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 |
|---|---|---|---|---|---|---|
WeierstrassCurve.Jacobian.Point.toAffine_of_Z_ne_zero | Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean | lemma toAffine_of_Z_ne_zero {P : Fin 3 → F} (hP : W.Nonsingular P) (hPz : P z ≠ 0) :
toAffine W P = .some ((nonsingular_of_Z_ne_zero hPz).mp hP) | F : Type u
inst✝ : Field F
W : Jacobian F
P : Fin 3 → F
hP : W.Nonsingular P
hPz : P z ≠ 0
⊢ toAffine W P = Affine.Point.some ⋯ | rw [toAffine, dif_pos ⟨hP, hPz⟩] | no goals | f782ef0757f09625 |
AlgebraicGeometry.StructureSheaf.toBasicOpen_surjective | Mathlib/AlgebraicGeometry/StructureSheaf.lean | theorem toBasicOpen_surjective (f : R) : Function.Surjective (toBasicOpen R f) | case intro.intro.intro.intro.intro.intro.intro
R : Type u
inst✝ : CommRing R
f : R
s : ↑((structureSheaf R).val.obj (op (PrimeSpectrum.basicOpen f)))
ι : Type u := ↥(PrimeSpectrum.basicOpen f)
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f
ah_ha : ∀ i ∈ t, ∀ j ∈ t, a ... | replace hn := Ideal.mul_mem_right f _ hn | case intro.intro.intro.intro.intro.intro.intro
R : Type u
inst✝ : CommRing R
f : R
s : ↑((structureSheaf R).val.obj (op (PrimeSpectrum.basicOpen f)))
ι : Type u := ↥(PrimeSpectrum.basicOpen f)
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f
ah_ha : ∀ i ∈ t, ∀ j ∈ t, a ... | a0032c84fc951902 |
MonoidHom.submonoidMap_surjective | Mathlib/Algebra/Group/Submonoid/Operations.lean | theorem submonoidMap_surjective (f : M →* N) (M' : Submonoid M) :
Function.Surjective (f.submonoidMap M') | case mk.intro.intro
M : Type u_1
N : Type u_2
inst✝¹ : MulOneClass M
inst✝ : MulOneClass N
f : M →* N
M' : Submonoid M
x : M
hx : x ∈ ↑M'
⊢ ∃ a, (f.submonoidMap M') a = ⟨f x, ⋯⟩ | exact ⟨⟨x, hx⟩, rfl⟩ | no goals | ef5cf861c8abc90e |
NumberField.mixedEmbedding.convexBodySumFun_eq_zero_iff | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean | theorem convexBodySumFun_eq_zero_iff (x : mixedSpace K) :
convexBodySumFun x = 0 ↔ x = 0 | K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : mixedSpace K
⊢ convexBodySumFun x = 0 ↔ x = 0 | rw [← forall_normAtPlace_eq_zero_iff, convexBodySumFun, Finset.sum_eq_zero_iff_of_nonneg
fun _ _ ↦ mul_nonneg (Nat.cast_pos.mpr mult_pos).le (normAtPlace_nonneg _ _)] | K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : mixedSpace K
⊢ (∀ i ∈ Finset.univ, ↑i.mult * (normAtPlace i) x = 0) ↔ ∀ (w : InfinitePlace K), (normAtPlace w) x = 0 | 8e6b87327805e297 |
IsLocalRing.closedPoint_mem_iff | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | theorem closedPoint_mem_iff (U : TopologicalSpace.Opens <| PrimeSpectrum R) :
closedPoint R ∈ U ↔ U = ⊤ | case mp
R : Type u
inst✝¹ : CommSemiring R
inst✝ : IsLocalRing R
U : TopologicalSpace.Opens (PrimeSpectrum R)
⊢ closedPoint R ∈ U → ⊤ ≤ U | exact fun h x _ => (specializes_closedPoint x).mem_open U.2 h | no goals | 41957f74368b388c |
CategoryTheory.Triangulated.Subcategory.W.shift | Mathlib/CategoryTheory/Triangulated/Subcategory.lean | lemma W.shift {X₁ X₂ : C} {f : X₁ ⟶ X₂} (hf : S.W f) (n : ℤ) : S.W (f⟦n⟧') | C : Type u_1
inst✝⁵ : Category.{u_2, u_1} C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : Preadditive C
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
S : Subcategory C
X₁ X₂ : C
f : X₁ ⟶ X₂
hf : S.W f
n : ℤ
⊢ S.W ((shiftFunctor C n).map f) | rw [← smul_mem_W_iff _ _ (n.negOnePow)] | C : Type u_1
inst✝⁵ : Category.{u_2, u_1} C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : Preadditive C
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
S : Subcategory C
X₁ X₂ : C
f : X₁ ⟶ X₂
hf : S.W f
n : ℤ
⊢ S.W (n.negOnePow • (shiftFunctor C n).map f) | 522dfa6c54def5f6 |
Set.iUnionLift_binary | Mathlib/Data/Set/UnionLift.lean | theorem iUnionLift_binary (dir : Directed (· ≤ ·) S) (op : T → T → T) (opi : ∀ i, S i → S i → S i)
(hopi :
∀ i x y,
Set.inclusion (show S i ⊆ T from hT'.symm ▸ Set.subset_iUnion S i) (opi i x y) =
op (Set.inclusion (show S i ⊆ T from hT'.symm ▸ Set.subset_iUnion S i) x)
(Set.incl... | α : Type u_1
ι : Sort u_2
β : Sort u_3
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
dir : Directed (fun x1 x2 => x1 ≤ x2) S
op : ↑T → ↑T → ↑T
opi : (i : ι) → ↑(S i) → ↑(S i) → ↑(S i)
hopi : ∀ (i : ι) (x y : ↑(S ... | subst hT' | α : Type u_1
ι : Sort u_2
β : Sort u_3
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⟩
dir : Directed (fun x1 x2 => x1 ≤ x2) S
opi : (i : ι) → ↑(S i) → ↑(S i) → ↑(S i)
opβ : β → β → β
h : ∀ (i : ι) (x y : ↑(S i)), f i (opi i x y) = opβ (f i x)... | 10339d735ec21489 |
MeasureTheory.integrableOn_Ioi_deriv_of_nonneg | Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | theorem integrableOn_Ioi_deriv_of_nonneg (hcont : ContinuousWithinAt g (Ici a) a)
(hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x) (g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x)
(hg : Tendsto g atTop (𝓝 l)) : IntegrableOn g' (Ioi a) | g g' : ℝ → ℝ
a l : ℝ
hcont✝ : ContinuousWithinAt g (Ici a) a
hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x
g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x
hg : Tendsto g atTop (𝓝 l)
hcont : ContinuousOn g (Ici a)
x : ℝ
hx : x ∈ Ioi a
h'x : a ≤ id x
y : ℝ
hy : y ∈ Ioc a (id x)
⊢ g' y = |g' y| | rw [abs_of_nonneg] | g g' : ℝ → ℝ
a l : ℝ
hcont✝ : ContinuousWithinAt g (Ici a) a
hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x
g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x
hg : Tendsto g atTop (𝓝 l)
hcont : ContinuousOn g (Ici a)
x : ℝ
hx : x ∈ Ioi a
h'x : a ≤ id x
y : ℝ
hy : y ∈ Ioc a (id x)
⊢ 0 ≤ g' y | 1f10c323da5db2cd |
RootPairing.Base.sub_nmem_range_root | Mathlib/LinearAlgebra/RootSystem/Base.lean | lemma sub_nmem_range_root [CharZero R] [Finite ι]
{i j : ι} (hi : i ∈ b.support) (hj : j ∈ b.support) :
P.root i - P.root j ∉ range P.root | ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : AddCommGroup N
inst✝² : Module R N
P : RootPairing ι R M N
b : P.Base
inst✝¹ : CharZero R
inst✝ : Finite ι
i j : ι
hi : i ∈ b.support
hj : j ∈ b.support
hij : j ≠ i
_i : Fintype ι
f : ι → ℤ := fu... | rwa [hf] | no goals | 5327338e738a9572 |
HasCompactSupport.convolutionExistsAt | Mathlib/Analysis/Convolution.lean | theorem _root_.HasCompactSupport.convolutionExistsAt {x₀ : G}
(h : HasCompactSupport fun t => L (f t) (g (x₀ - t))) (hf : LocallyIntegrable f μ)
(hg : Continuous g) : ConvolutionExistsAt f g x₀ L μ | 𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedAddCommGroup E'
inst✝⁹ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedSpace 𝕜 E'
inst✝⁵ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
in... | let u := (Homeomorph.neg G).trans (Homeomorph.addRight x₀) | 𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedAddCommGroup E'
inst✝⁹ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedSpace 𝕜 E'
inst✝⁵ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
in... | 1264fad810af6cce |
List.zipIdxLE_trans | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Sort/Lemmas.lean | theorem zipIdxLE_trans (trans : ∀ a b c, le a b → le b c → le a c)
(a b c : α × Nat) : zipIdxLE le a b → zipIdxLE le b c → zipIdxLE le a c | α : Type u_1
le : α → α → Bool
trans : ∀ (a b c : α), le a b = true → le b c = true → le a c = true
a b c : α × Nat
ab₂ : le a.fst b.fst = true
ba₂ : le b.fst a.fst = true
bc₂ : le b.fst c.fst = true
ab₁ : a.snd ≤ b.snd
cd₂ : le c.fst b.fst = false
ca₂ : ¬le c.fst a.fst = false
⊢ False | simp_all [trans _ _ _ (by simpa using ca₂) ab₂] | no goals | 4b08ab1e530cfafb |
Fin.cycleRange_zero | Mathlib/GroupTheory/Perm/Fin.lean | theorem cycleRange_zero (n : ℕ) [NeZero n] : cycleRange (0 : Fin n) = 1 | n : ℕ
inst✝ : NeZero n
⊢ cycleRange 0 = 1 | ext j | case H.h
n : ℕ
inst✝ : NeZero n
j : Fin n
⊢ ↑((cycleRange 0) j) = ↑(1 j) | 65b86f8bda73cf21 |
Field.Emb.Cardinal.iSup_filtration | Mathlib/FieldTheory/CardinalEmb.lean | theorem iSup_filtration : ⨆ j : Iio i, filtration j = filtration i | case coe.intro.intro
F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
rank_inf : Fact (ℵ₀ ≤ Module.rank F E)
inst✝ : Algebra.IsAlgebraic F E
a✝ : (Module.rank F E).ord.toType
hi : IsSuccPrelimit ↑a✝
j : (Module.rank F E).ord.toType
hj : j ∈ Iio a✝
⊢ (⇑b ∘ φ) j ∈ ↑(⨆ j, filtration ↑j) | refine le_iSup (α := IntermediateField F E) _ ⟨j⁺, ?_⟩ (subset_adjoin F _ ?_) | case coe.intro.intro.refine_1
F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
rank_inf : Fact (ℵ₀ ≤ Module.rank F E)
inst✝ : Algebra.IsAlgebraic F E
a✝ : (Module.rank F E).ord.toType
hi : IsSuccPrelimit ↑a✝
j : (Module.rank F E).ord.toType
hj : j ∈ Iio a✝
⊢ (⇑b ∘ φ) j ∈ ⇑b ∘ φ '' Iio (j⁺)
... | 2aa6c169ef4dbdc1 |
finprod_mem_union_inter' | Mathlib/Algebra/BigOperators/Finprod.lean | theorem finprod_mem_union_inter' (hs : (s ∩ mulSupport f).Finite) (ht : (t ∩ mulSupport f).Finite) :
((∏ᶠ i ∈ s ∪ t, f i) * ∏ᶠ i ∈ s ∩ t, f i) = (∏ᶠ i ∈ s, f i) * ∏ᶠ i ∈ t, f i | α : Type u_1
