + 0)) :=
tendsto_const_nhds.add ENNReal.tendsto_inv_nat_nhds_zero | case neg
α : Type u_1
inst✝ : MeasurableSpace α
m : OuterMeasure α
s : Set α
ms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), m t
hs : ¬ms = ⊤
this✝ : ∀ r > ms, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t < r
t : ℕ → Set α
hsub : ∀ (n : ℕ), s ⊆ t n
hm : ∀ (n : ℕ), MeasurableSet (t n)
hm' : ∀ (n : ℕ), m (t n) < ms + (↑n... | ff6f41c6fa9458e0 |
pNilradical_le_nilradical | Mathlib/FieldTheory/IsPerfectClosure.lean | theorem pNilradical_le_nilradical {R : Type*} [CommSemiring R] {p : ℕ} :
pNilradical R p ≤ nilradical R | case pos
R : Type u_1
inst✝ : CommSemiring R
p : ℕ
hp : 1 < p
⊢ pNilradical R p ≤ nilradical R | rw [pNilradical, if_pos hp] | no goals | f5aa52d6c9596eb4 |
uniformity_hasBasis_closed | Mathlib/Topology/UniformSpace/Basic.lean | theorem uniformity_hasBasis_closed :
HasBasis (𝓤 α) (fun V : Set (α × α) => V ∈ 𝓤 α ∧ IsClosed V) id | case intro.intro.intro
α : Type ua
inst✝ : UniformSpace α
t : Set (α × α)
h : t ∈ 𝓤 α
w : Set (α × α)
w_in : w ∈ 𝓤 α
w_symm : SymmetricRel w
r : w ○ w ○ w ⊆ t
⊢ closure w ⊆ w ○ w ○ w | rw [closure_eq_uniformity] | case intro.intro.intro
α : Type ua
inst✝ : UniformSpace α
t : Set (α × α)
h : t ∈ 𝓤 α
w : Set (α × α)
w_in : w ∈ 𝓤 α
w_symm : SymmetricRel w
r : w ○ w ○ w ⊆ t
⊢ ⋂ V ∈ {V | V ∈ 𝓤 α ∧ SymmetricRel V}, V ○ w ○ V ⊆ w ○ w ○ w | aadae38d36995a1c |
eq_pos_convex_span_of_mem_convexHull | Mathlib/Analysis/Convex/Caratheodory.lean | theorem eq_pos_convex_span_of_mem_convexHull {x : E} (hx : x ∈ convexHull 𝕜 s) :
∃ (ι : Sort (u + 1)) (_ : Fintype ι),
∃ (z : ι → E) (w : ι → 𝕜), Set.range z ⊆ s ∧ AffineIndependent 𝕜 z ∧ (∀ i, 0 < w i) ∧
∑ i, w i = 1 ∧ ∑ i, w i • z i = x | case intro.intro.intro.intro.intro.intro.refine_1
𝕜 : Type u_1
E : Type u
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s : Set E
x : E
t : Finset E
ht₁ : ↑t ⊆ s
ht₂ : AffineIndependent 𝕜 Subtype.val
w : E → 𝕜
hw₁ : ∀ y ∈ t, 0 ≤ w y
hw₂ : ∑ y ∈ t, w y = 1
hw₃ : t.centerMass w id = x
t' :... | exact Subset.trans (Finset.filter_subset _ t) ht₁ | no goals | c36d0bcdbb3360db |
ByteArray.get_extract_aux | Mathlib/.lake/packages/batteries/Batteries/Data/ByteArray.lean | theorem get_extract_aux {a : ByteArray} {start stop} (h : i < (a.extract start stop).size) :
start + i < a.size | case a
i : Nat
a : ByteArray
start stop : Nat
h : i < (a.extract start stop).size
⊢ min (stop - start) (a.size - start) ≤ a.size - start | exact Nat.min_le_right .. | no goals | 1f6abf8e0b73a81f |
NNReal.rpow_arith_mean_le_arith_mean2_rpow | Mathlib/Analysis/MeanInequalitiesPow.lean | theorem rpow_arith_mean_le_arith_mean2_rpow (w₁ w₂ z₁ z₂ : ℝ≥0) (hw' : w₁ + w₂ = 1) {p : ℝ}
(hp : 1 ≤ p) : (w₁ * z₁ + w₂ * z₂) ^ p ≤ w₁ * z₁ ^ p + w₂ * z₂ ^ p | case refine_1
w₁ w₂ z₁ z₂ : ℝ≥0
hw' : w₁ + w₂ = 1
p : ℝ
hp : 1 ≤ p
⊢ ∑ i : Fin (Nat.succ 0).succ, ![w₁, w₂] i = 1 | simp [hw', Fin.sum_univ_succ] | no goals | 601b73122c43db26 |
Right.sign_neg | Mathlib/Data/Sign.lean | theorem Right.sign_neg [AddRightStrictMono α] (a : α) :
sign (-a) = -sign a | α : Type u_1
inst✝³ : AddGroup α
inst✝² : Preorder α
inst✝¹ : DecidableRel fun x1 x2 => x1 < x2
inst✝ : AddRightStrictMono α
a : α
⊢ sign (-a) = -sign a | simp_rw [sign_apply, Right.neg_pos_iff, Right.neg_neg_iff] | α : Type u_1
inst✝³ : AddGroup α
inst✝² : Preorder α
inst✝¹ : DecidableRel fun x1 x2 => x1 < x2
inst✝ : AddRightStrictMono α
a : α
⊢ (if a < 0 then 1 else if 0 < a then -1 else 0) = -if 0 < a then 1 else if a < 0 then -1 else 0 | 059250f01c82c8ab |
ZMod.val_cast_eq_val_of_lt | Mathlib/Data/ZMod/Basic.lean | theorem val_cast_eq_val_of_lt {m n : ℕ} [nzm : NeZero m] {a : ZMod m}
(h : a.val < n) : (a.cast : ZMod n).val = a.val | case succ.zero
m : ℕ
nzm : NeZero (m + 1)
a : ZMod (m + 1)
nzn : NeZero 0
h : a.val < 0
⊢ a.cast.val = a.val | cases nzn | case succ.zero.mk
m : ℕ
nzm : NeZero (m + 1)
a : ZMod (m + 1)
h : a.val < 0
out✝ : 0 ≠ 0
⊢ a.cast.val = a.val | 4f9773579b5fa7c2 |
Nimber.invAux_mem_invSet_of_lt | Mathlib/SetTheory/Nimber/Field.lean | theorem invAux_mem_invSet_of_lt (ha : a ≠ 0) (hb : a < b) : invAux a ∈ invSet b | a b : Nimber
ha : a ≠ 0
hb : a < b
H : a.invAux * (1 + (b + a) * 0) ∈ b.invSet
⊢ a.invAux ∈ b.invSet | rwa [mul_zero, add_zero, mul_one] at H | no goals | b2282682ce79204c |
Polynomial.eq_zero_of_mul_eq_zero_of_smul | Mathlib/Algebra/Polynomial/AlgebraMap.lean | theorem eq_zero_of_mul_eq_zero_of_smul (P : R[X]) (h : ∀ r : R, r • P = 0 → r = 0) (Q : R[X])
(hQ : P * Q = 0) : Q = 0 | case h
R : Type u
inst✝ : CommSemiring R
P : R[X]
h : ∀ (r : R), r • P = 0 → r = 0
Q : R[X]
hQ : P * Q = 0
i : ℕ
hi : i > P.natDegree
⊢ P.coeff i • Q = 0 | rw [coeff_eq_zero_of_natDegree_lt hi, zero_smul] | no goals | 00213fbe9af3ea6e |
Submonoid.LocalizationMap.mk'_one | Mathlib/GroupTheory/MonoidLocalization/Basic.lean | theorem mk'_one (x) : f.mk' x (1 : S) = f.toMap x | M : Type u_1
inst✝¹ : CommMonoid M
S : Submonoid M
N : Type u_2
inst✝ : CommMonoid N
f : S.LocalizationMap N
x : M
⊢ f.toMap x * ↑1⁻¹ = f.toMap x | exact mul_one _ | no goals | a5098a6c4ff0cec0 |
Std.DHashMap.Internal.Raw₀.Const.getKey_modify | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/RawLemmas.lean | theorem getKey_modify (h : m.1.WF) [Inhabited α] {k k' : α} {f : β → β}
(hc : (Const.modify m k f).contains k') :
(Const.modify m k f).getKey k' hc =
if heq : k == k' then
k
else
haveI h' : m.contains k' | α : Type u
β✝ : α → Type v
m✝ : Raw₀ α β✝
inst✝⁵ : BEq α
inst✝⁴ : Hashable α
inst✝³ : LawfulBEq α
β : Type v
inst✝² : EquivBEq α
inst✝¹ : LawfulHashable α
m : Raw₀ α fun x => β
h : m.val.WF
inst✝ : Inhabited α
k k' : α
f : β → β
hc : (modify m k f).contains k' = true
heq : ¬(k == k') = true
⊢ m.contains k' = true | rwa [contains_modify _ h] at hc | no goals | 52799614b2339a72 |
collinear_insert_of_mem_affineSpan_pair | Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean | theorem collinear_insert_of_mem_affineSpan_pair {p₁ p₂ p₃ : P} (h : p₁ ∈ line[k, p₂, p₃]) :
Collinear k ({p₁, p₂, p₃} : Set P) | k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p₁ p₂ p₃ : P
