name stringlengths 3 112 | file stringlengths 21 116 | statement stringlengths 17 8.64k | state stringlengths 7 205k | tactic stringlengths 3 4.55k | result stringlengths 7 205k | id stringlengths 16 16 |
|---|---|---|---|---|---|---|
CategoryTheory.Quotient.lift_spec | Mathlib/CategoryTheory/Quotient.lean | theorem lift_spec : functor r ⋙ lift r F H = F | case h_map
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
r : HomRel C
D : Type u_3
inst✝ : Category.{u_4, u_3} D
F : C ⥤ D
H : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F.map f₁ = F.map f₂
⊢ autoParam (∀ (X Y : C) (f : X ⟶ Y), (functor r ⋙ lift r F H).map f = eqToHom ⋯ ≫ F.map f ≫ eqToHom ⋯) _auto✝ | rintro X Y f | case h_map
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
r : HomRel C
D : Type u_3
inst✝ : Category.{u_4, u_3} D
F : C ⥤ D
H : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F.map f₁ = F.map f₂
X Y : C
f : X ⟶ Y
⊢ (functor r ⋙ lift r F H).map f = eqToHom ⋯ ≫ F.map f ≫ eqToHom ⋯ | 9549d608fcca58b6 |
SymOptionSuccEquiv.encode_decode | Mathlib/Data/Sym/Basic.lean | theorem encode_decode [DecidableEq α] (s : Sym (Option α) n ⊕ Sym α n.succ) :
encode (decode s) = s | case inl
α : Type u_1
n : ℕ
inst✝ : DecidableEq α
s : Sym (Option α) n
⊢ encode (decode (Sum.inl s)) = Sum.inl s | simp | no goals | 35ba1e8a60769782 |
ContinuousMap.compactOpen_eq_generateFrom | Mathlib/Topology/ContinuousMap/SecondCountableSpace.lean | theorem compactOpen_eq_generateFrom {S : Set (Set X)} {T : Set (Set Y)}
(hS₁ : ∀ K ∈ S, IsCompact K) (hT : IsTopologicalBasis T)
(hS₂ : ∀ f : C(X, Y), ∀ x, ∀ V ∈ T, f x ∈ V → ∃ K ∈ S, K ∈ 𝓝 x ∧ MapsTo f K V) :
compactOpen = .generateFrom (.image2 (fun K t ↦
{f : C(X, Y) | MapsTo f K (⋃₀ t)}) S {t : S... | X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
S : Set (Set X)
T : Set (Set Y)
hS₁ : ∀ K ∈ S, IsCompact K
hT : IsTopologicalBasis T
hS₂ : ∀ (f : C(X, Y)) (x : X), ∀ V ∈ T, f x ∈ V → ∃ K ∈ S, K ∈ 𝓝 x ∧ MapsTo (⇑f) K V
f : C(X, Y)
K : Set X
hK : IsCompact K
U : Set Y
hU : IsOpen U
hfKU ... | intro x hx | X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
S : Set (Set X)
T : Set (Set Y)
hS₁ : ∀ K ∈ S, IsCompact K
hT : IsTopologicalBasis T
hS₂ : ∀ (f : C(X, Y)) (x : X), ∀ V ∈ T, f x ∈ V → ∃ K ∈ S, K ∈ 𝓝 x ∧ MapsTo (⇑f) K V
f : C(X, Y)
K : Set X
hK : IsCompact K
U : Set Y
hU : IsOpen U
hfKU ... | febdf58bf7ae25f1 |
map_inv_natCast_smul | Mathlib/Algebra/Module/Basic.lean | theorem map_inv_natCast_smul [AddCommMonoid M] [AddCommMonoid M₂] {F : Type*} [FunLike F M M₂]
[AddMonoidHomClass F M M₂] (f : F) (R S : Type*)
[DivisionSemiring R] [DivisionSemiring S] [Module R M]
[Module S M₂] (n : ℕ) (x : M) : f ((n⁻¹ : R) • x) = (n⁻¹ : S) • f x | case pos
M : Type u_3
M₂ : Type u_4
inst✝⁷ : AddCommMonoid M
inst✝⁶ : AddCommMonoid M₂
F : Type u_5
inst✝⁵ : FunLike F M M₂
inst✝⁴ : AddMonoidHomClass F M M₂
f : F
R : Type u_6
S : Type u_7
inst✝³ : DivisionSemiring R
inst✝² : DivisionSemiring S
inst✝¹ : Module R M
inst✝ : Module S M₂
n : ℕ
x : M
hR : ¬↑n = 0
hS : ↑n =... | suffices ∀ y, f y = 0 by simp [this] | case pos
M : Type u_3
M₂ : Type u_4
inst✝⁷ : AddCommMonoid M
inst✝⁶ : AddCommMonoid M₂
F : Type u_5
inst✝⁵ : FunLike F M M₂
inst✝⁴ : AddMonoidHomClass F M M₂
f : F
R : Type u_6
S : Type u_7
inst✝³ : DivisionSemiring R
inst✝² : DivisionSemiring S
inst✝¹ : Module R M
inst✝ : Module S M₂
n : ℕ
x : M
hR : ¬↑n = 0
hS : ↑n =... | acfa63a0edc35b17 |
IsPurelyInseparable.trans | Mathlib/FieldTheory/PurelyInseparable/Basic.lean | theorem IsPurelyInseparable.trans [Algebra E K] [IsScalarTower F E K]
[h1 : IsPurelyInseparable F E] [h2 : IsPurelyInseparable E K] : IsPurelyInseparable F K | case intro.intro.intro
F : Type u
E : Type v
inst✝⁶ : Field F
inst✝⁵ : Field E
inst✝⁴ : Algebra F E
K : Type w
inst✝³ : Field K
inst✝² : Algebra F K
inst✝¹ : Algebra E K
inst✝ : IsScalarTower F E K
q : ℕ
h2✝ : ∀ (x : K), ∃ n, x ^ q ^ n ∈ (algebraMap E K).range
h1 : ∀ (x : E), ∃ n, x ^ q ^ n ∈ (algebraMap F E).range
h✝ ... | obtain ⟨m, z, h1⟩ := h1 y | case intro.intro.intro.intro.intro
F : Type u
E : Type v
inst✝⁶ : Field F
inst✝⁵ : Field E
inst✝⁴ : Algebra F E
K : Type w
inst✝³ : Field K
inst✝² : Algebra F K
inst✝¹ : Algebra E K
inst✝ : IsScalarTower F E K
q : ℕ
h2✝ : ∀ (x : K), ∃ n, x ^ q ^ n ∈ (algebraMap E K).range
h1✝ : ∀ (x : E), ∃ n, x ^ q ^ n ∈ (algebraMap F... | db24f710628c1cfb |
Finset.Nonempty.norm_prod_le_sup'_norm | Mathlib/Analysis/Normed/Group/Ultra.lean | /-- Nonarchimedean norm of a product is less than or equal the norm of any term in the product.
This version is phrased using `Finset.sup'` and `Finset.Nonempty` due to `Finset.sup`
operating over an `OrderBot`, which `ℝ` is not.
-/
@[to_additive "Nonarchimedean norm of a sum is less than or equal the norm of any term ... | case cons.refine_2
M : Type u_1
ι : Type u_2
inst✝¹ : SeminormedCommGroup M
inst✝ : IsUltrametricDist M
s : Finset ι
f : ι → M
j : ι
t : Finset ι
hj : j ∉ t
hs✝ : t.Nonempty
IH : ∃ b ∈ t, ‖∏ i ∈ t, f i‖ ≤ ‖f b‖
h : ‖f j‖ ≤ ‖∏ i ∈ t, f i‖
⊢ ∃ a ∈ t, ‖f j * ∏ i ∈ t, f i‖ ≤ ‖f a‖ | exact ⟨_, IH.choose_spec.left, (norm_mul_le_max _ _).trans <|
((max_eq_right h).le.trans IH.choose_spec.right)⟩ | no goals | 3c2c7b0860a7c76e |
MeasureTheory.Lp.eLpNorm'_lim_eq_lintegral_liminf | Mathlib/MeasureTheory/Function/LpSpace/Basic.lean | theorem eLpNorm'_lim_eq_lintegral_liminf {ι} [Nonempty ι] [LinearOrder ι] {f : ι → α → G} {p : ℝ}
{f_lim : α → G} (h_lim : ∀ᵐ x : α ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))) :
eLpNorm' f_lim p μ = (∫⁻ a, atTop.liminf (‖f · a‖ₑ ^ p) ∂μ) ^ (1 / p) | α : Type u_1
G : Type u_6
m0 : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup G
ι : Type u_7
inst✝¹ : Nonempty ι
inst✝ : LinearOrder ι
f : ι → α → G
p : ℝ
f_lim : α → G
h_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))
⊢ eLpNorm' f_lim p μ = (∫⁻ (a : α), liminf (fun x => ‖f x a‖ₑ ^ p) at... | suffices h_no_pow : (∫⁻ a, ‖f_lim a‖ₑ ^ p ∂μ) = ∫⁻ a, atTop.liminf fun m => ‖f m a‖ₑ ^ p ∂μ by
rw [eLpNorm'_eq_lintegral_enorm, h_no_pow] | α : Type u_1
G : Type u_6
m0 : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup G
ι : Type u_7
inst✝¹ : Nonempty ι
inst✝ : LinearOrder ι
f : ι → α → G
p : ℝ
f_lim : α → G
h_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))
⊢ ∫⁻ (a : α), ‖f_lim a‖ₑ ^ p ∂μ = ∫⁻ (a : α), liminf (fun m => ‖f m a... | 0eb42e6f44204d59 |
CategoryTheory.Limits.preservesPushout_symmetry | Mathlib/CategoryTheory/Limits/Preserves/Shapes/Pullbacks.lean | /-- If `F` preserves the pushout of `f, g`, it also preserves the pushout of `g, f`. -/
lemma preservesPushout_symmetry : PreservesColimit (span g f) G where
preserves {c} hc := ⟨by
apply (IsColimit.precomposeHomEquiv (diagramIsoSpan.{v₂} _).symm _).toFun
apply IsColimit.ofIsoColimit _ (PushoutCocone.isoMk _)... | case ht.refine_2
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
G : C ⥤ D
W X Y : C
f : W ⟶ X
g : W ⟶ Y
inst✝ : PreservesColimit (span f g) G
c : Cocone (span g f)
hc : IsColimit c
⊢ PreservesColimit (span f g) G | infer_instance | no goals | a31b72fa98bb4cd6 |
Matrix.inv_mulVec_eq_vec | Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | lemma inv_mulVec_eq_vec {A : Matrix n n α} [Invertible A]
{u v : n → α} (hM : u = A.mulVec v) : A⁻¹.mulVec u = v | n : Type u'
α : Type v
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : CommRing α
A : Matrix n n α
inst✝ : Invertible A
u v : n → α
hM : u = A *ᵥ v
⊢ A⁻¹ *ᵥ u = v | rw [hM, Matrix.mulVec_mulVec, Matrix.inv_mul_of_invertible, Matrix.one_mulVec] | no goals | 973e1e1efd7de833 |
CategoryTheory.Functor.isContinuous_of_coverPreserving | Mathlib/CategoryTheory/Sites/CoverPreserving.lean | /-- If `F` is cover-preserving and compatible-preserving, then `F` is a continuous functor. -/
@[stacks 00WW "This is basically this Stacks entry."]
lemma Functor.isContinuous_of_coverPreserving (hF₁ : CompatiblePreserving.{w} K F)
(hF₂ : CoverPreserving J K F) : Functor.IsContinuous.{w} F J K where
op_comp_isShe... | case hunique.intro.intro.intro.intro
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F : C ⥤ D
J : GrothendieckTopology C
K : GrothendieckTopology D
hF₁ : CompatiblePreserving K F
hF₂ : CoverPreserving J K F
G : Sheaf K (Type w)
X : C
S : Sieve X
hS : S ∈ J X
x : FamilyOfElements (F.op ... | rw [H] | no goals | 343ebb90057ceb12 |
Polynomial.EisensteinCriterionAux.isUnit_of_natDegree_eq_zero_of_isPrimitive | Mathlib/RingTheory/EisensteinCriterion.lean | theorem isUnit_of_natDegree_eq_zero_of_isPrimitive {p q : R[X]}
-- Porting note: stated using `IsPrimitive` which is defeq to old statement.