M : Type u_5
inst✝ : CommMonoid M
f : α → M
s t : Set α
hs : (s ∩ mulSupport f).Finite
ht : (t ∩ mulSupport f).Finite
⊢ (∏ᶠ (i : α) (_ : i ∈ s ∪ t), f i) * ∏ᶠ (i : α) (_ : i ∈ s ∩ t), f i =
(∏ᶠ (i : α) (_ : i ∈ s), f i) * ∏ᶠ (i : α) (_ : i ∈ t), f i | rw [← finprod_mem_inter_mulSupport f s, ← finprod_mem_inter_mulSupport f t, ←
finprod_mem_union_inter hs ht, ← union_inter_distrib_right, finprod_mem_inter_mulSupport, ←
finprod_mem_inter_mulSupport f (s ∩ t)] | α : Type u_1
M : Type u_5
inst✝ : CommMonoid M
f : α → M
s t : Set α
hs : (s ∩ mulSupport f).Finite
ht : (t ∩ mulSupport f).Finite
⊢ (∏ᶠ (i : α) (_ : i ∈ s ∪ t), f i) * ∏ᶠ (i : α) (_ : i ∈ s ∩ t ∩ mulSupport f), f i =
(∏ᶠ (i : α) (_ : i ∈ s ∪ t), f i) * ∏ᶠ (i : α) (_ : i ∈ s ∩ mulSupport f ∩ (t ∩ mulSupport f)), f ... | 71a33776e9e8c40b |
Trivialization.contMDiffOn_symm | Mathlib/Geometry/Manifold/VectorBundle/Basic.lean | theorem Trivialization.contMDiffOn_symm (e : Trivialization F (π F E)) [MemTrivializationAtlas e] :
ContMDiffOn (IB.prod 𝓘(𝕜, F)) (IB.prod 𝓘(𝕜, F)) n e.toPartialHomeomorph.symm e.target | n : WithTop ℕ∞
𝕜 : Type u_1
B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝¹⁵ : NontriviallyNormedField 𝕜
EB : Type u_7
inst✝¹⁴ : NormedAddCommGroup EB
inst✝¹³ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝¹² : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹¹ : TopologicalSpace B
inst✝¹⁰ : ChartedSpace HB B
inst✝... | refine ⟨contMDiffOn_fst.congr fun x hx ↦ e.proj_symm_apply hx,
contMDiffOn_snd.congr fun x hx ↦ ?_⟩ | n : WithTop ℕ∞
𝕜 : Type u_1
B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝¹⁵ : NontriviallyNormedField 𝕜
EB : Type u_7
inst✝¹⁴ : NormedAddCommGroup EB
inst✝¹³ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝¹² : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹¹ : TopologicalSpace B
inst✝¹⁰ : ChartedSpace HB B
inst✝... | dd22632b78b42a61 |
Set.image_of_range_union_range_eq_univ | Mathlib/Data/Set/Image.lean | lemma image_of_range_union_range_eq_univ {α β γ γ' δ δ' : Type*}
{h : β → α} {f : γ → β} {f₁ : γ' → α} {f₂ : γ → γ'} {g : δ → β} {g₁ : δ' → α} {g₂ : δ → δ'}
(hf : h ∘ f = f₁ ∘ f₂) (hg : h ∘ g = g₁ ∘ g₂) (hfg : range f ∪ range g = univ) (s : Set β) :
h '' s = f₁ '' (f₂ '' (f ⁻¹' s)) ∪ g₁ '' (g₂ '' (g ⁻¹' s))... | α : Type u_5
β : Type u_6
γ : Type u_7
γ' : Type u_8
δ : Type u_9
δ' : Type u_10
h : β → α
f : γ → β
f₁ : γ' → α
f₂ : γ → γ'
g : δ → β
g₁ : δ' → α
g₂ : δ → δ'
hf : h ∘ f = f₁ ∘ f₂
hg : h ∘ g = g₁ ∘ g₂
hfg : range f ∪ range g = univ
s : Set β
⊢ h '' s = f₁ '' (f₂ '' (f ⁻¹' s)) ∪ g₁ '' (g₂ '' (g ⁻¹' s)) | rw [← image_comp, ← image_comp, ← hf, ← hg, image_comp, image_comp, image_preimage_eq_inter_range,
image_preimage_eq_inter_range, ← image_union, ← inter_union_distrib_left, hfg, inter_univ] | no goals | 484a7b56ad6ec5be |
Rat.sqrt_intCast | Mathlib/Data/Rat/Sqrt.lean | theorem sqrt_intCast (z : ℤ) : Rat.sqrt (z : ℚ) = Int.sqrt z | z : ℤ
⊢ sqrt ↑z = ↑(Int.sqrt z) | simp only [sqrt, num_intCast, den_intCast, Nat.sqrt_one, mkRat_one] | no goals | 9e70017356b010cb |
aemeasurable_withDensity_iff | Mathlib/MeasureTheory/Integral/LebesgueNormedSpace.lean | theorem aemeasurable_withDensity_iff {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[SecondCountableTopology E] [MeasurableSpace E] [BorelSpace E] {f : α → ℝ≥0}
(hf : Measurable f) {g : α → E} :
AEMeasurable g (μ.withDensity fun x => (f x : ℝ≥0∞)) ↔
AEMeasurable (fun x => (f x : ℝ) • g x) μ | case mp.intro.intro.htc
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : SecondCountableTopology E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
f : α → ℝ≥0
hf : Measurable f
g g' : α → E
g'meas : Measurable g'
hg' : g =ᶠ[ae (μ.withDensity f... | filter_upwards [ae_restrict_mem A.compl] | case h
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : SecondCountableTopology E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
f : α → ℝ≥0
hf : Measurable f
g g' : α → E
g'meas : Measurable g'
hg' : g =ᶠ[ae (μ.withDensity fun x => ↑(f x))] ... | b18b4573d54eb587 |
AlgebraicGeometry.exists_eq_pow_mul_of_is_compact_of_quasi_separated_space_aux_aux | Mathlib/AlgebraicGeometry/Morphisms/QuasiSeparated.lean | theorem exists_eq_pow_mul_of_is_compact_of_quasi_separated_space_aux_aux {X : TopCat}
(F : X.Presheaf CommRingCat) {U₁ U₂ U₃ U₄ U₅ U₆ U₇ : Opens X} {n₁ n₂ : ℕ}
{y₁ : F.obj (op U₁)} {y₂ : F.obj (op U₂)} {f : F.obj (op <| U₁ ⊔ U₂)}
{x : F.obj (op U₃)} (h₄₁ : U₄ ≤ U₁) (h₄₂ : U₄ ≤ U₂) (h₅₁ : U₅ ≤ U₁) (h₅₃ : U₅ ... | X : TopCat
F : TopCat.Presheaf CommRingCat X
U₁ U₂ U₃ U₄ U₅ U₆ U₇ : Opens ↑X
n₁ n₂ : ℕ
y₁ : ↑(F.obj (op U₁))
y₂ : ↑(F.obj (op U₂))
f : ↑(F.obj (op (U₁ ⊔ U₂)))
x : ↑(F.obj (op U₃))
h₄₁ : U₄ ≤ U₁
h₄₂ : U₄ ≤ U₂
h₅₁ : U₅ ≤ U₁
h₅₃ : U₅ ≤ U₃
h₆₂ : U₆ ≤ U₂
h₆₃ : U₆ ≤ U₃
h₇₄ : U₇ ≤ U₄
h₇₅ : U₇ ≤ U₅
h₇₆ : U₇ ≤ U₆
e₁ : ((y₁ |_ U... | dsimp only [TopCat.Presheaf.restrictOpenCommRingCat_apply] at e₁ e₂ ⊢ | X : TopCat
F : TopCat.Presheaf CommRingCat X
U₁ U₂ U₃ U₄ U₅ U₆ U₇ : Opens ↑X
n₁ n₂ : ℕ
y₁ : ↑(F.obj (op U₁))
y₂ : ↑(F.obj (op U₂))
f : ↑(F.obj (op (U₁ ⊔ U₂)))
x : ↑(F.obj (op U₃))
h₄₁ : U₄ ≤ U₁
h₄₂ : U₄ ≤ U₂
h₅₁ : U₅ ≤ U₁
h₅₃ : U₅ ≤ U₃
h₆₂ : U₆ ≤ U₂
h₆₃ : U₆ ≤ U₃
h₇₄ : U₇ ≤ U₄
h₇₅ : U₇ ≤ U₅
h₇₆ : U₇ ≤ U₆
e₁ :
(Concre... | b1dd395ddfad0c82 |
Algebra.FormallyUnramified.finite_of_free | Mathlib/RingTheory/Unramified/Finite.lean | /-- An unramified free algebra is finitely generated. Iversen I.2.8 -/
lemma finite_of_free [Module.Free R S] : Module.Finite R S | case a.a.a
R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : FormallyUnramified R S
inst✝¹ : EssFiniteType R S
inst✝ : Module.Free R S
I : Type u_3 := Module.Free.ChooseBasisIndex R S
b : Basis I R S := Module.Free.chooseBasis R S
f : I →₀ S
hf : elem R S = f.sum fun i x =>... | trans (x ⊗ₜ 1) * elem R S | R : Type u_2
S : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : FormallyUnramified R S
inst✝¹ : EssFiniteType R S
inst✝ : Module.Free R S
I : Type u_3 := Module.Free.ChooseBasisIndex R S
b : Basis I R S := Module.Free.chooseBasis R S
f : I →₀ S
hf : elem R S = f.sum fun i x => x ⊗ₜ[R] b ... | ad272845a5bade6c |
IsMulFreimanHom.prod | Mathlib/Combinatorics/Additive/FreimanHom.lean | @[to_additive]
lemma IsMulFreimanHom.prod (h₁ : IsMulFreimanHom n A₁ B₁ f₁) (h₂ : IsMulFreimanHom n A₂ B₂ f₂) :
IsMulFreimanHom n (A₁ ×ˢ A₂) (B₁ ×ˢ B₂) (Prod.map f₁ f₂) where
mapsTo := h₁.mapsTo.prodMap h₂.mapsTo
map_prod_eq_map_prod s t hsA htA hs ht h | α₁ : Type u_5
α₂ : Type u_6
β₁ : Type u_7
β₂ : Type u_8
inst✝³ : CommMonoid α₁
inst✝² : CommMonoid α₂
inst✝¹ : CommMonoid β₁
inst✝ : CommMonoid β₂
A₁ : Set α₁
A₂ : Set α₂
B₁ : Set β₁
B₂ : Set β₂
f₁ : α₁ → β₁
f₂ : α₂ → β₂
n : ℕ
h₁ : IsMulFreimanHom n A₁ B₁ f₁
h₂ : IsMulFreimanHom n A₂ B₂ f₂
s t : Multiset (α₁ × α₂)
hs :... | simpa using hsA.1 | no goals | 33dda5fafc4888d5 |
SimpleGraph.FarFromTriangleFree.lt_half | Mathlib/Combinatorics/SimpleGraph/Triangle/Basic.lean | lemma FarFromTriangleFree.lt_half (hG : G.FarFromTriangleFree ε) : ε < 2⁻¹ | α : Type u_1
𝕜 : Type u_3
inst✝³ : LinearOrderedField 𝕜
G : SimpleGraph α
ε : 𝕜
inst✝² : Fintype α
inst✝¹ : DecidableRel G.Adj
inst✝ : Nonempty α
hG : G.FarFromTriangleFree ε
hε : 1 ≤ ε * 2
hε₀ : 0 < ε
⊢ 2 * #⊤.edgeFinset < Fintype.card α ^ 2 | rw [edgeFinset_top, filter_not, card_sdiff (subset_univ _), card_univ, Sym2.card] | α : Type u_1
𝕜 : Type u_3
inst✝³ : LinearOrderedField 𝕜
G : SimpleGraph α
ε : 𝕜
inst✝² : Fintype α
inst✝¹ : DecidableRel G.Adj
inst✝ : Nonempty α
hG : G.FarFromTriangleFree ε
hε : 1 ≤ ε * 2
hε₀ : 0 < ε
⊢ 2 * ((Fintype.card α + 1).choose 2 - #(filter Sym2.IsDiag univ)) < Fintype.card α ^ 2 | b19427286740d73a |
BitVec.msb_umod | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem msb_umod {x y : BitVec w} :
(x % y).msb = (x.msb && (x < y || y == 0#w)) | case neg
w : Nat
x y : BitVec w
msb_x : x.msb = true
x_lt_y : ¬x < y
y_le_x : y.toNat ≤ x.toNat
hy : y.toNat ≠ 0
⊢ x.toNat % y.toNat < 2 ^ (w - 1) | by_cases msb_y : y.toNat < 2 ^ (w - 1) | case pos
w : Nat
x y : BitVec w
msb_x : x.msb = true
x_lt_y : ¬x < y
y_le_x : y.toNat ≤ x.toNat
hy : y.toNat ≠ 0
msb_y : y.toNat < 2 ^ (w - 1)
⊢ x.toNat % y.toNat < 2 ^ (w - 1)
case neg
w : Nat
x y : BitVec w
msb_x : x.msb = true
x_lt_y : ¬x < y
y_le_x : y.toNat ≤ x.toNat
hy : y.toNat ≠ 0
msb_y : ¬y.toNat < 2 ^ (w - 1... | a4b9057a3dee0f33 |
UniqueFactorizationMonoid.induction_on_coprime | Mathlib/RingTheory/UniqueFactorizationDomain/Multiplicative.lean | theorem induction_on_coprime {P : α → Prop} (a : α) (h0 : P 0) (h1 : ∀ {x}, IsUnit x → P x)
(hpr : ∀ {p} (i : ℕ), Prime p → P (p ^ i))
(hcp : ∀ {x y}, IsRelPrime x y → P x → P y → P (x * y)) : P a | case neg
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
P : α → Prop
a : α
h0 : P 0
h1 : ∀ {x : α}, IsUnit x → P x
hpr : ∀ {p : α} (i : ℕ), Prime p → P (p ^ i)
hcp : ∀ {x y : α}, IsRelPrime x y → P x → P y → P (x * y)
this✝ : DecidableEq α := Classical.decEq α