h : p₁ ∈ affineSpan k {p₂, p₃}
⊢ Collinear k {p₁, p₂, p₃} | rw [collinear_insert_iff_of_mem_affineSpan h] | k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p₁ p₂ p₃ : P
h : p₁ ∈ affineSpan k {p₂, p₃}
⊢ Collinear k {p₂, p₃} | 3e815615057728e7 |
MeasureTheory.lintegral_add_aux | Mathlib/MeasureTheory/Integral/Lebesgue.lean | theorem lintegral_add_aux {f g : α → ℝ≥0∞} (hf : Measurable f) (hg : Measurable g) :
∫⁻ a, f a + g a ∂μ = ∫⁻ a, f a ∂μ + ∫⁻ a, g a ∂μ :=
calc
∫⁻ a, f a + g a ∂μ =
∫⁻ a, (⨆ n, (eapprox f n : α → ℝ≥0∞) a) + ⨆ n, (eapprox g n : α → ℝ≥0∞) a ∂μ | case e_f.h
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
a : α
⊢ ⨆ a_1, (eapprox f a_1) a + (eapprox g a_1) a = ⨆ n, (⇑(eapprox f n) + ⇑(eapprox g n)) a | simp only [Pi.add_apply] | no goals | 31f99a3e9be8da34 |
Int.fdiv_add_fmod | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean | theorem fdiv_add_fmod (a b : Int) : b * a.fdiv b + a.fmod b = a | a b : Int
⊢ b * a.fdiv b + a.fmod b = a | rw [Int.add_comm] | a b : Int
⊢ a.fmod b + b * a.fdiv b = a | 5638c52f436fbe4d |
zpow_eq_one_iff_modEq | Mathlib/GroupTheory/OrderOfElement.lean | theorem zpow_eq_one_iff_modEq {n : ℤ} : x ^ n = 1 ↔ n ≡ 0 [ZMOD orderOf x] | G : Type u_1
inst✝ : Group G
x : G
n : ℤ
⊢ x ^ n = 1 ↔ n ≡ 0 [ZMOD ↑(orderOf x)] | rw [Int.modEq_zero_iff_dvd, orderOf_dvd_iff_zpow_eq_one] | no goals | decfc1182cbe6a39 |
MeasureTheory.sigmaFinite_restrict_sigmaFiniteSetWRT' | Mathlib/MeasureTheory/Decomposition/Exhaustion.lean | lemma sigmaFinite_restrict_sigmaFiniteSetWRT' (μ ν : Measure α) [IsFiniteMeasure ν] :
SigmaFinite (μ.restrict (μ.sigmaFiniteSetWRT' ν)) | α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure ν
this✝ : ∀ (n : ℕ), SigmaFinite (μ.restrict (μ.sigmaFiniteSetGE ν n))
f : ℕ × ℕ → Set α :=
fun p =>
(μ.sigmaFiniteSetWRT' ν)ᶜ ∪ spanningSets (μ.restrict (μ.sigmaFiniteSetGE ν p.1)) p.2 ∩ μ.sigmaFiniteSetGE ν p.1
e : ℕ ≃ ℕ × ℕ := Nat.pair... | rw [this, Set.compl_union_self] | no goals | 858f4f82daf7e411 |
tangentConeAt.lim_zero | Mathlib/Analysis/Calculus/TangentCone.lean | theorem tangentConeAt.lim_zero {α : Type*} (l : Filter α) {c : α → 𝕜} {d : α → E}
(hc : Tendsto (fun n => ‖c n‖) l atTop) (hd : Tendsto (fun n => c n • d n) l (𝓝 y)) :
Tendsto d l (𝓝 0) | 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
y : E
α : Type u_5
l : Filter α
c : α → 𝕜
d : α → E
hc : Tendsto (fun n => ‖c n‖) l atTop
hd : Tendsto (fun n => c n • d n) l (𝓝 y)
A : Tendsto (fun n => ‖c n‖⁻¹) l (𝓝 0)
B : Tendsto (fun n => ‖c n •... | have C : Tendsto (fun n => ‖c n‖⁻¹ * ‖c n • d n‖) l (𝓝 (0 * ‖y‖)) := A.mul B | 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
y : E
α : Type u_5
l : Filter α
c : α → 𝕜
d : α → E
hc : Tendsto (fun n => ‖c n‖) l atTop
hd : Tendsto (fun n => c n • d n) l (𝓝 y)
A : Tendsto (fun n => ‖c n‖⁻¹) l (𝓝 0)
B : Tendsto (fun n => ‖c n •... | 3c1ce69539c5157c |
Filter.coprod_inf_prod_le | Mathlib/Order/Filter/Prod.lean | theorem coprod_inf_prod_le (f₁ f₂ : Filter α) (g₁ g₂ : Filter β) :
f₁.coprod g₁ ⊓ f₂ ×ˢ g₂ ≤ f₁ ×ˢ g₂ ⊔ f₂ ×ˢ g₁ := calc
f₁.coprod g₁ ⊓ f₂ ×ˢ g₂
_ = (f₁ ×ˢ ⊤ ⊔ ⊤ ×ˢ g₁) ⊓ f₂ ×ˢ g₂ | α : Type u_1
β : Type u_2
f₁ f₂ : Filter α
g₁ g₂ : Filter β
⊢ f₁.coprod g₁ ⊓ f₂ ×ˢ g₂ = (f₁ ×ˢ ⊤ ⊔ ⊤ ×ˢ g₁) ⊓ f₂ ×ˢ g₂ | rw [coprod_eq_prod_top_sup_top_prod] | no goals | 68d8ee2d6078c99c |
MeasureTheory.Martingale.stoppedValue_ae_eq_restrict_eq | Mathlib/Probability/Martingale/OptionalSampling.lean | theorem stoppedValue_ae_eq_restrict_eq (h : Martingale f ℱ μ) (hτ : IsStoppingTime ℱ τ)
(hτ_le : ∀ x, τ x ≤ n) [SigmaFinite (μ.trim (hτ.measurableSpace_le_of_le hτ_le))] (i : ι) :
stoppedValue f τ =ᵐ[μ.restrict {x | τ x = i}] μ[f n|hτ.measurableSpace] | Ω : Type u_1
E : Type u_2
m : MeasurableSpace Ω
μ : Measure Ω
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : CompleteSpace E
ι : Type u_3
inst✝⁵ : LinearOrder ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : OrderTopology ι
inst✝² : FirstCountableTopology ι
ℱ : Filtration ι m
inst✝¹ : SigmaFiniteFiltration μ ℱ
τ... | refine Filter.Eventually.of_forall fun x hx => ?_ | Ω : Type u_1
E : Type u_2
m : MeasurableSpace Ω
μ : Measure Ω
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : CompleteSpace E
ι : Type u_3
inst✝⁵ : LinearOrder ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : OrderTopology ι
inst✝² : FirstCountableTopology ι
ℱ : Filtration ι m
inst✝¹ : SigmaFiniteFiltration μ ℱ
τ... | 7f55941689e2a746 |
Subadditive.apply_mul_add_le | Mathlib/Analysis/Subadditive.lean | theorem apply_mul_add_le (k n r) : u (k * n + r) ≤ k * u n + u r | case zero
u : ℕ → ℝ
h : Subadditive u
n r : ℕ
⊢ u (0 * n + r) ≤ ↑0 * u n + u r | simp only [Nat.cast_zero, zero_mul, zero_add] | case zero
u : ℕ → ℝ
h : Subadditive u
n r : ℕ
⊢ u r ≤ u r | 531212c23562281b |
Nat.odd_add | Mathlib/Algebra/Ring/Parity.lean | @[parity_simps] lemma odd_add : Odd (m + n) ↔ (Odd m ↔ Even n) | m n : ℕ
⊢ Odd (m + n) ↔ (Odd m ↔ Even n) | rw [← not_even_iff_odd, even_add, not_iff, ← not_even_iff_odd] | no goals | a9e35b52c4458a14 |
Batteries.RBNode.mem_insert_of_mem | Mathlib/.lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean | theorem mem_insert_of_mem {t : RBNode α} (ht : Balanced t c n) (h : v' ∈ t) :
v' ∈ t.insert cmp v ∨ cmp v v' = .eq | α : Type u_1
c : RBColor
n : Nat
v' : α
cmp : α → α → Ordering
v : α
t : RBNode α
ht : t.Balanced c n
h : v' ∈ t
⊢ v' ∈ insert cmp t v ∨ cmp v v' = Ordering.eq | match e : zoom (cmp v) t with
| (nil, p) =>
let ⟨_, _, h₁, h₂⟩ := exists_insert_toList_zoom_nil ht e
simp [← mem_toList, h₁] at h
simp [← mem_toList, h₂]; cases h <;> simp [*]
| (node .., p) =>
let ⟨_, _, h₁, h₂⟩ := exists_insert_toList_zoom_node ht e
simp [← mem_toList, h₁] at h
simp [← mem_toList, h₂]; rc... | no goals | bb0a162495bf6167 |
SSet.StrictSegal.isPointwiseRightKanExtensionAt.fac_aux₂ | Mathlib/AlgebraicTopology/SimplicialSet/Coskeletal.lean | lemma fac_aux₂ {n : ℕ}
(s : Cone (proj (op ⦋n⦌) (Truncated.inclusion 2).op ⋙ (Truncated.inclusion 2).op ⋙ X))
(x : s.pt) (i j : ℕ) (hij : i ≤ j) (hj : j ≤ n) :
X.map (mkOfLe ⟨i, by omega⟩ ⟨j, by omega⟩ hij).op (lift s x) =