(hu : IsPrimitive (p * q)) (hpm : p.natDegree = 0) : IsUnit p | R : Type u_1
inst✝ : CommRing R
p q : R[X]
hu : (p * q).IsPrimitive
hpm : p.natDegree = 0
⊢ IsUnit (p.coeff 0) | refine hu _ ?_ | R : Type u_1
inst✝ : CommRing R
p q : R[X]
hu : (p * q).IsPrimitive
hpm : p.natDegree = 0
⊢ C (p.coeff 0) ∣ p * q | 447194fa7b9cf99d |
Associates.exists_prime_dvd_of_not_inf_one | Mathlib/RingTheory/UniqueFactorizationDomain/FactorSet.lean | theorem exists_prime_dvd_of_not_inf_one {a b : α} (ha : a ≠ 0) (hb : b ≠ 0)
(h : Associates.mk a ⊓ Associates.mk b ≠ 1) : ∃ p : α, Prime p ∧ p ∣ a ∧ p ∣ b | α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a b : α
ha : a ≠ 0
hb : b ≠ 0
h : Associates.mk a ⊓ Associates.mk b ≠ 1
⊢ (Associates.mk a).factors ⊓ (Associates.mk b).factors ≠ 0 | contrapose! h with hf | α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a b : α
ha : a ≠ 0
hb : b ≠ 0
hf : (Associates.mk a).factors ⊓ (Associates.mk b).factors = 0
⊢ Associates.mk a ⊓ Associates.mk b = 1 | 4b4dfcc04fa14351 |
lt_mul_iff_one_lt_left' | Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean | theorem lt_mul_iff_one_lt_left' [MulRightStrictMono α]
[MulRightReflectLT α] (a : α) {b : α} : a < b * a ↔ 1 < b :=
Iff.trans (by rw [one_mul]) (mul_lt_mul_iff_right a)
| α : Type u_1
inst✝³ : MulOneClass α
inst✝² : LT α
inst✝¹ : MulRightStrictMono α
inst✝ : MulRightReflectLT α
a b : α
⊢ a < b * a ↔ 1 * a < b * a | rw [one_mul] | no goals | f17213218151fc57 |
reesAlgebra.fg | Mathlib/RingTheory/ReesAlgebra.lean | theorem reesAlgebra.fg (hI : I.FG) : (reesAlgebra I).FG | case h
R : Type u
inst✝ : CommRing R
I : Ideal R
s : Finset R
hs : Ideal.span ↑s = I
⊢ Algebra.adjoin R (⇑(monomial 1) '' ↑s) = Algebra.adjoin R ↑(Submodule.map (monomial 1) (Ideal.span ↑s)) | change
_ =
Algebra.adjoin R
(Submodule.map (monomial 1 : R →ₗ[R] R[X]) (Submodule.span R ↑s) : Set R[X]) | case h
R : Type u
inst✝ : CommRing R
I : Ideal R
s : Finset R
hs : Ideal.span ↑s = I
⊢ Algebra.adjoin R (⇑(monomial 1) '' ↑s) = Algebra.adjoin R ↑(Submodule.map (monomial 1) (Submodule.span R ↑s)) | 69fde803654cdf7f |
CompleteLattice.ωScottContinuous.sup | Mathlib/Order/OmegaCompletePartialOrder.lean | lemma ωScottContinuous.sup (hf : ωScottContinuous f) (hg : ωScottContinuous g) :
ωScottContinuous (f ⊔ g) | α : Type u_2
β : Type u_3
inst✝¹ : OmegaCompletePartialOrder α
inst✝ : CompleteLattice β
f g : α → β
hf : ωScottContinuous f
hg : ωScottContinuous g
⊢ ωScottContinuous (f ⊔ g) | rw [← sSup_pair] | α : Type u_2
β : Type u_3
inst✝¹ : OmegaCompletePartialOrder α
inst✝ : CompleteLattice β
f g : α → β
hf : ωScottContinuous f
hg : ωScottContinuous g
⊢ ωScottContinuous (SupSet.sSup {f, g}) | 43f8aa09f2e0bd75 |
Nat.Primes.summable_rpow | Mathlib/NumberTheory/SumPrimeReciprocals.lean | theorem Nat.Primes.summable_rpow {r : ℝ} :
Summable (fun p : Nat.Primes ↦ (p : ℝ) ^ r) ↔ r < -1 | case pos
r : ℝ
h : r < -1
⊢ Summable fun p => ↑↑p ^ r | exact (Real.summable_nat_rpow.mpr h).subtype _ | no goals | 1efd7ce0900dcc13 |
MeasureTheory.AEMeasurable.ae_eq_of_forall_setLIntegral_eq | Mathlib/MeasureTheory/Function/AEEqOfLIntegral.lean | theorem AEMeasurable.ae_eq_of_forall_setLIntegral_eq {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ)
(hg : AEMeasurable g μ) (hfi : ∫⁻ x, f x ∂μ ≠ ∞) (hgi : ∫⁻ x, g x ∂μ ≠ ∞)
(hfg : ∀ ⦃s⦄, MeasurableSet s → μ s < ∞ → ∫⁻ x in s, f x ∂μ = ∫⁻ x in s, g x ∂μ) :
f =ᵐ[μ] g | α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hfi : ∫⁻ (x : α), f x ∂μ ≠ ⊤
hgi : ∫⁻ (x : α), g x ∂μ ≠ ⊤
hfg : ∀ ⦃s : Set α⦄, MeasurableSet s → μ s < ⊤ → ∫⁻ (x : α) in s, f x ∂μ = ∫⁻ (x : α) in s, g x ∂μ
hf' : AEFinStronglyMeasurable f μ
hg' : AEFinStronglyM... | have := hg'.sigmaFinite_restrict | α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hfi : ∫⁻ (x : α), f x ∂μ ≠ ⊤
hgi : ∫⁻ (x : α), g x ∂μ ≠ ⊤
hfg : ∀ ⦃s : Set α⦄, MeasurableSet s → μ s < ⊤ → ∫⁻ (x : α) in s, f x ∂μ = ∫⁻ (x : α) in s, g x ∂μ
hf' : AEFinStronglyMeasurable f μ
hg' : AEFinStronglyM... | 928eae5949838171 |
numDerangements_tendsto_inv_e | Mathlib/Combinatorics/Derangements/Exponential.lean | theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) | case h
s : ℕ → ℝ := fun n => ∑ k ∈ Finset.range n, (-1) ^ k / ↑k.factorial
this : ∀ (n : ℕ), ↑(numDerangements n) / ↑n.factorial = s (n + 1)
⊢ HasSum (fun i => (-1) ^ i / ↑i.factorial) (exp ℝ (-1)) | exact expSeries_div_hasSum_exp ℝ (-1 : ℝ) | no goals | 368760d6fa6aef58 |
CategoryTheory.IsPushout.isVanKampen_iff | Mathlib/CategoryTheory/Adhesive.lean | theorem IsPushout.isVanKampen_iff (H : IsPushout f g h i) :
H.IsVanKampen ↔ IsVanKampenColimit (PushoutCocone.mk h i H.w) | case mp.refine_1
C : Type u
inst✝ : Category.{v, u} C
W X Y Z : C
f : W ⟶ X
g : W ⟶ Y
h : X ⟶ Z
i : Y ⟶ Z
H✝ : IsPushout f g h i
H : H✝.IsVanKampen
F' : WalkingSpan ⥤ C
c' : Cocone F'
α : F' ⟶ span f g
fα : c'.pt ⟶ (PushoutCocone.mk h i ⋯).pt
eα : α ≫ (PushoutCocone.mk h i ⋯).ι = c'.ι ≫ (Functor.const WalkingSpan).map ... | rw [(IsColimit.equivOfNatIsoOfIso (diagramIsoSpan F') c' (PushoutCocone.mk _ _ this)
_).nonempty_congr] | case mp.refine_1
C : Type u
inst✝ : Category.{v, u} C
W X Y Z : C
f : W ⟶ X
g : W ⟶ Y
h : X ⟶ Z
i : Y ⟶ Z
H✝ : IsPushout f g h i
H : H✝.IsVanKampen
F' : WalkingSpan ⥤ C
c' : Cocone F'
α : F' ⟶ span f g
fα : c'.pt ⟶ (PushoutCocone.mk h i ⋯).pt
eα : α ≫ (PushoutCocone.mk h i ⋯).ι = c'.ι ≫ (Functor.const WalkingSpan).map ... | 03b3318bb2e454c2 |
Asymptotics.isTheta_bot | Mathlib/Analysis/Asymptotics/Theta.lean | theorem isTheta_bot : f =Θ[⊥] g | α : Type u_1
E : Type u_3
F : Type u_4
inst✝¹ : Norm E
inst✝ : Norm F
f : α → E
g : α → F
⊢ f =Θ[⊥] g | simp [IsTheta] | no goals | cdefb79688fb2d6c |
norm_div_eq_norm_left | Mathlib/Analysis/Normed/Group/Basic.lean | @[to_additive]
lemma norm_div_eq_norm_left (x : E) {y : E} (h : ‖y‖ = 0) : ‖x / y‖ = ‖x‖ | E : Type u_5
inst✝ : SeminormedGroup E
x y : E
h : ‖y‖ = 0
⊢ ‖x / y‖ ≤ ‖x‖ | simpa [h] using norm_div_le x y | no goals | 4d6cda813de16e1c |
List.forIn'_pure_yield_eq_foldl | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Monadic.lean | theorem forIn'_pure_yield_eq_foldl [Monad m] [LawfulMonad m]
(l : List α) (f : (a : α) → a ∈ l → β → β) (init : β) :
forIn' l init (fun a m b => pure (.yield (f a m b))) =
pure (f := m) (l.attach.foldl (fun b ⟨a, h⟩ => f a h b) init) | m : Type u_1 → Type u_2
α : Type u_3
β : Type u_1
inst✝¹ : Monad m
inst✝ : LawfulMonad m
l : List α
f : (a : α) → a ∈ l → β → β
init : β
⊢ (forIn' l init fun a m_1 b => pure (ForInStep.yield (f a m_1 b))) =
pure
(foldl
(fun b x =>
match x with
| ⟨a, h⟩ => f a h b)
init l.at... | simp only [forIn'_eq_foldlM] | m : Type u_1 → Type u_2
α : Type u_3
β : Type u_1
inst✝¹ : Monad m
inst✝ : LawfulMonad m
l : List α
f : (a : α) → a ∈ l → β → β
init : β
⊢ ForInStep.value <$>
List.foldlM
(fun b x =>
match b with
| ForInStep.yield b => pure (ForInStep.yield (f x.val ⋯ b))
| ForInStep.done b =... | 242f86750c77fe00 |
BoxIntegral.unitPartition.mem_admissibleIndex_of_mem_box | Mathlib/Analysis/BoxIntegral/UnitPartition.lean | theorem mem_admissibleIndex_of_mem_box {B : Box ι} (hB : hasIntegralVertices B) {x : ι → ℝ}
(hx : x ∈ B) : index n x ∈ admissibleIndex n B | case intro.intro.intro.refine_2
ι : Type u_1
n : ℕ
inst✝¹ : NeZero n
inst✝ : Fintype ι
B : Box ι
x : ι → ℝ
hx : x ∈ B
l u : ι → ℤ
hl : ∀ (i : ι), B.lower i = ↑(l i)
hu : ∀ (i : ι), B.upper i = ↑(u i)
i : ι
⊢ (↑⌈↑n * x i⌉ - 1 + 1) / ↑n ≤ ↑(u i) | exact (mem_admissibleIndex_of_mem_box_aux₂ n (x i) (u i)).mp ((hu i) ▸ (hx i).2) | no goals | 1dc74fce82988f92 |
Set.MapsTo.iterate_restrict | Mathlib/Data/Set/Function.lean | theorem MapsTo.iterate_restrict {f : α → α} {s : Set α} (h : MapsTo f s s) (n : ℕ) :
(h.restrict f s s)^[n] = (h.iterate n).restrict _ _ _ | α : Type u_1
f : α → α
s : Set α
h : MapsTo f s s
n : ℕ
⊢ (restrict f s s h)^[n] = restrict f^[n] s s ⋯ | funext x | case h
α : Type u_1
f : α → α
s : Set α
h : MapsTo f s s
n : ℕ
x : ↑s
⊢ (restrict f s s h)^[n] x = restrict f^[n] s s ⋯ x | 46f3dbaf8f18289b |