P_of_associated : ∀ {... | letI : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid | case neg
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
P : α → Prop
a : α
h0 : P 0
h1 : ∀ {x : α}, IsUnit x → P x
hpr : ∀ {p : α} (i : ℕ), Prime p → P (p ^ i)
hcp : ∀ {x y : α}, IsRelPrime x y → P x → P y → P (x * y)
this✝¹ : DecidableEq α := Classical.decEq α
P_of_associated : ∀ ... | 2e303a16bbd0c49e |
Algebra.mem_ideal_map_adjoin | Mathlib/RingTheory/Polynomial/Ideal.lean | lemma _root_.Algebra.mem_ideal_map_adjoin {R S : Type*} [CommRing R] [CommRing S] [Algebra R S]
(x : S) (I : Ideal R) {y : adjoin R ({x} : Set S)} :
y ∈ I.map (algebraMap R (adjoin R ({x} : Set S))) ↔
∃ p : R[X], (∀ i, p.coeff i ∈ I) ∧ Polynomial.aeval x p = y | R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
y : ↥(adjoin R {x})
p : R[X]
hp : ∀ (i : ℕ), p.coeff i ∈ I
hp' : (aeval x) p = ↑y
⊢ y = ∑ i ∈ p.support, p.coeff i • ⟨(aeval x) (X ^ i), ⋯⟩ | trans ∑ i ∈ p.support, ⟨_, (C (p.coeff i) * X ^ i).aeval_mem_adjoin_singleton _ x⟩ | R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
y : ↥(adjoin R {x})
p : R[X]
hp : ∀ (i : ℕ), p.coeff i ∈ I
hp' : (aeval x) p = ↑y
⊢ y = ∑ i ∈ p.support, ⟨(aeval x) (C (p.coeff i) * X ^ i), ⋯⟩
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝... | a169c5a246991a31 |
Vector.mem_flatten | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Lemmas.lean | theorem mem_flatten {L : Vector (Vector α n) m} : a ∈ L.flatten ↔ ∃ l, l ∈ L ∧ a ∈ l | α : Type u_1
n : Nat
a : α
L : Array (Vector α n)
l : Vector α n
h₁ : l ∈ L
h₂ : a ∈ l
⊢ a ∈ l.toArray | simpa using h₂ | no goals | 534e3970864e927f |
PartialHomeomorph.MDifferentiable.symm_comp_deriv | Mathlib/Geometry/Manifold/MFDeriv/Atlas.lean | theorem symm_comp_deriv {x : M} (hx : x ∈ e.source) :
(mfderiv I' I e.symm (e x)).comp (mfderiv I I' e x) =
ContinuousLinearMap.id 𝕜 (TangentSpace I x) | 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpac... | have : mfderiv I I (_root_.id : M → M) x = ContinuousLinearMap.id _ _ := mfderiv_id | 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpac... | e6e797b0292c0c2a |
Equiv.succ_embeddingFinSucc_fst_symm_apply | Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | /-- Technical lemma used in the proof of `hasFTaylorSeriesUpTo_iteratedFDeriv`, to compare sums
over embedding of `Fin k` and `Fin (k + 1)`. -/
private lemma _root_.Equiv.succ_embeddingFinSucc_fst_symm_apply {ι : Type*} [DecidableEq ι]
{n : ℕ} (e : Fin (n+1) ↪ ι) {k : ι}
(h'k : k ∈ Set.range (Equiv.embeddingFin... | case intro
ι : Type u_4
inst✝ : DecidableEq ι
n : ℕ
e : Fin (n + 1) ↪ ι
j : Fin (n + 1)
h'k : e j ∈ range ⇑((Equiv.embeddingFinSucc n ι) e).fst
hj : j ≠ 0
⊢ (((Equiv.embeddingFinSucc n ι) e).fst.toEquivRange.symm ⟨e j, h'k⟩).succ = j | have : e j = (Equiv.embeddingFinSucc n ι e).1 (Fin.pred j hj) := by simp | case intro
ι : Type u_4
inst✝ : DecidableEq ι
n : ℕ
e : Fin (n + 1) ↪ ι
j : Fin (n + 1)
h'k : e j ∈ range ⇑((Equiv.embeddingFinSucc n ι) e).fst
hj : j ≠ 0
this : e j = ((Equiv.embeddingFinSucc n ι) e).fst (j.pred hj)
⊢ (((Equiv.embeddingFinSucc n ι) e).fst.toEquivRange.symm ⟨e j, h'k⟩).succ = j | 812eabf947ba927a |
Orientation.oangle_eq_zero_or_eq_pi_iff_right_eq_smul | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | theorem oangle_eq_zero_or_eq_pi_iff_right_eq_smul {x y : V} :
o.oangle x y = 0 ∨ o.oangle x y = π ↔ x = 0 ∨ ∃ r : ℝ, y = r • x | case pos
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : SameRay ℝ x (-y)
hx : x = 0
⊢ x = 0 ∨ ∃ r, y = r • x | simp [hx] | no goals | 71453a6198b623ff |
Matrix.fromBlocks_conjTranspose | Mathlib/Data/Matrix/Block.lean | theorem fromBlocks_conjTranspose [Star α] (A : Matrix n l α) (B : Matrix n m α) (C : Matrix o l α)
(D : Matrix o m α) : (fromBlocks A B C D)ᴴ = fromBlocks Aᴴ Cᴴ Bᴴ Dᴴ | l : Type u_1
m : Type u_2
n : Type u_3
o : Type u_4
α : Type u_12
inst✝ : Star α
A : Matrix n l α
B : Matrix n m α
C : Matrix o l α
D : Matrix o m α
⊢ (fromBlocks A B C D)ᴴ = fromBlocks Aᴴ Cᴴ Bᴴ Dᴴ | simp only [conjTranspose, fromBlocks_transpose, fromBlocks_map] | no goals | bc52fd4d1a3b85ae |
Squarefree.eq_zero_or_one_of_pow_of_not_isUnit | Mathlib/Algebra/Squarefree/Basic.lean | theorem Squarefree.eq_zero_or_one_of_pow_of_not_isUnit [Monoid R] {x : R} {n : ℕ}
(h : Squarefree (x ^ n)) (h' : ¬ IsUnit x) :
n = 0 ∨ n = 1 | R : Type u_1
inst✝ : Monoid R
x : R
n : ℕ
h : Squarefree (x ^ n)
h' : 2 ≤ n
⊢ IsUnit x | have : x * x ∣ x ^ n := by rw [← sq]; exact pow_dvd_pow x h' | R : Type u_1
inst✝ : Monoid R
x : R
n : ℕ
h : Squarefree (x ^ n)
h' : 2 ≤ n
this : x * x ∣ x ^ n
⊢ IsUnit x | 3bc2beb83f458233 |
DyckWord.firstReturn_add | Mathlib/Combinatorics/Enumerative/DyckWord.lean | @[simp]
lemma firstReturn_add : (p + q).firstReturn = if p = 0 then q.firstReturn else p.firstReturn | case neg
p q : DyckWord
h : ¬p = 0
⊢ (p + q).firstReturn = p.firstReturn | have u : (p + q).toList = p.toList ++ q.toList := rfl | case neg
p q : DyckWord
h : ¬p = 0
u : ↑(p + q) = ↑p ++ ↑q
⊢ (p + q).firstReturn = p.firstReturn | 0d1d3be3ac0640d4 |
sup_eq_half_smul_add_add_abs_sub' | Mathlib/Algebra/Order/Module/OrderedSMul.lean | lemma sup_eq_half_smul_add_add_abs_sub' (x y : β) : x ⊔ y = (2⁻¹ : α) • (x + y + |y - x|) | α : Type u_6
β : Type u_7
inst✝⁵ : DivisionSemiring α
inst✝⁴ : NeZero 2
inst✝³ : Lattice β
inst✝² : AddCommGroup β
inst✝¹ : Module α β
inst✝ : AddLeftMono β
x y : β
this : Invertible 2 := invertibleOfNonzero ⋯
⊢ x ⊔ y = 2⁻¹ • (x + y + |y - x|) | exact sup_eq_half_smul_add_add_abs_sub α x y | no goals | 13fe1fc5737c52c6 |
AList.keys_subset_keys_of_entries_subset_entries | Mathlib/Data/List/AList.lean | theorem keys_subset_keys_of_entries_subset_entries
{s₁ s₂ : AList β} (h : s₁.entries ⊆ s₂.entries) : s₁.keys ⊆ s₂.keys | α : Type u
β : α → Type v
s₁ s₂ : AList β
h : s₁.entries ⊆ s₂.entries
k : α
hk : k ∈ s₁.keys
this : DecidableEq α := Classical.decEq α
⊢ k ∈ s₂.keys | have := h (mem_lookup_iff.1 (Option.get_mem (lookup_isSome.2 hk))) | α : Type u
β : α → Type v
s₁ s₂ : AList β
h : s₁.entries ⊆ s₂.entries
k : α
hk : k ∈ s₁.keys
this✝ : DecidableEq α := Classical.decEq α
this : ⟨k, (lookup k s₁).get ⋯⟩ ∈ s₂.entries
⊢ k ∈ s₂.keys | 4921c9b7ae5a8cb0 |
IsOfFinOrder.finite_powers | Mathlib/GroupTheory/OrderOfElement.lean | @[to_additive]
lemma IsOfFinOrder.finite_powers (ha : IsOfFinOrder a) : (powers a : Set G).Finite | G : Type u_1
inst✝ : Monoid G
a : G
ha : IsOfFinOrder a
⊢ (↑(Finset.image (fun x => a ^ x) (Finset.range (orderOf a)))).Finite | exact Finset.finite_toSet _ | no goals | f75b7fda41622a54 |
List.Nodup.cyclicPermutations | Mathlib/Data/List/Rotate.lean | theorem Nodup.cyclicPermutations {l : List α} (hn : Nodup l) :
Nodup (cyclicPermutations l) | α : Type u
l : List α
hn : l.Nodup
⊢ l.cyclicPermutations.Nodup | rcases eq_or_ne l [] with rfl | hl | case inl
α : Type u
hn : [].Nodup
⊢ [].cyclicPermutations.Nodup
case inr
α : Type u
l : List α
hn : l.Nodup
hl : l ≠ []
⊢ l.cyclicPermutations.Nodup | 2e9cb67be0f29a76 |
dist_triangle_left | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | theorem dist_triangle_left (x y z : α) : dist x y ≤ dist z x + dist z y | α : Type u
inst✝ : PseudoMetricSpace α
x y z : α
⊢ dist x y ≤ dist x z + dist z y | apply dist_triangle | no goals | e8ea350bd4f3633e |
Algebra.Generators.H1Cotangent.δAux_ofComp | Mathlib/RingTheory/Kaehler/JacobiZariski.lean | lemma δAux_ofComp (x : (Q.comp P).Ring) :
δAux R Q ((Q.ofComp P).toAlgHom x) =
P.toExtension.toKaehler.baseChange T (CotangentSpace.compEquiv Q P
(1 ⊗ₜ[(Q.comp P).Ring] (D R (Q.comp P).Ring) x : _)).2 | case h_X
R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
T : Type uT
inst✝³ : CommRing T
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
Q : Generators S T
P : Generators R S
this✝ : AddCommGroup (T ⊗[S] Ω[S⁄R]) := inferInstance
this : IsScalarTower (Q.comp P).Ri... | simp only [map_mul, Hom.toAlgHom_X, ofComp_val, δAux_mul,
← @IsScalarTower.algebraMap_smul Q.Ring T, algebraMap_apply, Hom.algebraMap_toAlgHom,
id.map_eq_id, map_aeval, RingHomCompTriple.comp_eq, comp_val, RingHom.id_apply, coe_eval₂Hom,
IH, Derivation.leibniz, tmul_add, tmul_smul, ← cotangentSpaceBasis_apply,
... | case h_X
R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
T : Type uT
inst✝³ : CommRing T
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
Q : Generators S T
P : Generators R S
this✝ : AddCommGroup (T ⊗[S] Ω[S⁄R]) := inferInstance
this : IsScalarTower (Q.comp P).Ri... | 9950f156a5aba173 |
Std.Tactic.BVDecide.BVExpr.bitblast.mkOverflowBit.go_eq_carry | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Carry.lean | theorem go_eq_carry (aig : AIG α) (curr : Nat) (hcurr : curr ≤ w) (cin : Ref aig) (origCin : Ref aig)
(lhs rhs : RefVec aig w) (lhsExpr rhsExpr : BitVec w) (assign : α → Bool)
(hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, lhs.get idx hidx, assign⟧ = lhsExpr.getLsbD idx)