s.π.app (strArrowMk₂ (mkOfLe ⟨i, by omega⟩ ⟨j, by omega⟩ hij)) x | case intro.zero
X : SSet
inst✝ : X.StrictSegal
n : ℕ
s : Cone (proj (op ⦋n⦌) (Truncated.inclusion 2).op ⋙ (Truncated.inclusion 2).op ⋙ X)
x : s.pt
i : ℕ
hij : i ≤ i
hj : i ≤ n
this : mkOfLe ⟨i, ⋯⟩ ⟨i, ⋯⟩ ⋯ = ⦋1⦌.const ⦋0⦌ 0 ≫ ⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩
α : strArrowMk₂ (⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩) ⋯ ⟶ strArrowMk₂ (⦋1⦌.const ⦋0⦌ 0 ≫... | rw [lift, StrictSegal.spineToSimplex_vertex] | case intro.zero
X : SSet
inst✝ : X.StrictSegal
n : ℕ
s : Cone (proj (op ⦋n⦌) (Truncated.inclusion 2).op ⋙ (Truncated.inclusion 2).op ⋙ X)
x : s.pt
i : ℕ
hij : i ≤ i
hj : i ≤ n
this : mkOfLe ⟨i, ⋯⟩ ⟨i, ⋯⟩ ⋯ = ⦋1⦌.const ⦋0⦌ 0 ≫ ⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩
α : strArrowMk₂ (⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩) ⋯ ⟶ strArrowMk₂ (⦋1⦌.const ⦋0⦌ 0 ≫... | 764b0c7acb432811 |
SimplexCategoryGenRel.multiplicativeClosure_isGenerator_eq_top | Mathlib/AlgebraicTopology/SimplexCategory/GeneratorsRelations/Basic.lean | /-- A property is true for every morphism iff it holds for generators and is multiplicative. -/
lemma multiplicativeClosure_isGenerator_eq_top : generators.multiplicativeClosure = ⊤ | ⊢ generators.multiplicativeClosure = ⊤ | apply le_antisymm (by simp) | ⊢ ⊤ ≤ generators.multiplicativeClosure | ab632cedc2a52020 |
IsCyclotomicExtension.Rat.isIntegralClosure_adjoin_singleton_of_prime_pow | Mathlib/NumberTheory/Cyclotomic/Rat.lean | theorem isIntegralClosure_adjoin_singleton_of_prime_pow [hcycl : IsCyclotomicExtension {p ^ k} ℚ K]
(hζ : IsPrimitiveRoot ζ ↑(p ^ k)) : IsIntegralClosure (adjoin ℤ ({ζ} : Set K)) ℤ K | p : ℕ+
K : Type u
inst✝¹ : Field K
ζ : K
hp : Fact (Nat.Prime ↑p)
inst✝ : CharZero K
x : K
h : IsIntegral ℤ x
u : ℤˣ
n n✝ : ℕ
hcycl : IsCyclotomicExtension {p ^ (n✝ + 1)} ℚ K
hζ : IsPrimitiveRoot ζ ↑(p ^ (n✝ + 1))
B : PowerBasis ℚ K := IsPrimitiveRoot.subOnePowerBasis ℚ hζ
hint : IsIntegral ℤ B.gen
this : FiniteDimensi... | simpa using H | no goals | 076c1160c95c322f |
HomologicalComplex.mapBifunctor₂₃.ι_eq | Mathlib/Algebra/Homology/BifunctorAssociator.lean | lemma ι_eq (i₁ : ι₁) (i₂ : ι₂) (i₃ : ι₃) (i₂₃ : ι₂₃) (j : ι₄)
(h₂₃ : ComplexShape.π c₂ c₃ c₂₃ ⟨i₂, i₃⟩ = i₂₃)
(h : ComplexShape.π c₁ c₂₃ c₄ (i₁, i₂₃) = j) :
ι F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ i₁ i₂ i₃ j
(by rw [← h, ← h₂₃, ← ComplexShape.assoc c₁ c₂ c₃ c₁₂ c₂₃ c₄]; rfl) =
(F.obj (K₁.X i₁)).map (ιMapBif... | C₁ : Type u_1
C₂ : Type u_2
C₂₃ : Type u_4
C₃ : Type u_5
C₄ : Type u_6
inst✝²² : Category.{u_15, u_1} C₁
inst✝²¹ : Category.{u_17, u_2} C₂
inst✝²⁰ : Category.{u_16, u_5} C₃
inst✝¹⁹ : Category.{u_13, u_6} C₄
inst✝¹⁸ : Category.{u_14, u_4} C₂₃
inst✝¹⁷ : HasZeroMorphisms C₁
inst✝¹⁶ : HasZeroMorphisms C₂
inst✝¹⁵ : HasZeroM... | subst h₂₃ | C₁ : Type u_1
C₂ : Type u_2
C₂₃ : Type u_4
C₃ : Type u_5
C₄ : Type u_6
inst✝²² : Category.{u_15, u_1} C₁
inst✝²¹ : Category.{u_17, u_2} C₂
inst✝²⁰ : Category.{u_16, u_5} C₃
inst✝¹⁹ : Category.{u_13, u_6} C₄
inst✝¹⁸ : Category.{u_14, u_4} C₂₃
inst✝¹⁷ : HasZeroMorphisms C₁
inst✝¹⁶ : HasZeroMorphisms C₂
inst✝¹⁵ : HasZeroM... | 8b87b74eb56c47d3 |
Polynomial.degree_modByMonic_le_left | Mathlib/Algebra/Polynomial/Div.lean | theorem degree_modByMonic_le_left : degree (p %ₘ q) ≤ degree p | R : Type u
inst✝ : Ring R
p q : R[X]
⊢ (p %ₘ q).degree ≤ p.degree | nontriviality R | R : Type u
inst✝ : Ring R
p q : R[X]
a✝ : Nontrivial R
⊢ (p %ₘ q).degree ≤ p.degree | 20e50b3878cb55a3 |
Finset.card_toLeft_add_card_toRight | Mathlib/Data/Finset/Sum.lean | lemma card_toLeft_add_card_toRight : u.toLeft.card + u.toRight.card = u.card | α : Type u_1
β : Type u_2
u : Finset (α ⊕ β)
⊢ #u.toLeft + #u.toRight = #u | rw [← card_disjSum, toLeft_disjSum_toRight] | no goals | 0a1d52d2085c215e |
IntermediateField.botEquiv_def | Mathlib/FieldTheory/IntermediateField/Adjoin/Defs.lean | theorem botEquiv_def (x : F) : botEquiv F E (algebraMap F (⊥ : IntermediateField F E) x) = x | F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
x : F
⊢ (botEquiv F E) ((algebraMap F ↥⊥) x) = x | simp | no goals | 2f113cc31dff0030 |
LinearMap.ker_ne_bot_of_finrank_lt | Mathlib/LinearAlgebra/FiniteDimensional.lean | lemma ker_ne_bot_of_finrank_lt [FiniteDimensional K V] [FiniteDimensional K V₂] {f : V →ₗ[K] V₂}
(h : finrank K V₂ < finrank K V) :
LinearMap.ker f ≠ ⊥ | K : Type u
V : Type v
inst✝⁶ : DivisionRing K
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module K V
V₂ : Type v'
inst✝³ : AddCommGroup V₂
inst✝² : Module K V₂
inst✝¹ : FiniteDimensional K V
inst✝ : FiniteDimensional K V₂
f : V →ₗ[K] V₂
h : finrank K V₂ < finrank K V
h₁ : finrank K ↥(range f) + finrank K ↥(ker f) = finrank K V
h₂... | suffices 0 < finrank K (LinearMap.ker f) from Submodule.one_le_finrank_iff.mp this | K : Type u
V : Type v
inst✝⁶ : DivisionRing K
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module K V
V₂ : Type v'
inst✝³ : AddCommGroup V₂
inst✝² : Module K V₂
inst✝¹ : FiniteDimensional K V
inst✝ : FiniteDimensional K V₂
f : V →ₗ[K] V₂
h : finrank K V₂ < finrank K V
h₁ : finrank K ↥(range f) + finrank K ↥(ker f) = finrank K V
h₂... | 518682793e112d46 |
Submodule.mem_dualAnnihilator | Mathlib/LinearAlgebra/Dual.lean | theorem mem_dualAnnihilator (φ : Module.Dual R M) : φ ∈ W.dualAnnihilator ↔ ∀ w ∈ W, φ w = 0 | R : Type u
M : Type v
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
W : Submodule R M
φ : Dual R M
⊢ φ ∈ W.dualAnnihilator ↔ ∀ w ∈ W, φ w = 0 | refine LinearMap.mem_ker.trans ?_ | R : Type u
M : Type v
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
W : Submodule R M
φ : Dual R M
⊢ W.dualRestrict φ = 0 ↔ ∀ w ∈ W, φ w = 0 | 801085860b6ef13c |
LinearOrderedAddCommGroup.discrete_iff_not_denselyOrdered | Mathlib/GroupTheory/ArchimedeanDensely.lean | /-- Any linearly ordered archimedean additive group is either isomorphic (and order-isomorphic)
to the integers, or is densely ordered, exclusively. -/
lemma LinearOrderedAddCommGroup.discrete_iff_not_denselyOrdered (G : Type*)
[LinearOrderedAddCommGroup G] [Archimedean G] :