ContinuousMap.tendsto_of_tendstoLocallyUniformly | Mathlib/Topology/UniformSpace/CompactConvergence.lean | theorem tendsto_of_tendstoLocallyUniformly (h : TendstoLocallyUniformly (fun i a => F i a) f p) :
Tendsto F p (𝓝 f) | α : Type u₁
β : Type u₂
inst✝¹ : TopologicalSpace α
inst✝ : UniformSpace β
f : C(α, β)
ι : Type u₃
p : Filter ι
F : ι → C(α, β)
h : TendstoLocallyUniformly (fun i a => (F i) a) (⇑f) p
K : Set α
hK : IsCompact K
⊢ TendstoLocallyUniformlyOn (fun i a => (F i) a) (⇑f) p K | exact h.tendstoLocallyUniformlyOn | no goals | 2a076f002ba6ff4c |
norm_div_pos_iff | Mathlib/Analysis/Normed/Group/Basic.lean | theorem norm_div_pos_iff : 0 < ‖a / b‖ ↔ a ≠ b | E : Type u_5
inst✝ : NormedGroup E
a b : E
⊢ 0 < ‖a / b‖ ↔ a ≠ b | rw [(norm_nonneg' _).lt_iff_ne, ne_comm] | E : Type u_5
inst✝ : NormedGroup E
a b : E
⊢ ‖a / b‖ ≠ 0 ↔ a ≠ b | a5c810d9d84d6c6c |
SSet.Quasicategory.hornFilling | Mathlib/AlgebraicTopology/Quasicategory/Basic.lean | lemma Quasicategory.hornFilling {S : SSet} [Quasicategory S] ⦃n : ℕ⦄ ⦃i : Fin (n+1)⦄
(h0 : 0 < i) (hn : i < Fin.last n)
(σ₀ : Λ[n, i] ⟶ S) : ∃ σ : Δ[n] ⟶ S, σ₀ = hornInclusion n i ≫ σ | case succ.succ
S : SSet
inst✝ : S.Quasicategory
n : ℕ
i : Fin (n + 1 + 1 + 1)
h0 : 0 < i
hn : i < Fin.last (n + 1 + 1)
σ₀ : Λ[n + 1 + 1, i] ⟶ S
⊢ ∃ σ, σ₀ = hornInclusion (n + 1 + 1) i ≫ σ | exact Quasicategory.hornFilling' σ₀ h0 hn | no goals | 7effe73f43c70d45 |
WeierstrassCurve.Projective.negDblY_of_Z_eq_zero | Mathlib/AlgebraicGeometry/EllipticCurve/Projective.lean | lemma negDblY_of_Z_eq_zero [NoZeroDivisors R] {P : Fin 3 → R} (hP : W'.Equation P) (hPz : P z = 0) :
W'.negDblY P = -P y ^ 4 | R : Type r
inst✝¹ : CommRing R
W' : Projective R
inst✝ : NoZeroDivisors R
P : Fin 3 → R
hP : W'.Equation P
hPz : P z = 0
⊢ ⋯ - ⋯ - ⋯ * 0 ^ 4 + ⋯ * P y ^ 2 * 0 + ⋯ * 0 * P y * 0 ^ 2 + 9 * W'.a₂ ^ 2 * 0 ^ 4 - 8 * W'.a₂ ^ 2 * 0 * P y ^ 2 * 0 -
... | ring1 | no goals | 70a44ce1db5dd732 |
monotone_of_odd_of_monotoneOn_nonneg | Mathlib/Order/Monotone/Odd.lean | theorem monotone_of_odd_of_monotoneOn_nonneg {f : G → H} (h₁ : ∀ x, f (-x) = -f x)
(h₂ : MonotoneOn f (Ici 0)) : Monotone f | G : Type u_1
H : Type u_2
inst✝¹ : LinearOrderedAddCommGroup G
inst✝ : OrderedAddCommGroup H
f : G → H
h₁ : ∀ (x : G), f (-x) = -f x
h₂ : MonotoneOn f (Ici 0)
x : G
hx : x ∈ Iic 0
y : G
hy : y ∈ Iic 0
hxy : x ≤ y
⊢ f (-y) ≤ f (-x) | exact h₂ (neg_nonneg.2 hy) (neg_nonneg.2 hx) (neg_le_neg hxy) | no goals | fca0f3ebe0373e0a |
Finset.UV.kruskal_katona_helper | Mathlib/Combinatorics/SetFamily/KruskalKatona.lean | /-- The main Kruskal-Katona helper: use induction with our measure to keep compressing until
we can't any more, which gives a set family which is fully compressed and has the nice properties we
want. -/
private lemma kruskal_katona_helper {r : ℕ} (𝒜 : Finset (Finset (Fin n)))
(h : (𝒜 : Set (Finset (Fin n))).Sized... | case inr.intro.mk.intro
n r : ℕ
𝒜 : Finset (Finset (Fin n))
h : Set.Sized r ↑𝒜
usable : Finset (Finset (Fin n) × Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ∧ ¬IsCompressed t.1 t.2 𝒜) univ
husable : usable.Nonempty
U V : Finset (Fin n)
hUV : (U, V) ∈ usable
t : ∀ x' ∈ usable, #(U, V).1 ... | rw [mem_filter] at hUV | case inr.intro.mk.intro
n r : ℕ
𝒜 : Finset (Finset (Fin n))
h : Set.Sized r ↑𝒜
usable : Finset (Finset (Fin n) × Finset (Fin n)) :=
filter (fun t => Finset.UV.UsefulCompression t.1 t.2 ∧ ¬IsCompressed t.1 t.2 𝒜) univ
husable : usable.Nonempty
U V : Finset (Fin n)
hUV : (U, V) ∈ univ ∧ Finset.UV.UsefulCompression (... | ad1c271bd9c8d7ce |
Nat.frequently_atTop_modEq_one | Mathlib/NumberTheory/PrimesCongruentOne.lean | theorem frequently_atTop_modEq_one {k : ℕ} (hk0 : k ≠ 0) :
∃ᶠ p in atTop, Nat.Prime p ∧ p ≡ 1 [MOD k] | k : ℕ
hk0 : k ≠ 0
n : ℕ
⊢ ∃ b ≥ n, Prime b ∧ b ≡ 1 [MOD k] | obtain ⟨p, hp⟩ := exists_prime_gt_modEq_one n hk0 | case intro
k : ℕ
hk0 : k ≠ 0
n p : ℕ
hp : Prime p ∧ n < p ∧ p ≡ 1 [MOD k]
⊢ ∃ b ≥ n, Prime b ∧ b ≡ 1 [MOD k] | 3062ae9187c2fafe |
WittVector.ghostFun_natCast | Mathlib/RingTheory/WittVector/Basic.lean | theorem ghostFun_natCast (i : ℕ) : ghostFun (i : 𝕎 R) = i :=
show ghostFun i.unaryCast = _ by
induction i <;>
simp [*, Nat.unaryCast, ghostFun_zero, ghostFun_one, ghostFun_add, -Pi.natCast_def]
| p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : Fact (Nat.Prime p)
i : ℕ
⊢ WittVector.ghostFun i.unaryCast = ↑i | induction i <;>
simp [*, Nat.unaryCast, ghostFun_zero, ghostFun_one, ghostFun_add, -Pi.natCast_def] | no goals | 22bebc9c8a72ffc9 |
PrimeSpectrum.zeroLocus_empty_iff_eq_top | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | theorem zeroLocus_empty_iff_eq_top {I : Ideal R} : zeroLocus (I : Set R) = ∅ ↔ I = ⊤ | case mpr
R : Type u
inst✝ : CommSemiring R
I : Ideal R
⊢ I = ⊤ → zeroLocus ↑I = ∅ | rintro rfl | case mpr
R : Type u
inst✝ : CommSemiring R
⊢ zeroLocus ↑⊤ = ∅ | 3bed0d5e08a59c7f |
CategoryTheory.MorphismProperty.IsStableUnderBaseChange.mk' | Mathlib/CategoryTheory/MorphismProperty/Limits.lean | theorem IsStableUnderBaseChange.mk' [RespectsIso P]
(hP₂ : ∀ (X Y S : C) (f : X ⟶ S) (g : Y ⟶ S) [HasPullback f g] (_ : P g),
P (pullback.fst f g)) :
IsStableUnderBaseChange P where
of_isPullback {X Y Y' S f g f' g'} sq hg | C : Type u
inst✝¹ : Category.{v, u} C
P : MorphismProperty C
inst✝ : P.RespectsIso
hP₂ : ∀ (X Y S : C) (f : X ⟶ S) (g : Y ⟶ S) [inst : HasPullback f g], P g → P (pullback.fst f g)
X Y Y' S : C
f : X ⟶ S
g : Y ⟶ S
f' : Y' ⟶ Y
g' : Y' ⟶ X
sq : IsPullback f' g' g f
hg : P g
this : HasPullback f g
e : Y' ≅ pullback f g := ... | exact hP₂ _ _ _ f g hg | no goals | a7faae7a4ba1aa38 |
ConvexOn.le_right_of_left_le' | Mathlib/Analysis/Convex/Function.lean | theorem ConvexOn.le_right_of_left_le' (hf : ConvexOn 𝕜 s f) {x y : E} {a b : 𝕜} (hx : x ∈ s)
(hy : y ∈ s) (ha : 0 ≤ a) (hb : 0 < b) (hab : a + b = 1) (hfx : f x ≤ f (a • x + b • y)) :
f (a • x + b • y) ≤ f y | 𝕜 : Type u_1
E : Type u_2
β : Type u_5
inst✝⁵ : OrderedSemiring 𝕜
inst✝⁴ : AddCommMonoid E
inst✝³ : LinearOrderedCancelAddCommMonoid β
inst✝² : SMul 𝕜 E
inst✝¹ : Module 𝕜 β
inst✝ : OrderedSMul 𝕜 β
s : Set E
f : E → β
hf : ConvexOn 𝕜 s f
x y : E
a b : 𝕜
hx : x ∈ s
hy : y ∈ s
ha : 0 ≤ a
hb : 0 < b
hab : b + a = 1
... | exact hf.le_left_of_right_le' hy hx hb ha hab hfx | no goals | 1b3839f8b3fe1c83 |
EulerProduct.one_sub_inv_eq_geometric_of_summable_norm | Mathlib/NumberTheory/EulerProduct/Basic.lean | lemma one_sub_inv_eq_geometric_of_summable_norm {f : ℕ →*₀ F} {p : ℕ} (hp : p.Prime)
(hsum : Summable fun x ↦ ‖f x‖) :
(1 - f p)⁻¹ = ∑' (e : ℕ), f (p ^ e) | F : Type u_1
inst✝¹ : NormedField F
inst✝ : CompleteSpace F
f : ℕ →*₀ F
p : ℕ
hp : Nat.Prime p
hsum : Summable fun x => ‖f x‖
⊢ Summable fun n => f p ^ n | refine Summable.of_norm ?_ | F : Type u_1
inst✝¹ : NormedField F
inst✝ : CompleteSpace F
f : ℕ →*₀ F
p : ℕ
hp : Nat.Prime p
hsum : Summable fun x => ‖f x‖
⊢ Summable fun a => ‖f p ^ a‖ | bde940928d778afb |
List.cons_le_cons_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lex.lean | theorem cons_le_cons_iff [DecidableEq α] [LT α] [DecidableLT α]
[i₀ : Std.Irrefl (· < · : α → α → Prop)]
[i₁ : Std.Asymm (· < · : α → α → Prop)]
[i₂ : Std.Antisymm (¬ · < · : α → α → Prop)]
{a b} {l₁ l₂ : List α} :
(a :: l₁) ≤ (b :: l₂) ↔ a < b ∨ a = b ∧ l₁ ≤ l₂ | case h
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : LT α
inst✝ : DecidableLT α
i₀ : Std.Irrefl fun x1 x2 => x1 < x2
i₁ : Std.Asymm fun x1 x2 => x1 < x2
i₂ : Std.Antisymm fun x1 x2 => ¬x1 < x2
a b : α
l₁ l₂ : List α
h₁ : ¬b < a
h₂ : ¬Lex (fun x1 x2 => x1 < x2) l₂ l₁
h₃ : ¬a < b
⊢ a = b ∧ ¬Lex (fun x1 x2 => x1 < x2) l₂ l... | exact ⟨i₂.antisymm _ _ h₃ h₁, h₂⟩ | no goals | 75deba79ccf9ad38 |
Set.offDiag_union | Mathlib/Data/Set/Prod.lean | theorem offDiag_union (h : Disjoint s t) :
(s ∪ t).offDiag = s.offDiag ∪ t.offDiag ∪ s ×ˢ t ∪ t ×ˢ s | case h.mpr.inl.inr.intro
α : Type u_1
s t : Set α
h : Disjoint s t
x : α × α
h0 : x.2 ∈ s
h1 : x.2 ∈ t
h3 : x.1 = x.2
⊢ False | exact Set.disjoint_left.mp h h0 h1 | no goals | e368a686903f78bc |
cfcₙHom_eq_cfcₙ_extend | Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/NonUnital.lean | lemma cfcₙHom_eq_cfcₙ_extend {a : A} (g : R → R) (ha : p a) (f : C(σₙ R a, R)₀) :