(hright : ∀ (idx : Nat) (hidx : idx ... | α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
curr : Nat
hcurr : curr ≤ w
cin origCin : aig.Ref
lhs rhs : aig.RefVec w
lhsExpr rhsExpr : BitVec w
assign : α → Bool
hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := lhs.get idx hidx }⟧ = lhsExpr.getLsbD idx
hright : ∀ (idx : N... | omega | no goals | 90b5750be1dd4f8d |
NumberField.InfinitePlace.mk_eq_iff | Mathlib/NumberTheory/NumberField/Embeddings.lean | theorem mk_eq_iff {φ ψ : K →+* ℂ} : mk φ = mk ψ ↔ φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ | K : Type u_2
inst✝ : Field K
φ ψ : K →+* ℂ
⊢ mk φ = mk ψ ↔ φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ | constructor | case mp
K : Type u_2
inst✝ : Field K
φ ψ : K →+* ℂ
⊢ mk φ = mk ψ → φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ
case mpr
K : Type u_2
inst✝ : Field K
φ ψ : K →+* ℂ
⊢ φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ → mk φ = mk ψ | 1d2ef4e3545a7979 |
MeasureTheory.StronglyMeasurable.stronglyMeasurable_of_measurableSpace_le_on | Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | theorem stronglyMeasurable_of_measurableSpace_le_on {α E} {m m₂ : MeasurableSpace α}
[TopologicalSpace E] [Zero E] {s : Set α} {f : α → E} (hs_m : MeasurableSet[m] s)
(hs : ∀ t, MeasurableSet[m] (s ∩ t) → MeasurableSet[m₂] (s ∩ t))
(hf : StronglyMeasurable[m] f) (hf_zero : ∀ x ∉ s, f x = 0) :
StronglyMe... | case pos
α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
hs_m₂ : MeasurableSet s
g_seq_s : ℕ → α →ₛ E
hg_seq_ten... | ext1 y | case pos.h
α : Type u_5
E : Type u_6
m m₂ : MeasurableSpace α
inst✝¹ : TopologicalSpace E
inst✝ : Zero E
s : Set α
f : α → E
hs_m : MeasurableSet s
hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)
hf : StronglyMeasurable f
hf_zero : ∀ x ∉ s, f x = 0
hs_m₂ : MeasurableSet s
g_seq_s : ℕ → α →ₛ E
hg_seq_t... | d81af41568179792 |
FiniteField.roots_X_pow_card_sub_X | Mathlib/FieldTheory/Finite/Basic.lean | theorem roots_X_pow_card_sub_X : roots (X ^ q - X : K[X]) = Finset.univ.val | K : Type u_1
inst✝¹ : Field K
inst✝ : Fintype K
⊢ (X ^ q - X).roots = univ.val | classical
have aux : (X ^ q - X : K[X]) ≠ 0 := X_pow_card_sub_X_ne_zero K Fintype.one_lt_card
have : (roots (X ^ q - X : K[X])).toFinset = Finset.univ := by
rw [eq_univ_iff_forall]
intro x
rw [Multiset.mem_toFinset, mem_roots aux, IsRoot.def, eval_sub, eval_pow, eval_X,
sub_eq_zero, pow_card]
rw... | no goals | f38f049b4c3caab9 |
Algebra.smul_leftMulMatrix | Mathlib/LinearAlgebra/Matrix/ToLin.lean | theorem smul_leftMulMatrix {G} [Group G] [DistribMulAction G S]
[SMulCommClass G R S] [SMulCommClass G S S] (g : G) (x) :
leftMulMatrix (g • b) x = leftMulMatrix b x | R : Type u_1
S : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : Semiring S
inst✝⁶ : Algebra R S
m : Type u_3
inst✝⁵ : Fintype m
inst✝⁴ : DecidableEq m
b : Basis m R S
G : Type u_4
inst✝³ : Group G
inst✝² : DistribMulAction G S
inst✝¹ : SMulCommClass G R S
inst✝ : SMulCommClass G S S
g : G
x : S
⊢ (leftMulMatrix (g • b)) x =... | ext | case a
R : Type u_1
S : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : Semiring S
inst✝⁶ : Algebra R S
m : Type u_3
inst✝⁵ : Fintype m
inst✝⁴ : DecidableEq m
b : Basis m R S
G : Type u_4
inst✝³ : Group G
inst✝² : DistribMulAction G S
inst✝¹ : SMulCommClass G R S
inst✝ : SMulCommClass G S S
g : G
x : S
i✝ j✝ : m
⊢ (leftMulMa... | 34815c79788fab47 |
Asymptotics.isEquivalent_zero_iff_isBigO_zero | Mathlib/Analysis/Asymptotics/AsymptoticEquivalent.lean | theorem isEquivalent_zero_iff_isBigO_zero : u ~[l] 0 ↔ u =O[l] (0 : α → β) | α : Type u_1
β : Type u_2
inst✝ : NormedAddCommGroup β
u : α → β
l : Filter α
h : u =O[l] 0
⊢ ∃ s ∈ l, Set.EqOn u 0 s | exact ⟨{ x : α | u x = 0 }, isBigO_zero_right_iff.mp h, fun x hx ↦ hx⟩ | no goals | b727e8f3469d26e6 |
AntitoneOn.sum_le_integral | Mathlib/Analysis/SumIntegralComparisons.lean | theorem AntitoneOn.sum_le_integral (hf : AntitoneOn f (Icc x₀ (x₀ + a))) :
(∑ i ∈ Finset.range a, f (x₀ + (i + 1 : ℕ))) ≤ ∫ x in x₀..x₀ + a, f x | case h₂
x₀ : ℝ
a : ℕ
f : ℝ → ℝ
hf : AntitoneOn f (Icc x₀ (x₀ + ↑a))
k : ℕ
hk : k < a
⊢ x₀ + ↑(k + 1) ≤ x₀ + ↑a | simp only [add_le_add_iff_left, Nat.cast_le, Nat.succ_le_of_lt hk] | no goals | d351347e7bd629a9 |
BitVec.sshiftRight_add | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem sshiftRight_add {x : BitVec w} {m n : Nat} :
x.sshiftRight (m + n) = (x.sshiftRight m).sshiftRight n | case neg
w : Nat
x : BitVec w
m n i : Nat
a✝ : i < w
h₁ : ¬w ≤ i
h₂ : ¬n + i < w
⊢ (if m + (n + i) < w then x.getLsbD (m + (n + i)) else x.msb) =
if n + i < w then !decide (w ≤ n + i) && if m + (n + i) < w then x.getLsbD (m + (n + i)) else x.msb
else (x.sshiftRight m).msb | simp only [h₂, ↓reduceIte] | case neg
w : Nat
x : BitVec w
m n i : Nat
a✝ : i < w
h₁ : ¬w ≤ i
h₂ : ¬n + i < w
⊢ (if m + (n + i) < w then x.getLsbD (m + (n + i)) else x.msb) = (x.sshiftRight m).msb | af6792ce3f635bc7 |
MulAction.isPretransitive_compHom | Mathlib/Algebra/Group/Action/Hom.lean | /-- If an action is transitive, then composing this action with a surjective homomorphism gives
again a transitive action. -/
@[to_additive]
lemma isPretransitive_compHom {E F G : Type*} [Monoid E] [Monoid F] [MulAction F G]
[IsPretransitive F G] {f : E →* F} (hf : Surjective f) :
letI : MulAction E G := MulAct... | E : Type u_4
F : Type u_5
G : Type u_6
inst✝³ : Monoid E
inst✝² : Monoid F
inst✝¹ : MulAction F G
inst✝ : IsPretransitive F G
f : E →* F
hf : Surjective ⇑f
x✝ : MulAction E G := compHom G f
x y : G
⊢ ∃ g, g • x = y | obtain ⟨m, rfl⟩ : ∃ m : F, m • x = y := exists_smul_eq F x y | case intro
E : Type u_4
F : Type u_5
G : Type u_6
inst✝³ : Monoid E
inst✝² : Monoid F
inst✝¹ : MulAction F G
inst✝ : IsPretransitive F G
f : E →* F
hf : Surjective ⇑f
x✝ : MulAction E G := compHom G f
x : G
m : F
⊢ ∃ g, g • x = m • x | b302746ffeaa5463 |
IsPrimitiveRoot.exists_neg_pow_of_isOfFinOrder | Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean | theorem exists_neg_pow_of_isOfFinOrder [IsCyclotomicExtension {n} ℚ K]
(hno : Odd (n : ℕ)) {ζ x : K} (hζ : IsPrimitiveRoot ζ n) (hx : IsOfFinOrder x) :
∃ r : ℕ, x = (-ζ) ^ r := by
have hnegζ : IsPrimitiveRoot (-ζ) (2 * n) | n : ℕ+
K : Type u
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {n} ℚ K
hno : Odd ↑n
ζ x : K
hζ : IsPrimitiveRoot ζ ↑n
hx : IsOfFinOrder x
hnegζ : IsPrimitiveRoot (-ζ) (2 * ↑n)
k : ℕ
hkpos : 0 < k
hkn : x ^ k = 1
l : ℕ
hl : l ∈ k.divisors
hlroot : IsPrimitiveRoot x l
hlzero : NeZero l
this : NeZ... | rw [ha, pow_mul, hlroot.pow_eq_one, one_pow] | no goals | 3ad57181433c8001 |
ArithmeticFunction.vonMangoldt.not_summable_residueClass_prime_div | Mathlib/NumberTheory/LSeries/PrimesInAP.lean | /-- The function `n ↦ Λ n / n` restricted to primes in an invertible residue class
is not summable. This then implies that there must be infinitely many such primes. -/
lemma not_summable_residueClass_prime_div (ha : IsUnit a) :
¬ Summable fun n : ℕ ↦ (if n.Prime then residueClass a n else 0) / n | q : ℕ
inst✝ : NeZero q
a : ZMod q
ha : IsUnit a
H : Summable fun n => (if Nat.Prime n then residueClass a n else 0) / ↑n
key : Summable fun n => residueClass a n / ↑n
C : ℝ := ∑' (n : ℕ), residueClass a n / ↑n
H₁✝ : ∀ {x : ℝ}, 1 < x → ∑' (n : ℕ), residueClass a n / ↑n ^ x ≤ C
C' : ℝ
hC' : ∀ {x : ℝ}, x ∈ Set.Ioc 1 2 → (... | refine ⟨min (1 + (q.totient : ℝ)⁻¹ / (C + C') / 2) 2, ⟨?_, min_le_right ..⟩, ?_⟩ | case refine_1
q : ℕ
inst✝ : NeZero q
a : ZMod q
ha : IsUnit a
H : Summable fun n => (if Nat.Prime n then residueClass a n else 0) / ↑n
key : Summable fun n => residueClass a n / ↑n
C : ℝ := ∑' (n : ℕ), residueClass a n / ↑n
H₁✝ : ∀ {x : ℝ}, 1 < x → ∑' (n : ℕ), residueClass a n / ↑n ^ x ≤ C
C' : ℝ
hC' : ∀ {x : ℝ}, x ∈ S... | fc94f129713b76bf |
MeasureTheory.MemLp.toLp_eq_toLp_iff | Mathlib/MeasureTheory/Function/LpSpace/Basic.lean | theorem toLp_eq_toLp_iff {f g : α → E} (hf : MemLp f p μ) (hg : MemLp g p μ) :
hf.toLp f = hg.toLp g ↔ f =ᵐ[μ] g | α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
f g : α → E
hf : MemLp f p μ
hg : MemLp g p μ
⊢ toLp f hf = toLp g hg ↔ f =ᶠ[ae μ] g | simp [toLp] | no goals | 17ef0d2daaf5800a |
NNReal.rpow_le_self_of_le_one | Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | theorem rpow_le_self_of_le_one {x : ℝ≥0} {z : ℝ} (hx : x ≤ 1) (h_one_le : 1 ≤ z) : x ^ z ≤ x | x : ℝ≥0
z : ℝ
hx : x ≤ 1
h_one_le : 1 ≤ z
⊢ x ^ z ≤ x | rcases eq_bot_or_bot_lt x with (rfl | (h : 0 < x)) | case inl
z : ℝ
h_one_le : 1 ≤ z
hx : ⊥ ≤ 1
⊢ ⊥ ^ z ≤ ⊥
case inr
x : ℝ≥0
z : ℝ
hx : x ≤ 1
h_one_le : 1 ≤ z
h : 0 < x
⊢ x ^ z ≤ x | 3c26778a22def5f9 |
CategoryTheory.Functor.preservesFiniteColimits_of_preservesCokernels | Mathlib/CategoryTheory/Preadditive/LeftExact.lean | /-- A functor between preadditive categories which preserves kernels preserves all finite limits.