Nonempty (G ≃+o ℤ) ↔ ¬ DenselyOrdere... | case inl.intro
G : Type u_2
inst✝¹ : LinearOrderedAddCommGroup G
inst✝ : Archimedean G
this : (G ≃+o ℤ) → ¬DenselyOrdered G
h : G ≃+o ℤ
⊢ Nonempty (G ≃+o ℤ) ↔ ¬DenselyOrdered G | simpa [this h] using ⟨h⟩ | no goals | ef74c2b5003499f5 |
ContinuousMap.specializes_coe | Mathlib/Topology/CompactOpen.lean | @[norm_cast]
lemma specializes_coe {f g : C(X, Y)} : ⇑f ⤳ ⇑g ↔ f ⤳ g | X : Type u_2
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f g : C(X, Y)
⊢ ⇑f ⤳ ⇑g ↔ f ⤳ g | refine ⟨fun h ↦ ?_, fun h ↦ h.map continuous_coeFun⟩ | X : Type u_2
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f g : C(X, Y)
h : ⇑f ⤳ ⇑g
⊢ f ⤳ g | 2fe88ee65e32207f |
Matrix.updateCol_apply | Mathlib/Data/Matrix/RowCol.lean | theorem updateCol_apply [DecidableEq n] {j' : n} :
updateCol M j c i j' = if j' = j then c i else M i j' | m : Type u_2
n : Type u_3
α : Type v
M : Matrix m n α
i : m
j : n
c : m → α
inst✝ : DecidableEq n
j' : n
⊢ M.updateCol j c i j' = if j' = j then c i else M i j' | by_cases h : j' = j | case pos
m : Type u_2
n : Type u_3
α : Type v
M : Matrix m n α
i : m
j : n
c : m → α
inst✝ : DecidableEq n
j' : n
h : j' = j
⊢ M.updateCol j c i j' = if j' = j then c i else M i j'
case neg
m : Type u_2
n : Type u_3
α : Type v
M : Matrix m n α
i : m
j : n
c : m → α
inst✝ : DecidableEq n
j' : n
h : ¬j' = j
⊢ M.updateCo... | 4e29a7bc6bb264e0 |
CategoryTheory.Subpresheaf.equalizer_eq_iff | Mathlib/CategoryTheory/Subpresheaf/Equalizer.lean | lemma equalizer_eq_iff :
Subpresheaf.equalizer f g = A ↔ f = g | C : Type u
inst✝ : Category.{v, u} C
F₁ F₂ : Cᵒᵖ ⥤ Type w
A : Subpresheaf F₁
f g : A.toPresheaf ⟶ F₂
this : A ≤ Subpresheaf.equalizer f g ↔ f = g
⊢ Subpresheaf.equalizer f g = A ↔ f = g | rw [← this] | C : Type u
inst✝ : Category.{v, u} C
F₁ F₂ : Cᵒᵖ ⥤ Type w
A : Subpresheaf F₁
f g : A.toPresheaf ⟶ F₂
this : A ≤ Subpresheaf.equalizer f g ↔ f = g
⊢ Subpresheaf.equalizer f g = A ↔ A ≤ Subpresheaf.equalizer f g | 453d75f07cc12a5f |
CategoryTheory.prod_preservesConnectedLimits | Mathlib/CategoryTheory/Limits/Connected.lean | /-- The functor `(X × -)` preserves any connected limit.
Note that this functor does not preserve the two most obvious disconnected limits - that is,
`(X × -)` does not preserve products or terminal object, eg `(X ⨯ A) ⨯ (X ⨯ B)` is not isomorphic to
`X ⨯ (A ⨯ B)` and `X ⨯ 1` is not isomorphic to `1`.
-/
lemma prod_pre... | case h₁
C : Type u₂
inst✝³ : Category.{v₂, u₂} C
inst✝² : HasBinaryProducts C
J : Type v₂
inst✝¹ : SmallCategory J
inst✝ : IsConnected J
X : C
K : J ⥤ C
c : Cone K
l : IsLimit c
s : Cone (K ⋙ prod.functor.obj X)
m : s.pt ⟶ ((prod.functor.obj X).mapCone c).pt
L : ∀ (j : J), m ≫ ((prod.functor.obj X).mapCone c).π.app j =... | rfl | no goals | 7acbcc9df1349cf2 |
Real.tendsto_integral_gaussian_smul | Mathlib/Analysis/Fourier/Inversion.lean | lemma tendsto_integral_gaussian_smul (hf : Integrable f) (h'f : Integrable (𝓕 f)) (v : V) :
Tendsto (fun (c : ℝ) ↦
∫ w : V, ((π * c) ^ (finrank ℝ V / 2 : ℂ) * cexp (-π ^ 2 * c * ‖v - w‖ ^ 2)) • f w)
atTop (𝓝 (𝓕⁻ (𝓕 f) v)) | V : Type u_1
E : Type u_2
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : MeasurableSpace V
inst✝⁴ : BorelSpace V
inst✝³ : FiniteDimensional ℝ V
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : V → E
inst✝ : CompleteSpace E
hf : Integrable f volume
h'f : Integrable (𝓕 f) volume
v : V
A ... | simpa | no goals | 6e9a9f98d2dcf25f |
CategoryTheory.BasedFunctor.isHomLift_map | Mathlib/CategoryTheory/FiberedCategory/BasedCategory.lean | /-- For a based functor `F : 𝒳 ⟶ 𝒴`, and an arrow `φ` in `𝒳`, then `φ` lifts an arrow `f` in `𝒮`
if `F(φ)` does. -/
lemma isHomLift_map [IsHomLift 𝒴.p f (F.map φ)] : IsHomLift 𝒳.p f φ | 𝒮 : Type u₁
inst✝¹ : Category.{v₁, u₁} 𝒮
𝒳 : BasedCategory 𝒮
𝒴 : BasedCategory 𝒮
F : 𝒳 ⥤ᵇ 𝒴
R S : 𝒮
a b : 𝒳.obj
f : R ⟶ S
φ : a ⟶ b
inst✝ : 𝒴.p.IsHomLift f (F.map φ)
⊢ 𝒳.p.IsHomLift f φ | apply of_fac 𝒳.p f φ (F.w_obj a ▸ domain_eq 𝒴.p f (F.map φ))
(F.w_obj b ▸ codomain_eq 𝒴.p f (F.map φ)) | 𝒮 : Type u₁
inst✝¹ : Category.{v₁, u₁} 𝒮
𝒳 : BasedCategory 𝒮
𝒴 : BasedCategory 𝒮
F : 𝒳 ⥤ᵇ 𝒴
R S : 𝒮
a b : 𝒳.obj
f : R ⟶ S
φ : a ⟶ b
inst✝ : 𝒴.p.IsHomLift f (F.map φ)
⊢ f = eqToHom ⋯ ≫ 𝒳.p.map φ ≫ eqToHom ⋯ | 28d03dce1c0db115 |
SimpleGraph.Subgraph.restrict_coeSubgraph | Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) :
Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' | case verts.h
V : Type u
G : SimpleGraph V
G' : G.Subgraph
G'' : G'.coe.Subgraph
x✝ : ↑G'.verts
⊢ x✝ ∈ (Subgraph.restrict (Subgraph.coeSubgraph G'')).verts ↔ x✝ ∈ G''.verts | simp | no goals | 04bcafaed490095b |
IsBaseChange.ofEquiv | Mathlib/RingTheory/IsTensorProduct.lean | theorem IsBaseChange.ofEquiv (e : M ≃ₗ[R] N) : IsBaseChange R e.toLinearMap | case h.refl
R : Type u_1
M : Type v₁
N : Type v₂
inst✝⁴ : AddCommMonoid M
inst✝³ : AddCommMonoid N
inst✝² : CommSemiring R
inst✝¹ : Module R M
inst✝ : Module R N
e : M ≃ₗ[R] N
Q : Type (max v₁ v₂ u_1)
I₁ : AddCommMonoid Q
I₂ : Module R Q
g : M →ₗ[R] Q
I₄ : IsScalarTower R R Q
⊢ ∀ (y : N →ₗ[R] Q), (fun g' => ↑R g' ∘ₗ ↑e... | rintro y (rfl : _ = _) | case h.refl
R : Type u_1
M : Type v₁
N : Type v₂
inst✝⁴ : AddCommMonoid M
inst✝³ : AddCommMonoid N
inst✝² : CommSemiring R
inst✝¹ : Module R M
inst✝ : Module R N
e : M ≃ₗ[R] N
Q : Type (max v₁ v₂ u_1)
I₁ : AddCommMonoid Q
I₂ : Module R Q
I₄ : IsScalarTower R R Q
y : N →ₗ[R] Q
⊢ y = (↑R y ∘ₗ ↑e) ∘ₗ ↑e.symm | 4bf0cf3eb1d464b0 |
CategoryTheory.Localization.Monoidal.pentagon | Mathlib/CategoryTheory/Localization/Monoidal.lean | lemma pentagon (Y₁ Y₂ Y₃ Y₄ : LocalizedMonoidal L W ε) :
Pentagon Y₁ Y₂ Y₃ Y₄ | case intro.intro.intro.intro.intro.intro.intro.intro
C : Type u_1
D : Type u_2
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Category.{u_4, u_2} D
L : C ⥤ D
W : MorphismProperty C
inst✝² : MonoidalCategory C
inst✝¹ : W.IsMonoidal
inst✝ : L.IsLocalization W
unit : D
ε : L.obj (𝟙_ C) ≅ unit
Y₁ Y₂ Y₃ Y₄ : LocalizedMonoidal L W... | rw [← (L').map_comp_assoc, whiskerLeft_comp, μ_inv_natural_right_assoc, ← (L').map_comp_assoc] | case intro.intro.intro.intro.intro.intro.intro.intro