cfcₙHom ha f = cfcₙ (Function.extend Subtype.val f g) a | R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : S... | ext | case h
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
in... | 21b58a7c62c97b99 |
TensorProduct.equivFinsuppOfBasisRight_apply_tmul_apply | Mathlib/LinearAlgebra/TensorProduct/Basis.lean | lemma TensorProduct.equivFinsuppOfBasisRight_apply_tmul_apply
(m : M) (n : N) (i : κ) :
(TensorProduct.equivFinsuppOfBasisRight 𝒞) (m ⊗ₜ n) i =
𝒞.repr n i • m | R : Type u_1
M : Type u_3
N : Type u_4
κ : Type u_6
inst✝⁵ : CommSemiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
inst✝² : AddCommMonoid N
inst✝¹ : Module R N
inst✝ : DecidableEq κ
𝒞 : Basis κ R N
m : M
n : N
i : κ
⊢ ((equivFinsuppOfBasisRight 𝒞) (m ⊗ₜ[R] n)) i = (𝒞.repr n) i • m | simp only [equivFinsuppOfBasisRight_apply_tmul, Finsupp.mapRange_apply] | no goals | 90ef4be1340cec36 |
AffineSubspace.isConnected_setOf_sOppSide | Mathlib/Analysis/Convex/Side.lean | theorem isConnected_setOf_sOppSide {s : AffineSubspace ℝ P} {x : P} (hx : x ∉ s)
(h : (s : Set P).Nonempty) : IsConnected { y | s.SOppSide x y } | case intro
V : Type u_2
P : Type u_4
inst✝³ : SeminormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
s : AffineSubspace ℝ P
x : P
hx : x ∉ s
p : P
hp : p ∈ ↑s
this : Nonempty ↥s
⊢ IsConnected ((fun x_1 => x_1.1 • (x -ᵥ p) +ᵥ x_1.2) '' Set.Iio 0 ×ˢ ↑s) | refine (isConnected_Iio.prod (isConnected_iff_connectedSpace.2 ?_)).image _
((continuous_fst.smul continuous_const).vadd continuous_snd).continuousOn | case intro
V : Type u_2
P : Type u_4
inst✝³ : SeminormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
s : AffineSubspace ℝ P
x : P
hx : x ∉ s
p : P
hp : p ∈ ↑s
this : Nonempty ↥s
⊢ ConnectedSpace ↑↑s | e51bcb5e18670498 |
ZMod.dft_even_iff | Mathlib/Analysis/Fourier/ZMod.lean | /-- The discrete Fourier transform of `Φ` is even if and only if `Φ` itself is. -/
lemma dft_even_iff {Φ : ZMod N → ℂ} : (𝓕 Φ).Even ↔ Φ.Even | N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
h : ∀ {f : ZMod N → ℂ}, Function.Even f → Function.Even (𝓕 f)
⊢ Function.Even (𝓕 Φ) ↔ Function.Even Φ | refine ⟨fun hΦ x ↦ ?_, h⟩ | N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
h : ∀ {f : ZMod N → ℂ}, Function.Even f → Function.Even (𝓕 f)
hΦ : Function.Even (𝓕 Φ)
x : ZMod N
⊢ Φ (-x) = Φ x | 16cf2a81b38e6f67 |
DividedPowers.ext | Mathlib/RingTheory/DividedPowers/Basic.lean | theorem DividedPowers.ext (hI : DividedPowers I) (hI' : DividedPowers I)
(h_eq : ∀ (n : ℕ) {x : A} (_ : x ∈ I), hI.dpow n x = hI'.dpow n x) :
hI = hI' | case mk
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI' : DividedPowers I
hI : ℕ → A → A
h₀ : ∀ {n : ℕ} {x : A}, x ∉ I → hI n x = 0
dpow_zero✝ : ∀ {x : A}, x ∈ I → hI 0 x = 1
dpow_one✝ : ∀ {x : A}, x ∈ I → hI 1 x = x
dpow_mem✝ : ∀ {n : ℕ} {x : A}, n ≠ 0 → x ∈ I → hI n x ∈ I
dpow_add✝ : ∀ {n : ℕ} {x y : A}, x ∈ I → ... | obtain ⟨hI', h₀', _⟩ := hI' | case mk.mk
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
hI : ℕ → A → A
h₀ : ∀ {n : ℕ} {x : A}, x ∉ I → hI n x = 0
dpow_zero✝¹ : ∀ {x : A}, x ∈ I → hI 0 x = 1
dpow_one✝¹ : ∀ {x : A}, x ∈ I → hI 1 x = x
dpow_mem✝¹ : ∀ {n : ℕ} {x : A}, n ≠ 0 → x ∈ I → hI n x ∈ I
dpow_add✝¹ : ∀ {n : ℕ} {x y : A}, x ∈ I → y ∈ I → hI n (x... | 3b635b1fbc30763f |
Real.summable_exp_nat_mul_iff | Mathlib/Analysis/SpecialFunctions/Exp.lean | lemma summable_exp_nat_mul_iff {a : ℝ} :
Summable (fun n : ℕ ↦ exp (n * a)) ↔ a < 0 | a : ℝ
⊢ (Summable fun n => rexp (↑n * a)) ↔ a < 0 | simp only [exp_nat_mul, summable_geometric_iff_norm_lt_one, norm_of_nonneg (exp_nonneg _),
exp_lt_one_iff] | no goals | d02e87599149c19e |
Path.trans_range | Mathlib/Topology/Path.lean | theorem trans_range {a b c : X} (γ₁ : Path a b) (γ₂ : Path b c) :
range (γ₁.trans γ₂) = range γ₁ ∪ range γ₂ | case h
X : Type u_1
inst✝ : TopologicalSpace X
a b c : X
γ₁ : Path a b
γ₂ : Path b c
x : X
t : ℝ
ht0 : 0 ≤ t
ht1 : t ≤ 1
hxt : γ₂ ⟨t, ⋯⟩ = x
h : t = 0
⊢ { toFun := (fun t => if t ≤ 1 / 2 then γ₁.extend (2 * t) else γ₂.extend (2 * t - 1)) ∘ Subtype.val,
continuous_toFun := ⋯, source' := ⋯, target' := ⋯ }
⟨... | rw [coe_mk_mk, Function.comp_apply, if_pos le_rfl, Subtype.coe_mk,
mul_one_div_cancel (two_ne_zero' ℝ)] | case h
X : Type u_1
inst✝ : TopologicalSpace X
a b c : X
γ₁ : Path a b
γ₂ : Path b c
x : X
t : ℝ
ht0 : 0 ≤ t
ht1 : t ≤ 1
hxt : γ₂ ⟨t, ⋯⟩ = x
h : t = 0
⊢ γ₁.extend 1 = x | 4045b3b0d5c91c52 |
Submodule.basis_of_pid_aux | Mathlib/LinearAlgebra/FreeModule/PID.lean | theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type*} [AddCommGroup O] [Module R O]
(M N : Submodule R O) (b'M : Basis ι R M) (N_bot : N ≠ ⊥) (N_le_M : N ≤ M) :
∃ y ∈ M, ∃ a : R, a • y ∈ N ∧ ∃ M' ≤ M, ∃ N' ≤ N,
N' ≤ M' ∧ (∀ (c : R) (z : O), z ∈ M' → c • y + z = 0 → c = 0) ∧
(∀ (c : R) (z : O), z... | case neg.intro.intro.intro.refine_1.refine_2.intro
ι : Type u_1
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
inst✝² : Finite ι
O : Type u_4
inst✝¹ : AddCommGroup O
inst✝ : Module R O
M N : Submodule R O
b'M : Basis ι R ↥M
N_bot : N ≠ ⊥
N_le_M : N ≤ M
this : ∃ ϕ, ∀ (ψ : ↥M →ₗ[R] R... | refine ⟨-b, Submodule.mem_map.mpr ⟨⟨_, N.sub_mem zN (N.smul_mem b yN)⟩, ?_, ?_⟩⟩ | case neg.intro.intro.intro.refine_1.refine_2.intro.refine_1
ι : Type u_1
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
inst✝² : Finite ι
O : Type u_4
inst✝¹ : AddCommGroup O
inst✝ : Module R O
M N : Submodule R O
b'M : Basis ι R ↥M
N_bot : N ≠ ⊥
N_le_M : N ≤ M
this : ∃ ϕ, ∀ (ψ : ↥... | 48ee91e21f4317c3 |
NumberField.RingOfIntegers.HeightOneSpectrum.one_lt_absNorm | Mathlib/NumberTheory/NumberField/FinitePlaces.lean | /-- The norm of a maximal ideal is `> 1` -/
lemma one_lt_absNorm : 1 < absNorm v.asIdeal | K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
v : HeightOneSpectrum (𝓞 K)
h : absNorm v.asIdeal ≤ 1
⊢ Finite (𝓞 K ⧸ v.asIdeal) | exact (v.asIdeal.fintypeQuotientOfFreeOfNeBot v.ne_bot).finite | no goals | 849af9e987d28ab6 |
List.mem_range' | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Range.lean | theorem mem_range' : ∀{n}, m ∈ range' s n step ↔ ∃ i < n, m = s + step * i
| 0 => by simp [range', Nat.not_lt_zero]
| n + 1 => by
have h (i) : i ≤ n ↔ i = 0 ∨ ∃ j, i = succ j ∧ j < n | s step m : Nat
⊢ m ∈ range' s 0 step ↔ ∃ i, i < 0 ∧ m = s + step * i | simp [range', Nat.not_lt_zero] | no goals | 1c34e10b4040f331 |
ContractingWith.isFixedPt_fixedPoint_iterate | Mathlib/Topology/MetricSpace/Contracting.lean | theorem isFixedPt_fixedPoint_iterate {n : ℕ} (hf : ContractingWith K f^[n]) :
IsFixedPt f (hf.fixedPoint f^[n]) | α : Type u_1
inst✝² : MetricSpace α
K : ℝ≥0
f : α → α
inst✝¹ : Nonempty α
inst✝ : CompleteSpace α
n : ℕ
hf : ContractingWith K f^[n]
x : α := fixedPoint f^[n] hf
hx : f^[n] x = x
this✝ : ¬IsFixedPt f x
this : 0 < dist x (f x)
⊢ ↑K * dist x (f x) < dist x (f x) | simpa only [NNReal.coe_one, one_mul, NNReal.val_eq_coe] using (mul_lt_mul_right this).mpr hf.left | no goals | c937df74ec08dd68 |
FormalMultilinearSeries.changeOrigin_eval | Mathlib/Analysis/Analytic/ChangeOrigin.lean | theorem changeOrigin_eval (h : (‖x‖₊ + ‖y‖₊ : ℝ≥0∞) < p.radius) :
(p.changeOrigin x).sum y = p.sum (x + y) | case mk.mk.mk
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
p : FormalMultilinearSeries 𝕜 E F
x y : E
h : ↑‖x‖₊ + ↑‖y‖₊ < p.radius
radius_pos : 0 < p.rad... | exact p.nnnorm_changeOriginSeriesTerm_apply_le _ _ _ _ _ _ | no goals | ccc324cf6a7a6ae5 |
SimpleGraph.Subgraph.comap_monotone | Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) | case right
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
f : G →g G'
H H' : G'.Subgraph
h : H ≤ H'
v w : V
⊢ G.Adj v w → H.Adj (f v) (f w) → H'.Adj (f v) (f w) | intro | case right
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
f : G →g G'
H H' : G'.Subgraph
h : H ≤ H'
v w : V
a✝ : G.Adj v w
⊢ H.Adj (f v) (f w) → H'.Adj (f v) (f w) | b65a5ecf003263d1 |
Stonean.epi_iff_surjective | Mathlib/Topology/Category/Stonean/Basic.lean | /--
A morphism in `Stonean` is an epi iff it is surjective.