-/
lemma preservesFiniteColimits_of_preservesCokernels [HasFiniteCoproducts C] [HasCoequalizers C]
[HasZeroObject C] [HasZeroObject D]
[∀ {X Y} (f : X ⟶ Y), PreservesColimit (parallelPair f 0) F] : PreservesFiniteC... | C : Type u₁
inst✝¹⁰ : Category.{v₁, u₁} C
inst✝⁹ : Preadditive C
D : Type u₂
inst✝⁸ : Category.{v₂, u₂} D
inst✝⁷ : Preadditive D
F : C ⥤ D
inst✝⁶ : F.PreservesZeroMorphisms
inst✝⁵ : HasBinaryBiproducts C
inst✝⁴ : HasFiniteCoproducts C
inst✝³ : HasCoequalizers C
inst✝² : HasZeroObject C
inst✝¹ : HasZeroObject D
inst✝ : ... | exact preservesFiniteColimits_of_preservesCoequalizers_and_finiteCoproducts F | no goals | a5abf65f972ee903 |
Finset.expect_div | Mathlib/Algebra/BigOperators/Expect.lean | lemma expect_div (s : Finset ι) (f : ι → M) (a : M) : (𝔼 i ∈ s, f i) / a = 𝔼 i ∈ s, f i / a | ι : Type u_1
M : Type u_3
inst✝¹ : Semifield M
inst✝ : CharZero M
s : Finset ι
f : ι → M
a : M
⊢ (𝔼 i ∈ s, f i) / a = 𝔼 i ∈ s, f i / a | simp_rw [div_eq_mul_inv, expect_mul] | no goals | 2543fb3ab8ffd5cb |
AlgebraicGeometry.ΓSpecIso_obj_hom | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | theorem ΓSpecIso_obj_hom {X : Scheme.{u}} (U : X.Opens) :
(Scheme.ΓSpecIso Γ(X, U)).hom = (Spec.map U.topIso.inv).appTop ≫
U.toScheme.toSpecΓ.appTop ≫ U.topIso.hom | X : Scheme
U : X.Opens
⊢ (Scheme.ΓSpecIso Γ(X, U)).hom =
Scheme.Hom.appTop (Spec.map U.topIso.inv) ≫ Scheme.Hom.appTop (↑U).toSpecΓ ≫ U.topIso.hom | simp | no goals | 1431735a351881a8 |
CategoryTheory.isIso_prod_iff | Mathlib/CategoryTheory/Products/Basic.lean | theorem isIso_prod_iff {P Q : C} {S T : D} {f : (P, S) ⟶ (Q, T)} :
IsIso f ↔ IsIso f.1 ∧ IsIso f.2 | case mp
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
P Q : C
S T : D
f : (P, S) ⟶ (Q, T)
⊢ IsIso f → IsIso f.1 ∧ IsIso f.2 | rintro ⟨g, hfg, hgf⟩ | case mp.mk.intro.intro
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
P Q : C
S T : D
f : (P, S) ⟶ (Q, T)
g : (Q, T) ⟶ (P, S)
hfg : f ≫ g = 𝟙 (P, S)
hgf : g ≫ f = 𝟙 (Q, T)
⊢ IsIso f.1 ∧ IsIso f.2 | 31bd369438c946ae |
OnePoint.equivProjectivization_symm_apply_mk | Mathlib/Topology/Compactification/OnePointEquiv.lean | @[simp]
lemma equivProjectivization_symm_apply_mk (x y : K) (h : (x, y) ≠ 0) :
(equivProjectivization K).symm (mk K ⟨x, y⟩ h) = if y = 0 then ∞ else y⁻¹ * x | K : Type u_1
inst✝¹ : DivisionRing K
inst✝ : DecidableEq K
x y : K
h : (x, y) ≠ 0
⊢ (equivProjectivization K).symm (mk K (x, y) h) = if y = 0 then ∞ else ↑(y⁻¹ * x) | simp [equivProjectivization] | no goals | 480505e1c9e6ca5b |
Polynomial.irreducible_comp | Mathlib/FieldTheory/IntermediateField/Adjoin/Basic.lean | theorem _root_.Polynomial.irreducible_comp {f g : K[X]} (hfm : f.Monic) (hgm : g.Monic)
(hf : Irreducible f)
(hg : ∀ (E : Type u) [Field E] [Algebra K E] (x : E) (_ : minpoly K x = f),
Irreducible (g.map (algebraMap _ _) - C (AdjoinSimple.gen K x))) :
Irreducible (f.comp g) | K : Type u
inst✝ : Field K
f g : K[X]
hfm : f.Monic
hgm : g.Monic
hf : Irreducible f
hg :
∀ (E : Type u) [inst : Field E] [inst_1 : Algebra K E] (x : E),
minpoly K x = f → Irreducible (Polynomial.map (algebraMap K ↥K⟮x⟯) g - C (AdjoinSimple.gen K x))
hf' : f.natDegree ≠ 0
hg' : g.natDegree ≠ 0
H₁ : f.comp g ≠ 0
H... | have := Fact.mk hp₁ | K : Type u
inst✝ : Field K
f g : K[X]
hfm : f.Monic
hgm : g.Monic
hf : Irreducible f
hg :
∀ (E : Type u) [inst : Field E] [inst_1 : Algebra K E] (x : E),
minpoly K x = f → Irreducible (Polynomial.map (algebraMap K ↥K⟮x⟯) g - C (AdjoinSimple.gen K x))
hf' : f.natDegree ≠ 0
hg' : g.natDegree ≠ 0
H₁ : f.comp g ≠ 0
H... | 9c1ae09b4550aa94 |
Continuous.strictMono_of_inj | Mathlib/Topology/Order/IntermediateValue.lean | theorem Continuous.strictMono_of_inj {f : α → δ}
(hf_c : Continuous f) (hf_i : Injective f) : StrictMono f ∨ StrictAnti f | case pos
α : Type u
inst✝⁶ : ConditionallyCompleteLinearOrder α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : OrderTopology α
inst✝³ : DenselyOrdered α
δ : Type u_1
inst✝² : LinearOrder δ
inst✝¹ : TopologicalSpace δ
inst✝ : OrderClosedTopology δ
f : α → δ
hf_c : Continuous f
hf_i : Injective f
H : ∀ {c d : α}, c < d → StrictMon... | by_cases h : ∃ b : α, a ≠ b | case pos
α : Type u
inst✝⁶ : ConditionallyCompleteLinearOrder α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : OrderTopology α
inst✝³ : DenselyOrdered α
δ : Type u_1
inst✝² : LinearOrder δ
inst✝¹ : TopologicalSpace δ
inst✝ : OrderClosedTopology δ
f : α → δ
hf_c : Continuous f
hf_i : Injective f
H : ∀ {c d : α}, c < d → StrictMon... | a6f33dd3051e4aa2 |
AlgebraicGeometry.RingedSpace.basicOpen_res | Mathlib/Geometry/RingedSpace/Basic.lean | theorem basicOpen_res {U V : (Opens X)ᵒᵖ} (i : U ⟶ V) (f : X.presheaf.obj U) :
@basicOpen X (unop V) (X.presheaf.map i f) = unop V ⊓ @basicOpen X (unop U) f | case h.h.mp.intro
X : RingedSpace
U V : (Opens ↑↑X.toPresheafedSpace)ᵒᵖ
i : U ⟶ V
f : ↑(X.presheaf.obj U)
x : ↑↑X.toPresheafedSpace
hxV : x ∈ unop V
hx : IsUnit ((ConcreteCategory.hom (X.presheaf.germ (unop U) x ⋯)) f)
⊢ x ∈ ↑(unop V ⊓ X.basicOpen f) | exact ⟨hxV, i.unop.le hxV, hx⟩ | no goals | 95747598eb7ff13b |
Vitali.exists_disjoint_subfamily_covering_enlargement_closedBall | Mathlib/MeasureTheory/Covering/Vitali.lean | theorem exists_disjoint_subfamily_covering_enlargement_closedBall
[PseudoMetricSpace α] (t : Set ι)
(x : ι → α) (r : ι → ℝ) (R : ℝ) (hr : ∀ a ∈ t, r a ≤ R) (τ : ℝ) (hτ : 3 < τ) :
∃ u ⊆ t,
(u.PairwiseDisjoint fun a => closedBall (x a) (r a)) ∧
∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBa... | case inl
α : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α
x : ι → α
r : ι → ℝ
R τ : ℝ
hτ : 3 < τ
hr : ∀ a ∈ ∅, r a ≤ R
⊢ ∃ u ⊆ ∅,
(u.PairwiseDisjoint fun a => closedBall (x a) (r a)) ∧
∀ a ∈ ∅, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b) | exact ⟨∅, Subset.refl _, pairwiseDisjoint_empty, by simp⟩ | no goals | dde5a8ee9cd5465a |
Polynomial.expand_one | Mathlib/Algebra/Polynomial/Expand.lean | theorem expand_one (f : R[X]) : expand R 1 f = f :=
Polynomial.induction_on f (fun r => by rw [expand_C])
(fun f g ihf ihg => by rw [map_add, ihf, ihg]) fun n r _ => by
rw [map_mul, expand_C, map_pow, expand_X, pow_one]
| R : Type u
inst✝ : CommSemiring R
f : R[X]
r : R
⊢ (expand R 1) (C r) = C r | rw [expand_C] | no goals | 5f648ee3615f36f5 |
AlgebraicGeometry.isIntegral_appTop_of_universallyClosed | Mathlib/AlgebraicGeometry/Morphisms/Proper.lean | theorem isIntegral_appTop_of_universallyClosed (f : X ⟶ Y) [UniversallyClosed f] [IsAffine Y] :
f.appTop.hom.IsIntegral | case surj.hfinj
X Y : Scheme
f : X ⟶ Y
inst✝¹ : UniversallyClosed f
inst✝ : IsAffine Y
this✝¹ : CompactSpace ↑↑X.toPresheafedSpace
this✝ : UniversallyClosed (X.toSpecΓ ≫ Spec.map (Scheme.Hom.appTop f))
this : UniversallyClosed X.toSpecΓ
⊢ Function.Injective ⇑(ConcreteCategory.hom (Scheme.ΓSpecIso Γ(X, ⊤)).hom) | exact (ConcreteCategory.bijective_of_isIso (Scheme.ΓSpecIso Γ(X, ⊤)).hom).1 | no goals | 0f196145409a30ed |
gold_add_goldConj | Mathlib/Data/Real/GoldenRatio.lean | theorem gold_add_goldConj : φ + ψ = 1 | ⊢ (1 + √5) / 2 + (1 - √5) / 2 = 1 | ring | no goals | 3a020a00c6921c34 |
MeasureTheory.lintegral_iInf_directed_of_measurable | Mathlib/MeasureTheory/Integral/Lebesgue.lean | theorem lintegral_iInf_directed_of_measurable {mα : MeasurableSpace α} [Countable β]
{f : β → α → ℝ≥0∞} {μ : Measure α} (hμ : μ ≠ 0) (hf : ∀ b, Measurable (f b))
(hf_int : ∀ b, ∫⁻ a, f b a ∂μ ≠ ∞) (h_directed : Directed (· ≥ ·) f) :
∫⁻ a, ⨅ b, f b a ∂μ = ⨅ b, ∫⁻ a, f b a ∂μ | case refine_2
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
inst✝ : Countable β
f : β → α → ℝ≥0∞
μ : Measure α
hμ : μ ≠ 0
hf : ∀ (b : β), Measurable (f b)
hf_int : ∀ (b : β), ∫⁻ (a : α), f b a ∂μ ≠ ⊤
h_directed : Directed (fun x1 x2 => x1 ≥ x2) f
val✝ : Encodable β