C : Type u_1
D : Type u_2
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Category.{u_4, u_2} D
L : C ⥤ D
W : MorphismProperty C
inst✝² : MonoidalCategory C
inst✝¹ : W.IsMonoidal
inst✝ : L.IsLocalization W
unit : D
ε : L.obj (𝟙_ C) ≅ unit
Y₁ Y₂ Y₃ Y₄ : LocalizedMonoidal L W... | b35ed8989ed4d8c7 |
Int.Matrix.exists_ne_zero_int_vec_norm_le | Mathlib/NumberTheory/SiegelsLemma.lean | theorem exists_ne_zero_int_vec_norm_le
(hn : Fintype.card α < Fintype.card β) (hm : 0 < Fintype.card α) : ∃ t : β → ℤ, t ≠ 0 ∧
A *ᵥ t = 0 ∧ ‖t‖ ≤ (n * max 1 ‖A‖) ^ ((m : ℝ) / (n - m)) | case intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
A : Matrix α β ℤ
hn : m < n
hm : 0 < m
x : β → ℤ
hxT : x ∈ T
y : β → ℤ
hyT : y ∈ T
hneq : x ≠ y
hfeq : A *ᵥ x = A *ᵥ y
n_mul_norm_A_pow_e_nonneg : 0 ≤ (↑n * (1 ⊔ ‖A‖)) ^ e
i : β
j : Unit
⊢ ‖x i - y i‖ ≤ (↑n * (1 ⊔ ‖A‖)) ^ ... | rw [Int.norm_eq_abs, ← Int.cast_abs] | case intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
A : Matrix α β ℤ
hn : m < n
hm : 0 < m
x : β → ℤ
hxT : x ∈ T
y : β → ℤ
hyT : y ∈ T
hneq : x ≠ y
hfeq : A *ᵥ x = A *ᵥ y
n_mul_norm_A_pow_e_nonneg : 0 ≤ (↑n * (1 ⊔ ‖A‖)) ^ e
i : β
j : Unit
⊢ ↑|x i - y i| ≤ (↑n * (1 ⊔ ‖A‖)) ^... | 028ca7465c0ecd0f |
Fin.mem_piFinset_iff_last_init | Mathlib/Data/Fin/Tuple/Finset.lean | lemma mem_piFinset_iff_last_init :
f ∈ piFinset s ↔ f (last n) ∈ s (last n) ∧ init f ∈ piFinset (init s) | n : ℕ
α : Fin (n + 1) → Type u_1
f : (i : Fin (n + 1)) → α i
s : (i : Fin (n + 1)) → Finset (α i)
⊢ f ∈ piFinset s ↔ f (last n) ∈ s (last n) ∧ init f ∈ piFinset (init s) | simp only [Fintype.mem_piFinset, forall_fin_succ', init, and_comm] | no goals | 6e6f4d71efdc5485 |
Bornology.IsVonNBounded.image | Mathlib/Analysis/LocallyConvex/Bounded.lean | theorem IsVonNBounded.image {σ : 𝕜₁ →+* 𝕜₂} [RingHomSurjective σ] [RingHomIsometric σ] {s : Set E}
(hs : IsVonNBounded 𝕜₁ s) (f : E →SL[σ] F) : IsVonNBounded 𝕜₂ (f '' s) | E : Type u_3
F : Type u_4
𝕜₁ : Type u_6
𝕜₂ : Type u_7
inst✝⁹ : NormedDivisionRing 𝕜₁
inst✝⁸ : NormedDivisionRing 𝕜₂
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜₁ E
inst✝⁵ : AddCommGroup F
inst✝⁴ : Module 𝕜₂ F
inst✝³ : TopologicalSpace E
inst✝² : TopologicalSpace F
σ : 𝕜₁ →+* 𝕜₂
inst✝¹ : RingHomSurjective σ
inst✝ :... | simpa only [comp_def, image_smul_setₛₗ _ _ σ f] using hf₀.image_smallSets.comp hs | no goals | f6a2b97356c4d756 |
IsTranscendenceBasis.isAlgebraic | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | theorem IsTranscendenceBasis.isAlgebraic [Nontrivial R] (hx : IsTranscendenceBasis R x) :
Algebra.IsAlgebraic (adjoin R (range x)) A | ι : Type u_1
R : Type u_3
A : Type u_5
x : ι → A
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : Nontrivial R
hx : IsTranscendenceBasis R x
a : A
ai : AlgebraicIndependent R fun o => o.elim a x
h₁ : range x ⊆ range fun o => o.elim a x
h : range x = range fun o => o.elim a x
⊢ a ∈ range x | rw [h] | ι : Type u_1
R : Type u_3
A : Type u_5
x : ι → A
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : Nontrivial R
hx : IsTranscendenceBasis R x
a : A
ai : AlgebraicIndependent R fun o => o.elim a x
h₁ : range x ⊆ range fun o => o.elim a x
h : range x = range fun o => o.elim a x
⊢ a ∈ range fun o => o.e... | 58f042b3657fb5c4 |
DHashMap.Option.map_cast_apply | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Lemmas.lean | theorem Option.map_cast_apply {γ γ' : Type u} (h : γ = γ') (x : Option γ) :
Option.map (cast h) x = cast (congrArg Option h) x | case refl
γ : Type u
x : Option γ
⊢ Option.map (cast ⋯) x = cast ⋯ x | cases x <;> simp | no goals | 9dfbdfb0b3d53514 |
HurwitzZeta.isBigO_atTop_cosKernel_sub | Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean | /-- The function `cosKernel a - 1` has exponential decay at `+∞`, for any `a`. -/
lemma isBigO_atTop_cosKernel_sub (a : UnitAddCircle) :
∃ p, 0 < p ∧ IsBigO atTop (cosKernel a · - 1) (fun x ↦ Real.exp (-p * x)) | case h
a p : ℝ
hp : 0 < p
hp' : (fun t => HurwitzKernelBounds.F_nat 0 1 t - if 1 = 0 then 1 else 0) =O[atTop] fun t => rexp (-p * t)
t : ℝ
ht : 0 < t
⊢ ∀ (i : ℕ),
‖2 * Real.cos (2 * π * a * (↑i + 1)) * rexp (-π * (↑i + 1) ^ 2 * t)‖ ≤ 2 * HurwitzKernelBounds.f_nat 0 1 t i | intro n | case h
a p : ℝ
hp : 0 < p
hp' : (fun t => HurwitzKernelBounds.F_nat 0 1 t - if 1 = 0 then 1 else 0) =O[atTop] fun t => rexp (-p * t)
t : ℝ
ht : 0 < t
n : ℕ
⊢ ‖2 * Real.cos (2 * π * a * (↑n + 1)) * rexp (-π * (↑n + 1) ^ 2 * t)‖ ≤ 2 * HurwitzKernelBounds.f_nat 0 1 t n | 58d473e972d0ef4b |
MvQPF.Cofix.abs_repr | Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean | theorem Cofix.abs_repr {α} (x : Cofix F α) : Quot.mk _ (Cofix.repr x) = x | case mk
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
R : Cofix F α → Cofix F α → Prop := fun x y => abs y.repr = x
a✝ : (P F).M α
⊢ LiftR' (α.RelLast' R) ((TypeVec.id ::: Quot.mk Mcongr) <$$> MvQPF.abs (M.dest (P F) (repr (Quot.mk Mcongr a✝))))
(Quot.lift (fun x => (TypeVec.id ::: Quot.mk Mc... | simp only [Cofix.repr, M.dest_corec, abs_map, MvQPF.abs_repr, Function.comp] | case mk
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
R : Cofix F α → Cofix F α → Prop := fun x y => abs y.repr = x
a✝ : (P F).M α
⊢ LiftR' (α.RelLast' R)
((TypeVec.id ::: Quot.mk Mcongr) <$$>
(TypeVec.id ::: M.corec (P F) (MvQPF.repr ∘ dest)) <$$> dest (Quot.mk Mcongr a✝))
((TypeVe... | de50b1854d18340e |
List.getElem_replace | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean | theorem getElem_replace [LawfulBEq α] {l : List α} {i : Nat} (h : i < l.length) :
(l.replace a b)[i]'(by simpa) = if l[i] == a then if a ∈ l.take i then a else b else l[i] | α : Type u_1
inst✝¹ : BEq α
a b : α
inst✝ : LawfulBEq α
l : List α
i : Nat
h : i < l.length
⊢ (l.replace a b)[i] = if (l[i] == a) = true then if a ∈ take i l then a else b else l[i] | apply Option.some.inj | case x
α : Type u_1
inst✝¹ : BEq α
a b : α
inst✝ : LawfulBEq α
l : List α
i : Nat
h : i < l.length
⊢ some (l.replace a b)[i] = some (if (l[i] == a) = true then if a ∈ take i l then a else b else l[i]) | 1e8d30c6e947e3e9 |