-/
lemma epi_iff_surjective {X Y : Stonean} (f : X ⟶ Y) :
Epi f ↔ Function.Surjective f | X Y : Stonean
f : X ⟶ Y
h : Epi f
y : ↑Y.toTop
hy : ∀ (a : ↑X.toTop), (ConcreteCategory.hom f) a ≠ y
⊢ False | let C := Set.range f | X Y : Stonean
f : X ⟶ Y
h : Epi f
y : ↑Y.toTop
hy : ∀ (a : ↑X.toTop), (ConcreteCategory.hom f) a ≠ y
C : Set ((fun X => ↑X.toTop) Y) := Set.range ⇑(ConcreteCategory.hom f)
⊢ False | 333cfe532dfd4152 |
AkraBazziRecurrence.GrowsPolynomially.mul | Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean | protected lemma GrowsPolynomially.mul {f g : ℝ → ℝ} (hf : GrowsPolynomially f)
(hg : GrowsPolynomially g) : GrowsPolynomially fun x => f x * g x | f g : ℝ → ℝ
hf✝¹ : GrowsPolynomially f
hg✝¹ : GrowsPolynomially g
b : ℝ
hb : b ∈ Set.Ioo 0 1
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf✝ :
∀ᶠ (x : ℝ) in atTop,
∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|) u ∈ Set.Icc (c₁ * (fun x => |f x|) x) (c₂ * (fun x => |f x|) x)
c₃ : ℝ
hc₃_mem : c₃ > 0
c₄ : ℝ
hc₄_mem :... | ring | no goals | c399539f7cb9b401 |
MeasureTheory.Measure.eq_withDensity_rnDeriv | Mathlib/MeasureTheory/Decomposition/Lebesgue.lean | theorem eq_withDensity_rnDeriv {s : Measure α} {f : α → ℝ≥0∞} (hf : Measurable f) (hs : s ⟂ₘ ν)
(hadd : μ = s + ν.withDensity f) : ν.withDensity f = ν.withDensity (μ.rnDeriv ν) | case intro.intro
α : Type u_1
m : MeasurableSpace α
μ ν s : Measure α
f : α → ℝ≥0∞
hf : Measurable f
hs : s ⟂ₘ ν
hadd : μ = s + ν.withDensity f
this : μ.HaveLebesgueDecomposition ν
hmeas : Measurable (μ.rnDeriv ν)
hsing : μ.singularPart ν ⟂ₘ ν
hadd' : μ = μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)
⊢ ν.withDensity f... | obtain ⟨⟨S, hS₁, hS₂, hS₃⟩, ⟨T, hT₁, hT₂, hT₃⟩⟩ := hs, hsing | case intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
m : MeasurableSpace α
μ ν s : Measure α
f : α → ℝ≥0∞
hf : Measurable f
hadd : μ = s + ν.withDensity f
this : μ.HaveLebesgueDecomposition ν
hmeas : Measurable (μ.rnDeriv ν)
hadd' : μ = μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)
S : Set α
hS₁ : Measura... | fc3f06ec8038d87c |
Cycle.Chain.imp | Mathlib/Data/List/Cycle.lean | theorem Chain.imp {r₁ r₂ : α → α → Prop} (H : ∀ a b, r₁ a b → r₂ a b) (p : Chain r₁ s) :
Chain r₂ s | case HI
α : Type u_1
s : Cycle α
r₁ r₂ : α → α → Prop
H : ∀ (a b : α), r₁ a b → r₂ a b
a✝¹ : α
l✝ : List α
a✝ : Chain r₁ ↑l✝ → Chain r₂ ↑l✝
p : List.Chain r₁ a✝¹ (l✝ ++ [a✝¹])
⊢ List.Chain r₂ a✝¹ (l✝ ++ [a✝¹]) | exact p.imp H | no goals | f7c8f7679a6fc82c |
Std.DHashMap.Raw.Const.getKeyD_ofList_of_contains_eq_false | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/RawLemmas.lean | theorem getKeyD_ofList_of_contains_eq_false [EquivBEq α] [LawfulHashable α]
{l : List (α × β)} {k fallback : α}
(contains_eq_false : (l.map Prod.fst).contains k = false) :
(ofList l).getKeyD k fallback = fallback | α : Type u
inst✝³ : BEq α
inst✝² : Hashable α
β : Type v
inst✝¹ : EquivBEq α
inst✝ : LawfulHashable α
l : List (α × β)
k fallback : α
contains_eq_false : (List.map Prod.fst l).contains k = false
⊢ (ofList l).getKeyD k fallback = fallback | simp_to_raw using Raw₀.Const.getKeyD_insertMany_empty_list_of_contains_eq_false | no goals | d55c5bbc0972f7ce |
Ideal.mem_pointwise_smul_iff_inv_smul_mem | Mathlib/RingTheory/Ideal/Pointwise.lean | theorem mem_pointwise_smul_iff_inv_smul_mem {a : M} {S : Ideal R} {x : R} :
x ∈ a • S ↔ a⁻¹ • x ∈ S :=
⟨fun h => by simpa using smul_mem_pointwise_smul a⁻¹ _ _ h,
fun h => by simpa using smul_mem_pointwise_smul a _ _ h⟩
| M : Type u_1
R : Type u_2
inst✝² : Group M
inst✝¹ : Semiring R
inst✝ : MulSemiringAction M R
a : M
S : Ideal R
x : R
h : x ∈ a • S
⊢ a⁻¹ • x ∈ S | simpa using smul_mem_pointwise_smul a⁻¹ _ _ h | no goals | 02576b44d4ec2894 |
MeasureTheory.Measure.OuterRegular.of_restrict | Mathlib/MeasureTheory/Measure/Regular.lean | /-- If the restrictions of a measure to countably many open sets covering the space are
outer regular, then the measure itself is outer regular. -/
lemma of_restrict [OpensMeasurableSpace α] {μ : Measure α} {s : ℕ → Set α}
(h : ∀ n, OuterRegular (μ.restrict (s n))) (h' : ∀ n, IsOpen (s n)) (h'' : univ ⊆ ⋃ n, s n) :... | α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen (s n)
h'' : univ ⊆ ⋃ n, s n
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (s n)
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)... | intro n | α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen (s n)
h'' : univ ⊆ ⋃ n, s n
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (s n)
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)... | aae30475f9af97a9 |
Real.fourierCoeff_tsum_comp_add | Mathlib/Analysis/Fourier/PoissonSummation.lean | theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)}
(hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖)
(m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 f m | f : C(ℝ, ℂ)
hf : ∀ (K : Compacts ℝ), Summable fun n => ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight ↑n))‖
m : ℤ
e : C(ℝ, ℂ) := (fourier (-m)).comp { toFun := QuotientAddGroup.mk, continuous_toFun := ⋯ }
neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMap.restri... | simp_rw [eadd] | no goals | ec2dd51bd198a0fb |
AkraBazziRecurrence.GrowsPolynomially.eventually_atTop_nonneg_or_nonpos | Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean | lemma eventually_atTop_nonneg_or_nonpos (hf : GrowsPolynomially f) :
(∀ᶠ x in atTop, 0 ≤ f x) ∨ (∀ᶠ x in atTop, f x ≤ 0) | f : ℝ → ℝ
hf : GrowsPolynomially f
c₁ : ℝ
left✝¹ : c₁ > 0
c₂ : ℝ
left✝ : c₂ > 0
h : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
heq : c₁ = c₂
c : ℝ
hc✝ : ∀ᶠ (x : ℝ) in atTop, f x = c
hneg : c < 0
x : ℝ
hc : f x = c
⊢ f x < 0 | simpa only [hc] | no goals | 69674f7264744f71 |
Std.DHashMap.Internal.List.mem_alterKey_of_key_ne | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/List/Associative.lean | theorem mem_alterKey_of_key_ne {a : α} {f : Option (β a) → Option (β a)}
{l : List ((a : α) × β a)} (p : (a : α) × β a) (hne : p.1 ≠ a) :
p ∈ alterKey a f l ↔ p ∈ l | α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : LawfulBEq α
a : α
f : Option (β a) → Option (β a)
l : List ((a : α) × β a)
p : (a : α) × β a
hne : p.fst ≠ a
⊢ p ∈ alterKey a f l ↔ p ∈ l | rw [alterKey] | α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : LawfulBEq α
a : α
f : Option (β a) → Option (β a)
l : List ((a : α) × β a)
p : (a : α) × β a
hne : p.fst ≠ a
⊢ (p ∈
match f (getValueCast? a l) with
| none => eraseKey a l
| some v => insertEntry a v l) ↔
p ∈ l | c3b5d8c9bced3123 |
Set.range_list_getElem? | Mathlib/Data/Set/List.lean | theorem range_list_getElem? :
range (l[·]? : ℕ → Option α) = insert none (some '' { x | x ∈ l }) | case refine_2
α : Type u_1
l : List α
⊢ none ∈ range fun x => l[x]? | exact ⟨_, getElem?_eq_none_iff.mpr le_rfl⟩ | no goals | ac656803bb38f6d4 |
doublyStochastic_sum_perm_aux | Mathlib/Analysis/Convex/Birkhoff.lean | /--
If M is a scalar multiple of a doubly stochastic matrix, then it is a conical combination of
permutation matrices. This is most useful when M is a doubly stochastic matrix, in which case
the combination is convex.