h✝ : Nonempty β
inhabited_h : Inhabited β
this : ∀ ... | exact iInf_le (fun b => ∫⁻ a, f b a ∂μ) _ | no goals | 0fae92523fce0ee1 |
PiNat.mem_cylinder_iff_dist_le | Mathlib/Topology/MetricSpace/PiNat.lean | theorem mem_cylinder_iff_dist_le {x y : ∀ n, E n} {n : ℕ} :
y ∈ cylinder x n ↔ dist y x ≤ (1 / 2) ^ n | E : ℕ → Type u_1
x y : (n : ℕ) → E n
n : ℕ
hne : y ≠ x
this : (∀ i < n, y i = x i) ↔ n ≤ firstDiff y x
⊢ y ∈ cylinder x n ↔ dist y x ≤ (1 / 2) ^ n | simpa [dist_eq_of_ne hne] | no goals | 9da20a277a1de40c |
Stream'.interleave_tail_tail | Mathlib/Data/Stream/Init.lean | theorem interleave_tail_tail (s₁ s₂ : Stream' α) : tail s₁ ⋈ tail s₂ = tail (tail (s₁ ⋈ s₂)) | α : Type u
s₁ s₂ : Stream' α
⊢ s₁.tail ⋈ s₂.tail = (s₁ ⋈ s₂).tail.tail | rw [interleave_eq s₁ s₂] | α : Type u
s₁ s₂ : Stream' α
⊢ s₁.tail ⋈ s₂.tail = (s₁.head :: s₂.head :: (s₁.tail ⋈ s₂.tail)).tail.tail | 746bf88f1c06104a |
Stream'.Seq.head_eq_destruct | Mathlib/Data/Seq/Seq.lean | theorem head_eq_destruct (s : Seq α) : head.{u} s = Prod.fst.{u} <$> destruct.{u} s | α : Type u
s : Seq α
⊢ s.head = Prod.fst <$> s.destruct | unfold destruct head | α : Type u
s : Seq α
⊢ s.get? 0 = Prod.fst <$> (fun a' => (a', s.tail)) <$> s.get? 0 | 1e5d1a788db819d8 |
Std.Tactic.BVDecide.BVExpr.bitblast.blastConst.go_decl_eq | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Impl/Const.lean | theorem blastConst.go_decl_eq {aig : AIG α} (i : Nat) (s : AIG.RefVec aig i) (val : BitVec w)
(hi : i ≤ w) :
∀ (curr : Nat) (h1) (h2),
(go aig val i s hi).aig.decls[curr]'h2 = aig.decls[curr]'h1 | case isFalse
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
i : Nat
s : aig.RefVec i
val : BitVec w
hi : i ≤ w
res : AIG.RefVecEntry α w
h✝ : ¬i < w
hgo :
(let_fun hcurr := ⋯;
{ aig := aig, vec := hcurr ▸ s }) =
res
⊢ ∀ (curr : Nat) (h1 : curr < aig.decls.size) (h2 : curr < res.aig.dec... | dsimp only at hgo | case isFalse
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
i : Nat
s : aig.RefVec i
val : BitVec w
hi : i ≤ w
res : AIG.RefVecEntry α w
h✝ : ¬i < w
hgo : { aig := aig, vec := ⋯ ▸ s } = res
⊢ ∀ (curr : Nat) (h1 : curr < aig.decls.size) (h2 : curr < res.aig.decls.size), res.aig.decls[curr] = aig.... | a8695d661d5b3e87 |
Nat.sInf_mem | Mathlib/Data/Nat/Lattice.lean | theorem sInf_mem {s : Set ℕ} (h : s.Nonempty) : sInf s ∈ s | s : Set ℕ
h : s.Nonempty
⊢ sInf s ∈ s | rw [Nat.sInf_def h] | s : Set ℕ
h : s.Nonempty
⊢ Nat.find h ∈ s | b3a6adc09f8ed6ff |
ContinuousOn.aestronglyMeasurable | Mathlib/MeasureTheory/Integral/IntegrableOn.lean | theorem ContinuousOn.aestronglyMeasurable [TopologicalSpace α] [TopologicalSpace β]
[h : SecondCountableTopologyEither α β] [OpensMeasurableSpace α] [PseudoMetrizableSpace β]
{f : α → β} {s : Set α} {μ : Measure α} (hf : ContinuousOn f s) (hs : MeasurableSet s) :
AEStronglyMeasurable f (μ.restrict s) | case inl
α : Type u_1
β : Type u_2
inst✝⁴ : MeasurableSpace α
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
h : SecondCountableTopologyEither α β
inst✝¹ : OpensMeasurableSpace α
inst✝ : PseudoMetrizableSpace β
f : α → β
s : Set α
μ : Measure α
hf : ContinuousOn f s
hs : MeasurableSet s
this✝¹ : MeasurableSpac... | exact isSeparable_range <| continuousOn_iff_continuous_restrict.1 hf | no goals | 56f399750f10b410 |
MeasureTheory.setIntegral_eq_of_subset_of_ae_diff_eq_zero | Mathlib/MeasureTheory/Integral/SetIntegral.lean | theorem setIntegral_eq_of_subset_of_ae_diff_eq_zero (ht : NullMeasurableSet t μ) (hts : s ⊆ t)
(h't : ∀ᵐ x ∂μ, x ∈ t \ s → f x = 0) : ∫ x in t, f x ∂μ = ∫ x in s, f x ∂μ | case h
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
s t : Set X
μ : Measure X
ht : NullMeasurableSet t μ
hts : s ⊆ t
h't : ∀ᵐ (x : X) ∂μ, x ∈ t \ s → f x = 0
h : IntegrableOn f t μ
f' : X → E := AEStronglyMeasurable.mk f ⋯
x : X
hx : x ∈ t \ s → f x = ... | rw [← h'x h''x.1, hx h''x] | no goals | 223bc5488a731ffc |
zsmul_eq_mul | Mathlib/Data/Int/Cast/Lemmas.lean | @[simp] lemma _root_.zsmul_eq_mul (a : α) : ∀ n : ℤ, n • a = n * a
| (n : ℕ) => by rw [natCast_zsmul, nsmul_eq_mul, Int.cast_natCast]
| -[n+1] => by simp [Nat.cast_succ, neg_add_rev, Int.cast_negSucc, add_mul]
| α : Type u_3
inst✝ : NonAssocRing α
a : α
n : ℕ
⊢ ↑n • a = ↑↑n * a | rw [natCast_zsmul, nsmul_eq_mul, Int.cast_natCast] | no goals | 12aeea7c7406e69e |
DivisorChain.second_of_chain_is_irreducible | Mathlib/RingTheory/ChainOfDivisors.lean | theorem second_of_chain_is_irreducible {q : Associates M} {n : ℕ} (hn : n ≠ 0)
{c : Fin (n + 1) → Associates M} (h₁ : StrictMono c) (h₂ : ∀ {r}, r ≤ q ↔ ∃ i, r = c i)
(hq : q ≠ 0) : Irreducible (c 1) | case succ.refine_2.intro.mk
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
q : Associates M
hq : q ≠ 0
n : ℕ
hn : n + 1 ≠ 0
c : Fin (n + 1 + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i
i : ℕ
hi : i < n + 1 + 1
hb : c ⟨i, hi⟩ < c 1
⊢ c ⟨i, hi⟩ = ⊥ | cases i | case succ.refine_2.intro.mk.zero
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
q : Associates M
hq : q ≠ 0
n : ℕ
hn : n + 1 ≠ 0
c : Fin (n + 1 + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i
hi : 0 < n + 1 + 1
hb : c ⟨0, hi⟩ < c 1
⊢ c ⟨0, hi⟩ = ⊥
case succ.refine_2.intro.mk.succ... | 90ab663cd42221e6 |
Set.mul_subgroupClosure | Mathlib/Algebra/Group/Subgroup/Pointwise.lean | @[to_additive (attr := simp)]
lemma mul_subgroupClosure (hs : s.Nonempty) : s * closure s = closure s | G : Type u_2
inst✝ : Group G
s : Set G
hs : s.Nonempty
⊢ ⋃ a ∈ s, a • ↑(closure s) = ↑(closure s) | have h a (ha : a ∈ s) : a • (closure s : Set G) = closure s :=
smul_coe_set <| subset_closure ha | G : Type u_2
inst✝ : Group G
s : Set G
hs : s.Nonempty
h : ∀ a ∈ s, a • ↑(closure s) = ↑(closure s)
⊢ ⋃ a ∈ s, a • ↑(closure s) = ↑(closure s) | 142e456280bb9b90 |
contMDiffWithinAt_iff_source_of_mem_maximalAtlas | Mathlib/Geometry/Manifold/ContMDiff/Defs.lean | theorem contMDiffWithinAt_iff_source_of_mem_maximalAtlas
[IsManifold I n M] (he : e ∈ maximalAtlas I n M) (hx : x ∈ e.source) :
ContMDiffWithinAt I I' n f s x ↔
ContMDiffWithinAt 𝓘(𝕜, E) I' n (f ∘ (e.extend I).symm) ((e.extend I).symm ⁻¹' s ∩ range I)
(e.extend I x) | 𝕜 : Type u_1
inst✝¹¹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁸ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁷ : TopologicalSpace M
inst✝⁶ : ChartedSpace H M
E' : Type u_5
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpa... | simp_rw [ContMDiffWithinAt,
(contDiffWithinAt_localInvariantProp n).liftPropWithinAt_indep_chart_source he hx,
StructureGroupoid.liftPropWithinAt_self_source,
e.extend_symm_continuousWithinAt_comp_right_iff, contDiffWithinAtProp_self_source,
ContDiffWithinAtProp, Function.comp, e.left_inv hx, (e.extend I).left_... | 𝕜 : Type u_1
inst✝¹¹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁸ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁷ : TopologicalSpace M
inst✝⁶ : ChartedSpace H M
E' : Type u_5
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpa... | 424cdc085dbdcaaa |
Monotone.disjointed_add_one_sup | Mathlib/Algebra/Order/Disjointed.lean | protected lemma Monotone.disjointed_add_one_sup {f : ι → α} (hf : Monotone f) (i : ι) :
disjointed f (i + 1) ⊔ f i = f (i + 1) | α : Type u_1
ι : Type u_2
inst✝⁵ : GeneralizedBooleanAlgebra α
inst✝⁴ : LinearOrder ι
inst✝³ : LocallyFiniteOrderBot ι
inst✝² : Add ι
inst✝¹ : One ι
inst✝ : SuccAddOrder ι
f : ι → α
hf : Monotone f
i : ι
⊢ disjointed f (i + 1) ⊔ f i = f (i + 1) | simpa only [succ_eq_add_one i] using hf.disjointed_succ_sup i | no goals | fbabe1f722cfd0cd |
BitVec.toNat_ne_iff_ne | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem toNat_ne_iff_ne {n} {x y : BitVec n} : x.toNat ≠ y.toNat ↔ x ≠ y | case mpr
n : Nat
x y : BitVec n
h : x ≠ y
h_eq : x.toNat = y.toNat