CStarAlgebra.mul_star_le_algebraMap_norm_sq | Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Order.lean | lemma CStarAlgebra.mul_star_le_algebraMap_norm_sq {a : A} :
a * star a ≤ algebraMap ℝ A (‖a‖ ^ 2) | A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
this : a * star a ≤ (algebraMap ℝ A) ‖a * star a‖
⊢ a * star a ≤ (algebraMap ℝ A) (‖a‖ ^ 2) | rwa [CStarRing.norm_self_mul_star, ← pow_two] at this | no goals | 2fc1c022829c67a0 |
Filter.countable_biInf_eq_iInf_seq' | Mathlib/Order/Filter/CountablyGenerated.lean | theorem countable_biInf_eq_iInf_seq' [CompleteLattice α] {B : Set ι} (Bcbl : B.Countable)
(f : ι → α) {i₀ : ι} (h : f i₀ = ⊤) : ∃ x : ℕ → ι, ⨅ t ∈ B, f t = ⨅ i, f (x i) | case inr
α : Type u_1
ι : Type u_4
inst✝ : CompleteLattice α
B : Set ι
Bcbl : B.Countable
f : ι → α
i₀ : ι
h : f i₀ = ⊤
Bnonempty : B.Nonempty
⊢ ∃ x, ⨅ t ∈ B, f t = ⨅ i, f (x i) | exact countable_biInf_eq_iInf_seq Bcbl Bnonempty f | no goals | 1109f96365705c26 |
Finsupp.prod_dvd_prod_of_subset_of_dvd | Mathlib/Algebra/BigOperators/Finsupp.lean | theorem prod_dvd_prod_of_subset_of_dvd [AddCommMonoid M] [CommMonoid N] {f1 f2 : α →₀ M}
{g1 g2 : α → M → N} (h1 : f1.support ⊆ f2.support)
(h2 : ∀ a : α, a ∈ f1.support → g1 a (f1 a) ∣ g2 a (f2 a)) : f1.prod g1 ∣ f2.prod g2 | case h
α : Type u_1
M : Type u_8
N : Type u_10
inst✝¹ : AddCommMonoid M
inst✝ : CommMonoid N
f1 f2 : α →₀ M
g1 g2 : α → M → N
h1 : f1.support ⊆ f2.support
h2 : ∀ a ∈ f1.support, g1 a (f1 a) ∣ g2 a (f2 a)
⊢ ∏ a ∈ f1.support, g1 a (f1 a) ∣ ∏ x ∈ f1.support, g2 x (f2 x) | apply prod_dvd_prod_of_dvd | case h.h
α : Type u_1
M : Type u_8
N : Type u_10
inst✝¹ : AddCommMonoid M
inst✝ : CommMonoid N
f1 f2 : α →₀ M
g1 g2 : α → M → N
h1 : f1.support ⊆ f2.support
h2 : ∀ a ∈ f1.support, g1 a (f1 a) ∣ g2 a (f2 a)
⊢ ∀ a ∈ f1.support, g1 a (f1 a) ∣ g2 a (f2 a) | 0bb6f4c77891fac1 |
IsFundamentalDomain.QuotientMeasureEqMeasurePreimage_smulHaarMeasure | Mathlib/MeasureTheory/Measure/Haar/Quotient.lean | theorem IsFundamentalDomain.QuotientMeasureEqMeasurePreimage_smulHaarMeasure {𝓕 : Set G}
(h𝓕 : IsFundamentalDomain Γ.op 𝓕 ν) (h𝓕_finite : ν 𝓕 ≠ ⊤) :
QuotientMeasureEqMeasurePreimage ν
((ν ((π ⁻¹' (K : Set (G ⧸ Γ))) ∩ 𝓕)) • haarMeasure K) | G : Type u_1
inst✝¹² : Group G
inst✝¹¹ : MeasurableSpace G
inst✝¹⁰ : TopologicalSpace G
inst✝⁹ : IsTopologicalGroup G
inst✝⁸ : BorelSpace G
inst✝⁷ : PolishSpace G
Γ : Subgroup G
inst✝⁶ : Γ.Normal
inst✝⁵ : T2Space (G ⧸ Γ)
inst✝⁴ : SecondCountableTopology (G ⧸ Γ)
inst✝³ : Countable ↥Γ
ν : Measure G
inst✝² : ν.IsHaarMeasu... | have hμK : μ K = c := by simp [μ, haarMeasure_self] | G : Type u_1
inst✝¹² : Group G
inst✝¹¹ : MeasurableSpace G
inst✝¹⁰ : TopologicalSpace G
inst✝⁹ : IsTopologicalGroup G
inst✝⁸ : BorelSpace G
inst✝⁷ : PolishSpace G
Γ : Subgroup G
inst✝⁶ : Γ.Normal
inst✝⁵ : T2Space (G ⧸ Γ)
inst✝⁴ : SecondCountableTopology (G ⧸ Γ)
inst✝³ : Countable ↥Γ
ν : Measure G
inst✝² : ν.IsHaarMeasu... | 1a929710e5b9315b |
Filter.exists_eventually_atTop | Mathlib/Order/Filter/AtTopBot/Basic.lean | lemma exists_eventually_atTop {r : α → β → Prop} :
(∃ b, ∀ᶠ a in atTop, r a b) ↔ ∀ᶠ a₀ in atTop, ∃ b, ∀ a ≥ a₀, r a b | α : Type u_3
β : Type u_4
inst✝² : Preorder α
inst✝¹ : IsDirected α fun x1 x2 => x1 ≤ x2
inst✝ : Nonempty α
r : α → β → Prop
⊢ (∃ x y, ∀ b ≥ x, r b y) ↔ ∃ a, ∀ b ≥ a, ∃ b_1, ∀ a ≥ b, r a b_1 | exact exists_congr fun a ↦ .symm <| forall_ge_iff <| Monotone.exists fun _ _ _ hb H n hn ↦
H n (hb.trans hn) | no goals | 507b48c8588eff7a |
Int.natAbs_add_le | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/Order.lean | theorem natAbs_add_le (a b : Int) : natAbs (a + b) ≤ natAbs a + natAbs b | a b : Int
⊢ ∀ (a b : Nat), (subNatNat a b.succ).natAbs ≤ (a + b).succ | refine fun a b => subNatNat_elim a b.succ
(fun m n i => n = b.succ → natAbs i ≤ (m + b).succ) ?_
(fun i n (e : (n + i).succ = _) => ?_) rfl | case refine_1
a✝ b✝ : Int
a b : Nat
⊢ ∀ (i n : Nat), (fun m n i => n = b.succ → i.natAbs ≤ (m + b).succ) (n + i) n ↑i
case refine_2
a✝ b✝ : Int
a b i n : Nat
e : (n + i).succ = b.succ
⊢ -[i+1].natAbs ≤ (n + b).succ | 2617f5dcc449a695 |
List.getElem?_modifyHead | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/Modify.lean | theorem getElem?_modifyHead {l : List α} {f : α → α} {n} :
(l.modifyHead f)[n]? = if n = 0 then l[n]?.map f else l[n]? | α : Type u_1
l : List α
f : α → α
n : Nat
⊢ (modifyHead f l)[n]? = if n = 0 then Option.map f l[n]? else l[n]? | cases l with
| nil => simp
| cons hd tl => cases n <;> simp | no goals | fe3a222c0d61aa8a |
tendstoUniformlyOn_iff_tendstoUniformlyOnFilter | Mathlib/Topology/UniformSpace/UniformConvergence.lean | theorem tendstoUniformlyOn_iff_tendstoUniformlyOnFilter :
TendstoUniformlyOn F f p s ↔ TendstoUniformlyOnFilter F f p (𝓟 s) | case h
α : Type u
β : Type v
ι : Type x
inst✝ : UniformSpace β
F : ι → α → β
f : α → β
s : Set α
p : Filter ι
⊢ ∀ a ∈ 𝓤 β, (∀ᶠ (n : ι) in p, ∀ x ∈ s, (f x, F n x) ∈ a) ↔ ∀ᶠ (n : ι × α) in p ×ˢ 𝓟 s, (f n.2, F n.1 n.2) ∈ a | simp_rw [eventually_prod_principal_iff] | case h
α : Type u
β : Type v
ι : Type x
inst✝ : UniformSpace β
F : ι → α → β
f : α → β
s : Set α
p : Filter ι
⊢ ∀ a ∈ 𝓤 β, True | 1257e80c771b6e5a |
Vitali.exists_disjoint_covering_ae | Mathlib/MeasureTheory/Covering/Vitali.lean | theorem exists_disjoint_covering_ae
[PseudoMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α]
[SecondCountableTopology α] (μ : Measure α) [IsLocallyFiniteMeasure μ] (s : Set α) (t : Set ι)
(C : ℝ≥0) (r : ι → ℝ) (c : ι → α) (B : ι → Set α) (hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a))
(μB : ∀ a ∈... | α : Type u_1
ι : Type u_2
inst✝⁴ : PseudoMetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)
μB : ∀ a ∈ t, μ (cl... | by_contra h | α : Type u_1
ι : Type u_2
inst✝⁴ : PseudoMetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)
μB : ∀ a ∈ t, μ (cl... | 4466d2fc3e80cb3f |
Equiv.Perm.IsCycleOn.apply_mem_iff | Mathlib/GroupTheory/Perm/Cycle/Basic.lean | theorem IsCycleOn.apply_mem_iff (hf : f.IsCycleOn s) : f x ∈ s ↔ x ∈ s :=
⟨fun hx => by
convert hf.1.perm_inv.1 hx
rw [inv_apply_self], fun hx => hf.1.mapsTo hx⟩
| α : Type u_2
f : Perm α
s : Set α
x : α