This particular formulation is chosen to make the inductive step easier: we no longer need to
rescale... | R : Type u_1
n : Type u_2
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : LinearOrderedField R
h✝ : Nonempty n
d : ℕ
ih :
∀ m < d,
∀ (M : Matrix n n R) (s : R),
0 ≤ s →
(∃ M' ∈ doublyStochastic R n, M = s • M') →
#(filter (fun i => M i.1 i.2 ≠ 0) univ) = m →
∃ w, (∀ (σ : Equi... | exact single_le_sum (fun j _ => hM.1 i j) (by simp) | no goals | fe82d3c50ee8248f |
MeasureTheory.aestronglyMeasurable_condExpL1CLM | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean | theorem aestronglyMeasurable_condExpL1CLM (f : α →₁[μ] F') :
AEStronglyMeasurable[m] (condExpL1CLM F' hm μ f) μ | case refine_1
α : Type u_1
F' : Type u_3
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
f : ↥(Lp F' 1 μ)
c : F'
s : Set α
hs : MeasurableSet s
hμs : μ s < ⊤
⊢ AEStronglyMeasurable (↑↑((condExpL1CLM F' ... | rw [condExpL1CLM_indicatorConst hs hμs.ne c] | case refine_1
α : Type u_1
F' : Type u_3
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
f : ↥(Lp F' 1 μ)
c : F'
s : Set α
hs : MeasurableSet s
hμs : μ s < ⊤
⊢ AEStronglyMeasurable (↑↑((condExpInd F' hm... | 3f7d93520dda47e8 |
IntermediateField.Lifts.union_isExtendible | Mathlib/FieldTheory/Extension.lean | theorem union_isExtendible [alg : Algebra.IsAlgebraic F E]
[Nonempty c] (hext : ∀ σ ∈ c, σ.IsExtendible) :
(union c hc).IsExtendible := fun S ↦ by
let Ω := adjoin F (S : Set E) →ₐ[F] K
have ⟨ω, hω⟩ : ∃ ω : Ω, ∀ π : c, ∃ θ ≥ π.1, ⟨_, ω⟩ ≤ θ ∧ θ.carrier = π.1.1 ⊔ adjoin F S | F : Type u_1
E : Type u_2
K : Type u_3
inst✝⁵ : Field F
inst✝⁴ : Field E
inst✝³ : Field K
inst✝² : Algebra F E
inst✝¹ : Algebra F K
c : Set (Lifts F E K)
hc : IsChain (fun x1 x2 => x1 ≤ x2) c
alg : Algebra.IsAlgebraic F E
inst✝ : Nonempty ↑c
hext : ∀ σ ∈ c, σ.IsExtendible
S : Finset E
Ω : Type (max u_2 u_3) := ↥(adjoin... | simp_rw [carrier_union, iSup_range', eq] | F : Type u_1
E : Type u_2
K : Type u_3
inst✝⁵ : Field F
inst✝⁴ : Field E
inst✝³ : Field K
inst✝² : Algebra F E
inst✝¹ : Algebra F K
c : Set (Lifts F E K)
hc : IsChain (fun x1 x2 => x1 ≤ x2) c
alg : Algebra.IsAlgebraic F E
inst✝ : Nonempty ↑c
hext : ∀ σ ∈ c, σ.IsExtendible
S : Finset E
Ω : Type (max u_2 u_3) := ↥(adjoin... | f447a1810182010c |
frontier_univ_prod_eq | Mathlib/Topology/Constructions.lean | theorem frontier_univ_prod_eq (s : Set Y) :
frontier ((univ : Set X) ×ˢ s) = univ ×ˢ frontier s | X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set Y
⊢ frontier (univ ×ˢ s) = univ ×ˢ frontier s | simp [frontier_prod_eq] | no goals | b3fc2d5c16ff578b |
List.fst_lt_add_of_mem_enumFrom | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/Range.lean | theorem fst_lt_add_of_mem_enumFrom {x : Nat × α} {n : Nat} {l : List α} (h : x ∈ enumFrom n l) :
x.1 < n + length l | case intro
α : Type u_1
n : Nat
l : List α
i : Fin (enumFrom n l).length
h : (enumFrom n l).get i ∈ enumFrom n l
⊢ ((enumFrom n l).get i).fst < n + l.length | simpa using i.isLt | no goals | c1ce1a46fa439db5 |
MeasureTheory.IsSetSemiring.disjointOfUnion_props | Mathlib/MeasureTheory/SetSemiring.lean | theorem disjointOfUnion_props (hC : IsSetSemiring C) (h1 : ↑J ⊆ C) :
∃ K : Set α → Finset (Set α),
PairwiseDisjoint J K
∧ (∀ i ∈ J, ↑(K i) ⊆ C)
∧ PairwiseDisjoint (⋃ x ∈ J, (K x : Set (Set α))) id
∧ (∀ j ∈ J, ⋃₀ K j ⊆ j)
∧ (∀ j ∈ J, ∅ ∉ K j)
∧ ⋃₀ J = ⋃₀ (⋃ x ∈ J, (K x : Set (Set ... | case h.refine_6
α : Type u_1
C : Set (Set α)
J✝ : Finset (Set α)
hC : IsSetSemiring C
s : Set α
J : Finset (Set α)
hJ : s ∉ J
hind :
↑J ⊆ C →
∃ K,
(↑J).PairwiseDisjoint K ∧
(∀ i ∈ J, ↑(K i) ⊆ C) ∧
(⋃ x ∈ J, ↑(K x)).PairwiseDisjoint id ∧
(∀ j ∈ J, ⋃₀ ↑(K j) ⊆ j) ∧ (∀ j ∈ J, ∅ ∉ ... | simp only [iUnion_iUnion_eq_or_left, ht2, sUnion_union, ht2, K1] | case h.refine_6
α : Type u_1
C : Set (Set α)
J✝ : Finset (Set α)
hC : IsSetSemiring C
s : Set α
J : Finset (Set α)
hJ : s ∉ J
hind :
↑J ⊆ C →
∃ K,
(↑J).PairwiseDisjoint K ∧
(∀ i ∈ J, ↑(K i) ⊆ C) ∧
(⋃ x ∈ J, ↑(K x)).PairwiseDisjoint id ∧
(∀ j ∈ J, ⋃₀ ↑(K j) ⊆ j) ∧ (∀ j ∈ J, ∅ ∉ ... | defbf789029f4cb3 |
FractionalIdeal.isFractional_span_iff | Mathlib/RingTheory/FractionalIdeal/Operations.lean | theorem isFractional_span_iff {s : Set P} :
IsFractional S (span R s) ↔ ∃ a ∈ S, ∀ b : P, b ∈ s → IsInteger R (a • b) :=
⟨fun ⟨a, a_mem, h⟩ => ⟨a, a_mem, fun b hb => h b (subset_span hb)⟩, fun ⟨a, a_mem, h⟩ =>
⟨a, a_mem, fun _ hb =>
span_induction (hx := hb) h
(by
rw [smul_zero]
... | R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
s : Set P
x✝³ : ∃ a ∈ S, ∀ b ∈ s, IsInteger R (a • b)
a : R
a_mem : a ∈ S
h : ∀ b ∈ s, IsInteger R (a • b)
x✝² : P
hb : x✝² ∈ span R s
x y : P
x✝¹ : x ∈ span R s
x✝ : y ∈ span R s
hx : IsInteger R (a • x)
hy : IsIntege... | rw [smul_add] | R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
s : Set P
x✝³ : ∃ a ∈ S, ∀ b ∈ s, IsInteger R (a • b)
a : R
a_mem : a ∈ S
h : ∀ b ∈ s, IsInteger R (a • b)
x✝² : P
hb : x✝² ∈ span R s
x y : P
x✝¹ : x ∈ span R s
x✝ : y ∈ span R s
hx : IsInteger R (a • x)
hy : IsIntege... | 6aeb6df4d04dfba9 |
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos' | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean | theorem exists_subset_restrict_nonpos' (hi₁ : MeasurableSet i) (hi₂ : s i < 0)
(hn : ¬∀ n : ℕ, ¬s ≤[i \ ⋃ l < n, restrictNonposSeq s i l] 0) :
∃ j : Set α, MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ s j < 0 | case neg.h
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h : ¬s ≤[i] 0
hn : ∃ n, s ≤[i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
k : ℕ := Nat.find hn
hk₂ : s ≤[i \ ⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonpo... | exact restrictNonposSeq_subset _ hx | no goals | 37ca223bdee11c33 |
Mathlib.Tactic.LinearCombination'.eq_of_add | Mathlib/Tactic/LinearCombination'.lean | theorem eq_of_add [AddGroup α] (p : (a:α) = b) (H : (a' - b') - (a - b) = 0) : a' = b' | α : Type u_1
a a' b b' : α
inst✝ : AddGroup α
p : a - b = 0
H : a' - b' - (a - b) = 0
⊢ a' - b' = 0 | rwa [sub_eq_zero, p] at H | no goals | cd0168b15ad88650 |
CategoryTheory.ChosenFiniteProducts.prodComparisonBifunctorNatTrans_comp | Mathlib/CategoryTheory/ChosenFiniteProducts.lean | theorem prodComparisonBifunctorNatTrans_comp {E : Type u₂} [Category.{v₂} E]
[ChosenFiniteProducts E] (G : D ⥤ E) : prodComparisonBifunctorNatTrans (F ⋙ G) =
whiskerRight (prodComparisonBifunctorNatTrans F) ((whiskeringRight _ _ _).obj G) ≫
whiskerLeft F (whiskerRight (prodComparisonBifunctorNatTrans ... | case w.h.w.h
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : ChosenFiniteProducts C
D : Type u₁
inst✝³ : Category.{v₁, u₁} D
inst✝² : ChosenFiniteProducts D
F : C ⥤ D
E : Type u₂
inst✝¹ : Category.{v₂, u₂} E
inst✝ : ChosenFiniteProducts E
G : D ⥤ E
x✝¹ x✝ : C
⊢ ((prodComparisonBifunctorNatTrans (F ⋙ G)).app x✝¹).app x✝ ... | simp [prodComparison_comp] | no goals | f893aa99a63240f8 |
List.Duplicate.mono_sublist | Mathlib/Data/List/Duplicate.lean | theorem Duplicate.mono_sublist {l' : List α} (hx : x ∈+ l) (h : l <+ l') : x ∈+ l' | case cons₂
α : Type u_1
l : List α
x : α
l' l₁ l₂ : List α
y : α
h : l₁ <+ l₂
IH : x ∈+ l₁ → x ∈+ l₂
hx : x ∈+ y :: l₁
⊢ x ∈+ y :: l₂ | rw [duplicate_cons_iff] at hx ⊢ | case cons₂
α : Type u_1
l : List α
x : α
l' l₁ l₂ : List α
y : α
h : l₁ <+ l₂
IH : x ∈+ l₁ → x ∈+ l₂
hx : y = x ∧ x ∈ l₁ ∨ x ∈+ l₁
⊢ y = x ∧ x ∈ l₂ ∨ x ∈+ l₂ | 81309db93c0eb7b6 |
MeasureTheory.Content.innerContent_pos_of_is_mul_left_invariant | Mathlib/MeasureTheory/Measure/Content.lean | theorem innerContent_pos_of_is_mul_left_invariant [Group G] [IsTopologicalGroup G]
(h3 : ∀ (g : G) {K : Compacts G}, μ (K.map _ <| continuous_mul_left g) = μ K) (K : Compacts G)
(hK : μ K ≠ 0) (U : Opens G) (hU : (U : Set G).Nonempty) : 0 < μ.innerContent U | G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
h3 : ∀ (g : G) {K : Compacts G}, μ (Compacts.map (fun b => g * b) ⋯ K) = μ K
K : Compacts G
hK : μ K ≠ 0