⊢ False | apply h <| eq_of_toNat_eq h_eq | no goals | 94d07c4a50176edd |
Subalgebra.comap_map_eq | Mathlib/RingTheory/Adjoin/Basic.lean | theorem comap_map_eq (f : A →ₐ[R] B) (S : Subalgebra R A) :
(S.map f).comap f = S ⊔ Algebra.adjoin R (f ⁻¹' {0}) | case a.hab
R : Type uR
A : Type uA
B : Type uB
inst✝⁴ : CommSemiring R
inst✝³ : Ring A
inst✝² : Algebra R A
inst✝¹ : Ring B
inst✝ : Algebra R B
f : A →ₐ[R] B
S : Subalgebra R A
⊢ ⇑f '' (⇑f ⁻¹' {0}) ⊆ ↑(map f S) | exact (Set.image_preimage_subset f {0}).trans (Set.singleton_subset_iff.2 (S.map f).zero_mem) | no goals | a0a721642f7f2130 |
Matrix.fromBlocks_submatrix_sum_swap_left | Mathlib/Data/Matrix/Block.lean | theorem fromBlocks_submatrix_sum_swap_left (A : Matrix n l α) (B : Matrix n m α) (C : Matrix o l α)
(D : Matrix o m α) (f : p → l ⊕ m) :
(fromBlocks A B C D).submatrix Sum.swap f = (fromBlocks C D A B).submatrix id f | l : Type u_1
m : Type u_2
n : Type u_3
o : Type u_4
p : Type u_5
α : Type u_12
A : Matrix n l α
B : Matrix n m α
C : Matrix o l α
D : Matrix o m α
f : p → l ⊕ m
⊢ (fromBlocks A B C D).submatrix Sum.swap f = (fromBlocks C D A B).submatrix id f | ext i j | case a
l : Type u_1
m : Type u_2
n : Type u_3
o : Type u_4
p : Type u_5
α : Type u_12
A : Matrix n l α
B : Matrix n m α
C : Matrix o l α
D : Matrix o m α
f : p → l ⊕ m
i : o ⊕ n
j : p
⊢ (fromBlocks A B C D).submatrix Sum.swap f i j = (fromBlocks C D A B).submatrix id f i j | 56a6d057a3ca998f |
Subgroup.isComplement'_stabilizer | Mathlib/GroupTheory/Complement.lean | theorem isComplement'_stabilizer {α : Type*} [MulAction G α] (a : α)
(h1 : ∀ h : H, h • a = a → h = 1) (h2 : ∀ g : G, ∃ h : H, h • g • a = a) :
IsComplement' H (MulAction.stabilizer G a) | case intro
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
α : Type u_2
inst✝ : MulAction G α
a : α
h1 : ∀ (h : ↥H), h • a = a → h = 1
h2 : ∀ (g : G), ∃ h, h • g • a = a
g : G
h : ↥H
hh : h • g • a = a
hh' : (↑h * g) • a = a
⊢ ∃! x, ↑x.1 * ↑x.2 = g | refine ⟨⟨h⁻¹, h * g, hh'⟩, inv_mul_cancel_left ↑h g, ?_⟩ | case intro
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
α : Type u_2
inst✝ : MulAction G α
a : α
h1 : ∀ (h : ↥H), h • a = a → h = 1
h2 : ∀ (g : G), ∃ h, h • g • a = a
g : G
h : ↥H
hh : h • g • a = a
hh' : (↑h * g) • a = a
⊢ ∀ (y : ↑↑H × ↑↑(MulAction.stabilizer G a)), (fun x => ↑x.1 * ↑x.2 = g) y → y = (h⁻¹, ⟨↑h * g, hh... | 7aab5c5b20d3db73 |
CategoryTheory.PreGaloisCategory.connected_component_unique | Mathlib/CategoryTheory/Galois/Decomposition.lean | /-- Up to isomorphism an element of the fiber of `X` only lies in one connected component. -/
lemma connected_component_unique {X A B : C} [IsConnected A] [IsConnected B] (a : F.obj A)
(b : F.obj B) (i : A ⟶ X) (j : B ⟶ X) (h : F.map i a = F.map j b) [Mono i] [Mono j] :
∃ (f : A ≅ B), F.map f.hom a = b | C : Type u₁
inst✝⁶ : Category.{u₂, u₁} C
inst✝⁵ : GaloisCategory C
F : C ⥤ FintypeCat
inst✝⁴ : FiberFunctor F
X A B : C
inst✝³ : IsConnected A
inst✝² : IsConnected B
a : (F.obj A).carrier
b : (F.obj B).carrier
i : A ⟶ X
j : B ⟶ X
h : F.map i a = F.map j b
inst✝¹ : Mono i
inst✝ : Mono j
Y : C := pullback i j
u : Y ⟶ A :... | erw [Types.pullbackIsoPullback_inv_snd_apply (F.map i) (F.map j)] | no goals | 5ad7d80c4aeba386 |
MeasureTheory.integrable_condExpIndSMul | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean | theorem integrable_condExpIndSMul (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hs : MeasurableSet s)
(hμs : μ s ≠ ∞) (x : G) : Integrable (condExpIndSMul hm hs hμs x) μ | case refine_2
α : Type u_1
G : Type u_5
inst✝² : NormedAddCommGroup G
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
hs : MeasurableSet s
hμs : μ s ≠ ⊤
x : G
t : Set α
ht : MeasurableSet t
hμt : μ t ≠ ⊤
⊢ μ (s ∩ t) * ↑‖x‖₊ ≤ μ s * ↑‖x‖₊ | gcongr | case refine_2.bc.h
α : Type u_1
G : Type u_5
inst✝² : NormedAddCommGroup G
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
hs : MeasurableSet s
hμs : μ s ≠ ⊤
x : G
t : Set α
ht : MeasurableSet t
hμt : μ t ≠ ⊤
⊢ s ∩ t ⊆ s | 9b18d60f94da4571 |
HasStrictDerivAt.div_const | Mathlib/Analysis/Calculus/Deriv/Mul.lean | theorem HasStrictDerivAt.div_const (hc : HasStrictDerivAt c c' x) (d : 𝕜') :
HasStrictDerivAt (fun x => c x / d) (c' / d) x | 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
𝕜' : Type u_2
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
c : 𝕜 → 𝕜'
c' : 𝕜'
hc : HasStrictDerivAt c c' x
d : 𝕜'
⊢ HasStrictDerivAt (fun x => c x / d) (c' / d) x | simpa only [div_eq_mul_inv] using hc.mul_const d⁻¹ | no goals | c39b5ec1ea691af8 |
norm_iteratedFDerivWithin_comp_le_aux | Mathlib/Analysis/Calculus/ContDiff/Bounds.lean | theorem norm_iteratedFDerivWithin_comp_le_aux {Fu Gu : Type u} [NormedAddCommGroup Fu]
[NormedSpace 𝕜 Fu] [NormedAddCommGroup Gu] [NormedSpace 𝕜 Gu] {g : Fu → Gu} {f : E → Fu} {n : ℕ}
{s : Set E} {t : Set Fu} {x : E} (hg : ContDiffOn 𝕜 n g t) (hf : ContDiffOn 𝕜 n f s)
(ht : UniqueDiffOn 𝕜 t) (hs : Uniq... | case hC
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
Fu : Type u
inst✝³ : NormedAddCommGroup Fu
inst✝² : NormedSpace 𝕜 Fu
f : E → Fu
s : Set E
t : Set Fu
x : E
ht : UniqueDiffOn 𝕜 t
hs : UniqueDiffOn 𝕜 s
hst : MapsTo f s t
hx : x ∈ s
C D : ℝ
n ... | rw [this] | case hC
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
Fu : Type u
inst✝³ : NormedAddCommGroup Fu
inst✝² : NormedSpace 𝕜 Fu
f : E → Fu
s : Set E
t : Set Fu
x : E
ht : UniqueDiffOn 𝕜 t
hs : UniqueDiffOn 𝕜 s
hst : MapsTo f s t
hx : x ∈ s
C D : ℝ
n ... | 59e1e58bb3dc8aba |
Polynomial.natDegree_cyclotomic' | Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | theorem natDegree_cyclotomic' {ζ : R} {n : ℕ} (h : IsPrimitiveRoot ζ n) :
(cyclotomic' n R).natDegree = Nat.totient n | R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
ζ : R
n : ℕ
h : IsPrimitiveRoot ζ n
⊢ ∑ i ∈ primitiveRoots n R, (X - C i).natDegree = n.totient | simp only [IsPrimitiveRoot.card_primitiveRoots h, mul_one, natDegree_X_sub_C, Nat.cast_id,
Finset.sum_const, nsmul_eq_mul] | no goals | b4505e4eb5d9c1c1 |
Nat.add_lt_iff_lt_sub_right | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean | theorem add_lt_iff_lt_sub_right {a b c : Nat} : a + b < c ↔ a < c - b | a b c : Nat
⊢ a + b < c ↔ a < c - b | omega | no goals | f6985ef4b3c790e5 |
Nat.minFacAux_has_prop | Mathlib/Data/Nat/Prime/Defs.lean | theorem minFacAux_has_prop {n : ℕ} (n2 : 2 ≤ n) :
∀ k i, k = 2 * i + 3 → (∀ m, 2 ≤ m → m ∣ n → k ≤ m) → minFacProp n (minFacAux n k)
| k => fun i e a => by
rw [minFacAux]
by_cases h : n < k * k
· have pp : Prime n :=
prime_def_le_sqrt.2
⟨n2, fun m m2 l d => not_lt_of_ge l <| lt_of_lt... | n : ℕ
n2 : 2 ≤ n
x✝ : ℕ
k : ℕ := x✝
i : ℕ
e : x✝ = 2 * i + 3
a : ∀ (m : ℕ), 2 ≤ m → m ∣ n → x✝ ≤ m
⊢ Nat.minFacProp n (n.minFacAux x✝) | rw [minFacAux] | n : ℕ
n2 : 2 ≤ n
x✝ : ℕ
k : ℕ := x✝
i : ℕ
e : x✝ = 2 * i + 3
a : ∀ (m : ℕ), 2 ≤ m → m ∣ n → x✝ ≤ m
⊢ Nat.minFacProp n (if n < x✝ * x✝ then n else if x✝ ∣ n then x✝ else n.minFacAux (x✝ + 2)) | af1b898ad7dbf94b |
Finset.big_intersection | Mathlib/Combinatorics/Additive/VerySmallDoubling.lean | private lemma big_intersection {x y : G} (hx : x ∈ A) (hy : y ∈ A) :
2 * #A ≤ #((x • A) ∩ (y • A)) + #(A * A) | G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
A : Finset G
x y : G
hx : x ∈ A
hy : y ∈ A
this : #(x •> A ∪ y •> A) ≤ #(A * A)
⊢ 2 * #A = #(x •> A ∩ y •> A) + #(x •> A ∪ y •> A) | rw [card_inter_add_card_union] | G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
A : Finset G
x y : G
hx : x ∈ A
hy : y ∈ A
this : #(x •> A ∪ y •> A) ≤ #(A * A)
⊢ 2 * #A = #(x •> A) + #(y •> A) | 6f82c54018b437b2 |
ContMDiffWithinAt.mpullbackWithin_vectorField' | Mathlib/Geometry/Manifold/VectorField.lean | /-- The pullback of a `C^m` vector field by a `C^n` function with invertible derivative and
with `m + 1 ≤ n` is `C^m`.