hf : f.IsCycleOn s
hx : f x ∈ s
⊢ x ∈ s | convert hf.1.perm_inv.1 hx | case h.e'_5
α : Type u_2
f : Perm α
s : Set α
x : α
hf : f.IsCycleOn s
hx : f x ∈ s
⊢ x = f⁻¹ (f x) | 6d1ab221553255b0 |
LieModule.exists_nontrivial_weightSpace_of_isSolvable | Mathlib/Algebra/Lie/LieTheorem.lean | theorem exists_nontrivial_weightSpace_of_isSolvable
[IsSolvable L] [LieModule.IsTriangularizable k L V] :
∃ χ : Module.Dual k L, Nontrivial (weightSpace V χ) | k : Type u_1
inst✝¹¹ : Field k
L : Type u_2
inst✝¹⁰ : LieRing L
inst✝⁹ : LieAlgebra k L
V : Type u_3
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module k V
inst✝⁶ : LieRingModule L V
inst✝⁵ : LieModule k L V
inst✝⁴ : CharZero k
inst✝³ : Module.Finite k V
inst✝² : Nontrivial V
inst✝¹ : IsSolvable L
inst✝ : IsTriangularizable k L V... | let toEndo : L →ₗ[k] imL := LinearMap.codRestrict imL.toSubmodule (toEnd k L V)
(fun x ↦ LinearMap.mem_range.mpr ⟨x, rfl⟩ : ∀ x : L, (toEnd k L V) x ∈ imL) | k : Type u_1
inst✝¹¹ : Field k
L : Type u_2
inst✝¹⁰ : LieRing L
inst✝⁹ : LieAlgebra k L
V : Type u_3
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module k V
inst✝⁶ : LieRingModule L V
inst✝⁵ : LieModule k L V
inst✝⁴ : CharZero k
inst✝³ : Module.Finite k V
inst✝² : Nontrivial V
inst✝¹ : IsSolvable L
inst✝ : IsTriangularizable k L V... | 2a2773cc59ef943b |
LinearMap.commute_mulLeft_right | Mathlib/Algebra/Algebra/Bilinear.lean | theorem commute_mulLeft_right (a b : A) : Commute (mulLeft R a) (mulRight R b) | case h
R : Type u_1
A : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : NonUnitalSemiring A
inst✝² : Module R A
inst✝¹ : SMulCommClass R A A
inst✝ : IsScalarTower R A A
a b c : A
⊢ (mulLeft R a * mulRight R b) c = (mulRight R b * mulLeft R a) c | exact (mul_assoc a c b).symm | no goals | 96324f8d122f6f14 |
RingHom.ofLocalizationSpanTarget_iff_finite | Mathlib/RingTheory/LocalProperties/Basic.lean | theorem RingHom.ofLocalizationSpanTarget_iff_finite :
RingHom.OfLocalizationSpanTarget @P ↔ RingHom.OfLocalizationFiniteSpanTarget @P | case h.mpr.intro.intro
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
a✝ b✝ : Type u
c✝ : CommRing a✝
d✝ : CommRing b✝
e✝ : a✝ →+* b✝
h :
∀ (s : Finset b✝),
Ideal.span ↑s = ⊤ → (∀ (r : { x // x ∈ s }), P ((algebraMap b✝ (Localization.Away ↑r)).comp e✝)) → P e✝
s : Set b✝
hs : ... | exact h s' h₂ fun x => hs' ⟨_, h₁ x.prop⟩ | no goals | 34725f8250ff835b |
Nat.Partrec.Code.encode_ofNatCode | Mathlib/Computability/PartrecCode.lean | theorem encode_ofNatCode : ∀ n, encodeCode (ofNatCode n) = n
| 0 => by simp [ofNatCode, encodeCode]
| 1 => by simp [ofNatCode, encodeCode]
| 2 => by simp [ofNatCode, encodeCode]
| 3 => by simp [ofNatCode, encodeCode]
| n + 4 => by
let m := n.div2.div2
have hm : m < n + 4 | n : ℕ
m : ℕ := n.div2.div2
hm : m < n + 4
_m1 : (unpair m).1 < n + 4
_m2 : (unpair m).2 < n + 4
IH : (ofNatCode m).encodeCode = m
IH1 : (ofNatCode (unpair m).1).encodeCode = (unpair m).1
IH2 : (ofNatCode (unpair m).2).encodeCode = (unpair m).2
⊢ (match n.bodd, n.div2.bodd with
| false, false => (ofNatCode (unpair... | cases n.bodd <;> cases n.div2.bodd <;>
simp [m, encodeCode, ofNatCode, IH, IH1, IH2, Nat.bit_val] | no goals | 09c2a9545f6ced5d |
Int.bodd_neg | Mathlib/Data/Int/Bitwise.lean | theorem bodd_neg (n : ℤ) : bodd (-n) = bodd n | case negSucc
n : ℕ
⊢ (!n.bodd) = -[n+1].bodd | change (!Nat.bodd n) = !(bodd n) | case negSucc
n : ℕ
⊢ (!n.bodd) = !(↑n).bodd | c8fc577e5e9172f8 |
Polynomial.coeff_mul_natTrailingDegree_add_natTrailingDegree | Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean | theorem coeff_mul_natTrailingDegree_add_natTrailingDegree : (p * q).coeff
(p.natTrailingDegree + q.natTrailingDegree) = p.trailingCoeff * q.trailingCoeff | R : Type u
inst✝ : Semiring R
p q : R[X]
⊢ ∀ b ∈ antidiagonal (p.natTrailingDegree + q.natTrailingDegree),
b ≠ (p.natTrailingDegree, q.natTrailingDegree) → p.coeff b.1 * q.coeff b.2 = 0 | rintro ⟨i, j⟩ h₁ h₂ | case mk
R : Type u
inst✝ : Semiring R
p q : R[X]
i j : ℕ
h₁ : (i, j) ∈ antidiagonal (p.natTrailingDegree + q.natTrailingDegree)
h₂ : (i, j) ≠ (p.natTrailingDegree, q.natTrailingDegree)
⊢ p.coeff (i, j).1 * q.coeff (i, j).2 = 0 | d436c374a999220d |
GenContFract.of_den_mono | Mathlib/Algebra/ContinuedFractions/Computation/Approximations.lean | theorem of_den_mono : (of v).dens n ≤ (of v).dens (n + 1) | case inr.intro
K : Type u_1
v : K
n : ℕ
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
g : GenContFract K := of v
not_terminated : ¬g.partDens.TerminatedAt n
b : K
nth_partDen_eq : g.partDens.get? n = some b
⊢ (of v).dens n ≤ (of v).dens (n + 1) | have : 1 ≤ b := of_one_le_get?_partDen nth_partDen_eq | case inr.intro
K : Type u_1
v : K
n : ℕ
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
g : GenContFract K := of v
not_terminated : ¬g.partDens.TerminatedAt n
b : K
nth_partDen_eq : g.partDens.get? n = some b
this : 1 ≤ b
⊢ (of v).dens n ≤ (of v).dens (n + 1) | 1c98869d10e9f213 |
EuclideanGeometry.existsUnique_dist_eq_of_insert | Mathlib/Geometry/Euclidean/Circumcenter.lean | theorem existsUnique_dist_eq_of_insert {s : AffineSubspace ℝ P}
[HasOrthogonalProjection s.direction] {ps : Set P} (hnps : ps.Nonempty) {p : P} (hps : ps ⊆ s)
(hp : p ∉ s) (hu : ∃! cs : Sphere P, cs.center ∈ s ∧ ps ⊆ (cs : Set P)) :
∃! cs₂ : Sphere P,
cs₂.center ∈ affineSpan ℝ (insert p (s : Set P)) ∧... | V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
s : AffineSubspace ℝ P
inst✝ : HasOrthogonalProjection s.direction
ps : Set P
hnps : ps.Nonempty
p : P
hps : ps ⊆ ↑s
hp : p ∉ s
hu : ∃! cs, cs.center ∈ s ∧ ps ⊆ Metric.sphere cs.cen... | rcases hu with ⟨⟨cc, cr⟩, ⟨hcc, hcr⟩, hcccru⟩ | case intro.mk.intro.intro
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
s : AffineSubspace ℝ P
inst✝ : HasOrthogonalProjection s.direction
ps : Set P
hnps : ps.Nonempty
p : P
hps : ps ⊆ ↑s
hp : p ∉ s
this : Nonempty ↥s
cc : P
c... | 6d31314b3548c9e7 |
Complex.continuousWithinAt_log_of_re_neg_of_im_zero | Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | theorem continuousWithinAt_log_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) :