U : Opens G
hU : (↑U).Nonempty
this : (interior ↑U).Nonempty
s : Finset G
hs : K.carrier ⊆ ⋃ g ∈ s, (fun x => g * x)... | simpa only [Opens.iSup_def, Opens.coe_comap, Subtype.coe_mk] | no goals | 08eb0f8d74959da1 |
SmoothBumpCovering.embeddingPiTangent_ker_mfderiv | Mathlib/Geometry/Manifold/WhitneyEmbedding.lean | theorem embeddingPiTangent_ker_mfderiv (x : M) (hx : x ∈ s) :
LinearMap.ker (mfderiv I 𝓘(ℝ, ι → E × ℝ) f.embeddingPiTangent x) = ⊥ | ι : Type uι
E : Type uE
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : FiniteDimensional ℝ E
H : Type uH
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I ∞ M
inst✝¹ : T2Space M
inst✝ : Fintype ι
s : Set M
f : ... | apply bot_unique | case h
ι : Type uι
E : Type uE
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : FiniteDimensional ℝ E
H : Type uH
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I ∞ M
inst✝¹ : T2Space M
inst✝ : Fintype ι
s : Set... | a0f8631339390c09 |
Set.preimage_iUnionLift | Mathlib/Data/Set/UnionLift.lean | theorem preimage_iUnionLift (t : Set β) :
iUnionLift S f hf T hT ⁻¹' t =
inclusion hT ⁻¹' (⋃ i, inclusion (subset_iUnion S i) '' (f i ⁻¹' t)) | case h.mp
α : Type u_1
ι : Sort u_3
β : Type u_2
S : ι → Set α
f : (i : ι) → ↑(S i) → β
hf : ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j), f i ⟨x, hxi⟩ = f j ⟨x, hxj⟩
T : Set α
hT : T ⊆ iUnion S
t : Set β
x : ↑T
⊢ iUnionLift S f hf T hT x ∈ t → ∃ i x_1, f i x_1 ∈ t ∧ inclusion ⋯ x_1 = inclusion hT x | rcases mem_iUnion.1 (hT x.prop) with ⟨i, hi⟩ | case h.mp.intro
α : Type u_1
ι : Sort u_3
β : Type u_2
S : ι → Set α
f : (i : ι) → ↑(S i) → β
hf : ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j), f i ⟨x, hxi⟩ = f j ⟨x, hxj⟩
T : Set α
hT : T ⊆ iUnion S
t : Set β
x : ↑T
i : ι
hi : ↑x ∈ S i
⊢ iUnionLift S f hf T hT x ∈ t → ∃ i x_1, f i x_1 ∈ t ∧ inclusion ⋯ x_1 = i... | 553a23558fd28573 |
Ideal.Filtration.submodule_eq_span_le_iff_stable_ge | Mathlib/RingTheory/Filtration.lean | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F : I.Filtration M
n₀ : ℕ
F' : Submodule (↥(reesAlgebra I)) (PolynomialModule R M) :=
Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(F.N i))
hF : ∀ n ≥ n₀, I • F.N n = F.N (n + 1)
i✝ i : ℕ
... | refine Set.Subset.trans ?_ Submodule.subset_span | R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F : I.Filtration M
n₀ : ℕ
F' : Submodule (↥(reesAlgebra I)) (PolynomialModule R M) :=
Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(F.N i))
hF : ∀ n ≥ n₀, I • F.N n = F.N (n + 1)
i✝ i : ℕ
... | fba02d553ffa8877 |
AntilipschitzWith.hausdorffMeasure_preimage_le | Mathlib/MeasureTheory/Measure/Hausdorff.lean | theorem hausdorffMeasure_preimage_le (hf : AntilipschitzWith K f) (hd : 0 ≤ d) (s : Set Y) :
μH[d] (f ⁻¹' s) ≤ (K : ℝ≥0∞) ^ d * μH[d] s | case inl.inr.intro.inl
X : Type u_2
Y : Type u_3
inst✝⁵ : EMetricSpace X
inst✝⁴ : EMetricSpace Y
inst✝³ : MeasurableSpace X
inst✝² : BorelSpace X
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
f : X → Y
s : Set Y
hf : AntilipschitzWith 0 f
x : X
hx : x ∈ f ⁻¹' s
this : f ⁻¹' s = {x}
hd : 0 ≤ 0
⊢ 1 ≤ μH[0] s | exact one_le_hausdorffMeasure_zero_of_nonempty ⟨f x, hx⟩ | no goals | d3852939648a0879 |
CategoryTheory.Functor.mem_mapTriangle_essImage_of_distinguished | Mathlib/CategoryTheory/Triangulated/Functor.lean | lemma mem_mapTriangle_essImage_of_distinguished
[F.IsTriangulated] [F.mapArrow.EssSurj] (T : Triangle D) (hT : T ∈ distTriang D) :
∃ (T' : Triangle C) (_ : T' ∈ distTriang C), Nonempty (F.mapTriangle.obj T' ≅ T) | case intro.intro.intro.intro.intro.intro.intro.intro
C : Type u_1
D : Type u_2
inst✝¹⁴ : Category.{u_4, u_1} C
inst✝¹³ : Category.{u_5, u_2} D
inst✝¹² : HasShift C ℤ
inst✝¹¹ : HasShift D ℤ
F : C ⥤ D
inst✝¹⁰ : F.CommShift ℤ
inst✝⁹ : HasZeroObject C
inst✝⁸ : HasZeroObject D
inst✝⁷ : Preadditive C
inst✝⁶ : Preadditive D
i... | exact ⟨_, H, ⟨isoTriangleOfIso₁₂ _ _ (F.map_distinguished _ H) hT e₁ e₂ w⟩⟩ | no goals | 5c95eca80a040aef |
toIcoDiv_zsmul_add | Mathlib/Algebra/Order/ToIntervalMod.lean | theorem toIcoDiv_zsmul_add (a b : α) (m : ℤ) : toIcoDiv hp a (m • p + b) = m + toIcoDiv hp a b | α : Type u_1
inst✝ : LinearOrderedAddCommGroup α
hα : Archimedean α
p : α
hp : 0 < p
a b : α
m : ℤ
⊢ toIcoDiv hp a (m • p + b) = m + toIcoDiv hp a b | rw [add_comm, toIcoDiv_add_zsmul, add_comm] | no goals | df5425d9871ec943 |
MeasureTheory.OuterMeasure.IsMetric.borel_le_caratheodory | Mathlib/MeasureTheory/Measure/Hausdorff.lean | theorem borel_le_caratheodory (hm : IsMetric μ) : borel X ≤ μ.caratheodory | X : Type u_2
inst✝ : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
⊢ MeasurableSpace.generateFrom {s | IsClosed s} ≤ μ.caratheodory | refine MeasurableSpace.generateFrom_le fun t ht => μ.isCaratheodory_iff_le.2 fun s => ?_ | X : Type u_2
inst✝ : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : t ∈ {s | IsClosed s}
s : Set X
⊢ μ (s ∩ t) + μ (s \ t) ≤ μ s | 603b48ab3190787f |
Matrix.PosSemidef.fromBlocks₁₁ | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | theorem PosSemidef.fromBlocks₁₁ [Fintype m] [DecidableEq m] [Fintype n] {A : Matrix m m 𝕜}
(B : Matrix m n 𝕜) (D : Matrix n n 𝕜) (hA : A.PosDef) [Invertible A] :
(fromBlocks A B Bᴴ D).PosSemidef ↔ (D - Bᴴ * A⁻¹ * B).PosSemidef | case mpr
m : Type u_2
n : Type u_3
𝕜 : Type u_5
inst✝⁷ : CommRing 𝕜
inst✝⁶ : StarRing 𝕜
inst✝⁵ : PartialOrder 𝕜
inst✝⁴ : StarOrderedRing 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
⊢ (D - Bᴴ * A⁻¹ * B).PosSe... | refine fun h => ⟨h.1, fun x => ?_⟩ | case mpr
m : Type u_2
n : Type u_3
𝕜 : Type u_5
inst✝⁷ : CommRing 𝕜
inst✝⁶ : StarRing 𝕜
inst✝⁵ : PartialOrder 𝕜
inst✝⁴ : StarOrderedRing 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h : (D - Bᴴ * A⁻¹ * B).Pos... | 80a4e91088bca52a |
mul_inv_mul_cancel | Mathlib/Algebra/GroupWithZero/Basic.lean | theorem mul_inv_mul_cancel (a : G₀) : a * a⁻¹ * a = a | case pos
G₀ : Type u_2
inst✝ : GroupWithZero G₀
a : G₀
h : a = 0
⊢ a * a⁻¹ * a = a | rw [h, inv_zero, mul_zero] | no goals | c905b8d43fbe3da1 |
BitVec.le_zero_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem le_zero_iff {x : BitVec w} : x ≤ 0#w ↔ x = 0#w | case mp
w : Nat
x : BitVec w
h : x ≤ 0#w
⊢ x = 0#w | have : x ≥ 0 := not_lt_iff_le.mp not_lt_zero | case mp
w : Nat
x : BitVec w
h : x ≤ 0#w
this : x ≥ 0
⊢ x = 0#w | d0bc1f43385390a3 |
Topology.IsClosedEmbedding.preimage_closedPoints | Mathlib/Topology/JacobsonSpace.lean | lemma Topology.IsClosedEmbedding.preimage_closedPoints (hf : IsClosedEmbedding f) :
f ⁻¹' closedPoints Y = closedPoints X | X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
hf : IsClosedEmbedding f
⊢ f ⁻¹' closedPoints Y = closedPoints X | ext x | case h
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
hf : IsClosedEmbedding f
x : X
⊢ x ∈ f ⁻¹' closedPoints Y ↔ x ∈ closedPoints X | 7d8dbc1c3380d18e |
SimpleGraph.Walk.IsEulerian.even_degree_iff | Mathlib/Combinatorics/SimpleGraph/Trails.lean | theorem IsEulerian.even_degree_iff {x u v : V} {p : G.Walk u v} (ht : p.IsEulerian) [Fintype V]
[DecidableRel G.Adj] : Even (G.degree x) ↔ u ≠ v → x ≠ u ∧ x ≠ v | case h.e'_1.h.e'_3
V : Type u_1
G : SimpleGraph V
inst✝² : DecidableEq V
x u v : V
p : G.Walk u v
ht : p.IsEulerian
inst✝¹ : Fintype V
inst✝ : DecidableRel G.Adj
⊢ (G.incidenceFinset x).val.card = (Multiset.filter (fun e => x ∈ e) ↑p.edges).card | congr 1 | case h.e'_1.h.e'_3.e_a
V : Type u_1
G : SimpleGraph V
inst✝² : DecidableEq V
x u v : V
p : G.Walk u v
ht : p.IsEulerian
inst✝¹ : Fintype V
inst✝ : DecidableRel G.Adj
⊢ (G.incidenceFinset x).val = Multiset.filter (fun e => x ∈ e) ↑p.edges | 5fed760b6a554578 |
Finset.Colex.erase_le_erase_min' | Mathlib/Combinatorics/Colex.lean | /-- If `s ≤ t` in colex and `#s ≤ #t`, then `s \ {a} ≤ t \ {min t}` for any `a ∈ s`. -/
lemma erase_le_erase_min' (hst : toColex s ≤ toColex t) (hcard : #s ≤ #t) (ha : a ∈ s) :
toColex (s.erase a) ≤