Version within a set at a point, with a set used for the pullback possibly larger. -/
protected lemma _root_.ContMDiffWithinAt.mpullbackWithin_vectorField' {u : Set M}
(hV : ContMDiffWithinAt I' I'... | 𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝¹⁴ : TopologicalSpace H
E : Type u_3
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace H M
H' : Type u_5
inst✝⁹ : TopologicalSpace H'
E' : Type u_6
in... | have Y := (contMDiffWithinAt_iff_contMDiffWithinAt_nhdsWithin (by simp)).1 (hf.of_le hn) | 𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝¹⁴ : TopologicalSpace H
E : Type u_3
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace H M
H' : Type u_5
inst✝⁹ : TopologicalSpace H'
E' : Type u_6
in... | 9410fe422f25e41a |
CategoryTheory.GrothendieckTopology.arrow_stable | Mathlib/CategoryTheory/Sites/Grothendieck.lean | theorem arrow_stable (f : Y ⟶ X) (S : Sieve X) (h : J.Covers S f) {Z : C} (g : Z ⟶ Y) :
J.Covers S (g ≫ f) | C : Type u
inst✝ : Category.{v, u} C
X Y : C
J : GrothendieckTopology C
f : Y ⟶ X
S : Sieve X
h : Sieve.pullback f S ∈ J Y
Z : C
g : Z ⟶ Y
⊢ Sieve.pullback (g ≫ f) S ∈ J Z | simp [h, Sieve.pullback_comp] | no goals | 33c03517f82a4664 |
ProbabilityTheory.hasDerivAt_integral_pow_mul_exp | Mathlib/Probability/Moments/ComplexMGF.lean | /-- For `z : ℂ` with `z.re ∈ interior (integrableExpSet X μ)`, the derivative of the function
`z' ↦ μ[X ^ n * cexp (z' * X)]` at `z` is `μ[X ^ (n + 1) * cexp (z * X)]`. -/
lemma hasDerivAt_integral_pow_mul_exp (hz : z.re ∈ interior (integrableExpSet X μ)) (n : ℕ) :
HasDerivAt (fun z ↦ μ[fun ω ↦ X ω ^ n * cexp (z * ... | case intro.intro.intro.refine_4
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
z : ℂ
hz : z.re ∈ interior (integrableExpSet X μ)
n : ℕ
hX : AEMeasurable X μ
l u : ℝ
hlu : z.re ∈ Set.Ioo l u
h_subset : Set.Ioo l u ⊆ integrableExpSet X μ
t : ℝ := ((z.re - l) ⊓ (u - z.re)) / 2
h_pos : 0 < (z.re - l) ⊓ (u - z.r... | simp only [norm_mul, norm_pow, norm_real, Real.norm_eq_abs] | case intro.intro.intro.refine_4
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
z : ℂ
hz : z.re ∈ interior (integrableExpSet X μ)
n : ℕ
hX : AEMeasurable X μ
l u : ℝ
hlu : z.re ∈ Set.Ioo l u
h_subset : Set.Ioo l u ⊆ integrableExpSet X μ
t : ℝ := ((z.re - l) ⊓ (u - z.re)) / 2
h_pos : 0 < (z.re - l) ⊓ (u - z.r... | 39ea945347d2d3e2 |
AlgebraicGeometry.stalkMap_injective_of_isOpenMap_of_injective | Mathlib/AlgebraicGeometry/Morphisms/ClosedImmersion.lean | /-- If `f : X ⟶ Y` is open, injective, `X` is quasi-compact and `Y` is affine, then `f` is stalkwise
injective if it is injective on global sections. -/
lemma stalkMap_injective_of_isOpenMap_of_injective [CompactSpace X]
(hfopen : IsOpenMap f.base) (hfinj₁ : Function.Injective f.base)
(hfinj₂ : Function.Injecti... | X Y : Scheme
inst✝¹ : IsAffine Y
f : X ⟶ Y
inst✝ : CompactSpace ↑↑X.toPresheafedSpace
hfopen : IsOpenMap ⇑(ConcreteCategory.hom f.base)
hfinj₁ : Function.Injective ⇑(ConcreteCategory.hom f.base)
hfinj₂ : Function.Injective ⇑(ConcreteCategory.hom (Scheme.Hom.appTop f))
x : ↑↑X.toPresheafedSpace
⊢ Function.Injective ⇑(Co... | let φ : Γ(Y, ⊤) ⟶ Γ(X, ⊤) := f.appTop | X Y : Scheme
inst✝¹ : IsAffine Y
f : X ⟶ Y
inst✝ : CompactSpace ↑↑X.toPresheafedSpace
hfopen : IsOpenMap ⇑(ConcreteCategory.hom f.base)
hfinj₁ : Function.Injective ⇑(ConcreteCategory.hom f.base)
hfinj₂ : Function.Injective ⇑(ConcreteCategory.hom (Scheme.Hom.appTop f))
x : ↑↑X.toPresheafedSpace
φ : Γ(Y, ⊤) ⟶ Γ(X, ⊤) := ... | b31e93c621949e44 |
Rat.abs_def | Mathlib/Algebra/Order/Ring/Unbundled/Rat.lean | theorem abs_def (q : ℚ) : |q| = q.num.natAbs /. q.den | case inr
q : ℚ
hq : 0 ≤ q
⊢ q = ↑q.num.natAbs /. ↑q.den | rw [← num_divInt_den q, ← zero_divInt, Rat.divInt_le_divInt Int.zero_lt_one (mod_cast q.pos),
mul_one, zero_mul] at hq | case inr
q : ℚ
hq✝ : 0 ≤ q.num /. ↑q.den
hq : 0 ≤ q.num
⊢ q = ↑q.num.natAbs /. ↑q.den | 26d7b22d26c0017d |
HomotopyCategory.Pretriangulated.shift_distinguished_triangle | Mathlib/Algebra/Homology/HomotopyCategory/Pretriangulated.lean | lemma shift_distinguished_triangle (T : Triangle (HomotopyCategory C (ComplexShape.up ℤ)))
(hT : T ∈ distinguishedTriangles C) (n : ℤ) :
(Triangle.shiftFunctor _ n).obj T ∈ distinguishedTriangles C | C : Type u_1
inst✝² : Category.{u_3, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasBinaryBiproducts C
T : Triangle (HomotopyCategory C (ComplexShape.up ℤ))
hT : T ∈ distinguishedTriangles C
n : ℤ
⊢ (Triangle.shiftFunctor (HomotopyCategory C (ComplexShape.up ℤ)) n).obj T ∈ distinguishedTriangles C | obtain ⟨K, L, φ, ⟨e⟩⟩ := hT | case intro.intro.intro.intro
C : Type u_1
inst✝² : Category.{u_3, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasBinaryBiproducts C
T : Triangle (HomotopyCategory C (ComplexShape.up ℤ))
n : ℤ
K L : CochainComplex C ℤ
φ : K ⟶ L
e : T ≅ CochainComplex.mappingCone.triangleh φ
⊢ (Triangle.shiftFunctor (HomotopyCategory C (Comple... | ad0b91e80d9590e4 |
ContinuousMap.exists_extension_forall_mem_of_isClosedEmbedding | Mathlib/Topology/TietzeExtension.lean | theorem exists_extension_forall_mem_of_isClosedEmbedding (f : C(X, ℝ)) {t : Set ℝ} {e : X → Y}
[hs : OrdConnected t] (hf : ∀ x, f x ∈ t) (hne : t.Nonempty) (he : IsClosedEmbedding e) :
∃ g : C(Y, ℝ), (∀ y, g y ∈ t) ∧ g ∘ e = f | case out'.intro.intro.intro.intro
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : NormalSpace Y
f : C(X, ℝ)
t : Set ℝ
e : X → Y
hs : t.OrdConnected
hf : ∀ (x : X), f x ∈ t
hne : t.Nonempty
he : IsClosedEmbedding e
h : ℝ ≃o ↑(Ioo (-1) 1)
F : X →ᵇ ℝ := { toFun := Subtype.val ∘ ⇑h... | lift z to Ioo (-1 : ℝ) 1 using Icc_subset_Ioo (h x).2.1 (h y).2.2 hz | case out'.intro.intro.intro.intro.intro
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : NormalSpace Y
f : C(X, ℝ)
t : Set ℝ
e : X → Y
hs : t.OrdConnected
hf : ∀ (x : X), f x ∈ t
hne : t.Nonempty
he : IsClosedEmbedding e
h : ℝ ≃o ↑(Ioo (-1) 1)
F : X →ᵇ ℝ := { toFun := Subtype.va... | b5717c30218ed2f5 |
AffineSubspace.wSameSide_iff_exists_left | Mathlib/Analysis/Convex/Side.lean | theorem wSameSide_iff_exists_left {s : AffineSubspace R P} {x y p₁ : P} (h : p₁ ∈ s) :
s.WSameSide x y ↔ x ∈ s ∨ ∃ p₂ ∈ s, SameRay R (x -ᵥ p₁) (y -ᵥ p₂) | R : Type u_1
V : Type u_2
P : Type u_4
inst✝³ : LinearOrderedField R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x y p₁ : P
h : p₁ ∈ s
⊢ s.WSameSide x y ↔ x ∈ s ∨ ∃ p₂ ∈ s, SameRay R (x -ᵥ p₁) (y -ᵥ p₂) | constructor | case mp
R : Type u_1
V : Type u_2
P : Type u_4
inst✝³ : LinearOrderedField R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x y p₁ : P
h : p₁ ∈ s
⊢ s.WSameSide x y → x ∈ s ∨ ∃ p₂ ∈ s, SameRay R (x -ᵥ p₁) (y -ᵥ p₂)
case mpr
R : Type u_1
V : Type u_2
P : Type u_4
inst✝³ : Linear... | b828f1ac322a3263 |
EisensteinSeries.auxbound1 | Mathlib/NumberTheory/ModularForms/EisensteinSeries/UniformConvergence.lean | lemma auxbound1 {c : ℝ} (d : ℝ) (hc : 1 ≤ c ^ 2) : r z ≤ ‖c * (z : ℂ) + d‖ | case mk
c d : ℝ
hc : 1 ≤ c ^ 2
z : ℂ
hz : 0 < z.im
⊢ r ⟨z, hz⟩ ≤ ‖↑c * ↑⟨z, hz⟩ + ↑d‖ | have H1 : z.im ≤ √((c * z.re + d) ^ 2 + (c * z).im ^ 2) := by
rw [Real.le_sqrt' hz, im_ofReal_mul, mul_pow]
exact (le_mul_of_one_le_left (sq_nonneg _) hc).trans <| le_add_of_nonneg_left (sq_nonneg _) | case mk
c d : ℝ
hc : 1 ≤ c ^ 2
z : ℂ
hz : 0 < z.im
H1 : z.im ≤ √((c * z.re + d) ^ 2 + (↑c * z).im ^ 2)
⊢ r ⟨z, hz⟩ ≤ ‖↑c * ↑⟨z, hz⟩ + ↑d‖ | e8fac847c19bb038 |
MeasureTheory.Measure.exists_positive_of_not_mutuallySingular | Mathlib/MeasureTheory/Decomposition/Lebesgue.lean | theorem exists_positive_of_not_mutuallySingular (μ ν : Measure α) [IsFiniteMeasure μ]
[IsFiniteMeasure ν] (h : ¬ μ ⟂ₘ ν) :
∃ ε : ℝ≥0, 0 < ε ∧
∃ E : Set α, MeasurableSet E ∧ 0 < ν E
∧ ∀ A, MeasurableSet A → ε * ν (A ∩ E) ≤ μ (A ∩ E) | α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h : ¬μ ⟂ₘ ν
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → ((1 / (↑n + 1)) • ν) (t ∩ f n) ≤ μ (t ∩ f n)
hf₃ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → μ (t ∩ (f n)ᶜ)... | have hA₃ (n : ℕ) : μ A ≤ (1 / (n + 1) : ℝ≥0) * ν A := by simpa using hA₂ n univ .univ | α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h : ¬μ ⟂ₘ ν
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → ((1 / (↑n + 1)) • ν) (t ∩ f n) ≤ μ (t ∩ f n)
hf₃ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → μ (t ∩ (f n)ᶜ)... | 48437d57402f42cd |
CategoryTheory.Subobject.factors_self | Mathlib/CategoryTheory/Subobject/FactorThru.lean | theorem factors_self {X : C} (P : Subobject X) : P.Factors P.arrow :=
(factors_iff _ _).mpr ⟨𝟙 (P : C), by simp⟩
| C : Type u₁
inst✝ : Category.{v₁, u₁} C
X : C
P : Subobject X
⊢ 𝟙 (underlying.obj P) ≫ (representative.obj P).arrow = P.arrow | simp | no goals | 96b029f47280c095 |
Monoid.exists_orderOf_eq_exponent | Mathlib/GroupTheory/Exponent.lean | theorem exists_orderOf_eq_exponent (hG : ExponentExists G) : ∃ g : G, orderOf g = exponent G | G : Type u
inst✝ : CommMonoid G
hG : ExponentExists G
he : exponent G ≠ 0
hne : (Set.range orderOf).Nonempty
hfin : (Set.range orderOf).Finite
t : G
ht : orderOf t = sSup (Set.range orderOf)
p : ℕ
hp : Nat.Prime p
k : ℕ := (orderOf t).factorization p
hpe : k < (exponent G).factorization p
hk : k = (orderOf t).factoriza... | apply Nat.pow_succ_factorization_not_dvd (hG.orderOf_pos <| t ^ p ^ k).ne' hp | no goals | c9d54a30c09c3ae9 |
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