ContinuousWithinAt log { z : ℂ | 0 ≤ z.im } z | case convert_1
z : ℂ
hre : z.re < 0
him : z.im = 0
⊢ ‖z‖ ≠ 0 | lift z to ℝ using him | case convert_1.intro
z : ℝ
hre : (↑z).re < 0
⊢ ‖↑z‖ ≠ 0 | fa596a77b673a65a |
sup_eq_prod_inf_factors | Mathlib/RingTheory/DedekindDomain/Ideal.lean | theorem sup_eq_prod_inf_factors [DecidableEq (Ideal T)] (hI : I ≠ ⊥) (hJ : J ≠ ⊥) :
I ⊔ J = (normalizedFactors I ∩ normalizedFactors J).prod | case a
T : Type u_4
inst✝² : CommRing T
inst✝¹ : IsDedekindDomain T
I J : Ideal T
inst✝ : DecidableEq (Ideal T)
hI : I ≠ ⊥
hJ : J ≠ ⊥
H : normalizedFactors (normalizedFactors I ∩ normalizedFactors J).prod = normalizedFactors I ∩ normalizedFactors J
this : (normalizedFactors I ∩ normalizedFactors J).prod ≠ 0
⊢ ∀ p ∈ nor... | intro p hp | case a
T : Type u_4
inst✝² : CommRing T
inst✝¹ : IsDedekindDomain T
I J : Ideal T
inst✝ : DecidableEq (Ideal T)
hI : I ≠ ⊥
hJ : J ≠ ⊥
H : normalizedFactors (normalizedFactors I ∩ normalizedFactors J).prod = normalizedFactors I ∩ normalizedFactors J
this : (normalizedFactors I ∩ normalizedFactors J).prod ≠ 0
p : Ideal T... | 1476ff204afbb5d8 |
HurwitzZeta.completedCosZeta_eq | Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean | lemma completedCosZeta_eq (a : UnitAddCircle) (s : ℂ) :
completedCosZeta a s =
completedCosZeta₀ a s - 1 / s - (if a = 0 then 1 else 0) / (1 - s) | a : UnitAddCircle
s : ℂ
⊢ completedCosZeta a s = completedCosZeta₀ a s - 1 / s - (if a = 0 then 1 else 0) / (1 - s) | rw [completedCosZeta, WeakFEPair.Λ, sub_div, sub_div] | a : UnitAddCircle
s : ℂ
⊢ (hurwitzEvenFEPair a).symm.Λ₀ (s / 2) / 2 - (1 / (s / 2)) • (hurwitzEvenFEPair a).symm.f₀ / 2 -
((hurwitzEvenFEPair a).symm.ε / (↑(hurwitzEvenFEPair a).symm.k - s / 2)) • (hurwitzEvenFEPair a).symm.g₀ / 2 =
completedCosZeta₀ a s - 1 / s - (if a = 0 then 1 else 0) / (1 - s) | 9ab6209aeb503fcb |
Function.FromTypes.curry_uncurry | Mathlib/Data/Fin/Tuple/Curry.lean | theorem curry_uncurry (f : Function.FromTypes p τ) : curry (uncurry f) = f | case zero
τ : Type u
p : Fin 0 → Type u
f : FromTypes p τ
⊢ curry f.uncurry = f | rfl | no goals | 40e0e88e80620c42 |
MeasureTheory.Measure.dmaSMul_apply | Mathlib/MeasureTheory/Group/MeasurableEquiv.lean | lemma dmaSMul_apply (μ : Measure A) (g : Gᵈᵐᵃ) (s : Set A) :
(g • μ) s = μ (DomMulAct.mk.symm g • s) | G : Type u_1
A : Type u_2
inst✝⁵ : Group G
inst✝⁴ : AddCommGroup A
inst✝³ : DistribMulAction G A
inst✝² : MeasurableSpace A
inst✝¹ : MeasurableSpace G
inst✝ : MeasurableSMul G A
μ : Measure A
g : Gᵈᵐᵃ
s : Set A
⊢ μ (⇑(MeasurableEquiv.smul (DomMulAct.mk.symm g)⁻¹) ⁻¹' s) = μ (DomMulAct.mk.symm g • s) | congr 1 | case h.e_6.h
G : Type u_1
A : Type u_2
inst✝⁵ : Group G
inst✝⁴ : AddCommGroup A
inst✝³ : DistribMulAction G A
inst✝² : MeasurableSpace A
inst✝¹ : MeasurableSpace G
inst✝ : MeasurableSMul G A
μ : Measure A
g : Gᵈᵐᵃ
s : Set A
⊢ ⇑(MeasurableEquiv.smul (DomMulAct.mk.symm g)⁻¹) ⁻¹' s = DomMulAct.mk.symm g • s | 36ac8d9734e64a06 |
quadraticChar_neg_one | Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean | theorem quadraticChar_neg_one [DecidableEq F] (hF : ringChar F ≠ 2) :
quadraticChar F (-1) = χ₄ (Fintype.card F) | F : Type u_1
inst✝² : Field F
inst✝¹ : Fintype F
inst✝ : DecidableEq F
hF : ringChar F ≠ 2
h : (quadraticChar F) (-1) = if (-1) ^ (Fintype.card F / 2) = 1 then 1 else -1
⊢ (quadraticChar F) (-1) = χ₄ ↑(Fintype.card F) | rw [h, χ₄_eq_neg_one_pow (FiniteField.odd_card_of_char_ne_two hF)] | F : Type u_1
inst✝² : Field F
inst✝¹ : Fintype F
inst✝ : DecidableEq F
hF : ringChar F ≠ 2
h : (quadraticChar F) (-1) = if (-1) ^ (Fintype.card F / 2) = 1 then 1 else -1
⊢ (if (-1) ^ (Fintype.card F / 2) = 1 then 1 else -1) = (-1) ^ (Fintype.card F / 2) | 17e3c888f93db25a |
Multiset.mem_sym2_iff | Mathlib/Data/Multiset/Sym.lean | theorem mem_sym2_iff {m : Multiset α} {z : Sym2 α} :
z ∈ m.sym2 ↔ ∀ y ∈ z, y ∈ m :=
m.inductionOn fun xs => by simp [List.mem_sym2_iff]
| α : Type u_1
m : Multiset α
z : Sym2 α
xs : List α
⊢ z ∈ Multiset.sym2 ⟦xs⟧ ↔ ∀ y ∈ z, y ∈ ⟦xs⟧ | simp [List.mem_sym2_iff] | no goals | 7f564004a58c0507 |
MeasureTheory.AEFinStronglyMeasurable.ae_eq_zero_of_forall_setIntegral_eq_zero | Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean | theorem AEFinStronglyMeasurable.ae_eq_zero_of_forall_setIntegral_eq_zero {f : α → E}
(hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
(hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0)
(hf : AEFinStronglyMeasurable f μ) : f =ᵐ[μ] 0 | case refine_2
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : α → E
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, f x ∂μ ... | rw [Measure.restrict_restrict hs] | case refine_2
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : α → E
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, f x ∂μ ... | 0f1e714a9354b744 |
Cardinal.isUnit_iff | Mathlib/SetTheory/Cardinal/Divisibility.lean | theorem isUnit_iff : IsUnit a ↔ a = 1 | case inr.intro.ha
a : Cardinal.{u}
h : ∀ (y : Cardinal.{u}), a ∣ y
ha : a ≠ 0
t : Cardinal.{u}
ht : a * t = 1
⊢ 1 ≤ a | exact one_le_iff_ne_zero.mpr ha | no goals | e34b9b8b494b68cd |
IsPGroup.isNilpotent | Mathlib/GroupTheory/Nilpotent.lean | theorem IsPGroup.isNilpotent [Finite G] {p : ℕ} [hp : Fact (Nat.Prime p)] (h : IsPGroup p G) :
IsNilpotent G | case intro.hstep
G✝ : Type u_1
inst✝² : Finite G✝
p : ℕ
hp : Fact (Nat.Prime p)
val✝ : Fintype G✝
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Nontrivial G
ih : ∀ (β : Type u_1) [inst : Fintype β], card β < card G → ∀ [hG : Group β], IsPGroup p β → IsNilpotent β
hG✝ : Group G
h : IsPGroup p G
hcq : card (G ⧸ center G) < car... | have hnq : IsNilpotent (G ⧸ center G) := ih _ hcq (h.to_quotient (center G)) | case intro.hstep
G✝ : Type u_1
inst✝² : Finite G✝
p : ℕ
hp : Fact (Nat.Prime p)
val✝ : Fintype G✝
G : Type u_1
inst✝¹ : Fintype G
inst✝ : Nontrivial G
ih : ∀ (β : Type u_1) [inst : Fintype β], card β < card G → ∀ [hG : Group β], IsPGroup p β → IsNilpotent β
hG✝ : Group G
h : IsPGroup p G
hcq : card (G ⧸ center G) < car... | ca07fccb134adeea |
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