toColex (t.erase <| min' t <| card_pos.1 <| (card_pos.2 ⟨a, ha⟩).trans_le hcard) | case inr.intro.intro.intro.inr.inl
α : Type u_1
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : #s ≤ #t
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
hwt : m ∈ t
hws : m ∉ s
hw : ∀ ⦃a : α⦄, m < a → (a ∈ s ↔ a ∈ t)
haw : a < m
⊢ { ofColex := s.erase a } ≤ { ofColex := t.erase m } | have : erase t m ⊆ erase s a := by
rintro b hb
rw [mem_erase] at hb ⊢
exact ⟨(haw.trans_le <| min'_le _ _ hb.2).ne',
(hw <| hb.1.lt_of_le' <| min'_le _ _ hb.2).2 hb.2⟩ | case inr.intro.intro.intro.inr.inl
α : Type u_1
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : #s ≤ #t
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
hwt : m ∈ t
hws : m ∉ s
hw : ∀ ⦃a : α⦄, m < a → (a ∈ s ↔ a ∈ t)
haw : a < m
this : t.erase m ⊆ s.erase a
⊢ { ofColex := s.erase a } ≤ { ofColex := t.erase m... | 94805d4e030ffc34 |
isLUB_Ioo | Mathlib/Order/Bounds/Basic.lean | theorem isLUB_Ioo {a b : γ} (hab : a < b) : IsLUB (Ioo a b) b | γ : Type v
inst✝¹ : SemilatticeInf γ
inst✝ : DenselyOrdered γ
a b : γ
hab : a < b
⊢ IsLUB (Ioo a b) b | simpa only [dual_Ioo] using isGLB_Ioo hab.dual | no goals | 7075546f82949525 |
Estimator.improveUntilAux_spec | Mathlib/Order/Estimator.lean | theorem Estimator.improveUntilAux_spec (a : Thunk α) (p : α → Bool)
[Estimator a ε] [WellFoundedGT (range (bound a : ε → α))] (e : ε) (r : Bool) :
match Estimator.improveUntilAux a p e r with
| .error _ => ¬ p a.get
| .ok e' => p (bound a e') | case pos
α : Type u_1
ε : Type u_2
inst✝² : Preorder α
a : Thunk α
p : α → Bool
inst✝¹ : Estimator a ε
inst✝ : WellFoundedGT ↑(range (bound a))
e : ε
r : Bool
h : p (bound a e) = true
⊢ match
if True then pure e
else
match improve a e, ⋯ with
| none, x => Except.error (if r = true then none else som... | exact h | no goals | 66dc4c70c5091567 |
Dynamics.coverMincard_univ | Mathlib/Dynamics/TopologicalEntropy/CoverEntropy.lean | lemma coverMincard_univ (T : X → X) {F : Set X} (h : F.Nonempty) (n : ℕ) :
coverMincard T F univ n = 1 | X : Type u_1
T : X → X
F : Set X
h : F.Nonempty
n : ℕ
⊢ coverMincard T F univ n = 1 | apply le_antisymm _ ((one_le_coverMincard_iff T F univ n).2 h) | X : Type u_1
T : X → X
F : Set X
h : F.Nonempty
n : ℕ
⊢ coverMincard T F univ n ≤ 1 | 1832f6ac09cf7d05 |
KaehlerDifferential.span_range_derivation | Mathlib/RingTheory/Kaehler/Basic.lean | theorem KaehlerDifferential.span_range_derivation :
Submodule.span S (Set.range <| KaehlerDifferential.D R S) = ⊤ | case intro.mk.refine_4.intro
R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x✝ : S ⊗[R] S
hx : x✝ ∈ ideal R S
this : x✝ ∈ Submodule.span S (Set.range fun s => 1 ⊗ₜ[R] s - s ⊗ₜ[R] 1)
r : S
x : S ⊗[R] S
hx₁ : x ∈ ideal R S
hx₂ : (ideal R S).toCotangent ⟨x, hx₁⟩ ∈ Submodule.span S (Set.r... | exact ⟨((KaehlerDifferential.ideal R S).restrictScalars S).smul_mem r hx₁,
Submodule.smul_mem _ r hx₂⟩ | no goals | c1fdba8f7521637c |
schnirelmannDensity_finset | Mathlib/Combinatorics/Schnirelmann.lean | /-- The Schnirelmann density of any finset is `0`. -/
lemma schnirelmannDensity_finset (A : Finset ℕ) : schnirelmannDensity A = 0 | A : Finset ℕ
ε : ℝ
hε : 0 < ε
hε₁ : ε ≤ 1
⊢ ∃ n, 0 < n ∧ ↑(#(filter (fun a => a ∈ ↑A) (Ioc 0 n))) / ↑n < ε | let n : ℕ := ⌊#A / ε⌋₊ + 1 | A : Finset ℕ
ε : ℝ
hε : 0 < ε
hε₁ : ε ≤ 1
n : ℕ := ⌊↑(#A) / ε⌋₊ + 1
⊢ ∃ n, 0 < n ∧ ↑(#(filter (fun a => a ∈ ↑A) (Ioc 0 n))) / ↑n < ε | 8e68ea4bfdcfb76e |
Polynomial.iterate_derivative_C_mul | Mathlib/Algebra/Polynomial/Derivative.lean | theorem iterate_derivative_C_mul (a : R) (p : R[X]) (k : ℕ) :
derivative^[k] (C a * p) = C a * derivative^[k] p | R : Type u
inst✝ : Semiring R
a : R
p : R[X]
k : ℕ
⊢ (⇑derivative)^[k] (C a * p) = C a * (⇑derivative)^[k] p | simp_rw [← smul_eq_C_mul, iterate_derivative_smul] | no goals | e6a4491999f3fb6b |
CategoryTheory.Limits.Cofork.app_zero_eq_comp_π_left | Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean | theorem Cofork.app_zero_eq_comp_π_left (s : Cofork f g) : s.ι.app zero = f ≫ s.π | C : Type u
inst✝ : Category.{v, u} C
X Y : C
f g : X ⟶ Y
s : Cofork f g
⊢ s.ι.app zero = f ≫ s.π | rw [← s.app_one_eq_π, ← s.w left, parallelPair_map_left] | no goals | d64991ddd51fc3ff |
LaurentSeries.single_order_mul_powerSeriesPart | Mathlib/RingTheory/LaurentSeries.lean | theorem single_order_mul_powerSeriesPart (x : R⸨X⸩) :
(single x.order 1 : R⸨X⸩) * x.powerSeriesPart = x | case neg
R : Type u_1
inst✝ : Semiring R
x : R⸨X⸩
n : ℤ
h : 0 ≠ x.coeff n
⊢ order x ≤ n | exact order_le_of_coeff_ne_zero h.symm | no goals | 29116b940ffcf48a |
Rat.substr_num_den' | Mathlib/Data/Rat/Lemmas.lean | theorem substr_num_den' (q r : ℚ) :
(q - r).num * q.den * r.den = (q.num * r.den - r.num * q.den) * (q - r).den | q r : ℚ
⊢ (q - r).num * ↑q.den * ↑r.den = (q.num * ↑r.den - r.num * ↑q.den) * ↑(q - r).den | rw [sub_eq_add_neg, sub_eq_add_neg, ← neg_mul, ← num_neg_eq_neg_num, ← den_neg_eq_den r,
add_num_den' q (-r)] | no goals | e22aecd803aed83c |
map_le_nonZeroDivisors_of_injective | Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean | theorem map_le_nonZeroDivisors_of_injective [NoZeroDivisors M₀'] [MonoidWithZeroHomClass F M₀ M₀']
(f : F) (hf : Injective f) {S : Submonoid M₀} (hS : S ≤ M₀⁰) : S.map f ≤ M₀'⁰ | case inr.intro.intro
F : Type u_1
M₀ : Type u_2
M₀' : Type u_3
inst✝⁴ : MonoidWithZero M₀
inst✝³ : MonoidWithZero M₀'
inst✝² : FunLike F M₀ M₀'
inst✝¹ : NoZeroDivisors M₀'
inst✝ : MonoidWithZeroHomClass F M₀ M₀'
f : F
hf : Injective ⇑f
S : Submonoid M₀
hS : S ≤ M₀⁰
h✝ : Nontrivial M₀
x : M₀
hx : x ∈ ↑S
hx0 : f x = 0
⊢ ... | exact zero_not_mem_nonZeroDivisors <| hS <| map_eq_zero_iff f hf |>.mp hx0 ▸ hx | no goals | c454728866afa604 |
iUnion_Iic_eq_Iio_of_lt_of_tendsto | Mathlib/Topology/Order/OrderClosed.lean | theorem iUnion_Iic_eq_Iio_of_lt_of_tendsto {ι : Type*} {F : Filter ι} [F.NeBot]
[ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [ClosedIicTopology α]
{a : α} {f : ι → α} (hlt : ∀ i, f i < a) (hlim : Tendsto f F (𝓝 a)) :
⋃ i : ι, Iic (f i) = Iio a | case intro
α : Type u
ι : Type u_1
F : Filter ι
inst✝³ : F.NeBot
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : ClosedIicTopology α
f : ι → α
i : ι
hlt : ∀ (i_1 : ι), f i_1 < f i
hlim : Tendsto f F (𝓝 (f i))
⊢ False | exact (hlt i).false | no goals | 01d4e3e5ecc81e53 |
Pell.n_lt_a_pow | Mathlib/NumberTheory/PellMatiyasevic.lean | theorem n_lt_a_pow : ∀ n : ℕ, n < a ^ n
| 0 => Nat.le_refl 1
| n + 1 => by
have IH := n_lt_a_pow n
have : a ^ n + a ^ n ≤ a ^ n * a | a : ℕ
a1 : 1 < a
n : ℕ
IH : n < a ^ n
this : a ^ n + a ^ n ≤ a ^ n * a
⊢ n + 1 < a ^ n + a ^ n | exact add_lt_add_of_lt_of_le IH (lt_of_le_of_lt (Nat.zero_le _) IH) | no goals | e47a66206aecfba3 |
solvableByRad.induction | Mathlib/FieldTheory/AbelRuffini.lean | theorem induction (P : solvableByRad F E → Prop)
(base : ∀ α : F, P (algebraMap F (solvableByRad F E) α))
(add : ∀ α β : solvableByRad F E, P α → P β → P (α + β))
(neg : ∀ α : solvableByRad F E, P α → P (-α))
(mul : ∀ α β : solvableByRad F E, P α → P β → P (α * β))
(inv : ∀ α : solvableByRad F E, P ... | F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
P : ↥(solvableByRad F E) → Prop
base : ∀ (α : F), P ((algebraMap F ↥(solvableByRad F E)) α)
add : ∀ (α β : ↥(solvableByRad F E)), P α → P β → P (α + β)
neg : ∀ (α : ↥(solvableByRad F E)), P α → P (-α)
mul : ∀ (α β : ↥(solvableByRad F E)), P... | suffices ∀ α : E, IsSolvableByRad F α → ∃ β : solvableByRad F E, ↑β = α ∧ P β by
intro α
obtain ⟨α₀, hα₀, Pα⟩ := this α (Subtype.mem α)
convert Pα
exact Subtype.ext hα₀.symm | F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
P : ↥(solvableByRad F E) → Prop
base : ∀ (α : F), P ((algebraMap F ↥(solvableByRad F E)) α)
add : ∀ (α β : ↥(solvableByRad F E)), P α → P β → P (α + β)
neg : ∀ (α : ↥(solvableByRad F E)), P α → P (-α)
mul : ∀ (α β : ↥(solvableByRad F E)), P... | 856cf0182c98d000 |
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